Chemical Reactions and Their Control on the Femtosecond Time Scale: 20th Solvay Conference on Chemistry

ADVANCES IN CHEMICAL PHYSICS VOLUME 101 EDITORIAL BOARD BRUCE,J. BERNE,Department of Chemistry, Columbia University, N...

Author: Pierre Gaspard | Irene Burghardt

11 downloads 773 Views 41MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

ADVANCES IN CHEMICAL PHYSICS VOLUME 101

EDITORIAL BOARD BRUCE,J. BERNE,Department of Chemistry, Columbia University, New York, New York, U.S.A. KURTBINDER,Institut fur Physik, Johannes Gutenberg-Universit Mainz, Mainz, Germany A. WELFORD CASTLEMAN, JR., Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania, U.S.A. DAVID CHANDLER, Department of Chemistry, University of California, Berkeley, California, U.S.A. M. S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford, U.K. WILLIAM T. COFFEY,Department of Microelectronics and Electrical Engineering, Trinity College, University of Dublin, Dublin, Ireland F. FLEMING CRIM, Department of Chemistry, University of Wisconsin, Madison, Wisconsin, U.S.A. ERNEST R. DAVIDSON, Department of Chemistry, Indiana University, Bloomington, Indiana, U.S.A. GRAHAMR. FLEMING,Department of Chemistry, The University of Chicago, Chicago, Illinois, U.S.A. KARLF. FREED, The James Franck Institute, The University of Chicago, Chicago, Illinois, U.S.A. PIERRE GASPARD, Center for Nonlinear Phenomena and Complex Systems, Brussels, Belgium ERICJ. HELLER,Institute for Theoretical Atomic and Molecular Physics, HarvardSmithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A. ROBIN M. HOCHSTRASSER, Department of Chemistry, The University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A. R. KOSLOFF,The Fritz Haber Research Center for Molecular Dynamics and Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel RUDOLPH A. MARCUS, Department of Chemistry, California Institute of Technology, Pasadena, California, U.S.A. G. Nrco~is,Center for Nonlinear Phenomena and Complex Systems, Universith Libre de Bruxelles, Brussels, Belgium THOMAS P. RUSSELL,Department of Polymer Science, University of Massachusetts, Amherst, Massachusetts DONALD G. TRUHLAR, Department of Chemistry, university of Minnesota, Minneapolis, Minnesota, U.S.A. JOHND. WEEKS,Institute for Physical Science and Technology and Department of Chemistry, University of Maryland, College Park, Maryland, U.S.A. PETERG. WOLYNES, Department of Chemistry, School of Chemical Sciences, University of Illinois, Urbana, Illinois, U.S.A.

Advances in CHEMICAL PHYSICS Chemical Reactions and Their Control on the Femtosecond Time Scale XXth Solvay Conference on Chemistry Edited by

PIERRE GASPARD Center for Nonlinear Phenomena and Complex Systems Universiti Libre de Bruxelles Brussels, Belgium

and

IRENE BURGHARDT Institut fur Physikalische und Theoretische Chemie der Universitiit Bonn Bonn, Germany

Series Editors

I. PRIGOGINE

STUART A. RICE

Center for Studies in Statistical Mechanics and Complex Systems The University of Texas Austin, Texas and International Solvay Institutes Universitk Libre de Bruxelles Brussels, Belgium

Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois

VOLUME 101

AN INTERSCIENCE@ PUBLICATION

NEW YORK

JOHN WILEY & SONS, INC.

CHICHESTER

WEINHEIM

BRISBANE

SINGAPORE TORONTO

A NOTE TO THE READER This book has been electronically reproduced from digital information stored at John Wiley & Sons, Inc. We are pleased that the use of this new technology will enable us to keep works of enduring scholarly value in print as long as there is a reasonable demand for them. The content of this book is identical to previous printings.

This text is printed on acid-free paper. An InterscienceB Publication Copyright 0 1997 by John Wiley Sc Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Library of Congress Catalog Number: 58-9935 ISBN 0-471-18048-3 Printed in the United States of America 10 9 8 7 6 5 4 3 2

ADMINISTRATIVE COUNCIL OF THE INSTITUTS INTERNATIONAUX DE PHYSIQUE ET DE CHIMIE founded by E. SOLVAY J. SOLVAY, Prksident

F. BINGEN,Vice Prksident & Trksorier G. NICOLIS, Secrituire D. JANSSEN A. JAUMOTTE

F. LAMBERT J.-M. RRET 3.

REISSE

L.WIJNS

I. PRIGOGINE, Directeur I. ANTONIOU, Directeur-adjoint

SCIENTIFIC COMMITTEE IN CHEMISTRY OF THE INSTITUTS INTERNATIONAUX DE PHYSIQUE ET DE CHIMIE founded by E. SOLVAY S . A. EWE,Prisident, The University of Chicago, U.S.A. M.EIGEN,Max Planck Institute, Gottingen, Germany K. FUKUI,Institute for Fundamental Chemistry, Kyoto, Japan B. HESS,Max Planck Institute, Dortmund, Germany V

vi

ADMINISTRATIVE COUNCIL AND SCIENTIFIC COMMITEE

E. KATCHALSKI-KATZIR, The Weizmann Institute of Science, Rehovot, Israel J.-M. LEHN,Universit6 de Strasbourg, France W. N. LIPSCOMB, Harvard University, Cambridge, U.S.A. V. PRELOG, ETH Zurich, Switzerland Lord TODD,University of Cambridge, United Kingdom A. BELLEMANS, Secrc%uire, Universit6 Libre de Bruxelles, Belgium

PARTICIPANTS H. R. H.the PRINCE LAURENT of Belgium, President of the Royal Institute for the Sustainable Management of Natural Resources and the Promotion of Clean Technologies P. W. BRUMER, University of Toronto, Ontario, Canada I. BURGHARDT, Universith Libre de Bruxelles, Belgium G. CASATI,University of Milan, Italy L. S. CEDERBAUM, Universitit Heidelberg, Germany M. CHERGUI, Universitk de Lausanne, Switzerland M. S. CHILD,University of Oxford, United Kingdom M. DESOUTER-LECOMTE, UniversitC de Likge, Belgium V. ENGEL,Universitat Wurzburg, Germany U. EVEN,Tel-Aviv University, Israel R. W. FIELD,Massachusetts Institute of Technology, Cambridge, U.S.A. The University of Chicago, U.S.A. G. R. FLEMING, P.GASPARD, Universitk Libre de Bruxelles, Belgium D. GAUYACQ, UniversitC de Paris-Sud, Orsay, France G. GERBER, Universitit Wurzburg, Germany M. GODEFROID, Universith Libre de Bruxelles, Belgium H. HAMAGUCHI, The University of Tokyo, Japan P.-H. HEENEN, UniversitC Libre de Bruxelles, Belgium M. HERMAN, Universith Libre de Bruxelles, Belgium Benno HESS,Max Planck Institute, Heidelberg, Germany Bemd. A. HESS,Universit5t Bonn, Germany J. JEENER, Universitk Libre de Bruxelles, Belgium R. JOST,Service National des Champs Intenses, Grenoble, France Ch. JUNGEN, UniversitC de Paris-Sud, Orsay, France M. E. KELLMAN,University of Oregon, Eugene, U.S.A vii

...

Vlll

PARTICIPANTS

A. KIRSCH-DE MESMAEKER, Universitk Libre de Bruxelles, Belgium T. KOBAYASHI, The University of Tokyo, Japan B. KOHLER, The Ohio State University, Columbus, U.S.A. F. KONG,Chinese Academy of Sciences, Beijing, China V. S. LETOKHOV, Russian Academy of Sciences, Moscow, Russia R. D. LEVINE,The Hebrew University, Jerusalem, Israel J. LIEVIN,Universiti Libre de Bruxelles, Belgium M. LOMBARD[,tJniversit6 Joseph Fourier, Grenoble, France J.-C. LORQUET, Universitk de Likge, Belgium P. MANDEL, Universiti Libre de Bruxelles, Belgium J. MANZ,Freie Universit2t Berlin, Germany R. A. MARCUS, California Institute of Technology, Pasadena, U.S.A. J.-L. MARTIN,Ecole Polytechnique, Palaiseau, France W. H. MILLER,University of California, Berkeley, U.S.A. S. MUKAMEL, University of Rochester, U.S.A. D. M. NEUMARK, University of California, Berkeley, U.S.A. H. J. NEUSSER, Technische Universitiit Munchen, Garching, Germany G. NICOLIS,Universitk Libre de Bruxelles, Belgium T. OKADA,Osaka University, Japan J.-P. PIQUE,Universitk Joseph Fourier, Grenoble, France E. POLLAK, The Weizmann Institute of Science, Rehovot, Israel I. FRIGOGINE, Universitk Libre de Bruxelles, Belgium M. QUACK,ETH Zurich, Switzerland H. RABITZ,Princeton University, U.S.A. J. REISSE,Universiti Libre de Bruxelles, Belgium F. REMACLE, Universiti de Likge, Belgium S. A. RICE,The University of Chicago, U.S.A. J.-P. RYCKAERT, Universitk Libre de Bruxelles, Belgium K. SCHAFFNER, Max Planck Institute, Mulheim an der Ruhr, Germany R. ScHiNKE, Max Planck Institute, Gottingen, Germany E. W. SCHLAG, Technische Universitiit Munchen, Garching, Germany

PARTICIPANTS

M.SHAPIRO, The Weizmann Institute of Science, Rehovot, Israel

University of Oxford, United Kingdom T. P. SOFTLEY, D. J. TANNOR, The Weizmann Institute of Science, Rehovot, Israel J. TROE,UniversiGt Gottingen, Germany M.VANDER AUWERAER, Katholieke Universiteit Leuven, Belgium L. Wosm, Freie UniversiGt Berlin, Germany K. YAMANOUCHI, The University of Tokyo, Japan K. YAMASHITA, The University of Tokyo, Japan A. H.ZEWAIL, California Institute of Technology, Pasadena, U S A . J.-Y. ZHOU,Zhongshan University, Canton, China

ix

SCIENTIFIC SECRETARY OF THE CONFERENCE P. GASPARD, Conference C h a i m n I. BURGHARDT D. DAEMS B. GREMAUD

S. R. JAIN

KARLSON D. MEI A.

A. SUAREZ

P.VAN EDEVAN DER PALS

ADMINISTRATIVE SECRETARY OF THE CONFERENCE M.ADAM

A X . BLONDLET N. GALLAND M.KIES P. KINET

M.n E t N

M. KUNEBEN I. SAVERINO J. TACHELET S. WELLENS xi

GROUP PHOTOGRAPH OF THE PARTICIPANTS IN THE XXTH SOLVAY CONFERENCE ON CHEMISTRY 1. I. Prigogine 2. S. A. Rice 3. R. A. Marcus 4. E. W. Schlag 5. 1. Troe 6. M. Quack 7. L. Woste 8. P. Gaspard 9. H. Rabitz 10. Benno Hess 11. G. R. Fleming 12. A. H.Zewail 13. A. Karlson 14. 1. Burghardt 15. D. Mei 16. D. Daems 17. E. Pollak 18. T. Okada 19. T. Kobayashi 20. M. S. Child 21. D. M.Neumark

22. K. Yamanouchi 23. S. Mukamel 24. G. Casati 25. R. D. Levine 26. G. Gerber 27. P. W. Brumer 28. P. van Ede van der Pals 29. S. R. Jain 30. B. Grimaud 31. A. Sukez 32. M. Shapiro 33. J. Manz 34. R. W. Field 35. M. Herman 36. D. Gauyacq 37. M. Lombardi 38. M. E. Kellman 39. H. J. Neusser 40. B. Kohler 4 1. R. Jost 42. H.Hamaguchi

M. Godefroid D. J. Tannor L. S. Cederbaum M. Chergui V. Engel Ch. Jungen J.-L. Martin 50. F. Remacle 51. F. Kong 52. U. Even 53. M.Van der Auweraer 54. J. LiCvin 55. J.-P. Pique 56. R. Schinke 57. T. Softley 58. Bemd A. Hess 59. K. Yamashita 60.J.-Y. Zhou 6 I. K. Schaffner 62. L. Michaille

43. 44. 45. 46. 47. 48. 49.

...

Xlll

CONTRIBUTORS TO VOLUME 101 A. BARTANA, Department of Physical Chemistry and The Fritz Haber

Research Center, The Hebrew University, Jerusalem, Israel

T. BAUMERT, Physikalisches Institut, Universitit Wurzburg, Wurzburg, Germany C. BECK,Max-Planck-Institut fiir Stromungsforschung, Gottingen, Germany R. S. BERRY,Department of Chemistry, The James Franck Institute, The University of Chicago, Chicago, Illinois V. BONACIC-KOUTECKY, Walter Nernst-Institut Humbold-Universitiit zu Berlin, Berlin, Germany P. BRUMER, Chemical Physics Theory Group and The Ontario Laser and Lightwave Research Centre, University of Toronto, Toronto, Canada

I. BURGHARDT, Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems, Universitk Libre de Bruxelles, Brussels, Belgium Z. CHEN,Chemical Physics Theory Group and The Ontario Laser and Lightwave Research Centre, University of Toronto, Toronto, Canada M. CHERGUI, Institut de Physique Exp6rimentale Facult; des Sciences, BSP Universite de Lausanne, Lausanne-Dorigny, Switzerland M. CHO,Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts H. CHOI,Department of Chemistry, University of California Berkeley, California, and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California C. CIORDAS-CIURDARIU, Department of Chemistry, University of Rochester, Rochester, New York A. J. DOBBYN,Theory, Computational Science and Computing Division, Daresbury, Warrington, Cheshire, England R. W. FIELD,Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts G. R. FLEMING, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois xv

CONTRIBUTORS TO VOLUME 101

xvi

H. FLOTHMANN, Max-Planck-Institut fur Stromungsforschung, Gottingen, Germany P. GASPARD, Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems, Universit6 Libre de Bruxelles, Brussels, Belgium J. GAUS,Walter Nemst-Institut, Humbold-Universitat zu Berlin, Berlin, Germany

G. GERBER,Physikalisches Institut, UniversiGt Wurzburg, Wurzburg, Germany

I. GOLUB,Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel J. HELBING, Physikalisches Institut, Universitit Wurzburg, Wurzburg, Germany A. HISHIKAWA, Department of Pure and Applied Sciences, The University of Tokyo, Tokyo, Japan H. ISHIKAWA, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts M. P. JACOBSON, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts C. JEANNIN, Institut de Physique Expirimentale, BSP Universit6 de Lausanne, Lausanne-Dorigny, Switzerland T. JOO, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois

CH.JUNGEN, Laboratoire Aim6 Cotton du CNRS, Universit6 de Paris-Sud, Orsay, France

H.-M. KELLER,Max-Planck-Institut fur Stromungsforschung, Gottingen, Germany V. KHIDEKEL, Department of Chemistry, University of Rochester, Rochester, New York

M.V. KOROLKOV, Belarus Academy of Sciences, Institute of Physics, Republic of Belarus

R. KOSLOFF,Department of Physical Chemistry and The Fritz Haber Research Center, The Hebrew University, Jerusalem, Israel TH. LEISNER,Institut fur Experimentalphysik, Freie Universitit Berlin, Berlin, Germany


xvii

V. S. LETOKHOV, Institute of Spectroscopy, Russian Academy of Sciences, Troitzk, Moscow Region, Russia R. D. LEVINE,The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem, Israel, and Department of Chemistry and Biochemistry, University of California Los Angeles, Los Angeles, California S. R. MACKENZIE, Physical and Theoretical Chemistry Laboratory, Oxford, United Kingdom J. MANZ,Institut fur Physikalische und Theoretische Chemie, Freie Universitiit Berlin, Berlin, Germany R. A. MARCUS, Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California F. MERKT,Laboratorium fur Physikalische Chemie, Zurich, Switzerland W. H. MILLER, Department of Chemistry, University of California, Berkeley, California, and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California Department of Chemistry, University of California, BerkeD. H. MORDAUNT, ley, California S. MUKAMEL, Department of Chemistry, University of Rochester, Rochester, New York R. NEUHAUSER, Institut fur Physikalische und Theoretische Chemie, Technische Universitit Miinchen, Garching, Germany D. M. NEUMARK, Department of Chemistry, University of California, Berkeley, California, and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California

H. J. NEUSSER, Institute fur Physikalische und Theoretische Chemie, Tech-

nische Universitit Munchen, Garching, Germany J. P. OBRIEN, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts K. OHDE,Department of Pure and Applied Sciences, The University of Tokyo, Tokyo, Japan D. L. OSBORN, Department of Chemistry, University of California, Berkeley, California, and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California G. K. PARAMONOV, Belarus Academy of Sciences, Institute of Physics, Minsk, Republic of Belarus

xviii


M. T. PORTELLA-OBERLI, Institut de Physique Ex+rimentale, BSP Universitt de Lausanne, Lausanne-Dorigny, Switzerland W. F. POLIK,Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts H. RABITZ,Department of Chemistry, Princeton University, Princeton, New Jersey B . RJZISCHL-LENZ, Institut f i r Physikalische und Theoretische Chemie, Freie Universitit Berlin, Berlin, Germany S. A. RICE,Department of Chemistry, The James Franck Institute, The University of Chicago, Chicago, Illinois D. ROLLAND, Physical and Theoretical Chemistry Laboratory, Oxford, United Kingdom

H. RUPPE,Institut fur Experimentalphysik, Freie Universi~tBerlin, Berlin, Germany

S. R u n , Institut fur Experimentalphysik, Freie Universitit Berlin, Berlin, Germany R. SCHINKE, Max-Planck-Institut Fur Stromungsforschung, Gottingen, Germany

E. W. SCHLAG, Institut fur Physikalische und Theoretische Chemie, Technische Universitat Munchen, Munchen, Gemany E. SCHREIBER, Institut fur Experimentalphysik, Freie Universitat Berlin, Berlin, Germany M. SHAPIRO, Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel A. SHNITMAN, Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel I. SOFER, Department of Energy and Environment, The Weizmann Institute of Science, Rehovot, Israel T. P. SOFTLEY,Physical and Theoretical Chemistry Laboratory, Oxford, United Kingdom S. A. B. SOLINA, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts M. STUMPF, Physical and Theoretical Chemistry Laboratory, Oxford, United Kingdom D. J. TANNOR, Department of Chemical Physics, The Weizrnann Institute of Science, Rehovot, Israel


xix

J. "ROE,Institut fiir Physikalische Chemie, Universitat Gottingen, Gottingen,

Germany ST.VAJDA,Institut fiir Experimentalphysik, Freie Universitilt Berlin, Berlin, Germany R. DE VIVIE-RIEDLE, Institut fiir Physikalische und Theoretische Chemie, Freie Universitiit Berlin, Berlin, Germany S. WOLF,Institut f i r Experimentalphysik, Freie Universitilt Berlin, Berlin, Germany L. Wosm, Institut fur Experimentalphysik, Freie UniversiGt Berlin, Berlin, Germany K. YAMANOUCHI, Department of Pure and Applied Sciences, The University of Tokyo, Tokyo, Japan A. YOGEV,Department of Energy and Environment, The Weizrnann Institute of Science, Rehovot, Israel Arthur Amos Noyes Laboratory of Chemical Physics, CaliA. H. ZEWAIL, fornia Institute of Technology, Pasadena, California

INTRODUCTION Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field. I. PRIGNINE STUART A. RICE

xxi

The XXth Solvay Conference on Chemistry was held at the Free University of Brussels from November 28 to December 2, 1995. It gathered 65 participants, including scientists from the Free University of Brussels, and has consisted of 17 reports, 11 communications, and 9 discussion sessions. This volume contains the papers and discussions presented at the Conference. During the past decade, an unprecedented time resolution has been achieved in the study of chemical reactions thanks to the development of femtosecond lasers. This remarkable progress currently allows one to observe the dynamics of molecules on the intrinsic time scale of their vibrations, dissociative motions, and electronic excitations. In addition, this new laser technology has concretized the possibility of the coherent control of chemical reactions as well as of more general quantum systems. It is to these problems of fundamental importance and interest that this XXth Solvay Conference has been devoted. Special emphasis was placed on the new perspectives opened at this challenging and promising new frontier of science, including possibilities of technological applications. We wish to express our thanks to Ilya Prigogine, Director of the Solvay Institutes, to Grigoire Nicolis, Secretary of the Solvay Instibtes, and to Stuart A. Rice, Chairman of the Scientific Committee in Chemistry of the Solvay Institutes, for their support and advice in the preparation of this Conference. Special thanks are due to Mr. Jacques Solvay for the active interest he has taken in the Conference. It is also our great pleasure to thank Nadine Galland, as well as the other members of the Scientific and Administrative Secretaries, for their essential role in the organization and running of the Conference and in the preparation of the Proceedings. The Conference has been financially supported by the International Institutes of Physics and Chemistry, founded by E. Solvay and as an Advanced Research Meeting by a grant of the DG XI1 of the European Commission, which are gratefully thanked.

P. GASPARD I. BURGHARDT

xxiii

CONTENTS xxxi

OPENING REMARKS J. Solvay

FEMTOCHEMISTRY: FROMISOLATED MOLECULES TO CLUSTERS FEMTOCHEMISTRY: CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

3

A. H.Zewail Discussion on the Report by A. H.Zewail

COHERENT CONTROL WITH FEMTOSECOND LASERPULSES

47

T. Baumert, J. Helbing, and G. Gerber Discussion on the Report by G. Gerbet

GENERAL DISCUSSION ON FEMTOCHEMISTRY: FROM ISOLATED MOLECULES TO CLUSTERS FEMTOCHEMISTRY:

83

FROMCLUSTERS TO SOLUTIONS

SIZE-DEPENDENT ULTRAFAST RELAXATION PHENOMENA IN METAL CLUSTERS

101

R. S. Berry, K Bonu&5Kouteck$ J. Gaus, Th. Leisner, H. Ruppe, S. Rutz, E. Schreiber, St. Vajak, S. Wolf;L. Woste, J. Manz, B. Reischl-Lenz, and R. de Vivie-Riedle Discussion on the Report by L Woste FEMTOSECOND CHEMICAL DYNAMICS IN CONDENSED PHASES

141

G. R. Fleming, T Joo, and M. Cho Discussion on the Report by G. R. Fleming

FEMTOSECOND LASERCONTROL ULTRAFAST DIFFRACTION

OF ELECTRON BEAMSFOR

185

V S. Letokhov Discussion on the Communication by K S. Letokhov xxv

xxvi

CONTENTS

GENERAL DISCUSSION ON SOLUTIONS

FEMTOCHEMISTRY:

FROMCLUSTERS TO

193

LASER CONTROL OF CHEMICAL REACTIONS ON THE CONTROL OF QUANTUM MANY-BODY DYNAMICS: APPLICATION TO CHEMICAL REACTIONS

PERSPECTIVES

213

S.A. Rice

Discussion on the Report by S.A. Rice

EXPERIMENTAL OBSERVATION OF LASER CONTROL: ELECTRONIC BRANCHING IN THE PHOTODISSOCIATION

OF N a 2

285

A. Shnitmun, I. Sofer, I. Golub, A. Yogev,M. Shapiro, Z. Chen, and I? Brumer Discussion on the Communication by M. Shupiro

COHERENT CONTROL OF BIMOLECULAR SCATTERING

295

R Brumer and M. Shapiro LASERHEATING, COOLING, AND TRANSPARENCY OF INTERNAL DEGREES OF FREEDOM OF MOLECULES

301

D. J. Tannor, R. Koslofi and A. Bartana Discussion on the Communication by D. J. Tannor RAMIFICATIONS

DYNAMICS

OF

FEEDBACK FOR CONTROL OF QUANTUM

315

H. Rabitz Discussion on the Communication by H.Rabitz

THEORY OF LASERCONTROL OF VIBRATIONAL TRANSITIONS AND PULSES CHEMICAL REACTIONS BY ULTRASHORT INFRARED LASER

327

M. V Korolkov, J. Manz, and G. K. Paramonov Discussion on the Communication by J. Manz

TIME-FREQUENCY AND COORDINATE-MOMENTUM WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

S. Mukamel, C. Ciordas-Ciurdariu, and V Khidekel GENERAL DISCUSSION ON REACTIONS

LASERCONTROL OF CHEMICAL

345

373

xxvii

CONTENTS INTRAMOLECULAR SOLVENT

DYNAMICS

DYNAMICS AND RRKM THEORY OF CLUSTERS

39 1

R. A. Marcus Discussion on the Report by R. A. Marcus

HIGH-RESOLUTION SPECTROSCOPY DYNAMICS

AND INTRAMOLECULAR

409

H. J. Neusser and R. Neuhauser Discussion on the Report by H. J. Neusser GENERAL DISCUSSION ON

INTRAMOLECULAR

REGULAR AND IRREGULAR FEATURESIN SPECTRA AND DYNAMICS INTRAMOLECULAR

DYNAMICS

449

UNIMOLECULAR

DYNAMICS IN THE FREQUENCY DOMAIN

463

R. W Field, J. I! O’Brien, M. I! Jacobson, S. A. B. Solina, W E Polik, and H. Ishikuwa OF CLASSICAL PERIODIC ORBITS AND CHAOS IN EMERGENCE INTRAMOLECULAR AND DISSOCIATION DYNAMICS

49 1

€? Gaspard and I. Burghardt

GENERAL DISCUSSION ON REGULAR AND IRREGULAR FEATURES IN UNIMOLECULAR SPECTRA AND DYNAMICS

583

MOLECULAR RYDBERG STATESAND ZEKE SPECTROSCOPY 607

ZEKE SPECTROSCOPY E. W Schlag Discussion on the Report by E. W Schlag

SEPARATION OF R m SCALES IN THE DYNAMICS OF HIGH MOLECULAR RYDBERGSTATES

625

R. D. Levine GENERAL DISCUSSION ON MOLECULAR RYDBERG STATES AND

ZEKE SPECTROSCOPY:

PART

I

647

xxviii

CONTENTS

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REAmlONS USING ZEKE SPECTROSCOPY

667

T F! Sofley, S. R. Mackenzie, E Merkt, and D. Rolland Discussion on the Report by T. l? Sofley QUANTUM DEFECTTHEORY OF THE DYNAMICS OF MOLECULAR RYDBERG STATES

701

Ch. Jungen Discussion on the Report by Ch. Jungen SUBPICOSECOND STUDY OF BUBBLE FORMATION UPON STATE EXCITATION IN CONDENSED RAREGASES

RYDBERG

711

M.-T Portella-Oberli, C. Jeannin, and M. Chergui Discussion on the Communication by M. Chergui GENERAL DISCUSSION ON MOLECULAR RYDBERG STATES AND ZEKE SPECTROSCOPY: PARTXI

719

TRANSITION-STATE SPECTROSCOPY AND PHOTODISSOCIATION PHOTODISSOCIATION SPECTROSCOPY AND DYNAMICS OF THE VINOXY(CH2CHO) RADICAL

729

D. L. Osbom, H. Choi, and D. M. Neumurk Discussion on the Report by D. M. Neumrk RESONANCES IN UNIMOLECULAR DISSOCIATION: FROM MODE-SPECIFIC TO STATISTICAL BEHAVIOR

745

R. Schinke, H.-M. Keller, H. Flothmann, M. Stumpf; C. Beck, D. H. Mordaunt, and A. J. Dobbyn Discussion on the Report by R. Schinke PHOTODISSOCIATING SMALL POLYATOMIC MOLECULES IN THE VUV REGION: RESONANCES IN THE l ~ + - l C +BANDOF OCS

789

K. Yamanouchi,K. Ohde, and A. Hishikawa Discussion on the Communicationby K. Yamanouchi PHASE AND AMPLITUDE IMAGING OF BY SPECTROSCOPIC MEANS

EVOLVING WAVEPACKETS

M. Shapiro Discussion on the Communication by M. Shapiro

799

CONTENTS

GENERAL DISCUSSION ON TRANSITION-STATE SPECTROSCOPY AND PHOTODISSOCIATION REACTION

xxix 809

RATE THEORES

RECENT ADVANCES IN STATISTICAL ADIABATIC CHANNEL DYNAMICS CALCULATIONS OF STATE-SPECIFIC DISSOCIATION

819

J. Troe Discussion on the Report by J. Tme

QUANTUM AND SEMICLASSICAL THEORIES OF CHEMICAL REACTION RATES

853

M! H. Miller Discussion on the Report by M! H . Miller FEMTOSPECTROCHEMISTRY: NOVEL POSSIBILITIES WITH RESOLUTION THREE-DIMENSIONAL (SPACE-TIME)

873

K S. btokhov Discussion on the Communication by F S. Letokhov TO KING

ACADEMIC SESSION AT THE CASTLE OF LAEKEN: bSENTATI0N ALBERT

889

MODERNPHOTOCHEMISTRY

889

S. A. Rice

FEMTOCHEMISTRY

892

A. H. Zewail

CONCLUDING REMARKS

S.A. Rice and K S. Letokhov

893

AUTHORINDEX

899

SUBJECT INDEX

927

OPENING REMARKS J. Solvay President, Administrative Council, International Institutes for Physics and Chemistry It is a pleasure and an honor to welcome the participants to this XXth Solvay Conference on Chemistry. The subject of your investigations is photochemistry, with particular emphasis on chemical reactions on the femtosecond time scale. When browsing through the abstracts for the conference that we have received, I find that they often refer to quantum mechanics as applied to the understanding of chemical reactions. The theory of quanta has been a central preoccupation of the Solvay Conference since the first Solvay Conference on Physics, which took place in 1911. On that occasion, the fundamental discrepancies between classical theory and experimental data, which appeared at the beginning of the century, were discussed by major figures of modem science like Marie Curie, Albert Einstein, Max Planck, Henri Poincark, and many others. Today, the theory of quanta has led to many applications, among which are the femtosecond lasers that can be used for the control of quantum systems. This modem laser technology allows the detailed observation of different types of molecular motions on ultrashort time scales with unprecedented resolution in energy. Knowing from personal experience the difficulty of control in industrial applications,the control of chemical reactions at the level of the femtosecond inspires my admiration for the range of applications of quantum theory. In turn, these new results appear particularly challenging for the theory of quanta. A major issue is to understand the relationships between the ultrashort and the long-time scales as well as between the microscopic and macroscopic properties of chemical reactions. In this context, the new possibilities of controlling microsystems highlight the fundamental paradoxes of quantum mechanics, where probabilistic features coexist so closely with coherent behaviors. Might the new frontier at ultrashort time scales contribute to understanding at a deeper level the role of quantum mechanics in irreversible processes such as chemical reactions? Such questions were discussed in 1962 at the XIIth Solvay Conference on Chemistry, entitled “Energy Transfer in Gases,” which gathered G. Herzberg, R. Karplus, R. G. W. Nonish, xxxi

xxxii

OPENING REMARKS

J. C. Polanyi, G. Porter, 0. K. Rice, N. N. Semenov, N. B. Slater, A. R. Ubbelohde, and E. P. Wigner, among others. Today, these questions remain as puzzling and important as ever. The recent advances in modem technology continue to open new opportunities for the observation of chemical reactions on shorter and shorter time scales, at higher and higher quantum numbers, in larger and larger molecules, as well as in complex media, in particular, of biological relevance. As an example of open questions, the most rapid reactions of atmospheric molecules like carbon dioxide, ozone, and water, which occur on a time scale of just a few femtoseconds, still remain to be explored. Another example is the photochemistry of the atmospheres of nearby planets like Mars and Venus or of the giant planets and their satellites, which can help us to understand better the climatic evolution of our own planet. We hope that this Conference will contribute to solving these fundamental questions. In creating the Solvay Institutes of Physics and Chemistry, the hope of Ernest Solvay has been that international meetings at the highest level should foster progress through reports and discussions. My wife and I are looking forward to having most of you for a buffet dinner at home this evening. You will be able to tell me if this conference indeed fulfills the expectations of Ernest Solvay. I would like to express my gratitude to the European Commission, which provided substantial support in the financing of this Conference.

ADVANCES IN CHEMICAL PHYSICS VOLUME 101

FEMTOCHEMISTRY: FROM ISOLATED MOLECULES TO CLUSTERS

FEMTOCHEMISTRY: CHEMICAL REACTION DYNAMICS AND THEIR CONTROL A. H.ZEWAIL Arthur Amos Noyes Laboratory of Chemical Physics California Institute of Technology Pasadena, California

CONTENTS 1. Introductory Remarks 11. Concept of Coherence and the Evolution to Ferntochemistry

A. B. C. D.

Coherence and Dephasing Coherence Control by Phase-Coherent Pulses Coherence in the States of Isolated Molecules: IVR Coherence in Orientation: Molecular Structures E. Coherence in Reactions: Wavepackets and Nuclear Motions F. Coherence in Solvation: Clusters and Dense Fluids G. Coherence Control of Wavepackets: Reactive and Nonreactive Systems H. Coherence in Electron Diffraction: Complex Molecular Structures 111. Prototype Systems: Uni- and Bimolecular Reactions A. Resonances in Unimolecular Reactions B. Barrier Reactions: Saddle-Point Transition State C. Bimolecular Reactions: Ground-State Dynamics D. Complex Organic Reactions E. Electron Transfer Reactions F. Tautomerization Reactions of DNA Models IV. Scope of Reactions Studied V. Concluding Remarks Bibliography References

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale. XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

3

4

A. H. ZEWAIL

I. INTRODUCTORY REMARKS The International Institutes of Physics and Chemistry were founded by Ernest Solvay at the beginning of this century. The Solvay Conferences in Brussels have played an essential role in the history of physics, as remarked by one of the founders of quantum mechanics, Werner Heisenberg. The first Solvay Conference on Physics in 1911 became famous for its discussions on the birth of quantum mechanics, a marked departure from classical concepts, by Marie Curie, Albert Einstein, Max Planck, Henri Poincarb, and many others. The XXth Solvay Conference on Chemistry was devoted to “chemical reactions and their control on the femtosecond time scale.” The conference followed the wonderful tradition of appreciation of scientists and scientific discoveries. We owe our appreciation to Ilya Prigogine, Director of the Solvay Institutes, Stuart Rice, Chairman of the Scientific Committee, and Pierre Gaspard, the Conference Scientific Secretary and Organizer. I had the honor to review the field, as described by the title of this chapter, I would like to take this opportunity here to focus on some concepts that were essential in the development of femtochemistry: reaction dynamics and control on the femtosecond time scale. The following is not an extensive review, as many books and articles have already been published [l-121 on the subject, but instead is a summary of our own involvement with the development of ferntochemistry and the concept of coherence. Most of the original articles are given in a recent two-volume book that overviews the work at Caltech [5], up to 1994. Reaction dynamics on the femtosecond time scale are now studied in all phases of matter, including physical, chemical, and biological systems (see Fig. 1). Perhaps the most important concepts to have emerged from studies over the past 20 years are the five we summarize in Fig. 2. These concepts are fundamental to the elementary processes of chemistry-bond breaking and bond making-and are central to the nature of the dynamics of the chemical bond, specifically intramolecular vibrational-energy redistribution, reaction rates, and transition states. In a classical Bohr orbit, the electron makes a completejourney in 0.15 fs. In reactions, the chemical transformation involves the separation of nuclei at velocities much slower than that of the electron. For a velocity -lo5 cm/s and a distance change of lo-* cm (1 A), the time scale is 100 fs. This is a key concept in the ability of femtochemistry to expose the elementary motions as they actually occur. The classical picture has been verified by quantum calculations. Furthermore, as the deBroglie wavelength is on the atomic scale, we can speak of the coherent motion of a single-moleculetrajectory and not of an ensemble-averaged phenomenon. Unlike kinetics, studies of dynamics require such coherence, a concept we have been involved with for some time.

5

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL Gas phase molecular beam Barrier reactions Dissociation Electron transfer Proton transfer Vander Waals reactions Eimolecular reactions Rydberg reactions Organometallic reactions Exchange reactions lsomerization reactions Cleavageladdition Abstract reactions Norrish reactions Elimination reactions ~

Clusters Acid-base reactions Step-wise solvation Metal clusters

Surfaces Femto-STM dynamics Desorption

Caging reactions Harpoon reactions Excimers Exciplexes Polymer reactions Semiconductor reactions

Femtosecond Resolution in Chemistry

-

&

Dense fluids Dissociation reactions Recombination reactions Dephasing Vibrational relaxation

-

‘I:

Biology

Liquids Coherent dissociation Geminate recombination Dephasing Proton transfer Electron transfer Vibrational relaxation Barrierless reactions Bimolecular reactions Ionic reactions Solvation dynamics Friction dynamics Polarization (kerr) ~

Control

t*

U*(d

I

~

Biological systems Ligand-myoglobin Protein dynamics Bacteriorhodopsin Light harvesting Pigment-protein complexes Photosynthetic reaction centers

Figure 1. Schematic indicating the different phases studied with femtosecond resolution and the area of control studied by spatial (r*), temporal (t*), phase (a*), or potential-energy (v*)manipulation.

6

A.

H.ZEWAIL

SOME CONCEPTS IN FEMTOCHEMISTRY Concept of Time Scales and Atomic-Scale

Electronmotion Nudear motion DeBroglie wavelength

Concept of Coherence and Single-Molecule

State, orientation, wave packet coherence

UncertaintyprincipIeMdwherence Single-molecule, not ensemble, trajectory Dynamics, not kinetics Complex systems, robustnessof phenomena

Concept of Physical and Chemical Time Scales

Bond breaking & Bond making time scales Spreadingin space & dephasing time scales Analogy with Ti and Tz

Freezing structures at fs resolution Probing with mass spectrometry, LIP, m,

Probing Transition States and Intermediates

PE, SEF' and absorption Reaction mechanisms,bonding, structures

~~

Concept of Controlling Reactivity Figure 2.

Reaction wave packet wntml Time &space control Phase conml PES control

Summary of some key concepts in ferntochemistry discussed in text.

What coherence is, how it can be probed, and why it is robust in molecular systems are some fundamental issues of concern here. In what follows, we discuss these issues and then provide examples for a number of different classes of reactions. 11. CONCEPT OF COHERENCE AND THE

EVOLUTION TO FEMTOCHEMISTRY

Twenty years ago, the concept of coherence in molecular systems was new. In the beginning, and certainly within the chemistry community, the relevance

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

7

and importance of a notion such as dephasing in molecular and reaction dynamics was suspect! Coherence as a theoretical concept was fully developed in the physics and optics literature, especially following the invention of lasers and the development of nonlinear optics. In chemistry, however, the concept was first appreciated only in the context of nuclear magnetic resonance (NMR) research.

A. Coherence and Dephasing In the early days of studying molecular coherence (Fig. 3), the focus was on the advancement of new techniques and on the observation of the phenomenon of dephasing. In the 1970s, nanosecond lasers were used to form a coherent superposition of ground and excited electronic states, and the resulting polarization was monitored to measure the coherence of the ensemble. At Caltech, we made these pulses by the “switching” of a single-mode laser, with particular emphasis on the generation of laser acoustic-diffraction pulses. Coherent transients were observed in solids and in gases, both in “bulbs” and in molecular beams (Fig. 4).

La

ry=

1

/

*

I

,+I

I

Figure 3. Coherence, induced between two states (lowest and first-harmonic wave functions) and the nature of the hybrid superposition, which evolves with time.

t :ppy - I 8

A.

H.ZEWAIL lntenrity

Emission intensity

900

IOV

&I

300

100

1%

20

loo

180

260

350

340

I - 0

0

1

2

3

4

5

300

6

6aou 1

7

Figure 4. Coherent transients observed in gases and molecular beams. Shown are the photon echo (detected by spontaneous emission), the free induction decay, and T i for different pressures (iodine gas and beam).

The importance of these results is manifested in the ability of separating homogeneous from inhomogeneous dephasings in complex molecular systems. Previously, the true homogeneous dynamics were often masked in the apparent absorption line shapes, occurring on a time scale four orders of magnitude slower than would have been inferred from the absorption spectra. One significant development in this area, which we used later for phase control, was the detection of coherence on the incoherent spontaneous emis-


9

sion of molecules using a pulse train (in phase!) to convert coherence (xy plane of the rotating frame) to population (kz axis).

B. Coherence Control by Phase-Coherent Multiple Pulses

In 1980, with W. Warren, we decided to extend these multiple-pulse experiments on molecular systems to include a prescribed phase of each pulse, opening the way for optical phase control and for the optical analogue of multiple-pulse NMR spectroscopy. The multiple-pulse phase-coherent sequences were generated using the laser acoustic diffraction method mentioned in the above section. This way, we observed the photon echo with a pulse sequence, such as XXX-XXZ (in the rotating-frame notation), and we could use a y pulse to lock the polarization (photon locking), making it possible to control dephasing due to collisions, as demonstrated for iodine (Fig. 5). Other pulse sequences, including composite pulses, were developed, both theoretically and experimentally, to control the fluorescence of inhomogeneous ensemble as well as the coherence (photon echo) in an inhomogeneous system (Fig. 5). These techniques of control using phase-coherent pulses have been revisited and extended to the picosecond (and more recently to the femtosecond) time domain by several groups, and represent an active field of current interest (see Ref. 1).

C. Coherence in the States of Isolated Molec~~ies: IVR The concepts of dephasing and coherence were introduced to the field of isolated molecular dynamics, also in 1980, via the following question: Can a single, isolated large molecule with many degrees of freedom exhibit intramolecular dephasing by developing its own heat bath? Using molecular beams and picosecond pulses, we were (pleasantly) surprised to observe coherence among the enormously large number of vibrational states in the anthracene molecule. We observed not only the coherence among a restricted number of levels in a bath of other states but also the phase of the oscillatory motion (Fig. 6). These studies were extended to different energies and different molecules and were generalized to include the effect of rotations and rotational-vibrational couplings on coherence. The phenomenon of intramolecular vibrational-energy redistribution (IVR) was now on solid ground, as its nature and origin could be related to the concept of phase coherence in the hybridized eigenstutes of the isolated molecule. From these and other studies, we could define different regions of IVR (restricted and dissipative) in molecules, and this same behavior was found in many other molecules studied in different laboratories. Such observation of “coherent motion” in isolated molecular systems triggered our interest in molecular reaction dynamics in the 1980s and the 1990s.

10

A.

H.ZEWAIL

x x x xxx m..-m Tz 941 f 17 as

L 100 200 300 400 500

:rnL I

PE

Aru- 1.00

20 30 Rcuurc. mTorr

10

T,me. nr

h

P,:60,-3~-60,

P,:1 q - 1%

0

PI: 200, 1

2

3

4

11)

05

0

Figure 5. Composite pulse trains for control of fluorescence and echo. Also shown are the echoes (iodine gas) and T I and T2 at different pressures.


11

Restricted (Coherent)

IVR

I

I

0

I

I

I

i 6

I

T i (nsec)

Dissipative .

IVR

Fast Slow

N

I

0

I

I

1

I

Time (nsec)

17

I

I

I

7

Figure 6. Restricted and dissipative IVR observed for an anthracene beam at a rotational temperature of -3 K. Note the in-phase and out-of-phase nature of the coherent oscillations (top).

D. Coberence in Orientation: Molecular Structures In extending the studies of vibrational coherence to rotational coherence in isolated molecules, we formulated the concept of rotational recurrences (echoes!), which led to rotational coherence spectroscopy. A polarized picosecond (and later femtosecond) pulse was used to orient a molecular ensemble (Fig. 7). The molecules then rotate freely with different speeds

12

A. H. ZEWAIL

OrientationalAliQnment & Structural Determination

t=o

Time

b

Classical Motion

tU)

Figure 7. (a) Concept of time-dependent alignment as a method for structural determination. Top: Initial alignment at t = 0, dephasing, and recurrence of alignment at later times. Bottom: Classical motion of a rigid prolate symmetric top. (b) Structures of stilbene and tryptamine-water complex from rotational coherence spectroscopy; transients are shown. [see ref. 131.

(depending on the angular momentum), but despite the difference of their speeds, they realign again at well-defined times. This recurrence, or echo, gives direct information on the molecular structure, namely the moment of inertia along the principal axes. Figure 7 gives two examples of using rotational coherence spectroscopy to determine the molecular structure; there are now more than 100 structures obtained by this method. The approach has also been used to study ground- and excited-state structures. (For a recent review, see Ref. 13.)

w

L

I

0

I

I

AA

T i m (ns)

I

2

I

3

m r y

1

4

(b) Figure 7. (Continued)

I

Structures from Rotatlonal Coherence Spectroscopy

14

A. H. ZEWAIL

E. Coherence in Reactions: Wavepackets and Nuclear Motions The nature of coherence, when created on the femtosecond time scale, opened up the possibility of making localized nuclear wavepackets in molecules and reactions. This was achieved and probed in both nonreacfive and reactive systems. In this field of femtochemistry, one is studying the fundamentals of bond breaking and bond making. Initially, the elementary femtosecond dynamics for bound molecules, for elementary unimolecular reactions, and for bimolecular reactions were the focus used to establish the methodology and the synthesis of wavepackets in molecular systems. The reaction trajectories were also observed, and the spreading (dephasing) of the wavepacket and its recurrence or echo was seen. These femtochemical real-time probings of the elementary dynamics of the nuclear motion have been extended over the last 10 years to different phases (gases, molecular beams, clusters, surfaces, liquids, solids, and biological systems) and to many classes of reactions (elementary, more complex, solvated, etc.). Some examples are given below in Section 111, and these and other studies (see Section IV) are detailed in the collected works on femtochemistry mentioned above [5] and in some recent reviews [7].

F. Coherence in Solvation: Clusters and Dense Fluids The concept of coherent nuclear motion was extended to studies of solvation, as the time scale of solute-solvent collisions could be of the same magnitude or longer as the time scale of the wavepacket motion. Thus, on the femtosecond time scale, the system can be “frozen” in time and its interaction with the solvent can be mapped out by probing the evolution of the dynamics at longer times. The approach we took is that of stepwise solvation in two regimes of the dynamics. Studies of elementary and complex reactions were made in (1) molecular beams (clusters) and in (2) dense Auids (high pressures). Figure 8 shows the wavepacket motion of iodine in different argon solvent densities (up to 1900 atrn); elsewhere (see references below), we discuss this system up to 3000 atm. Of particular interest were the dephasing and predissociation rates as a function of solvent density, the mean vibrational frequency of the packet, and the molecular dynamics in the solvent. Studies in clusters of argon (Fig. 9) were made and compared to those in dense fluids without regular solvent structures. Complex reactions such as acid-base, isomerization, and electron and proton transfers have been examined similarly.

G. Coherence Control of Wavepackets: Reactive and Nonreactive Systems With the concept of molecular coherence well established, we wrote a review making the point that with ultrashort pulses we should be able to control


4

0

0

1

b p

12

2

3

n-4ddW.p

4

5

15

1

100

~~.115

cm-’

Figure 8. Coherence and solvation in dense fluids. The damping of coherence by dephasing is clearly manifested for iodine in helium at different pressures. Fourier transforms are to the right.

molecular dynamics [14]. With the success we had in probing such dynamics and the knowledge of multiple-pulse techniques (see above), we decided to focus some effort on the use of femtosecond multiple pulses to control the dynamics of a chemical reaction. In Fig. 10, we show how two successive pulses with well-defined phase angle can add constructively or destructively

16

A. H. ZEWAIL

-

Time. P

Figure 9. Femtosecond dynamics of an elementary reaction (12 21) in solvent (Ar) cages. The study was made in clusters for two types of excitation: to the dissociative A state and to the predisswiative B state. The potentials in the gas phase govern a much different time scale for bond breakage (femtosecond for A state and picosecond for B state). Based on the experimental transients, three snapshots of the dynamics are shown with the help of molecular dynamics simulations at the top. The bond breakage time, relative to solvent rearrangement, plays a crucial role in the subsequent recombination (caging) dynamics. Experimental transients for the A and B states and molecular dynamics simulations are shown.

-

the population of a wavepacket. Such an approach was exploited to turn on and @the yield of a chemical reaction: Xe + 12 XeI + I, as illustrated in Fig. 10. We extended the same approach to a unimolecular reaction, the dissociation of NaI, where the yield in a given channel can be changed by the control from the transition-state region (Fig. 11). The use of different pulses


17

Figure 10. Top: Multiple-pulse (femtosecond) preparation of wavepackets in molecular iodine. Two pulses were used for excitation, with a well-defined delay time (or phase angle), and one pulse for probing (middle). The phase of the wavepacket motion relative to time zero is shown on the left with the in-phase and out-of-phase feature of the oscillation (note the similarity to the IVR problem). Because of this known phase difference, wavepacket population can be “added or “subtracted” with multiple pulses as shown (at right) experimentally and theoretically. Bottom: Femtosecond control of a reaction yield: Xe + I2 XeI + I. The potential along three coordinates (1-1. I2 ... Xe, and Xe ... I) is shown. The product yield XeI was monitored using the chemiluminescence (right, bottom). The change in the yield of XeI with the delay between the initiating and controlling pulses is shown in the panel at the bottom left. The wavepacket motion of I2 is also shown and clearly tracks the modulation in the product yield. For this bimolecular reaction, studied in a gas cell, the controlling pulse acts as a “switch” to lift the wavepacket to the “harpoon region” of the reaction, where entry to product Xel is made.

-.

A. H.ZEWAIL

18 Potentid energy. Id cm-’ 30r I

1

4

3101620/589/(589)

0 1

blk At variable

-30

-

310/620/589/(589) 3101620/-/(589)

x , q : Affucd

0

5

10 15 htunuc~earseparation.

Femtosecond pulse sequence: XI x.

A

0

I

2

3

Time delay, ps

x;

Figure 11. Reaction channel control in the unimolecular dissociation of NaI. The potentials (a), the pulse sequence, and the results (b)are shown. The pulse A, controls the relative yield in the two channels: Na + I and Na* + I.

to manipulate the packet on such potentials is at the heart of the Rice-Tannor scheme of control discussed at the conference (see the chapters by these authors). There are other schemes discussed at the conference and for recent reviews see the excellent articles by Wilson’s and Manz’s groups [15, 21.

H. Coherence in Electron Diffraction: Complex Molecular Structures As a closing example of the powerful application of the concept of coherence in structures and dynamics, we point to its importance in obtaining molecular structural changes with time using ultrafast electron diffraction (UED) (Fig. 12). The UED technique has been developed, so far with -1-ps resolution (Fig. 12). We have reported recently that the introduction of rotational orientation (Section D above) to the diffraction in real time can provide a “three-dimensional image” of the structure, instead of the conventional two-


19

& CLCI

0

2

4

6

8

1

0

s. inverse A

0

&

1

2Intcrnuckcar 3 separation. 4 A5

1

2

Iodine

0

2

4

6

8

s,invtrse A

'c

0

3

4

5

,in

Y

b-8 Molcc~larsample (isotropic)

Detector

Figure 12. Coherence and UED. Shown are the experimental diffraction results (top) and the coherent alignment (bottom).

20

A. H.ZEWAIL

dimensional molecular scatterings (Fig. 12) observed in the gas phase. These methods complement the spectroscopic approach outlined above and promise numerous applications, especially in complex systems.

III. PROTOTYPE SYSTEMS: UNI- AND BIMOLECULAR REACTIONS A. Resonances in Unimolecular Reactions One example that illustrates the methodology and the concept of femtosecond transition-state spectroscopy (FTS)is the dissociation reactions of alkali halides. For these systems, the covalent (M+ X,where M denotes the alkali atom and X the halogen) potential and the ionic (M+ + X-)potential cross at an internuclear separation larger than 3 A. The reaction coordinate therefore changes character from being covalent at short distances to being ionic at larger distances. The reaction occurs by a harpoon mechanism and has a large cross section because of the involvement of the ionic potential. In studying this system, the first femtosecond pulse takes the ion pair M+X- to the covalent branch of the MX potential at a separation of 2.7 The activated complexes [MX]*$ ,following their coherent preparation, increase their internuclear separation and ultimately transform into the ionic [M+ . . X-]*form. With a series of pulses delayed in time from the first one the nuclear motion through the transition state and all the way to the final M + X products can be followed. The probe pulse examines the system at an absorption frequency corresponding to either the complex [M - - XI** or the free atom M. Figure 13 gives the observed transients of the NaI reaction for the two detection limits. The localized wavepacket and its motion is shown as calculated from ab initio quantum dynamics. The resonance along the reaction coordinate describes the oscillatory motion of the wavepacket when the activated complexes are monitored at a certain internuclear separation. The steps describe the quantized buildup of free Na, with separations matching the resonance period of the activated complexes. Not all of the complexes dissociate on every outbound pass, since there is a finite probability that the I atom I internuclear separation reaches can be harpooned again when the Na the crossing point at 7 A. The complexes survive for about 10 oscillations before completely dissociating to products. When adjusting the position of the crossing point by changing the difference in the ionization potential of M and the electronegativity of X (e.g., NaBr), the survival of the complex changes (NaBr complexes, e.g., survive for only one period). The results in Fig. 13 illustrate additionally some important features of the dynamics. The oscillatory motion is damped in a quantized fashion due to

A.

e . 1

21


Theoretical: Quantum wavepacket motion

Potential-energysurfaces

I Covalent

. I Ionic

r 0

1 : Internuclearseparation (A)

u 0

20

gR.=6.93iIO

6

10

20

15

Internuclear separation (i)

Experimental

g/ -2

-1

0

1

2

3

4

Tlme delay (ps)

5

6

7

8

Figure 13. Femtosecond dynamics of dissociation (NaI) reaction. Bottom: Experimental observations of wavepacket motion, made by detection of the activated complexes ”all** or the free Na atoms. Top: Potential energy curves (left) and the “exact” quantum calculations (right) showing the wavepacket as it changes in time and space. The corresponding changes in bond character are also noted: covalent (at 160 fs), covalent/ionic (at 500 fs), ionic (at 700 fs), and back to covalent (at 1.3 ps).

bifurcation of the wavepacket at the crossing point of the covalent and ionic potentials, as shown in the quantum calculation given in the figure. In fact, it is this damping that provides important parts of the dissociation dynamics, namely the reaction time, the probability of dissociation, and the extent of covalent and ionic characters in the bond. These observations and their anal-

22

A.

H.ZEWAIL

yses have been discussed in more detail elsewhere, and other systems have been examined similarly. Numerous theoretical studies of these systems have been made to test quantum, semiclassical and classical descriptions of the reaction dynamics and to compare theory with experiment.

B. Barrier Reactions: Saddle-Point Transition State

The simplest system for addressing the dynamics of barrier reactions is of the type [ABA]$ -+ AB + A. This system is the half-collision of the A + BA full collision (see Fig. 14). It involves one symmetrical stretch (Qs),one and one bend (4);it defines a barrier along the asymmetrical stretch reaction coordinate. The “lifetime” of the transition state over a saddle point near the top of the barrier is the most probable time for the system to stay near this configuration. It is simply expressed, for a one-dimensional reaction coordinate (frequency w ) near the top of the barrier, as = l / w . For values of tiw from 50 to 500 cm-’ ,T $ ranges from 100 to 10 fs. In addition to this motion, one must consider the transverse motion perpendicular to the reaction coordinate, with possible vibrational resonances, as discussed below. The IHgI system, representing this class of reactions, was one of the first reactions studied in femtochemistry [5]. The activated complexes [IHgI]*$, for which the asymmetric (translational) motion gives rise to vibrationally cold (or hot) nascent HgI, were prepared coherently at the crest of the energy barrier (Fig. 14). The barrier-descent motion was then observed using series of probe pulses. As the bond of the activated complex breaks during the descent, both the vibrational motion (-3oOfs) of the separating diatom and the mtutional motion (-2.3 ps) caused by the torque can be observed. These studies of the dynamics provided the initial geometry of the transition state, which was found to be bent, and the nature of the final torque, which induces rotations in the nascent HgI fragment. Classical and quantum molecular dynamics simulations show the important features of the dynamics and the nature of the force acting during bond breakage. Two snapshots are shown in Fig. 14. The force controls the remarkably persistent coherence in products, a feature that was unexpected, especially in view of the fact that all trajectory calculations are normally averaged (by Monte Carlo methods) without such coherences. Only recently has theory addressed this point and emphasized the importance of the transverse force, that is, the degree of anharmonicity perpendicular to the reaction coordinate. The same type of coherence along the reaction coordinate, first observed in 1987 by our group, was found for reactions in solutions, in clusters, and in solids, offering a new opportunity for examining solvent effects on reaction dynamics in the transition-state region.

tea),


23

Potential energy surfaces

II

I*

HgI + I

I + HgI

Theoretical: Quantum wave packet motion t=Ots

tr600fs

(a1 Figure 14. (a)Potential-energy surfaces, with a trajectory showing the coherent vibrational motion as the diatom separates from the I atom. nkto snapshots of the wavepacket motion (quantum molecular dynamics calculations) are shown for the same reaction at t = 0 and r = 600 fs. (b) Femtosecond dynamics of barrier reactions, IHgI system. Experimental obser-

vations of the vibrational (femtosecond) and rotational (picosecond) motions for the barrier (saddle-point transition state) descent, [IHgI]** --r HgI(vib, rot) + I, are shown. The vibrational coherence in the reaction trajectories (oscillations) is observed in both polarizations of FTS.The rotational orientation can be seen in the decay of FTS spectra (parallel) and buildup of FTS (perpendicular) as the HgI rotates during bond breakage (bottom).

A. H. ZEWAIL

24 Experimental

1

vibrational motion (fs)

0.8

-m

& 0.E

.-tn

-0 Q

$ 0

0.4

0.2

o FTS (parallel polarization) 0 FTS (perpendicular polarization) -Rotational alignment

C -500

I

0

I

500

I

I

I

I

I

I

1000 1500 2000 2500 3000 3500

Figure 14. (Conrinued)

Even more surprising was the fact that this same phenomenon was also found to be robust and common in biological systems, where wavepacket motion was found in the twisting of a bond (e.g., in rhodopsin and bacteriorhodopsin), in the breakage of a bond (e.g., in ligand-myoglobin systems), in photosynthetic reaction centers, and in the light-harvesting antenna of purple bacteria. The implications as to the global motion of the protein are fundamental to the understanding of the mechanism (coherent vs. nonstatistical


25

energy or electron flow) and such new observations are triggering numerous theoretical studies in these biological systems. For the wavepacket motion in dissociation and barrier reactions discussed above, there have been recent studies of the same (or similar) systems in solutions, and the results are striking. Sundstrom and co-workers have observed the wavepacket motion for the twisting process in a barrierless isomerization reaction in solutions. Their findings give a direct view of the motion and examine the nature (coherent vs. diffusive) of the coupling to the solvent during the reaction. This wavepacket-type behavior indicates the persistence of coherence along the reaction coordinate and provides the time scales for intramolecular motion and solvation. Hochstrasser's group has shown for Hg12 in ethanol solutions that the HgI is formed in a coherent state, similar to the observation we made in the gas phase. Their study is rich with information regarding solvated wavepacket dynamics, relaxation in the solvent, and the effective Potential Energy Surface (PES). Of particular interest is the study of solvent-induced relaxation of nascent fragments.

C. Bimolecular Reactions: Ground-State Dynamics

Real-time clocking of abstraction reactions was first performed on the I-H/COz system for the dynamics on the ground-state PES [5]:

H + C02

[HOCO]* --t CO + OH

Two pulses were used, the first to initiate the reaction and the second delayed to probe the OH product. The decay of [HOCO]* was observed in the buildup of the OH final fragment in real time. The two reagents were synthesized in a van der Waals complex. The results established that the reaction involves a collision complex and that the lifetime of [HOCO]* is relatively long, about a picosecond. Wittig's group [13 has recently reported accurate rates with subpicosecond resolution, covering a sufficiently large energy range to test the description of the lifetime of [HOCO]' by an RRKM theory. Recent crossed-molecular beam studies of OH and CO, from the group in Perugia, Italy, have shown that the angular distribution is consistent with a long-lived complex. Vector correlation studies by the Heidelberg group addressed the importance of the lifetime to IVR and to product state distributions. The molecular dynamics calculations (with ab initio potentials) by Clary, Schatz, and Zhang are also consistent with such lifetimes of the complex and provide new insight into the effect of energy, rotations, and resonances on the dynamics of [HOCO]' . This is one of the reactions in which both theory and experiment have been examined in a very critical and detailed manner.

26

A. H. ZEWAIL

For exchange reactions, the femtosecond dynamics of bond breaking and bond making were examined in the following system: Br + I2 -+[BrII]$ -+BrI + I The dynamics of this Br + 12 reaction (Fig. 15) were resolved in time by detecting the BrI with the probe pulses using laser-induced fluorescence. The reaction was found to be going through a sticky collision complex lasting tens of picoseconds. More recently, McDonald’s group has monitored this same reaction using multiphoton ionization mass Spectrometry and found the rise of I (and I;!) to be similar to the rise of BrI (Fig. 15). They proposed a picture of the PES for the dynamics, including the involvement of the Br* + I, surface. With molecular dynamics simulations, comparison [5] with the experimental results showed the trapping of trajectories in the [BrII]* potential well; the complex is a stable molecular species on the picosecond time scale. Gruebele et al. drew a simple analogy between collision (Br + 12) and half-collision (hv + 12) dynamics based on the change in bonding and employed frontier orbitals to describe it (see Ref. 5). The PES may involve avoided crossings. Other bimolecular reactions of complex systems, such as those of benzene and iodine and acid-base reactions, have also been studied. Currently, we are examining the inelastic and reactive collision of halogen atoms with polyatomics (e.g., CH3I). Other groups at the National Institutes of Science and Technology and at the University of Southern California have studied a new class of reactions: 0 + CHq [CH3OH]* CH3 + OH and H + ON;! HO + N2 or HN + NO.

-

-.

-+

D. Complex Organic Reactions One of the most well-studied addition/elimination reactions, both theoretically and experimentaily,is the ring opening of cyclobutane to yield ethylene or the reverse addition of two ethylenes to form cyclobutane (Fig. 16). Such is a classic case study for a Woodward-Hoffmann description of concerted reactions. The reaction, however, may proceed directly through a transition state at the saddle region of an activation barrier or it could proceed with a diradical intermediate, beginning with the breakage of one u bond to produce tetramethylene, which in turn passes through a transition state before yielding final products. The concept, therefore, besides being important to the definition of diradicals as stable species, is crucial to the fundamental nature of the reaction dynamics: a concerted one-step process vs. a two-step process with an intermediate. Experimental and theoretical studies have long focused on the possible existence of diradicals and on the role they play in affecting the processes


27

E? a, 5

a,

Experimental

Theoretical

zz nme delay (ps)

'2

-

3

4

S o

6

7

Br-I d i E t a n e e (A)

Figure 15. Femtosecond dynamics of the Br + 12 BrI + I exchange reaction. Here, the collision complex is long lived, zC = 53 ps. As shown by the molecular dynamics, the [BrIIIS complex is trapped in the transition-state region; the reaction may also involve avoided crossings (see text).

of cleavage, closure, and rotation. The experimental approach is based primarily on studies of the stereochemistry of reactants and products, chemical kinetics, and the effect of different precursors on the generation of diradicals. The time "clock" for rates is internal, inferred from the rotation of a single bond, and is used to account for any retention of stereochemistry from reac-

A. H.ZEWAIL

28

"6'

Diradical intermediate

Parent

diradical

t o 3,O

e

,

3.5

, 10 oo(

-A

5 OO(

0

0

0.2

1 1 1 1 , 1 0.4

0.6

0.8

R (nm)

Figure 1. The IBr potential energy curve. The arrows depict the one- and three-photon pathways that interfere with and whose phase difference is used to control the product selection. (From Ref. 27.)

223

CONTROL OF QUANTUM MANY-BODY DYNAMICS

)

Figure 2. Contour plot of the percentage of Br*(2Plp) in the photodissociation of IBr from an initial bound state in the vibrationless ground state with j i = 1 and mi = 0. (From Ref. 27.)

tion of the molecular angular momentum along the z axis is fixed, and Fig. 3 shows the results obtained when the projection of the angular momentum along the z axis is randomized for fixed angular momentum. These results clearly demonstrate that, in principle, the ratio of the amounts of products in this branching reaction can be controlled over a considerable range by varying the intensities and relative polarization of the one- and three-photon fragmentation pathways . An example of one photon-three photon continuous-wave (CW) interference control of the product distribution in the branching photofragmentation and photoionization reactions

HI HI

--c --c

HI'+e H+I

has been reported by Zhu et al. [15]. Figure 4 displays the HI and I yields

S. A. RICE

224 36

28

6 rc,

21

I

8

I4

7

Figure 3. Contour plot of the percentage of Br*(2Pl,z) in the photodissociation of IBr from an initial bound state in the vibrationless ground state with j i = 42 and m-averaged. (From Ref. 27.1

(measured by the ion concentrations HI+ and I+) as a function of the phase difference between the three-photon and one-photon fields (proportional to the pressure of hydrogen through which the two beams pass, taking advantage of the difference in refractive indices at the frequencies w3 and w I ). Note that the signals from HI+ and I+ are out of phase (about 150"), which is the signature of the phase-modulated control of the yields of the products in this branching reaction. There are many different variants of the Brumer-Shapiro control scheme; the reader is referred to the original publications for discussions of these variants. Bmmer and Shapiro have also discussed the selection rules relevant to interference control of product selectivity and the influence of sources of incoherence on the quality of interference control achievable [26].


I'

14

i

O

L

1

2

3

Internuclear Distance (A) (a1 Figure 4. ( a ) Potential-energy diagram for HI, with arrows showing the one- and threephoton paths whose interference is used to control the ratio of products formed in the branching reactions HI I+ + e and HI H + I. (b) Modulation of the HI' and I+ signals as a function of phase difference between the one- and three-photon pathways (proportional to the HZ pressure in the cell used to phase shift the beams). (From Ref. 15.)

-.

-.

226

S. A. RICE

'p .-V

> c

0

I I

0.I25

1

2

3

4

5

6

7

8

H, Pressure (Torr) (b) Figure 4.

(Continued)

IV. THE TANNOR-RICE-KOSLOFT-RABITZ METHOD In 1985 Tannor and Rice [4] showed that it is possible to control the relative yields of products in a branching chemical reaction by varying the interval between an initial pump pulse of radiation that transfers amplitude from the ground-state potential-energy surface to an excited-state potential-energy surface and a later dump pulse of radiation that transfers amplitude in the opposite direction. They represented the search for the optimal dump pulse shape for a given pump pulse shape and total number of photons as a problem in the calculus of variations. The pump-dump scheme was first formulated using second-order perturbation theory to describe the time evolution of the wavepacket amplitude on the ground and excited potential-energy surfaces. This formulation leads to a nonlinear integral equation for the optimal pulse shape. The utility and power of optimal control theory for the calculation of the temporal shape and spectral content of the pulse that maximizes the yield of the specified product were first recognized by Rabitz and coworkers [7, 81 and, independently and a little later, by Kosloff et al. 161.


227

Reformulations and/or extensions of this work have been reported by several groups. Because of the transparency of interpretation, it is convenient to start with the original Tannor-Rice perturbation theory treatment of control of molecular dynamics. We consider a molecule that can undergo fragmentation to produce two products: ABC ABC

-+

-+

BC+A AB+C

For simplicity, the fragmentation is assumed to occur on the ground-state potential-energy surface. If the bond lengths and bond angles of the molecule are different in an excited state than in the ground state, the potential-energy surface for that excited state will be translated and rotated relative to the ground-state potential-energy surface. Consequently, amplitude placed on the excited-state potential-energy surface by a photoinduced Franck-Condon transition will not be stationary; it is the evolution of the amplitude with increasing time that is used to alter the probability of formation of the two products of the reaction. The Hamiltonian for the system we are considering can be represented as a 2 x 2 matrix of operators:

and the time-dependent Schriidinger equation reads

We assume that at t = 0 the system is in the ground state, so

Equation (4.2) can be transformed into two coupled integral equations for the amplitudes on the ground- and excited-state potential-energy surfaces, namely,

228

S . A. RICE

(4.3)

To second order in perturbation theory (the weak-field regime) we have

which contains the field only to second order. Equation (4.4) has the following simple interpretation: The amplitude $g(0)evolves on the ground-state potential-energy surface from t = 0 to t = t 1 . At f 1 the ground-state amplitude is transferred, by a vertical transition, to the excited-state potential-energy surface, on which it propagates until t2. At t 2 the amplitude on the excitedstate surface is transferred, by a vertical transition, back to the ground-state surface, on which it evolves until t. Because the fields E(t1) and E(t2) are distributed in time, we must integrate over 41 the instants at which the up and down transitions can occur. Let FA and Pc be the projection operators onto the exit channels for products BC + A and AB + C, respe5tively. Given $&), the probability for forming product A is limt- & g ( t ) l P A I$&)). We can gain insight into how pulse shape and amplitude propagation are related by consideration of a simple example. Suppose, starting from an initial state with wave function $', we wish to populate a particular discrete state of the molecule, with wave function $f,by a two-photon process. Given the shape of the pump pulse El(t,), what shape must the dump pulse E2(t2) have to maximize the population of the target state? Tannor and Rice showed that for given El(tl)

that is, the optimal dump pulse shape is matched to the convolution of the


229

excitation pulse shape with the wavepacket propagated on the excited-state potential-energy surface. A similar result is found when the final state is in the continuum, although in that case the formal representation of the dump pulse shape involves a Fredholm integral equation of the second kind. To determine how effective pump-dump pulse separation is as a tool for altering the ratio of concentrations of products in a branching reaction, Tannor et al. [5] studied a model system with a ground-state potential-energy HH + D and surface resembling that for the branching reactions HHD HHD DH + H. The model calculations did not include the effects of molecular rotation. We show in Fig. 5 the results of that calculation. Clearly, the ratio of product concentrations can be varied over a considerable range by altering the time between the pump and dump pulses. The pulse delay control scheme has been demonstrated by Baumer and co-workers [171 with respect to the competition between ionization and dis-

-

-

I

Quantum branching ratios

I= Channel 1 0 = Channel 2

210

410

610

810

1010

Stimulation time (au)

-

Figure 5. Product yield as a function of pulse separation for the model branching reaction patterned after HHD + HH + D and HHD DH + H. Channel 1 corresponds to formation of D.

230

S. A. RICE

4.5

+

(D

+N

z"

3.0

1.5

I 0

I

*

I

2000

loo0 Pump-probe delay (fs)

Figure 6. Ratio of the concentrations Na2+/Na+ as a function of pulse delay for the competing reactions Naz --c Na2' + e and Na2 + Na+ + Na + e. (From Ref. 17.)

-

sociative ionization of Na2, namely Na2 Na2+ + e versus Na2 Na' + Na + e, by varying the time delay between the first and second pulses. Figure 6 displays the ratio of concentrations Na2+/Na+ found in their experiments. It is easy to see that if one wishes to use the Tannor-Rice scheme to generate a large concentration of a particular reaction product, it is necessary to have most of the wavepacket amplitude on the excited-state potential-energy surface simply and compactly distributed over that product exit channel on the ground-state potential-energy surface. However, in the typical case, the evolution of the wavepacket on the excited-state surface generates a very complicated distribution of amplitude; hence a simple dump pulse cannot efficiently transfer amplitude to the exit channel on the ground-state surface. We can determine what initial amplitude distribution on the excited-state surface will evolve to the amplitude distribution over the exit channel that we seek by integrating the Schriidinger equation backward with the desired final amplitude distribution as an initial condition. The result of this calculation is an initial distribution of amplitude so complicated that it cannot be created by a Franck-Condon transition from the ground state. Nevertheless, this calculation conveys an important message. If, instead of using separated pump and dump pulses, we use a temporally and spectrally shaped field, it should be possible to continuously transfer amplitude back and forth between the --+


23 1

ground-state and excited-state surfaces as the wavepackets move about on these surfaces. It is this observation that led Kosloff et al. [6] to develop the formalism for optimal pulse shaping. Consider again the system studied by Tannor and Rice. We now seek to maximize

at some specified time t = T. Here, J is a functional of the radiation field E(t); hence the maximization is to be carried out with respect to variation of the functional form of E(t), that is, its temporal shape and spectral content. The time T is chosen such that the outgoing wavepacket amplitude on the excitedstate surface is beyond any barriers in the ground-state surface exit channel for the particular product selected. When properly chosen, J is independent of T. The formalism used to calculate the pulse shape that maximizes J is optimal control theory. This formalism can be considered to be an extension of the calculus of variations to the case where the constraints include differential equations. In general, the constraints expressed in the form of differential equations express the restriction that the amplitude must always satisfy the Schrodinger equation. In addition, there can be a variety of other constraints, such as a restriction on the total energy in the pulse or on the shape of the pulse. These constraints are accounted for by the method of Lagrange multipliers, which modify the objective functional (4.6) and thereby permit the calculation of the unconstrained maximum of the modified objective functional. When the only constraints are satisfaction of the Schrdinger equation and limitation of the pulse energy, the modified objective functional can be written in the form

where

232

S. A. RICE

The variation of 5, is taken with respect to E and J,.We require that 67 = 0 for all 6 Re E, 6 Im E, and 6$. Then one finds that we must simultaneously satisfy

subject to

and

a*

i A -= H $

(4.1 1)

J,(O) = *o

(4.12)

%,

at

subject to

Equation (4.9) is the equation of motion of the Lagrange multiplier that restricts the solution to satisfy the Schriidinger equation; it is to be solved subject to the final-state condition (4.10). Equation (4.11) is the Schrodinger equation for our system; it is to be solved subject to the initial condition (4.12). The field that results from these calculations is given by

with

Note that the calculation of the optimal pulse shape is a double-ended boundary-value problem: J , is known at t = 0 while x is known at t = T. This aspect of the calculation of the optimal pulse shape mirrors the considerations advanced concerning the competition between spreading of the wavepacket as it moves on the potential-energy surface and the use of interference between pump and dump fields to counteract that spreading.


233

The solution to Eqs. (4.9)-(4.14) must, in general, be obtained by numerical analysis. The reader is referred to the literature for the techniques used. We consider, as an example, calculation of the optimal pulse shapes for generation of products for (essentially) the same model system studied by Tannor, Kosloff, and Rice (see above). Some of the results [6] of the calculations are shown in Figs. (7) and (8).

.075 h

c

0.5 1 e

s Q)

0.3

0

;F

f

-.025

- .05C

300 600 900 1200 1500

1 .a

E

B z

b

a

J

- .5

0 300 600 900 1200 1500

Time (a.u.)

0.6;

0.3: -0.01

J

10

C

375

750

x 10' Time (a.u.)

113

J

150

Figum 7. (a) Pulse sequence resulting from the optimization of the control field for the HH + D and HHD reaction surface patterned after that for the branching reactions HHD DH + H.This pulse sequence is intended to maximize the formation of D. (b) The Husimi transform of the pulse sequence shown in (a). (c) Time dependence of the norms of the groundstate and excited-state populations as a result of application of the pulse sequence shown in (a). Absolute value of the ground-state wave function at 1500 au (37.5 fs) propagated under the pulse sequence shown in (a). shown superposed on a contour diagram of the groundstate potential-energy surface. (From D. J. Tannor and Y. Jin, in Mode Selective Chemistry, B. Pullman, J. Jortner, and R. D. Levine, Eds. Kluwer, Dordrecht, 1991.)

-

+

234

S. A. RICE

0.5 ?

3

0.3

0.1

c 0 a -.l U

E

LI

. 0 7 5 -

a

300 600 900 1200 1500

0

Time (a.u.)

-

b 5 1 0. 300. 600. 900.1200. 1500 Time (a.u.)

I

1.00 1

-0.00

- .3

-

OOo

375

750

x 10' Time ( a u )

113

,501

Figure 8. ( a ) Pulse sequence resulting from optimization of the control field to generate H in the same reaction as studied in Fig. 6. (b) The Husimi transform of the pulse sequence shown in (a).(c) Time dependence of the norms of the ground-state and excited-state populations as a result of application of the pulse sequence shown in (a). Absolute value of the ground-state wave function at 1500 au (37.5 fs) propagated under the pulse sequence shown in (a), shown superposed on a contour diagram of the ground-state potential energy surface. (From D. I. Tannor and Y. Jin, in Mode Selecrive Chemistry, B. Pullman, J. Jortner, and R. D. Levine, Eds. Kluwer, Dordrecht, 1991.)

In general, the results of the calculations establish that it is possible to guide the reaction to preferentially form one or the other product with high yield. Note that, unlike the original Tannor-Rice pump-dump scheme, in which the pulse sequences that favor the different products have different temporal separations, the complex optimal pulses occupy about the same time window. Indeed, the optimal pulse shape that generates one product is very crudely like a two-pulse sequence, which suggests that the mechanism of the enhancement of product formation in this case is that the time delay between the pulses is such that the wavepacket on the excited-state


235

potential-energy surface is located over the ground-state exit channel and has outgoing momentum when the second pulse stimulates transfer to the ground-state surface. On the other hand, the optimal pulse shape that generates the other product has a more complex structure and, in fact, indicates there is continuous excitation and deexcitation, albeit with amplitudes that vary with time. The mechanism of enhancement of product formation in this case appears to be cyclical repetition of the transfer of amplitude between the two potential-energy surfaces in a fashion that minimizes the destructive interference in the transfer of the wavepacket placed over the exit channel on the ground state. The plots displayed in Figs. 7 and 8 do not show the phases of the optimally shaped pulses; these are likely significantly different. Finally, we note that the optimally shaped pulses convert a large fraction of the reactant to the desired products, unlike the simple Tannor-Rice scheme, which generates high selectivity of product formation but only small yields of products. We will return to the issue of complete transformation of a reactant to a selected product later in this review. Wilson and co-workers [28-321 have proposed a slightly different formulation of the calculation of the field required for optimal control of a quantum many-body system. This formulation is more general than the analysis described above in the sense that it is based on the equation of motion for the density matrix. The advantage that the density matrix representation of the system affords is the ability to study mixed states of the system, such as are characteristic of a thermal ensemble of molecules. Most of the applications of control theory to molecular dynamics reported by Wilson and co-workers are based on the assumption that the yield is determined by linear response theory, which restricts the cases that can be considered to those with weak fields. This version of the control theory is less general than the analysis described above. Indeed, this version of the control theory is very similar to the second-order perturbation theory analysis of Tannor and Rice. For example, with (4.6) as target functional and (4.9) as the constraint on the pulse energy, the yield is determined by the solution of an eigenvalue problem in which the optimal field is the eigenfunction belonging to the maximum eigenvalue of a Fredholm linear integral equation whose kernel depends on the evolution of the amplitude is an undriven system. Specifically, to second order in the field strength the density matrix is given as

where, for a system with two electronic states,

236

S . A. RICE

(4.16)

and 60is the (Liouville space) propagator associated with the field-free system Hamiltonian. The interpretation of (4.15) is very much like the interpretation of (4.4). The initial state of the system is represented by &), and it propagates freely until t l , at which time an interaction with the field occurs (e.g., a photon is absorbed). The excited system then propagates freely from r 1 to 12, at which time a second interaction with the field occurs (e.g., there is stimulated emission). The integrations account for all possible times at which the two interactions of the molecule with the field can occur, With (4.6) as the target functional the optimal control field is determined by

In this weak-field limit the relationship between the yield of product and the optimal field is

(4.18)

We call the reader’s attention to the similarity between Eqs. (4.17) and (4.5). The result obtained by Wilson and co-workers is more general that the Tannor-Rice result in that the latter calculates the field that maximizes the product yield for a pump pulse with given shape whereas the former makes no restriction concerning the shape of the pump pulse and does not assume that the pump and dump pulses can be distinguished from one another. Wilson and co-workers have also considered optimal control of molecular dynamics in the strong-field regime using the density matrix representation of the state of the system 1321. This formulation is also substantially the same as that of Kosloff et al. [6] and that of Pierce et al. [8, 91. Kim and Girardeau [33] have treated the optimization of the target functional, subject to the constraint specified by (4.8), using the Balian-Veneroni [34]variational method. The overall structure of the formal results is similar to that we have already described.


237

V. GENERIC CONDITIONS FOR CONTROL OF QUANTUM DYNAMICS We now describe a generic formalism for active control of a molecule coupled to the radiation field. That is, we examine how the control conditions for a variety of circumstances can be expressed in terms of the phase of the external field and the phase of the relevant dynamical variables. For simplicity, we consider a simple case, namely, when only two electronic states of the molecule play roles in the reaction dynamics; we take these to be the ground electronic state and the first excited electronic state. The radiation that couples the two surfaces is the means of control. The internal state of the molecule is defined by the density operators i,,j E g, e, where g and e denote the ground and excited states, respectively. The combined density operator describing the state of the system can be represented as

where the symbol 8 denotes the-tensor product, f i j is a projection operator on surfacej E g, e, and the & are raising and lowering operators that transfer amplitude from one surface to another. The first two terms in Eq. (5.1) represent the state of the molecules with population on the ground and excited surfaces, whereas the last two terms represent the electronic coherence induced by the radiation field. The Hamiltonian of the system consists of the sum of internal Hamiltonians and an interaction term

where the zero-order Hamiltonian is given by Ho=

fi* fig B i g+ H e63 Pe = 8 1 + vgig + veie 2m II

(5.3)

Of course, each of the terms in (5.3) is a function of the internal coordinates of the molecule. For the case we consider, the interaction terms in (5.2), which control the transfer of amplitude between the two electronic manifolds, only contain the radiative coupling term

V , = -F @ {i+E(t)+ kE*(t)}

(5.4)

238

S. A. RICE

where is the transition dipole moment operator and E(t) represents a semiclassical time-dependent radiation field. It is via control of the spectral composition, the time profile of the field amplitude, and the phase of the field that we can control the evolution of the molecule. Although not accounted for in the Hamiltonian considered thus far, when intramolecular coupling of electronic manifolds is included in the Hamiltonian, radiationless transitions within the molecule can be included in the group of dynamical processes to be controlled. The evolution of the molecule is described by the generalized Liouville-von Neumann equation 135, 361

where is an operator representing the dissipative coupling of the system to background states. Equation (5.5) describes the dynamics of an open quantum mechanical system under the assumption that the evolution operator d:fines a dynamical semigroup [36-391. The source of the dissipative term LD; is the reduction of the combined system and bath dynamics to the dynamics of the system only. Th: semigroup formalism provides an explicit form for the dissipative operator Lo, but we shall not need that detailed form. Note that the first term in (5.5) describes the unitary dynamics supported by the Hamiltonian. The equation of motion of an explicitly time-dependent operator is given as

whe;e the first term represents the explicit time depeFdence of the operator A, the second term the Hamiltonian evolution of A, and the third term the Heisenberg representation version of the dissipative superoperator in Eq. (5.5). The mechanism by which control of the dynamical evolution of our model molecule is achieved is the alteration, by variation of the external field, of population and energy transfers between its two electronic states. This mechanism is, in a sense, analogous to the control of transformation of the equilibrium states of a macroscopic system by altering population and energy transfers between macroscopic states via variation of external parameters. Accordingly, it is interesting to examine the exchange of energy between the molecule and the external field and to relate that energy exchange to


239

alteration in the populations of the molecular states. The rate of change of energy is, using the equation of motion described above,

(5.7) since [h,k]= 0. Equation (3.1) is a version of the first law of thermodynamics [40-43], written in terms of the time rate of change of the energy and the power,

which is the time derivative of the work, and the heat flow,

dQ = (iEh) dt

(5.9)

With these definitions, the power absorbed from the field into the system becomes (5.10) In (5.10), @ Q i+) is the expectation value of the instantaneous transition dipole moment; variation of its value provides the means for controlling the molecular evolution. For the system under consideration, with only two electronic potentialenergy surfaces, the conservation of population implies that d N g + dN, = 0

(5.11)

The flow of population from one electronic surface to the other can then be calculated using Eq.(5.6): (5.12) Ignoring nonradiative couplings between the ground- and excited-state sur-

240

S. A. RICE

faces implies setting LDP, = 0, whereupon the ground-state surface population change becomes * * A

-d N - ,- - 2 Im(G 63 5+) E(t)) dt h

(5.13)

The flow of energy from the ground state can also be calculated when it is assumed that the rate of electronic dephasing is smallA(i.5.,LEk, = 0) and/or the rate of pure vibrational dephasing is small (i.e., LZH, = 0). These conditions apply when the rate of relaxation to equilibrium is small relative to the rate of loss of phase coherence. Under these conditions (5.14)

Consider the case that the external field is pulsed so E(t) = 0 when t = f=. Energy balance requires that

which is a formal statement of the fact that the power uptake can be distributed into the ground and excited states as yell as into energy stored by the interaction. The boundary term -2Re{G@S+) *E(t)},which goes to zero when deriving Eq. (5.15), represents the transient loading of the system. The energy balance equation is one of the main tools of the thermodynamic representation of the control process. The spatial derivative of the loading term represents the internal force that the electromagnetic field exerts on the molecule. This force is (5.16)

where x represents the internal coordinates of the molecule. This interpretation of the interaction between an external field and a molecule leads to the possibility of applying a directional force on the molecule. Molecular transfer processes can be promoted either by controlling the field E ( t ) or its time derivative. We note that the transfer equations (5.13) and (5.14) have similar structure, namely, each contains the imaginary part


24 1

of a product of a molecular expectation value (X) and the field E(t). Equation (5.10) has an analogous structure; transfer is controlled by the real part of the product of a molecular expectation value and the time derivative of the field. For convenience we rewrite (5.13) in the form (5.17)

where $p is the phase angle of the instantaneous dipole moment and +E is the phase angle of the radiation field. The overall phase angle in (5.17) is the sum of the phase angle of the induced polarization of the molecule and the phase angle of the polarization of the light. In a similar way (5.18)

and (5.19) where +,,H is the phase angle of

GH, @ $+). We also have (5.20)

+,,.

where is the phase angle of (@;/ax) @ g+). Equations (5.17)-(5.20) clearly show that the most important control parameter for transfer of energy or population between the energy surfaces of a molecule is the phase angle. Consider the case when the external field is a monochromatic circularly polarized pulse, IE(r)l = A exp(-iwt), where A is a slowly varying envelope function. For this pulse the phase angle of dE/& is rotated by 1~/2from the direction of E. From Eqs. (5.17) and (5.18) we then find d% P=-Aw dt

(5.21)

so the power absorbed is proportional to the population transfer. When more general pulse shapes are used, the character of the control can be active or passive. For active control conditions the phase angle is given as

242

S. A. RICE

4~ 4' = +

4p

+4E

{0

for maximum energy transfer for maximum energy emission

r

(5.22)

U for maximum positive population transfer

=

{-:

(5.23) for maximum negative population transfer

U -

maximum energy transfer to the ground-state surface for maximum energy removal from the ground-state surface

(5.24) $1' -k

6E=

0 ?r

for maximum positive force for maximum negative force

(5.25)

Active control of population transfer using the control relation displayed in Eq. (5.23) has been demonstrated experimentally by Sherer et al. [18]. In this experiment gaseous 12 was irradiated with two short (femtosecond) laser pulses; the first pulse transfers population from the ground-state potential-energy surface to the excited-state potential-energy surface, thereby creating an instantaneous transition dipole moment. The instantaneous transition dipole moment is modulated by the molecular vibration on the excitedstate surface. At the proper instant, when the instantaneous transition dipole moment expectation value is maximized, a second pulse is applied. The direction of population transfer is then controlled by changing the phase of the second pulse relative to that of the first pulse. Another interesting possibility that emerges from examination of Eqs. (5.17H5.25) is the induction of stimulated emission; this is predicted to occur when the combined phase angle in Eq. (5.22) is u.In principle, then, provided the phase relation is right, a pulsed laser can be built from an ensemble of molecules without the usual condition of population inversion. For some purposes, passive control of molecular evolution can be more important than active control; one such case occurs when we wish to prevent transfer of population or energy. The phase angle relations are

4,, + 4 k = +r

for zero total energy transfer

4,, + 4E= 0, ?r for zero population transfer

(5.26)

(5.27)


&, +

= 0, T

+ t$E = +r

243

for zero change in the ground-state energy (5.28) for zero force

(5.29)

Examination of the passive control conditions in Eqs. (5.26H5.29) shows that there are two values of the sum of phase angles for which zero transfer occurs. In principle, then, one can simultaneously block the transfer of, say, the energy and select the direction of the transfer of the population. One particularly interesting case is the definition of the phase angles for zero total power absorption. Since no energy is absorbed or emitted from the field these conditions define laser catalysis [a]. A note of caution must be inserted at this point. It appears, at first sight, that there is a meaning that can be attached to the absolute phase of the field and to the phases of the molecular expectation values. However, it must be remembered that the phase of the molecular quantity is induced by the radiation field prior to the present time. Therefore all phases must be related to the phase of a previous pulse that synchronizes the molecular clock with the field clock. With this synchronization it is possible to understand how quantum mechanical interference between events induced in the past propagates and can be used to control energy and/or population transfer at a later time. The generalization to the control of the dynamics of a molecule with n electronic states is straightforward. For the purpose of deducing the control conditions we will examine the extreme case in which every possible pair of these electronic states is connected via the radiation field and a nonzero transition dipole moment. If the molecule is coupled to a radiation field that is a superposition of individual fields, each of which is resonant with a dipole allowed transition between two surfaces, the density operator of the system can be represented in the form

yhere kii is the projection operator onto surface i, i E 1, .. . , n, and $, and

qjare lowering and raising operators.

The various transfer equations for the control of molecular dynamics can

be worked out as before, leading to the following phase angle conditions for

active and passive control:

244

S . A. RICE

$ij

+ &..

0 for maximum energy absorption T for maximum energy emission

=

(5.31)

for I c i I n, where Ei, is the resonant electric field component between states i and j;

‘pli

+ ’Eli =

{

for maximum positive population transfer for maximum negative population transfer

$r

(5.32)

where Eli is the resonant field component between the ground state and excited state i; ’ $T

-

‘ p ~ ‘ ~ i j= +

{

+

for maximum energy transfer to ground-state surface (5.33) for maximum energy removal from ground-state surface

0 for maximum positive force r for maximum negative force

(5.34)

The above relations define the conditions for concurrent control of population and energy transfers between all of the states of the system that are connected by dipole allowed transitions. It is unlikely that a situation that complicated will ever be encountered. In the n-state molecule language, typically, not all pairs of states of the molecule are connected with nonzero transition dipole moments. In the skeleton spectrum language, there is usually only a small subset of dipole coupled “doorway states.” In both cases, of course, when only some pairs of states are coupled with nonzero transition dipole moments, the appropriate control conditions are simplified. We now consider, as an example, the construction of a globally optimym control field that affects the transfer of a ground-state quantity 8 = (A,) with minimum power consumption under the restriction of zero population transfer between the two electronic states of the system. As before, this field is obtained by varying 9 at a specific final time T with the following constraints: (a) The evolution of the system is governed by the Liouville-von Neumann equation (5.5). (b) There is zero population transfer so dN, = 0.


(c) The power consumption is bounded by (4.9).

245

Taking account of the constraints by the method of Lagrange multipliers, the functional to be minimized takes the form

8*= Tr[i, 8 f',$(t)]

+

1: [($ dt Tr

- I$)

h + h(E(t)(']

(5.35)

where h is an operator Lagrange multiplier and X is a scalar Lagrange multiplier. The variation of 8*is with respect to ;and IEI. The condition dN,/dt = 0 determines the phase of the optimal field through Eq. (5.27). It therefore is omitted from the variation. Taking the variation of (5.35) and integrating by parts leads to the following equations: (a) A forward equation for the density operator,

a; = L'; A

at

(5.36)

subject to the initial condition $ = b(0). (b) A backward equation for the Lagrange operator h, dt

(5.37)

subject to the final condition h ( T ) = A, 03 b,. The dissipative part of (5.37) is symmetric in time, meaning that dissipation takes place in the forward as well as in the backward evolution. (c) A condition on the field:

Equation (5.38)can be interpreted as the scalar product of a forwardmoving density and a backward-moving time-dependent operator. The optimal field at time t is determined by a time-dependent objective function propagated from the target time T backward to time t . A first-order perturbation approach to obtain a similar equation for optimal chemical control in Liouville space has been derived in a different method by Yan et al. [28].

246

S . A. RICE

In dissipative dynamics, the backward-propagating target operator decays into a stationary operator, and therefore, i*b(--) = 0. This leads to loss of control, as can be seen from Eq. (5.38). If the goal of the control process is the reduction of the epergy on the ground-state potential-enepy surface, the target operator is H,(T). When the control field is weak, B(t) can be expanded in powers of the field. For the energy reduction scheme under consideration the target fynction then becomes time dependent and is just the Schrodinger operator H,. Thus far we have not made explicit use of the phase constraint that defines the control of a particular dynamical process. Although we have use! a c9nstraint on the energy while minimizing, via the variational calculus, (Ag@Pg) on the ground-state surface, this procedure yields only the variational solution for the amplitude of the electric field; the variational solution for the field phase is lost. However, the phase of the field is constrained by the condition on dNg/dt through Q. (5.27). In general, if the goal is to minimize some dynamical function without consideration of the changes of any other observables, then we do not need to explicitly specify any of the phase relationships exhibited by the field. We now examine the formalism needed to explicitly include a constraint on the phase of the field in the optimization procedure [20]. Consider the case where we try to minimize the ground surface energy under the condition of zero population transfer, for which we have the phase relation

To incorporate (5.39) as a constraint in the variational calculation of the optimal field, we represent the electric field in the form E(t) = A&) exp[i+~(t)] and the objective functional in the form

where XI and A2 are two scalar Lagrange multipliers. We note that the time average of 4~ + 4p vanishes but the time average of ( 4 +~4J2 is positive definite, hence the form that a ears in (5.40). Taking the variation of 8 with respect to i , AE and ~#JE leads to the

P


247

folloying equation for the phase, in addition to (5.36), (5.37), and (5.38)for

PO), B W ,and A&):

(5.41)

Equation (5.41) explicitly describes how the time evolution of the phase angles must vary so as to minimize the value of (&c + c#J,,)~to satisfy the constraint of zero population transfer between potential-energy surfaces. Note that control of the field phase is critically influenced by the backward propagation of the target function B(t).

VI. HOW MUCH CONTROL OF QUANTUM MANY-BODY DYNAMICS IS ATTAINABLE? Thus far our examination of the quantum mechanical basis for control of many-body dynamics has proceeded under the assumption that a control field that will generate the goal we wish to achieve (e.g., maximizing the yield of a particular product of a reaction) exists. The task of the analysis is, then, to find that control field. We have not asked if there is a fundamental limit to the extent of control of quantum dynamics that is attainable; that is, whether there is an analogue of the limit imposed by the second law of thermodynamics on the extent of transformation of heat into work. Nor have we examined the limitation to achievable control arising from the sensitivity of the structure of the control field to uncertainties in our knowledge of molecular properties or to fluctuations in the control field arising from the source lasers. It is these subjects that we briefly discuss in this section. We consider, first, whether it is in principle possible to control the quantum dynamics of a many-body system. The goal of such a study is the establishment of an existence theorem, for which purpose it is necessary to distinguish between complete controllability and optimal control of a system. A system is completely controllable if an arbitrary initial state can be transformed, without loss to other states, into an arbitrary final state at some time T. A system is strongly completely controllable if an arbitrary initial state can be transformed, without loss to other states, into an arbitrary final state at a specified time T. Optimal control theory designs a field, subject to specified constraints, that guides the evolution of an initial state of the system to be as close as possible to the desired final state at time T. The work of Huang, Tarn, Clark et al. [46-SO] deals with a general for-

248

S. A. RICE

mulation of the complete controllability problem for quantum many-body dynamics and includes an existence proof that establishes sufficient (but not necessary) conditions for complete co~trol.They consider a system [45] that can be described by the Hamiltonian Ho in the absence of control fields; the system is assumed to have a discrete, but not necessarily bounded, spectrum (e.g., a harmonic oscillator). To control the dynamical evolution of the system, external fields are applied and the Schrodinger equation has the form

where the uk are real functions of the time and the f i k are linear Hermitian operators. Note that the U k needed to control the evolution of the state of the system depend on the state of the system, so Eq. (6.1) is, in fact, strongly nonlinear. The Huang-Tam-Clark theorem [47] is developed for the case that the fik are independent of the time and the control amplitudes U k are piecewise constant functions of the time. The proof involves many technical details; we will be concerned only with a loose paraphrasing of the results. Huang et al. show that, for a system with a discrete spectrum, under the conditions stated for the control field amplitudes and the corresponding operators, one can always find a set of field amplitudes that will guide the evolution of an initial state0' to come arl$rarily close to a chosen final state at some time T. If the set of operators H k generates an infinite dimensional Lie algebra, an infinite set of switchings of the piecewise constant fields is required to achieve the final state. However, even though an infinite set of switchings is needed to reach the limit of complete transformation of the initial state to the final state, a very large fraction of that transformation may be achieved in a small n%mber of switchings. If the Lie algebra generated by the set of operators H k has finite dimension, it can be shown that a system with a discrete nondegenerate spectrum is completely controllable in the sense that an arbitrary initial state can be transformed into an arbitrary final state at some later time. The scope of the Huang-Tam-Clark theorem is not strictly restricted to systems with a discrete spectrum, although only one very simple example [47]of control of a system with a continuous spectrum has been discussed. Rarnakrishna et al. [5 I] have studied the controllability of quantum manybody dynamics of systems with a finite number of levels from a point of view that is somewhat different from that used by Huang et al. [47].For


249

the purpose of investigating controllability, Huang et al. interpret (6.1) as an infinite dimensional bilinear system. Ramakrishna and co-workers [5 I] instead express the Schrodinger equation with included control field in terms of the eigenstates of an operator of interest. This approach yields, for a finite set of states, a finite dimensional bilinear control representation. We refer the reader to the original publication for the technical details of the analysis. A loose paraphrasing of the results of Ramakrishna et al. is that in a system with a finite number of nondegenerate discrete levels it is always possible to completely control the evolution of an arbitrary initial state to a selected final state. This result confirms the inference drawn by Tersigni et al. [52] from a study of the optimal control fields that transform various initial states to selected final states in a model five-level system. Shapiro and Brumer [53]have examined a system in which the eigenstates of the Hamiltonian are subdivided into three sets, with dimensions Mo,M I and M2,and ask if it is possible to transform a specified initial state of a system that lies in the subset of states with dimension M O into a specified final state of the system that lies in the subspace of states with dimension M 1 without passage through the states of the system that lie in the subspace with dimension M2.It is shown that if Mz 2 Mo stringent restrictions are required to prevent involvement of the states in M2 in the specified transformation. This inability to direct the evolution of the state of the system away from a specified set of substates does not contradict the Huang-Tam-Clark theorem, since that theorem does not admit constraints on the evolution pathway of the state of the system. Establishing the complete controllability of the quantum dynamics of a many-body system is an important backdrop to the development of practical algorithms for generating that control. However, the extant existence theorems give no hint as to how such algorithms can be formulated or how the controllability is influenced by constraints on the applied fields and/or on the evolution pathways that can be used. It is just the latter issues that play a central role in the optimal control theory analysis of the guided evolution of quantum many-body dynamics. It is important to note that setting up a calculation of the optimal control field for the transformation of an initial state of a system into a particular final state of that system does not guarantee that such a field exists. And, it must be remembered, that even when the optimal control field can be found, it does not, in general, provide complete control, since the latter implies that the norm of the difference between the final state reached with the optimal field and the target final state can be made arbitrarily small with respect to variation across admissible controls, not merely a minimum. Peirce et al. [54], Zhao and Rice [55], and Demiralp and Rabitz [56) have studied the existence of optimal control fields with respect to quan-

250

S. A. RICE

tum dynamics. For the case that the control field is bounded and can be expressed as an integral operator of the Hilbert-Schmidt type, the following results have been obtained: (i) Peirce et al. [54] proved that for a spatially bounded quantum system, which necessarily has spatially localized states and a discrete spectrum, optimal control of the evolution of a state is possible. (ii) Zhao and Rice [55] adapted the analysis of Pierce et al. [54] to show that in a system with both discrete and continuous states optimal control of evolution in the subspace of discrete states is possible. (iii) Zhao and Rice [55] also showed that evolution can be optimally controlled in the subset of continuum states that can be transformed to be L2 integrable by a complex rotation (wavepacket states that can be transformed to surrogate localized states). (iv) Demiralp and Rabitz [56] have shown that, in general, there is a denumerable infinity of solutions to a well-posed problem of control of quantum dynamics; the solutions can be ordered in quality according to the magnitude of the minimum of the objective functional J (see Section IV) that is achieved. In all of these cases what is established is the existence of a control field that will minimize the objective functional J . If the value of the minimum attained is zero, the system is completely controllable; if not, the optimal solution defines the maximum attainable control of the evolution of the system for the given set of control functions. The preceding analyses assume that we possess complete knowledge of the molecular Hamiltonian of the system we wish to control and that no disturbances will generate fluctuations in the control field used to guide a particular state-to-state evolution. In practice, neither of these assumptions is completely correct. Except for the smallest molecules we have imperfect knowledge of many-body potential-energy surfaces, of state-to-state coupling energies, and so on, and real experiments are always subject to a variety of mechanical, electronic, and optical disturbances. It is therefore desirable to design a control field that will generate the desired transformation of an initial state to a final state while being robust with respect to uncertainties in the molecular Hamiltonian and to disturbances that generate control field fluctuations. Rabitz and co-workers [57-591 have devoted considerable effort to devising a method for reducing the sensitivity of the field that controls a stateto-state transformation to various uncertainties and disturbances. The general scheme they develop introduces the constraints implied by the existence of the disturbances into the functional J and then uses minimax


25 1

analysis to determine the optimal control field. T h ~ s , ~the i f effects of the disturbances are collected in a pseudo-Hamiltonian Hdist(x,t , u(t), w(x,t)), which can be a function of any or all of the molecular coordinates x, the field u(r), the disturbance w(t), and the time, the optimal field is determined from min,max,J(u(t), w(x,t) subject to the dynamical constraint that the Schradinger equation be satisfied and a constraint on the disturbances of the form F(w(x,t ) ) = 0. The minmax point Urn(?) and the function W m ( X , t) satisfy

subject to the constraints mentioned, with urn(?)the best optimal field and wm(x,t ) the worst disturbance. Clearly, J(u,(t), w,(x, t)) determines the best control field for the worst disturbance, and hence is a very conservative representation of the limitation of the extent of control generated by disturbances. Zhang and Rabitz I57,SSl have applied this analysis to several simple cases. The results obtained illustrate that robust control fields can be designed under a variety of conditions and for a range of types and magnitudes of disturbances; they also suggest that there can be circumstances in which the magnitude of a disturbance is sufficient to destroy the possibility of control of the quantum dynamics. All of the analyses described above are used in a predictive mode. That is, given the molecular Hamiltonian, the sources of the external fields, the constraints, and the disturbances, the focus has been on designing an optimal control field for a particular quantum dynamical transformation. Given the imperfections in our knowledge and the unavoidable external disturbances, it is desirable to devise a control scheme that has feedback that can be used to correct the evolution of the system in real time. A schematic outline of the feedback scheme starts with a proposed control field, applies that field to the molecular system that is to be controlled, measures the success of the application, and then uses the difference between the achieved and desired final state to design a change that improves the control field. Two issues must be addressed. First, does a feedback mechanism of the type suggested exist? Second, which features of the overall control process are most efficiently subject to feedback control? Judson and Rabitz [60] have provided a numerical demonstration of an existence theorem for feedback control in the guiding of the evolution of the state of a system. The example they consider is the transfer of 100% of the population from the vibrationless ground rotational state of KCl to the vibrationless state withj = 3, m = 0, by a suitable field. The novel idea they exploit is to use the population transfer generated by a trial field as input to an adaptive learning algorithm for comparison with the desired popula-

252

S . A. RICE

tion transfer. The learning algorithm then defines changes in the trial field that are intended to decrease the difference between the population transfer generated by the new trial field and the desired population transfer. The specific learning algorithm they adopt is the so-called genetic algorithm, by means of which an analogy is exploited between the search for fields that enhance population transfer and survival of the fittest in a Darwinian evolution. This analogy is exploited by starting the search for the control field with a large family of fields constructed from pulse sequences; the initial pulse sequences can be defined in any fashion, even by use of a random-number generator. After application of the initial family of fields to the molecular system, all fields except the two that generate the largest population transfer are rejected. The two selected fields form the basis for a new set of fields generated by rearranging the pulses of the selected fields in all possible ways. Then the new family of trial fields is applied to the molecular system and all except the two fields that generate the largest population transfer are discarded. This process is continued until the desired population transfer is achieved. We consider the Judson-Rabitz result to be extremely important because it establishes that feedback can be used to correct the evolution of an applied field to become an optimal control field. However, the manner in which the feedback is implemented, namely by what is equivalent to a very rapid random search of the appropriate parameter space, gives no clues concerning the relationships between the physical parameters of the system and the optimal field needed for a particular transformation of the state of the system. A different demonstration of the value of feedback in the generation of an optimal control field has been reported by Amstrup et al. [59]. They studied the optimal control of a pump-dump experiment involving CsI (see Section IV). In this example the population is first transferred to the excited (dissociative) state and then, after a delay, transferred back to the ground state. The goal is to prepare a large ground-state population of molecules with some large internuclear separation. The pump and dump fields were separately optimized, assuming that either the time between pulses or the amplitudes of the pulses were uncertain. A chirp representation of the electromagnetic field was used, since this is experimentally realizable and has only a few parameters. A “generalized genetic algorithm” was used to generate the feedback for driving the evolution of the field, with special attention paid to the role of uncertainties in pulse delay and pulse amplitudes. The results obtained show that uncertainties in pulse delay and pulse amplitudes can be effectively mitigated by the use of feedback. Finally, we remark that the use of an optimal control field to enhance achievement of a particular quantum dynamical transformation usually increases the efficiency of that transformation by several orders of magni-


253

tude. And, it is usually the case that the optimal control field has a complicated structure. However, a good approximation to the major features of the optimal control field can usually be generated from a few Fourier components of that field, and in test cases it is found that this approximate field only degrades the efficiency by of order 50%.That degradation is acceptable when the gain in efficiency obtained from the optimal control field is 104, as in the pumpdump enhancement of the photodissociation of 12 studied by Amstrup et al. [61].

VII. REDUCED SPACE ANALYSES OF THE CONTROL OF QUANTUM DYNAMICS The analyses of control of the quantum dynamics of a many-body system given above are not, in principle, restricted by the complexity of the system. Nevertheless, the calculations required to design an optimal control field or to identify competing pathways between the same initial and final states become more difficult very rapidly as the number of degrees of freedom increases. For that reason done it is desirable to have a reduced space formulation of the control process. In addition, our intuition about chemical reactions is rooted in the similarities between functional group properties across a variety of molecules. In this sense much of chemistry is “local,” and we expect a suitable reduced-space representation to be applicable to the design of control process. In this section we sketch three reduced representations of quantum dynamics that have been suggested for use in the design of optimal control fields.

A. Reduced Representation in State Space We first examine the quantum dynamics of a system whose properties are represented by an n-state spectrum. In general, these states cannot be uniquely associated with individual degrees of freedom of the system. Indeed, we must expect that the amplitude associated with a state will, typically, depend on many (possibly all) of the degrees of freedom of the system. We seek a representation of the population dynamics of this n-state system in terms of the properties of a surrogate system with fewer states. The states of the surrogate system must, of course, be defined in terms of the n states of the full system, so we are really developing an alternative representation of the n-state system. However, we expect the formal relations that define the reduction of n to, say, rn states to be of value in guiding the generation of accurate approximations to the dynamics of the full n-state system. Tang et al. [20] have analyzed a reduction procedure using the Schrijdinger representation of the dynamics of the n-state system. The molecular wave function of the n-state system can be written as a superposition

254

S . A. RICE

of the eigenfunctions $ j , with coefficients C j ( t ) e - i w j ' . Substitution of this expansion into the Schrijdinger equation yields the equation of motion for the coefficients: (7.1)

where V j k = ( U j l V l U k ) and W k j = W k - W j . We assume that the molecule-field coupling is dominated by the dipole transition interaction and represents the resonant continuous electric field e(t) in the form

Adopting the rotating-wave approximation (RWA) and introducing the detuning frequency A o j k = W j k - y j k and the Rabi frequency M j k = - p j k l 4 0 l / k we find cj(t) +i

E

ke (I,

Mjk(t)Ck(t)

(7.3)

...,n )

where the M j k ( t ) are the elements of the n x n time-dependent matrix

If all states of the system are strongly coupled to each other, the system dynamics can only be described by completely solving the above equations. However, it is extremely unlikely that this is the case. Rather, it is commonly found that some pairs of states are strongly coupled and other pairs of states are weakly coupled. Then we expect that the population transfers among strongly coupled states dominate the system dynamics and that it should be possible to study the n-state system dynamics in the subspace of strongly coupled states with a correction from the influence of the weakly coupled states.


255

The reduction schemes used by Tang et al. [20] to define the surrogate fewer state system follows the method proposed by Shore [62]. The scheme has a cFmpact form when we introduce two orthogonal projection operators P and Q and work in the frequency domain instead of the time domain. The time evolution matrix for the n-state system dynamics, U(t), and its Fourier transform, G(w), satisfy the following equations:

U(t) = 0;

(wI - M ) G ( w ) = 1

(7.5)

where w is the frequency-domain variable and 1 is the n x n unit matrix. Let F be the projection operator onto the subspace composed of the states having stronger cyuplings within which we try to approximate the system dynamics, and let Q be the projection operator onto the remaining states. We are :Interested in evaluating the matrix elements of G(w ) within the subspace of P states in the frequemy dcmain. Multiplying both sides of the equation for G ( w ) in Eq. (7.5) by P + Q = 1, we find

Further multiplication by k from the right and Q from the left, followed by some rearrangement, yields &(w)F

= Q [ Q ( w l - M)Q]-'Q&G(u)F

(7.7)

Note that we require the inverse of the matrix (01 ; M ) within theAQsub; space. Now multiplying both sides of Eq.17.6) by P, substituting Q G ( u ) P from Eq. (7.7), and again multiplying by P on both sides yields P{wl

-

PMF - PM&Q(wl- M)Q]-'QMF}PG(w)b = P

(7.8)

We now write M ( w ) = PMi, + PMQ[Q(wl- M)Q]-'QMF

(7.9)

which i,s a representation of the frequency-domain time evolution operator within P space: FG(w)F = F[ol- M(w)]-'P

(7.10)

256

S. A. RICE

From (7.10) we can get the time evolution operator bU(t)k by use of a Fourier transform. These localized operators permit the construction of thos? portions of the time-evolved state v e c t p that lie within the subspace of P states. The influence of the remaining Q states occurs through the action of the operator M(w). The preceding analysis is just a transformation of one representation of the n-state problem to another representation. To be useful, the new representation must admit simplifying approximations not suggested by the original representation. One such approx(mation is to replace the frequency variable w in M(o) in (7.10) by a typical P space eigenfrequency, say MY. We thereby obtain the frequency-independent effective operator

M = bMF + kMQ[Q(oY - M)Q]-'Q&

(7.11)

Viewed in the time domain, the replacement of M ( w ) by M washes out the details of the time variation within Q space. For this appryximation to be useful, all strongly coupled states should be included in the P space !nd the Q space should not include any states that couple strongly to the P space (weak coupling assumption). We now find that the population dynamics of the m levels within the P space is governed by the equations of motion kim

(7.12) k e ( I , ...,m )

We now connect the analysis given above with the equation of motion displayed in Eq.(5.5). That equation of motion follows from subdivision of a system into an open subsystem S and a complementary reservoir R. When the coupling between S and R is weak, the evolution of the open system S, due to the internal dynamics of S and the interaction with the reservoir R, can be described in density matrix form by Eq. (5.5). Now writing

we find (7.14)


257

which we require to satisfy the semigroup condition [38, 391 =

Al

(7.15)

+

Hence i is a semigroup generator. Returning to the formal reduction procedure described at the beginning of this section, we note again that the operator M ( w ) incorporates all of the dynamics associated with evolution of the n-state system, and the formalism merely reorganizes the exact representation of the n-state population transfer ciynamics. The formalism, as such, does not demand that the m states in the P subspace are strongly coupled to each other and htt: the n - M states in the Q subspace are weakly coupled to those in-the P subspace. The use of a typical unperturbed eigenfrequency of the P subspace, to replace the variable frequency w in Mid),by virtue of washing out the details of the time variation within the Q subspace, generates the separation of the total system into a strongly coupled subsystem that is weakly coupled to a reservoir. In general, we expect this approximation will lead to a loss of time revepibility, and hence can be used to obtain an explicit form for the operator LD. Given (7.12) it is straightforward to obtain the corresponding density matrix form of the equation of motion. The time depeFdence of the density matrix element for the mth-state population in the P subspace is found to be

~7,

A comparison of (7.16) and (5.5) yields

where, in the RWA,io has the representation

258

S . A. RICE

Tang et al. [20] have examined the population dynamics in a three-level system, and its representation in a surrogate two-level system, to test the scheme outlined above. In the model system considered state 3 is weakly coupled with states 1 and 2, so that population transfer between states 1 and 2 should dominate the dynamics, with only a small contribution from population transfer to and from state 3. The coupling of state 3 with states 1 and 2 was taken to be one-tenth of the coupling between states 1 and 2, that is, Mi3 = M23 = M12/10 = - 1/10. Using the formalism sketched above, the exact system dynamics is governed by the coupled equations of motion for the three states, t l ( t ) = 0 . 5 i ! ~ ~ (+t o) . ~ c ~ ( ~ ) I &(r) = 0 . 5 i [ ~ , ( +t )0.1~3(t)l

C3(r) = o . s ~ [ o . ~ c ~+( ~ o .)I c ~ ( ~ ) I

(7.19)

and the approximate system dynamics is governed by the two coupled equations of motion for the two surrogate states,

C1 ( t ) = i[0.o05c1( t )+ 0.505~2(t)l c * ( r ) = i [ 0 . 5 0 5 ~ ~+( to.oo5c2(t)l )

(7.20)

The values of (Cl(t)I2and IC2(t)I2 obtained from (7.19) and (7.20) are compared in Figs. 9 and 10. The amplitudes and periods of the temporal evolution predicted by the two approaches to the system dynamics are seen to agree quite well. The differences seen in the amplitudes shown in Fig. 9 are a consequence of the replacement of the exact eigenfrequ%nciesof the Rabi frequency matrix with a typical eigenfrequency from the P subspace. Kaluza and Muckerman [63] have suggested a reduced representation in state space that is similar in spirit to that proposed by Tang et al. [20] but that also has some distinctive differences from their analysis. Kaluza and Muckerman [63] examine the states of a molecule populated by pulsed laser multiphoton excitation; they call these states “primitive bright states.” The primitive bright states are not, in general, eigenstates of the Hamiltonian. A set of active bright states is constructed by orthogonalization of the states generated by successive application of a pulsed field, at discrete time intervals, to the initial state of the molecule. Then the overlap matrix of the states so generated is calculated and diagonalized, and only that subset of the active bright states with largest overlap is selected to describe the time evolution of the state of the molecule. Kaluza and Muckerman have illustrated their method of analysis by examining the selective excitation of acetylene. Some of their results are shown in Fig. 11. Clearly, the Kaluza-Muckerman

259


0.8

-

0.6

-

0.4

-

0.2

0

3

9

6

12

time

Figure 9. Population of the ground-state surface obtained from the exact three-state dynamics (-) and surrogate two-state dynamics (-). The initial conditions are Cl(0) = 1 and C2(0) = C3(O) = 0.

reduction scheme is quite accurate and is useful for calculation of control fields.

B. Reduced Representation in Coordinate Space A very perceptive treatment of chemical reaction dynamics, called the reaction path Hamiltonian analysis, states that the reactive trajectory is determined as the minimum energy path, and small displacements from that path, on the potential-energy surface [64-711. The usual analysis keeps the full dimensionality of the reacting system, albeit with a focus on motion along and orthogonal to the minimum energy path. It is also possible to define a reaction path in a reduced dimensionality representation. The definition of a reduced dimensionality reaction path starts with the full Cartesian coordinate representation of the classical N-particle molecular Hamiltonian,

c p' 3N

H(p,x) =

i= 1

2mi

+ " ( X I , . .. , X 3 N )

(7.21)

where x and p are the 3N-dimensional coordinate and conjugate momentum

260

S. A. RICE

0

3

6

9

12

time Figure 10. Population of the first excited-state surface obtained from the exact three-state and surrogate two-state dynamics (-). The initial conditions are Ci(0) = 1 dynamics (-) and Cz(0) = C3(O) = 0.

vectors. Let a = ( a , , .. ., ~ 3 be~ a )vector on the reaction path. Then the potential-energy function V(x) can be expanded near the reaction path in powers of x - a. When only terms to second order are retained,

V(x) = V(a) + VV(a) . (x - a) + ;(x

- a)

- F - (x - a) + . ..

(7.22)

where F is the force constant matrix. Because the displacement vector x - a is orthogonal to VV(a) in the 3N-dimensional vector space, the linear term in Eq. (7.22) vanishes. For simplicity, and because they are not of interest for our purposes, the motions corresponding to molecular rotations and translation of the center of mass are removed by use of the projector

FP = (1 - P*) * F (1 - PT)

(7.23)

-8

-2:

v)

Figure 11. Comparison between the expectation values of several observables pertaining to the state of the linear HCCH molecule at the end of the pulse for various transform-limited pulses. A 91-state active bright-state basis approximation is compared with the result of a full calculation (labeled DVR). ( a ) Total energy at the end of the pulse. (6) Norm of the wavepacket. ( c ) Kinetic energy in the CH, stretching motion. ( d ) The CH, bond length. (e) Kinetic energy in the CC stretching motion. (f)Expectation value of the CC distance. (From Ref. 63.)

- 5\o:

26 1

262

S. A. RICE

where iRT is the projection operator for the translational and rotational a normal-mode analysis is carried motions. Following the application of iRT, out. For a polyatomic reactant with many degrees of freedom the numerical calculations required to execute the program outlined above can easily achieve a scale that is impossible to handle even with a vectorized parallel processor supercomputer. The simplest approximation that reduces the scale of the numerical calculations is the neglect of some subset of the internal molecular motions, but this approximation usually leads to considerable error. A more sophisticated and intuitively reasonable approximation [72, 731 is to reduce the system dimensionality by placing constraints on the values of the internal molecular coordinates (instead of omitting them from the analysis). Assuming that most of the atomic displacements in the reactant molecule are small, the obvious choices for constraints on the internal coordinates not directly participating in the reaction are fixed values of the bond lengths and bond angles (say, f in number). The technical details concerning the construction of mathematical representations of these constraints and their incorporation into the appropriate projection operators can be found elsewhere [24]. The result is, formally,

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64]:

As usual, the Qk and wk (k = 1, ... , 3N-f - 7) are the normal coordinates and the corresponding normal-mode frequencies. The kinetic energy is then

where the Bkf are coupling constants.


263

Suppose the reactive polyatomic molecule of interest can undergo unimolecular reaction to form several products, and we imagine carrying out a constrained reaction path analysis for each of the product channels. To carry out the analysis of a particular constrained reaction path, Zhao and Rice adopted a system-bath model [74] in which the reaction path coordinate defines the system and all other coordinates constitute the bath. The use of this representation permits the elimination of the bath coordinates, which then increases the efficiency of calculation of the optimal control field for motion along the reaction coordinate. Miller and co-workers [64,671 have shown that a canonical transformation of the reaction path Hamiltonian yields the form

(7.27)

with the Gkl coupling constant between the normal modes k and 1. After expansion of the first term in (7.27)to terms of quadratic order, and assuming vibrational adiabaticity, it is found that 2

H eff -- ps + 2

3N-f-7

k= I

(7.28)

264

S . A. RICE

with E the total system energy. We note that this effective Hamiltonian treats the bath as a set of linearly shifted harmonic oscillators. The harmonic bath coordinates can be eliminated by use of the vibrational adiabatic approximation [75] or, alternatively, the use of a shifted linear oscillator basis set to construct the Hamiltonian in the matrix representation, which renders the Hamiltonian matrix function a function of only the system reaction coordinates s. To use the reaction coordinate representation in the optimal control theory analysis, Zhao and Rice suppose that the ground- and excited-state potentialenergy surfaces are similar enough to each have product channels that yield, respectively, ground and excited states of the same product. The key approximation they introduce is the assumption that, using the same coordinates, the reaction paths on the two surfaces are roughly parallel. If that assumption is valid, the projection of the ground-state reaction path on the upper surface will usually lie in the valley that defines the upper surface reaction path, and vice versa. By “in the valley” we mean that the projected amplitude is in the neighborhood of the reaction path, although likely displaced to higher energy. We then expect that it will be possible to design a control field that transfers amplitude back and forth between the two potentialenergy surfaces and forces the amplitude to follow the general direction of the reaction path, although not the detailed reaction path itself. ClearIy, this approximation leads to decrease in the efficiency of product formation from that achievable with a field optimized with respect to the full space of the potential-energy surface. The calculations required to determine the (optimal) field that maximizes the yield of a particular product are similar to those we have described earlier, except for the added complexity of working in the reaction path representation. The major change is that the reaction path Hamiltonian replaces the full Hamiltonian in defining the constraint on the system dynamics. Looked at in general terms, the scheme Zhao and Rice have proposed [74] should be applicable, by construction, to the class of reactive systems in which the reaction paths on the upper and lower potential energy surfaces are similar. Actually, we expect this scheme to be applicable to a larger class of systems, since all that is required of the upper potential energy surface is that it not have a structure with a valley that is everywhere orthogonal to the projection of the lower state reaction path. However, the more similar are the contours of the two potential energy surfaces, the more efficient will be the optimization of the selected product yield. We then expect, if the condition described in the following paragraph is satisfied, that the scheme can be used to design a field that optimizes the product selectivity from the reaction of a polyatomic molecule. At present there are no reported tests of the accuracy of the Zhao-Rice


265

reduction scheme. Clearly, the reaction path formalism replaces the true potential energy surface with a simpler surface. The most important question raised by the use of this representation, which involves approximations, is whether it is sufficiently accurate to preserve the phase relations that characterize the transfer of amplitude between the true upper and lower potential energy surfaces.

C. Reduction by Factorization: Time-Dependent Hartme Approximation

The formulation of the calculation of the optimal control field that guides the evolution of a quantum many-body system relies, basically, on the solution of the time-dependent Schrodinger equation. Messina et al. [25] have proposed an implementation of the calculation of the optimal control field for an n-degree-of-freedom system in which the Hartree approximation is used to solve the time-dependent Schriidinger equation. In this approximation, the n-degree-of-freedom wave function is written as a product of n single-degree-of-freedom wave functions, and the factorization is assumed to be valid for all time. Messina et al. consider a system with two electronic states Jg) and )e). The system is partitioned into a subset of degrees of freedom that are to be controlled, labeled Z, and a background subset of degrees of freedom, labeled x; the dynamics of the 2 subset, which is to be controlled, is treated exactly, whereas the dynamics of the n subset is described with the time-dependent Hartree approximation. The formulation of the calculation is similar to the weak-response optimal control theory analysis of Wilson et al. described in Section IV [28-321. The solution of the time-dependent Schrodingerequation for this system can be represented in the form

The reduced density matrix for the system is defined by

with (7.3 1)

and similarly for j e ( Z ,Z’, t). Following the procedure sketched in Section IV, it is found that the weak-response optimal control field is determined by

266

S. A. RICE

(7.32) J to

where

A ( t f ) Tr[&;,(tf)]

dt dt’ Ec(t)MFd(t,t’)E:(t’)

=

(7.33)

is the target yield,

dZ dZ‘&(Z,Z’)

d x $bo’*(Z,x,tf - t)$bo)(Z’,x,tf

- t’)

(7.34) is the reduced material response function, and E&) is a slowly varying complex field from which the high-frequency component oeg(the resonance frequency) has been removed. The Hartree approximation is now introduced for the purpose of calculating the reduced material response function (7.34). The fashion in which this is carried out is by writing the overall bath Hamiltonian as a sum of the Hamiltonians for each bath degree of freedom, which implies that the overall bath wave function is a product of the wave functions for each bath degree of freedom. It can then be shown that

(M7d(t,~’))TDH = Q ( t , t‘)

J

dZ dZ‘ A,(Z, Z’)&(Z, tf

- t)&.(Z’, tf - t’)

with 8 the phase angle

(7.36)


267

Messina et al. [25] test the time-dependent Hartree reduced representation with a simple two-degree-of-freedom model consisting of the I2 vibration coupled to a one-harmonic-oscillator bath. The objective function is a minimum-uncertainty wavepacket on the B state potential curve of 12. Figure 12, which displays a typical result, shows that this approximate representation gives a rather good account of the short-time dynamics of the system.

VIII. THE CONTROL OF DYNAMICS-INVERSE SCATTERING DUALITY Thus far we have examined the determination of a field that will control the quantum many-body dynamics of a system when all that is specified is the initial and final states of the system and the constraints imposed by the equations of motion and physical limitations on the field. When posed in this fashion, the calculation of the control field is an inverse problem that has similarities to the determination of the interaction potential from scattering data. Despite the similarities, the mathematical methods used are very different. Because only the end points of the initial-to-final state transforma-

C

H

0)

=I

I&

18.0:

Figure 12. Magnitude of the response function for the stretched I2 target state. The solid line is the result of an exact calculation; the dotted line is the result of the use of the Hartree approximation. The parameter c is the coupling constant between the I2 molecule and the bath oscillator. (a) Bath oscillator frequency of 50 cm' . (b) Bath oscillator frequency of 100 cm' . (From Ref. 25.)

268

S. A. RICE

tion are specified, it is necessary to use a variational approach to calculate the optimal field that generates that transformation. Note, however, that a given applied field will uniquely determine the time evolution of the expectation value of an observable. This observation suggests that if, in addition to the initial and final states, extra information in the form of the trajectory of an observable is specified, the calculation of the field should become very much simpler. This approach is called inverse control and is obviously complementary to the optimal control approach with which we have been concerned to this point. The theoretical basis for inverse control of the quantum dynamics of a many-body system was established by Ong et al. [481. These investigators established the sufficient and necessary conditions for the existence and uniqueness of the solution to the inverse-control equations. The application of the inverse-control analysis to molecular problems has been advanced by the work of Rabitz and co-workers [76-781. The general concepts of inverse control are simple to grasp, Suppose we specify, a,priori, the time dependence of the expectation value of some observable, (0) = y ( t ) . The Schrodinger equation for the system with applied control field is just

and the equation of motion for an observable is


269

Since y ( t ) is known by definition, (8.2) may be directly inverted to calculate

u(t) except at points at which ( [ O , H l ] )= 0:

Indeed, even when ( [ b , H l ] )= 0 there is a prescription focf calculating u(t) from an equation for a higher order time derivative of (0) [76]. We note that the structure of (8.3) defines a form of feedback of information from the system to the field since the evaluation of the expectation values of the commutators requires knowledge of the system wave function. The methodology has been extended to the case where the field to be calculated is constrained to best reproduce the trajectories of several expectation values [77]. Rabitz and co-workers have provided interesting examples of the use of the procedure described to determine the field that drives a prescribed time evolution of the internuclear separation of a nonrotating diatomic molecule, a field designed by use of energy tracking that dissociates the HF molecule, and a field designed by use of competitive atomic acceleration tracking to selectively break the stronger bond in a nonrotating linear triatomic molecule. What is arguably the most interesting potential application of the analysis inherent in (8.3) is the suggestion, by Lu and Rabitz [78],that tracking be used to invert laboratory data to obtain, for example, the potentialenergy curve of a nonrotating diatomic molecule. In this case the laboratory data are measured energy-level positions and Franck-Condon factors for transitions between levels, which can be used to synthesize the trajecl($(0)(9i)(2 exp(-iEit/Fi). Application of this methodology tory y(r) = to multidimensional potential-energy surfaces is also, in principle, possible. The results just described establish the existence of a correspondence between the control of molecular dynamics when only the initial and final states are specified and the determination of a field that generates a defined trajectory of an observable. The distinction between the approaches arises from the point of view taken. This correspondence also provides, in a sense, a connection between the modem view of the control of molecular dynamics and the intuitive “driven bond-breaking” view.

xi

270

S. A. RICE

IX. coNcLusIoNs This overview was designed to illustrate that control of quantum manybody dynamics is in principle both possible and experimentally feasible. At present, the analyses described are likely to be most important as tools for learning more about molecular dynamics and for testing concepts advanced to describe aspects of molecular dynamics. The cost of photons is sufficiently great that it seems unlikely that commercial chemical syntheses can be based on the enhanced selectivity of product yield generated by control fields. However, there are likely other practical applications. For example, Kuriziki et al. [79] have shown how interference effects can be used to generate a fast semiconductor optical switch, and this prediction has been verified [80]. Since the underlying principles of control theory as applied to quantum dynamics are very broadly applicable, it is likely that many other applications will be developed in the near future. Indeed, as laser technology improves and as our understanding of complex molecules advances, it is likely that control of quantum many-body dynamics will become a mainstream component of chemical physics.

Acknowledgments The work described in this chapter could not have been carried out without the intellectually stimulating interactions I have had with my co-workers David Tannor, Ronnie Kosloff, Pierre Gaspard, Sam Tersigni, Andras Lorincz, Bjame Amstrup, Roger Carlson. and Meishan Zhao. I have also had many fruitful discussions with Paul Brumer, Moshe Shapiro, Kent Wilson, Herschel Rabitz, Jeff Cina, and Graham Fleming. This research was supported by grants from the National Science Foundation.

References M. Shapiro and P. Brumer, J. Chem. Phys. 84,4103 (1986). M. Shapiro and P. Brumer, Acc. Chem. Res. 22,407 (1989). P.Brumer and M. Shapiro, Ann. Rev. Phys. Chem. 43,257 (1992). D. J. Tannor and S. A. Rice, J. Chem. Phys. 83,5013 (1985). 5. D. J. Tannor, R. Kosloff, and S. A. Rice, J. Chem. Phys. 85,5805 (1986). 6. R. Kosloff, S. A. Rice, P. Gaspard, S. Tersigni, and D. J. Tannor, Chem. Phys. 139,201

1. 2. 3. 4.

(1 989). 7. S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988). 8. A. P. Peirce, M. A. Dahleth, and H. Rabitz, Phys. Rev. A 37, 4950 (1988). 9. A. P. Peirce, M. A. Dahleh, and H. Rabitz, Phys. Rev. A 42, 1065 (1990). 10. S. Shi and H. Rabitz, Comp. Phys. Commun. 63,71 (1991). 11. P.Gross, D. Neuhauser, and H. Rabitz, J. Chem. Phys. 96, 2834 (1992). 12. S. M. Park, S-P. Lu, and J. Gordon, J. Chem. Phys. 94, 8622 (1991). 13. S-P.Lu, S. M. Park, Y. Xie, and R. J. Gordon, J. Chem. Phys. 96,6613 (1992). 14. V. Kleiman, L. Zhu, X. Li, and R. J. Gordon, J. Chem. Phys. 102,5863 (1995).


27 1

IS. L. Zhu, V. Kleiman, X. Li, S-P. Lu. K. Trentelman, and R. J. Gorden, Science, 270, 77

(1995). 16. V. Kleiman, L. Zhu, J. Allen, and R. J. Gordon, J. Chem. Phys., 103, 10800 (1995). 17. T. Baumet, R. Thalweiser, V. Weiss, and G. Gerber, Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie. Weinheim 1994. 18. N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991). 19. B. Amstrup, R. J. Carlson, A. Matro, and S. A. Rice, J. Phys. Chem. 95, 8019 (1991). 20. H. Tang, R. Kosloff, and S. A. Rice, J. Chem. Phys. 104, 5457 (1996). 21. P. Gross, H. Singh, H.Rabitz, K. Mease, and G. M. Huang, Phys. Rev. A 47,4593 (1993). 22. Y. Chen, P. Gross, V. Ramakrishna, and H. Rabitz, J. Chem. Phys. 102, 8001 (1995). 23. Z-M. Lu and H. Rabitz, J. Phys. Chem. 99, 13731 (1995). 24. M. Zhao and S. A. Rice, SPIE 2124,246 (1994). 25. M. Messina, K. R. Wilson, and J. L. Krause, J. Chem. Phys., 104, 173 (1996). 26. P. Brumer and M. Shapiro, in Molecules in Laser Fields, A. D. Bandrauk, Ed., Dekker, New York, 1994. 27. C. K. Chan, P. Brumer, and M. Shapiro, J. Chem. Phys. 94, 2688 (1991). 28. Y.J. Yan, R. E. Gillilan, R. M. Whitnell, K. R. Wilson, and S. Mukamel, J. Chem. Phys. 97,2320 (1993). 29. J. L. Krause, R. M. Whitnell, K. R. Wilson, Y. J. Yan, and S. Mukamel, J, Chem. Phys. 99,6562 (1993). 30. B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, R. M. Whitnell, K.R. Wilson, V. V. Yakolev, Y. J. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993). 31. J. L. Krause, R. M. Whitnell, K. R. Wilson, and Y. Y. Yan, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1994. 32. J. L. Krause, M. Messina, K. R. Wilson, and Y. J. Yan, J. Phys. Chem. 99, 13736 (1995). 33. K. G. Kim and M. D. Girardeau, Phys. Rev. A 52, 891 (1995). 34. R. Balian and M. Veneroni, Phys. Rev. Lett. 47, 1353 (1981). 35. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955. 36. G. Lindblad, Commun. Math. Phys. 48, 119 (1976). 37. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976). 38. R. Alicki and K. Landi, Quantum Dynamical Semigroups and Applications, Springer-Verlag, 1987. 39. E. B. Davis, Quantum Theory of Open Systems, Academic, London, 1976. 40. A. Banana, R. Kosloff, and D. Tannor, J. Chem. Phys. 99, 196 (1993). 41. R. Kosloff, J. Chem. Phys. 80, 1625 (1984). 42. E. Geva and R. Kosloff, J. Chem. Phys. 96,3054 (1992); J. Chem. Phys. 97,4398 (1992); Phys. Rev. E 49, 3903 (1994). 43. S. Sieniutycz and P. Salamon, Eds., Finite Time Thermodynamics and Thennoeconomics, Advances in Thermodynamics, Vol. 4, Taylor and Francis, 1991. 44. M. Sargent 111, M. 0. Scully, and W. E. Lamb, Jr., Laser Physics, Addison-Wesley, Reading, MA, 1974.

272

S. A. RICE

45. J. W. Clark, in Condensed Matter Theories, Vol. 11, E. V. Ludena, Ed., Nova Scientific Publishers, Cammack, New York, 1996. 46. T. J. Tam,G. Huang, and J. W. Clark, Math. Modelling 1, 109 (1980). 47. G. M. Huang, T.J. Tam,and J. W. Clark, J. Math. Phys. 24, 2608 (1983). 48. C. K. Ong, G. M. Huang, T. J. Tam, and J. W. Clark, Systems Theory 17, 335 (1984). 49. J. W. Clark, C. K. Ong, T. J. Tam, and G. M. Huang, Math. Systems Theory 18,33 (1985). 50. T. J. Tam, J. W. Clark, and G. M. Huang, in Modeling and Control of Systems, A. Blaquiere, Ed., Springer, Berlin, 1995. 51. V. Ramakrishna, M. V. Salapaka, M. Dahleh, H. Rabitz, and A. Peirce, Phys. Rev. A 51, 960 (1995). 52. S. Tersigni, P. Gaspard, and S. A. Rice, J. Chem. Phys. 93, 1670 (1990). 53. M. Shapiro and P. Brumer, J. Chem. Phys. 103,487 (1995). 54. A. P. Peirce, M.A. Dahleh, and H. Rabitz, Phys. Rev. A 37, 4950 (1988). 55. M. Zhao and S. A. Rice, J. Chem. Phys. 95, 2465 (1991). 56. M. Demiralp and H.Rabitz, Phys. Rev. A 47, 809 (1993). 57. H. Zhang and H. Rabitz, Phys. Rev. A 49,2241 (1994). 58. H. Zhang and H. Rabitz, J. Chem. Phys. 101,8580 (1994). 59. B. Amstrup, G. 1. Toth, G. Szabo, H. Rabitz, and A. Lorincz, J. Phys. Chem. 99, 5206 (1995). 60. R. S . Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 (1992). 61. B. Amstrup, R. J. Carlson, A. Matro, and S. A. Rice, J. Phys. Chem. 95, 8019 (1991). 62. B. W. Shore, The Theory of Coherent Atomic Excitation, Vol. 2, Wiley, New York, 1990. 63. M. Kaluza and J. T. Muckeman, Chem. Phys. Lett. 239, 161 (1995). 64. W. H. Miller, N. C. Handy, and J. E. Adams, J. Chem. Phys. 72,99 (1980). 65. W. H. Miller, J. Phys. Chem. 87, 3811 (1983). 66. W. Miller, in The Theory of Chemical Reaction Dynamics, D. C. Clary, Ed., Reidel, Boston, 1986, p. 27. 67. W. Miller, in Potential Energy Sulface and Dynamics Calculations, D. G. Truhlar, Ed., Plenum, New York, 1981, p. 243. 68. D. G. Truhlar, A. D. Isaacson, R. T. Skodje, and B. C. Garrett, J. Phys. Chem. 86,2252 (1982). 69. D. G. Truhlar, A. D. Isaacson, and B. C. Garrett, in Theory of Chemical Reaction Dynamics, Vol. 4, M. Baer, Ed., CRC, Boca Raton, FL, 1985, p. 65. 70. B. C. Garrett and D. G. Truhlar, in Potential Energy Surface and Dynamics Calculations, D. G. Truhlar, Ed., Plenum, New York, 1981, p. 897. 71. D. G. Truhlar, F. B. Brown, R. Steckler, and X . Isaacson, NATO AS1 Sex C 170, 285 (1986). 72. D. Lu, M.Zhao, and D. G. Truhlar, J. Compr. Chem. 12,376 (1991). 73. D. Lu and D. G. Truhlar, J. Chem. Phys. 99,2723 (1993). 74. N. Makri and W. H. Miller, J. Chern. Phys. 86, 1451 (1987). 75. B. C. Garrett and D. G. Truhlar, J. Phys. Chem. 86, 1136 (1982). 76. P. Gross, H.Singh, H. Rabitz, K. Mease, and G. M.Huang, Phys. Rev. A 47,4593 (1993). 77. Y.Chen, P. Gross, V. Ramakrishna, and H. Rabitz, J. Chem. Phys. 102, 8001 (1995).


273

78. Z-M. Lu and H. Rabitz, J. Phys. Chem. 99, 13731 (1995). 79. G. Kurizki, M. Shapiro, and P.Brumer, Phys. Rev. B 39, 3435 (1989).

80.

E.Dupont, P.B. Corkum, H. C. Liu, M. Buchanan, and 2. R. Wasitewski, Phys. Rev. k t t . 74,3596 (1995).

DISCUSSION ON THE REPORT BY S. A. RICE Chairman: M. Quack

B. Kohler: I would like to make a comment about the experimental status of shaped pulse control. A series of experiments have been performed in Prof. Kent Wilson’s laboratory that directly validate the use of tailored light fields for controlling molecular quantum dynamics. In one experiment we sought to control the evolution of a vibrational wavepacket in the bound region of an electronically excited state of the 12 molecule [I]. Specifically, we chose to produce a focused, minimumuncertainty wavepacket centered about a particular internuclear separation at a particular time after the application of a control pulse. To achieve this target, it is necessary to overcome the wavepacket spreading that is usually observed when an ultrashort but otherwise untailored pulse prepares a nuclear wavepacket on an anharmonic potentialenergy surface. Optimal control theory showed that the best light field in the linear response limit for achieving our target consists of a single, approximately Gaussian-shaped pulse with substantial phase modulation in the form of a pronounced negative-frequency chirp. This pulse was subsequently synthesized as well as possible in the laboratory, and the resultant wavepacket focusing at the target time was confirmed by monitoring the total fluorescence induced by a second, variabiy delayed probe pulse. These results demonstrate the feasibility of using tailored light fields designed by optimal control theory to control the quantum evolution of molecules. Finally, I would like to point out that the ability to produce focused wavepackets and thereby localize a molecule in a chosen region of phase space at a chosen target time can play a significant role in controlling the outcome of chemical reactions using two or more light pulses as originally proposed by Tannor and Rice 121. Our results on iodine demonstrate for the first time that “active” localization with shaped pulses is experimentally achievable. In fact, in the most recent experiments carried out in San Diego a two-pulse sequence has been used to control the predissociation of gaseous sodium iodide. In this experiment the second pulse was timed to transfer a wavepacket that had been focused by an initial, tailored control pulse to a higher

274

S. A. RICE

potential-energy surface and thereby reduce the amount of neutral product formation due to predissociation.

1. B. Kohler, V. V. Yakovlev, J. Che, 1. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Y. Yan, Phys. Rev. Letr. 74, 3360 (1995).

2. D. J. Tannor and S. A. Rice, J. Chem. Phys. 83,5013 (1985).

J. Manz: Prof. S. A. Rice has given a comprehensive survey and classification of various theoretical strategies for laser control, together with important experimental examples. The most fundamental distinction is between (i) the strategies of Brumer and Shapiro [l], who employ phase control of interfering excitation pathways from reactants to products using continuous-wave lasers, and (ii) the strategies of Tannor et al. [Z] and Rabitz [3], who employ ultrashort laser pulses to drive the system coherently from the reactant to the product channels (see current chapter). I would like to point to some additional work pioneered by Paramonov and Savva already in 1983 [4] and developed further in our group [5] for laser control of vibrational transitions and isomerizations (see the survey by Korolkov et al., this volume). Furthermore, we have demonstrated the possibility of laser control of reaction rates in Ref. 6. These theoretical studies [MI employ (series of) infrared (IR) femtosecond/picosecond laser pulses in order to drive the nuclear wavepacket from the reactant to the product on a single electronic state, typically the ground state. In contrast, Tannor et al. [2] employ ultraviolet/visible (UV/VIS) femtosecond laser pulses inducing electronic transitions. We have also developed an IR laser pulse variant of optimal control theory [7],similar to the theory pioneered by Rabitz et al. [3]. All strategies [4-71 employ ultrashort infrared laser pulses; they may be classified, therefore, as variants of strategy (ii), according to the classification scheme of S. A. Rice. My question to Prof. B. Kohler (as representative of the group of K. R. Wilson) is whether he would agree with S. A. Rice’s classification that puts the technique of K. R.Wilson et al. [8] into strategy (ii)? What are the fundamental analogies and what are the differences between their approach [8] and the Tannor-Rice-Kosloff-Rabitz approach (see Refs. 2 and 3 and current chapter)? Finally, I should like to point to another strategy (iii) of laser control by vibrationally mediated chemistry that is achieved by IR + UV continuous-wave (CW) multiphoton transitions (see the pioneering papers by Letokhov [9] and sequel theoretical developments [101 and experimental applications [1 11). 1 . M. Shapiro and P. Brumer, J. Chem. Phys. 84, 4103 (1986); P. Brumer and M. Shapiro, Acc. Chem. Res. 22,407 (1994).

2. D. J. Tannor and S. A. Rice, 1. Chem. Phys. 83, 5013 (1985); D. J. Tannor, R.


275

Kosloff, and S. A. Rice, J. Chem. Phys. 85, 5805 (1986); D. J. Tannor and S. A. Rice, Adv. Chem. Phys. 70,441 (1988). 3. S. Shi, A. Woody, and H.Rabitz, J. Chem. Phys. 88, 6870 (1988); W. S. Warren, H.Rabitz. and M. Dahleh, Science 259, 1581 (1993). 4. G. K. Paramonov and V. A. Savva, Phys. Lett. A 97, 340 (1983); G. K. Paramonov, V. A. Savva, and A. M. Samson, Infrared Phys. 25, 201 (1985); G. K. Paramonov. Phys. Lett. A 152, 191 (1991); G. K. Paramonov, in Femtosecond Chemistry, Vol. 2, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 671. 5. J. E. Combariza, B. Just, J. Manz, and G. K. Paramonov, J. Chem. Phys. 95, 10355 (1991); M. V. Korolkov, Yu. A. Logvin, and G. K. Paramonov, J. Chem. Phys., in press; M. V. Korolkov and G. K. Paramonov, Phys. Rev. A, submitted; M. V. Korolkov, G. K. Paramonov, and B. Schmidt, J. Chem. Phys., in press; M. V. Korolkov, J. Manz, and G. K. Paramonov, J. Chem. Phys., in press; M. V. Korolkov, J. Manz, and G. K. Paramonov, J. Phys. Chem., in press. 6. T. Joseph and J. Manz, Molec. Phys. 58, 1149 (1986). 7. W. Jakubetz, J. Manz, and H.-J. Schreiber, Chem. Phys. Lett. 165, 100 (1990); W. Jakubetz, E. Kades, and J. Manz, J. Phys. Chem. 97, 12609 (1993). 8. B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakolev, Y. J. Yan, and S. Mukamel. J. Phys. Chem. W, 12602 (1993); J. L. Krause, R. M. Whitnell, K. R. Wilson, and Y. J. Yan, in Femtosecond Chemistry, Vol. 2, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 743; B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner. R. M. Whitnell, and Y. J. Yan, Phys. Rev. Lett. 74, 3360 (1995). 9. V. Letokhov, Science 180,451 (1973). 10. E.Segev and M. Shapiro, J. Chem. Phys. 77,5604 (1982); V. Engel and R. Schinke, J. Chem. Phys. 88, 6831 (1988); D. G. Imre and J. Zhang, J. Chem. Phys. 89, 139 (1989).

11.

F. F. Cnm, Science 249, 1387 (1990).

B. Kohler: The approach of Wilson and co-workers to the control of quantum dynamics within the linear response approximation has, I believe, much in common with the perturbation theory results of Rice and others. But as Prof. Rice points out in his report, by recasting the control problem in terms of the dynamics of a density matrix in phase space, the Wilson group’s formulation permits the treatment of mixed states as well as the treatment of external interactions with the environment (i.e., solvent interactions) in a particularly straightforward manner. In addition, no restrictions are imposed on the structure of the electric field of the light (other than the requirement that the field be zero outside of a temporal interaction window), and the solutions are globally optimal in the weak-field sense. Three interesting aspects of the optimal fields that were computed in San Diego for several timedomain control scenarios are (1) the tailored fields are relatively simple; (2) the control is quite robust to changes to the tailored fields [ 11; and

276

S. A. RICE

(3) these weak-field solutions may be used to control a large molecular population either by scaling their intensity or by using them as a starting point in an iterative search for strong-field solutions [2]. 1. J. L. Krause, R. M. Whitnell, K. R. Wilson, Y. Yan, and S. Mukamel, J. Phys. 99, 6562 (1993); B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, Whitnell, K. R. Wilson, V. V. Yakovlev, Y. Yan, and S . Mukamel, J. Phys. 97, 12602 (1993). 2. J. L. Krause, M. Messina, K. R. Wilson, and Y. Yan, J. Phys. Chem. 99, (1995).

Chem.

R. M. Chem. 13736

S. A. Rice: I agree with Prof. Kohler that the use of a density matrix formalism by Wilson and co-workers generalizes the optimal control treatment based on wave functions so that it can be applied to, for example, a thermal ensemble of initial states. All of the applications of that formalism I have seen are based on perturbation theory, which is less general than the optimal control scheme that has been developed by Kosloff, Rice, et al. and by Rabitz et al. Incidentally, the use of perturbation theory is not to be despised. Brumer and Shapiro have shown that the perturbation theory results can be used up to 20% product yield. Moreover, from the point of view of generating an optimal control field, the perturbation theory result can be used as a first guess, for which purpose it is very good. S. Mukamel: The phase-space formulation of coherent control made using the density matrix and its Wigner representation generalizes the wave-function-based formulation in several important aspects: It allows performing calculationsat finite temperatures,the development of reduced descriptions, and the incorporation of more general constraints and “objectives.” In the perturbative limit (which is not essential to this formulation) the results were explained using molecular nonlinear (multitime) response functions that do not depend on the driving field. In addition, this approach provides an important semiclassical intuition and analytical expressions for the globally optimized solution. S. A. Rice: I agree with Prof. Mukamel’s remarks. P. W.Brumer: I would tend to agree with Prof. Rice that producing tankerloads of molecules by coherent control methods is probably far-fetched due to the cost of photons. However, we all realize that the possibility exists of producing expensive molecules in this way, or possibly catalysts, where small quantities suffice. However, I do not agree with Prof. Rice’s suggestion that the Brumer-Shapiro schemes, as the optimal control schemes, become more difficult to apply as the molecules increase in size. At the moment, barring the question of the role of overlapping levels, where


277

the overlap is due to the natural linewidths, larger molecules should behave the same way, with respect to coherent control, as small ones. In particular, note that many of our proposed control scenarios provide the experimentalist with a clear-cut statement of which parameters need to be varied to achieve control. Further, they tend to utilize relatively simple laser pulse (or CW) characteristics. Thus, our approach would apply to larger molecules in the same way as to smaller molecules; that is, the experimentalist needs only to vary the indicated parameters (e.g., laser intensities, phases, etc.) and search for control in this parameter space. I note, however, the caveat above: Our work to date has ignored the possibility of overlapping levels due to the high density of molecular states coupled with the background radiation field. We expect to examine these cases shortly.

M.Quack I should like to point out that, in spite of the pessimism expressed by Stuart Rice (and in the discussion by Paul Brumer) concerning possible applications of reaction control on a large-scale chemical production basis, one may have a much more optimistic outlook in the following sense: One can think of a future use of the reaction control on biomolecules (enzymes, DNA, RNA, etc.) in order to design specific biomolecules by selective reactions. Currently such design is carried out by “wet biochemistry” techniques. Future design would perhaps be by physical (laser) techniques. The large-scale chemical production occurs then subsequently by ordinary biomolecular multiplication and use in biotechnological processes. The cost of the photons (or the whole initial process) in generating the modified biocatalysts would play no role in such a scheme. The potential advantage of physical methods in molecular design would be their anticipated better general applicability. Of course, the exact form of the methods of physical (laser chemical) molecule design may differ from what we discuss now. I often like to make the analogy with the first magnetic resonance experiments in atomic beams following Stern and Gerlach and I. Rabi. At that time one would not have predicted today’s general use of magnetic resonance in organic, inorganic, and biochemical analytical laboratories (and in medicine). Yet this is exactly what happened, but of course not with molecular beams. Thus, future methods of physical (laser) reaction control may look a bit different from what is discussed now. One must distinguish principles from practical realization.

S. A. Rice: I do not wish to play the role of cynic with respect to the potential commercial use of active control of product selection in a reaction, but I suspect the path to such commercial exploitation

278

S. A. RICE

of the basic ideas and to the creation of viable applications has many potholes and detours, some likely minor and some likely serious. I find Prof. Quack’s suggestions (and enthusiasm) for possible amplification schemes that decrease the cost of active control to be stimulating, and I hope they will prove to be workable. M. Shapiro: I am more optimistic about the possible practical applications of coherent control. In particular, if one stays within the confines of perturbation theory, using two-path control, there exists a generic expression, in the form of a quadratic polynomial, for the probability of observing a given product. Provided one can identify spectroscopically the two intermediate states needed for this type of control, one need not do any calculation. One can hone in on the desired field parameters by simply measuring the product yield at three independent combinations of field parameters. In this way one completely determines the quadratic polynomial coefficients in the known generic form, thereby knowing the behavior of the system for all sets of field parameters, including the ones that maximize the probability of observing a given product or quantum state. S. A. Rice: I agree with the optimism of Prof. Shapiro concerning the broad range of opportunities for use of optimal control methods. The point I wished to make concerning large molecules is that their spectra are complex because there are many degrees of freedom. It is likely that states that yield different products are intermingled. With a two-path interference control scheme these must be identified so that a small change in excitation wavelength does not access unwanted species, as was seen by Gordon even for the simple molecule HI. There is a corresponding problem with the optimal control scheme, since it is difficult to visualize the dynamics in the full dimensionality of the potential-energy surface. Accordingly, I believe we need a reduced description to find our way through the complicated dynamics or to locate the states that produce the desired products. R.A. Marcus: I have a question for Prof. Rice: Do most industrial photochemical processes involve chain reactions? If so, one suspects that they will be less susceptible to “coherent control.” Are there some industrial photochemical processes not involving free radical chains? I gather that you hold the view, which seems reasonable, that the opportunities for coherent control are greater in devices such as electronic switches and generally in telecommunications? S. A. Rice: I think the first industrial applications of control techniques such as I have discussed will be for optoelectronic switching and, possibly, other optoelectronic devices. The use of control methods


279

to synthesize high-value complex molecules, assuming we learn how to predict the general features of the control field and develop suitable feedback methods to generate the actual field needed, is in my opinion not yet in sight. I understand that with modem biotechnology amplification methods it is in principle possible to generate a very special complex intermediate by use of optimal control methods and then use those amplification tools for production. I know nothing about the necessary techniques, costs, and so on, so I leave speculation concerning that matter to others.

K. Schafber: Prof. Marcus’s inquiry whether there are any examples of industrial photochemical nonradical chain production processes can be answered positively, although one has to admit that the usage of this sector of synthetic photochemistry has been quite limited, much to the disappointment of organic photochemists. The actual progress in applied photochemical technology can be learned from publications by Roberts et al. [I] and Braun et al. [2]. The most frequent arguments against photochemical production, whenever alternatives to photochemical processes are at hand, concern the high cost of electric energy, the need to invest in new plant facilities (arguments that occasionally are debatable), and the lack of experience in the design of lamp sources and reaction vessels. These latter arguments are unfortunately true to an often embarrassing extent. The fact is that even in large chemical companies the know-how in design of photochemical plants is frequently still in its infancy. Savings in ever-increasing costs of working up procedures owing to substantial decreases in pollutants very often do not seem to counterbalance the arguments against photochemistry. As a general rule, production quantities for photoreactions with quantum yields of < 1 are limited to around 100 tons per year, and photoreactor design tends to limit the size to units of about 20 kW. As a consequence, productions concentrate mostly on small batches of highly prized intermediates and end products, predominantly in pharmaceutical and perfume industries. Photophysical and photochemical selectivities are often provided for by the structural complexity of relatively large organic molecules, the phototransformationsof which most often are independent of excitation wavelength (practically all preparative reactions are carried out in solution and occur from the lowestlying excited state of a given spin multiplicity; such cases have not been addressed by Prof. Rice). For example, internal conversion to ground state and fluorescence may be the only depletion channels competing T I intersystem with a reaction from the SI excited state, or SI crossing may be predominant and one uni- or bimolecular triplet pro-

-

280

J. HIJMANS AND TH. HOLLEMAN

12. Prigogine, I., Bellemans, A., and Naar-Colin, C., J . Chem. Phys. 26, 751 (1957). 13. Prigogine, I., The Molecular Theory of Solutions, North-Holland Publ. Comp., 1957, Chapters XVI and XVII. 14. Mathot, V., Comfit. rend. Riunion sur les Changements de Phases, Paris, 1952, p. 115. 15. Prigogine, I., Trappeniers, N., and Mathot, V., Discussions Faraday SOC. 15, 93 (1953);J. Chem. Phys. 21, 559-60 (1953). 16. Lennard-Jones, J. E., and Devonshire, A. F., Proc. Roy. SOC., A163,63 (1937) and A164,l (1938). 17. See for instance Guggenheim, E. A., Mixtures, Oxford University Press, 1952, Chapter X. 18. Hijmans, J., Physica, 27, 433 (1961). 19. Holleman, Th., and Hijmans, J., Physica, 28, 604 (1962). 20. American Petroleum Institute, Research Project 44, Pittsburg, 1953, Selected values of physical and thermodynamic properties of hydrocarbons and related compounds, Table 20d. 21. Reference 20, Table 2Om. 22. Waddington, G., and Douslin, D. R.J. Am. Chem. SOC. 69,2274 (1947); Osbome, N. S., and Ginnings, D. C.,J. Res. Nut. Bur. Stand., 39,453 (1947); Waddington, G., Todd, S. S., and Hufhan, H. M., J . Am. Chem. SOC. 69, 22 (1947). 23. Reference 20, Table 20k. 24. McGlashan, M. L., and Potter, D. J. B.,Joint Conference on Thermal and Transport Properties of Fluids, London, 1957, session 1, paper 10. 25. Boelhouwer, J. W. M., Physica 26, 1021 (1960). 26. Boelhouwer, J. W. M., Physica 34, 484 (1967). 27. Flory, P. J., Orwoll, R. A,, and Vrij, A., J. Am. Chem. SOC.86, 3507 (1964). 28. Brensted, J. N., and Koefoed, J., Kgl. Danske Videnskab. Selskab. Mat. Fys. Medd. 22, No. 17, 1 (1946). 29. Hijmans, J., Mol. Phys. 1, 307 (1958). 30. Longuet-Higgins, H. C., Discussions Faraday SOC.15, 73 (1953). 31. Dixon, J. A . , J . Chem. Eng. Data 4, 289 (1959). 32. See for instance : CarathCodory, C., Conformal Representation, Cambridge University Press, 1952, Chapters 10 and 11. 33, Desmyter, A., and van der Waals, J. H., Rec. Trav. Chim. 77, 53 (1958); van der Wads, J. H., private communication. 34. Holleman, Th., Physica 29, 585 (1963). 35. Hijmans, J., and Holleman, Th., Mol. Phys. 4, 91 (1961). 36. Van der Waals, J. H., and Hermans, J. J., Rec. Truer. Chim. 69, 949 (1950); van der Wads, J . H., Thesis, Groningen, 1950. 37. McGlashan, M. L., and Morcom, K. W., Trans. Faraday SOC. 57, 907 (1961). 38. Bhattacharyya, S. N., Patterson, D., and Somcynsky, T., Physica 30, 1276 (1964): 39. Holleman, Th., Physica 31, 49 (1965). ,

I


281

J. Manz 1. In this chapter Prof. Rice has advocated two techniques that should be useful for evaluations of optimal fields for laser control of chemical reactions: (i) reduced space of eigenstates for representations of nuclear wavepackets and (ii) the use of effective reaction coordinates. Both techniques have already been used for efficient evaluations of reaction probabilities in model reactions. See, for example, Ref. 1 for the prediction of population inversion and Ref. 2 for the demonstration of rather strong deviations of chemical reactions from the reaction path, specifically in the case of hydrogen transfer reactions. 2. Very recently, Jakubetz et al. have extended the applications of our variant of Rabitz’s theory of optimal control by IR femtosecond/picosecond laser pulses [3] from vibrational transitions to isomerizations, specifically for the HCN + CNH reaction [4]. 3. Prof. S. A. Rice has pointed to another experimental verification of the Tannor-Rice-Kosloff scheme, carried out by Prof. G. R. Fleming. I would like to ask Prof. Fleming whether he could explain to us his experiment, that is, how are the two pump and control laser pulses used to control the branching ratio of competing chemical products? 1. J. Manz and H. H. R. Schor, Chem. Phys. LRtt. 107, 549 (1984). 2. B. Hartke and J. Manz, J. Am. Chem. Soc. 110,3063 (1988).

3. S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988); W.Jakubetz, J. Manz, and H.-J. Schreiber, Chem. Phys. Lett. 165, 100 (1990); W. Jakubetz, E. Kades, and J. Manz, J. Phys. Chem. 97, 12609 (1993). 4. W. Jakubetz, B. L. Lan, and V. Parasuk, in Femtosecond Chemistry and Physics of Ultrafast Processes, M. Chergui, Ed., World Scientific, Singapore, 1996; W. Jaku-

betz and B. L. Lan, in preparation.

S. A. Rice: I agree with Prof. Manz that reduced representations

of many-body dynamics have been introduced many times in other

problems. Such representations are common in statistical mechanics; for example, the Boltzmann equation describes the time evolution of a gas at the level of the single-particle distribution function, due to collisions that are representative of some of the properties of the twoparticle dynamics, and with neglect of all higher order dynamics. In general, a reduced representation is an exact transfonnation of the Nbody dynamics from one formalism to another formalism. The value of the reduced representation is that it suggests new approximations that could not easily be envisaged in the original N-body formulation; I expect the same advantage to apply to reduced representations used in control theory.

282

S. A. RICE

G. R. Fleming: In reply to Prof. Manz let me say that our experiments [J. Chem. Phys. 93, 856 (1990); 95, 1487 (1991); 96, 4180 (1992)l were the first femtosecond experiments in which the optical phase was explicitly controlled. The trick amounts to figuring out how to vary the delay in integer (integer +; or integer +$) multiples of the wavelength. D. J. Tannor: I would like to point out that the Scherer-Fleming wavepacket interferometry experiment is very different from the Tannor-Rice pump-dump scheme, in that it exploits optical phase coherence of the laser light (optical phase coherence translates into electronic phase coherence between the wavepackets on different potential surfaces). However, there was a paragraph in the first paper of Tannor and Rice [J. Chem. Phys. 83,5013 (1985), paragraph above 4. (ll)] that did in fact discuss the role of optical phase and suggested the possibility of experiments of the type performed by Scherer and Fleming. M. Quack: Prof. Rice has given a superb review and insight into the field. In a sense, it is a wonderful complement to his lecture on dynamics of molecules in Schliersee, published just 20 years ago [ 13. I would like to ask Prof. Rice how he would fit in the scheme proposed many years ago by Tom George, who suggested manipulating effective Bom-Oppenheimer potential surfaces for bimolecular reactions with strong laser fields in order to achieve reaction control. I realize that this was very hypothetical, requiring very intense fields that would presumably lead to ionization, but it still seems to me interesting just as a concept. 1 . S. A. Rice, in Excited States, Vol. 2, E. C. Lin, Ed., Academic, New York, 1975, p. 111.

S. A. Rice: The coupled matter-radiation system considered in the control schemes can, indeed, be studied from the point of view of dressed potential-energy surfaces, as suggested by the remark by Prof. Quack. We find it more convenient to use the equivalent point of view of continuous transfer of amplitude back and forth between the undressed potential-energy surfaces, because the formalism we have developed calculates the temporally and spectrally shaped field for that dynamical representation. P. W. Brumer: Tom George did indeed do significant work in control of reactions. However, if I recall correctly, his approach required extremely high field strengths and, as such, induced undesirable processes that competed with control (e.g., molecule ionization).


283

G. Gerber: The w3 = 3 0 and w scheme has been experimentally explored by Dan Elliott using CW laser radiation and by Bob Gordon using nanosecond radiation. I would like to know the viewpoint of Prof. Rice about what we would learn additionally by using ultrafast laser pulses. In ultrashort laser excitation, the individual levels under consideration are coherently coupled. S. A. Rice: To answer the question by Prof. Gerber, I should emphasize that the key element of the Brumer-Shapiro scheme is the exploitation of interference between two pathways connecting the same initial and final states. Unless the excitation sources connect the same initial and final states, and not different ones for the two pathways, the interference effect is compromised. The use of short laser pulses for the two excitation pathways has the potential disadvantage of providing bandwidth for the excitation of other pathways in addition to the desired pathways; hence the pulse must be kept long enough to avoid this possibility.

EXPERIMENTAL OBSERVATION OF LASER CONTROL: ELECTRONIC BRANCHING IN THE PHOTODISSOCIATION OF Na2 A. SHNITMAN, I. SOFER*, I. GOLUB, A. YOGEV* and M. SHAPIRO** Departments of Chemical Physics and *Energy and Environment The W e i m n n Institute of Science Rehovot, Israel 76100, 2. CHEN and P. BRUMER

Chemical Physics Theory Group and The Ontario Laser and Lightwave Research Centre University of Toronto, Toronto MSS IAI, Canada.

Control over the product branching ratio in the photodissociation of Naz into Na(3s) + Na(3p), and Na(3s) + Na(3d) is demonstrated using a two-photon incoherent interference control scenario. Ordinary pulsed nanosecond lasers are used and the Na;! is at thermal equilibrium in a heat pipe. Results show a depletion in the Na(3d) product of at least 25% and a concomitant increase in the Na(3p) yield as the relative frequency of the two lasers is scanned. We report the experimental observation of laser control over a branching photochemical reaction. The reaction studied is the 2-photon dissociation of **Communication presented by M. Shapiro Advances in Chemical Physics. Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-47 1 - 18048-3 0 1997 John Wiley & Sons, Inc.

285

286

SHNITMAN et al.

the Na2 molecule at energies where one Na atom is in its ground state and one Na atom is in the 3p, 4s or 3d states, i.e.

Na2

2hv ---t

Na(3s)+ Na(3p), Na(3s)+ Na(4s), Na(3s)+ Na(3d).

Control is demonstrated over the Na(3d), Na(3p) branching ratio. Achieving laser control over dynamical processes has been a longstanding goal of both physicists and chemists. Recent theoretical work [1]-[3] has shown that this goal may be achieved by manipulating quantum interferences, an area of research known as coherent control. Experimental verification of the basic principles of coherent control have followed [4]-[ 121 showing, for example, that total ionization rates can be coherently modulated [6]-[9] and that current directionality can be phase-controlled [7]-[ 121. However, there has only been one very recent report [131 of the primary aim of coherent control: to successfully manipulate integral yields into diferent competing product channels. Here we present an experimental demonstration of such control. Our approach is based upon our recent theoretical prediction [I41 that laser induced continuum structure (LICS) [15] can give rise to final channel selectivity. In this arrangement one gives structure to the continuum by optically dressing it with a bound state. We showed theoretically [ 141 that if we dress the continuum with an initially unpopulated bound state using a laser field of frequency w2, while exciting a populated bound state to this dressed continuum using a laser field of frequency w 1 , then a quantum interference arises whose destructive or constructive character depends upon the final channel. (An illustration of this scenario as it applies to Na2 is shown in Fig. 1). Theoretical studies [14] further showed that the character of this interference depends on the relative frequency between the two light fields, and that selectivity between the Na(3p) and Na(3d) channels [Eq.(I)] can be achieved by varying w1 or w2. This effect is virtually independent of the relative phase between the two light fields, i.e. the light fields need not be coherent. Thus, although the control depends on quantum interference, these interferences are not destroyed by incoherence of the incident laser radiation. The fact that this control scenario does not require laser coherence makes it especially attractive for laboratory use since generally available, non-transform limited, nsec dye lasers can be used. In our experiment we use two dye lasers pumped by a frequency-doubled Nd-Yag laser. One dye laser, whose frequency w2 was tuned between 13,312 crn-land 13,328 cm-I, was used

287

F'HOTODISSOCIATION OF Na2

.15 ?

2

W

v1

.lo-

4

a)

3 .05-

8

4

0

a

0-

XIC,

Figure 1. Incoherent Interference Control (IIC) scheme and potential energy curves for Na2. This scheme is composed of a 2 w i photon process proceeding from an initial state, assigned here as ( u = 5, J = 37). via the u = 35, J = 36, 38 levels, belonging to the interacting A' /311u electronic states, and a one 0 2 photon dresses the continuum with the (initially unpopulated) u = 93, J = 36 and u = 93, J = 38 levels of the A ' /311, electronic states.

xu

xu

to dress the continuum with a vib-rotational state of the A' C,/311umixed electronic state [16] of N g . The other dye laser, whose frequency W I was fixed at 17,474.12 cm-', was used to induce a 2-photon dissociation of the u = 5 , J = 37 ground state of Na2, through intermediate resonances (assigned as u = 35, J = 38 and u = 35, J = 36) of the A' C,/311, mixed state. Our W I and w2 pulses, both of -5 nsec duration with the stronger amongst them ( w 2 ) having an energy of -3.5 d, were made to overlap in a heat pipe containing Na vapor at 370 - 410°C. Spontaneous emission from the excited Na atoms [Na(3d) -+ Na(3p) and Na(3p) -+ Na(3s)l resulting from the Na2 photodissociation, was detected and dispersed in a spectrometer and a detector with a narrow bandpass filter. Figure 2 shows experimental Na(3d) and Na(3p) emission as a function of w2 at a fixed w 1 . Each point represents an average over a few hundred laser shots, each chosen to have an w 2 pulse energy which deviates by less than 5% from 3.5 mJ. At these energies we estimate our pulse intensity to be -10' Watts/cm2. We see that when the Na(3d) yield dips, the Na(3p) yield peaks, in accordance with theoretical expectation [171. The controlled modulation of the Na(3p)/Na(3d) branching ratio is seen to exceed 30%.

288

SHNITMAN et al.

-

Figure 2. Experimental Na(3d) fluorescence (solid), and Na(3p) fluorescence (dashed), (both uncalibrated), for the Na2 Na(3s) + Na(3d). Na(3p) IIC scenario whose details are given in Fig. 1, as a function of the w:! frequency. The w l frequency is fixed at 17,474.12 cm-'.

The theoretical calculations [ 171 of the Na(3d) yield resulting from photodissociation of a single initial Na2 bound state are presented in Fig. 3 and contrasted with the experimental results of Fig. 2. The same is done in Fig. 4 for the Na(3p) yield. We see two major Na(3d) dips, accompanied by Na(3p) peaks, in good agreement with the experiment. The calculations were done for the initially unpopulated u = 93, J = 31 and u = 93, J = 33 levels of the mixed A' /311, electronic state, accessed via the u = 33, J = 31 and u = 33, J = 33 intermediate resonances by the 2w 1 photon process. These u, J values differ slightly from those which we experimentally assigned (u = 35 and J = 36 and 38), but the lineshapes were found to change very little with small changes in u,J. Considerng the uncertainties in the theoretical potentials used [ 17, 181, the agreement between theory and experiment (especially in the Na(3d) signal) is impressive. Additional computations [ 141 suggest that the observed experimental sub-structures may be due to the excitation of numerous additional, as yet unassigned, thermally populated vib-rotational Na2 energy levels. The Na(3p) experimental signal is superimposed on a high background due to population of the Na(3p) state by emission from the Na(3d) and Na(4s) states, and due to direct population of Na(3p) from of the Na2 molecule by an 0 1 + 0 2 absorption (not possible energetically for the Na(3d) channel).

xu

289

PHOTODISSOCIATION OF Na2 NaZ-Na+Na(3d);

13310

13315

Theory a n d experiment

13320 13325 oz (cm-')

13330

-

13335

Figure 3. Comparison of the experimental and theoretical Na2 Na(3s) + Na(3d) yields as a function of 0 2 . In the calculation. an intermediate u = 33, J = 31, 33 resonance is used and 0 1 is fixed at 17,720.7 cm-.'. The intensities of the two laser fields are I(w1) = 1 . 7 2 ~lo8 Watts/cm2 and I(w2) = 2 . 8 4 ~10' Watts/cm2. The w2 frequency axis of the calculated results was shifted by -1.5 cm-' in order to better compare the predicted and measured lineshapes.

We also had to overcome radiation trapping effects by monitoring the Na(3p) -+ Na(4s) emission off line-center. The results shown in Figs. 2 and 4 are obtained by subtracting the contribution of these processes from the observed Na(3p) signal. To do so we calibrated the contribution from the Na(3d)

-

2

P

m

68

?

"* 13315

13i20

I

13325 o2 (cm-')

I

I

13330

Figure 4. Comparison of the experimental and theoretical Na2 as a function of 0 2 . with parameters, as in Fig. 3.

13335 -c

Na(3s) + Na(3p) yields

290

SHNITMAN et al.

Na(3p) emission via a separate experiment, where we monitored the Na(3p) signal resulting from the direct 2-photon excitation of the Na(3d) state. The direct w I + w2 contribution was accounted for by measuring the Na(3p) signal at different w I + w2 intensities. Because we saturate the one photon w 1 resonance, the dependence of the w I + w2 process was found to be linear in the w2 intensity. Hence, determining the slope and intercept of this linear dependence allowed us to subtract out its contribution at the experimental W I and w 2 intensities. To confirm that the observed Na(3d) dip and Na(3p) peak structures are indeed due to incoherent interference control, i.e., the interference between the w1 and the w2 induced optical processes, we ran the following checks: 1. We verified that what we are seeing is a strong field effect by changing the w2 power. Reducing the power by a factor of -50 resulted in the complete vanishing of the dip/peak structure. 2. We verified that the observed structures are due to the combined action of the two lasers by delaying the W I pulse relative to the w2 pulse. A delay of 339 nsec, guaranteeing no overlap between the pulses, completely eliminated the Na(3d) dips. 3. Real time measurements of the rise and decay of the Na(3d) and Na(3p) signals were performed. If the observed structures are due to an accidental secondary transfer of population from the Na(3d) to the Na(3p) state then such a mechanism would be reflected in the time dependence of the Na(3d) signal. Specifically, if a (collisional or other relaxational) mechanism was in effect at one 0.9 frequency and not at another, thus giving rise to the observed Na(3d) dip and Na(3p) peak at that particular w2 frequency, we would see a faster decay of the Na(3d) signal at that frequency relative to the other. Our findings show an identical decay curve for the Na(3d) signal at all w2 frequencies probed. The only thing which changes is the area under the decay curve. This indicates that it is the actual production of the Na(3d) [Na(3p)] state which is affected by changing w2, and not its subsequent decay [buildup]. 4. We shifted the w I frequency (by -0.19 cm-' and by -0.40 cm-') and examined the effect of such shifts on the dependence of the Na(3d) and Na(3p) yields on w2. Since the 2-photon w 1 absorption is mediated by a (saturated) intermediate I-photon resonance, the w 2 dependence of the Na(3d) and Na(3p) structures should move by an amount equal to the W I shift. This is indeed the case, as demonstrated in Fig. 5, where the updependent structures are seen to red shift, respectively by 0.23 cm-' and 0.37 cm-' . These values are, within our frequency resolution of fo.04cm-' ,in perfect accord with the above expectations. A similar

29 1

PHOTODISSOCIATION OF Na2 I

I

I

I

, The Na(3d) fluorescence as a function of w 2 for three different w1 frequencies. The lowest trace corresponds to an w l value of 17,474.12 Em-'. The upper traces result from red shifting the w l frequency by .I9 cm-I and .40 cm-l. We observe a red shift in the w 2 dependence of the Na(3d) yield of, respectively, .23 cm-l and .37 cm-', each being, within our experimental uncertainty of identical to the respective o 1 shift.

shift of the Na(3p) peaks was also observed, thus verifying the optical origin of the effect. In summary, we have experimentally demonstrated laser control of a branching photochemical reactions using quantum interference phenomena. In addition we have overcome two major experimental obstacles to the general implementation of optical control of reactions: (a) we have achieved control using incoherently related light sources, and (b) we have affected control in a bulk, thermally equilibrated, system. Acknowledgments This work was supported by the Minerva Foundation, Germany, by the Israel Academy of Sciences and by the U.S. Office of Naval Research under contract number N00014-90-J-1014.

References 1. For recent reviews, see M. Shapiro and P. Brumer, Int. Reviews Phys. Chem. 13, 187 (1994); P. Brumer and M.Shapiro, Ann. Rev. Phys. Chem. 43, 257 (1992). See also, P. Brumer and M.Shapiro, Chem. Phys. Lett. 126, 541 (1986). 2. See, D. J. Tannor, and S. A. Rice, Adv. Chem. Phys. 70, 441 (1988); R. Kosloff, S. A. Rice, P.Gaspard. S. Tersigni, and D. J. Tannor, Chem. Phys. 139,201 (1989); S. Shi and

292

SHNITMAN et al.

H. Rabitz, Chem. Phys. 139, 185 (1989); Y. Yan, R. E. Gillilan, R. M. Whitnell and K. R. Wilson, J. Phys. Chem. 97, 2320 (1993). 3. A. D. Bandrauk, J-M. Gauthier and J. F. McCann, Chem. Phys. Lett. 200,399 (1992); M. Yu. Ivanov, P. B. Corkum and I? Dietrich, Laser Physics 3, 375 (1993). 4. C. Chen, Y-Y. Yin, and D. S. Elliott, Phys. Rev. Lett. 64, 507 (1990); ibid, 65, 1737 ( 1990). 5. S. M. Park, S-P. Lu, and R. 3. Gordon, J. Chem. Phys. 94, 8622 (1991); S-P. Lu, S. M. Park, Y. Xie, and R.J. Gordon, J. Chem. Phys. 96,6613 (1992). 6. V. D. Kleiman, L. Zhu, X. Li and R. G. Gordon, J. Chem. Phys. 102,5863 (1995). 7. G. Kurizki, M. Shapiro, and P. Brumer, Phys. Rev. B 39, 3435 (1989). 8. 8. A. Baranova, A. N. Chudinov, and B. Ya Zel’dovitch, Opt. Comm., 79, 116 (1990). 9. Y-Y. Yin, C. Chen, D. S. Elliott, and A. V. Smith, Phys. Rev. Lett. 69, 2353 (1992). 10. E. Dupont, P. B. Corkum, H.C . Liu, M. Buchanan and 2. R. Wasilewski, Phys. Rev. Letters 74, 3596 (1995). 11. B. Sheeny, B. Walker and L. F. Dimauro, Phys. Rev. Lett. 74,4799 (1995). 12. Y.-Y. Yin, R. Shehadeh, D. Elliott, and E. Grant, Proceedings, OSA Annual Meeting, Dallas (1994). 13. L. Zhu, V. Kleiman, X. Li, S. P. Lu, K. Trentelman and R. J. Gordon, Science 270, 77 (1995). 14. Z. Chen, M. Shapiro and P. Brumer, Chem. Phys. Letters 228,289 (1994); J. Chem. Phys. 102, 5683 (1995). 15. See, for example, P. L. Knight, M.A. Lauder and B. J. Dalton. Phys. Rep. 190, 1 (1990). and references therein; 0. Faucher, D. Charalambidis, C. Fotakis, J. Zhang and P.Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993). 16. Z. Chen, M. Shapiro and P. Brumer, J. Chem. Phys. 98, 8647 (1993); J. Chem. Phys. 98, 6843 (1993). 17. Theoretical calculations reported here are for CW excitation. Pulsed laser based computational studies are ongoing. 18. The potential curves and the relevant electronic dipole moments are from I. Schmidt, Ph.D. Thesis, Kaiserslautern University, 1987.

DISCUSSION ON THE COMMUNICATION BY M. SHAPIRO Chairman: M. Quack B. Kohler: Prof. Shapiro, could you add some “nonarbitrary” units

to the y-axis of the data from your interesting Na2 control experiment?

What percentage change in the yield are you able to obtain? M. Shapiro: The change in the Na(3d)/Na(3p) branching ratio is typically 30%-40%.We cannot give absolute units for the yield of each channel because we have not calibrated the fluorescence emission signal. M. S. Child: It seems to me that control by changing the frequency

PHOTODISSOCIATIONOF Na2

293

of the laser is quite different from other control schemes. How does this differ in principle from one photon control by turning from one electronic state to another? M.Shapiro: The method of incoherent interference control used in our experiment is completely general and allows us to use Fano type interferences to control final states even if such lines do not naturally exist. Of course if the molecule accommodates you (as FNO)you do not need this but this is a rare situation. For other situations, especially when the material continuum is slowly varying and tiny changes in the laser frequencies in the one photon transition have absolutely no effect on the product ratios, our method allows for control, via the optical induction of resonances, in complete generality.

COHERENTCONTROLOF BIMOLECULAR SCATTERING P. BRUMER* Chemical Physics Theory Group, Department of Chemistry University of Toronto Toronto, Canada

M.SHAPIRO Department of Chemical Physics The Weizmann Institute of Science Rehovot, Israel

CONTENTS I. Introduction II. Control of Collisions References

I. INTRODUCTION Coherent control of molecular processes has seen enormous progress in the last half decade. Advances have been summarized in a number of recent reviews (see S. A. Rice, this volume; see also Refs. 1 and 2). Consideration *Communication presented by I? Brumer Advances in Chemical Physics, Volume 101: Chemical Reacrions and Their Control on the Femtosecond Time Scale, XXrh Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

295

296

P. BRUMER AND M. SHAPIRO

of these papers shows that the vast majority of control papers address the problem of controlling bound-molecule dynamics or photodissociation. By contrast, bimolecular processes, which form the bulk of interesting chemistry, has only received theoretical attention in one paper [3]. In that paper we showed that a variant of the two-level pump-dump control scenario [4] could be used to control bimolecular reactions, Unfortunately, the controlled yield above the reaction threshold was small compared to the natural reactive probability. As such, the proposed scheme was useful primarily below the threshold for reaction. In this chapter we briefly summarize the essence of our recent work [ 5 ] , which introduces a coherent control strategy to control bimolecular collisions, Computations designed to examine the range of control possible with this scenario are currently being carried out [6]. The essential principle of coherent control in the continuum is to create a linear superposition of degenerate continuum eigenstates out of which the desired process (e.g., dissociation) occurs. If one can alter the coefficients a; of the superposition at will, then the probabilities of processes, which derive from squares of amplitudes, will display an interference term whose magnitude depends upon the a;. Thus, varying the coefficients a; allows control over the product properties via quantum interference. This strategy forms the basis for coherent control scenarios in which multiple optical excitation routes are used to dissociate a molecule. It is important to emphasize that interference effects relevant for control over product distributions arise only from energetically degenerate states [7], a feature that is central to the discussion below. Consider now the bimolecular reaction B + C F + G, where B, C, F, and G are atoms or molecules. Here the goal is to control the reactive versus nonreactive probabilities. Creating the required superposition of degenerate eigenstates of B + C means creating continuum states with a welldefined relationship between the internal energies of B, C and the B, C relative translational energy. Optical control of the kinetic energy of a colliding pair can in principle be achieved by excitations of continuum states while a collision takes place [3]. However, such control is difficult because the time spent by the colliding partners in the region where they strongly interact is very short (typically 0.1-1 ps). Indeed, there are very few examples in which direct optical excitation of continuum states in the strong-interaction region (the so-called transition-state spectroscopy) has been achieved [81. A practical way of forming superposition states of correlated scattering states, thereby achieving control, is the topic of this chapter. Space limitations prevent more than a sketch; details can be found elsewhere [5].

-

COHERENT CONTROL OF BIMOLECULAR SCATTERING

297

II. CONTROL OF COLLISIONS The approach we advocate is straightforward. Consider the bimolecular process

B(i) + C ---t F + G

(1)

Here we assume that C is an atom, and we have explicitly indicated the internal state of molecule B using the label i. The energy of this state is given by E = EC(O) + €B(i) + EF"(i), where EC(O) is the internal energy of C, E B ( ~ is ) the internal energy of B(i), and E;"(i) is the center-of-mass kinetic energy in the reactant channel, which we have labeled the 0 channel. For two different eigenstates, comprised of B(I) + C and B(2) + C, to be of the same energy requires

Thus, to attain control, we wish to create a superposition of states of B(1) + C and B(2) + C:

with kinetic energies satisfying Eiq. (2). Here l i , X ) with X = B, C are eigenstates, of energy e x ( i ) , of the internal Hamiltonians he and hc of B and C. The IEF(i)) denote plane waves describing the free motion of B relative to C, that is, (RIEF"(i)}I exp(iki . R), where lkil = { 2 p ~ ~ E y ( i ) ) ' / * / h and p ~ =c mBW/(mB + W) is the reduced mass of the BC pair. Then, in a manner analogous to unimolecular control, varying ai will alter the product distributions [ 5 ] . Control over the ai and production of the desired superposition states can be achieved by several routes. One nice way is to utilize the reactants from an earlier photodissociation step, altering the ui by any of a number of coherent control scenarios [2] for this prereactive step. Consider then preparing In, /3) via a "prereactive" stage in which an adduct AB, made up of a structureless atom A and the molecular fragment B, is photodissociated. The AB is assumed to be initially in a pure state of energy Eg and the photodissociation is carried out with a coherent source. Under these circumstances photodissociation produces B in a linear combination of internal states. For

298


convenience we assume two such internal states:

AB

--. ho

A+B(i)

i = 1,2

(4)

The total energy in the photolysis stage, denoted EAB, is EAB = E, + ho, where w is the frequency of the photolysis laser. Hence, following the photodissociation process, fragment B is described by In,B) =

C ajli,B)IEg"(i))

i = 1.2

where

where ~ A =B mAmg/(mA + m g ) is the reduced mass of the AB pair and m A and me are the masses of the A and B fragments. Preparing &. ( 5 ) is not, however, sufficient to produce the desired In,@ state [Eq.(3)] since one needs contributions to In, 0) to be of the same energy. This can be achieved by colliding B and C at a specific angle. To see this, consider the following kinematic argument in order to choose 6 , the incident angle of C, and x , the direction of the photolysis fragment B (both defined with respect to V A Bthe , AB center-of-mass velocity vector). We denote the velocity of the jth particle in the AB center of mass system by u, and its laboratory velocity by v,. For particle B we have laboratory velocities v g ( I), vB(2) and corresponding center-of-mass velocities uB( I), ug(2), where, following the photolysis prereactive step, the velocity of the B fragment in the AB center-of-mass system is given, according to EQ.(6), as

We choose is

uc collinear with ue(i) so that the relative BC velocity, UB&,

We focus only on the plus solution, the minus sign giving nonphysical results. Using Eq. (8), the kinetic energy of B relative to C is given as

COHERENT CONTROL OF BIMOLECULAR SCATTERING Ekin

i

6 ()=

ic~Bcv2,c(i) = ; P B c { ~ ~ + u 3 i ) + 2ucue(i))

299

(9)

Imposing the condition in Eq. (2) gives, using Eqs. (2), (7),and (9), that

where ~ C - A B = mC(mA + mB)/(mA

+ mB + mc). Knowing UC, we have that

and

X

sin 8 = uc sin VC

(12)

This then provides the desired 8 and x to ensure that the elements of the superposition state [Q. (3)] are degenerate. The above formalism can be readily extended to general superposition states of the form

i,1

with E F ( i , I ) = E - ~ c ( 1 )- € ~ ( i )Here, . the I E F ( i , 1)) states are plane waves describing the free motion of B relative to C [(RJEp(i,l)) = exp(iki,.R)], where fkirl = {2jtBCEp(i, Z))’/2/h. That is, we can show that such a superposition leads to interference and hence to the possibility of control over the reaction cross sections. However, the experimental realization of such states presents a serious challenge. First, it is entirely unclear how to produce general controllable ail. Second, a typical experiment produces wave functions where the eigenstates of HB, hc, and Kp are uncorrelated and spread out over a considerable energy range. Under these circumstances any quantum interference terms are negligible compared to the uncontrolled contributions. Indeed, the inability to create controllable states such as Eq. (1 3) should serve to emphasize the insight required to design adequate scenarios. Note added in proof: Careful consideration has shown that the above approach pays insufficient attention to the motion of the center of mass

300


associated with the B-C collision when B is prepared as described above. Appropriate consideration of this aspect of the collision shows that an alternate scenario is necessary to ensure that quantum interference, and hence control, survives. See references [5] and [6] for details.

References 1. 2. 3. 4. 5. 6. 7. 8.

P. Brumer and M. Shapiro, Sci. Am. 272, 56 (1995). M. Shapiro and P. Brumer, Int. Rev. Phys. Chem. 13, 187 (1994). J. L. Krause, M. Shapiro, and P. Brumer, J. Chem. fhys. 92, 1126 (1990). D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985); T. Seideman, M. Shapiro, and P.Brumer, J. Chem. Phys. 90, 7132 (1989). M. Shapiro and P. Brumer, Phys. Rev. Lett., 77, 2574 (19%). D. Holmes, P. Brumer, and M. Shapiro. J. Chem. Phys. 105,9162 (1996). P. Brumer and M. Shapiro, Chem. Phys. 139, 221 (1989). B. A. Collings, J. C. Polanyi, M. A. Smith, A. Stolow, and A. W. Tarr, Phys. Rev. Lett. 59, 2551 (1987).

LASER HEATING, COOLING, AND TRANSPARENCY OF INTERNAL DEGREES OF FREEDOM OF MOLECULES D.J. TANNOR* Department of Chemical Physics Weizmann Institute of Science Rehovot Israel R. KOSLOFF AND A. BARTANA Department of Physical Chemistry and the Fritz Haber Research Center The Hebrew University Jerusalem Israel

CONTENTS I. Introduction

11. Instantaneous Dipole Moment: Generalized Einstein B Coefficient 111. Vibrational Heating Using Nondestructive Optical Cycling

IV. Nonevaporative Cooling References

I. INTRODUCTION We are in the infancy of a new era in the physical sciences. It is probable that in the coming years we will learn general strategies for manipulating the *Communication presented by D. J. Tannor Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

301

302

D. J. TANNOR. R. KOSLOFF AND A. BARTANA

precise quantum states of atoms and molecules using laser light [l-5, 181 (see also S. A. Rice, Perspectives on the Control of Quantum Many-Body Dynamics: Application to Chemical Reactions). The possibilities inherent in manipulation go far beyond simply the excitation of particular quantum states that might otherwise be hard to access but, in addition, to controlling the phase coherence of superposition states in order to achieve desired chemical and physical purposes. This phase coherence can at times be responsible for the most peculiarly quantum effects and at other times for the building of quantum mechanical wavepackets and the recovery of classical-like behavior. One of the new areas to exploit the assumption of optical phase coherence is quantum computing 161. The key concept is that in a set of N two-level systems there are 2'"' basis functions that can be manipulated independently using the laser and that are available for computation, as compared with the N elements that are up or down in conventional computing. A second new area is laser-induced transparency: An ordinarily opaque material can become transparent in the presence of laser light [7]. Again, the idea is simple: If the laser prepares a coherent superposition of states that connect via opposite signs to the excited state, the net absorption will be zero. This same concept is behind lasing without population inversion and population inversion without lasing [8]. Moreover, the same basic concept of preparing a coherently trapped superposition state underlies one of the most successful methods of atom cooling [9, 101. Here, a superposition of translational states is created that is dark for u = 0 and not dark for nonzero u's. As atoms get cooled u 0 and then they no longer interact with the light! This cooling mechanism has allowed cooling to 2 pK, below the limit dictated by random recoil from spontaneous emission. In this brief chapter we discuss two applications of phase tailored pulses to the manipulation of molecules: vibrational heating without demolition and laser cooling of internal molecular degrees of freedom. In both these applications the laser field is used to maintain orthogonality of a lower and upper electronic wavepacket, and as a result there is no absorption. In some sense this work can be viewed as a generalization of the principle of laserinduced transparency and its relatives, discussed in the previous paragraph, from three-level systems to multilevel wavepackets. However, there is a richness in the molecular multilevel case that is absent in the atomic case. The coordinate dependence of the wavepacket in general has a nontrivial spatial dependence that can be manipulated while maintaining the conditions of zero absorption. 3

LASER HEATING, COOLING, AND TRANSPARENCY

303

II. INSTANTANEOUS DIPOLE MOMENT: GENERALIZED EINSTEIN B COEFFICIENT The time-dependent Schriidinger equation for a two-electronic-state system with transition dipole moment p can be written as

The subscripts g and e refer to the ground and excited electronic state indices, respectively. The term Hg/e refers to the Born-Oppenheimer Hamiltonian for the ground/excited electronic state, respectively; $g,e is the wave function (wavepacket) associated with the ground/excited Bom-oppenheimer potential-energy surface. The two electronic states are coupled by the transition dipole moment p , which interacts with the electric field e ( t ) associated with the laser pulse. Complex values of the field are considered admissible by associating the real and the imaginary parts of the field with the two independent polarizations of the laser light perpendicular to its direction of propagation. It is straightforward to show that Eq. (1) leads to the following equation for the rate of change in excited-electronic-statepopulation:

The quantity peg = (+e\p\+g) is the instantaneous (complex) transition dipole moment, also called the polarization. It is a generalizationof the constant peg, which is essentially the Einstein B coefficient. In the standard Einstein treatment the B coefficients for absorption and stimulated emission are identical. In fact, however, peg is complex and not equal to pge; moreover, it is time dependent. An example of the use of Eq. (2) is provided by the wavepacket interferometry experiments of Scherer, Fleming et al. [ 111. These workers have demonstrated that the phase of the light can be used to control constructive versus destructive interference of wavepackets in the excited electronic state. An alternative way of interpreting their experiment is that the phase of the second pulse relative to the first determines the direction of population transfer between the two electronic states. In the spirit of the present discussion, absorption versus stimulated emission is being controlled by the choice of phase of the light relative to the instantaneous p g e $ peg! Since the direction of population transfer is not determined in this case by population inversion

304

D. J. TANNOR, R. KOSLOFF AND A. BARTANA

alone, it is a short step to consider the possibility of lasing without population inversion and population inversion without lasing using phase-locked laser pulse sequences. It is possible to view Eq. (2) as a generalization of the third of the optical Bloch equations, that is, the equation for population transfer. This leads us to consider analogies between coherence phenomena in two-level systems and the case where there is a wavepacket on each of two electronic state potential-energy surfaces. Moreover, the Feynman-Vernon-Hellwarth (WH) geometric picture, which is so useful for two-level systems [121, may be expected to be useful in the wavepacket context as well. In particular, the applications below to heating and cooling can be viewed as wavepacket generalizations of photon locking in two-level systems [13].

III. VlBRATIONAL HEATING USING NONDESTRUCTIVE OPTICAL CYCLING Several years ago, Nelson posed the following problem [14]: Is it possible to design a pulse sequence, using an impulsive stimulated Raman mechanism, to give large-amplitude vibrational motion on the ground electronic state without creating significant amounts of excited electronic state population? The problem with excited-state population is twofold: It complicates the interpretation of the experimental results, and it is often a precursor to dissociation or ionization. It is simple to see that the condition

where C(t) is some real envelope function (positive or negative) guarantees the condition of zero electronic population transfer [19]. The equation for the energy of the wavepacket on the ground state is given by

The strategy is then to adjust the sign of C(t) in such a way that the vibrational energy on the ground state will monotonically increase. Numerical experiments show that this works beautifully, both for weak fields and strong fields; for strong fields the heating is simply faster (see Figs. l a d ) [19]. It is interesting to contemplate the possibility of extending these ideas


305

T

Figure 1. (a) The FVH diagram illustrating photon locking in two-level system. Pseudopolarization (r) vector begins at z = - I , corresponding to all population in ground state, and precesses about field (a)vector (which initially points along x axis) according to equation dr/dt = x r. When r vector reaches x-y plane ( r / 2 pulse), phase of field is changed by b / 2 . This rotates field vector by 90".bringing it into alignment with r vector and making further precession impossible. Physical manifestation of this geometric picture is that although resonant field continues to be applied, there is no further population transferred between two levels. (b)Potential-energystructure of model. Frequencies of ground and excited surfaces are 1.0 and 0.8, respectively. Excited surface shifted by 7.0 units of energy and 3.0 units of distance, which leads to vertical distance of 10.6 units. Dipole operator has slope of 1. Coherent wave function with energy of 1.0 used as initial state on ground surface. Ground and excited absolute values of wave function shown after first exciting pulse (not to scale). (c) Change in population on ground state ( A N g )(dotted line), change in ground-state energy (AE,) (dashed line). and real part of field e (solid line) as function of time. Note monotonic increase in groundstate energy, coming in bursts at times that pulse is on. This is weak-field case; strong-field case has same qualitative behavior but rise in energy is faster and pulse shapes more sharply modulated. ( d ) Power spectrum of pulse in (c). Frequency of vertical transition 9; spectrum shows dips at excited-state vibrational resonances, consistent with condition of zero absorption.

to more than two electronic states. For example, if there is a third higher electronic state that is strongly coupled to the second electronic state, is it still possible to apply the excitation without demolition procedure? The answer when there are three states is manifestly yes; if there are many more excited states, the situation is not clear.

IV. NONEVAPORATIVE COOLING One of the most exciting areas in atomic physics in the last 10 years has been laser cooling of translational motion in atoms to 10-3-10-9 K [9, 10, 151. Several variations on cooling of atomic translational motion have now been proposed, but the generic scheme is as follows: Two monochromatic laser beams are propagated, one along and one opposite an atomic beam; by tuning the frequency of the laser to the red of resonance with a sharp

306


30.

20.

10.

0.0

I

-7.0

1

1

-3.0

1.o

time (C)

Figure 1. (Continued)

I

5.0

9.0


307

0.0

0

(4

Figure 1. (Conrinued)

electron.,: transition in the atom, those atoms propagating faster an^ therefore having a larger Doppler shift absorb light and are slowed down by the momentum imparted by the photon. Slowly shifting the frequency of the light into resonance progressively slows down more and more of the atoms. An added effect comes from a combination of level splitting in the presence of the light, together with a light-induced polarization gradient, leading to a so-called Sisyphus effect [9]. The cooling limit of this process is dictated by the random recoil imparted by the spontaneous emission of the photon; however, even this limit can be overcome by a method of velocity-dependent coherent population trapping [lo]. It is generally believed that the same schemes for translational cooling cannot be applied to molecules. The reasons are quickly perceived: molecules, even the smallest of them, have a high density of internal states coming primarily from rotations and vibrations, although also from electronic and spin degrees of freedom. The atomic cooling scheme relies on the validity of a two-level system picture, so that a single frequency red shifted from resonance will cool the entire population. In a molecule with a

308


congested energy-level spectrum, red shifting from one particular resonance line will cool only those molecules in that one level; then molecules in each level will have to be cooled separately. If there are thousands or millions of internal states occupied, this is quite a challenge. In addition, red shifting away from one level may entail shifting into resonance with another level. Recently, Bahns et al. [161 suggested a strategy for cooling molecules that involves the progressive cooling of rotations, then translations, and finally vibrations. Probably the most challenging of these stages is the first, the cooling of rotations. The Connecticut group proposes to use the molecule itself to generate the laser light at the characteristic absorption frequencies of the rotational manifolds. When this multitude of frequencies impinges on the rotationally hot molecule, it will lead to optical absorption followed by spontaneous emission. If the frequency corresponding to resonant excitation out of the ground rovibrational state is blocked, the population in this state can only grow, through spontaneous emission, and never diminish. One can imagine a variation on this scheme in which all frequencies are at first blocked and then unblocked from lowest to highest according to some schedule. An alternative approach to vibrational cooling uses shaped pulses, again based on photon locking 1171. The goal is to cool an initial thermal rotational or vibrational distribution down to 0 K. The strategy, as in the heating scenario, is to use the excited electronic state as an optical lever for cooling without transferring a significant amount of population upstairs. Formally, the biggest difference from the heating is that the equations must be cast in terms of the density operator to describe the absence of coherence in the initial population and the subsequent incomplete coherence. We begin with some defining equations for the operators when there are two coupled potential surfaces. The density operator is given by p = pg (8 P ,

+ p, @ P , ipi (8 s,

+pt

0s-

where Pele are projection operators for the ground- and excited-state surfaces, respectively, and S, are the raising and lowering operators from one surface to another. The Hamiltonian is given by

H = Ho + V(t) where

(7)


309

and the surface Hamiltonians H, and H, are functions of the internal coordinates. Moreover,

The evolution of the density operator is described by the Liouville-von Neumann equation

The equation for population change is now given by 1

d N R - _- . dt A

(p 63 (S+E - SA*))

where ,&t is the phase of the instantaneous dipole and is the phase angle of the radiation field. Moreover, the equation for the change in ground-state energy is now

where t $ , ~ is the phase angle of ( p H R C3 S+).For zero population transfer, choose E = C(r)+((p BS-)), where C(t) is a real function of time and +(X) = X / l X ) = eid' is the phase factor. Under the condition of zero population transfer the change of energy on the ground state becomes

The condition on the function C(t) that will lead to a monotonic decrease in energy is


310

T +

Imag

--

Q.

L

0

Real

0.5

h

3

cEgz witout dissipation <Eg> with dissipation

0.49

-

I

--------I

9)

Y

6k

0.48

9)

w

C

0.47

0.46

0

10

20

30

40

T h e (fs) (b)

Figure 2. (a) Phase-angle diagram for zero mass transport with monotonic decrease in ground-state energy. Key observation is that zero mass transport condition requires that g u + @ , = 0, n; this determines %e within a sign. Except for pathological case, there will always be one choice of sign that gives %,H + gt < 0, leading to monotonic decrease in ground-state energy; other choice of sign leads to monotonic increase in ground-state energy. (b) Ground surface energy in presence of the cooling pulse as function of time, with and without dissipation. (c) Phase-space density of initial density operator p g ( 0 ) ( Wigner distribution function in position-momentum phase space). Upper panel: stereoscopic projection. Lower panel: contour map. (d) Phase-space display of final density operator without dissipation pg(tf ). Upper panel: stereoscopic projection. Lower panel: contour map.

LASER HEATING, COOLING, AND TRANSPARENCY 6.

4. 2.

0

1

1.5

2

R

(A) (d)

Figure 2. (Continued)

2.5

311

312

D. J. TANNOR. R. KOSLOFF AND A. BARTANA

These equations are the density matrix analogues of the heating equations in the previous section. Figures 2u-d (see pages 3 10-3 11) show that this strategy is effective in producing monotonic cooling. However, the numerical results indicate that in absolute terms the temperature drop is not dramatic. It turns out that entropy limits the degree of cooling attainable. These entropy considerations, together with alternative laser cooling strategies, will be presented in a separate publication.

Acknowledgment This work was supported by a grant from the U.S. Office of Naval Research.

References 1. A. H. Zewail, Phys. Today 33, 27 (1980). 2. P. Brumer and M. Shapiro, Sci. Am. (March 1995). 3. W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1993). 4. D. J. Tannor, in Molecules in Laser Fields, A. Bandrauk, Ed., Dekker, New York, 1994. 5 . B. Kohler, J. Krause, F. Raksi, K. R. Wilson, R. M. Whitnell, V. V. Yakovlev, and Y. Yan, Accr. Chem. Res. 28, 133 (1995). 6. D. P. DiVincenzo, Science 270,255 (1995). 7. K.J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66,2593 (1991); J. E. Field, K. H. Hahn, and S. E. Hams, Phys. Rev. Lett. 67,3062 (1991). 8. M. 0. Scully and M. Fleischhauer, Science 263, 337 (1994); Phys. Today, p. 17 (May 1992). 9. C. N. Cohen-Tannoudji and W. D. Phillips, Phys. Today, p. 33 ( a t . 1990). 10. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988). 11. N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991). 12. L. Allen and J. H.Eberly, Optical Resonance and Two-Level Arums, Dover, New York, 1987. 13. E. T. Sleva, I. M. Xavier, Jr., and A. H. Zewail, J. Opt. Soc. Am. B 3,483 (1986). 14. K. Nelson, in Mode Selective Chemistry, Vol. 24, Jerusalem Symposium on Quantum Chemistry and Biochemistry, by J. Jortner, Ed., Kluwer Academic, Hingham, MA, 1991, p. 527. 15. A. Kastberg, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and P. S . Jessen, Phys. Rev. Lett. 74, 1542 (1995). 16. J. Bahns, P. Gould, and W. Stwalley, ACS Absa (April 1995). 17. A. Bartana, R. Kosloff, and D. J. Tannor, J . Chem. Phys. 99, 196 (1993).


313

18. R. Kosloff, S . A. Rice, P. Gaspard, S. Tersigni, and D. J. Tannor, Chem. Phys. 139, 201 (1989). 19. R. Kosloff, A. D. Hammerich, and D. Tannor. Phys. Rev. Leu. 69, 2172 (1992).

DISCUSSION ON THE COMMUNICATION BY D. J. TANNOR Chairman: M. Quack D. M. Neumark We are interested in generating coherent vibrational motion in negative ions, which typically do not have bound excited electronic states. Does your Impulsive Stimulated Raman Scattering (ISRS) scheme work if the excited state is not bound? D. J. Tannor: I think there is a good chance that the scheme, or some variation, may work even with a dissociative intermediate state. The reason is that the excited-state population is small; in some sense it is “virtual” in the old sense of Raman scattering. Moreover, when strong fields are used, the validity of the original Born-Oppenheimer surfaces as a good zero-order picture generally breaks down. In other words, with strong fields, there is no a priori reason to believe the excited-state dynamics will remain dissociative. It would be very interesting for us to do the calculation. J. Manz: Prof. D. Tannor has demonstrated the possibility to achieve laser cooling (or heating) of molecules in the electronic ground state. These effects are achieved, however, by laser-induced transitions to and from the electronic excited state. As a consequence, one may end up with an ensemble of molecules where part of them are cooled (or heated) in the electronic ground state, whereas the rest may be found in the electronic excited state with a rather uncontrolled “mixed” distribution of nuclear states. I would like to ask Prof. Tannor to give us some details about the relative fraction of molecules in the excited state, about their distributions of vibrational states, and about their role in view of the original purpose of laser cooling or heating. D. J. Tannor: In reply to Prof. Manz, I have to emphasize that there are three quantities to examine: 1. Norm (population) in the excited electronic state 2. Energy in the excited electronic state 3. Entropy in the excited electronic state

The scheme guarantees that the norm in the excited state is maintained constant (e.g., at 2%. to avoid sample destruction). The excitedstate energy does not necessarily rise; there is no conservation rule that

314


states the sum of the energy in the ground and excited electronic states must be conserved, since the light can carry away energy. However, the entropy in the ground and excited states is in fact conserved, and hence the entropy in the excited state must increase as the entropy in the ground state decreases (in the cooling scenario). This in fact puts limitations on the ability to cool, depending on the excited-electronicstate density of internal vibrational/rotational states.

RAMIFICATIONS OF FEEDBACK FOR CONTROL OF QUANTUM DYNAMICS H. RABITZ Department of Chemistry Princeton University Princeton, New Jersey CONTENTS I. Introduction

11. The Ubiquitous Role of Feedback A. Feedback in the Design of Molecular Controls

B. Feedback in the Laboratory Control of Molecular Dynamics C. Feedback in the Inversion of Molecular Dynamics 111. Conclusion References

I. INTRODUCTION Recent years have seen a flurry of activity in both the theoretical and experimental aspects of control over molecular processes [l] (see also S. A. Rice, “Perspectives on the Control of Quantum Many-Body Dynamics: Application to Chemical Reactions,” this volume). Most of the emphasis has been on the use of optical fields as a means for control, although other approaches can be envisioned in special circumstances 121. The key underlying principle of the overall subject is the achievement of control through the manipulation of quantum wave interferences [ l , 31, although full control will surely not be lost in the incoherent regime. The topic of molecular control is rich in detail as well as in potential appliAdvances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, Xyth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

315

316

H. RABITZ

cations. The purpose of this chapter is to emphasize a few common points of generic interest to molecular control theory and its laboratory implementation. In particular, it will be argued that the principle of feedback underlies both control field design and the implementation of control in the laboratory, especially for complex systems. The notion of feedback leads to a compact nonlinear form for the control Schrodinger equation. In turn, this mode of thinking naturally suggests an important spin-off application of control for inversion (i.e., learning about molecular Hamiltonians from laboratory data). This chapter will present a common framework for all of these topics.

II. THE UBIQUITOUS ROLE OF FEEDBACK The notion of quantum feedback control naturally suggests a closed-loop process in the laboratory to stabilize or guide a system to a desired state. In addition, feedback is important in the design of molecular controls. These points will be made clear below, starting with considerations of design followed by a discussion of its role in the laboratory and finally leading to feedback concepts for the inversion of laboratory data. A. Feedback in the Design of Molecular Controls

Here, we consider the design of an optical electric field ~ ( t interacting ) through a molecular dipole moment p to achieve control over quantum molecular evolution. Various approaches have been taken to the design of e ( t ) to achieve molecular objectives. The most general technique is to pose the task as an optimal control problem [l, 41 (see also S. A. Rice, this volume), where the goal is to design the field e ( t ) to meet the objective in balance with a variety of possible competing deleterious processes. Each physical problem will have its own transcription of this competition. As we a priori do not know the control e ( t ) , it is not sufficient just to consider solving the Schrodinger equation alone,

where Ho is the molecular Hamiltonian and *O is the assumed known initial condition. The optimal design problem, in its simplest form, reduces to solving Eiq. (l), along with the equation d at

ih - += [Ho - pc(t)]+

FEEDBACK FOR CONTROL OF QUANTUM DYNAMICS

317

where t#J can be thought of as a Lagrange multiplier, having a final condition * T ) = XOQ(T). Here, T is the final target time when we desire the observable (Q(T)lOl\k(T))to take on the value 0,where the objective operator is 0. The coefficient X is the target deviation h = ((*(T)lOlq(T))- 0).Equations (1) and (2) are accompanied by an additional relation for the desired control field,

W))

c (1) = R (W)II.L I

(3)

Equations (1) and (2) are deceptively simple. However, their real nature is revealed by substitution of Q. (3) directly into Eqs. (1) and (2), which yields two coupled cubically nonlinear Schrodinger-type equations. In addition, Eq. (2) has a final condition that depends on the as yet unknown solution * ( T ) to Eq. (1) at the target time T. It can be shown that there are generally multiple solutions to these equations [ 5 ] , with each one corresponding to a local optimal field c ( t ) , producing a particular value (\k(T)lOl\k(T))for the target goal. Thus, we may also view these equations as corresponding to a nonlinear eigenvalue problem, where X is the eigenvalue indicating the quality of the achieved control. Various numerical illustrations of these equations have been performed for a variety of applications [6],with encouraging results for eventual laboratory implementation. Our purpose here is to examine the general structure of these equations. The role o f + deserves special attention. Given the form of Eqs. (1) and (2), appears to be on equal footing with the physical state 9'. However, t#J has the special role of a feedback function, as it is driven by the final condition, which is proportional to the target deviation A. Thus, the solution to Eq. (2) feeds back information on the quality of the achieved final state to in turn alter the control field in Eq. (3), which then manipulates the evolving state in Eq. (1). This convoluted feedback process generally necessitates solving Eqs. (l), (2), and (3), iteratively, but most interesting is the simple observation that a formal solution of Eqs. (2) and (3) followed by substitution into Eq. (1) yields the final effective Schrodinger equation

The dependence of the control field c(\k, A) on the state \k could be quite complex, as a solution of Eqs. (2) and (3) would suggest. In general, this relation may only be revealed numerically. Furthermore, the dependence of the control field on the target deviation eigenvalue h emphasizes the inherent role of feedback in the control design process. This relation also indicates the nonlinear eigenvalue role of X.

318

H.RABITZ

Although little is known about the general nature of the solutions to Eq. (4),some revealing special cases have been examined. In particular, for socalled tracking control 171, where the objective is to follow a given evolution (O(t))= (*(t)lO(*(t)) to the target value 0 , then one may explicitly show that

Here, substitution of Eq. (5) into Eq. (4) produces an integrable nonlinear Schrodinger equation for the state \k. The field exactly meeting the objective is given by substituting \k(t),from solving Eq.(4), back into Eq. (5). Another intriguing case is for the control of a free atom, where Ho is just the kineticenergy operator. In this case, one has the classic nonlinear SchrGdinger equation

where M is the mass of the atom, y is a coupling parameter, and motion in one dimension is considered. Here, the control field is ~ ( xt,) = (!P(x, t ) I 2 . At first sight, it might appear to be impossible to create a field that explicitly depends on the spatial and temporal dependence of the evolving probability density, but a special case with potentially important significance may be practical (81, as mentioned in Section 1I.B. The full ramifications and exploitation of the feedback nature of Eq. (4) has not been explored for design purposes. Although tracking control gives a specific realization of the feedback nature of the field, even there, many variations on the theme arise [7].It may also be possible to use intuition to guide the form of €(*,A) to achieve good quality control, once again keeping in mind the special role of feedback for steering the currently evolving wavepacket to the final desired state. The physical clarity of the feedback process and the potential for computational savings suggest that analysis of this matter may be especially interesting for molecular control.

B. Feedback in the Laboratory Control of Molecular Dynamics

If the various notions and concepts in Section 1I.A can be exercised to yield , serious problems remain. First, precise knowledge a control field ~ ( t )two of the molecular Hamiltonian H is often seriously lacking, although there is typically a qualitative understanding of virtually any Hamiltonian. Sec-


319

ond, even in those cases where the Hamiltonian is known and the design equations may be numerically solved to high accuracy, the resultant control ) be subject to some error upon its generation in the laborafield ~ ( twill tory. Given that the inherent molecular control mechanism is manipulation of wave interference [ 1, 31, then the degree of anticipated tolerance to these errors and uncertainties may be low in many cases. This type of circumstance is exactly the regime where laboratory feedback control is nautrally applied in analogous traditional engineering applications [9]. Although there are many nuances to laboratory feedback control, the essential features can be easily understood [lo]. One would typically start with a zero-order design ~ ( tfrom ) the efforts outlined in Section 1I.A and implement the field design in the laboratory for application to the molecular sample. At this stage, the laboratory observation of the target deviation X would then be fed into a learning algorithm, which would suggest a new control ~ ( t )and , the process would be repeated as many times as necessary until convergence occurs. The notion of feedback suggested here is distinct from that often used in traditional engineering control, where real time feedback is employed. Here, we refer to feedback in the sense of a sequential learning process. Naturally, a key issue is the rate of learning or convergence, and the rate may depend on many factors, including the sophistication of the learning algorithm, the flexibility inherent in the laboratory control field adjustments, and the nature and quality of the laboratory observations [lo]. In the most difficult cases, it may be necessary to perform intermediate-state measurements in order to garner information about where the system has evolved to, short of reaching the desired target state. Computer simulations of this learning control process have been very encouraging for its eventual success in the laboratory, and perhaps most intriguing is the suggestion that the feedback process may be self-starting under the best of conditions [lOa] and thus not even requiring a trial design. Precise knowledge of the created control field is not necessary to implement feedback control. It is only necessary that the control laser be stable and reliable, such that its performance does not wander while going from one experiment to the next. Another important point is that the control field frequency bandwidth will not need to become ever broader, as larger molecules are considered for control. The bandwidth should saturate rather rapidly with molecular size, and this suggests that the same apparatus capable of achieving control over simple molecules may be readily adapted to complex ones. We also suggest that laboratory feedback will be the only way to reliably investigate the feasibility of control over polyatomic molecules, as theoretical limitations may prevent a proper analysis. Most encouraging is the fact that all of these issues have analogues in traditional engineering control, where they are successfully exploited almost in a routine fashion [9]. Hopefully, it

320

H. RABITZ

will not be too long before the same capability is present in the molecular control domain. A special case of feedback control arises in consideration of the nonlinear Schrodinger equation (6). A capability now exists for creating travelingwave-like potentials U(x - uf), where u is a characteristic velocity [ll]. Under these conditions, it may be shown that Eq. (6) can be transformed into a stationary Schrodinger equation dependent on y = x - uf admitting a solitonic solution [8]. This situation suggests that it might be possible to create a quantum soliton in the laboratory. However, the realization of such a soliton calls for a special understanding in the realm of feedback control. The potential is the probability density l\k(y)1*, which could, in principle, be measured through ultrafast X-ray or electron beam diffraction [ 121. Thus, one could envision the generation and observation of a feedback-controlled quantum soliton [8]. The nondispersive nature of the wavepacket does not violate any principles of quantum mechanics, as the inherent time dependence of the potential exactly compensates for the naturat tendency of the wavepacket to spread. Such an application of quantum control would be intriguing and, possibly, of some practical consequences.

C. Feedback in the Inversion of Molecular Dynamics As commented in Section IIB, we typically do not know the Hamiltonian H to high precision. A major interest in chemical physics continues to be performance of experiments and computations to better refine molecular Hamiltonians. Some studies have suggested that the control concepts presented above can be turned around and applied to address this important goal of learning about Hamiltonian structure [13]. This viewpoint is especially related to molecular tracking embodied in Eq. (5). For inversion, the goal is to determine some portion of HO(particularly the potential) through laboratory-observed knowledge of the data track (O(t)),which in turn will satisfy Heisenberg's equation of motion

This equation may be viewed as an integral equation, where the unknown is the operator Ho, with the data on the left-hand side being known. Equations (5) and (7)are analogous to each other, except that in Eq. (5) the unknown portion of the Hamiltonian corresponding to the control field was explicitly solved for. In Eq. (7), we obtain an integral equation for Ho serving this purpose. The analogy goes further, in that the right-hand side of Eq. ( 5 ) depends on the unknown evolving state 'k, as does the kernel of the integral


32 1

term on the right-hand side of Eq. (7). Thus, we can create an inversion algorithm based on simultaneously solving Eqs. (7)and (8): iA

a

- \k = H o ~ at

This algorithm may be viewed as “direct,” since the solution will yield the sought-after Hamiltonian Ho. Feedback is present here in that the information learned about the operator Ho in Eq. (7)will depend on the evolving state analogous to the same way that the field depends on the state in Eqs. (3), (4)- or (5). A host of issues need to be explored to establish the full viability of this tracking approach to laboratory data inversion, but preliminary examination of the algorithm has been most encouraging [ 131. It has been referred to as a tracking inversion algorithm, since the data (O(t))are tracked through Q. (7)to yield the sought-after Hamiltonian Ho.This concept naturally admits ultrafast pumpprobe data but is not restricted to it. In particular, it would in principle be possible to generate a “synthetic” track by Fourier transforming traditional continuous-wave spectral data, and a relation analogous to Eq. (7) will result. This prospect is especially interesting, as it would allow for the utilization of abundant high-quality continuous-wave data in a natural time-dependent framework for inversion. It may turn out that one of the most important legacies of molecular control is its linkage to inversion of laboratory data to learn about molecules. The full embodiment of this concept again embraces feedback, as a relay of control and inversion can be envisioned to learn about molecules in the most efficient manner. 111. CONCLUSION

This chapter has emphasized the special and central role that feedback plays in virtually all aspects of control over molecular quantum phenomena. In terms of applications, the manipulation of chemical reactions still stands as a prime historical objective. However, other rich applications abound. For example, the growing interest in the field of quantum computing is a potentially exciting area [ 141, and any practical realization of quantum computers will surely entail control over quantum phenomena. Other unforeseen applications may also lie ahead.

Acknowledgments The author acknowledges support from the Army Research Office and the National Science Foundation.

322

H. RABlTZ

References 1. W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1993). 2. P. Gross, V. Ramakrishna, E. Vilallonga, H. Rabitz, M. Littman, S. A. Lyon, and M. Shayegan, Phys. Rev. E 49, 11100 (1994). 3. P. Brumer and M. Shapiro, Ann. Rev. Phys. Chem. 43, 257 (1992). 4. A. Peirce, M. Dahleh, and H. Rabitz, Phys. Rev. A 37,4950 (1988). 5 . M. Demiralp and H. Rabitz, J. Marh. Chem. 16, 185 (1994). 6. R. Kosloff, S. Rice, P. Gaspard, S. Tersigni, and D. Tannor, Chem.Phys. 139,201 (1989); Y.Yan, R. Gillilan, R. Whitnell, K. Wilson, and S. Mukamel, J. Chem. Phys. 97, 2320 (1993); T. Szak6cs. B. Amstrup, P. Gross, R. Kosloff, H. Rabitz, and A. Liirincz, Phys. Rev. A 50, 2540 (1994). 7. P. Gross, H. Singh, H. Rabitz, K. Mease, and G. M. Huang, Phys. Rev. A 47,4593 (1993). 8. M.Demiralp and H. Rabitz, “Feedback Controlled Quantum Solitons,” to be published. 9. K. J. Astrom and B. Wittenmark, Adaptive Conrrof (Addison-Wesley, 1995). 10. (a) R. S. Judson and H . Rabitz, Phys. Rev. Lett. 68,1500 (1992); (b) G. J. T&h, A. Lorincz, and H. Rabitz, J. Cbern. Phys. 101,3715 (1994). 11. R. Graham, M. Schlautmann, and P. Zoller, Phys. Rev. A 45, R19 (1992). 12. M. Dantos, S . Kim, J. Williamson, and A. Zewail, J. Phys. Chem. 98, 2782 (1994). 13. 2.-M. Lu and H. Rabitz, J. Phys. Chem. 99, 13731 (1995). 14. T. Slator and H. Weinfurter, Phys. Rev. Lett. 74, 4087 (1995).

DISCUSSION ON THE COMMUNICATION BY H. RABITZ Chairman: M. Quack T. Kobayashi 1. What is the origin of the nonlinearity introduced in the Schrodinger equation represented by a potential proportional to I$ 12? 2. How can such a potential be introduced experimentally? 3, Where does the soliton propagate? 4. Is the feedback expected to be very fast?

H. Rabitz: In general, nonlinearity arises in quantum control through the design or feedback process. In either case, nonlinearity enters as quantum control is inherently an inverse problem. The physical objective is prescribed, and we seek to find a piece of the Hamilwould tonian (i.e., the electric field). A nonlinearity of the form be introduced in the laboratory by measuring the probability density through, perhaps, electron or X-ray diffraction and feeding that observation back in order to create a potential of that form. Special forms for such potentials are becoming feasible to generate in the laboratory.


323

The soliton considered here might correspond to an atom translating in an optical field or perhaps an exciton in a solid. Under these conditions, the feedback control does not necessarily have to be ultrafast, as the propagation and velocity are at our disposal.

R. W. Field: Prof. Rabitz, I like the idea of “sending out a scout” to map a local region of the potential-energy surface. But I get the

impression that the inversion scheme you are proposing would make no use of what is known from frequency-domain spectroscopy or even from nonstandard dynamical models based on multiresonance effective Hamiltonian models. Your inversion scheme may be mathematically rigorous, unbiased, and carefully filtered against a too detailed model of the local potential, but I think it is naive to think that a play-andlearn scheme could assemble a sufficient quantity of information to usefully control the dynamics of even a small polyatomic molecule. In our conversation after the discussion, I became at least willing to believe that we both believe in simplified models. However, you present your control schemes in a way that feels dismissive of all previous efforts by spectroscopists to extract information from spectra. Having in advance a simple picture of what a molecule wants to do and also knowing why the molecule wants to do it seems superior to a picture designed on-the-fly on a purely numerical basis.

H. Rabitz A key motivation in developing the inverse algorithm was to draw a firm linkage between control field design and potential determination. Both are inverse problems aiming to determine a portion of the Hamiltonian, and it is most interesting that a common mathematical formulation now exists for both applications. In particular, the Schrodingerequation for the wavepacket is coupled to an integral equation for the potential (or the control field). Focusing on potential inversion, the temporal data is a source term in the integral equation. The form of the potential extracted from the equation depends on the representation chosen as most appropriate in a given case. In particular, the potential could be sought in coordinate space or as matrix elements in a state space. The wavepacket “scout” wandering about on the surface or in a state space will naturally identify what may be learned regarding the potential. A temporal picture is an inherent component of this analysis. The inverse algorithm is a direct marching process to reveal more about the potential as time evolves and the wavepacket samples more of the potential. Virtually all inverse problems are ill-posed, in that the finite available data will not permit a full determination of the potential. There-

324

H. RABITZ

fore, one needs to regularize (stabilize) the inversion by (1) assuring that the potential is not sought in regions poorly sampled by the data and (2) the finite amount of data is supplemented as necessary by a priori physical knowledge about the potential. The algorithm naturally assures point 1 by only extracting the potential in regions where the wavepacket is significant. A variety of input information can be used to handle point 2. It is essential not to overly constrain the inversion to just give a potential rather than the true potential; at the same time a constraint that is too weak will leave the inversion algorithm unstable, which may render a nonunique potential. In some cases it has been found that merely asking for the potential surface to be smooth, in a differential sense, is sufficient to stabilize the inversion. The goal is to seek stabilization with the mildest criteria possible and then let the algorithm pin down the potential in the regions permitted by the data. Naturally many practical implementation issues arise, including the need to solve the dynamical equations, at least in the regions of importance sampled by the data. In this regard there is a classical mechanical analogue of the coupled dynamical and integral equations. Exploitation of classical inversion may be important, at least as a first step to define the potential in polyatomic cases. The key point at this time is that the new formulation provides a rigorous foundation to build upon for achieving direct practical inversions of temporal and spectroscopic data.

R. W.Field: The goal is not merely to represent the spectrum or the dynamics but to be able to create reduced-dimension pictures that are intelligible to mortals. Pictures lead to insight. Insight leads to more effective control strategies. H. Rabitz: The goals of Hamiltonian information extraction are clear, but prior means for this purpose are generally unsatisfactory in many respects. Various sources of data are available for exploitation. In some applications, reduced models will suffice, while for others, only high accuracy detailed potentials can meet the needs. What is necessary is a rigorous inversion algorithm that can incorporate appropriate physical constraints to reliably extract the Hamiltonian information. D. J. Tannor: Prof. Rabitz, from the time-domain formulation of conventional electronic absorption spectroscopy, we know that the information content in the wavepacket autocorrelation function is identical to that in the high-resolution spectrum. Yet it is clear that the wavepacket autoconelation function only directly probes the times at


325

which the wavepacket returns to its initial position and hence is a direct probe only of the Franck-Condon region. We know the amount of time the wavepacket spends away from the Franck-Condon region, but not where it goes. In what way does your tracking algorithm interrogate the wavepacket a b u t where it went when it left the Franck-Condon region?

H.Rabitz: The information in the recurrence time alone is minimal. However, the temporal structure of the recurrence signal contains detailed information on the surface explored by the wandering scout wavepacket during its excursion. Further experiments may be necessary to follow (i.e., track) the wavepacket through its excursion over the potential surface. Such pump-probe experiments go beyond conventional spectroscopy.

THEORY OF LASER CONTROL OF VIBRATIONAL TRANSITIONS AND CHEMICAL REACTIONS BY ULTRASHORT INFRARED LASER PULSES M.V. KOROLKOV, J. MANZ,* and G. K. PARAMONOV Freie Universitat Berlin Institut fur Physikalische und Theoretische Chemie Berlin, Germany

CONTENTS I. Introduction 11. Models and Techniques 111. Applications A. Individual Vibrational-State-to-Vibrational-State Transitions B. Series of Vibrational Transitions C. Vibrational Transitions in Competition with Dissipative Processes D. Above-Threshold Dissociation E. Isomerization IV. Conclusions References

I. INTRODUCTION The theory of laser control of chemical reactions may be classified into two

different domains: Laser control by continuous wave (CW) lasers and by laser pulses. The former includes, for example, the strategies of (i) vibrationally mediated chemistry [ 11 and (ii) coherent superpositions of independent excitation routes [2]; for experimental demonstrations see (i) Ref. 3 and *Communication presented by J. MQnZ Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scaie, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, 1. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

327

328

M. V. KOROLKOV, J. MANZ, AND G. K. PARAMONOV

(ii) Ref. 4, respectively. The latter involves the theories for selective transitions by series of (iii) ns [S] or (iv) ps/fs [6] ultraviolet (UV) or visible (VIS) laser pulses, by (v) chirped laser pulses [71, and by (vi) optimal Control [S]; for experimental verifications see (iii) Ref. [9], (iv) Ref. [lo], and (v) Ref. [7]. For surveys, see Ref. 11 as well as several other fundamental contributions in this volume (S. A. Rice, Chapter 6, this volume) or elsewhere [121. The purpose of this chapter is to present a brief survey of the theoretical work of our group on another strategy, that is, laser control of vibrational transitions by single IR laser pulses in the femtosecond/picosecond time domain [13] or by selective series of such pulses [14] with applications to isolated molecules or radicals in the gas phase [13, 141 or in an environment (151 and with extensions to selective photodissociation or predissociation [16] and isomerizations [14, 171; see also the review [18] and complementary work on laser control by IR femtosecond/picosecond laser pulse [19, 201. The model systems include simple pseudo-one-dimensional (Id) ones [13-191 as well as more complex (2d, 3d) ones [18,20,21]. In our strategy, all of these applications and extensions, that is, series of vibrational transitions, photodissociation, and unimolecular reactions, are essentially decomposed into sequences of selective vibrational-state-to-vibrational-state or vibrational-state-to-continuum-state transitions, with possible extensions to continuum-state-to-continuum-state transitions (223, which are achieved by corresponding sequences of IR femtosecond/picosecond laser pulses. It will be helpful, therefore, to consider first the fundamental quantum effect of state-to-state transitions induced by individual IR femtosecond/picosecond laser pulses, before presenting extensions to multiple transitions or chemical reactions induced by series of IR femtosecond/picosecond laser pulses. Accordingly, the chapter is organized as follows: In Section 11, we give a brief survey of the fundamental models and theories, followed by theoretical applications to vibrational transitions in closed [13, 141 (Section IILA, B) and open [ 151 (Section 1II.C) systems with extensions to photodissociation [161 (Section 1II.D) and isomerization [ 171 (Section 1II.E). The conclusions are in Section IV. 11. MODELS AND TECHNIQUES The original theory of individual state-selective vibrational transitions induced by IR femtosecond/picosecond laser pulses has been developed by Paramonov and Savva et al. for single laser pulses [13] (see also Ref. 23) followed by more general extensions to series of IR femtosecond/picosecond laser pulses in Refs. 14 and 24. For illustration, let us consider two simple, one-dimensional model systems that are assumed to be decoupled from any

CONTROL BY ULTRASHORT INFRARED PULSES

329

other inter- or intramolecular (vibrational, rotational, etc.) degrees of freedom. Our first example is a Morse oscillator tailored to the OH bond in HOD, (adapted from Ref. 25; see also Refs. 14-16, 21, 23, and 24). This model will serve to explain our strategies for IR femtosecond/picosecond laser-pulse-induced vibrational transitions from initial to final states [e.g., OH(Ui) OH(uf)] and for photodissociation [OH(ui) 0 + HI. It will also serve as reference for more complex systems (e.g., state-selective vibrational transitions in HOD) [18, 211 or OH coupled to other intra- and intermolecular vibrations, which may be represented by a thermal bath [151. The second one-dimensional example is a model system with a shallow, slightly asymmetric double-well potential tailored to substituted semibullvdenes (SBVs) (adapted from Ref. 26). This model will be used to illustrate IR femtosecond/picosecond laser control of unimolecular isomerizations, specifically the Cope rearrangement of the substituted SBV [26]. In brief, the two model systems will be called OH and SBV; they should be considered as representatives for similar systems [ 14, 18-21, 271. The OH and SBV potential-energy surfaces V ( q ) versus bond or reaction coordinates q, together with the vibrational levels E , and eigenfunctions +,,(q), are shown in Figs. 1 and 2, respectively. Our first task will be to design optimal IR femtosecond/picosecond laser pulses for vibrational transitions,

-

-

is an isolated system from the initial (t = 0) state hi to the final (t = t p ) one &, [tp is the duration of the laser pulses and E = exp(-iq) denotes an irrelevant phase factor]. For this purpose, we solve the corresponding timedependent Schodinger equation using the semiclassical dipole approximation and neglecting all other couplings,

subject to the initial condition (1). The standard notations in Eq. (2) denote the molecular Hamiltonian H = T + V and dipole operator p as well as the electric field of the IR femtosecond/picosecond laser pulse, € ( t )= €0 . cos w t * s ( t )

(3)

where €0, w , and s ( t ) are the amplitude, frequency, and shape function. As a convenient example, we employ sin2 pulses,


3 30 20000

m-----OH

10000

0

-5

/-

- 0 +

I

II II

1

H

1

.

-10000

> P a

c

W

-20000 -

\

n n

1 ” T - f

-30000

-40000 0

1

.2

3

qlao

4

5

6

Figure 1. Morse potential V(q), vibrational levels Eu, and wave functions &(q) for the model OH (adapted from Ref. 14). The arrows indicate various selective vibrational transitions as well as above-threshold dissociations (ATDs) induced by IR femtosecond/picosecond laser pulses, as discussed in Sections 1II.A-1II.D; see Figs. 3-5. Horizontal bars on the mows mark multiple photon energies hw of the laser pulses; cf. Table I. The resulting ATD spectrum is illustrated by the insert above threshold.


33 1

2500 2000 l-l

'E 0

1500

\

h

g G

1000

500 0 -100

-50

50

0

9/Pm

100

Figure 2. Double-well potential V ( q )with corresponding vibrational levels Eu and wave functions &(q) for the model 2,6-dicyanoethylmethylsemibullvalene(SBV) (adapted from Ref. 26). The reaction coordinate q indicates the Cope rearrangement of the model SBV from the reactant (R) isomer versus the transition state f to the product (P) isomer. Vertical arrows f P by two IR femtosecond/picosecond indicate the laser control of the isomerization R laser pulses: cf. Fig. 6 and Table I. +

s ( t ) = sin*

+

( 7) for I I 0

t

tp

(4)

The dipole function p for OH is modeled as a Mecke function (adapted from Ref. 28), for SBV we assume the ubiquitous linear relation p = f . e q with effective chargef e and scaling factorf. Equation (2) can be solved either

-

-

332


directly using fast Fourier transform (m) propagation methods [29] or by expanding $4)in terms of vibrational eigenstates &,

and transforming Fq. (2) into the matrix version ihC(t) = [ H - p ( t ) ] * c(t)

(6)

which is then solved, for example, by the Runge-Kutta method [30]. In this representation, the transition (1) is rewritten as C,(t

= 0 ) = 6",i

--t

C U ( t = t p )= 6 , ,

*

f

(7)

or in terms of populations

we have that PU(t= 0) = 6,,

Pu(t = t p )= 6,,

--+

(9)

These selective transitions (l), (7),and (9) may be achieved by proper optimization of the parameters € 0 and w, as described elsewhere [13, 18, 211. Extensions to IR femtosecond/picosecond laser-pulse-induced dissociation or predissociation have been derived in Ref. 16, using either the direct or the indirect solutions of the Schrdinger equation (2); the latter requires extensions of the expansion (5) from bound to continuum states [16,31]. (The consistent derivation in Ref. 16 is based on s. Flugge in Ref. 31). The same techniques can also be used for IR femtosecond/picosecond laser-pulse-induced isomerization as well as for more complex systems that are two dimensional, three dimensional, and so on, at the expense of increasing numerical efforts due to the higher dimensionality grid representations of the wavepackets $ ( t ) or the corresponding expansions (5) (see, e.g., Refs. 18, 20, and 21). Extensions from the preceding ideal, isolated systems to others that are coupled to an environment are quite demanding and nontrivial 1321 because the IR femtosecond/picosecond laser pulse has to achieve the selective vibrational transition (9) in competition against nonselective processes such as dissipation. For simulations, we employ the equation of motion for the reduceddensity operator


333

The first term on the right-hand side corresponds to Eq. (2), whereas the second term describes dissipative effects that are induced in the system due to its coupling to the environment. The latter is modeled, as usual [32, 331, as the thermal (temperature T) ensemble of harmonic oscillators, with nonlinear coupling A ' Q(q) F ( { q k } ) between the system and the thermal bath, expressed in terms of nonlinear molecular and linear environment coupling operators Q(q) and F ( { q k } ) . As shown in Ref. 15, it is important to describe the dissipative term in Eq. (10) by making use of the non-Markovian expression e

with time correlation function (F(t')F(O)); compare with the Markovian approximation in Ref. 32. The latter may be appropriate for other purposes, for example, IR absorption spectroscopy with traditional CW lasers [33]. In this chapter, we employ the algebraic, vibrational state representation of J2q. (10) [compare with 3.(6)]. Proper optimizations of the IR picosecond laser pulse parameters should then achieve the vibrational transition (9) as well as possible, where PU(t)= p u U ( t ) .

III. APPLICATIONS

A. Individual Vibrational-State-to-Vibrational-Statelkansitions

As our first example, we consider individual vibrational-state-to-vibrationalstate transitions in the model OH (see Fig. 1); specifically,

and OH(U~= 5 ) +OH(U~= 10)

334

M. V. KOROLKOV, 1. MANZ, AND G. K. PARAMONOV

The results are shown in Fig. 3 and Table I. Apparently, optimal IR femtosecond/picosecond laser pulses with durations tp = 500 fs may induce nearly perfect transitions (12), (13) in the model OH. Similar examples are documented in Refs. 13, 18, and 23. A detailed discussion of the derivation of the optimal laser parameters, depending on the vibrational level Eu and the transition dipole matrix elements puw, is also given in Refs. 13, 18, and 23. Suffice it here to say that in many (but not all) cases the optimal frequency w is close (but not identical) to the resonance frequency,

250 0

-250

-500

1

1

0

0.5 1 T I M E (ps)

1.5

0

0.5

1 T I M E (ps)

1.5

1

0.8

0.6 0.4

0.2 0 -

5)

-.

Figure 3. Selective vibrational transitions OH@, = O)-OH(uf = 5 ) and OH(vi = OH(vf = 10) induced by two individual IR femtosecond/picosecond laser pulses. The

electric fields E ( t ) and the population dynamics &(t) are shown in panels ( a ) and (b),respectively. Sequential combination of the two individual laser pulses yields the overall transition OH(u = 0) --c OH(u = 5 ) OH(uf = 10);cf. Fig. 1 and Table I. For the isolated system, the population of the target state Pv= &) is constant after the series of iR femtosecond/picosecond laser pulses, t > 1 ps.

-.

335

CONTROL BY ULTRASHORT INFRARED PULSES TABLE I Laser Parameters for Vibrational Transitions in Model Systems OH and SBV Transition

-

OH(u; = 0) + OH(U~ = 5)a OH(u; = 5) OH(U~ = 10)' OH(Q = 10) + OH(U~= 15)' OH(Q = 15) + 0 + H' SBV(U~ = 0) + SBV(U~ = 6)' SBV(U~ = 6) + SBV(uf = I)'

I , (ps)

u12m (cm-')

0.5 0.5

3426.2 2524.6 1625.5 822.5 1252.8 1185.0

1.o

0.5 1.o

1.o

EO

(MV/cm-') 400.8 214.6 148.1 177.3 32.8/f 30.51'

10 (TW/cm2)

2 12Ae 61 .O 29.0 41.7 1.451f 1.251f

=See Figs. 1, 3, and 4. bSee Figs. 1 and 5. 'See Figs. 2 and 6. df is the scaling factor for the dipole function p = f e q of SBV. eIntensities are calculated as follows: 10 = icoce$ they can be reduced by making use of series of vibrational transitions.

with rather high demands for the accuracy, typically 6w/w < lW3. In contrast, the system offers several choices of optimal electric field amplitudes €0, with rather moderate requirements for the accuracies, say, BEO/EO< 0.1. In practice, one should employ the smallest possible choice of €0 in order to avoid competing nonselective processes (e.g., power broadening at larger electric field strengths). The resulting optimal IR femtosecond/picosecond laser pulses may be interpreted as generalized if-pulses [19].

B. Series of Vibrational Wansitions Combination of several individual vibrational transition (Section 1II.A) yields a selective sequence of vibrational transitions induced by series of IR femtosecond/picosecond laser pulses. For example, the two individual transitions (12). (13) may be combined to the sequence

OH(U= 0 )

-

OH(U= 5 ) -+OH(U= 10)

(15)

and the selective overall transition (15 ) is achieved simply by the corresponding series of individual IR femtosecond/picosecond laser pulses; see Fig. 3. Similar series of state-selective vibrational transitions induced by series of IR femtosecond/picosecond laser pulses are documented in Refs. 14, 17-19, and 24; see also Refs. 18, 20, and 21 for applications to more complex sys-

336

M. V. KOROLKOV, J. MANZ, AND G . K. PARAMONOV

tems such as HOD. In practice, it may be quite advantageous to employ such series of state-selective IR femtosecond/picosecond laser pulses, because their individual components may have smaller field amplitudes, in comparison with single pulses. On the other hand, series of nonoveralpping IR femtosecond/picosecond laser pulses (such as those shown in Fig. 3) may imply rather long durations of the overall excitation process (in comparison with the duration of their individual components), and this may enhance the probability of competing processes such as IVR. Gratifyingly, however, exceedingly long overall durations may be reduced by partial overlaps of the pulses; see Refs. 14, 18, 19, and 24 and compare, for example, Figs. 3 and 4.

C. Vibrational Transitions in Competition with Dissipative Processes

Next let us extend the strategy of selective sequences of vibrational transitions by series of IR femtosecond/picosecond laser pulses from closed systems (Section IILB) to open ones. For direct comparison with the results shown in Fig. 3, we consider the case of the transition (15) in competition with dissipation to a thermal bath. The results are shown in Fig. 4. Here we employ essentially the same sequence of IR femtosecond/picosecond laser pulses as in Section 1II.B without retuning of the laser parameters but with partial overlaps in order to reduce the overall duration and, therefore, the probability of competing dissipative processes. As a consequence,these laser pulses yield rather successful population transfer even in the open system, in close analogy to the isolated system, that is, the populations of the intermediate and final vibrational levels OH(u = 5 ) and OH(u = 10) approach maximal values of 0.97 and 0.88, respectively, corresponding to the target transition (15). In contrast with the isolated system, however, these maximum populations have rather short lifetimes (400fs) because the coupling to the environment induces competing relaxations to lower levels, for example, OH(u= 1 0 ) + 0 H ( ~ =9 ) - + 0 H ( ~ =8)-

..-

(16)

The decay process (16) is clearly visible in Fig. 4, in particular after the end of the laser pulses (t > 750 fs); ultimately, the population P, approach the thermal Boltzmann distribution (T = 300 K in Fig. 4).

D. Above-Threshold Dissociation

The strategy of multiple vibrational transitions induced by series of IR femtosecond/picosecond laser pulses may also be extended to photodissociation or predissociation. This is documented in Fig. 5 for the case

OH(U= 10)-+OH(u= 15)-O+H

(17)


-500 I

0

337

1 T I M E (ps)

1.5

1

1.5

0.5

1

-

z 0 +

0.8

3

0.4

-

0 a

0.2

-

v)

e (

4 4

0.6 -

0, Y

0

0

v=o

4

0.5

T I M E (PSI

Figure 4. Series of IR femtosecond/picosecond laser pulses for the sequence of vibrational transitions OH(u = O)+OH(u = S)-OH(u = 10) for the model OH coupled to a thermal heat bath; cf. Fig. 1 and Table I. The notations are as in Fig. 3. The laser pulses compete against dissipation. After the pulses, the coupled system relaxes toward a Boltzrnann distribution (T= 300 K).

which may be considered as an extension of the vibrational transitions (15) to photodissociation; see also Fig. 1. For the present purpose, the most important effect is achieved by the final IR femtosecond laser pulse, which converts the bound state OH(u = 15) into a distribution of continuum states representing 0 + H. In Ref. 16 it has been shown that one can control the resulting populations of continuum states; for example, by proper optimization of the final IR femtosecond/picosecond laser pulse, it is possible to produce nearly monoenergetic products 0 + H, or several wavepackets with specific energies separated by the IR femtosecond laser photon energy h a , and so on.

M. V. KOROLKOV, J. MANZ, AND G . K. PARAMONOV

-300

0

0.5

1 T I M E (ps)

I .5

0

0.5 1 T I M E (ps)

1.5

1

0.8

0.6 0.4

0.2 0

Figure 5. Series of IR femtosecond/picosecond laser pulses for the sequence of transitions OH(u = 10)-OH(u = 15) -0 + H for the isolated model OH; cf. Fig. 1 and Table I. The notations are as in Fig. 3; populations Pwell(r) = P J r ) and Pcont(t)= IP,ll(t) indicate the total populations of bound and continuum states embedded in the potential well and above the dissociation threshold, respectively. The resulting spectrum of ATD is shown in Fig. I .

c$Lo

The resulting phenomena of above-threshold dissociation (ATD) resemble those of above-threshold ionization (ATI); see Ref. 34.

E. Isomerization

Our final example provides the extension from multiple vibrational transitions (Section II1.B) to isomerization. Specifically, we consider the model Cope rearrangement of 2,6-dicyanoethylmethylsemibullvalene(SBV) from the reactant (R) via the transition state d to the product (P) isomer; see Fig. 2. The system has been designed by Quast with specific substitutions


339

that cause a slightly asymmetric potential-energy surface with rather shallow potential-energy barrier along the reaction coordinate q; for details see Ref. 35. Infrared femtosecond/picosecond laser pulse control of the isomerization is achieved by two subsequent vibrational transitions, similar to expression (15) but with the second transition causing stimulated emission rather than absorption:

SBV(U= 0)

-

SBV(U= 6) +SBV(U= 1)

cf. Fig. 2. The first transition excites the reactant ground state SBV(u = 0) to the delocalized state SBV(u = 6) with energy just above the potential barrier. The second pulse deexcites the intermediate state SBV(u = 6) into the product ground state SBV(u = 1). The corresponding selective IR femtosecond/picosecond laser pulses and population dynamics are shown in Fig. 6. Apparently, the series of two IR femtosecond/picosecond laser pulses induces nearly perfect control of the model isomerization. Similar controls of model reactions are documented in Refs. 14 and 17.

IV. CONCLUSIONS Control of vibrational transitions, ATD, and AT1 by series of IR femtosecondfpicosecond laser pulses is a promising strategy. It should be a challenge for our experimental partners to verify this strategy by applications to real systems. The recent development of powerful IR femtosecond laser pulses [36] is an important step toward this goal. Simultaneously, there are also challenging tasks for theoreticians, for example, further developments of the present strategy to chemical reactions in open systems. Most important, perhaps, will be extensions and tests of our strategy for even more complex multidimensional model systems, including rotations. Specifically, it will be fascinating to see whether the strategy can achieve selective overall transitions in systems with rather large level densities, for example, with vibrational level spacings smaller than the spectral widths of the IR femtosecond/picosecond laser pulses. For practical purposes, it will also be important to compare the present strategy [ 13-24] with others [1-12] (see also S. A. Rice, Chapter 6, this volume). For example, series of IR femtosecond/picosecond laser pulses may be considered as special cases of optimal control [8] subject to the restriction of simple pulse shapes, and this restriction may turn out to be advantageous for experimental implementations but disadvantageous for the aim of designing optimal laser pulses with rather low intensities (compare, e.g., Refs. 21 and 8). For the specific purpose of laser control of isomerization, it is also interesting to compare the efficiency of the present strategy, which employs two IR

340

3w


-50

0

0.5

1

1.5

0

0.5

1

1.5

T I M E (ps)

1 cn

x

0.8

e

0.6

Ll

0.4

0 Y

4

5

a 0 a

0.2 0

T I M E (ps)

-.

Figure 6. Series of IR femtosecond/picosecond Laser pulses for the sequence of vibrational transitions SBV(u = 0) +SBV(u = 6) SBV(u = 1) for laser control of the Cope rearrangement of the model substituted semibullvalene (SBV) shown in Fig. 2 (adapted from Ref. 26). The notations are as in Fig. 3. The electric field is scaled by the scaling factorf of the effective charge associated with the dipole function = f . e . q.

femtosecond/picosecond pump-and-dump laser pulses (see Section IILE), with the strategy of V I S j W femtosecond pump-and-dump laser pulses [6]. We consider the recent experimental verifications [101 of the latter as encouragement for equivalent implementations of our own strategy. Work along these lines is in progress.


34 1

Acknowledgments We thank our previous coauthors [13-17. 22-27] for their substantial contributions to the development of the strategy of laser control of vibrational transitions and unimolecular reactions by IR femtosecond/picosecond laser pulses. Specifically, we are grateful to M. Dohle for preparing Figures 2 and 6 for this survey (adapted from Ref. 26). Generous financial support by Volkswagen-Stiftung through project I/69348 and Fonds der Chemischen Industrie is also gratefully acknowledged. The computer simulations have been carried out on HP 750 workstations at Freie Universitit Berlin.

References 1. V. S. Letokhov, Science 180,451 (1973); E. Segev and M. Shapiro. J. Chem. Phys. 77, 5604 (1982); V. Engel and R. Schinke, J. Chem. Phys. 88,6831 (1988); D. G. Imre and J. Zhang, Chem. Phys. 139,89 (1989). 2. M. Shapiro and P. Brumer, J. Chem. Phys. 84, 4103 (1986); M. Shapiro and P. Brumer, Acc. Chem. Res. 22,407 (1989); P. Brumer and M. Shapiro, Ann. Rev. Phys. Chem. 43,

257 (1992). 3. R. V. Ambartsumian and V. S . Letokhov, Chem. Phys. Lett. 13,446 (1972); I. N. Knyazev, Yu. A. Kudriavtsev, N. P. Kazmina, V. S. Letokhov, and A. A. Sarkisian, Appl. Phys. 17, 427 (1978); P. F. Zittel and D. D. Little, J. Chem. Phys. 7,713 (1979); M.D. Likar, J. E. Baggott, A. Sinha, T. M. Ticich, R. L. Vander Wal, and F. F. Crim, J. Chem. SOC.Faraday Trans. 2, 84, 1483 (1988). D. Hausler, P. Andresen, and R. Schinke, J. Chem. Phys. 87, 3949 (1987); F. F. Crim, Science 249, 1387 (1990); S. S. Brown, H. L.Berghout, and F. F. Crim, J. Chem. Phys. 102,8440 (1995). 4. L. Zhu, V. Kleiman, X.Li, S. P. Lu, K. Trentelman, and R. J. Gordon, Science 270, 77 (1995). V. Kleiman, L. Zhu, J. Allen, and R. J. Gordon, J. Chem. Phys. 103, 10800 (1995). 5. G. W. Coulston and K. Bergmann. J. Chem. Phys. 96, 3467 (1992). 6. D. J. Tannor and S. A. Rice, J. Chem. Phys. 83,5013 (1985); S. A. Rice, D. J. Tannor, and R. Kosloff, J. Chem. SOC. Faraduy Trans. 2, 82, 2423 (1986); D. J. Tannor, R. Kosloff, and S. A. Rice, J. Chem. Phys. 85, 5805 (1986); S. A. Rice, J. Phys. Chem. 90, 3063 (1986). 7. J. L. Krause, R. M. Whitnell, K. R. Wilson, and Y. Yan, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995; J. L. Krause, M. Messina, K. R. Wilson, and Y. J. Yan, J. Phys. Chem. 99, 13736 (1995). 8. S. Shi, A. Woody, and H.Rabitz, J. Chem. Phys. 88, 6870 (1988). S. Shi and H. Rabitz, J. Chem. Phys. 92, 2927 (1990); M. Dahleh, A. P. Peirce, and H.Rabitz, Phys. Rev. A 42, 1065 (1990). W. Iakuktz, J. Manz, and H . J . Schreier, Chem. Phys. Lett. 165, 100 (1990); W. Jakubetz, E. Kades, and J. Manz, J. Phys. Chem. 97, 12609 (1993). 9. U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, J. Chem. Phys. 92,5363 (1990); K. Bergmann, S. Schiemann, and A. Kuhn, Phys. Bl. 48, 907 (1992). 10. E. P. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, Nature 355, 66 (1992); A. H. Zewail, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 15; T. Baumert, R. Thalweiser, U. Weiss, and G. Gerber, in Femtosecond Chemistry, J. Manz and L.Woste, Eds. Verlag Chemie, Weinheim 1995, p. 397. 11. V. S. Letokhov, Appl. Phys. B 46,237 (1988); F. P. Schafer, Appl. Phys. E 46, 199 (1988); A. D. Bandrauk, Ed., Atomic and Molecular Processes with Short Intense Laser Pulses,

342


Plenum, New York, 1988; A. D. Bandrauk, E. Aubanel, and S. Chelkowski, in Femrosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim 1995, p. 731. 12. B. Hartke, E. Kolba, J. Manz, and H. H. R. Schor, Be,: Bunsenges. Phys. Chem. 94, 1312 (1990); E. Kolba and 3. Manz, Faraday Discuss. Chem. Soc. 91, 369 (1991). 13. G. K. Paramonov and V. A. Savva, Phys. Lett. A 97,340 (1983). G. K. Paramonov, V. A. Savva, and A. M. Samson, Infrared Phys. 25,201 (1985); Z. E. Dolya, N. B. Nazarova, G. K. Paramonov, and V. A. Savva, Chem. Phys. Lett. 145,499 (1988). 14. G. K. Paramonov and V. A. Savva, Chem. Phys. Lert. 107,394 (1984); J. E. Combariza, B. Just, J. Manz, and G. K. Paramonov, J. Phys. Chem. 95, 10351 (1991); B. Just, J. Manz, and G . K. Paramonov, Chem. Phys. Lett. 193,429 (1992). 15. M. V. Korolkov and G. K. Paramonov, Phys. Rev. A, 55 589 (1997). 16. M. V. Korolkov, Yu. A. Logvin, and G. K. Paramonov, J. Phys. Chem., 100 8070 (1996); M. V. Korokov, G . K. Paramonov, and B. Schmidt, J. Chem. Phys., 105 1862 (1996). 17. J. E. Combariza, 1. Manz, and G. K. Paramonov, Faraday Discuss. Chem. SOC.91 358 (1991); J. E. Combariza, C. Daniel, B. Just, E. Kades, E. Kolba, J. Manz, W. Malisch, G. K. Paramonov, and B. Warmuth, in Isotope Eflecrs in Gas-Phase Chemistry, J. E. Kaye, Ed., ACS Symposium Series, 1992, p. 310; M. V. Korolkov, J. Manz, and G. K. Paramonov, J. Chem. Phys., 105 10874 (1996). 18. G. K. Paramonov, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chernie, Weinheim, 1995, p. 671. 19. H. P. Breuer, K. Dietz, and M. Holthaus, J. Phys. B 24, 1343 (1990); H. P. Breuer and M. Holthaus, J. Phys. Chem. 97, 12634 (1993); M. Holthaus, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 713; M. Holthaus and B. Just, Phys. Rev. A 49, 1950 (1994). 20. W. Gabriel and F? Rosmus, J. Phys. Chem. 97, 12644 (1993); B. Amstrup and N. E. Henriksen, J. Chem. Phys. 97,8285 (1992). 21. J. Manz and G. K. Paramonov, J. Phys. Chem. 97, 12625 (1993). 22. T.Joseph and J. Manz, Molec. Phys. 57, 1149 (1986). 23. G. K. Paramonov and V. A. Savva, Optics Commun. 52, 69 (1984); G. K. Paramonov, Chem. Phys. Lett. 169,573 (1990); G. K. Paramonov, Phys. Lett. A 152, 191 (1991); G. K. Paramonov, Chem. Phys. 177, 169 (1993); W. Jakubetz, B. Just, J. Manz, and H.-J. Schreier, J. Phys. Chem. 94, 2294 (1990). 24. B. Just, Ph.D. Thesis, Universitat Wurburg, 1993. 25. J. S . Wright and D. J. Donaldson, Chem. Phys. 94, 15 (1985). 26. M. Dohle, J. Manz, G. K. Paramanov, and H. Quast, Chem. Phys. 197, 91 (1995); M. Dohle, J. Manz, and G. K. Paramonov, Bet Bunsenges. Phys. Chem. 99,478 (1995). 27. J. E. Combariza, S. Gortler, and J. Manz, Faraday Discuss. Chem. Soc. 91, 362 (1991); J. E. Combariza, S. Gonler, B. Just, and J. Manz, Chem. Phys. Lett. 195, 393 (1992); J. Manz, G. K. Paramonov, M. Poligek, and C. Schutte, Israel. J. Chem. 34, 115 (1994); W. Jakubetz, B. L. Lan, and V.Parasuk, in Femtosecond Chemistry and Physics of Ultrafast Processes, M. Chergui, Ed., World Scientific, Singapore, 1996, p. 86. 28. R. Mecke, 2. Elektrochem. 54,38 (1950). 29. R. Kosloff, J. Phys. Chem. 92,2087 (1988); R. Kosloff, Annu. Rev. Phys. 45, 145 (1994); N. Balakrishnan, C. Kalyanararnan, and N. Sathyamurthy, Phys. Rep., 280,79 (1997). 30. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes, Cambridge University Press, New York, 1986.


343

31. S. Flugge, Practical Quantum Mechanics, Springer-Verlag, New York 1974; A. Erdilyi, Ed., Higher Transcendental Functions, Vols. 1 and 2, Bateman Manuscript Project, McGraw-Hill, New York, 1953; A. Matsumoto, J. Phys. B: At. Mol. Opt. Phys. 21, 2863 (1988). 32. 0. Kuhn, D. Malzahn, and V. May, fnl. J. Quant. Chem. 57, 343 (1996). 33. F. Neugebauer, D. Malzahn, and V. May, Chem. Phys. 201,151 (1995);C. Scheurer and P. Saalfrank, J. Chem. Phys. 104,2869 (1996); P. A. Apanasevich, Principles of Interaction of Light with Matter, Minsk, Nauka i Thechnika, 1977 (in Russian); M. Schreiber, 0. Kuhn, and V. May, J. Lumin. 58, 85 (1994); C. Scheurer and P. Saalfrank, Chem. Phys. Lett. 245,201 (1995). 34. Y.R. Shen, in Fundamental Systems in Quantum Optics, Les Houches, Session LIII, J. Dalibard, J.-M. Raimond, and J. Zinn-Justin, a s . , North-Holland, Amsterdam 1992, p. 1049; M.Gavrila, Atoms in Intense Laser Fields, Acadmic, New York, 1992. 35. H. Quast, A. Witzel, E.-M. Peters, K. Peters, and H.G. von Schnenng, Chem. Ber. 125, 2613 (1992); A. Witzel, Ph.D. Thesis, Universitat Wurzburg, 1994; T. Dietz, Ph.D. Thesis, Universitat Wurzburg, 1995. 36. F. Seifert, V. Petrov, and M.Woerner, Opr. Lett. 19, 2009 (1994).

DISCUSSION ON THE COMMUNICATION BY J. MANZ Chairman: M. Quack S. A. Rice: What is the source of dissipation?

J. Manz: The surrounding molecules, for example.

TIME-FREQUENCY AND COORDINATE-MOMENTUM WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY S. MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

Department of Chemistry University of Rochester Rochestel; New York

CONTENTS I. Introduction 11. Correlation Function Expression for Spontaneous Light Emission 111. Wigner Wavepackets in Phase Space: The Doorway-Window Picture IV. Nuclear Wavepackets in Pump-Probe Spectroscopy V. Extension to Heterodyne-Detected Four-Wave Mixing Appendix A: Time- and Frequency-Gated Autocorrelation Signals Appendix B: The Signal and the Optical Polarization Appendix C: Four-Point Correlation Function Expression for Fluorescence Spectra Appendix D. Phase-Space Doorway-Window Wavepackets for Fluorescence Appendix E: Doorway-Window Phase-Space Wavepackets for Pump-Robe Signals References

I. INTRODUCTION In the analysis of linear and nonlinear optical spectroscopies, the electric fields and optical gates are commonly represented by their amplitudes. Similarly, the material system is represented by an amplitude as well, the wave function. However, optical signals are given by products of such amplitudes.

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Contml on the Femtosecond Erne Scale. XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

345

346

S. MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

For the material system we need two amplitudes, representing the bra and ket used in the calculation of the polarization matrix element, whereas the number of field amplitudes depends on the order of the particular nonlinear process. An improved intuitive picture can be obtained by grouping the field amplitudes in pairs using a mixed temporal-spectral (Wigner) representation. This often highlights much more clearly the roles of spectral and temporal features of the field and provides valuable information about the signal (e.g., its time-dependent spectrum). Similarly, the pair of wave function amplitudes can be used to form the density matrix, and a Wigner transform with respect to space brings it to a mixed coordinate-momentum (phase-space) representation that closely resembles the classical phase-space distribution.* In this chapter we show how such mixed Wigner representations can be used effectively to compute and interpret optical signals. The development of a mixed time-frequency representation in which both characteristics of the field and the response function are highlighted is currently receiving considerable attention. This activity is triggered by the rapid progress in pulse-shaping techniques, which made it possible to control the temporal profiles as well as the phases of optical fields with a remarkable accuracy [1-4]. These developments have further opened up the possibility of coherent control of dynamics in condensed phases [5-71. Mixed time-frequency measurements were first introduced in acoustics in the analysis of sound and speech [8]. The frequency-resolved optical gating (FROG)technique [9, 101 enables one to measure the FROG spectrograms that contain both temporal and spectral information about the signal. This spectrogram can then be inverted, and the intensity and phase of a pulse is obtained. However, the correspondence between FROG signals and the temporal and spectral profiles of the field is not always unique [7, 91. In this chapter we use the Wigner distribution, known also as the chronocyclic representation [11-15]. The interest in using this distribution, whose properties have been known for a long time, for the description of optical fields was resuscitated in the last few years, when it became apparent that it not only is a theoretical construct but also can be actually retrieved from nonlinear experiments [ 121, providing, therefore, a method for characterizing a signal completely, both in the time and frequency domain. The easiest way to obtain a mixed time-frequency picture (spectrogram) is to pass the signal through a spectral and temporal filter centered at some controlled frequency and time prior to its detection [16]. Phase-space wavepackets for nuclear motions have been applied to the interpretation of nonlinear optical measurements using the Liouville space *The electric field can also be transformed in coordinate space, but this will not be considered here.

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

347

representation of the response function 1171. We shall demonstrate how such wavepackets can be used in the analysis of fluorescence and pumpprobe spectroscopies. In Section I1 we study spontaneous light emission. We consider a setup in which the signal emitted by the sample passes through both a spectral and a temporal gate before it reaches the detector. We show that the detected signal can be written as a convolution of the bare autocorrelation function and a joint characteristic function of both gates. We then develop a correlation function expression for the bare signal and show that this expression is identical to but more compact than the one obtained by calculating the photon emission rate in Liouville space [17]. In Section I11 we rewrite the material response function in phase space using the Wigner representation for the external fields, the gate, and the doorway and window wavepackets. In Section IV we apply the same formalism to develop the doorway-window picture for pumpprobe spectroscopy. Finally, in Section V we show how our pumpprobe expressions may be generalized and applied to the calculation of heterodyne-detected four-wave mixing spectroscopies.

11. CORRELATION FUNCTION EXPRESSION FOR SPONTANEOUS LIGHT EMISSION We consider a single molecule whose interaction with an external radiation field E(r, t) is given by

where V is the molecular dipole operator and r is its position. (The following expressions hold for a real dilute sample made of noninteracting molecules. For simplifying the notation we consider a single molecule.) While discussing the properties of light using a classical description, the formalism is recast as close as possible to the quantum form by introducing the complex analytical signal E(r,t), which may be obtained from the real field E(r, t ) by [ 181

By construction, the Fourier transform of the analytical signal contains only positive frequencies. We further assume that the field comprises a finite number of pulses with wave vectors k,. We then have

348

S. MUKAMEL, C.CIORDAS-CIURDARIU, AND V. KHIDEKEL

E(r, t ) =

i

E,(t)eikjr

+ C.C.

where !€,(t) and I , ( t ) are the complex-valued analytic field and polarization, respectively. The field envelopes Ej(t) can be further represented in the form Ej(t) = gj(t)e-iwj',where g,(t) is a slowly-varying envelope and o, is the carrier frequency. In the simplest (homodyne) detection scheme one measures the time-integrated intensity of the field generated by the sample in a given direction k,: (2.3)

Time-integrated detection is commonly u.: xi in wave-mixing experiments, such as photon echoes [19]. However, time-integrated photon echo signals do not contain sufficient information to establish the complete form of the spectral density responsible for optical dephasing [20]. Additionally, most valuable microscopic information may be obtained by time-gated (or timeresolved) detection [21, 221, achieved by overlapping the total response with a narrow gate pulse, which provides the temporal profile of the signal. A number of more elaborate techniques, which provide a richer information, including spectral characteristics of the signal, have been developed. The second-order autocorrelation function (Ej*(tl)T.j(t*)) allows one to obtain more information about the field El81: its amplitude as well as the phase. This can be viewed as the time-resolved spectrum of the light pulse, namely the time-dependent strength of its different frequency components. The autocorrelation signal can be observed by passing the field through two gating devices, a spectral gate whose transmission function is centered at wo and a time gate centered at to. The measured signal (the total energy received by the detector) is given by Eq. (2.3), except that Ej(t) now denotes the gated field Ej. The resulting gated autocorrelation signal becomes a function of both wo and to, thus retaining its temporal and spectral information:


349

In Appendix A we show that the gated signal I ~ ~ ~ , , 00) f t " can , be recast in the form

where

[-- @*(t - $)$(t 00

? ~ " t ~a) ( t ,=

J

+ i7))eiwrd7

is the ideal (bare) autocorrelation signal, corresponding to a gate with an infinite temporal and spectral resolution, and +(r', w'; to, wo) is the joint gate function that depends on the transmission functions of both gates as well as the order in which they are applied. This function is calculated in Appendix A. The bare signal is not positive definite and it may even assume negative values. This is not the case for the gated signal, which is written as the squared amplitude (of the gated field) and is therefore guaranteed to be nonnegative. The gate function +(t', w'; to, w g ) is usually localized in both time and frequency. However, it cannot be made infinitely narrow in both variables due to fundamental uncertainty of the Fourier transform, &At 2 1. It then follows that mathematically the ideal gate + ( t , w ; t o , w o ) = &(to - t ) x 6(wo - (J) cannot be realized. Physically, however, an ideal gate is possible when both the temporal and spectral profiles of the gate are narrower than certain relevant material scales. The detected signal is then virtually identical to the bare signal. An example of such a limiting case is the snapshot limit of pump-probe spectroscopy [171. By expanding @*(t - ;7@(t + to second order in the external field, the autoc_orrelationsignal IAuro(t,w ) gives the spontaneous light emission, denoted I s ~ ~ a). ( t , In Appendix C we express the polarization in terms of a response function convoluted with the external field. By invoking the rotating-wave approximation, we get

i))

350


where

is the four-point equilibrium correlation function of the dipole operator. V(t) are operators in the interaction picture, evolving in time with respect to the material Hamiltonian H o (with no external field)

V(t) = exp(iHot)Vexp(-iHot)

(2.9)

Hereafter we set h = 1 (unless its inclusion is needed for clarity). The Liouville space path diagram corresponding to this correlation function is shown in Fig. 1. An alternative way to calculate the SLE spectrum is to expand the molecular density matrix to second order in the field and compute the time-dependent photon emission rate. The resulting expression is [23]

This formula, unlike Q.(2.7),maintains a complete bookkeeping of the time ordering of the various interactions with the radiation field [17] and has six terms; the Liouville space paths corresponding to three of them are shown in Fig. 2, and the paths for the complex-conjugate terms are obtained by interchanging the left and the right portions of each path. It can be easily shown that Eqs. (2.7) and (2.10) are identical. Indeed, the three-dimensional integration domain in (2.7)can be divided into six subdomains, each corresponding to a term of (2.10) upon a proper change of time variables:


35 1

-t

Figure 1. Liouville space diagram corresponding to the only term that contributes to the spontaneous light emission from a two-level system within the rotating-wave approximation [Eq.(2.7)]. Here (8) and le) denote the ground and the excited states, respectively.

(2.1 1)

(a) (b) (c) Figure 2. Liouville space paths for the three terms of Eq. (2.10).

352


rO

Figure 3. Diagrams showing how to divide the triple integral in (2.7) to get the six terms of (2.10).Domains (a),(b),and (c)correspond to the three Liouville space paths given in Fig. 2 and domains ( d ) , (e), and (f)to the complex-conjugate paths.

These integration domains are depicted in Fig. 3. The first three terms in (2.11) represent the third, the second, and the first terms in (2.10), respectively. A simple substitution of time variables, as specified in Table I, finally recovers the six terms of (2.10). We now rewrite Eq. (2.7) by introducing the Wigner distribution for the external field as well,

Wdt,a)=

I

m

E*(? - $7)E(t + 4 7 ) e i W ' d7

(2.12)

-m

Substituting (2.12) into (2.7), we obtain the autocorrelation signal (2.5), where the bare signal is given by TABLE I Correspondence between Time Arguments of Eq. (2.7) and Six Terms of Eq. (2.10)


353

We have thus expressed the autocorrelation signals using the Wigner representation for both the external fields and the gate. The molecular properties are contained in the response function F(4).In the next section, we show how when the incoming external pulses and the detection gate are temporally well separated, we can use the Wigner representation for the material system as well.

III. WIGNER WAVEPACKETS IN PHASE SPACE: THE DOORWAY-WINDOW PICTURE Assuming that the incoming pulses and the gated emission are temporally well separated, we can recast the general expression for the SLE presented in the previous section in a different form that lends itself more easily to physical interpretation. This will allow us to separate the process into the preparation of a doorway wavepacket by the pump field, a subsequent propagation of this wavepacket for a specified period, and finally observing it through a window wavepacket created by the detection device. By using the Wigner phase-space function for the doorway and the window wavepackets, we obtain an “all-Wigner” representation of the signal. We first rewrite FQ.(2.13) by making the following change of variables 71 + r2 -+ 27’, 71 - 7 2 + 7 + r”. The assumption that the pulses are well separated temporally allows us to extend all time integrals to infinity, resulting in

~ ( ~ -) 7’( t+ W ,t + 47, t 2

i7, t -

7’

- 47’’)

WE(t- r’, w 1)

(3.1)

By introducing a complete basis set in the coordinate representation, we show in Appendix D that the SLE signal can be recast in the form

354

S. MUKAMEL, C. CIORDAS-CIURDAFW, AND V. KHIDEKEL

with D,(xx’;to) =

(3.3) we(XX’; 00)=

1

27r

j1

-no

dt dr dw @ ( t ,o;0, oo)eiu’

Here De(xx’;0) is the doonvay wavepacket representing the molecular density matrix created by the external field at t” = 0. It then propagates for the delay period to resulting in D,(XX‘;to). Here, W,(xx’; 0 0 ) is the window wavepacket created by the gate and represents the density matrix responsible for emission at frequency WO. (Note that the time delay evolution is included now in the doorway wavepacket so that the window does not depend on to.) The signal is obtained by calculating the overlap of these two wavepackets in Liouville space. A more intuitive picture of these wavepackets can be obtained by switching to the Wigner (phase-space) representation

and the signal [Eq. (3.2)1 assumes the form

Now the overlap of these wavepackets is calculated in phase space. Note that this is a fully quantum mechanical picture, and no classical approximations were made. Our only assumption is that the excitation and gating processes are well separated. This form, however, allows the development


355

of semiclassical approximations for computing the doorway and the window wavepackets. When the Wigner distributions of both the external field and the gate function are fast compared with the time scale of nuclear dynamics and have a narrow spectrum compared with the dephasing rate, we can set W&”,wl) = 6(t”) - 6(wl - w,) and @(t,w;O,oo) = 6(t) - 6(w - 0 0 ) [here we is the frequency of the external field E(t)]. Although, as stated earlier, such a form is mathematically not possible, it can still be used approximately since the “narrowness” in time and in frequency are with respect to different molecular quantities. We can then calculate the integrals with respect to t”, 0 1 for the doorway wavepacket and t, w for the window wavepacket and obtain the snapshot limit of both wavepackets: co

d7” eiofT”

D:o)(xx‘; to) =

J

--

The ideal snapshot spectrum is obtained by substituting Eqs. (3.6) and (3.7) in Eq. (3.2).

IV. NUCLEAR WAVEPACKETS IN PUMP-PROBE SPECTROSCOPY

In this section we apply the same formalism to pumpprobe spectroscopy, where one measures the absorption of a probe pulse E 2 ( t ) by a molecule excited by a pump pulse El(t). The pumpprobe signal can be written as

J

--

where 4 ( t ) is the polarization induced in the sample by the external electric field. We start by expanding the polarization to first order in the probe amplitude E?(t):

356


J

-oo

where S(')(t,7) is the response function linear in the probe [171, S")(t, 7 ) = i8(t - 7)([Pl(f),P~(T)])

- 7 ) is the Heaviside step function and !P'(t) is the polarization operator in the Heisenberg picture, calculated with respect to the Hamiltonian H', which includes the pump field but excludes the probe:

O(t

The pumpprobe signal can then be written as

This formula resembles Eq.(2.6) for the autocorrelation signal. We can further expand S ( ' ) ( t ,7 ) to second order in the pump field and express the result in terms of the four-point correlation function (2.8) (see Appendix E). Similar to the correlation measurements discussed in the previous sections, we can define the doorway and window wavepackets and write the signal as their overlap in phase space. The derivation is presented in Appendix E, and we have

ZPP(t0,

wo) =

Jf

dx dr'[W,(xx'; w0)De(xx'; to) + W,(xx'; wo)Dg(xx'; to)] (4.3)

where to is the delay between the pump and probe pulses and wo is the carrier frequency of the probe. Upon transforming from the coordinate to the Wigner phase-space representation, we can write (4.3)as


357

The wavepackets D,,Wj,j = g, e, can be written in the coordinate representation as

(4.6)

* jjjy- d t d7 dw W&, w)eiwr

W,(xx‘; wo) = 27r

The first term in Eq. (4.3) is reminiscent of Eq.(3.2) for the spontaneous emission spectrum. It represents a doorway wavepacket created by the pump in the excited state, which is then detected by its overlap with a window. The only difference is that the role of the gate in determining the window in SLE is now played by the probe Wigner function 0 2 . In addition, the pumpprobe signal contains a second term that does not show up in fluorescence. This term represents a wavepacket created in the ground state (a “hole”) that evolves in time as well and is finally determined by a different window W, [24]. In the snapshot limit, defined in the preceding section, we have

358

S.MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

J

--

V. EXTENSION TO HETERODYNE-DETECTED FOUR-WAVE MIXING The description of pumpprobe signals presented in the preceding section can be immediately generalized to heterodyne-detected transient grating spectroscopy as well as to other four-wave mixing techniques. Heterodyne detection involves mixing the scattered field with an additional heterodyne field E4(t). The signal in the k, direction can then be written in terms of the polarization P&) as

Consider the signal emitted in the direction ks = k3 - k2 + kl. The polarization !P&) is given by &. (El) except that now the two fields that excite the sample are different. We therefore get

* - t 3 - t2 - ti) x E3(t - t3)E,(t - t 3 - t2)E2(t

x E3(t - t3)Z?(t - t 3 - t2)El(t - r3 - t 2 - t i )

(5.2)


359

Here the time ordering of the El and E2 fields can be arbitrary; we only assume that the field 2 3 comes after El and E 2 and does not overlap with them. We can then follow the calculations of pumpprobe signals in Appendix E and introduce the joint Wigner distribution for the fields El and E2 and for the field fE3 and 2 4 : am

am

Substituting this into (5.1), we get

The doorway-window picture applies here as well. We can use Eqs. (4.5X4.8) provided we replace the pump wavepacket W I by W ~and I the probe wavepacket W 2 by W 4 3 .

APPENDIX A: TIME- AND FREQUENCY-GATED AUTOCORRELATION SIGNALS Autoconelation signals may be observed by passing the field through two gating devices, a spectral and a time gate. Each gate has its transmission function (in the time and frequency domain) centered at 0 0 and to, respectively. The field passing through each of these gates becomes

Here F,(w; W O ) and !Fr(f; to) are the spectral and temporal transmission functions and and E are the fields before and after passing through the gate, respectively. We shall denote them the bare and the gated signal, and throughout this appendix bare quantities will be denoted by a tilde. Note that the order in which the two gates are applied is important, even though both

360

S. MUKAMEL. C . CIORDAS-CIURDARIU,AND V. KHIDEKEL

devices are linear and independently controlled. From a practical point of view, to attain optimal time and frequency resolution, it is advantageous to apply first the gate with the less ideal transfer function [25]. Below we will consider both configurations. The measured signal (the total energy received by the detector) is given by Eq. (2.3), except that the field now passes through gating devices prior to its detection. Therefore, the signal becomes a function of both 00 and to, thus retaining its temporal and spectral information. The autocorrelation signal is defined by Eq. (2.4). We will show in Appendix B that within the slowly varying amplitude approximation the measured field is proportional to the polarization induced in the sample by the external electric fields [Eq. (B4)]. We therefore have

It was shown elsewhere [26] that the gated signal Z ~ ~ ~ ~00) ( t is 0 given , by Eq. (2.5) together with (2.6). The transmission function is expressed in terms of the Wigner functions for the gates,

W$(t,0)=

1

0

Y;(W - $,J)Ts(w

+ $o’)e-io’r dw’

If the spectral gate is applied first, we have

where the subscript st indicates that the signal first passes through the spectral gate and then through the time gate. For the reverse order of the gates,

we have

These two equations can also be written in another form using the definition


36 1

of the Wiper function, we then have

as&’,w’;

I_., m

to, wo) =

dtl!F,(t; to)12Ws(t - t’, w’; wo)

(A31

Assuming specific forms of the time and spectral gates, we get the signal measured in different experimental setups. If the time gate is infinitely short, F1(f;fo)= &t - to), we obtain the spectrograms discussed in Ref. [16]. If the signal passes only through the time gate, then the gate acts like the reference pulse in the FROG configuration. The joint gate function is centered around the frequency wo and the time to and acts as a filter on the bare signal ?. As an example we shall consider the case when the spectral gate is given by the Fabri-Perot 6talon [16] and the time gate is exponential,

Note that in this case the transmission functions (A5) and (A6) depend on t, to and w, wo only through the differences t - to and w - w g and the joint gate function #sr(t’, w’; to, 0 0 ) only depends on T = f’ - to and Q = w’ - 00. The joint gate function for the gates ( A 3 and (A6), calculated by (A3), assumes the form T>O

In the case y>> I’ the above expression simplifies and can be written in the compact form

362

S. MUKAMEL, C.CIORDAS-CIURDARIU, AND V. KHIDEKEL

For the reverse order of gates, when the signal passes first through the time gate and then through the spectral one, we have

x { [Q - y(y +

r)]cos 2QT - (2y + r)sin 2QITI }

More elaborate gating profiles may be obtained using pulse-shaping techniques [2, 31.

APPENDIX B: THE SIGNAL AND THE OPTICAL POLARIZATION We relate here the scattered field to the polarization induced in the sample by external fields. The radiated field is the solution of the Maxwell wave equation

V x V x z(r, t ) + - I

a2

.2

at2

t)=

4?r

--

c2

a2

g(r, t ) at2

(Bl)

Here g(r,t) is the analytic polarization, which is defined in terms of the real polarization Ztr)(r,t ) in the same way as the analytic signal was defined in terms of the real signal [Eq.(Bl) is usually written for the real field and polarization; however, it follows from their definition that the analytic quantifies satisfy the same equation]. When the analytic polarization P(r, t) is given, the wave equation is linear and inhomogeneous and can be soived exactly in a closed form. The general solution of E!q. B1 is [17]

E(r, t ) =

where

I

dr’ dt‘ G(r - r‘; t - t’)g(r’, t’)


363

and

We are interested in far-field limit measurements, that is, when the size of the material system is much smaller than its distance from the detector. In this limit, the usual approximation one makes is [27]

lr-r’l=r-r’cos 8 where 8 is the angle between r (the vector giving the position of the observation point) and r‘ (the variable of integration over the volume of the material). In this approximation the Green function G(r - r; w) becomes w2

Gje(r- r‘; w ) = 2 (a,/ c

F)

e i ( a / c ) ( r - r’cos

-

e)

r

We now use this form for the Green function in Eq. (B2), integrating with respect to w and using the relation

J

2e-iCd(f - fo)

d2 dw = -27r 6(t - to) dt2

We then integrate with respect to t’ and obtain

2‘R Ej(r,t)= -C2Y

(6,l-

F)I

dr’

c

Since we are interested in the far-field case, we can approximate in the integrand r - rf cos 0 = r, and therefore, integrate over r’ and obtain the following relation between the scattered field and the polarization:

S . MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

364

where ztot(t) is the total polarization of the material. Although we have obtained a formula that takes into account retardation effects, we will neglect them in what follows. We also invoke the slowly-varying amplitude approximation setting B ( t ) = - w 2 p ( t ) . Finally, we can write E,(r,t) =

2uw -

c2r

Assuming that the polarization Pl(r‘, t) has the form of a plane wave along the z direction,

we can integrate over z’ in (B3) and get Ej(r, t ) =

2uw -

rkc2r

where =

j J,

dx’ dy’ p, (x’, y’, t )

APPENDIX C: FOUR-POINT CORRELATION FUNCTION EXPRESSION FOR J?LUORESCENCE SPECTRA We consider a molecular system interacting with an external electric field and described by the Hamiltonian

H = Ho - VE(r, t )

(C1)

where HO is the electronic and nuclear molecular Hamiltonian and the dipole operator V represents its interaction with the field. Throughout this section E(r, r ) denotes the real field. The autocorrelation function of the polarization operators (&r1)&f2)) is expressed in terms of the dipole operator as


365

where V&) denotes the dipole operator whose evolution is governed by the Hamiltonian H:

V H ( t )= U(t,--)V(--)Ut

--

( t , --)

U ( t ,t’)

= exp

{ j: i

H(t)dt}

--

The autocorrelation function (C2) is characterized by three intervals of evolution: from to t l , from f l to 1 2 , and from t2 to (Fig. 4a). It is inconvenient to use the finite interval ( t l , t2), since the corresponding time integrals will have (variable) finite limits. A more practical way to analyze (C2) is to extend the interval (tl ,t2) to and factorize the evolution operator U(t1,t2) as U(--,t2)Ut (--, t i ) (Fig. 4(b). We then have

--

Perturbative expansion with respect to the external electric field yields,

with

366


(a)

(b)

Figure 4. ' h o ways to describe the time-evolution of the dipole operators: single ( a ) and double (b) Liouville space diagrams.

Here V ( t ) is the dipole moment given by Eiq. (2.9). For n = 2 the subscriptsj, I , m,i take the values 0, 1, 2 (so that their sum is at most 2), and the result is expressed in terms of the four-point correlation functions:


367

where

We now consider a two-level system and invoke the rotating-wave approximation, in which we neglect highly oscillatory terms. We can now calculate the evolution operator Uo(t,r’) explicitly, using the Hamiltonian of the ground and excited states of the two-level system. If w is close to the transition frequency of the two-level system, then the only term that will not is the term with C?&. Keeping contain highly-oscillating factors e’(O this term only, we get

This formula can be easily rewritten for the analytic field and polarization. Substitution into Eq. (2.6) finally results in Eq. (2.7).

APPENDIX D: PHASE-SPACE DOORWAY-WINDOW WAVEPACKETS FOR FLUORESCENCE We now derive the expression for the fluorescence signal in terms of the doorway and window wavepackets instead of the four-point correlation function. We start with Eq.(3.1) and write the four-point correlation function F(4) explicitly as the trace with respect to the equilibrium density matrix. We then use the cyclic invariance of the trace and obtain 1 + p, t - 7’ + i7”) = Tr[V(t - $7)V(t + ;7)V(t - 7’ + $‘‘)V(t

~ ( ~’ 7’ ( t- :r”, t -

$7,t

- 7’- $7”))

(Dl)

We now write the equilibrium density matrix in the coordinate represen-

368


tation and insert in (Dl) the unity operator dx’Ix‘)(x’l. We then have F‘4’(t - 7’

1 1 I ’ I - p”, r - 97, t + 27, t - 7 + 97

’/

)

We assume that the external field is peaked at the time -to and the temporal gate at the time t = 0. Changing in the doorway wavepacket the variable t” = t + to - 7’, we can rewrite equation (D2) as F‘4’(t - 7’ -

iT”,

t - 47,t + 47, t - 7’

+ i7”)

where

is the doorway wavepacket created by the external field, and

is the window wavepacket. The response function is then calculated by the overlap of the window wavepacket with the doorway wavepacket propagated for the time to. These formulas can be rewritten in the Heisenberg picture, pD(Xt x’; f ” , 7”)

vege-iHg(r”/2)poe-iHs(~”/2) v

= (xle iHe(r”/2 - I” + 10)

ge

e i H e ( ~ ” / 2 + r” - 10)

lx’)

P w(x, x’; t , 7 )

eiffg(r/2) iH - (xle-i/fe(r/2 - ?)v eg e

(7/2)v e - i H e ( 7 / 2 + r ) ge

1 4 .

We now return to the expression for the gate signal (2.5) and substitute the bare signal (3.1) with the four-point correlation function F(4)replaced with the doorway and window wavepackets as in (D3). We then separate the four time integrals into two groups: the integrals with respect to t” and 7” for the doorway wavepacket and with respect to t and 7 for the window wavepacket. We can also change all limits of integration to infinity; this is


369

possible since we assumed that the Wigner function of the incoming field WE and the gate function CP are well separated in time. We then obtain Eqs. (3.2H3.4).

APPENDIX E: DOORWAY-WINDOW PHASE-SPACE WAVEPACKETS FOR PUMP-PROBE SIGNALS We shall calculate here the pumpprobe signal (4.1) using the doorway window wavepackets representation. The polarization P ~ ( tto) third order in the external field is given in Ref. 17 and is shown to be expressed in tenns of the four-point correlation function (2.8):

where R:3’(t3,t2,tl) = F‘4’(t - t3 - t 2 - tl,t,t - t3.t R‘3’ 2 (t3,

- t3 - t 2 )

t2,tl) = F‘4’(t- t3 - t2, t , t - t3, t - t 3 - t 2 - t i )

R(3) 3 (t3, t2, ti) =

F‘4’(t - t 3 , t , t - t 3

- t2,t

- t3 - t2 - ti)

R,( 3 )( t 3 , t 2 , t l ) = F(4’(t- 13 - t 2 - t i , t - t 3 - t 2 , t - t 3 , t )

Substituting this into (4.1) and introducing the Wigner representation for the pump and the probe fields as was done for the fluorescence, we get

370

S. MUKAMEL. C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

Similar to the Auoresence discussed in Appendix D, we can define doorway and window wavepackets and write the signal as their phase-space overlap. Assuming that the delay time between the probe and the pump pulses is to, we obtain Eqs. (4.5)-(4.8).

Acknowledgments

The support of the National Science Foundation and the Air Force Office of Scientific Research is gratefully acknowledged.

References 1. D. J. Tannor and S . A. Rice, J. Phys. Chem. 83, 5013 (1985). 2. A. M. Weiner, Prog. Quantum Electron. 19, 161 (1995). 3. J. T. Fourkas, L. Dhar, K. A. Nelson, and R. Trebino, J. Opt. SOC. Am. B 12, 155 (1995). 4. N. F. Scherer, L. D. Zeigler, and G. R. Fleming, J. Chem. Phys. 96, 5544 (1992). 5. W. S. Warren, H.Rabitz, and M. Dahleh, Science 259, 1581 (1993). 6. B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakovlev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993). 7. B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, and K . R. Wilson, Phys. Rev. Lett 74,3360 (1995). 8. W. Koenig, H. K. Dunn, and L. Y. Lacy, J. Acoust. SOC. Am. 18, 19 (1946). 9. D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 2 (1993). 10. R. Trebino and D. J. Kane, J. Opt. SOC.Am. A 10, 1101 (1993). 1 I . J. Paye, IEEE J. Quantum Elecrron. 28, 2262 (1992). 12. J . Paye, in Ultrafast Phenomena, Vol. 9, P. F. Barbara, W. H.Knox, G. A. Mourou, and A. H. Zewail, Eds., Springer-Verlag, New York 1994. 13. L. Cohen, Proc. IEEE 77,941 (1989). 14. L. Mandel and E. Wolf, Eds., Selected Papers on Coherence and Fluctuations of Light, with Bibliography, Dover, 1970. 15. M. G. Raymer, M. Beck, and D. T. Smithey, Phys. Rev. Lett. 70, 1244 (1993). 16. H. Stolz, lime-Resolved Light Scattering from Excitons, Springer, Verlag, Berlin; New York 1994. 17. S. Mukamel, Principles ofNonIinear Optical Spectroscopy, Oxford University Press, New York 1995. 18. H. M. Nussenzweig, Introduction to Quantum Optics, Gordon & Breach, London; New York 1973. 19. E. T. J. Nibbering, D. A. Wiersma, and K. Duppen, Phys. Rev. Lett. 66,2464 (1991). 20. D.C. Amett, P. Vohringer, R. A. Westervelt, M. 3. Feldstein, and N. F. Scherer, in Ultrafast Phenomena, Vol. 9, P. F. Barbara, W. H. Knox, G. A. Mourou, and A. H. Zewail, Eds., Springer-Verlag, Berlin 1994. 21. W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, Chem.Phys. Letf. 238,l (1995). 22. P. Vohringer, D. C. Amett, T.-S. Yang, and N. F. Scherer, Chem. Phys. Lett. 237, 387 ( 1995). 23. S. Mukamel, Adv. Chem. Phys. 70 (Part I), 165 (1988).


37 1

24. Y. J. Yan and S. Mukamel, Phys. Rev. A 41, 6485 (1990). 25. V. Wong and 1. A. Walmsley, J. Opt. Soc. Am. B 12, 1491 (1995). 26. S. Mukamel, C. Ciordas-Ciurdariu, and V. Khidekel, IEEE J. Quanrum Electron., 32, I278 ( 1996). 27. J. D. Jackson, Classical Elecrrodynarnics, Wiley, New York, 1975.

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

GENERAL DISCUSSION ON LASER CONTROL OF CHEMICAL REACTIONS Chairmun: M. Quack

V. Engel: Prof. Manz, 1want to come back to the question of dissipation. We learned from the talk of Prof. Fleming that coherences persist for a while in a liquid surrounding, but not too long. The control schemes we heard about rely on quantum interference or coherence. I would like to know: What are the perspectives of applying the control schemes to chemical reactions in a liquid? J. Ma-: Prof. V. Engel’s question points to the virtues of using (i) ultrashort laser pulses for controlling chemical reactions, in comparison with (ii) continuous-wave lasers. Compare, for example, the strategies of (i) Tannor et al. [l], Rabitz [2], Combariza et al. [3] (see also Korolkov et al., “Theory of Laser Control of Vibrational Transitions and Chemical Reactions by Ultrashort Infrared Laser Pulses,’’ this volume) versus (ii) the strategies of Brumer and Shapiro [4] or Letokhov [5] and others [6]. See the review by S. A. Rice (this volume). Intuitively, one may assume that the advantage of using ultrashort laser pulses should be that they can be faster and, therefore, “beat” competing dissipativeprocesses such as intramolecular vibrational redistribution (IVR). This argument is supported, certainly, by the observation of relatively long coherence lifetimes of reactive wavepackets even in the condensed phase, as has been demonstrated beautifully by G. R.Fleming et al. (“Femtosecond Chemical Dynamics in Condensed Phases,” this volume) and A. H. Zewail[7]. However, this advantage may be valid only for rather strong laser fields, because, otherwise, for weak laser fields, there is a theorem by Brumer and Shapiro saying that time-dependent and time-independent fields will achieve equivalent molecular transitions [81. 1. D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985); D. J. Tannor, R. Kosloff, and S. A. Rice, J. Chem. Phys. 85, 5805 (1986); D. J. Tannor and S. A. Rice, Adv. Chem. Phys. 70, 441 (1988). 2. S. Shi, A. Woody, and H. Rabitz. J. Chem. Phys. 88,6870 (1988); W. S. Warren. H. Rabitz, and M. Dahleh, Science 259, 1581 (1993); W. Jakubetz, J. Manz, and H.-J. Schreiber, Chem. Phys. Lett 165, 100 (1990): W. Jakubetz, E. Kades, and J. Manz, J. Phys. Chem. 97, 12609 (1993).

373

374

GENERAL DISCUSSION

3. J. E. Combariza, B. Just, J. Manz, and G. K. Paramonov, J. Chem. Phys. 95, 10355 ( 1991). 4.

M. Shapiro and P. Brumer, J. Chem. Phys. 84, 4103 (1986); P. Brumer and M. Shapiro, Acc. Chem. Res. 22,407 (1994).

5 . V. Letokhov, Science 180,451 (1973). 6. E. Segev and M. Shapiro, J. Chem. Phys. 77,5604 (1982); V. Engel and R. Schinke, J. Chem. Phys. 88,6831 (1988); D. G. Imre and J. Zhang, J. Chem. Phys. 89, 139 (1989); M. D. Likar, J. E. Baggott, A. Sinha, T. M.Ticich, R. L. Vander Wal, and F. F. Crim, J. Chem. Suc. Faraday Trans. I1 84, 1483 (1988); F. F. Crim, Science 249, 1387 (1990). 7. A. H. Zewail, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 15; R.Zadoyan, Z. Li, P. Ashjian, C. C. Martens, and V. A. Apkarian, Chem. Phys. LPtf. 218, 504 (1994). 8. M. Shapiro and P. Brumer, J. Chem. Phys. 84, 540 (1985).

M. Shapiro: In relation to the problem discussed by Profs. Engel and Manz, I should point out that, if there are fast dephasings, one definitely would have to use fast pulses. Graham Fleming’s data suggest, however, that coherences persist for much longer times than anticipated in the past. In order to overcome decoherence, we can consider adiabatic passage techniques (cf., e.g., Bergmann, Reuss, van Linden van der Heuvel, and Neusser) whose advantage is that you can completely empty the ground state, irrespective of the exact pulse shape, provided that the area under the pulse d t E ( t ) p l exceeds the limit imposed by the adiabatic condition. L. Woste: Prof. Manz, when you shape potential-energy curves or surfaces with high-intensity IR laser fields, don’t you believe that these field strengths can also induce electronic transitions leading to entirely different states and potential-energy surfaces, so that the proposed shaping process loses relevance? M. V. Korolkov, J. Maw, and G. K. Paramonov:* We are fully aware of the danger of using exceedingly intense IR picosecond/femtosecond laser pulses for control of molecular vibrations or reactions. As a rule derived from laser interactions with atoms, the intensities should be below the Keldish limit [ 11,

[I

2 *Comment presented by J. Manz.

16Z2

LASER CONTROL OF CHEMICAL REACTIONS

ZKeldish =

2 2

375

W/cm2

where Es is the ionization potential in electron-volts. Otherwise the laser may ionize the atoms, yielding a large displacement ro in the presence of an electric field with amplitude E Oand carrier frequency w [2]:

where I is in watts per square centimeters, h is in nanometers, and is in reciprocal centimeters. We wish to point out, however, that, to the best of our knowledge, there are so far no systematic investigations of similar Keldish-type limits for molecules. We have carried out exploratory test calculations for a model system that indicate that even stronger intensities may not be sufficient for ionization by ultrashort infrared laser pulses (see Fig. 1). In any case, one should always try to keep the laser intensities, specifically the amplitudes

or the corresponding amplitudes of the laser field strength as small as possible. There are several ways (i), (ii) or optimal conditions (iii), (iv), (v) for this purpose: (i) Prolongation of the pulse duration. (ii) Separation of a single IR femtosecond/picosecond laser pulse into a series of IR femtosecond/picosecond pulses [3]. (iii) Applications to systems with rather strong variations of the dipole function ~ ( q along ) the vibrational or reaction coordinate (4).Note that the semiclassical molecule-laser interaction operator is

376

GENERAL DISCUSSION (

INITIAL ELECTRON STATE

a)

...._.___.___.._.._____ ---.... E, = - 1 2 . 9 e V

-10

- 20

.'!

-30

-5

-2.5

(

0

0

COORDINATE

b)

100

2.5

( A )

LASER PULSE

200 T I M E

300 (fs)

400

( c ) ELECTRON MOTION

I .5

5

500

I

0.5

0 -0.5 -1 -1.5

'I

0

I loo

200 T I M E

300

(fs)

( d ) ELECTRON

400

I

SO0

MOTION

-5

-10

-IS

-

I

0

Es=-12.9eV

100

200 T I M E

300 IK&jish) ultrashort 1R laser pulses may not cause ionization; that is, the simple estimates (1)-(4) [ l , 21 are not applicable.


377

that is, large variations of p(q) may allow use of rather small values of field amplitudes ZO and, therefore, intensities [cf. Eq. (511. (iv) Applications to vibrational transitions between close (say Av = 1, 2, 3, ... , not 10, 20, 30, ...) vibrational levels. (v) For applications to isomerizations, items (iii) and (iv) imply that the reaction should proceed across rather shallow barriers of the potential energy surface, and the reactants and products should have rather different (e.g., with opposite signs) dipole moments. (Seealso Korolkov et al., “Theory of Laser Control of Vibrational Transitions and Chemical Reactions by Ultrashort Infrared Laser Pulses,” this volume; supported by DFG.) 1. L. V. Keldish, Sov. Phys. JETP 20, 1307 (1965); S. Augst, D. D. Meyerhofer, D. Strickland, and S . L. Chin, J. Opt. Soc. Am. B 8, 858 (1991); P. Dietrich and P. B.

Corkum, J. Chem. Phys. 97,3187 (1992). 2. A. D. Bandrauk. in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 261. 3. J. E. Combariza, B. Just, 1. Manz, and G. K. Paramonov, J. Chern. Phys. 95, 10351 (1991); B. Just, J. Manz, and G. K. Paramonov, Chem. Phys. Lett. 193,429 (1992).

M. Quack Somewhat related to the very nice isomerization scheme used by Dohle and co-workers [l], I would like to make a more general comment. In connection to “control” in dynamics I would like to take here a general point of view in terms of symmetries (see Scheme 1): We would start with control of some symmetries in an initial state and follow their time dependence. This can be used as a test of fundamental symmetries, such as parity, P, time reversal symmetry, T, CP, and CPT,or else we can use the procedure to discover and analyze certain approximate symmetries of the molecular dynamics such as nuclear spin symmetry species [2], or certain structural vibrational, rotational symmetries [3]. I would like to draw attention here to some work on chiral molecules, which allows very fundamental tests of symmetries in physics and chemistry. The experiment outlined in Scheme 2 [4] allows us to generate, by laser control, states of well-defined parity in molecules, which are ordinarily left handed (L) or right handed (R) chiral in their ground states. By watching the time evolution of parity, one can test for parity violation and I have discussed in detail [MI how parity violating potentials AEpv might be measured, even if as

378

GENERAL DISCUSSION Control of symmetry of initial state

Time dependence of symmetry properties?

Test of fundamental symmetries in nature (such as P, T, CP, CPT) or Test of approximate symmetries of dynamics

Scheme 1. Control of symmetries in dynamics.

T

w

4

h

rn

>

W

4 ->

R Scheme 2. Control of parity and chirality in the scheme of Ref. 4. L


379

small as J/mol. Time scales corresponding to such symmetry violation are hours to days. On the other hand, one can also generate a very short-lived time-dependent chiral excired state, even if the potential has a minimum at an achiral geometry. If the excited-state potential is harmonic, for example, one may derive interesting results on chirality in relation to harmonic oscillator dynamics [7]. This includes femtosecond evolution of chirality and control of stereomutation in chiral molecules [7]. On the most fundamental level, we have shown how this experimental scheme might be used for a fundamental test of CPT symmetry violation [8]. While still somewhat hypothetical, at present, this would constitute the most sensitive currently proposed test on CPT symmetry. The sensitivity expressed as a baryon mass difference Am between particles and antiparticles (with mass m) would be of the order 181. The best currently proposed other experiment of Am/m = is on antihydrogen spectroscopy at CERN (not yet carried out) with Am/m = and the best existing result for the proton-antiproton pair is h / m I [9]. Relating to the introductory comments by Prof. Prigogine on irreversibility on Tuesday, one might consider time reversal symmetry violation and indeed CPT symmetry violation as a more fundamental approach to this problem. One could then envisage that the second law is not just a de fact0 violation of time reversal symmetry but a de lege violation in the terminology of Refs. 5 and 6. Figure 1 shows the entropy evolution of CF2HCl on the femtosecond time scale, showing relaxation toward maximum entropy. With time reversal symmetry, the time reversed dynamic state would return to zero entropy after the appropriate time. This is the current status of experimental knowledge. However, one might envisage that on long time scales for complex systems time reversal symmetry may not apply and entropy stays near the maximum, as drawn roughly as a second option into the figure. 1. M. Dohle, J. Manz, and G . K. Paramonov, Be,: Bunsenges. Physik. Chem. 99,478 (1995). 2. M. Quack, Mol. Phys. 34, 471 (1977).

3. A. Beil, D. Luckhaus, R. Marquardt, and M. Quack, J. Chem. Soc. Furuduy Discuss. 99, 49 (1994); M. Quack, J. Mol. Srrucr. 347, 245 (1995). 4. M. Quack, Chem. Phys. Lett. 132, 147 (1986).

5. M. Quack, Angew. Chemie 101,588 (1989); Angew. Chemie Inr. Ed. Engl. 28,571 (1 989). 6. M. Quack, in Femtosecond Chemistry, J. Manz and L. Waste, Eds., Verlag Chemie, Weinheim, 1995, Chapter 27. p. 781.

380 t/ps

Figure 1. Time-dependent entropy for the three strongly coupled CH stretching and bending vibrations in CF2HCI[6].

t/ps


381

7. R. Marquardt and M . Quack, J. Chem. Phys. 90, 6320 (1989); J. Phys. Chem. 98, 3486 (1994); Z Phys. D 36,229 (1996). 8. M. Quack, Verhund. DPG VI 28,244 (1993); Chem. Phys. Lett. 231,421 (1994). 9. G. Gabrietse, D. Phillips, W. Quint, H. Kalinowsky, G. Rouleau, and W. Jhe, Phys. Rev. Letf. 74,3544 (1995); J. Groebner, H. Kalinowsky, D. Phillips, W. Quint, and G. Gabrielse, Verhund. DPG VI 28, 315 (1993).

M.E. Kellman: The idea by Prof. Quack of using molecular experiments to test fundamental symmetries is very interesting. In connection with the Na3 pseudorotation experiments we heard about yesterday, have you considered testing permutation symmetry of identical particles, that is, the Pauli principle and nuclear spin statistics? M. Quack: The violation of the principle of nuclear spin symmetry conservation [l] could be seen in a similar scheme as I discussed for parity, but, in contrast to parity violation, it can also be seen by more standard spectroscopic techniques (and has been seen repeatedly). On the other hand, one might also look for violations of the Pauli principle, which in fact we have done [2]. However, it seems unlikely to find such a violation (and nothing of that kind has ever been found), although in principle one must allow even for such a phenomenon. 1. M. Quack, Mol. Phys. 34, 477 (1977). 2. M. Quack, J. Mol. Strucr. 292, 171 (1993); and unpublished results.

M. S. Child: I would like to ask Prof. Quack to what extent tunnelling between the two enantiomers might affect these conclusions? M. Quack Tunneling will be completely suppressed if AEpv >> A&,,. Even if that is not the case, if M,, is negligible, tunneling may be exceedingly slow, as was already discussed by Hund in 1927 (see Refs. 1 and 2 and references cited therein). 1. M.Quack, Angew. Chemie 101,588 (1989); Angew. Chemie Inr. Ed. Engl. 28,571

( 1998). 2. M . Quack, in Ferntosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, Chapter 27, p. 781; R. Marquardt and M. Quack, 2 Phys. D 36, 229 (1996).

J. Maw: Prof. M. Quack has just presented to us a fascinating strategy for laser control of chemical enantiomers with different parities [ 13. A complementary approach has been suggested earlier by Brumer and Shapiro, based on their general strategy of laser control [2]. I would like to ask Prof. M.Shapiro whether he could comment on his approach in comparison with that of M.Quack. 1.

M. Quack, in Femtosecond Chemistry, J. Manz and L. Woste. Eds., Verlag Chemie,

382

GENERAL DISCUSSION

Weinheim, 1995, Chapter 27, p. 781, R. Marquardt and M. Quack, Z. Phys. D 36, 229 (1996). 2. M. Shapiro and P. Brumer, J. Chem. Phys. 95, 8658 (1991).

M. Shapiro: Our approach [M.Shapiro and P. Brumer, “Controlled Photon Induced Symmetry Breaking: Chiral Molecular Products from Achiral Precursors” J. Chem. Phys. 95, 8658 (1991)], differs from that of Quack in that we show how to generate chirality (from achiral precursors). T. Kobayashk I would like to make the comment that an interesting application of wavepacket control [I] is phonon squeezing in molecular systems and the creation of the Schrodinger cat state. It was theoretically predicted that there are several mechanisms that lead to squeezing of phonon states. It was found earlier that a sudden frequency change during an electronic Franck-Condon transition leads to special quantum mechanical statistics, called squeezing [2-9], of the molecular vibrations [10-121. A state is termed “squeezed” if some of its characteristics have less noise than the corresponding quantum noise of the vacuum state. The concept of squeezing turned out to be very fruitful in basic research and implies a lot of promising practical possibilities. The above-mentioned mechanism of squeezing the vibrational state prompted some controversial discussion in the literature [13-16]. The phenomenon is caused by the change of the frequency of the molecular vibration provided that the transition takes place in a fraction of time negligibly small as compared with the vibrational period. Recently we have shown that phonon squeezing, connected to the finite duration of the excitation pulse, occurs even in the absence of frequency change. This effect would be rather common in ultrashort laser pulse experiments [17-19]. Non-transform-limited pulses, either chirped or incoherent, are very useful in high time resolution spectroscopy 1201. For phonon squeezing, chirped pulses can also be used as shown in Figs. 1 and 2. With a chirped pulse even a Schrodinger cat state can be obtained, as shown in Fig. 2b. Advantages of using a chirped pulse are also shown in Fig. 2b. As can be seen from the figure, the chirped pulse applied to a periodically oscillating system is equivalent to a double pulse. In such a way, a well controlled chirped pulse may serve a purpose equivalent to a pulse train. The Wiper function corresponding to Fig. 2b is shown in Fig. 3. This means that the coherent states are interfering with each other. In such a way we can control a wavepacket by using a chirped pulse instead of an ultrashort pulse.

0.15

0.1 1

10

100

0.3

0.35 W

0.4

1

1.

chirp rate

0.25

I

I !

I I 1

P

Ph

0.5

t =25T

0.45

0.6

(~=0.0437)

0.55

Figure 1. Uncertainty of the quadrature AX: of the phonons (squeezing occurs if AX becomes less than unity) after a resonant Franck-Condon transition induced by a chirped pulse of moderate duration (u = 0.0437~) as a function of the chirp parameter w, which is in the units of the phonon frequency w. The electron-lattice constant is supposed to be g = 5 . The markers a-d refer to Fig. 2.

0.2

.I

..

I I

no-frequency-change case using chirpea long pulse

pulse width = 40 phonon periods (long pulse)

Figure 2. The Q-functions of phonon states after the electronic transition induced by differently chirped pulses. ( a ) The Q-function at the minimal AX+,that is, maximal squeezing (the chirp parameter w = 0.399; see marker a in Fig. 1). (b) The AX+ is in its next maximum. (c) An intermediate state between a maximum and minimum. As we consider lower and lower chirp parameters, the maxima and minima become less prominent ( d ) ,approaching the number state with equal distribution along a circle.

u = 0.0437 o


385

Figure 3. Wigner function corresponding to Fig. 26.

As a method to control wavepackets, alternative to the use of ultrashort pulses, I would like to propose use of frequency-modulated light. Since it is very difficult to obtain a well-controlled pulse shape without any chirp, it is even easier to control the frequency by the electro-optic effect and also by appropriate superposition of several continuous-wave tunable laser light beams. I . Physics Today 430) (1990), Special Issue on Dynamics of Molecular Systems.

2. D. F. Walls, Nature 306, 141 (1983).

3. R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).

4. 5. 6. 7.

M. C. Teich and B. E. Saleh, Quantum Opt. I, 153 (1989). L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett 57, 2520 (1986). W. Schleich and I. A. Wheeler, Nafure 326, 574 (1987). J. Gea-Banacloche, R. R. Schilcher, and M. S. Zubairy, Phys. Rev. A 38, 3514 (1988).

8. J. Janzky and Y. Yushin, Phys. Rev. A 36, 1288 (1987). 9. C. K. Hong and L. Mandel, Phys. Rev. A 32,974 (1985). 10. S. Reynaud, C. Fabre, and E. Giacobino, J. Opf. Soc. Am. B 4, 1520 (1987). 11. J. Janzky and Y. Yushin, Optics Commun. 59, 151 (1986). 12. R. Graham, J. Mod. Opt. 43, 873 (1987). 13. H.-Y. Fan and H. R. Zaidi, Phys. Rev. A 37,2985 (1988). 14. 1. Janzky and Y. Yushin, Phys. Rev. A 39,5445 (1989). 15. H.-Y. Fan and H. R. Zaidi, Phys. Rev. A 39, 5447 (1989). 16. Xi Ma and W . Rhodes, Phys. Rev. A 39, 1941 (1989). 17. J. Janzky. T. Kobayashi, and An. V. Vinogradov, Optics Commun. 76, 30 (1990). 18. W. T. Pollard, S.-Y. Lee, and R. A. Mathies, J. Chem. Phys. 92,4012 (1990). 19. J. Janzky and An. V. Vinogradov, Phys. Rev. Left. 64,2771 (1990). 20. T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, Appl. Phys. B 47, 107 (1988).

386

GENERAL DISCUSSION

S. R. Jain: When Prof. Rice talks about optimal control schemes, his Lagrange function follows a time-reversed Schrxinger equation. Is it assumed in the variational deduction that the Hamiltonian is time reversal invariant; that is, is it always diagonalizable by orthogonal transformations? S. A. Rice: Yes, there is time reversal invariance. S. Mukamel: I would like to make a comment regarding interference effects in quantum and classical nonlinear response functions (1, 21. Nonlinear optical measurements may be interpreted by expanding the polarization P in powers of the incoming electric field E. To nth order we have

Quantum mechanically, the nonlinear response function S(")(t,r,, is given by a combination of 2" terms representing all possible "left" and "right" actions of the various commutators divided by h". The various terms are given by an (n+ 1)-order correlation function of the dipole operator V (with different time arguments). These terms interfere, and this gives rise to many interesting effects, such as new resonances. The (l/tz)"factor indicates that individual correlation functions do not have an obvious classical limit; however the observables which are given by proper combinations of correlation functions are analytic functions of ti. The quantum linear response function is given by

. . . , 71)

The classical linear response function can be written using the fluctuation-dissipation theorem as a single term,

Unlike the quantum response (2), which contains an interference of two Liouville space paths, the classical expression (3) contains no inter-


387

ference and may be directly computed using classical trajectories that sample the initial density matrix. However, classical nonlinear response functions do involve interference between two or more terms. The second-order quantum and classical response functions are given by

and x, are the coordinates and momenta. The matrix where VJ = W/&,

M , which relates small deviations 6xj to 6Xk at different times, is known as the stability matrix. Its elements are defined as

The stability matrix carries the necessary information related to the vicinity of the trajectory and provides an efficient numerical procedure for computing the response function. It plays an important role in the field of classical chaos; the sign of its eigenvalues (related to the Lyapunov exponents) controls the chaotic nature of the system. Interference effects in classical response functions have a different origin than their quantum counterparts. For each initial phase-space point we need to launch two trajectories with very close initial conditions. [For S@) we need n trajectories.] The nonlinear response is obtained by adding the contributions of these trajectories and letting them interfere. It is interesting that the linear response does not depend on M.Nonlinear spectroscopy should therefore be a much more sensitive probe for classical chaos than linear spectroscopy. 1.

S. Mukamel, Principles of Nonlinear Optical Spectroscopy,Oxford University Press,

New York. 1995. 2. S. Mukamel, V. Khidekel, and V. Chemyak, Phys. Rev. E 53, 1 (1996).

388

GENERAL DISCUSSION

S. A. Rice: I would appreciate hearing the comments of Prof. Mukamel concerning the approach to the classical limit from the point of view of his formalism. Is that approach analytic? S. Mukamel: The nth-order response function is given by a combination of 2" correlation functions divided by A". In the classical limit, the combination of correlation functions (but not each one individually) behaves as tiR and ti cancels. It is then possible to expand the response function analytically in h. As long as we expand a physical quantity (i.e., a response function) rather than a correlation function, the result will be analytic. E. Pollak: I am not sure I understand the remark by Prof. Mukamel that chaos does not express itself in the linear response regime. The fluctuations of the exact quantum response about the classical will be described in terms of the Gutzwiller summation and will thus reflect the chaos. S. Mukamel: While there are some signatures of chaos in the linear response, my point is that the nonlinear response carries much more direct and sensitive information. The reason is that the stability matrix enters the nonlinear response directly, reflecting interference of initially close trajectories. Such interference is absent in the linear response. P. W. Brumer: Prof. Mukamel has emphasized that in examining objects as they approach the classical limit one sees essential singularity behavior or not depending upon the object. I would like to add to this remark by indicating that we have recently successfully completed a program designed to demonstrate the emergence of classical mechanics from quantum mechanics for dynamics, be it integrable or chaotic. This approach is an extensive generalization of our earlier papers [C. Jaffe and P. Brumer, J. Chem. Phys. 82, 2330 (1985); C. Jaffe, S. Kanfer, and P. Brumer, Phys. Rev. Len. 54, 8 (19931, where correspondence requires establishing a relationship, as h 0, of the eigenvalues and eigenfunctions of the quantum Liouville operator with the eigenvalues and eigenfunctions of the classical Liouville operator. In addition to producing a completely general approach to classical-quantumcorrespondence,our approach [J. Wilkie and P. Brumer, J. Chem. Phys., submitted] shows that the classical limit emerges by the elimination of essential singularities.

-

INTRAMOLECULAR DYNAMICS

SOLVENT DYNAMICS AND RRKM THEORY OF CLUSTERS R. A. MARCUS Noyes Laboratory of Chemical Physics California Institute of Technology Pasadena, Calgomia

CONTENTS 1. Introduction 11. Microcanonical Solvent Dynamics Modified RRKM Theory A. One-Coordinate Type Treatment B. Vibrational Assistance Treatment 111. Discussion References

I. INTRODUCTION In this chapter we consider the problem of reaction rates in clusters (microcanonical) modified by solvent dynamics. The field is a relatively new one, both experimentally and theoretically, and stems from recent work on welldefined clusters [l, 21. We first review some theories and results for the solvent dynamics of reactions in constant-temperaturecondensed-phase systems and then describe two papers from our recent work on the adaptation to microcanonical systems. In the process we comment on a number of questions in the constant-temperature studies and consider the relation of those studies to corresponding future studies of clusters. A brief review for constant-temperaturecondensed-phase systems is given in Fig. 1. The field of solvent dynamics has grown so extensively that it is

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXrh Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

39 1

392

R. A. MARCUS Solvent Dynamics & Chendcal Rcaciians

Memory

Gmte-Hynes 1980

TrOe

Hochstrasser

I

BT.

Langer

Zusman

1980

Wolynes 1983 Hyncs 1986 Ripsdormer 1987

Refs.

16

McCanunon 1983

Ligand AgmonHopfield 1983

BarrierlessBT. Flemingcral

ET. BerezhLowsLi Sumi1990 1990 Marcus E k f l U N C . 1983 1986

Pollak 1986

Sumi jwm+ 1991

&O-

Quantum

k(x)

Barbara, Weaver 1991-2

51)

Y~~hlhara, Rasaiah 1994

Yoshihara

1991

kwlxm Barbara, Weaver. Yoshihara, Simon

Figure 1. Brief survey of some developments in the solvent dynamics field.

difficult for recent reviews [3-51 to keep pace. Only some representative articles are cited below. The classic paper is solvent dynamics, due to Kramers, appeared in 1940 [6]. Subsequently, apart from some isolated works in the physics literature, such as Langer’s generalization to many coordinates in 1969 [7], there was relatively little follow-up, and particularly little in the chemical literature, until around 1980. The subsequent developments can be classified as being largely of three types: (1) those, like Kramers’s, that are one-coordinate treatments; (2) their many-coordinate extensions; and (3) treatments having one slow coordinate, the remainder being fast coordinates, appropriately averaged.

Kramers’s equation, it may be recalled, is [6]

where P( p, q, t ) is the probability density in phase space, q the reaction coor-

SOLVENT DYNAMICS AND RRKM THEORY OF CLUSTERS

393

dinate, p its conjugate momentum, U(q) the potential energy function, and ( the frictional coefficient. Kramers’s theory for reactions in liquids, which includes both the inertial and the overdamped limits, was extended by Grote and Hynes [8]to include a memory effect, namely, a frequency-dependentfriction f(w). A number of other one-dimensionalextensionshave also been made [9]. Some of the ideas were tested experimentally by many investigators 4, 5 , 10, 11. Another pioneering one-coordinate extension of Kramers’s analysis was made for electron transfer reactions by Zusman [ 121 and Alexandrov [121(1980), and further illuminating developments were made by a number of researchers [13-15]. When there are no relevant “vibrationally assisting” coordinates (examples are mentioned later), a one-coordinate approach suffices. The second approach, a multidimensional one, was given by Langer [7]. Other multidimensional developments were many [16-18]. McCammon [ 171 discussed a variational approach (1983) to seek the best path for crossing the transition-statehypersurface in multidimensionalspace and discussed the topic of saddle-point avoidance. Further developments have been made using variational transition state theory, for example, by Pollak [ 181. The third, and perhaps now the currently major, approach for treating the experimental data on electron transfer reactions assumes that there is one slow coordinate, with the remaining coordinates being fast. The equation used, or coupled with one for the back reaction or further extended by making D a D(t), is

ap

---

at

(

-D-ax a -+-ap ax k a ~””) ax -k(X)P(X)

(1.2)

and is obtained by an averaging over or adiabatic eEimination of the fast (vibrational) variables. Here, P(X)is the probability density along the slow coordinate X, D is a diffusion constant in this X space, G(X)is the free energy to reach any X from the equilibrium value of X,X = 0 for the reactant, and k(X)is a rate constant at any given X for crossing the barrier. The motion along X is, as seen in (1.2), treated as overdamped. Using Eq. (1.2) Agmon and Hopfield (Ref. 19; cf. Ref. 23) treated the dissociation of a ligand from a heme in a protein (1983), and Sumi and Marcus [20] treated electron transfer reactions (1986). For electron transfers the previous (one-coordinate) treatments neglected the very common case that solute vibrations play a major role (vibrational assistance) in the transfer when there are significant changes in vibrational geometry, for example, in bond lengths. The use of Eq. (1.2) removes that defect. Beginning around 1990 Berezhovskii, Zitserman, and co-workers introduced a number of treatments of the type based on Eq. (1.2) [21].

394

R. A. MARCUS

The various treatments in the literature based on Q. (1.2) have differed primarily in two respects: (1) the expression for the rate constant k ( X ) in Eq. (1.2) is specific for the process and so may differ from process to process and (2) the technique for solving Eq. (1.2) differs. For example, Agmon and Hopfield [19] solved Eq. (1.2) numerically, as did Nadler and Marcus [22], Agmon and co-workers [23], and others. Sumi and Marcus [201 introduced, instead, a decoupling approximation, which depended on there being a difference in time scales for the reaction and for the solvent fluctuations. (An excellent summary and an extension of their work is given in Rasaiah and Zhu [24].) Berezkhovskii et al. introduced an approximation that divided the X space into two parts, separated by a value of X at which the escape time 7esc(X) equals the solvent relaxation time T ~ ~ ~[21]. ( X Fleming ) and co-workers treated barrierless electronic energy transfer (1983) [25] and the corresponding barrierless electron transfer (1990) [26]. For electron transfers various extensions and experimental tests of the Sumi-Marcus treatment have been introduced. They include a quantum version for k ( X ) (of particular importance in the “inverted region”) [24, 27, 281, inclusion of forward and reverse reactions [24], and the use of a time-dependent D(t) to allow for several relaxation times [24,27,29]. Other experimental tests or extensions have also been introduced [30]. In experimental tests, effects such as the dynamic Stokes shift of fluorescence in these polar systems have been especially invaluable in providing necessary data for relaxation in these electron transfer systems [3 11. Numerical solutionsby Yoshihara and coworkers [29] and by Barbara and co-workers [27], permitting the inclusion of a D(t), have been important. Use of a D(t) in general had been made in the work of Hynes (141 (cf. Ref. 32). Analyses of solvent dynamics, accompanied by computer simulations, of Chandler and co-workers [33] and of Maroncelli, Fleming, and their co-workers [34] have provided further insight. Inertial effects on solvent dynamics, using a formalism of Mukamel and co-workers [353, have been incorporated by Barbara and co-workers [36]. Further relevant solvent dynamics theoretical analyses [37] and measurements using ultrafast laser spectroscopy have also been described [38]. Earlier theoretical studies of dynamical spectral shifts of solutes had been made by Bakhshiev, Mazurenko, and their co-workers [39] and references cited therein. In the case of electron transfer reactions, besides data on the dynamic Stokes shift and ultrafast laser spectroscopy, data on the dielectric dispersion E ( W ) of the solvent can provide invaluable supplementary information. In the case of other reactions, such as isomerizations, it appears that the analogous data, for example, on a solvent viscosity frequency dependence q ( w ) , or on a dynamic Stokes fluorescence shift may presently be absent. Its absence probably provides one main source of the differences in opinion [5, 40-433 on solvent dynamics treatments of isomerization.


395

In the next section we summarize two treatments of microcanonical systems [2, 441, one of the steady-state Kramers’ one-coordinate type and one including vibrational assistance. An earlier approach to the problem was given by Troe [45].

II. MICROCANONICAL SOLVENT DYNAMICS MODIFIED RRKM THEORY A. One-Coordinate Qpe Treatment The Kramers-type equation corresponding to Eq.( 1 . 1 ) and adapted in Ref. 2 to the microcanonical case for a system with coordinate q and its conjugate momentum p is

where S,(q) is the local vibrational entropy at q. T,[=l/(aS,/aE,)] is the local microcanonical vibrational temperature and is a function of q, and P is the probability distribution in (q, p) space. The adaptation was such as to pennit the equilibrium microcanonical phase-space distribution function to satisfy the equation identically [2]. In the steady-state approximation, solving Eq. (2.1) for the rate constant at the given total energy E yields [ 2 ]

where a t denotes T,d2S,/dq2 at q = qt and is an effective barrier frequency. The ~ R R K Mis the RRKM (microcanonical)rate constant at the given energy E.

B. Vibrational Assistance Treatment A second procedure, based on the vibrational assistance model for calculating the solvent-dynamics-modified rate, is given in Ref. 44. The reaction4iffusion equation, adapted from &. (1.2), is, for the case where the back reaction is neglected, given by (2.3).The more complete treatment, where the back reaction (recrossings) is included, is given in Ref. 44:

R. A. MARCUS

396

The adaptation is such as to pennit the equilibrium microcanonical distribution for the slow coordinate X to be a solution (2.3) when k(X) = 0. The S,(X) in EQ. (2.3) is the vibrational entropy change needed to reach X from

x=o:

(2.4)

[the constant does not affect the aS,/aX appearing in Eq. (2.3)J; p is the density (i.e., the number per unit energy) of quantum states of the reactant, and p(X) is that density at X per unit X. It is given by [44] 2

Px

=

[2(E - U(X)- Em)]”*

(2.5)

rn

where Px is the momentum conjugate to X. Throughout we use a massweighted unit for X for notational brevity. Here, Em is the energy of the mth quantum state of the reactant for all coordinates but X and U(X)is the potential energy at X at the local equilibrium value of the remaining coordinates. It is readily verified [44]that (2.5) satisfies jp(X)dX = 1, where the integral is over the reactant’s region of X space. We give in Fig. 2 a schematic plot showing contours on which a vibrational entropy S , ( q , X ) is constant, q is a fast coordinate, and the line C is the transition state in the (X, q ) space. This S , ( q , X ) can be defined as in an equation similar to (2.4) in terms of a k g In p(q,X), p ( q , X ) being the local density of study, that is, the number per unit energy per unit q and per unit X. While this plot is not used in the derivation, it can be visually helpful. The k(X)in Eq. (2.3) is given by an RRKM-like expression for the-given X [44]:

where N ( X ) is the local number of quantum states (per unit X) for the given X, along the transition state, with energy equal to or less than E and is given by (2.7), and x is a coordinate that is a projection (defined in Ref. 44) on


397

Product Region

Slow coordinate X Reactant Region

-

I

r

Figure 2. Schematic entropic surface as a function of a slow coordinate X and a fast coordinate q. Here, S is a saddle point, the line C is the transition state in this (X,q ) space, and X, lies at its intersection with the X axis.

the transition state space. We have

Here, En is the energy of the nth quantum state of the transition state for all coordinates but X and Q,the reaction coordinate, V , ( X ) is the potential energy in the transition state at the point X and at the position of minimum potential energy with respect to all other coordinates in the transition state, and px is the momentum conjugate to n. From the expressions for k(X) and N ( X ) it can be shown that the usual RRKM expression obtains when the diffusion along X is rapid, that is, when P(X)has its equilibrium value Peq(X)[=p ( X ) / p ] : We have

398

R. A. MARCUS

and so from (2.6) and (2.7)

where the limits on the integral in (2.9) are at the end-points p x = 0. [The system is now near the X where U,(X)is a minimum and so x has become a vibration.] The second equality in (2.9) arises because the integral there can be written as an integral over one cycle of the x motion and so equals $ p xdx/h. The latter is a constant, which we write as I + Semiclassically, 1 is an integer for any x quantum state. We now have in (2.9) what can be shown [44]to be a sum over the quantum states (for all coordinates but the reaction coordinate) with energy equal to or less than E. The sum is denoted by N ( E ) , and one obtains the usual RRKh4 expression. It remains now to solve Eq. (2.3). Here, there are various approaches, depending on the conditions. When a non-steady-state solution is required, one can introduce the decoupling approximation of Sumi and Marcus, if there is the difference in time scales mentioned earlier. Or one can integrate Eq. (2.3) numerically. For the steady-state approximation either Ekj. (2.3) can again be solved numerically or some additional analytical approximation can be introduced. For example, one introduced elsewhere [44]is to consider the case that most of the reacting systems cross the transition state in some narrow window (XI,XI f +A), narrow compared with the X region of the reactant [e.g., the interval (O,X,) in Fig. 21. In that case the k(X) can be replaced by a delta function, R(XI)A6(X - XI). Equation (2.3) is then readily integrated and the point XI is obtained as the X that maximizes the rate expression. The A is obtained from the width of the distribution of rates in that system [44]. A simple expression for the rate constant results:

i.

where X,,, is the X that maximizes krate(X)and hence minimizes the reaction time l/krate(X). That time appears in the final expression in Ref. 44 as (2.1 1)


399

where the diffusion-controlled and activation-controlled rate constants are given by

and (2.13) [When Xma,occurs at the end-point X,, there is a minor change in procedure, and the system now crosses the transition state in the interval (X,,X,- A), Equations (2.11H2.13) are again obtained.] The result in (2.11) that the reaction time is the sum of two other times is fairly common in the general reaction4ffusion literature, in which a steadystate approximation is used and there is a diffusion toward a sink followed B -+ C, with by reaction at that sink. For example, in the scheme A forward and reverse rate constants kl and k:! for the first step (equilibrium constant K = k,/k2) and rate constant k j for the last step, a steady-state l/kl), which has the same funcapproximation for B yields 1/krate(l/k3K)+( tional form as (2.11). The more complete expression, which allows for the back reaction (recrossings), has a slightly more complicated structure [44].

111. DISCUSSION At present the body of data on reactions in clusters is insufficient to test the above two microcanonical approaches. For electron transfers in solution it seems clear that the vibrational assistance approach, stemming from Eq. (1.2), with its extensions mentioned earlier, is the one that has been the most successful [27-301. For slow isomerizations Sumi and Asano have pointed out that an analysis based on Eq. (1.2) was again needed [40]. An approach based on FQ. (1.1) or on its extension to include a frequency-dependent friction, they noted, led to unphysical correlation times [40]. In investigations of fast isomerizations the most commonly studied system has been the photoexcited trans-stilbene [5,41-43,46]. Difficulties encountered by a one-coordinate treatment for that system have been reported [4, 81. Indeed, coherence results for photoexcited cis-stilbene have shown a coupling of a phenyl torsional mode to the torsional mode about the C=C bond [42, 471. Other investigatorshave used systems that are more apt to represent a one-

400

R. A. MARCUS

coordinate-like behavior, for example, the isomerization of “stiff-stilbene,” where the phenyl groups are tethered further to the respective carbon atoms of the double bond [48], and the conformational change of binaphthyl [lo]. In the last study a microviscosity was introduced into the Kramers formula to obtain the friction coefficient instead of using the bulk viscosity of the solvent, and was inferred from rotational relaxation or translational diffusion coefficient (f = ksT/D in mass-weighted units). In that case the expected Kramers function of q (via the Stokes-Einstein equation) was obeyed for the one-coordinate model. Use of the bulk viscosity led, instead, to an observed fractional dependence on q [ 101. A microfriction inferred from a rotational or translational relaxation time has been similarly successfully used by various other investigators [41, 43, 49, 501. An approximate molecular expression (“ballpark”) for a t for a cluster was given in Ref. 2. Many questions in the analysis of solvent dynamics effects for isomerizations in solution have arisen, such as (1) when is a frequency-dependent friction needed; (2) when does a change of solvent, of pressure, or of temperature change the barrier height (i.e., the threshold energy), and (3) when is the vibrational assistance model needed, instead of one based on Q. (I. 1) or its extensions? In the case of electron transfers in solution there appears to be a greater cohesiveness of views, and the need for vibrational assistance is well established for reactions accompanied by vibrational changes (e.g., changes in bond lengths). A detailed analysis of the experiments could be made because of the existence of independent data, which include X-ray crystallography, EXAFS, resonance Raman spectra, time-dependent fluorescence Stokes shifts, among others. One may inquire as to what this experience with solutions suggests for the study of reactions in clusters. In the case of electron transfers supplementary information, such as time-dependent fluorescence Stokes shift in clusters, would again be helpful. Equation (2.3) can be modified to include a D(t), as in the isothermal case, if needed from the results of such data. For isomerizations, also, it would be useful to have, for solutions or clusters, detailed analogous data such as the above Stokes shift. However, because of the low intensity of such a fluorescence in this case, such data appear to be absent or scarce. The questions that have arisen in regard to isomerizations in solution also apply to isomerizations in clusters. One of these questions, which has now been addressed, is the threshold energy and its variation with size of the cluster. Data on the threshold energies were obtained in the microcanonical study by Heikal et al. [l]. New questions, however, also arise for clusters: How rapidly is the energy transferred between the solute and the solvent molecules in the cluster? Outside the threshold region a reduction in k,,,(E) with

r,


401

increasing cluster size was observed experimentally 11. A principal question is whether this reduction is due to the increase, with increased cluster size, in the number of coordinates which can share the excess energy or whether it is due to increased frictional effects by the solvent molecules, or both. If the solvent molecules outside the first solvent layer in the cluster have little effect on the frictional forces, then this question can be addressed by comparing the reaction rates with those clusters that contain more than one solvent layer. Again, if instead of using stilbene one replaces one of the phenyl groups attached to the C=C double bond, the solvent viscosity effects should be less and the energy-sharing role of the extra coordinates more readily discerned. Yet again, microcanonical studies with molecules deliberately chosen, as in the solution case, to favor a one-coordinate approach would also be of particular interest. It is clear that the study of solvent dynamics in solution has proved to be a rich field. A number of questions remain to be resolved, and the study of clusters can open new avenues.

Acknowledgments It is a pleasure to acknowledge the support of this research by the National Science Foundation and the Office of Naval Research.

References 1. A. A. Heikal, S. H. Chong, J. S . Baskin, and A. H. Zewail, Chem. Phys. Left. 242, 380

(1995). 2. R. A. Marcus, Chem. Phys. Lett. 244, 10 (1995). 3. P. Hbggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62,251 (1990). 4. Various articles in Activafed Barrier Crossing, G . R. Fleming and P. Hinggi, Eds., World Scientific, Singapore, 1993. 5. J. Schroeder and J. Troe, Ann. Rev. Phys. Chem. 38, 163 (1987); Ref. 4, p. 206. 6. H. A. Kramers, Physica 7,284 (1940). 7. J. S. Langer, Ann. Phys. (NZ)54,528 (1969). 8. R. F. Grote and J. T.Hynes, J. Chem. Phys. 73, 2715 (1980); P. Hiinggi and F. Mojtabai, Phys. Rev. A 26, 1168 (1982); B. J. Gertner, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 90,3537 (1989); R. Mohrschladt, J. Schroeder, D. Schwartzer, J. Troe, and P. Vohringer, J. Chem. Phys. 101,7566 (1994). 9. H. C. Brinkman, Physica 22,29, 149 (1956); C. Blomberg, Physica MA, 49 (1977); B. Carmeli and A. Nitzan, Phys. Rev. A 29, 1481 (1984). 10. R. M. Bowman, K. B. Eisenthal, and D. P. Millar, J. Chem. Phys. 89,762 (1988); R.M. Bowman and K. B. Eisenthal, Chem. Phys. Left.155,99 (1989). 11. S. P. Velsko and G. R. Fleming, J. Chern. Phys. 76, 3553 (1982); S. P. Velska, D. H. Waldeck, and G. R. Fleming, J. Chem. Phys. 78, 249 (1983); G. Rothenberger, D. K. Negus, and R. M. Hochstrasser, J. Chem. Phys. 79,5360 (1983); R. Giniger, D. Huppert, and E. M. Kosower, Chem. Phys. Left. 118,240 (1985).

402

R. A. MARCUS

12. L. D. Zusman, Chem. Phys. 49,295 (1980); ibid., 80,29 (1983); 1. V. Alexandrov Chem. Phys. 51, 449 (1980). 13. D. F. Calef and P. G. Wolynes, J. Phys. Chem. 87,3387 (1983); J. Chem. Phys. 78,470 (1983); A. B. Helman, Chem. Phys. 79, 235 (1983). 14. J. T. Hynes, J. Phys. Chem. 90,3701 (1986); C . van der Zwan and J. T. Hynes, J. Phys. Chem. 89,4181 (1985); C . van der Zwan and J. T. Hynes,J. Phys. Chem. 89,4181 (1985); A. Garg, J. N. Onuchic, and V. Ambegeokar, J. Chem. Phys. 83,4491 (1985). 15. 1. Rips and 3. Jortner, J. Chem. Phys. 87, 2090, 6513 (1987); ibid., 88, 818 (1988). 16. R. Landauer and J. A. Swanson, Phys. Rev. 121, 1668 (1961); H. A. Weidenmuller and J . 3 . Zhang, J. Srur. Phys. 34, 191 (1984); A. Nitzan and Z. Schuss, Ref. 4, p. 42; M. M.

Klosek-Dygas, B. M. Hoffman, B. J. Matkowsky, A. Nitzan, M. A. Ratner, and Z. Schuss, J . Chem. Phys. 90, 1141 (1989); E. Pollak, J . Chem. Phys. 85, 865 (1986); H. Grabert, Phys. Rev. Lett. 61,1683 (1988); E. Pollak, H. Grabert, and P. Hhggi, J. Chem. Phys. 91, 4073 (1989); T. Fonseca, H. J. Kim, and J. T. Hynes, J. Photochem. Phorobiol. A: Chem. 82, 67 (1 994); R. Graham, J. Sfat. Phys. 60, 675 (1 990); R. F. Grote and J. T. Hynes, J. Chem. Phys. 74,4465 (1981). 17. M. Berkowitz, J. D. Morgan, and J. A. McCammon, J. Chem. Phys. 79,5563 (1983); S. H. Northrup and J. A. McCammon, J. Chem. Phys. 78,987 (1983). 18. S. Tucker and E. Pollak, J. Star. Phys. 66, 975 (1992); A. M. Frishman, A. M. Berezhkovskii, and E. Pollak, Phys. Rev. E 49, 1216 (1994); A. M. Berezhkovskii, A. M. Frishman, and E. Pollak, J. Chem. Phys. 101,4778 (1994); E. Pollak, Ref. 4, p. 5. 19. N. Agmon and J. J. Hopfield, J. Chem. Phys. 78,6947; ibid., 80,592 (1984) (E); 79,2042

( 1983). 20. H. Sumi and R. A. Marcus, J. Chem. Phys. 84,4894 (1986). 21. A. M.Berezhkovskii and V. Yu. Zitserman, Physica A 166, 585 (1990); ibid., 187, 519 (1992); Chem. Phys. 157, 141 (1991); A. M. Berezhkovskii, S. A. Dudko. and V. U. Zitserman, Chem. Phys. 187, 275 (1994). 22. W. Nadler and R. A. Marcus, J. Chem. Phys. 86, 3096 (1987). 23. N. Agmon and R. Kosloff, J. Phys. Chem. 91, 1988 (1987); S. Rabinovich and N. Agmon, Chem. Phys. Lett. 182, 336 (1991). 24. J. C. Rasaiah and J. J. Zhu, J. Chem. Phys. 98, 1213, (1993); J. J. Zhu and J. C. Rasaiah, J. Chem. Phys. 96, 1435 (1992); ibid., 95, 3325 (1991). M. Morillo and R. J. Cukier, J. Chem. Phys. 89, 6736 (1988). 25. B. Bagchi, G. R. Fleming, and D. W. Oxtoby, J. Chem. Phys. 78,7375 (1983). 26. B. Bagchi and G. R. Fleming, J. Phys. Chem. 94, 9 (1990). 27. G. C. Walker, E. Akesson, A. E. Johnson, N. E. Levinger, and P. F. Barbara, J. Phys. Chem. 96, 3728 (1992); G. C. Walker, P. F. Barbara, S. K. Doom, Y. Dong, and J. T. Hupp, J. Phys. Chem. 95, 5712 (1991); E. Akesson, G. C. Walker, and P. F. Barbara, J. Chem. Phys. 95,4188 (1991). 28. D. K. Phelps and M. J. Weaver, J. Phys. Chem. 96,7187 (1992). 29. K. Yoshihara, K. Tominaga, and Y. Nagasawa, Bull. Chem. Soc. Jn. 68, 696 (1995); Y. Nagasawa, A. P. Yartsev, K. Tominaga, P. B. Bisht, A. E. Johnson, and K. Yoshihara, J. Phys. Chem. 99, 653 (1995); Y. Nagasawa, A. P. Yartsev, K. Tominaga, A. E. Johnson, and K. Yoshihara, J. Chem. Phys. 101,5717 (1994); K. Yoshihara, Y. Nagasawa, and A. Yartsev, J. Mol. Liq. 65/66, 59 (1995). 30. J. D. Simon and R. Doolen, J. Am. Chem. Soc. 114,4861 (1992); M. V. Basilevsky and

SOLVENT DYNAMICS AND RRKM THEORY O F CLUSTERS

403

G. E. Chudinov, Z. Phys. Chem. 171, 1 (1991); S. G. Su and J. D. Simon, J. Chem. Phys. 89,908 (1988).

31. M. Cho, S.J. Rosenthal, N. F. Scherer, L. D. Ziegler, and G. R. Fleming, J. Chem. Phys. 96, 5033 (1993); G. van der Zwan and J. T. Hynes, J . Phys. Chem. 89,4181 (1985); R. Jiminez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, Narure 369, 471 (1994); S. J. Rosenthal, X. Xie, M. Du, and G. R. Fleming, J. Chem. Phys. 95, 4715 (1991). 32. T. Fonseca, J. Chem. Phys. 91, 2869 (1989). 33. D.Chandler, J. Stat. Phys. 42,49 (1986); D. C. Chandler, J. Chem. Phys. 68,2959 (1978); R. A, Kuharski, 1. S. Bader, D. Chandler, M. Sprik, M. L. Klein, and R. W. Impey, J. Chem. Phys. 89,3248 (1988); I. S. Bader and D. Chandler, Chem. Phys. Lett. 157,501 (1989); J. S. Bader, R. A. Kuharski, and D. Chandler, J. Chem. Phys. 93,230 (1990). 34. M. Maroncelli, J. Maclnnis, and G. R. Fleming, Science 243, 1674 (1989); M. Maroncelli, J. Chem. Phys. 94,2084 (1991); S. Roy and B. Bagchi, J. Chem. Phys. 99, 1310 (1993); S. J. Rosenthal, R. Jiminez, G. R. Fleming, P. V. Kumar. and M. Maroncelli, J. Mol. Liq. 60, 25 (1994); M. Maroncelli and G. R. Fleming, J. Chem. Phys. 86, 6221 (1987); J. Gardecki, M. L. Horng, A. Papazyan, and M.Maroncelli, J. Mol. Liq. 65/66,49 (1995). 35. Y. J. Yan, M. Sparpaglione, and S. Mukamel, J. Phys. Chem. 92,4842 (1988). 36. K. Tominga, D. A. V. Kliner. A. E. Johnson, N. E. Levinger, and P. F. Barbara, J. Chem. Phys. 98, 1228 (1993). 37. Y. J. Yan and S. Mukamel, Phys. Rev. A 41,6485 (1990); L. E. Fried and S. Mukamel, J. Chem. Phys. 93,932 (1990). 38. Y. J. Chang and E. W. Castner, Jr., J. Chem. Phys. 99,7289 (1993); and refs cited therein; 39.

M.Cho et al., Ref. 31. N. G. Baklshiev, Opf. Specfr. 16,446 (1969); Yu. T.Mazurenko and V. S . Udaltsov, Opt.

Spect,: 44,417 (1978). 40. H. Sumi and T. Asano, Chem. Phys. Left. 240, 125, 1995; J. Chem. Phys. 102, 9565 (1995); H. Sumi, J. Phys. Chem. 95,3334 (1991); T. Asano, H. Furuta, and H. Sumi, J. Am. Chem. SOC. 116, 5545 (1994); H. Sumi, J. Mol. Liq. 65/66, 65 (1995). 41. Y.-P. Sun, J. Saltiel, N. S. Park, E. A. Hoburg, and D. H. Waldeck, J. Phys. Chem. 95, 10336 (1991); J. Saltiel and Y.-P. Sun, J. Phys. Chem. 93,6246, 8310 (1989). 42. R. J. Sension, S. T. Repinec, A. Z. Szarka, and R. M. Hochstrasser, J. Chem. Phys. 98, 6291 (1993); A. Z. Szarka, N. Pugliano, D. K. Palit, and R. M. Hochstrasser, Chem. Phys. Lett. 240,25 (1995).

43. D. M. Zeglinski and D. H. Waldeck, J. Phys. Chem. 92, 692 (1988); D. H. Waldeck, Chem. Rev. 91,415 (1991). 44. R. A. Marcus, J. Chem. Phys., 105,5446 (1996). 45. I. Schroeder and I. Troe, Ref. 4, p. 206, see refs. 1, 2, and 5 in that article. 46. D. Raftery, R. J. Sension, and R. M. Hochstrasser, Ref. 4, p. 163. 47. S. Pedersen, L. Banares, and A.

H.Zewail, J. Chern. Phys. 97,8801 (1992).

48. F.E. Doany, E. J. Heilweil, E. J. Moore, and R. M. Hochstrasser, J. Chem. Phys. 80,201 (1984). 49. M. Lee, A. J. Bain, P. J. McCarthy, C. H. Han, J. N. Haseltine, A. B. Smith 111, and R. M. Hochstrasser, J. Chem. Phys. 85, 4341 (1986). 50. S. H. Courtney, S. K. Kim, S. Cononica, and G. R. Fleming, J. Chem. Phys. 78, 249 (1983).

404

R. A. MARCUS

DISCUSSION ON THE REPORT BY R. A. MARCUS Chairman: E. Pollak J. Tree: In our own work measuring energy-specific excited stilbene lifetimes in stilbene-hexane clusters, we found for larger clusters that there is no isomerization. We interpreted this as evidence for “boiling off’ of the cluster partners, removing energy from stilbene and thus suppressing the isomerization. This is an alternative to increasing the effective size of the reacting molecule. Which interpretation do you favor? We also clearly showed that the barrier for isomerization is decreased by the first cluster partners, as evidenced in our studies of the thermal reaction between gas and liquid phases. R. A. Marcus: I understand that the observed effect of a decrease in rate constant k(E) with increasing cluster size at a fixed E was not due to a boiling off of the hexane molecules (and hence to a reduced E), but I refer to my colleague, Ahmed Zewail, for an answer to your question. A. H. Zewail: My answer to Prof. Troe is that, in our experiment, already the cluster with one solvent shows the shift in Eo. As for the boiling off of solvent molecules in larger clusters this is a nontrivial problem that we have considered in our paper. Based on the analysis of the translational energy and the kinetics, we concluded that the exponential decays (rates) are determined by the isomerization [see Chem. Phys. Lett. 242, 380 (199511. In any event, only one solvent molecule (at most) can be evaporated for the available energy studied experimentally; recall that the binding energy of hexane is relatively large. H. Hamaguchi: I would like to comment on the stilbene photoisomerization in solution. We recently found an interesting linear relationship between the dephasing time of the central double-bond stretch vibration of S1 trans-stilbene, which was measured by time-resolved Raman spectroscopy, and the rate of isomerization in various solutions. Although the linear relationship has not been established in an extensive range of the isomerization rate, I can point out that the vibrational dephasing time measured by Raman spectroscopy is an important source of information on the solvent-inducedvibrational dynamics relevant to the reaction dynamics in solution. R. A. Marcus: It is good to hear about that; certainly one needs all types of information to be incorporated. D. M. Neumark Prof. Marcus, your theoretical treatment was motivated by experimental studies of isomerization in clusters with


405

very few solvent molecules (n = I , 2). How appropriate is your theory to these small clusters? In particular, can one discuss concepts such as viscosity and solvent friction in small clusters?

R. A. Marcus: The experiments involved hexane rather than argon and went from n = 0 to n = 5 hexane molecules. In Chem. Phys. Lett.

244, 10 (1993, I considered two limiting cases for the energy sharing of the trans-stilbene with the modes of the solvent molecules. Experiments comparing results for one and two shells of solvent molecules in the cluster may provide information on which limiting model might be the more appropriate. Previous experiments on trans-stilbene in solvents suggest a rapid energy sharing. In the above article I also gave a rough expression relating the “viscosity” or friction for the cluster to molecular properties, but I am sure it can be improved upon.

B. Hess: The structure and function of solvent components constituting the active site of enzymic reactions represent an exciting puzzle in protein chemistry. In an active pocket, a restricted cluster number is given by the small set of amino acid residues, mostly hydrophobic, in the nearest neighborhood to the ligand and its reaction partners. As Prof. Marcus pointed out, Frauenfelder and his collaborators studied experimentally the effect of solvent viscosity on protein dynamics. In case of CO myoglobin they could show that over a wide range in viscosity the transition rates in heme-CO are inversely proportional to the solvent viscosity and can consequently be described by the Kramers equation [I]. A complementary study was carried out to explore the effect of viscosity on the photocycle of bacteriorhodopsin. Here again the Kramers equation in a modified form was found to be useful [ 2 ] . Most recently, the photodissociation of carbon monoxide myoglobin was studied in crystals at liquid helium temperatures by two different groups [3]. Schlichting et al. [4]could show that CO dissociation leads to tilting of the proximal histidine and a decompression and motion of the F-helix toward its junction with the E-helix. These conformational changes are linked to an increase of the enthalpic barrier decreasing the association rate coefficient. It was speculated that the energy stored in this conformation of the residue and its neighbors is released during structural fluctuations associated with ligand escape. These and other observations (see also Ref. 5) illustrate the necessity to extend the theoretical approach of Prof. Marcus’s theory to the domain of intramolecular interactions in protein dynamics. I would be glad if he could comment on this development. 1. D.Beece, L. Eisenstein, H. Frauenfelder, D. Good, M. C. Morgan, L. Reinisch, A. H. Reynolds, L. B. Sorensen, and K. T. Yue, Biochemistry 18, 3421 (1979).

406

R. A. MARCUS

2. D. Beece, S. F. Bowne, J. Czege, L. Eisenstein, H. Frauenfelder, D. Good, M. C. Marden, J. Marque, P. Ormos, L. Reinisch, and K. T. Yue, Phorochem. PhorobioZ. 43, 171 (1981). 3. G. Petsko, Nature 371, 740 (1994). 4. I. Schlichting, J. Berendzen, G. N. Phillips, Jr., and R. M. Swwet, Nature 371, 8008 ( 1994). 5. H. Akiyama, T. Kakitani, Y. Imamoto, Y. Shichida, and Y. Hatano, J. Phys. Chem. 99,7147 (1995).

R. A. Marcus: Concerning the issue raised by Prof. Benno Hess, a number of treatments of the “solvent dynamics” of chemical reactions in proteins or liquid solvents assume one slow (X)and one or more fast coordinates. The theories then differ in the nature of how a rate constant k(X)depends on X. Agmon and Hopfield, for example, used a k(X)specific for the ligand-heme dissociation process they were considering. Sumi and I used a k(X) specific for the electron transfer reactions we were considering [ 11. In today’s talk, which concerns an isomerization, I used for k ( X ) the analog of an RRKM rate constant appropriate to it. These various treatments have in common the same differential equation for the probability distribution P(X)along X.They differ in the process considered and in the nature of solution. There is still much to be done, particularly for systems that deviate from single time-exponential behavior, to use some of the existing numerical solutions, to test various analytical approximations, and to develop new analytical approximations. It was interesting to hear from Prof. Hess about the new and detailed structural information becoming available for the protein systems, and extending the theory will be an interesting problem. 1. H. Sumi and R. A. Marcus, J. Chem. Phys. 84,4894 (1986).

W. Hebel: I have a rather general question for Prof. Marcus. You are discussing the complicated solvent dynamics of molecular clusters. Does this also include large biomolecules such as proteins in aqueous solutions? Could you perhaps comment on how far research has gone in analyzing and understanding the interaction of biopolymers in aqueous solvents? R. A. Marcus: Even though solvents and solvent-solute interactions or interactions with a protein can be very complicated and the resulting motion can be highly anharmonic, under a particular condition there can be a great simplification because of the many coordinates (perhaps analogous to the central-limit theorem in probability theory).


407

The condition is that of linear response of the solvent or protein, for example, that the change in dielectric polarization of the solvent be proportional to the change in charge of a solute. With this condition fluctuations give rise to a quadratic expression for free-energy changes. This simplification ultimately led, in the case of electron transfer reactions in appropriate atoms or group transfer, to new predicted relationships among rate constants of different reactions. The linear response approximation for the electron transfer systems was subsequently also tested by various investigators by computer simulations of solvent and of proteins. J. Troe: Professor Marcus, you were mentioning the 2D SumiMarcus model with two coordinates, an intra- and an intermolecular coordinate, which can provide “saddle-point avoidance.” I would like to mention that we have proposed multidimensional intramolecular Kramers-Smoluchowski approaches that operate with highly nonparabolic saddles of potential-energy surface [Ch. Gehrke, J. Schroeder, D. Schwarzer, J. Troe, and F. Voss, J. Chem. Phys. 92, 4805 (1990)]; these models also produce saddle-point avoidances, but of an intramolecular nature; the consequence of this behavior is strongly nonArrhenius temperature dependences of isomerization rates such as we have observed in the photoisomerization of diphenyl butadiene. R. A. Marcus: I used the words saddle-point avoidance, incidentally, to conform with current terminology in the literature. More generally, one could have said, instead, avoidance of the usual (quasi-equilibrium) transition-state region (ie., the most probable region if viscosity effects were absent). E. Pollak: In relation to the point discussed by Profs. Troe and Marcus, we have shown that those cases considered as saddle-point avoidance are consistent with variational transition-state theory (VTST). If one includes solvent modes in the VTST, one finds that the variational transition state moves away from the saddle point; the bottleneck is simply no longer at the saddle point. A. H. Zewail: I have a question for Prof. Marcus concerning the fact that, in the bulk solvation problem, there are two regimes for the description of solvation, the continuum model and the detailed molecular dynamics. Do you expect that in clusters the friction model will change as the number of solvent molecules changes from small to large? R. A. Marcus: In Chem. Phys. Lett. 244, 10 (1995), a very rough approximate hard-sphere model used for liquids was mentioned to relate the frictional coefJicient to the pair distribution function in the cluster.

408

R. A. MARCUS

D. J. Tannor: One would think that as one adds more and more layers of solvent one is introducing irreversible decay of the correlation function of the solute-solvent coupling. The main physical content of the Grote-Hynes expression for the rate constant is that contributions from this correlation function that are slow compared with the time scale for reaction do not really contribute to the reaction rate. This suggests that by starting with a description of only the first solvent shell and introducing shorter and shorter solvent memory, one will see a transition that resembles that of adding more and more solvent shells. R. A. Marcus: About the problem raised by Prof. Tannor, there are a number of questions to be resolved, such as energy migration to the solvent in the cluster, the detailed dynamical effect of successive layers of solvent in larger clusters, and comparing with cluster experiments of other suitably chosen reactants, for example, RlRzC = CR3R4, where R3 and R4 are so small that the molecule has no frictional effect in solution. Such an isomerization has previously been studied in liquids. In the present chapter several experiments are suggested to disentangle the various factors influencing the energy-dependent rate contants. H. Hamaguchi: What information do you have concerning the structure and dynamics of the hexane-dressed stilbene molecule? How flexible or how rigid is the structure? R. A. Marcus: I personally do not have the information. A. H. Zewail: To provide a partial answer to the question of Prof. Hamaguchi, the structure of the 1:1 stilbene-hexane species was determined with the help of rotational coherence spectroscopy. For higher clusters we used atom-atom model potentials and deduced structures.

HIGH-RESOLUTION SPECTROSCOPY AND INTRAMOLECULAR DYNAMICS H. J. NEUSSER* and R. NEUHAUSER Znstitut f i r Physikalische und Theoretische Chemie Technische llniversitat Miinchen Garching, Germuny

CONTENTS I. Introduction 11. Intramolecular Dynamics in Electronically Excited S1 State of Benzene A. Mechanism of Intramolecular Dynamics in Polyatomic Molecular System B. Intramolecular Dynamics in Benzene 1. States at Low Excess Energy 2. Dynamic Behavior of States at Intermediate Vibrational Excess Energy C. Influence of Van der Waals Bonded Noble-Gas Atoms on Intramolecular Dynamics 111. Laser-Driven Population Dynamics and Coherent Ion Dip Spectroscopy A. Introduction B. Incoherent Population Dynamics C. Coherent Population Dynamics D. Coherent Population Dynamics for Special Pulse Sequences E. Coherent Ion Dip Pulse Sequence F. Experimental Results 1. Experimental Setup 2. Experimental Procedure of Coherent Ion Dip Spectroscopy 3. spectra IV. Intramolecular Dynamics of High Rydberg States in Polyatomic Molecules A. General Remarks B. Experimental C. Experimental Results V. Conclusion *Report presented by H. J. Neusser Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond lime Scale, XXth Solvay Conference on Chemistry. Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

409

410

H. J. NEUSSER AND R. NEUHAUSER

I. INTRODUCTION In this work we discuss the intramolecular dynamics initiated by the interaction of light with isolated polyatomic molecules and van der Waals complexes. As is well known, in many polyatomic systems the intramolecular dynamics induced in this way precedes the chemical reaction and leads to an energy randomization within the numerous internal degrees of freedom of the system. This justifies the application of statistical theories that have been successfully applied to describe the unimolecular decay behavior of these systems [l, 21. In order to find precise information on the coupling mechanism and the coupling strength, experiments are required that lead to a sufficient selection of individual molecular states. Thus the experiments discussed in this work are performed with nanosecond light pulses providing the highest resolution of several tens of megahertz possible for their duration, that is, Fourier transform limited and thus coherent light pulses. We will present results on the intramolecular dynamics in different energy regimes, that is, the electronically excited state (Sl), the electronic ground state (SO), and in high Rydberg states with various kinds of couplings and a resulting different intramolecular dynamic behavior. Even in a molecule the size of benzene the resolution achieved in this way is sufficient to investigate the dynamic behavior of individual rotational states. For this it is necessary to eliminate the Doppler broadening of the rovibronic transitions. Two methods have been applied: (i) the elimination of Doppler broadening in a Doppler-free two-photon-transition and (ii) the reduction of Doppler broadening in a molecular beam. Measurements of the dynamic behavior have been performed in the frequency [3] and time domain [4].We will briefly summarize the results from high-resolution measurements and discuss the conclusions on the intramolecular decay mechanism. Then it will be discussed how the intramolecular dynamics is influenced by the attachment of an Ar or Kr atom to the benzene molecule, leading to a weakly bound van der Waals complex. If intense coherent light pulses interact with a molecular system, a special population dynamics in a three-level system results that is important and helpful for the spectroscopy of high vibrational levels in the electronic ground state of molecules [5, 61. In this work we present the features of a coherent high resolution technique leading to the sensitive spectroscopy of ground-state levels in molecules and van der Waals complexes. Exploiting the characteristics of the coherent excitation, we are able to detect intramolecular couplings not only in the excited S1 state but also in the electronic ground state. First examples are presented for benzene and applications to van der Waals complexes are discussed.

HIGH-RESOLUTION SPECTROSCOPY AND DYNAMICS

41 1

II. INTRAMOLECULAR DYNAMICS IN

ELECTRONICALLY EXCITED S1 STATE OF BENZENE In this section we present experimental results for the lifetime of individual rovibronic states of benzene at different excess energies in the S, electronic state. In this way the dependence of the lifetime of the states on their excess energy and their rotational quantum number is studied. A general model for the underlying coupling mechanism is presented, and the influence of a van der Waals bound noble-gas atom on the intramolecular dynamics is investigated.

A. Mechanism of Intramokcular Dynamics in Polyatomic Molecular System Absorption of light in a molecule via electric dipole interaction normally leads to the excitation of a zero-order state, typically a rovibrational Bom-Oppenheimer (BO) level within the first excited singlet SIstate, These levels are usually not eigenstates of the molecular system since they are coupled by the kinetic-energy operator to the ground-state electronic potential surface, by the spin-orbit coupling operator to the lowest triplet state (interstate coupling), and/or by anharmonic coupling or Coriolis coupling operators to other rovibrational states within the same electronic potential surface (intrastate coupling) [7, 81. At modest vibrational energies of the electronic ground state SOthe interstate coupling does not exist and intrastate coupling is prevailing. On the other hand, in the energy range of high Rydberg states (see Section IV) the density of electronic states is much higher than the density of vibrational and even rotational states and interstate coupling is the dominating mechanism. In the S, state both inter- and intrastate coupling takes placed. In a polyatomic molecule consisting of more than 10 atoms, the density of coupled states p within the SO ground electronic state, and sometimes the T I triplet state, is so high that coupling (with matrix element V) to a large number of states may occur ( p >> V - I ) . This condition approaches the statistical limit for the density of coupled states. Hence, coherent excitation leading to a defined phase of the complete set of eigenstates that contribute to a given BO state results in the initial production of this zero-order state. This initial character is consequently lost by dephasing of the prepared packet of eigenstates. Since the density of states is so high, no recurrence occurs on the time scale of the experiment and a single exponential decay of the prepared population will be observed (large-molecule limit [9]). These conditions produce a Lorentzian line shape for a continuous-wave (CW) spectroscopic experiment that uses high spectral resolution and a small coherence width. By contrast, the density of coupled states within the electronically excited SI state

412


(or the ground state SO) of a polyatomic molecule depends strongly on the vibrational energy. At low vibrational excess energy and for weak interstate coupling, only one proper state might be located within the range of the intrastate coupling matrix element, and coupling to only a single rovibronic state occurs. The conditions produce a repulsion of the two eigenstates, a situation frequently observed in the spectra of small molecules (small-molecule limit [lo]). The coherent excitation of the two coupled eigenstates results in an oscillatory behavior of the two populations (quantum beats) Ill, 121. Depending on the magnitude of the coupling matrix element, nanosecond pulses or even subnanosecond pulses are required for this coherent excitation. For higher excess energies with increasing density of states a more complicated time behavior is expected.

B. Intramolecular Dynamics in Benzene As an example we consider the prototype molecule benzene (C6€&)consisting of 12 atoms and possessing 30 vibrational degrees of freedom. A fluorescence quantum yield of SIbenzene smaller than unity (0.2) was observed under collision-free conditions on the time scale of fluorescence [ 13-15], and it was concluded that a nonradiative electronic decay takes place that is irreversible on the time scale of fluorescence and occurs in the statistical limit [16, 171. As zero-order €30states we take rovibronic states in the lowest singlet and triplet electronic states. Each state may be coupled to others by the different coupling mechanisms mentioned above. There is, however, a hierarchy of the coupling strengths with interstate coupling being weaker than intrastate coupling in benzene and many other molecules. At the energy of the benzene S, state (38086 cm-') the density of vibrational states in SO is so high (=1015 l/cm-') that coupling to the electronic ground state is in the statistical limit that results in irreversible internal conversion. The situation is not so clear for the coupling to the triplet state. There the density of states is considerably smaller (=lo6 l/cm-') due to the smaller excess energy (SI-TI energy gap: 8464 cm-I) and it is not a priori certain that coupling to triplet states really occurs in the statistical limit. Finally, the situation for coupling within the S1 potential surface is completely different. For excess energies less than 2500 cm-I, the density of states is sufficiently low that the coupling is expected to be in the small-molecule limit; that is, spectral lines involving coupling states are expected to be split rather than broadened. At very high excess energies, where a corresponding high density of states occurs, coherent excitation is expected to lead to a fast irreversible dephasing of the vibrational states within Sg and a significant broadening of spectral lines [ 181.


413

1. States at Low Excess Energy

The 14’ state of C6H6 at 1571 cm-’ vibrational energy above the zero point of S1 can be excited through two-photon absorption. Doppler-free CW spectra with thousands of well-separated rovibronic lines of 15 MHz width were recorded at room temperature. Individual transitions were assigned up to J’ = K’ = 10 [19] and the frequencies of 90% of these lines could be well fitted by a semirigid symmetric top model. The remaining lines are shifted from the expected position or split into two components separated by some gigahertz [20].The appearance of these isolated perturbations is due to the fact that the “light” 14’ state is coupled locally to “dark” states within SI through higher order rotation-vibration couplings [3]. From emission spectra of the single eigenstates we were able to determine the identity of the three dark states involved [21]. Pulsed excitation allowed the decay of single rotational states of the 14’ vibronic state to be measured. The decay was found to be singly exponential in all cases investigated [4, 211. For states that are not perturbed by a coupling within SI(unperturbed states) the decay time was found to be independent of the molecular rotation, whereas for perturbed states the decay times are shorter due to the admixture of the dark states, which decay faster. Most likely these dark states are combination states containing quanta of low-frequency out-of-plane vibrations that are strongly coupled to the So ground or T I triplet state and undergo a faster electronic nonradiative relaxation process than the excited “bright” states. The rotationally independent decay of the “unperturbed” states points to an electronic nonradiative relaxation process to T I in the statistical limit. To summarize, the rotationally resolved two-photon spectra at low vibrational excess energy show isolated perturbations that can be understood as rovibrational couplings in the strong limit to discrete background states in SI.This coupling causes a mixing of the vibrational character of the resulting quasi-eigenstates and therefore a rotational dependence of the nonradiative decay, which otherwise would not depend on the rotation of the molecule. The behavior of states excited through one-photon transition (i.e., differing symmetry and parity) is found to be qualitatively the same. Since Dopplerfree one-photon experiments were performed in a molecular beam at low rotational temperatures of 2 K only low J’, K’ values are observed [22]. In this narrow energy range a coupling to dark background states is not very likely. No perturbation was observed in the 6’ band at the low excess energy of 522 cm-I . The lifetime measured for various rotational states up to J‘ = 7, K’ = 6 was found to be independent of the rotational quantum number excited. This demonstrates that the excited rovibronic states do not couple to background states, as is expected for the low J’, K’ quantum num-

414


bers and the small excess energy with its low density of background states in S,. The decay is faster than that of the 14’ state at more than twice the excess energy. This has been explained by the energy gap model and different Franck-Condon factors for the Inter Systems Crossing (ISC) process

P31.

2. Dynamic Behavior of States at Intermediate Vibrational Excess Energy

For higher excess energies the density of background vibrational states increases. This causes a different dynamic behavior of the excited light zeroorder states. As an example we briefly describe the results for the 14’ l 2 state of C& at an excess energy of 3412 cm-’ [24].In the blue part (low rotational energy) of the qQ-branch of the 14’ band only (unshifted) K’ = 0 lines were seen in the fluorescence excitation spectrum, whereas further to the red (higher rotational energy) the only observed structure seemed to be due to lines with K’ = J’. This finding was interpreted as due to predominant coupling in the weak limit of 14’l 2 to a strongly diffusely broadened background state that allows only a minority of uncoupled (or weakly coupled) states to survive in the fluorescence excitation spectrum. The measured strong dependence of the collisionless linewidth on the J quantum number shown in Fig. 1 was supposed to be due to the rotational dependence of the coupling matrix element, that is, parallel Coriolis coupling for the blue part and perpendicular Coriolis coupling for the red part of the band [3]. A detailed theoretical description in this and other bands of benzene has been successfully performed using an artificial intelligence model for the description of the coupling pathways [25, 261. It was also possible to measure the decay times and the homogeneous collisionless widths of the K’ = 0 lines in the blue part of the 14’ l2 band. The decays were all found to be singly exponential within the experimental accuracy. The values of the decay rates determined from the time-resolved and frequency-resolved experiments agreed quite well, indicating that in our experiment single quantum states were observed. This was one of the first examples of a combined measurement of the dynamic behavior of a large molecular system in both the time and frequency domains. Similar results were found €or the one-photon band 6Ali band at 3287 cm-’ excess energy. This corroborates the model of a rotationally dependent intramolecular dynamics due to intrastate coupling [3, 4, 271. C. Infiuence of Van der Waals Bonded Noble-Gas Atoms on Intramolecular Dynamics When we attach an Ar atom to the benzene molecular plane, a weakly bound van der Waals complex is formed. Experiments with high-resolution rotation-


. .

L

..5

-

* .

JK= 20

..

-100

-.

12.7 MHz

-:

.

I

415

I

I

l

1 :

I

l

0

I

I

I

I

+loo

A u tMHzl

Figure 1. Linewidths of different rotational transitions in the 14A1; vibronic band of benzene measured with Doppler-free two-photon absorption. The observed strong dependence on the quantum number J of the rotational angular momentum is evidence for a rotationally dependent intramolecular coupling process. (Taken from Ref. 3.)

ally resolved ultraviolet (UV) spectroscopy in supersonic cooled molecular beams clearly show that the Ar atom is located on the Cg axis of benzene at a distance of 3.58 A from the molecular plane [28, 291. A decrease of the average van der Waals distance to 3.52 A after excitation to the electronically excited SIstate is found. In benzene-Ar three low-frequency intermolecular

416


modes (two bending, 31.16 cm-' ; one stretching vibration, 40.1 cm-I [30, 311) are added to the 30 vibrational degrees of freedom of benzene. The binding energy of the benzene-fir van der Waals complex in the SIstate is about 360 cm-', as has been found from mass-selective pulsed field threshold ionization measurements [32, 331. This is in reasonable agreement with recent results from ab initio calculatioy [34-361. For benzenes4= an average van der Waals distance of 3.68 A in SI[37] and a binding energy of less than 435 cm-I was measured [32, 331. When the benzene-Ar complex is excited to the low lying vibrational state 6l, we find a small red shift of 21 cm-' of this band from the respective transition in bare benzene that is mainly caused by the red shift of the zeropoint energy. Even for this low-lying vibrational state the excess energy of 522 cm-' is larger than the dissociation energy of 360 cm-' of the complex. Thus, in principle, coupling to the van der Waals modes and dissociation is a new relaxation channel in the complex. It is interesting to investigate whether the intermolecular channel can compete with the intramolecular relaxation processes in the bare molecule at this excess energy. In Fig. 2 the highresolution spectra of the 6; band in benzene-& leading to the 6' state at an excess energy of 522 cm-I and the 6Al; band leading to an excess energy of 1444 cm-' are shown. They were measured for benzene-Ar complexes produced in a supersonic molecular beam of benzene seeded in Ar gas under high pressure. Supersonic expansion leads to a low rotational temperature of 2 K. The observed linewidth of 120 MHz corresponds to the experimental linewidth given by the laser bandwidth and the residual Doppler broadening in the skimmed molecular beam. There is no additional broadening due to dynamic processes faster than nanoseconds. Even for an excess energy of 1444 em-', which is more than four times the binding energy, the complex is stable on the nanosecond time scale and the intramolecular dynamics is not affected drastically. Furthermore, no perturbations are observed in these bands, which would point to additional coupling paths in the van der Waals complexes due to the increased density of states. To discover smaller specific effects on the intramolecular dynamics after attachment of an Ar atom to the benzene molecule, we performed lifetime measurements of single rovibronic states in the 6; band of the benzene-Ar and the benzene-84 Kr complex. No dependence of the lifetime on the quantum number within one vibronic band was found [38]. This is in line with the results in the bare molecule and points to a nonradiative process in the statistical limit produced by a coupling to a quasi-continuum, for example, the triplet manifold. In Fig. 3 the decay curves of benzene and benzene-Ar are shown after excitation of the same individual rotational state in the 6' vibrational state. Though the rotational quantum numbers . I ; , = 44 and the vibrational quan-

a

I "

LY

-

Figure 2. Rotationally resolved REMPI spectra of the 6; and 6h1b vibronic bands of the benzene-Ar complex. No broadening of the lines in the 6AlA band is observed showing that even for an excess energy of 1444 cm-I, which is more than four times the binding energy, the complex is stable on the nanosecond time scale. (Taken from Ref. 29.)

L

417

418

H. J. NEUSSER AND R. NEUHAUSER I

Time [ns] Figure 3. Comparison of the fluorescence decay curves after selective excitation of the 6 ‘ , J’ = K’ = 4 rovibronic state of benzene and benzene-Ar. Note the shorter lifetime of the same rovibronic state in the complex.

tum numbers are identical in both cases, a pronounced shortening of the lifetime from 86 ns in benzene to 53 ns in benzene-& is found. The measurement of the lifetime of the J ; , = 22 state in benzene-& leads to an even shorter lifetime of less than 10 ns. Most likely the lifetime is in the range of a few nanoseconds since the exciting laser pulse has a pulse duration of 7 ns and no broadening of the lines is seen in the spectra [37]. We can exclude a predissociation process [39] responsible for the decrease of the lifetime for three reasons. (i) Dispersed emission spectra did not show any indication of emission from the fragment monomer [40]. Thus no dissociation occurs on the time scale of the fluorescence emission. (ii) The additional excitation of the van der Waals stretching vibration in benzene-Ar does not lead to a further decrease of the lifetime. (iii) The stronger decrease of the lifetime of the 6l state in benzene-Kr would not be expeced for a predissociation process since the benzene-& complex is more strongly bound and has only a slightly higher density of states since the frequencies of the three van der Wads modes do not differ very much from that of benzene-Ar 1411.

After exclusion of a predissociation process responsible for the lifetime shortening in complexes of benzene with noble gases, we consider the external heavy-atom effect on the intersystems crossing rate as the origin of the lifetime shortening [421. The strong decrease of the lifetime in the


419

benzene-= complex supports this explanation. An ambient heavy atom leads to stronger spin-orbit coupling in the aromatic molecule. External heavy-atom effects have been studied in the condensed phase. Experiments have been performed on benzene molecules in a noble-gas matrix at low temperatures [43,441 and on tetracene-noble gas complexes [45]. Here we present a first example where the external heavy-atom effect is studied for a specific rovibronic state in a van der Waals system with defined position of the heavy atom and known van der Waals vibrations. In conclusion, we have found that the intramolecular dynamics in the benzene molecule at low excess energy is not strongly influenced by the additional three vibrational degrees of freedom of the benzene-Ar complex. The coupling of the excited intramolecular modes to the low-frequency intermolecular modes is weak. The observed 40% decrease of the lifetime of the 6' state does not depend on the individual excited rotation and points to an external heavy-atom effect as the source of the lifetime shortening observed for the same selectively excited rovibronic state.

JII. LASER-DRIVEN POPULATION DYNAMICS AND COHERENT ION DIP SPECTROSCOPY

A. Introduction As pointed out in the previous sections, a major goal of this work is the investigation of intramoleculardynamic processes leading to an energy redistribution after excitation of defined quantum states of the molecule. Of great importance for this is the population dynamics that is introduced by the laser light itself, especially by the use of intense Fourier-transform-limited laser pulses. Here the laser light modifies the quantum mechanical system in such a way that not only the molecular system but the whole quantum mechanical system containing the molecular Hamiltonian as well as the electric field interaction Hamiltonian has to be considered [46]. This combined system can be modified by changing the interaction conditions, which is not the case for excitation with weak laser pulses leaving the molecular system unperturbed. This offers the possibility to prepare a system in a special state leading to an effective control of the intramolecular process [4749]. Here we will show that the laser-driven population dynamics makes feasible new spectroscopic techniques for the investigation of high-lying vibrational states in the electronic ground state of polyatomic molecules. We present a brief discussion of the population dynamics encountered in the molecular excitation process using intense coherent nanosecond laser pulses. The typical population dynamics is shown to lead to the new spectroscopic technique of coherent ion dip spectroscopy (CIS), which is useful for the investigation of the

420


intramolecular dynamics in molecules and van der Waals complexes [5,6, 301.

B. Incoherent Population Dynamics In most experiments in the frequency domain as well as in the time domain the population dynamics can be described in terms of rate equations based on transition probabilities depending on the coupling strength and levels of different density [50]. Neglecting the polarization properties of the laser light, the transition probability can be written as a product of the square of the transition matrix element times the laser light field strength and the level density of the final level. In this way absorption cross sections and stimulated and spontaneous emission coefficients have been deduced from quantum mechanical calculations. In Fig. 4 a typical lambda-type three-level system is shown that is realized, for example, in stimulated emission pumping

12>

I -

-

A

Cou ling

13a>

Decay

13>

dvv===

II> Figure 4. Level scheme of a lambda-type double-resonance eperiment. The pump laser pulse couples the initial level 1 to a single rovibronic intermediate level 2 in the electronically excited S1 state. The intermediate level 2 is coupled by the dump laser pulse to the vibrationally excited level 3 in the electronic ground state So. Ionization from level 2 is possible by absorption of an additional photon from the intense dump laser pulse. The coupling of the final level 3 to a single dark state (level 3 4 is indicated and discussed in the text.


42 1

(SEP) experiments (for a review see Ref. 51). This excitation scheme has been used for the excitation of high-lying vibrational staes in the electronic ground state of a series of molecules to investigate the intramolecular redistribution processes occumng at high internal energies [52]. Population in level 1 is transfered to level 2 by absorption of a photon provided by the pump laser while the dump laser laser light can cause stimulated emission into level 3, which is going to be investigated. The possibility of a fast decay of level 3 [e.g., by intramolecularenergy redistribution (IVR) in the statistical limit] or coupling of level 3 to single dark states (e.g., IVR in the intermediate case) is neglected for a moment. The rate equations for this three-level system read

where ni denotes the population in level i, 1p.d the intensity of the pump-anddump laser, and Bij, A, the Einstein coefficients. No decay in this three-level system is considered. The time-dependent incoherent population dynamics can be obtained in the usual way by solving the rate equations as a system of coupled linear differential equations with special initial conditions. For CW laser fields and for laser pulses that are long compared to the typical relaxation rates, (quasi)stationary solutions can be found. For short pulses a time-dependent solution of the system of differential equations [Eq. (l)] is necessary. In Fig. 5 the typical population dynamics in a three-level system is shown for temporally slightly overlapping intense laser pulses and 100% population in level 1 before laser interaction.The intense laser light field used in this calculation saturates all transitions. First the pump laser pulse causes a nearly equal distribution of population in levels 1 and 2. Then the dump laser pulse applied after the pump laser pulses causes stimulated emission into level 3, leading to nearly 25% population in levels 2 and 3 and about 50% population in level 1. If only the pump laser frequency is in resonance with the 2 c 1 transition and the dump laser frequency is not in resonance with the transition connecting levels 2 and 3, the population is distributed in equal parts between levels 1 and 2 and 50% of the population will be found in level

422


dump pulse

li

-5 2.0

pump pulse

0.0

.-35

-

3

Q

1.o 0.8 0.6 0.4

0.2 0.o

t Y ? -

-20.0 0.0 20.0 Time [ns]

Figure 5. Typical incoherent population dynamics within the level scheme of Fig. 4 for fully saturated resonant transitions calculated from the rate equations (1).

2. This behavior is exploited in SEP experiments [SI] where the lowering of the population of level 2 for double-resonance conditions is probed by laser-induced fluorescence (LIF) or ion detection (ion dip experiments) by ionizing the molecules in level 2 with a third laser pulse. It is obvious from the rate equations that no dip depth larger than 50% of the maximum offresonant signal can be obtained as long as no fast decays of the final levels must be considered. (However, for fast-decaying final levels deeper dips can be expected and the dip depth has been used for an estimate of the decay rate [53].) In the following we want to show how this 50% limit can be overcome by the use of intense Fourier-transform-limitednanosecond laser pulses. In this case the rate equations do not describe the population dynamics correctly, and a solution of the density matrix equation of the system is necessary. The resulting population dynamics allows the detection of weak transitions resulting from the coupling of the optically accessible bright states to dark states and the investigation of the coupling strength and energetic position of the unperturbed transitions.

C. Coherent Population Dynamics

When a molecular system interacts with intense Fourier-transform-limited


423

laser pulses, the full Hamiltonian of the molecular system and the field interaction must be considered and solved as a whole. The interaction Hamiltonian Hinteraction must not be treated by time-dependent perturbation theory but has to be included into the complete Hamiltonian of the system:

After transformation into the interaction picture and application of the rotating-wave approximation [46, 50, 541 the population dynamics can be calculated numerically by solving the time-dependent three-level Schriidinger equation or (if phenomenological relaxation rates are considered) by solving the density matrix equation (3) for the molecular system. The density matrix equation is given by

Here p is the density matrix for all molecular states in the three-level system depicted in Fig. 4, and all incoherent relaxation terms caused, for example, by collisions, spontaneous emission, or decay in a (quasi)continuum are incorporated in the relaxation matrix rrelax. For a three-level system the Hamiltonian in the interaction picture H i in Rotating Wave Approximation is given in matrix representation by

The Rabi frequencies 61p.d are given by

where pP,d is the transition matrix element for the pump- and dump-transition and Ep,d the amplitude of the laser field strength of the pump- and dumplaser pulse. For simplicity different polarizations of the laser light fields are not included in this discussion, although interesting applications have made use of the polarization dependence [55-591. Solution of Eq.(3) yields values for the diagonal matrix elements of the density matrix p and thus the timedependent population in every level. In a general approach it is possible to show that a coherent treatment

424


of the population dynamics is necessary if the Rabi frequency exceeds all intramolecular decay rates and typical inverse-phase relaxation times of the laser light [50].

D. Coherent Population Dynamics for Special Pulse Sequences Figures 6a-c show the population dynamics encountered in a three-level system (see Fig. 4) interacting resonantly with two Fourier-transform-limited laser pulses with three different delay times between the two pulses. The calculation was done assuming that the chosen Rabi frequencies fulfill the relation ( Q p , d > l/pulse duration) in all three cases. This relation ensures that the typical time for a Rabi oscillation of the population in an isolated two-level system is shorter than the pulse duration. Ionization from level 2 was introduced as a fast laser intensity-dependent decay of level 2 [6, 601, and resonant laser frequencies were assumed. In the upper parts of Figs. 6a-c the time-dependent Rabi frequencies of both laser pulses are shown for different delays. In all cases the dump laser pulse has a higher Rabi frequency than the pump laser pulse and twice its duration. Note that the Rabi frequency is proportional to the laser field strength and therefore to the square root of the pulse intensity. In the lower part the population dynamics for the three different pulse sequences is shown. The part of population transferred to the ionization continuum is indicated by a strong line.

6 3

I.o 0.0

0.6

$ 0.4

n 0.2

0.o -20.0 0.0 20.0 Time [ns]

-20.0 0.0 20.0 Time [ns]

Time Ins]

Figure 6. Coherent population dynamics calculated using the density matrix equation (3) for different delays (a+) of the laser pulses. Upper part: Time evolution of the Rabi frequencies of both laser pulses. Lower part: Calculated time evolution of the level populations for three different delays.


425

(a) Pump Laser Pulse before Dump Laser Pulse. This pulse sequence is equivalent to the pulse sequence used in most SEP experiments. The damped Rabi oscillation of the population between levels 1 and 2 clearly indicates the coherent character of the interaction. Most of the population is ionized during the first laser pulse because of the high laser intensities. Although the dump laser is in resonance, ionization is very efficient and no prominent ion dip will be observed in this case.

(b) Dump Laser Pulse Overlapping with Pump Laser Pulse. In this situation the oscillatory behavior vanishes and nearly no change in level population is observed. The ionization signal vanishes for resonance conditions and a nearly 100% ion dip is expected. This pulse sequence is used in the CIS experiments described below. (c) Pump Laser Pulse after Dump Laser Pulse. With this pulse sequence an effective population transfer to the final level 3 is possible over a wide range of delay times and Rabi frequencies. This behavior was exploited in population transfer experiments using the stimulated Raman rapid adiabatic passage technique (STIRAP), which has been analyzed in detail analytically and numerically [60-64].

E. Coherent Ion Dip Pulse Sequence As shown in Fig. 6b, for a dump laser pulse overlapping with the pump laser pulse no net population transfer occurs. It is particularly interesting that the intermediate level 2 is not significantly populated at any time although level 3 is weakly populated during the interaction. This surprising population dynamics can be exploited to check whether the dump laser frequency is in resonance with the 2 -+ 3 transition and thus the double-resonance condition is fulfilled: As in SEP experiments it is possible to monitor the population of level 2 either by fluorescence from level 2 or by ionization after absorption of an additional photon (see Fig. 4). In a simple model the ionization process from level 2 can be introduced by a time-dependent decay rate of level 2 [6, 601 that is proportional to the intensity of the laser pulses, whereas the fluorescence is only proportional to the population in level 2 after interaction with both laser pulses [54]. For an off-resonant dump laser frequency and a resonant pump laser frequency a high ion current is observed, whereas for double-resonance condition no ion current can be measured due to the negligible population of level 2. In Fig. 7a the calculated ion current and fluorescence signal is shown using the parameters of our recent work [6].The pump laser frequency was kept in resonance with the transition 2 t 1 and the dump laser frequency was scanned across the 2 -+ 3 transition. (It should be mentioned that for

426


0.40

1

P- 0.20

.

1

~

l

.

I

.

I

.

I

.

I

.

J

0 c

m

3

0.10

ion Current

I

-8.0

-8.0

-6.0

.

I

.

-4.0

1

.

-2.0 0.0

.

2.0

-6.0 -4.0 -2.0 0.0

.

*

.

4.0

2.0 4.0 Dump Laser Detuning [GHz]

-

I

.

6.0

6.0

8.0

8.0

Figure 7. ( a ) Three-level system: Calculated ion dip and fluorescence dip spectra with 1 transition. Ionization is treated as pump laser frequency tuned to resonance with the 2 an intensity-dependent decay of level 2. The sharp dip with nearly 100% depth indicates the coherent character of the excitation. (b) Four-level system: Different from (a) level 3 is now coupled to a dark state, yielding a splitting of level 3 in two states separated by 0.016 cm-' that are well resolved in the calculated ion dip spectrum.


427

the calculation the more realistic model of Ref. 6 for the interaction process was used. Here not only the time dependence of the Rabi frequencies and ionization decay rates but also a radial Gaussian distribution of the laser intensities and the dependence of the transition moments on the rn-quantum number of the orientation of the molecule in a laboratory fixed axis were included.) The calculated spectrum shows two interesting aspects. (i) For the given Rabi frequencies the fluorescence dip is less pronounced than the ionization dip. This behavior is due to the fact that ionization takes place mainly in regions with high Rabi frequencies because of the quadratic dependence of the ionization rate on the laser field strength, whereas the fluorescence signal originates from all spatial regions interacting with the laser pulses. (ii) While the linewidth of the dips [full width at half maximum (FWHM)] is large (several gigahertz), they are very sharp at the minimum. This allows us to determine the exact frequency position of the 2 --+ 3 transition. The linewidth of the dip depends strongly on the Rabi frequencies and ionization rates. High ionization rates and low Rabi frequencies yield small but narrow dips. When we apply the coherent technique to study intramolecular processes occurring in the final state 3, it is important to investigate how an intramolecular coupling or a dynamic process of level 3 affects the population dynamics and whether small spiittings can be observed. In Fig. 7b the calculated ionization signal for the same laser intensities and time delays as used in Fig. 7a is shown. Now, however, level 3 is coupled to a dark state 3a as depicted in Fig. 4. The state 3a is assumed to be dark since the transition moment 2 ---c 3 is very small due to small Franck-Condon factors or for symmetry reasons. The coupling is assumed to lead to two mixed eigenstates with an energetic separation of 0.016 cm-' that is larger than the laser bandwidth. As demonstrated by the calculated CIS spectrum of Fig. 7b, both states can be resolved even for intensity conditions leading to a broad CIS signal. Finally, we want to emphasize an interesting result of the numerical calculation that has been proven experimentally. As shown in Fig. 6, there exist two pulse sequences (b, c) leading to a small population of level 2. In case (c) most of the population is transferred to level 3 while in case (b) nearly all of the population remains in the initial level. In coherent ion dip experiments case (b) is used as it provides deeper dips due to the more effective suppression of ionization. Using higher laser intensities would allow us to achieve nearly 100% ion dips also in case (c); however, for off-resonant conditions the ion current would be smaller by an order of magnitude than in the pulse sequence of case (b) and dips are more difficult to detect.

428

H. 3. NEUSSER AND R. NEUHAUSER

F. Experimental Results In the following section a brief survey of our recent experimental results obtained with CIS is presented. 1. Experimental Setup

The experimental setup (Fig. 8) is similar to the one used in previous CIS experiments [6]. Both laser pulses are provided by two pulsed amplified CW ring lasers operating with Coumarin 102 and Fthodamine 110 dye, respectively. Amplification of the CW light in two three-stage amplifier systems and frequency doubling of the nearly Fourier-transform-limited visible laser light pulses yield UV pulses with energies of 400 pJ, pulse durations of 15 ns (FWHM), and a bandwidth of 70 MHz. The two counterpropagating laser beams are focused down to a common focus of less than 0.5 mm diameter. The first laser pulse exciting the molecule or the van der Waals complex from the electronic ground state to a single rovibronic state in the electronically excited Sl state is attenuated by a factor of 50 to reduce direct photoionization by absorption of two photons of this pulse. The counterpropagating narrow-band light pulses interact with the molecules in the center of a cooled molecular beam expanding from a reservoir with 2% benzene seeded in Ar at a backing pressure of 2 bars through a nozzle with a 300-pm orifice [22]. A conical skimmer (1.5 mm diameter) collimates the beam and reduces the Doppler width below the laser linewidth, and an experimental saturation broadened resolution of 200 MHz is achieved in a resonance-enhanced two-photon ionization (REMPI) experiment with the frequency of the second pulse not in resonance with the 2 3 transition. The produced ions are mass analyzed in a time-of-flight (TOW mass spectrometer and detected with multichannel plates.

-.

2. Experimental Procedure of Coherent Ion Dip Spectroscopy

In Fig. 9 the ion current calculated as described above is shown as a function of the detunings of the pump-and-dump laser frequency in a two-dimensional plot. A cut along the pump laser frequency axis for a nonresonant dump laser frequency yields REMPI spectra. The same cut for a resonant dump laser frequency show a “splitting” of the 2 1 transition that can be explained in the intuitive picture of a dynamic Stark effect. (For a closer discussion of this point, see Refs. 6 and 30). A cut along the dump laser frequency

-

Ar ton Laser

Kr ton Laser

I

Beam

Molecular

Figure 8. Scheme of the experimental setup of the CIS experiment. Two Fourier-transform-limited nanosecond laser pulses with different frequencies are interacting with cold molecules or van der Waals complexes in a skimmed supersonic molecular beam.

cw - Dye Laser

cw - Dye Laser

I

1

TOF~

430


Ion Current

Figure 9. Calculated ion current spectra for different detunings of the pump laser and the dump laser frequency. The two-dimensional plot includes REMPI spectra as well as CIS spectra.

axis with resonant pump laser frequency yields CIS spectra with ion dips. The process of measuring CIS spectra is indicated by the solid arrow in Fig. 9: In a first step the dump laser frequency is tuned out of resonance with the downward transition 2 3 and the pump laser frequency is scanned, yielding a rotationally resolved REMPI spectrum of the investigated vibronic band. In a second step the pump laser frequency is fixed on top of a single rotational line in the selected vibronic band and the dump laser frequency is scanned. (In Fig. 9, this means that the pump laser frequency is fixed on top of the ridge and the dump laser frequency is scanned along the ridge, as indicated by the arrow.) For double-resonance conditions, when the dump laser frequency is in resonance with the 2 -+ 3 transition, the ion current vanishes. In the two-dimensional plot this leads to the valley crossing the ridge in the middle of the plot.

--.

3. Spectra

Coherent ion dip spectroscopy has been shown to be a versatile tool for the investigation of high-lying intramolecular vibrations in the ground state of molecules and of intermolecular vibrations of van der Waals complexes.

-

(a) Intramolecular Coupling in Benzene Molecule. Incoherent SEP and CIS spectra have been measured for benzene (C6H6 and C6D6). Pre62 viously, we investigated the down transitions 6' + 12 and 6'

HIGH-RESOLUTIONSPECTROSCOPY AND DYNAMICS

431

in C6H6 in an incoherent SEP experiment with high-resolution pump but low-resolution dump laser [65]. Some rotational resolution was achieved in this way. Applying the high-resolution CIS technique to the 6' --c 62 transition leads to a complete resolution of all close-lying rotational transitions. For the 6' -+ 12 transition additional lines were observed in our previous SEP experiments. These were explained by a coupling of the 12 level through a Darling-Dennison [66] resonance to the close-lying 52 state. The CIS spectrum of the same transitions shows the well-resolved Darling-Dennison resonance but also a hitherto unknown splitting of the higher energetic component whose origin has not been explained yet. The measured spectrum resembles the theoretical spectrum in Fig. 7 b and demonstrates the feasibility of CIS for the sensitive highly resolved detection of intramolecular couplings. (b) CIS of Intermolecular Vibrations of Benzene-Ar Complex. The CIS technique has been used to detect the weak transitions to high-lying intermolecular van der Waals vibrational states in the benzene-Ar complex with rotational resolution. In Fig. 10, as an example the CIS spectra of the SI,6's' -SO, 12s' and SI,6's' +SO, l2b2 transitions in the c6D6-A~complex are shown. Here the pump laser frequency was fixed on top of the R44 transition of the S1 state of the benzene-Ar complex and the dump laser frequency scanned in the region where allowed rotationally resolved downward transitions are expected. (Here s denotes the totally symmetric intermolecular stretching mode, b the degenerate bending mode, and l b in Fig. 10 the vibrational angular momentum of the degenerate bending mode). Four prominent dips appear in the ion dip spectra that can be assigned to the four allowed rotational transitions in each case. Additional CIS spectra have been measured via different van der Waals vibrational states in the SI electronic state to a series of van der Waals vibrational states in the electronic ground state SOof the complex. In Fig. 11 an overview of all investigated transitions in the benzene (C6&)-Ar complex is given demonstrating the high sensitivity of CIS. Seven hitherto unknown transitions up to a van der Waals excess energy of 130 cm-' have been observed [30], and their rotationless frequency positions have been determined with a high accuracy of 0.03 cm-' . Recent experiments concentrate on the investigation of strongly perturbed bands like the 6's' band in c6D6-A~ using two-dimensional CIS, scanning both lasers independently as depicted in the calculation shown in Fig. 9. A further goal is to determine the rotational constants of the different van der Waals states. The effective rotational constants represent a value averaging over the vibrational motion and thus information about the vibra-

432


100 Oh-

50 % -

0 Yo-

61b2(Ib-O), (J'=K'=4,+/) 1-

+

6,b2(/':=0)

100 %-

50 % -

0 %-

Dump Frequency

-

I )

Figure 10. Experimental CIS spectra of the 6's1(J' = K' = 4,+1)-62sl (upper trace) 62b2(lL = 0) transition (lower trace) in transition and the 6'b2(1; = 0).(J' = K' = 4, +1) benkene-Ar ( C 6 b .Ar). Four dips in each spectrum are shown indicating single rovibrational transitions with different Hoed London factors. The iodine spectrum on top of each spectrum was synchronously recorded for absolute frequency calibration.

tional wave function. Theory predicts a strong mixing of the stretching and bending vibrations for higher states in benzene-Ar [67,681. Experimental evidence for mixing of the intermolecular bending and stretching modes has been found in the S1 state of para-difluorobenzene-Ar by an analysis of the


Van der Waals internal energy 0 20 40 60 80

100

433

[CHI]

m. Figure 11. Investigated van der waals transitions in the benzene-t\r (CgH6 . Ar) complex. The high sensitivity of the CIS allows the detection of weak transitions to hitherto unknown high-lying van der Waals states in the electronic ground state yielding precise information about their frequency.

rotational constants obtained from rotationally resolved REMPI spectra of different van der Waals states [69, 701.

IV. INTRAMOLECULAR DYNAMICS OF HIGH RYDBERG STATES IN POLYATOMIC MOLECULES A. General Remarks Sections I1 and I11 focused on the intramolecular dynamics occumng in low electronic potential surfaces (SO,SI, T I ) .For higher electronic valence states and low Rydberg states a drastic shortening of lifetimes due to fast dynamic processes has been observed. Qpical time constants in these energy regions are found to be in the subpicosecond range [71, 721. At very high excitation energies, close to the ionization continuum, there exist electronic states resembling the Rydberg states of hydrogen that are expected to be long-lived from the scaling laws [73]. The Rydberg series consist of electronic states with a single electron excited into a quasi-classical orbit around the positively charged molecular core with high Rydberg quantum number n. In this case the energetic separation of neighbored electronic states (i.e., Rydberg states with different hydrogen quantum numbers n, 1) becomes smaller than the spacing of vibrational or even rotational levels of the molecular core. This situation is depicted in the level scheme of Fig. 12 with three Ryd-

434


Figure 12. Level scheme of the rotationally resolved high-n Rydberg experiment. A first narrow-band laser pulse excites the molecule from the electronic ground state So into a single rotational state in the electronically excited SI state. The frequency of the second laser pulse is scanned to obtain the rotationally resolved Rydberg spectrum shown in Fig. 13.

berg series converging to different rotational energy levels of the molecular core. The term N + denotes the rotational angular momentum of the molecular core. (Note that the other relevant rotational quantum numbers, such as the projection K + of the core angular momentum N + on the symmetry axis of a symmetric top molecule, are omitted for simplification.) In this section we address the question of whether high Rydberg states can be selectively excited in a polyatomic molecule by high-resolution techniques to investigate their dynamic behavior, High Rydberg states play an important role in spectroscopic techniques utilizing pulsed-field ionization 1731for the investigation of the threshold ionization processes. Recent developments include the detection of electrons with zero kinetic energy (ZEKE) [74, 751 and the mass-analyzed detection of threshold ions (MATI) [76, 771 after pulsed-field ionization of high long-lived Rydberg states. While these techniques have been successfully applied to a variety of molecules and clusters leading to spectroscopic and dynamic information on the ionic ground state, little is known about the nature of the high Rydberg states of these systems involved in the excitation process. A major point of interest has been the discrepancy between the lifetimes found from extrapolation of the linewidth of low Rydberg levels and results for high Rydberg levels deduced from unresolved pulsed-field ionization experiments [78-801. This discrepancy has initiated the discussion of different mechanisms leading to a lengthening of the lifetime of high-n Rydberg states [81-881.


435

B. Experimental

The experimental setup used for the investigation of high-n Rydberg states is similar to that used in our coherent ion dip experiments (see Fig. 8). The main difference is that no external field is applied during excitation with the two overlapping laser pulses. Typically 100 ns after excitation of the molecule by the two laser pulses a pulsed electric field of 100 V/cm is turned on for 10 ps to field ionize the high-lying Rydberg levels. The produced ions are mass analyzed in a TOF mass spectrometer and detected with multichannel plates. Because of the lower laser power and smaller transition strength to the Rydberg series as compared to transitions investigated in CIS experiments (see Section III), coherent effects can be neglected. In Fig. 12 a typical doubleresonance excitation scheme is shown. In a first step the molecule is excited to a single rotational state in the electronically excited S1 state by absorption of a photon with energy hv 1 provided by one of the pulsed amplified single6’, J’ = K’ = 1(-1) mode CW laser systems shown in Fig. 8 (e.g., into the SI, state of benzene (C6D6). The symbols J’ and K’ denote the symmetric top quantum numbers and - I the lower lying level of the degenerate 6’ vibration of benzene split by Coriolis coupling [22]). By absorption of a photon with energy hv2 from the second pulsed amplified narrow-band laser system, the molecule is excited into individual states in the Rydberg series converging to different limits above the lowest ionization energy (IE) with N + = 0. Keeping the first laser frequency V I in resonance with a single rovibronic transition frequency to the S1 electronic state and scanning the frequency vq of the second laser yield the Rydberg spectrum presented in Fig. 13. The doubleresonance excitation scheme with sub-Doppler resolution in both excitation steps ensures that Rydberg series excited from different intermediate rovibronic levels do not overlap in the high-n region.

C. Experimental Results In Fig. 13 the Rydberg spectrum of benzene (C6D6) pumped via the lower component of the Coriolis-split SI, 6 ’ ,J’ = K’ = 1, ( - I ) intermediate state is shown demonstrating the resolution of Rydberg series for n > 100. A more detailed investigation of the spectrum reveals several weak series overlapping with the main series. All series can be identified throughout the complete measured spectra [89] and fit very accurately to the Rydberg formula

with a quantum defect p c 0.01. In Eq. (6) IE denotes the series limit and Rknzenethe mass-corrected Rydberg constant for benzene.


C,D,

I d n=89

Pumped via 610,P11

74555.0

74565.0

74575.0

Total Energy [cm-’1 Figure 13. Rydberg spectrum of Q D 6 obtained by pumping via the P I1 transition to the SI,6 ’ , .I’= K’ = 1 , ( - I ) rovibronic intermediate state. Note that Rydberg states up to n > 100 are resolved.

In Fig. 14 the magnified central part of the spectrum shown in Fig. 13 is presented together with the positions of the Rydberg peaks of two series calculated using Eq. (6). The assignment of the series limits leads to accurate values of the lowest adiabatic ionization energy of the benzene (C6D6) molecule and the rotational constants of its cation 1891. The width of the Rydberg peaks is about 1 GHz and larger by a factor of 10 than expected from the laser resolution. It is caused by Stark splitting of the n levels induced by small permanent residual stray fields present in our apparatus. Individual Rydberg peaks can no longer be resolved when the Stark splitting becomes larger than the distance of two neighbored Rydberg peaks of a series. In a simple model the Stark splitting increases with n2, and the distance of the Rydberg peaks decreases with l/n3. For the highest resolvable n = 110 we find from l/n3 = 3Fn2 an upper limit for the residual field of 50 mV/cm. When the pulsed-field ionization signal of a single excited Rydberg peak is measured as a function of the delay time between the extraction field pulse

437


,

CI

C

Q)

f

0

C

0

:

- 74563.0

74565.0

L

I

74567.0

74569.0

Total Energy [cm-’1

Figure 14. Magnified central part of the Rydberg spectrum of C& shown in Fig. 13. The filled circles indicate the positions for two Rydberg series converging to different core rotational states of the ion.

and the laser pulse, a decay of the signal is found that can be interpreted as a “lifetime” of the Rydberg level. Varying the delay in the 30-1000 ns range we found a multiexponential decay behavior with a fast decay component of some 10 ns and very long lived components. In recent experiments we were able to show that even for n = 58 there are Rydberg states that survive 45 ps delay time. This decay time is longer by a factor of 40 than extrapolated from the 500-fs lifetime [72] of low Rydberg states with n = 5 using the scaling law. Several models have been proposed to explain this lengthening of the Rydberg lifetime [82-881. All models include external effects such as I-mixing through the Stark effect in an electric field or rn-mixing through collisions. In these models fast optically accessible light zero-order states with small I-quantum number are coupled to dark zero-order states with larger Z-quantum number similar to the intrastate coupling scheme of the S, state discussed in Section Il. However, while in the S1 state the optically accessible light state is long-lived and coupling occurs to short-lived dark background states, in the Rydberg region the light states are short-lived and coupling occurs to long-lived dark states. Thus a lengthening of the lifetime is expected for Rydberg states rather than a shortening as in the SIstate.

438


This coupling leads to another interesting aspect concerning the measured decay time of the pulsed-field ionization signal: The rotationally resolved Rydberg states presented in Figs. 13 and 14 are states with a single n but consist of a manifold of Stark states that can be expressed as linear combinations of zero-order Rydberg states with the same molecular core rotational quantum numbers but different angular momentum quantum numbers 1 of the Rydberg electron. These zero-order Rydberg states may exhibit different optical accessibilities and different decay times. As the observed Rydberg peaks in the spectra are caused by the overlap of different Stark states with different decay times and optical accessibilities, this can result in a nonexponential decay behavior, as investigated theoretically in Ref. 88, and is expected to lead to changes of the “line shape” of the Rydberg peak measured for different delay times of the pulsed field as demonstrated in a recent high resolution experiment [89].

V. CONCLUSION We have shown that the investigation of the state-selected intramolecular dynamics of polyatomic molecules and their van der Waals complexes is now feasible in different energy regimes up to the ionization energy. At low excess energies (522 em-’) in SIthe intramolecular dynamics in the benzene molecule can be described by intrastate coupling of pairs of rovibronic levels in S1,leading to quasi-eigenstates. The zero-order levels are coupled to a quasi-continuum of levels in the T I or SOstate (interstate coupling), leading to an exponential decay. As the dark zero-order states have a shorter lifetime than the optically accessible light zero-order states, the intrastate coupling that is found to be induced by Coriolis forces results in a shortening of the lifetime of the quasi-eigenstates. When the intrastate couplings to short-lived dark background states become more frequent, this leads to a further shortening of the lifetime and a rapid decrease of the fluorescence quantum yield at higher excess energy. The intramolecular dynamics at high excess energies is not substantially affected in the benzene-& complex by the attachment of the Ar atom. Predissociation of the complex does not play a major role, though internal energy exceeds the dissociation energy. Evidence is found for an enhancement of the intersystem crossing rate by the external heavy-atom effect. We have presented a new technique for the investigation of intramolecular couplings in the electronic ground state SO. The new technique of CIS is based on the special population dynamics induced by the coherent excitation of a three-level system with two narrow-band Fourier-transform-limited laser pulses. It allows the investigation of high-lying intermolecular vibrational states in the electronic ground state of van der Waals complexes. These


439

states are found to be strongly mixed by a coupling between the bending and stretching van der Waals modes. In this way, in the low-energy region basic information on the relevant coupling mechanisms is obtained, leading to an energy flow within and between the subsets of intramolecular and intermolecular modes in a weakly bound van der Waals complex. In high Rydberg states (n > lOO), which were resolved for a large polyatomic molecule in the present work, the intramolecular dynamics is governed by interstate coupling between a manifold of close-lying electronic levels. The intramolecular dynamics in this energy range is influenced by external effects: Small electric stray fields present in every apparatus utilizing field ionization for Rydberg state detection lead to an externally induced mixing of many Rydberg states with identical principal quantum number n but different 1 angular momentum quantum numbers. This results in a lengthening of the lifetime as compared to the value expected for the zero-order optically accessible light state with low 1 quantum number and is an important characteristicof pulsed-field ionization experiments. Internal intramolecular couplings due to the frequent crossing of close-lying Rydberg series in polyatomic molecules may be observable after suppression of external fieldinduced effects. In conclusion in this work we have presented basic information on the nature of the coupling processes leading to the intramolecular dynamics in isolated molecules. This information is useful for the understanding of the origin and mechanism of the fast femtosecond energy flow in high valence and low Rydberg states.

Acknowledgments The authors thank R. Sussmann for helpful discussions. Financial support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

References R. A. Marcus and 0. K. Rice, J. Phys. Colloid Chem. 55, 894 (1951). D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Ann. Rev. Phys. Chem. 32,267 (1981). E. Riedle and H. J. Neusser, J. Chem. Phys. 80,4686 (1984). U. Schubert, E. Riedle, and H. J. Neusser, J. Chem. Phys. 84, 5326 (1986). R. Sussmann, R. Neuhauser, and H. J. Neusser, J. Chem. fhys. 100,4784 (1994). 6. R. Neuhauser, R. Sussmann, and H. J. Neusser, Phys. Rev. Lett. 74, 3141 (1995). 7. K. F. Freed, in Topics in Applied Physics, F. K. Fong, Ed.,1976, Springer, Berlin. 8. P. Avouris, W. M. Gelbart, and M. A. El-Sayed, Chem. Rev. 77, 793 (1977). 9. M. Bixon and J. Jortner, J. Chem. Phys. 50,4061 (1969). 10. M. Lefebvre-Brion and R. W. Field, Comments Af. Mol. Phys. 11, 37 (1981). 11. J. Jortner and S. Berry, J. Chem. Phys. 48,2757 (1968). 12. P. M. Felker and A. H. Zewail, Chem. Phys. Lett. 102, 113 (1983). 1. 2. 3. 4. 5.

440


13. E. M. Anderson and G. B. Kistiakowski, J. Chem. Phys. 48,4787 (1968). 14. A. E. Douglas and C. W. Mathews, J. Chem. Phys. 48,4788 (1968). 15. C. S. Parmenter and A. M. White, J. Chem. Phys. 50, 1631 (1969). 16. G. W. Robinson, J. Chem. Phys. 47, 1967 (1967). 17. J. Jortner, S. A. Rice, and R. M. Hochstrasser, Adv. Photochem. 7, 149 (1969). 18. J. H. Callomon, J. E. Parkin, and R. Lopez-Delgado, Chem. Phys. Lett. 13, 125 (1972). 19. U. Schubert, E. Riedle, H. J. Neusser, and E. W. Schlag, Israel J. Chem. 30, 197 (1990). 20. E. Riedle, H. Stepp, and H.J. Neusser, Chem. Phys. Lett. 110,452 (1984). 21. U. Schubert, E. Riedle, and H. J. Neusser, J. Chem. Phys. 90, 5994 (1989). 22. E. Riedle, Th. Knittel, Th. Weber, and H. J. Neusser, J. Chem. Phys. 91, 4555 (1989). 23. L. Wunsch, H. J. Neusser, and E. W. Schlag, Chem. Phys. Lett. 32, 210 (1975). 24. E. Riedle, H. J. Neusser, and E. W. Schlag, J. Phys. Chem. 91,4847 (1982). 25. A. Helman and R. A. Marcus, J. Chem. Phys. 99,5002 (1993). 26. A. Helman and R. A. Marcus, J. Chem. Phys. 99,5011 (1993). 27. U. Schubert, E. Riedle, H. J. Neusser, and E. W. Schlag, J. Chem. Phys. 84,6182 (1986). 28. Th. Weber, A. von Bargen, E. Riedle, and H. J. Neusser, J. Chem. Phys. 92,90 (1990). 29. H. J. Neusser, R. Sussmann, A. M. Smith, E. Riedle, and Th. Weber, Bec Bunsenges. Phys. Chem. 96, 1252 (1992). 30. R. Sussmann, R. Neuhauser, and H. J. Neusser, J. Chem. Phys. 103, 3315 (1995). 31. E. Riedle, R. Sussmann, Th. Weber, and H. J. Neusser, J. Chem. Phys., 104, 865 (1996). 32. H. Krause and H. J. Neusser, J. Chem. Phys. 99, 6278 (1993). 33. H. J. Neusser and H. Krause, Chem. Rev. 94, 1829 (1994). 34. P. Hobza, 0. Bludsky, H. L. Selzle, and E. W. Schlag, J. Chem. Phys. 97, 335 (1992). 35. N. B. Horin, U. Even, and 1. Jortner, J . Chem. Phys. 91,331 (1989). 36. Th. Brupbacher, J. Makarewicz, and A. Bauder, J. Chem. Phys. 101,9736 (1994). 37. Th. Weber, E. Riedle, H. J. Neusser, and E. W. Schlag, Chem. Phys. Latt. 183, 77 (1991). 38. Th. Maushart, Diplom Thesis, Fachbereich fur Chemie, Biologie und Geowissenschaften der Technischen Universitat Munchen, 1990. 39. G. E. Ewing, J. Phys. Chem. 90, 1790 (1986). 40. T. A. Stephenson and S. A. Rice, J. Chem. Phys. 81, 1083 (1984). 41. Th. Weber, Ph.D. Thesis, Fachbereich fur Chemie, Biologie und Geowissenschaften der Technischen Universitat Munchen, 1991. 42. M.Kasha, J. Chem. Phys. 20,71 (1952). 43. G. W. Robinson and R. P. Frosch, J. Chem. Phys. 38, 1187 (1963). 44. Y. P. Hsu and P. M.Johnson, 1. Chem. Phys. 59, 136 (1973). 45. A. Amirav, U. Even, and J. Jortner, Chem. Phys. Let?. 67,9 (1979). 46. N. B. Delone and V. I? Krainov, Afoms in Strong Light Fields, Springer-Verlag, Berlin, 1980. 47. M. Shapiro and P. Brumer, J. Chem. Phys. 84,4103 (1986). 48. Z. Chen, M. Shapiro, and P. Brumer, Chem. Phys. Lett. 228,289 (1994). 49. D. J. Tannor and S . A. Rice, J. Chem. Phys. 83, 5013 (1985).


44 1

50. V. S. Letokhov and V. P. Chebotayev, Nonlinear Laser Spectroscopy, Springer, Berlin, 1977.

5 1. Special issue, Molecular Spectroscopy and Dynamics by Stimulated-Emission Pumping, (J. Opt. SOC.Am. B 7, 1990). 52. C. E. Hamilton, J. L. Kinsey, and R. W. Field, Ann. Rev. Phys. Chem. 37,493 (1986). 53. T. Ebata, M. Furukawa, T. Suzuki, and M. Ito, J. Opt. SOC.Am. B 7 , 1890 (1990). 54. B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, New York, 1990. 55. W. L. Meerts, I. Ozier, and J. T. Hougen, J. Chem. Phys. 90,4681 (1989). 56. H. G. Rubahn, E. Konz, S. Schiemann, and K. Bergmann, Z Phys. D 22,401 (1991). 57. Y.B. Band and P. S. Julienne, J. Chem. Phys. %, 3339 (1992). 58. A. F. Linskens, N. Dam, J. Reuss, and B. Sartakov, J. Chem. Phys. 101,9384 (1994). 59. R. Neuhauser and H. 1. Neusser, J. Chem. Phys. 103,5362 (1995). 60. G. W. Coulston and K. Bergmann, J. Chem. Phys. 96,3467 (1992). 61. F. T. Him, Phys. Lett. 99A, 150 (1983). 62. U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, J. Chem. Phys. 92, 5362 (1990). 63. A. Kuhn, G. W. Coulston, G. 2.He, S. Schiemann, and K. Bergmann, J. Chem. Phys. 96, 4215 (1992). 64. Y. B. Band and 0. Magnes, J. Chem. Phys. 101,7528 (1994). 65. Th. Weber, E. Riedle, and H. J. Neusser, J. Opt. SOC.Am. B 7 , 1875 (1990). 66. B. T. Darling and D. M. Dennison, Phys. Rev. 57, 128 (1940). 67. Ad van der Avoird, J. Chem. Phys. 98, 5327 (1993). 68. E. Riedle and Ad van der Avoird. J. Chem. Phys., 104, 882 (1996). 69. R. Sussmann, R. Neuhauser, and H. J. Neusser, Chem. Phys. Lert. 229, 13 (1994). 70. R. Sussmann and H. J. Neusser, J. Chem. Phys. 102,3055 (1995). 71. J. M. Wiesenfeld and B. I. Greene, Phys. Rev. Lett. 51, 1745 (1983). 72. R. L. Whetten, S. G. Grubb, C. E. Otis, A. C. Albrecht, and E. R. Grant, J. Chem. Phys. 82, 1I15 (1985). 73. F. B. Dunning, in Rydberg Stares of Atoms and Molecules, R. F. Stebbings and F. B. Dunning, Eds., Cambridge University Press, 1983. 74. G. Reiser. W. Habenicht, K. Muller-Dethlefs, and E. W. Schlag, Chem. Phys. Lett. 152, 119 (1988). 75. F. Merkt and T. P. Softley, Phys. Rev. A 46, 302 (1992). 76. L. Zhu and P. M. Johnson, J. Chem. Phys. 94, 5769 (1991). 77. H. Krause and H. J. Neusser, J. Chem. Phys. 97,5923 (1992). 78. U. Even, R. D. Levine, and R. Bersohn, J. Phys. Chem. 98,3472 (1994). 79. F. Merkt, J. Phys. Chem. 100, 2623 (1994). 80. S. T. Pratt, J. Chem. Phys. 98, 9241 (1993). 81. C.Bordas, P.F. Brevet, M. Broyer, J. Chevaleyre, P. Labastie, and J. P.Petto, Phys. Rev. Lett. 60, 917 (1988). 82. W. A. Chupka, J. Chem. Phys. 98,4520 (1993). 83. D. Bahatt, U. Even, and R. D. kvine, J. Chem. Phys. 98, 1744 (1993). 84. W.A. Chupka, J. Chem. Phys. 99,5800 (1993).

442

H. 1. NEUSSER AND R. NEUHAUSER

85. E. Rabani, L. Y. Baranov, R. D. Levine, and U. Even, Chem. Phys. Lett. 221,473 (1994); F. Remacle and R. D. Levine, Chem. Phys. Letters 257, 1 1 1 (1996). 86. M.J. J. Vrakking and Y. T. Lee, Phys. Rev. A 51, R894 (1994). 87. M.J. J. Vrakking and Y. T. Lee, J. Chem. Phys. 102, 8818 (1995). 88. M.Bixon and J . Jortner, L Phys. Chem. 99,7466 (1995). 89. R. Neuhauser, K. Siglow, H. J. Neusser, J. Chem. Phys. 106,896 (1997).

DISCUSSION ON THE REPORT BY H. J. NEUSSER Chairman: E. Pollak

J. Manz: Prof. H. J. Neusser has presented to us beautiful highresolution spectra of medium-size molecules and clusters such as benzene and C6H6- At (see current chapter). The individual lines have been assigned to individual rovibronic eigenstates of the systems, and their widths have been interpreted in terms of various intramolecular processes between zero-order states (e.g., Coriolis coupling, anharmonic couplings between bright and dark states, and so on). Now let me be the advocate of the molecule, which does, in fact, not know anything about zero-order states. Instead, it has molecular eigenstates, at least in the ideal case of decoupling from the radiation field. Ultrahigh resolution spectroscopyof these eigenstates should yield spectra with zero linewidth, in contrast with the experimental results. I would like to ask, therefore, for an interpretation of the width of the observed spectral lines, not in terms of zero-order states, but in terms of molecular eigenstates. My guess is that the observed spectral width of molecular eigenstates can only be due either to couplings to the radiation field or to continuum states due to (pre-) dissociation but not to Coriolis coupling or anharmonic couplings of vibrational states. H. J. Neusser: As an approximation I separated intrastate coupling within the S1 state from the interstate coupling to the TI or So state. Eigenstates resulting from intrastate coupling are eigenstates with respect to the S1 state (“quasi-eigenstate”). From the J , K independent exponential decay it is concluded that the interstate coupling in benzene is in the statistical limit and thus leads to a Lorentzian line shape. In a completely isolated molecule with a finite density of triplet states, the individual eigenstates could be resolved (i) if the spontaneous lifetime is very long so that the zero-order states are not broadened and do not overlap and (ii) for a very high resolution of the exciting laser light. In real systems, like benzene in our experiment, the spontaneous lifetime


443

and collisions (and the limited spectral resolution) lead to a broadening of lines and make the observation of real eigenstates impossible. M. Quack: Relating to the discussion by Neusser and Manz, I should like to point out that in highly excited states of polyatomic molecules (such as benzene) the true spectrum is a continuum, and it thus makes little sense to talk about “true” discrete molecular eigenstates. This is rather a useful, simplified picture. The continuum nature arises from spontaneous emission [important in the IR, when the average distance between levels is smaller than the spontaneous emission linewidth (about cm-’ in the IR emission)]; there is furthermore a continuum due to possible dissociation channels in most polyatomic molecules. Finally, there is always coupling to the outside world by collisions and ultimately gravitation. Thus, while there are several useful levels of approximate descriptions of polyatomic molecules, the ultimately correct one would be a time-dependent state in the continuum, not an “eigenstate” of the energy. I have given a related discussion concerning IVR (intramolecular vibrational redistribution) in 1981

PI.

1. M. Quack, Faraday Disc. Chem. SOC. 71, 359 (1981).

L. Woste: Prof. Neusser, you are able to select specific rotational states at very high specific Rydberg state quantum numbers n. So your orbiting electron becomes very slow, even slower than the molecular rotation. Do you see any chance to bring both into phase, causing stroboscopic effects? H.J. Neusser: In the energy regime this condition is fulfilled for a close spacing of the two Rydberg series converging to different rotational quanta of the cation. I have shown in one of my slides that a frequent crossing of Rydberg series is expected for a large molecule with small rotational constants. At the crossing point the Rydberg lines from two series can be close to each other. This leads to a coupling of the two series. In particular, this crossing of two rotational Rydberg series with small An becomes possible. R. W. Field: Concerning the question by Prof. Woste about the possibility of efficient energy transfer between ion-core rotation and the Rydberg electron, Prof Neusser mentioned following two Rydberg series that converge to two different rotational levels of the ion. He also noted the location of the An = 1,2,3,. . . series of level crossings between the two series. In fact, these level crossings correspond to the situation where the Rydberg orbit period is equal to the rotational period, twice the rotational period, three times, and so on. The

444


classical period corresponds to h/AE, where AE is the quantum level spacing. V. Engel: Prof. Neusser, you mentioned the technique of Stimulated Raman Rapid Adiabatic Passage STIRAP, which allows for the coherent transfer of vibrational population selectively. Is the “technique” not another very efficient and experimentally verified scheme of coherent control? H. J. Neusser: Choosing a special pulse sequence of the dump and the pump laser pulse leads to a complete blocking of the population transfer in the CIS experiment or else makes it very efficient. We can say that a special channel is open or closed, that is, controlled by the experimental parameter. This is similar to STIRAP experiments. However, it was shown by Band and Magnes [l] that the adiabatic passage population transfer in STIRAP experiments does not represent a solution of an optimal control problem. 1. Y. B. Band and 0. Magnes, J. Chem. Phys. 101,7528 (1994).

T. Kobayashi: Let me ask two questions of Prof. Neusser: 1. Is it possible to observe a shift in coherent Raman scattering in the three-level system with A-type coupling? We have done an experiment to obtain a femtosecond Raman gain spectrum in polydiacetylenes. The Raman spectrum is shifted to the red under increased , amplification peak signal is pump ( w , ) intensity. By changing ~ 2 the to be shifted to lower frequency. If the optical Stark effect is observed, then, in principle, it should be possible to observe the effect of a high field on the coherent Raman spectrum (see Fig. 1).

a1

7

02

II 01

w2

__

amplification OfW

weak field

I

1 1

/ -

01

---_ Figure 1.

9

01

amplificationofq ’~mmgainsignd

strong field

-

2. Why is the dump frequency exactly resonant with the (2)

13)


445

transition (see Fig. 2)? Why were Mowllow triplets not observed? In other words, why is only the splitting observed, without evidence of the triplet?

Figure 2.

H. J. Neusser 1. Concerning the first question of Prof. Kobayashi, I think that shifts of Raman lines by interaction with other levels than the investigated ones should be observable in the Raman process. A pronounced shift is expected, for example, if one of the laser frequencies is close to a resonant multiphoton transition to another rovibronic state. 2. Refemng to the second question, let me point out that in a CIS experiment we do not monitor the fluorescence from the split level 2 to the split level 3, which would lead to an equally spaced triplet in the fluorescence spectrum.

The CIS “trapping experiment” monitors the population in level 2 by an incoherent ionization step. In the weak-field limit of the pump laser this can be compared to an experiment detecting the fluorescence from level 2 to a spectator state (here level 1) at the field-free resonance frequency between levels 1 and 2. In such an experiment no fluorescence is expected at this frequency due to the strong shift of level 2 for resonant dump laser frequency.

K. Yamanouchi: I would like to make the comment that, in the Rydberg series of a molecule, as the principal quantum number n increases, the energy spacings between adjacent Rydberg states become close to those between vibrational levels. In such a situation, an effective energy exchange could occur between the motion of a Rydberg electron and the vibrational motion of the ion core. If the total energy is above the ionization threshold, vibrational excitation would cause autoionization in an efficient way, such that the vibrational quantum number decreases in a stepwise manner and the vibrational energy is transferred to the Rydberg electron, resulting in a relatively large nega-

446


tive change in u, that is, Au = Uion - my& where URyd and Uion represent vibrational quantum numbers of the Rydberg state and of the ion produced after the autoionization, respectively. Recently, we investigated the Rydberg states of a HgNe van der Waals dimer by an optical-optical double-resonance method and found autoionization channels associated with Av = -3, -2 besides Av = - 1 in the energy region just above the ionization threshold. This observation may be interpreted in terms of an efficient energy exchange between a Rydberg electron and an ion-core vibration. This type of energy exchange in an autoionization process may correspond with the behavior of a kicked rotator in classical mechanics, which is known to exhibit chaos. It would be worthwhile to consider an autoionization process of a simple diatomic molecule in its Rydberg states to understand experimentally the essential dynamics of a quantum system, whose classical counterpart exhibits chaos. H. J. Newer: In relation to the comment by Prof. Yamanouchi, we should notice that an efficient interaction of the Rydberg electron with vibrations of the core is expected for small vibrational frequencies. Benzene as a rigid molecule has relatively large vibrational frequencies of more than 300 cm-' . An efficient coupling is expected for van der Waals complexes (e.g., the benzene-Ar complex) with low van der Waals vibrational frequencies of about 30 cm- I . T. P. Softley: The energy flow in high Rydberg states between electronic and nuclear motions is of active interest with regard to lifetime lengthening. Do you see any evidence in your high Rydberg spectra of benzene for coupling between Rydberg series converging to different rotational states of the ion, either under the lowest field conditions or when higher fields are applied? H. J. Neusser: For a selected intermediate J;, state we observe a couple of Rydberg series; for example, for J k , = l I we can identify two series under minimum residual field conditions. When we apply a stationary electric field of 300 mV/cm, additional series appear that are coupled by the electric field. All series have different limits representing different rotational states of the benzene cation. At present we cannot say whether the coupling observed under minimum residual field conditions is induced by the small stray field or by field-free intramolecular coupling. M.S. Child: The first part of the talk by Prof. Neusser concerned intramolecular interaction between the SI and T I states of benzene, and the second part referred to the high Rydberg spectrum. Each member of this singlet series must have a corresponding triplet, such that


447

the singlet-triplet energy separation is predicted on general grounds to decrease as l/n3. Is there any possibility of observing intersystem crossing at relatively low n values? If so, variation of n would provide a method for tuning through the vibrational manifold of the triplet. H. 3. Neusser: In this regard, I should say that, up to now, the assignment of Rydberg triplet states has not been possible. In our highresolution spectra only Rydberg series with a small quantum defect have been observed. It appears to be questionable whether triplet series become visible through spin-orbit coupling, which is expected to be small in a molecule like benzene. Furthermore, we expect that intersystem crossing plays a minor role in the nonradiative relaxation of high Rydberg states in benzene.


GENERAL DISCUSSION ON INTRAMOLECULAR DYNAMICS Chainnun:E. Pollak

V. S. Letokhov: Let me make two comments. My first comment is about terminology and the second about the possibility to laser control the intramolecular vibrational distribution (IVR) rate. 1. Prof. H. Neusser introduced the term “coherent ionization dip.” Talking about the coherence in an ionization process we should distinguish two different schemes. One scheme is based on two-frequency coherent control of a two-level A scheme (see Fig. l), which was discussed by Prof. H.Neusser.

Here coherence occurs only for three low-lying quantum states. The 449

450

GENERAL DISCUSSION

effect of blocking of the population of excited level 3 is related to a well-known effect in laser optical physics: “coherent population trapping.” Two coherent fields of frequencies w 1 and w 2 create a coherent superposition of atomic states 1 and 2. Under proper conditions the atom in such a superposition state cannot absorb radiation and will stay in the low-lying states. The ionization channel has no coherence. Ionization serves as a probing of the population of excited state 3 only. Another scheme exploits rhe coherence in ionization. Let us consider the following scheme (see Fig. 2).

Figure 2.

Suppose we have two close intermediate states, 2 and 3. In a two-€requency (a1 , w2) laser field the atom has two nondistinguishable channels of photoionization. Depending on the phase relationship, the quantum interference of the two channels can be “constructive” or “destructive.” In the latter case, the yield of photoionization can be greatly reduced. This effect is related to the well-known “Fano resonances” in atomic photoionization spectra. Also the interference will generate

P

VI c

Nonresonant modes

Stochastitation onset

I

\ ?'

1

I

1

-

Broadening 7 of absorption band

Frequency shift 6 due to intermode anharmonicity

001

IR Absorption spectrum

iI

I1

'

\ I

I

0

Frequency shift due to intramode anharmonicity

0

j!+l

$

Figure 3. Model of IR MP E/D of plyatomic molecules.

Resonant mode

0-

Resonant 4= modeselective interaction with discrete vibrational-rotational transit ions

*--->

*----+

Ferrni Range resonances, of high-density transition to vibrational stochastization

Quasi-resonant noncoherent mode-nonselective interaction with systems of coupled vibrational modes

Dissociation limit

Range of transient vibrational overexcitation

Dissociation of weakest bond

452

GENERAL DISCUSSION

12

2

13,800 20,700 Vibrational energy, E (crn-l) Figure 4. Theoretical dependence of rate of IVR as a function of vibrational 6900

energy of model molecule CFC12Br with given anharmonicity constant Xi,k (from Ref. 1).

quantum beats of the ionization probability, with the frequency of the energy splitting of levels 2 and 3. 2. Let me raise the following question: Can we control or modify the IVR rate by intense femtosecond laser pulses? For the laser coherent control of a chemical reaction we need ultrashort laser pulses with duration T~
40

Figure 9. Vibrational level spacing for HCP 2 C,AzG(u2) = C(0,u2 + l,O)G(O, u2 1,O) plotted (open circles) vs. the bending quantum number m.Also plotted l(open squares) is the reduced A2C. A2G(9) - 2 x 2 2 ~ 2 which , would be a horizontal line for a Morse oscillator.

R. W. FIELD et al.

486 0.70

.

0

d

10

20

30

40

ua

Figure 10. E ( o , ~ ~ , vs. o)

u2

for HCP. Values plotted are partially deperturbed [S].

direction and increases by 10% to B o , ~ , =o 0.705 cm-’ and then reverses direction again to B0,42,0 = 0.695 cm-’ (see Fig. 10). Although these changes in B may seem small to a nonspectroscopist, one should keep in mind that complete removal of the H at constant rcp would cause B to increase by only 18%! The rapid change in B o , ~ ~is, ?probably due to neglect of Coriolis interaction of (0, u2, 0)I = 0 levels with (1, u2 - 3, 0)E = 1 levels. Near the eH(Cp) bond angle of the HCP H HPC saddle point, WHX decreases from - 5uhnd to slightly less than 3 W k n d and then increases to slightly more than 3 W k n d [16]. When the WHX : @bend ratio crosses through 3 : 1, the Coriolis effects on Fffreach their maximum value and then change sign, exactly as is observed in the spectrum. This Coriolis hypothesis is borne out by the u2 dependence of all of the other “vibrational fine structure” constants (42,g22, yli) [5]. All of these constants are nearly independent of u2 up to u2 = 36, where they suddenly begin to change rapidly. The q 2 constant, which controls the A1 = 2 interaction between 1 = 0 and 1 = 2 components of the same (u, ,u2, u g )vibrational state, is affected by the same A1 = +1 Coriolis interaction that caused the rapid . 10 and 11 show the nearly identical u:! depenvariation of B o , ” ~ , oFigures ,~. 12, which is a plot of q2(u2) versus B o , ~ , o . dence of q2 and B O , , , ~Figure shows that despite the erratic u2 dependences of both q2 and B, the q 2 versus B plot is linear. This indicates that the changes in both constants originate in the same Coriolis interaction! The onset of sudden variations in vibrational fine structure is one of the most sensitive indicators of a change in resonance structure. The magnitudes of fine-structure parameters are determined by second-order perturbation theory (a Van Vleck or “contact” transformation) [17]. The energy denominators in these second-order sums over states are approximately independent of EVIBas long as the o 1 :w2 : .. .W 3 N - 6 resonance structure is conserved.

INTRAMOLECULAR DYNAMICS IN THE FREQUENCY DOMAIN

487

2.2,

1.2L

0

-

... . .

(3.11)

where r = 1, 2, .. , is the repetition number of the prime period. As a function of energy, the period is constant near the minimum of the potential and increases as the energy approaches the dissociation threshold, with the potential progressively broadening. At the threshold, the periodic orbit turns into the separatrix between bounded and scattering trajectories, which is known to have an infinite period [14]. Separatrices also exist if the potential has equilibrium points other than a minimum. If the potential has a maximum, the corresponding separatrix is at the origin of a so-called homoclinic bifurcation of the periodic orbit. The effect of the separatrices can be observed in the vibrogram representation. In one-dimensional systems, the amplitude Apr of the periodic orbits is constant with respect to the repetition number r, as shown by (2.34), when F = 1, so that recurrences due to repetitions of the prime orbit are observable.

a.

1. Morse-Type Model for I z ( 2 'C)

Figure 2 depicts the vibrogram corresponding to the dynamics on the ground state of iodine, modeled by a Morse potential with the equilibrium distance r = 2.67 A and the dissociation energy D = 12,542 cm-' [14, 1081. The periodic orbit and its repetitions clearly appear in the vibrogram. The classical

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

525

1345

0

0

E [cm-'1

12600

Figure 2. Vibrogram of a Morse model of 1 2 ( i 'C) calculated with the function (3.7) with c = loo0 em-'. The solid lines are the classical periods given by (3.11).

periods obtained from (3.11) have been indicated and show a nice agreement. The period of the time recurrences increases near the dissociation threshold as expected; note, however, that, due to the finite number of energy levels, the classical recurrences tend to disappear near the threshold as a consequence of overlap between periods and of the almost-periodic fluctuations. 2. Experimental Vibrogram of NaZ by Zewail and Co-workers [I151 As a second example, consider the experimental observation by Zewail and co-workers of time recurrences in NaI, as recorded in a pump-dump experiment [ 1151. In such experiments, the pump pulse creates a wavepacket in the potential with a variable mean energy, similar to the wavepacket (3.6) except that the distribution is not generally Gaussian. In the semiclassical limit, the time recurrences occur at the periods of the emerging classical orbits. The corresponding vibrogram analysis has been carried out in Ref. 115, and Fig. 3 shows the resulting experimental vibrogram, where the lengthening of the period can be observed. Here, the potential is composed of two surfaces with an avoided crossing. Moreover, one of the surfaces extends above the dissociation threshold, which affects the amplitudes of the periodic orbits since a coupling to the continuum occurs by predissociation.

E. Triatomic Molecules

More complicated behaviors are expected for triatomic molecules (i.e., for three-body problems). In general, the analysis is facilitated by the fact that

526

P. GASPARD AND I. BURGHARDT

5000

2500

0

, . 1 4ooo 8000 loo00

2000

6OOo

E [cm-'1

Figure 3. Experimental vibrogram of NaI obtained by Zewail and co-workers [115]. The triangles give the prime periods and the diamonds their repetitions. The solid line is a fit of the experimental points. (Experimental data from Ref. 115.)

Born-Oppenheimer potentials associated with bound electronic states feature minima around which the surface is nearly harmonic. Therefore, the motion is expected to be regular at low energies, as for coupled harmonic oscillators. Anharmonicities start to play a role as the energy increases. The anharmonicities cause the nearby energy levels to repel each other due to resonances of Fermi or Darling-Dennison type. A single such resonance may create local modes of vibration besides the normal modes without affecting the classical integrability, as we discussed earlier. Such phenomena can be pinpointed in the vibrogram from the analysis of the periodic orbits [14,110]. In triatomic species, the number of vibrational degrees of freedom is 4 for a linear molecule like CS2 and 3 for a nonlinear molecule like N02. Obviously, the search of periodic orbits is significantly more complicated than in diatomic molecules. Nevertheless, tools have been developed to obtain the shortest among the periodic orbits, involving in particular the identification of elliptic islands in Poincart5 surfaces of section, the use of integrability at low energies, and the search for commensurabilities among the frequencies. Let us discuss some details of the analysis for two particular molecules.

1. CS&?'C,') This molecule is currently being studied in the laboratory of Pique in Grenoble [ 116, 1171. In the ground state, the molecule of carbon disulfide is collinear and presents a Fermi resonance like carbon dioxide.


527

The vibrogram of the experimental data has been interpreted by Gaspard et al. [14], who used an effective model Hamiltonian proposed by Pique et al. [ 1171 on the basis of a Dunham expansion including the Fermi resonance between the symmetric stretching mode and the degenerate bending mode. The classical counterpart of the model Hamiltonian is integrable. Therefore, the Berry-Tabor trace formula applies and the periodic orbits can be labeled by integers (nl,ny2,n3), which give the numbers of periods of the individual modes in the full period. In the analysis of the bulk periodic orbits, a simplification occurs for the bending oscillations. Because the Hamiltonian of a linear molecule depends quadratically on the angular momentum variable L2, the time derivative of vanishes with L2, in contrast to the conjugated angle x2 given by x 2 = &&l the time derivatives of the other angle variables, which are essentially equal to bi = oi. Therefore, the subsystem L2 = 0 always contains bulk periodic orbits that are labeled by (nl,n;, n3). In the analysis of the periodic orbits, it is convenient to first study the edge periodic orbits of relevant 1F , 2F, or 3F subsystems rather than to start with the bulk periodic orbits for which all the F = 4 degrees of freedom are active. The amplitudes of the edge orbits are significantly smaller than those of the bulk periodic orbits, as explained in Section 11. Nevertheless, they help to localize the bulk periodic orbits because the edge periodic orbits form a skeleton for the bulk phase-space dynamics in integrable systems. The periods of the three modes of CS2 taken individually are respectively equal to TY = 2?r/wl = 50 fs for the symmetric stretch, T i = 27r/w2 = 83 fs for the bend, and 7'; = = 21 fs for the asymmetric stretch. Since the asymmetric stretching mode 3 is of high frequency, we may expect that many bulk periodic orbits (nl,n:,n3) have periods close to the period of each edge periodic orbit (n~,ni,-)with n3 = (w3/wl)nl f: ( ~ 3 / 0 2 ) n 2 . Since w3 is large with respect to both the other frequencies, a number of closely neighboring commensurabilities are possible. Following this argument, we may conclude that an important role is played by the subsystem in which only the symmetric stretching and the bending modes are active, as studied in Ref. 14. This study [ 141 has shown that a period-doubling bifurcation associated with the Fermi resonance occurs in this subsystem at the energy E = 3061.3 cm-' (with Eep= 0).Below the Fermi bifurcation, there exist edge periodic orbits of normal type, which are labeled by (nl,nT2,-)normal. At the Fermi bifurcation, a new periodic orbit of type (2, lo, -)Fermi appears by period doubling around a period of 2T1 = 100 fs. This orbit is surrounded by an elliptic island that forms a region of local modes in phase space. Therefore, another family of edge periodic orbits of local type are born after the F e d bifurcation that may be labeled by the integers (nl,ni,-)lWa1. They are distinct

528


from the periodic orbits of normal type because a homoclinic orbit (also called separatrix) separates the normal from the local orbits. The experimental vibrogram shows an important recurrence around 160 fs, which may be assigned to the edge periodic orbit (3,2*, Recently, the vibrogram analysis has been carried out by Michaille et al. [113] on the basis of another model proposed by Joyeux [ 1181 as well as on an ab initio potential fitted to the experimental data of Pique [1191. Essentially the same classical periodic orbits appear in the different models at low energies. In the same context, let us add that Joyeux has recently applied the Berry-Tabor trace formula to a 2F Fermi-resonance Hamiltonian model of CS2 [120] and carried out a classical analysis of several related resonance Hamiltonians [121]. At higher energies, a transition around 13,000 cm-I above the first vibrational level has been observed when the asymmetric stretching mode starts to interact with the other modes 171. Further anharmonic resonances may be expected in this region, which are the object of current studies.

This nonlinear molecule has been studied by several groups [5J and, recently, by Jost and co-workers 16, 1111. The ground electronic state of the molecule 2 2A1 appears very harmonic and may be modeled by a Dunham expansion. The excited electronic state 2B2 has its minimum about 10,000 cm-' above the minimum of the ground electronic surface. In the region above this minimum, the vibrational states of both electronic surfaces are intknnixed by nonadiabatic couplings beyond the Bom-oppenheimer leading approximation [52-541, which induce Wignerian repulsions between the vibrational levels. The spacing and intensity statistics have been obtained by Jost and co-workers [6] that obey respectively the Wigner and Porter-Thomas distributions, as predicted by random-matrix theories. The overlap between both electronic surfaces is therefore a mechanism of formation of irregular spectra that is extremely strong. The intermixing between both electronic surfaces explains the quenching of radiative transition rates, as discussed by Koppel et al. [53]. The vibrogram has also been obtained by Jost and co-workers in the region of 12,000-15,000 cm-*, where the vibrational spectrum is irregular [ 11I]. The main recurrence occurs at 47 fs and multiples thereof and corresponds to the bending frequency of about 714 cm-'. Thus, the spectrum is dominated by the bending motion in this regime of N02. A conical intersection is expected above 10,000 cm-' that has not yet been rigorously identified. This conical intersection also creates important anharmonicities in the fully diagonalized effective Harniltonians of Weigert

A


529

and Littlejohn [54] that should destroy the near harmonicity of the lower surface and give rise to classically chaotic behavior.

F. Tetra-atomic Molecules

In tetra-atomic molecules that are linear, the number of degrees of freedom is F = 7,which again considerably complicates the analysis. Nevertheless, the dynamics of such systems can turn out to be tractable if the anharmonicities may be considered as perturbations with respect to the harmonic zero-order Hamiltonian. In such cases, the regular classical motions remain dominant in phase space as compared with the chaotic zones, and the edge periodic orbits of subsystems again form a skeleton for the bulk periodic orbits. Recent works by Herman et al. and Field et al. have focused on molecules of the family of acetylene, in particular CzHD [112) and C2H2 (see Refs. 122 and 123 and Field et al., “Intramolecular Dynamics in the Frequency Domain,” this volume). These linear molecules have three stretching modes, 1, 2, and 3, and two doubly degenerate bending modes, trans 4 and cis 5. Isotopic effects appear particularly striking in the vibrational dynamics, as shown in the comparative study of the dynamics of the above isotopomers. In the energy range 0-16,000 cm-’, the vibrational Hamiltonian of this molecule can be modeled by a Dunham expansion without anharmonic resonances of the classical form [ 1121

(3.12)

The classical periodic orbits of this completely integrable system can be obtained by solving Eqs. (2.31). The action space can be systematically scanned to search for approximate commensurabilities between the frequencies. The harmonic periods of the individual modes are respectively TY = 10.09 fs, T i = 2 ~ / 0 2 = 18.19 fs, T ! = 2 ~ / 0 3 = 13.07 fs, T t = ~?F/WI 2?r/w4 = 65.79 fs, and Tg = ~ ? F / W S= 50.48 fs. In this integrable system, the bulk periodic orbits can be labeled by the integers (n, n2, n3, n?, ny ), as explained in Section 11. As in CS2 [ 141, the frequencies of the angles x4 and xs of bending defined by (3.12) are proportional to L4 and L5 so that periodic orbits always exist


530

SA

-

.Y(t 200

100

0

0

4000

8000

12000

E [cm-11 Figure 4. Vibrogram of C2HD calculated with 6 = 2000 cm-* from all the vibrational energy levels predicted by the Dunham expansion corresponding to the Hamiltonian (3.12) obtained by Herman and co-workers by fitting to high-resolution spectra [I 121. The periods of the bulk periodic orbits of Table I obtained numerically for the classical Hamiltonian (3.12) are superimposed as circles. On the right-hand side, the main labels ( n 4 , n 5 ) of the periodic orbits are given.

for which m4 = m5 = 0. Because of their low frequency, the bending modes determine the main recurrences of the vibrogram depicted in Fig. 4. These main recurrences at 200,250,400, and 450 fs correspond to the commensurabilities (n4,n5) = (3,4), (4,5), (6,8), (7,9), respectively (see Fig. 4). Table I shows a set of bulk periodic orbits associated with these recurrences. Here, again, several bulk periodic orbits correspond to the same commensurabilities specified only by (n4,n5) because the stretching modes are much faster than the bending modes. It is also important to notice that the periodic orbit (20, 11, 15, 3 O , 4') at about 200 fs is the shortest of the bulk periodic orbits and that all the shorter recurrences are due to edge periodic orbits in subsystems withf < F = 7. According to the Berry-Tabor formula, the amplitudes of these edge periodic orbits are necessarily smaller than the ones of the bulk periodic orbits [cf. (2.38)], which is in agreement with the amplitudesobserved in the vibrogram. This effect can be qualitatively described as a stroboscopic effect of Lissajous type [ 108,116,1171. The Berry-Tabor formula offers a systematic explanation of this effect in terms of the periodic orbits. 2.

12C2H2 (X'C')

An effective Hamiltonian for this system has been obtained recently by Herman and co-workers based on the high-resolution Fourier spectroscopy of


53 1

Table I Recurrences of Vibrogram of C2HD and Their Periodic-Orbit Assignment [ 1121

-200

(20, 11, 15, 3'. 4')

-250

(25, 14, 19, 4', 5') (25, 14, 20, 4', 5')

-400

(40, 22, 30,6', (40, 22, 31, 6', (41, 22, 31, 6', (45, 25, 34, 7', (45, 25, 34, 7', (45, 25, 35, 7'. (45, 25, 36,.'7

-450

8') 8') 8') 9') 9') 9') 9')

IOOO-

5000250010002000350050002000800500-

the spectrum up to 12,000 cm-' [123]. Contrary to its isotopomer C*HD, acetylene presents important anharmonic resonances, due to the fact that the harmonic frequencies oi allow closely neighboring commensurabilities, which causes many perturbations among the vibrational levels. Actually, this molecule has been the object of an early study that showed the presence of Wignerian repulsion at high energies around 25,000 cm-' [3]. The effective Hamiltonian by Abbouti Temsamani and Herman [123] is composed of a diagonal part given by a Dunham expansion with all the x's and g's as well as y z ~ and , of a nondiagonal part including the following resonances [ 1231: 0 0

0

vibrational Z-doubling 45/45; bending Darling-Dennison interaction 44/55; stretch-bend anharmonic resonances 3/245, 1/244, 1/255, 14/35, 33/1244; and stretching Darling-Dennison interaction 11/33.

The work of Kellman [27] shows that the Hamiltonian of Ref. 123 preserves the constants of motion:

with ii; = 5;+ i d i , which are the remaining good quantum numbers of the vibrational dynamics. These quantum numbers label superpolyads in which


532

the vibrational levels are intermixed so that (u1, vz, u3, 1155) are no longer good quantum numbers. The vibrogram of acetylene is depicted in Fig. 5, where we observe important differences with respect to the vibrogram of CzHD. The main recurrences of the vibrogram can be interpreted by the commensurabilities between the harmonic periods, which are here TY = 27r/w1 = 9.53 fs, 7': = . 453.65 fs, and T i = 2 ~ / ~16.57 2 fs, T! = 2 1 ~ / ~=39.76 fs, T: = 2 ~ / ~ = 27r/ws = 44.67 fs. In particular, the very strong recurrences at 50 fs and multiples thereof are due to commensurabilities between the stretching modes l and 3 and either of the bending modes 4 or 5. The longer recurrences are also due to commensurabilities among the trans and cis bending modes 4 and 5. The recurrences at 275, 320, and 380 fs can be assigned to commensurabilities ( q , n 5 ) = (5, 6), (6, 7), (7, a), respectively. We notice that the bending recurrences of C2HD and C2H2 are different. The only common recurrence (4, 5 ) corresponds to a real periodic orbit in C2HD but to a complex periodic orbit in the 2F bending subsystem of C2Hz (see Fig. 5). Another difference is

&, 1

500 400

300 200 100

1

0 1

0

I

I I

'

*

.

- !

4000

E [cm']

aooo

12000

Figure 5. Vibrogram of C2H2 calculated with e = 2000 cm-' from all the vibrational energy levels predicted by the effective Hamiltonian of Abbouti-Temsamaniand Herman [ 1231 obtained by fitting to high-resolution spectra. The period-energy diagram of some periodic orbits of the correspondingclassical Hamiltonian, given by (3.14) for the bending subsystem, is superimposed. The labels (n4,n5)refer to the edge periodic orbits (-, -, -,n!,n:) represented by solid lines when real and by short dashed lines when complex. The transition between real and complex periodic orbits is depicted by long dashed lines. The vertical dashed line marks the energy of the bifurcation as explained in the text. The bottom solid line around 10 fs is the edge periodic orbit (1, 1, -, -) while the next one around 50 fs is the edge periodic orbit (5. -, 5 , lo, -).

-.


533

that the bending recurrences' amplitudes decrease around 7000-8000 cm- I above the minimum. In order to give a more quantitative interpretation, we analyzed the classical motion. The classical Hamiltonian is a function of the following actionangle variables: HCl(Jfl,J 2 , J 3 , J4, J s , L 4 , LS.81 - 0 3 , 81- 8 2 - 2 8 4 , 0 4 - 0 5 , x 4 - X S ) , so that (3.13) are also classical constants of motion. As another consequence, the dynamics can be reduced to a 4F subsystem that determines the motion of the 7F system. This 4F subsystem contains itself many subsystems with a smaller number of active degrees of freedom. In order to assign the recurrences around 50 and 100 fs, we considered the 3F subsystem J 2 = 4 = J S = 0, which is completely integrable. The Hamiltonian of this subsystem is of the form H c , ( J 1 , J 3 , J 4 , 0 1 - 0 3 ) . The stretching Darling-Dennison interaction is particularly intense with k l 1 / 3 3 = -102.816 cm-' so that the bifurcation leading to local modes occurs at very low energies. The edge periodic orbit (5, -, 5, lo, -) has been calculated numerically for comparison with the corresponding recurrence of the vibrogram. We observe in Fig. 5 a very good agreement. The integrability of this subsystem explains that the recurrence persists up to higher energies. The bending recurrences have been analyzed by focusing on the 4F bending subsystem J1 = J 2 = J 3 = 0, which we suppose forms a skeleton for the bulk periodic orbits of the full system. Certainly, this assumption has the consequence that the action variables J 4 and J S are forced to take larger Values at fixed energy than would be the case if the energy was distributed on all the degrees of freedom. Accordingly, the anharmonicities are stronger in the subsystem J I = J2 = J 3 = 0 than in the full system at similar energies. We should keep in mind these differences between edge and bulk orbits in the following discussion. The Hamiltonian of the subsystem has the form

which preserves the constants of motion, P,I = J 4 + J5 and &,,I = 4 + L5. so that the motion of this 4F subsystem is driven by a 2F subsystem, which allows us to represent the motion by two-dimensional Poincar6 mappings. With the further constraint that L 4 = L5 = 0 and x 4 = x 5 initially, the action-

534


angle variables L4, L5, ~ 4 and , x 5 remain constant in time. Under these conditions, the dynamics reduces to the 1F subsystem in the variables [04 - 05, ~ ( Js J5)], which drives the angle O4 + 05. Periodic orbits corresponding to (n4,n5) can thus be obtained by integration of the one-degree-of-freedom subsystem. The bifurcation diagram of these periodic orbits is depicted in Fig. 5, where the analogy with CS;! appears clearly (cf. Fig. 52 of Ref. 14). The periodic orbits are generated in bifurcations along lower and upper borders defined by the periods of the 1F subsystems J4 = 0 and J5 = 0, respectively. A pitchfork bifurcation occurs at E = 4348.6 cm-' where local modes arise from J5 = 0. The progression of periodic orbits [-, -, -,n:,(nq + l)'] with n4 = 4, 5, 6, 7, 8, ... appears in nice agreement with the recurrences of the vibrogram, which confirms that these recurrences are due to bending. Moreover, the Poincari mappings of (3.14) at values of PCl fixed by the existence of (5,6) and L4 + L5 = 0 show the presence of classically chaotic motions with a bifurcation at E = 6900 cm-* (see Fig. 6). At this bifurcation, the periodic orbit (5,6) becomes unstable because one of its Lyapunov exponents turns positive, as shown in Fig. 7. The periodic orbit (6.7) destabilizes by a similar scenario around E = 7200 cm-I. These results show that the interaction between the bending modes leads to classically chaotic behaviors that destabilize successively the periodic orbits. For the bulk peri-

()

t.-'~ -

---.. ...... .

.... ... . .... ... . . . . . . . .. .. . . .. . . ......- .... . . .._- ._

-

.

-:.:.d

, ... , .._. . . ...., , .. _. .... . ... --.

Figure 6. Phase portraits of (3.14) in the Poincark surface of section x4 - x5 = 0 in the bending subs stem R - Lz = 0 in which the periodic orbit (n4,ng)= (5, 6) exists: ( a ) E = 5022.15 cm-', P = 6.97; (b)E = 7040,92 cm-'. P = 9.84; (c) E = 8060,88 cm-*, P = 11.31. (From Ref. 114.)

S‘E 0

2

W

UI

r

r

h

Y

11-

S’E-

S’E 0 Y

S’E-

odic orbits, for which the bending anharmonicities are weaker as explained above, we may expect that this chaotic destabilization occurs at higher energies. The transition to classical chaos can explain why the amplitudes of the main recurrences decrease, as observed in the vibrogram. The presence of a transition is further confirmed by Wignerian repulsions leading to irregular spectra in the superpolyads corresponding to the highest bending excitations [114]. This destabilization of the bending recurrences is due to the combined anharmonic resonances of vibrational l-doubling and of bending Darling-Dennison types, which arise because the frequencies of the normal

536

0.02

I

I

‘

-

-

s

n

x

0.0 I

-

0

.

4000

*

I

6000

E [ern-*]

8000

lo000

Figure 7. Lyapunov exponents of the periodic orbits (n4, n5) = ( 5 , 6 ) , (6,7)in the bending fs-’. (From Ref. 114.) subsystem of Hamiltonian (3.14). The numerical error is -5 x

modes are closer to the corresponding resonance conditions in C2H2 than in C*HD.* The vibrogram analysis [ 1081 based on the dispersion fluorescence spectrum of acetylene by Solina et al. [125] reveals a recurrence around 50 fs from 0 to 12,000 cm-I, which is similar to the one in the previous analysis. However, an important recurrence appears at 70 fs at higher energies from 4000 to 16,000 cm-’, which is caused either by anharmonic period lengthening or by a transition to a slower regime at higher energies.

G. Synthesis The vibrogram reveals the emerging periodic orbits of the vibrational dynamics. The periodic-orbit contribution to the spectrum concerns an energy scale larger than the mean spacing but small enough to “see” quantization effects. The recurrences of the vibrogram characterize the intramolecular dynamics in the time domain. We may attribute the presence of time recurrences to an inefficient vibrational energy exchange in the system, which is in agreement with the fact that the amplitudes of the recurrences are most prominent in classically integrable systems. Conversely, when the dynamics becomes classically chaotic, the recurrences tend to disappear, which provides evidence of a rapid energy exchange between the different vibrational degrees *Let us note that the energy range studied here is well below the vinylidene barrier at 17,300 cm-I or even the vinylidene minimum at 15,400 cm-I [124].


537

of freedom. The role of strong anharmonic resonances in the increase of intramolecular vibrational relaxation has been discussed by Quack [ 1261, in particular. Here, we obtain a more detailed evidence and understanding of such mechanisms in terms of the instability of the emerging periodic orbits of the vibrational dynamics. We may speculate that the time recurrences predicted by the theoretical vibrograms of acetylene could be observed in femtosecond pump-dump experiments. In this regard, let us remark that, very recently, energy-time plots similar to the ones we describe here have been experimentally obtained for rubidium Rydberg wavepackets [1271. Concerning the structures of the energy spectrum below the mean spacing, the above discussion shows the existence of two mechanisms by which the energy spectra may become irregular:

1. On a single electronic surface, by accumulation of anharmonic resonances that cause the intermixing of the vibrational levels within the superpolyads. 2. On two overlapping electronic surfaces, by the intermixing of the vibrational levels due to the nonadiabatic coupling beyond the leading Bom-oppenheimer approximation. Near the minimum of the ground electronic surface, the anharmonicities generally play the role of perturbations so that the spectra are regular and the first mechanism is weak. It should become more important when the potential surface deviates significantly from the parabolic shape. The second mechanism, by contrast, may have a very marked effect, as illustrated by the example of NO2 [5, 61. The formation of irregular spectra by the two preceding mechanisms has implications for intramolecular vibrational and electronic relaxation. In the work by Jortner et al. [128], electronic relaxation processes are modeled by the coupling of an isolated level to a dense quasicontinuum of levels, which leads to approximately exponential time evolution. Here, in NOz, for instance, the coupling occurs between two sets of levels that are sparsely distributed, a scenario that was also envisaged by Jortner [128]. The new results obtained since 1970 show that the coupling between sparsely distributed levels leads to Wignerian statistical repulsion and a Porter-Thomas distribution of intensities, as modeled by random-matrix theories. In sparse irregular spectra, the time evolution of the intramolecular relaxation may be more complicated than an approximate exponential, as evidenced by the recurrences in the vibrograms. Against this background, we may envisage that irregular spectra may also appear due to the intermixing between the rotational levels of several overlapping vibrational bands. Such a mechanism would involve the

538


rotational-vibrational interactions, which are expected to be strong, especially in the case of floppy molecules. In view of the recent results on irregular spectra observed in Rydberg atoms, we may also expect that similar phenomena exist in Rydberg molecules, as already suggested a few years ago by Lombardi et al. [129].

IV. OPEN SYSTEMS A. Energy and Time Domains If the quantum system is open (e.g., of scattering type), its energy spectrum will be continuous. Far from being featureless, the continuous spectrum shows structures that are interpreted in terms of the analytic singularities of the S matrix, that is, poles and branch cuts. The poles define resonances that correspond to metastable states characterized by a quantized energy but a finite lifetime, which is given by the inverse of the resonance half-width. Molecular resonances have been much studied recently and have been systematically used to interpret the dissociation dynamics of unimolecular reactions [9]. In open systems the discrete spectrum of the resonances plays a role similar to the spectrum of energy levels in bound systems. The global structures appearing in the spectrum provide a characterization of the dynamics. In particular, regular sequences of equally spaced resonances may be interpreted in terms of a recurrence period in the time domain, as shown by Heller [130], Pack [131], and others. On the other hand, irregular sequences arise in systems with several interfering recurrence periods, as we will discuss below. Thus, we can distinguish between regular and irregular spectra of resonances as well as between sparse and dense spectra of resonances. Since the resonances are distributed in a complex energy surface, we find spectra that extend not only along the real direction but also along the imaginary direction, as revealed in numerical studies, for example, for the inverted harmonic potential and the Eckart potential 19, 24, 1321. The resonance spectrum reflects the openness of the potential. We may distinguish between different types of potentials:

1. Weakly open potentials with a quasi-bounded region that is separated from the exit channels by potential barriers. This class of systems can be studied, for instance, with the R-matrix theory of Wigner [133] according to which the quasi-bounded region is first treated as a closed system with arbitrary boundary conditions at the bottlenecks, where the wave functions are thereafter matched with those of the exit channels. In such systems, we expect the resonances to have a number of properties in common with discrete energy levels, while the lifetimes are relatively long. The lifetimes are determined, on the one hand, by tunneling below the barrier and, on the


539

other hand, by the number of open channels above the barrier. In this case, we expect a dense spectrum of resonances that is either regular or irregular depending on whether the potential gives rise to anharmonicities (cf. our previous discussion). 2. Strongly open potentials that have no energy minimum but instead a saddle point, a maximum, or possibly no equilibrium point. In such potentials, the dissociation process is direct and very fast so that the lifetimes are very short. If there is no equilibrium point, it is possible that no resonance exists. We may expect sparse spectra that are regular, or irregular depending on the spectrum of interfering periodic orbits. Apart from this classification, there are systems like Rydberg molecules where it is essential to take into account surface hopping between several coupled potential surfaces. As the work of Heller and co-workers [ 1301 has shown, the energy spectrum is complementary to the time autocorrelation function by a Fourier transform relation. The continuous nature of the energy spectrum implies the decay of the autocorrelation function. At intermediate times, the decay is given in terms of a superposition of exponentials determined by the lifetimes of the resonances. The real parts of the energies determine the periods of recurrences in the autocorrelation function. In the semiclassical method, these recurrences can be interpreted as emerging periodic orbits, as discussed below. At long times, the decay is controlled by the fine structures in the energy spectrum and, in particular, by the behavior of the spectrum near its thresholds. Near these energies, where the kinetic energy is minimal, the spectrum is associated with the slowest translational motions of the dissociation fragments in the exit channels. Hence, the decay obeys power laws that are of quasiclassical character.

B. Unimolecular Dissociation Rates: RRKM Theory and Distribution of Resonances

Around a fixed energy E, the average reaction rate is given by the famous RRKM formula, which can be derived from both quasiclassical and quanta1 considerations [71, 721. In the context of the Wigner R-matrix theory [133], the rate is given by the sum of the half-widths of all the open channels. The rate is thus the product of the number v ( E ) of open channels and the rate per = l/hn,,(E), where h is the Planck constant. The average channel zCchannel(E) reaction rate is obtained as [ 134, 1351


540

in the case of potentials with nearly harmonic minima, where E$ is the minimum energy at the bottleneck, Eo IE* is the minimum energy of the potential in the quasi-bounded region, and F is the number of internal degrees of freedom. This formula is very powerful and can even explain the general behavior of the resonances in systems with other types of potentials. For Fdimensional well potentials we have %(E)- (E - E* ) ( F - ')/2/(E- E o ) ( ~2)/2 since the number of open channels increases as v ( E ) (E- E* ) ( F - ')I2 while n,,(E) - (E - E o ) ( ~ - ~In) /billiards, ~. the minimum energies are equal, E* = Eo, so that the decay rate is equal to %(E)- h independently of the number of degrees of freedom. This behavior has been numerically observed, in particular, for disk scattering systems. Formally, the formula (4.1) applies not only to deep potentials of type 1 but also to potentials of type 2 without minimum. In such a case, both extrema again coincide at the same energy, E* = Eo, but the average dissociation rate is constant, as observed for potentials with a near-harmonic saddle. Miller has shown that the number of open channels v ( E ) should be replaced by the sum of tunneling probabilities associated with the modes of the transition state when the energy approaches the barrier E = E$ 1136). Besides the average behavior of the rate, recent works have focused on the fluctuations in the distribution of resonances around the average rate. Random-matrix theories have been used to explain these fluctuations in the dense spectra of quasi-bounded systems. Wignerian repulsions are predicted and observed along the real energy axis. Along the imaginary axis, the Gaussian model by Porter and Thomas predicts chi-square probability distributions with parameter v for the reduced half-widths*:

-

d

Prob{c < x } = dx

y/2 - 1 2~/2r(~/2)

which is derived for v open channels contributing equally to the total halfwidth (1371. Let us emphasize that the Porter-Thomas distribution is here applied to the resonances of the molecular Hamiltonian in the absence of a radiation field. In the case of NO2 mentioned in Section 111, the same distribution with v = 1 was applied, by contrast, to the radiative linewidths of the molecular Hamiltonian [5, 61. Miller and co-workers performed a systematic analysis of the distribution of decay rates for the SO electronic state, as well as of the S1 - SO cou*The reduced half-widths are defined as the half-widths divided by the velocity associated with the resonance, I$ = r,,/u,,.


541

pling matrix elements in deuterated formaldehyde, D2CO [4,1381. In this molecule, the dissociation occurs slightly above the barrier of the SO state, which has been studied by Stark level-crossing spectroscopy between the SO and the S, states by Polik et al. [138]. This analysis revealed that the coupling between the electronic states Si and SOinvolves only a small number of open channels, which is almost insensitive to the electric field: Y = 1.6 and dv/d!E = -0.02 (kV/cm)-' [138]. On the other hand, the effective number of open channels responsible for the decay of formaldehyde in the SO state is observed to be equal to Y = 3.8 as obtained by extrapolation to zero electric field at an energy of about 0.5 kcal/mol above the barrier located at 80.6 kcal/mol. This effective number depends on the electric field according to d v / d z = +0.28 (kV/cm)-' [138]. Recently, Miller and co-workers have obtained a generalized form of the distribution of unimolecular decay rates for the case of coupled open channels contributing with unequal partial half-widths [139].Further results have also recently been obtained in the statistical theory of reactions where the possibility of algebraic decay besides the RRKM exponential decay has been discussed [140].* If the number of open channels increases with energy, the Porter-Thomas distribution (4.2)shows that the imaginary parts of the resonance energies no longer accumulate near the real axis like in the case v = 1 and, to a lesser extent, in the case v = 2. For v >> 1, the resonances tend to move below the real axis, leaving an empty region. This general behavior is confirmed by semiclassical theories based on the inequalities (2.18H2.19)that actually predict the formation of such a gap empty of resonances below the real axis [14].This gap exists in strongly open systems of type 2, which we shall focus on below.

C. Dissociation on Potentials with a Saddle: Classical Properties The prototype potential surface invoked in chemical kinetics is a two-dimensional surface with a saddle equilibrium point and two exit channels at lower energies. The classical and quantal dynamics of such surfaces has been the object of many studies since the pioneering works by Wigner and Polanyi. Recent advances in nonlinear dynamical systems theory have provided powerful tools, such as the concepts of bifurcations and chaos, to investigate the classical dynamics from a new point of view and to perform the semiclassical *In this context, it should be pointed out that an algebraic decay has also been numerically observed in classical Coulomb-type models of atomic autoionization processes by Bliimel [141]. This might turn out to be relevant for Rydberg molecules. which also represent Coulomb-type systems. For the recent observation of algebraic decays in Rydberg atoms, see Ref. 142.


542

quantization of the dissociation process. These new results provide a unifying scheme to interpret a broad set of observations on ultrafast processes and to predict the distribution of resonances in the nonseparable systems under investigation. We will start by describing the relevant aspects of the classical dynamics, introducing the concepts of the invariant set and the repeller, the Smale horseshoe and its symbolic dynamics, and the bifurcations at their origin [191. We then turn to the semiclassical quantization based on the Gutzwiller trace formula and the zeta functions. We proceed to show how this new theoretical framework allows us to explain the distribution of resonances in several molecules like HgI,, CO;?,and H3, to calculate their lifetimes, and to provide a synthesis with respect to previous work. We notice that the theory described here is issued from a previous work by Gaspard and Rice in which disk scattering systems were used as a vehicle for the study of unimolecular fragmentation 1331. The recent results obtained by Burghardt and Gaspard show the remarkable generality of these considerations in the context of ultrashort dissociation processes [lo, 141. I.

Classical Dynamics: The Repeller

Let us consider a triatomic molecule on an antibonding Bom-Oppenheimer potential surface with a saddle equilibrium point. Normal-mode analysis shows that the equilibrium point is characterized by real and imaginary frequencies. If the molecule is linear, the frequencies of the (nondegenerate) symmetric stretching and the (doubly degenerate) bending modes are typically real while the frequency of the (nondegenerate) asymmetric stretching mode (which corresponds to the reaction coordinate) is imaginary. The imaginary part is given by the positive Lyapunov exponent of the equilibrium point at the threshold energy [cf. (2.9) and (2.10)]. In nonlinear molecules, we typically have two instead of three real frequencies [23]. We focus on the nonlinear dynamics for the collinear configuration which we treat as part of the full multidimensional system. This is to be contrasted with two-degree-of-freedom models where the molecule is assumed to be frozen in some angular configuration, such that the bending degree of freedom is excluded from the dynamics. In our analysis, bending is taken into account in terms of linearized dynamics, which allows us to extend the results for the collinear situation to the full three-dimensional system. The restriction we must be aware of is that the three-dimensional system may have periodic orbits that are not of collinear type. For symmetric molecules XYX in the collinear configuration, the Hamiltonian is of the form


543

where rl and r;?are the two distances between the central nucleus Y and the outer nuclei X's, while pxy = (mi'+ m;')-I is the reduced mass of the XY system. We will focus on such symmetric species but indicate throughout how the analysis extends to nonsymmetric XYZ-type molecules. At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. Although unstable, this periodic orbit is an example of classical motion which leaves the molecule bounded. Other periodic and nonperiodic trajectories of this kind may exist at higher energies. The set of all the trajectories of a given energy shell that do not lead to dissociation under either forwardor backward-time propagation is invariant under the classical flow. When all trajectories belonging to this invariant set are unstable, the set is called the repeller R [19, 33, 35, 481. There also exist trajectories that approach the repeller in the future but dissociate in the past, which form the stable manifolds of the repeller Ws(R). Reciprocally, the trajectories that approach the repeller in the past and dissociate in the future form the unstable manifolds W , ( R ) (see Fig. 8). All other trajectories spend only a finite time in the scattering region. They are referred to as scattering orbits, and they constitute the vast majority

P

9 Figure 8. Schematic representation of a chaotic repeller and its stable W, and unstable W, manifolds in some Poincari surface of section ( q , p ) together with one-dimensional slices along the line L of typical escape time function I+.

544


of trajectories. The phase-space volume occupied by the repeller is actually equal to zero so that a typical trajectory has a vanishing probability to be on the repeller or on its stable or unstable manifolds. Therefore, the trajectories have probability 1 to be of scattering type. Nevertheless, the repeller and its stable and unstable manifolds play a CNcia1 role in guiding and delaying the scattering trajectories. Indeed, evidence of the repeller can be obtained by comparing the time delay associated with the escape of different scattering trajectories from the transition-state region. To this end, we construct numerically the escape-time function, which gives the time taken by a trajectory to escape from a certain region 'B surrounding the scattering region [35]: T+(q,p) = max{T > 0 : @k(q, p) E B

for t

E

[O, T [ }

(4.4)

where @; denotes the flow on the energy shell H(q,p) = E and (q,p) is some initial condition on this energy shell, usually taken in a two-dimensional Poincar6 surface of section. Equation (4.4)is a function that becomes arbitrarily large when the initial condition comes close to the stable manifolds of the repeller. Indeed, on the stable manifold, the time to escape will be infinite. The singularities of the forward escape-time function (4.4) can thus be used to reveal the stable manifolds of the repeller. Conversely, the unstable manifolds are uncovered by the backward escape-time function obtained by reversing time [35],

which has its singularities on the unstable manifolds. Since the trajectories of the repeller are located at the intersections of the stable and unstable manifolds we can construct the repeller from the escape-time function: We take the sum of the absolute values of the forward and backward escape-time functions and identify the intersections of the singularities of this sum [lo],

By this method, we have been able to study the repeller, in particular for the systems HgI, and CO,. Let us add that the stable and unstable manifolds play the very important role of separatnces between reacting and nonreacting trajectories [25]. The invariant set undergoes bifurcation sequences in the course of which its topology and its stability are modified. These bifurcations are responsible


545

for the transition from periodicity at low energies to chaos at higher energies. The following bifurcation scenarios are typically observed in saddletype potential surfaces. 2. Bifircation Scenarios Associated with Transition to Chaos As we said, the repeller is composed of only a single unstable periodic orbit at energies slightly above the saddle energy. For dissociative electronic surfaces of molecules like HgI,, COz, and H3, several bifurcation scenarios have been observed, which appear to be generic in the case of triatomic species with nuclei of similar mass, or of the light-heavy-light type. One famous scenariohas been first observed by Pechukas et al. in H3 [1431. These authors discussed the bifurcations in the context of reaction rate theory where the bottleneck of the reaction is identified with a dividing surface of minimal flux, that is, which should separate as well as possible the reacting from the nonreacting scattering orbits. A special role has been assigned to several periodic orbits of the invariant set of trapped trajectories, denoted periodicorbit dividing surfaces (PODSs). Each of these special periodic orbits has then been used to calculate reaction rates according to the variational reaction rate theory. Now in the context of our above discussion on the repeller, these special periodic orbits appear as part of the invariant set. This suggests that one might reconsider the general method of calculating variational reaction rates by taking into account a priori all the periodic orbits on the same footing and finding dynamical criteria to distinguish between them. One of the criteria is the stability of the periodic orbits. Another criterion, of topological nature, will be given below, which shows that in fully chaotic regimes the PODSs, as the shortest periodic orbits, generate all trajectories of the repeller by topological combination. As we show below, the PODSs turn out to be the most important periodic orbits in the bifurcation scenarios leading to chaos. Thus dynamical systems theory provides a reinterpretation of the role of these periodic orbits, which allows a new approach to the analysis of dissociation. We have shown elsewhere that the different bifurcation scenarios can be conveniently discussed in terns of area-preserving mappings generated by the action function [ 101

and which are of the form [14]

546

P. GASPARD AND 1. BURGHARDT

qn+l =

as --- 4 n + Pn+ I aPn+I

(4.8)

Such Hamiltonian mappings are generated by a Poincark surface of section transverse to the orbits of the flow. Thus, v(4) plays the role of a potential function for the motion perpendicular to the periodic orbit. Note that the mapping takes into account the nonseparability of the dynamics. The theory of bifurcations shows that the different types of bifurcations can be described in terms of normal forms, which represent local expansions of the dynamics around the bifurcating periodic orbit [ 19, 32, 491. The purpose of the above mapping is to describe the successive bifurcations of the symmetric-stretch periodic orbit, starting from low energies above the saddle point. Appropriate truncation of the Taylor series of the potential u(q)around q = 0, which corresponds to the location of the symmetric-stretch orbit, provides us with the normal forms of the bifurcations [la]. The bifurcations relevant for the dissociation dynamics under discussion can be described by truncating at the sixth order in q, (4.9)

Let us remark that the mapping should be invariant under the reflection q -+ -q for symmetric molecules so that K = v = u = 0 in this case. (a) Supercritical or Direct Antipitchfork Bifurcation in Symmetric Molecules XYX (See Fig. 9). In one of the simplest scenarios, as observed

in HgI,, the unstable periodic orbit of symmetric-stretch type undergoes a supercritical antipitchfork bifurcation. The orbit becomes neutrally stable of elliptic type at a critical energy E,. Just above this critical energy, the symmetric-stretch periodic orbit is embedded in an elliptic island. The pitchfork bifurcation gives birth to two unstable periodic orbits that exist above the bifurcation. These two periodic orbits are bordering the elliptic island and are the PODSs identified by Pechukas and Pollak [143]. They correspond to nonsymmetric vibrational motion of the molecule with one bond being stretched out further than the other. Since the molecule is symmetric, two distinct such periodic orbits must exist. We shall denote as 1 and 2 these new periodic orbits of shortest period born in the antipitchfork bifurcation, while the symmetric-stretch periodic orbit is called 0 [lo]. The periodic orbits l and 2 inherit the instability lost by the symmetric-stretch orbit


2

0

1

547

I\

Ed

Figure 9. Supercritical antipitchfork bifurcation scenario for symmetric XYX molecules; on the left, bifurcation diagram in the plane of energy versus position; in the center, typical phase portraits in some Poincari section in the different regimes; on the right, the fundamental periodic orbits in position space.

0 in the bifurcation, that is, they are hyperbolic (without reflection).* Their stable and unstable manifolds extend into the exit channels and form separatrices between nonreacting trajectories, which are immediately repelled by 1 or 2 back into the channels, and other trajectories that pass beyond the PODSs 1 and 2. The fate of these trajectories is still uncertain because of the homoclinic tangle [191between the inner branches of the stable and unstable manifolds of 1 and 2, which constitute partial separatrices between scattering orbits and orbits of the elliptic island. This homoclinic tangle forms a small chaotic zone surrounding the elliptic island. This chaotic zone as well as the interior of the elliptic island contain many periodic and nonperiodic orbits, but they are in general of longer periods than the periodic orbits 0, 1, and 2. The fact that the antipitchfork bifurcation is supercritical implies that all these new trapped orbits of the invariant set are born above the bifurcation *The cases of hyperbolic-withaut-reAectionand hyperbolic-with-reflection stability have to be distinguished. In both cases, the trajectories in the neighborhood of the periodic orbit trace out hyperbolic paths in the Poincak section, but if the stability is hyperbolic with reflection, the trajectories cross over between the branches of the hyperbola on each iteration.

548


so that the periodic orbit 0 is the only orbit of the repeller below the bifurcation (unless other global bifurcations with successive sub- and supercntical bifurcations have taken place below the antipitchfork bifurcation we are considering). At still higher energies, the elliptic island undergoes a typical cascade of bifurcations in which subsidiary elliptic islands of periods 6, 5, 4,3 are successively created, which leads to the global destruction of the main elliptic island to the benefit of the surrounding chaotic zone. The cascade ends with a period-doubling bifurcation at Ed, above which the periodic orbit 0 is hyperbolic with reflection, and the main elliptic island has disappeared [ 1451. Subsidiary elliptic islands of very small area continue to exist until a last homoclinic tangency occurs at Ehr, above which all the trapped orbits of the invariant set are unstable of saddle type. The system is then fully chaotic. According to this scenario, the invariant set may contain quasiperiodic motion for energies E, < E < Ehr, while the main elliptic island exists only for E, c E < Ed < Eh,. The interval Ehr - E, turns out to be small as compared with the energy interval above Eh,, where full chaos has set in and the invariant set is a repeller. The above scenario is accounted for by the normal form (4.9)truncated at fourth order in q with K = v = u = p = 0 and x c 0, taking p as the bifurcation parameter, which increases with energy ( p thus plays a similar role as the total energy in the actual Hamiltonian dynamics). The antipitchfork bifurcation occurs at pa = 0. The fixed points of the mapping (4.8)are given by p = 0 and du/dq = 0. Since the potential is quartic, there are either one or three fixed points that correspond to the shortest periodic orbits 0, 1, and 2 of the flow. (b) Subcritical Antipitchfork Bifurcation in Symmetric Molecules XYX (See Fig. 10). A different scenario is observed in H3 [143] as well as COz [146], where the unstable periodic orbit 0 stabilizes in a subcritical antipitchfork bifurcation at E, that is preceded by the birth of the periodic orbits 1 and 2 in two symmetric tangent bifurcations at E,, < E,. (Let us notice that a subcritical antipitchfork bifurcation is an inverted pitchfork bifurcation.) Therefore, the unstable symmetric-stretch orbit 0 alone constitutes the repeller only before the tangent bifurcations giving birth to 1 and 2. These tangent bifurcations occur at some distance from the periodic orbit 0. At each of the tangent bifurcations two new periodic orbits are generated, one of which is neutrally stable and embedded in a small elliptic island while the other is hyperbolic (without reflection) and can be identified with either 1 or 2. Therefore, slightly above the tangent bifurcation, there exist five periodic orbits of shortest period, as first observed by Pechukas and Pollak [143]: 1 and 2, which are hyperbolic, and 1’ and 2’, which are elliptic immedi-


549

P

Figure 10. Same as Fig. 9 for the subcritical antipitchfork bifurcation scenario for symmetric XYX.

ately above the tangent bifurcation. These orbits are ordered in space as 2 c 2 ' < 0 < 1'< 1. As the energy increases in the interval Ell < E < E,, the orbits I' and 2' progressively shift toward the symmetric-stretch orbit 0 and merge at the subcritical antipitchfork bifurcation. Just below this bifurcation, 1' and 2' are elliptic while 0 is still hyperbolic (without reflection). Between Err and E,, the periodic orbits 1' and 2' may either remain of elliptic type or become hyperbolic in the energy interval [Ed&,Ed&'] such that E,, C Ed8 c Ed&* C Ell. Above the subcritical antipitchfork bifurcation E, c E, the symmetricstretch periodic orbit 0 is elliptic and surrounded by a main elliptic island. From there onward, the bifurcation scenario is similar to the previous case as energy increases. The main elliptic island undergoes a cascade of bifurcations ending with a period doubling at Ed above which the periodic orbit 0 is hyperbolic (with reflection). At higher energies, there is a last homoclinic tangency Eh, leading to complete chaos. In this scenario, quasiperiodic motions exist for energies E,, c E c Ehr with a main elliptic island being present for E,, < E, < E c Ed < Eht. Here, again, the interval Eh, - Ell turns


550

out to be small as compared with the energy interval above E h t , in which the chaos is complete. This scenario is described by the normal form (4.9) with K = v = u = 0, x > 0, and p < 0. The antipitchfork bifurcation occurs at pa = 0, but it is preceded by a pair of tangent bifurcations at prr= x2/4p < 0.For prr< p < pa = 0, five fixed points exist that correspond to the five PODSs 2, 2’, 0, l’, and 1 of the flow. (c) Tangent Bifurcation in Nonsymmetric Molecules XYZ (See Fig. 11). If the molecule is not symmetric, as is the case for the transition complexes of FH2 and He12 studied by Poll& et al. 11431,the flow is not symmetric under reflection through the central nucleus Y, which is no longer compatible with the pitchfork bifurcation scenario. In this case, one of the simplest scenarios is given by (4.9) with Y = u = p = 0 but K 9 0, which breaks the reflection symmetry under q -+ -4, In this case, the antipitchfork bifurcation is replaced by a tangent bifurcation. Below the tangent bifurcation, there exists the unstable periodic orbit of hyperbolic type that is the analogue of the symmetric-stretch periodic orbit 0 in the previous scenarios. This orbit may continue to exist without changing its stability through the

2

0”

1

I

I

Figure 11. Same as Fig. 9 for the tangent bifurcation scenario for nonsymmetric XYZ.


55 I

tangent bifurcation: It is simply shifted to the right- or left-hand side, where it turns into the analogue of either 1 or 2, depending on the sign of K . At some distance from this unstable periodic orbit, a tangent bifurcation occurs in which a pair of periodic orbits is born, one of which is elliptic and the analogue of 0 above the bifurcation, while the other is hyperbolic (without reflection) and the analogue of either 2 or 1. Above this main tangent bifurcation, the three shortest periodic orbits 2 < 0 c 1 exist, and the elliptic periodic orbit 0 is embedded in a main elliptic island that undergoes a cascade of bifurcations essentially similar to the previous ones. Therefore, the major difference between the nonsymmetric and the symmetric (supercritical) cases is the smooth exchange of roles between the central and one of the bordering shortest periodic orbits. In the mapping (4.9) with v = u = p = 0 and x c 0, this main tangent bifurcation, which replaces the antipitchfork bifurcation, occurs at p, = ~ ( - x K 2/4)'/3, which is shifted from zero to positive values when K does not vanish. Other more complicated scenarios may occur if x is close to zero and changes its sign, in which case higher order terms should be included to describe the bifurcation scenario. To conclude, the normal form (4.9) allows us to classify the different types of bifurcations leading to the emergence of a chaotic repeller, to the description of which we now turn. As soon as more than one unstable periodic orbit exist in the invariant set, the stable and unstable manifolds of these periodic orbits may form homoclinic intersections. According to a theorem by Birkhoff, chaotic behaviors may exist in the flow under these circumstances [19]. However, as long as not all of the stable and unstable manifolds intersect transversally, but certain manifolds present homoclinic tangencies, other theorems, in particular by Shil'nikov and Gavrilov [147], show that elliptic periodic orbits exist, This means that tiny elliptic islands sustaining quasiperiodic motions persist. The dynamics is then a mixture of chaotic and regular motions, which requires a special treatment both classically and quantum mechanically. Now a very remarkable property of the systems we are considering here is that all the stable and unstable manifolds do intersect transversally at sufficiently high energies above the transition region, that is, above the last homoclinic tangency. Thus the energy windows where the motion is fully chaotic turn out to be very broad. This allows us to apply without further approximation the Gutzwiller trace formula and the zeta function quantization. In the fully chaotic regime, the repeller, which is highly unstable, can be constructed as explained above in terms of the sum of absolute values of the forward and backward escape-time functions, which displays the folding

552


structures formed by the stable and unstable manifolds. In this way, it is possible to confirm the transversality of the intersections between the stable and unstable manifolds, which guarantees that the system is purely hyperbolic.

3. Fully Chaotic Regime: Smale Horseshoes The key concept in understanding the fully chaotic regimes of classically chaotic systems has been introduced by Smale [148]and is a repeller with the shape of a horseshoe. The dynamics on the Smale horseshoe is described by a global Poincark mapping of the type (4.7)-(4.9), for instance with x = u = p = 0, which defines the area-preserving Hknon map. Horseshoe structures are formed by a mechanism of stretching and folding in phase space: Due to the nonlinear classical motion, a given phase-space region stretches out and then folds back onto the initial domain. This mechanism lets an infinite number of trapped orbits appear (see Fig. 12). In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. Ail the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet A = {0,1,2,. . . ,M - 1 }. The other orbits of the repeller are in one-to-one correspondencewith symbolic sequences constructed by concatenation of the symbols,

In particular, the periodic orbits are in correspondence with finite sequences such as wlw2...wp of period p. The periodicity occurring in the symbol sequences translates into the periodicity of the corresponding trajectory crossing the Poincar6 surface of section. The concatenation is performed without constraint on the successive symbols so that the motion on the repeller corresponds to a Bernoulli random process. The regions around the fundamental periodic orbits are successively visited in a random fashion without memory of the previous fundamental periodic orbit visited. As a consequence, the periodic orbits proliferate exponentially with their period, as described by (2.17). The topological entropy per symbol is equal to hrop= In M. In the horseshoe that has two branches, the number of fundamental peri-


553

Figure 12. Formation of a Smale horseshoe with (a)two branches, (b) three branches.

odic orbits is M = 2. Smale has also proposed variants of the horseshoe in which the number of branches is higher. In our context, Burghardt and Gaspard have discovered that horseshoes with three branches describe the repeller at high energies in collinear models of HgI, [lo]. The three fundamental orbits are the symmetric-stretch orbit 0 together with the off-diagonal periodic orbits 1 and 2, which are born in the

554


bifurcations described above. In the fully chaotic regime, the periodic orbit 0 is hyperbolic with reflection, while 1 and 2 are hyperbolic without reflection. All the other periodic and nonperiodic orbits of the repeller are topological combinations of these three fundamental ones according to the rule (4.10). Our recent works have shown that this result is of broad generality and should be expected for a general class of symmetric triatomic molecules with atoms of similar mass, or of light-heavy-light type. In particular, we have recently observed the same three-branch horseshoe in a collinear model of C02 [146], and the comparison with the results by Pechukas, Pollak, and Child [I431 strongly suggests that the repeller of dissociating H3 is also a three-branch horseshoe. The formation of three-branch horseshoes in scattering systems can be modeled by a global mapping like (4.7) and (4.8) with the potential [lo]

-

a model we proposed elsewhere with 6 = = 0. If p = S = 0, the poten-4,as applies to symmetric XYX molecules; tial is symmetric under q in the general case, (4.11) models nonsymmetric XYZ species. We should emphasize here, that the occurrence of the Smale horseshoe is not restricted to symmetric molecules but may extend to nonsymmetric molecules like F H 2 or HC12 [143], in which case the horseshoe has a nonsymmetric form. Other classically chaotic scattering systems have been shown to have repellers described by a symbolic dynamics similar to (4.10). One of them is the three-disk scatterer in which a point particle undergoes elastic collisions on three hard disks located at the vertices of an equilateral triangle. In this case, the symbolic dynamics is dyadic (it4 = 2) after reduction according to C3v symmetry. Another example is the four-disk scatterer in which the four disks form a square. The C,, symmetry can be used to reduce the symbolic dynamics to a triadic one based on the symbols {0,1,2}, which correspond to the three fundamental periodic orbits described above [ 141. In the presence of reflection symmetry with respect to the diagonal of the potential-energy surface, as in symmetric molecules or in the four-disk scatterer, Burghardt and Gaspard have shown that a further symmetry reduction can be performed in which the symbolic dynamics still contains three symbols A’ = {O, +, - } (101. The orbit 0 is the symmetric-stretch periodic orbit as before. The orbit + is one of the off-diagonal orbits 1 or 2 while - represents a half-period of the asymmetric-stretch orbit 12. Note that the latter has also been denoted the hyperspherical periodic orbit in the literature. The central feature of the symbolic dynamics is that it provides a complete classification of the periodic orbits: Once the symbolic dynamics has been


555

established on the basis of the phase-space structure, no periodic orbit can be missed. It is remarkable that this scheme indeed appears to provide the key to the dynamics of a whole class of dissociative species over a broad energy range.

D. Dissociation on Potentials with a Saddle: Semiclassical Quantization 1. Quantization in Periodic Regime Just above the saddle energy, the quantization can be performed by the usual perturbation theory applied to scattering systems as described by Miller and Seideman [24]. This equilibrium poirit quantization uses Dunham expansions of the form (2.8) with imaginary coefficients. This method is valid for relatively low-lying resonances above the saddle, up to the point where anharmonicities become so important that the Dunham expansion is no longer applicable (see the discussion in Section 1I.B). The regime of validity of the Dunham expansion, that is, the regular or periodic regime, coincides with the energy range where the repeller consists of a single unstable periodic orbit. Indeed, it is the onset of the bifurcations we described above (which go along with the resonance conditions mentioned in Section II.B) that causes the divergence of the coefficients in the Dunham series. Thus, for the regular regime, equilibrium point quantization can be compared with periodic-orbit quantization for the symmetric-stretch periodic orbit. For the latter, we are applying the quantization condition given in terms of the zeta function (2.21), which is limited to the leading semiclassical approximation. In the presence of a single periodic orbit for the collinear dynamics of a triatomic molecule, the total zeta function factorizes into zeta functions for states of given bending excitation (u, 1 ) with u = 0, 1, 2, . .. and 1 = u, u - 2, ... , -u and of given quantum number rn = 0, 1,2, . . . [lo, 1491 leading to as many quantization conditions, ZvdE) = 1 -

exp[(i/A)S(E) - i(7r/2)p - 2 ~ i p ( E ) ( + u l)] =O lA(E)I '/2A(E)'"

(4.12)

where S ( E ) is the action of the symmetric-stretch periodic orbit at energy E. Besides its action, the unstable periodic orbit is characterized by its stability eigenvalue A ( E ) under collinear infinitesimal perturbations transverse to the direction of the orbit as well as by the doubly degenerate stability eigenvalue exp[i2xp(E)] under infinitesimal perturbations of bending type. Here, we assume that the periodic orbit is unstable with respect to asymmetric stretching perturbations lA(E)I > 1 and neutrally stable with respect to bending perturbations characterized by the rotation number p(E). Moreover,


556

the motion of the asymmetric stretching perturbations is also characterized by the so-called Maslov index, which is the winding number of the stable or unstable manifolds during a prime period of the periodic orbit 1301. (The Maslov index is constant with respect to all classically continuous parameters such as E as long as the topology of the periodic orbit does not change.) All these quantities can be obtained as functions of energy E by numerical integration of the classical Hamiltonian equations. A polynomial fitting of the numerical data for S(E), A(E), and p ( E ) can then be carried out to obtain the analytic dependence on E. Rotation can in principle be included in this scheme by calculating the periodic orbits for a rotating molecule. The resonances are then obtained by searching for the complex zeros of the zeta functions (4.12) in the complex surface of the energy. Assuming that the action is approximately linear, S ( E , J ) = T(E - E$ ), while the stability eigenvalues are approximately constant near the saddle energy ES , the quantization condition (4.12) gives the resonances [101 Enm"l ES

+T

2rAp

(u + 1) - i

ti ln(lA)'/2A") T

(4.13)

These zeros are thus labeled in terms of the expected quantum numbers for a triatomic molecule: n is the principal quantum number corresponding to the progression in symmetric stretching excitation; u and I are the quantum numbers for the bending excitations. They appear in (4.12) in the harmonic approximation for the bending excitations. We further have the quantum number m associated with the unstable asymmetric stretching excitations. The result (4.13) shows that the lifetime of the resonances decreases with increasing value of the Lyapunov exponent X = ( 1 / T ) In IAl as well as with the quantum number m of asymmetric stretch. The corresponding resonant states have n nodes along the symmetric stretching direction and rn nodes along the asymmetric stretching direction. The result (4.13) suggests that the excitation of an m = 1 state instead of an m = 0 state from vibrational states with corresponding symmetries on a lower electronic surface should lead to a drastic reduction of the lifetime, approximately by a factor 3. The derivation of (4.13) shows that the equilibrium point quantization and the periodic-orbit quantization can be compared term by term. This comparison shows that the periodic-orbit quantization is able to take into account the anhannonicities in the direction of symmetric stretch. However, the anharmonicities are neglected in the other directions transverse to the periodic orbit. Their full treatment requires the calculation of A corrections to the Gutzwiller trace formula, as shown elsewhere [14]. It should be noted that this periodic-orbit quantization is no longer valid in


557

the immediate neighborhood of the first bifurcation where A approaches the value 1 for the bifurcating orbit, and a uniform approximation is required to take into account the nonlinear stability properties of the periodic orbit [49, 501. Indeed, equilibrium point quantization turns out to have a better behavior than periodic-orbit quantization as the region of the bifurcation is approached.

2. Quantization in Transition Regime For molecules composed of atoms with similar mass, as well as light-heavy-light species, the transition region, which covers the bifurcations leading up to the fully chaotic regime, turns out to have a very small extension in energy, which might even be smaller than the spacing between two resonances of successive principal quantum number n. In this case, the resonant states extend in mock phase space over a volume that is larger than the volume of the elliptic island around the symmetric-stretch periodic orbit. Under such circumstances, the properties of the resonant states are essentially determined semiclassically by the classical dynamics outside the elliptic island in regions that are far from the regions of validity of linearized stability analysis. The dynamics in these regions affects the local dynamics only at the antipitchfork bifurcation, which requires the knowledge of the nonlinear stability properties of the periodic orbit in order to obtain the semiclassical quantum amplitudes, as discussed in Section 1I.G. By contrast, in heavy-light-heavy molecules such as HMuH, ClHCI, or IHI, a very extended elliptic island exists in the classical phase space [150]. In such cases, the elliptic island may be the support of several metastable states that can be obtained by Bohr-Sommerfeld quantization. Their lifetime is determined by dynamical tunneling from inside the elliptic island to the outside regions. 3. Periodic-Orbit Quantization in Fully Chaotic Regime

Periodic-orbit theory provides the unique semiclassical quantization scheme for nonseparable systems with a fully chaotic and fractal repeller. AS we mentioned in Section 11, the different periodic orbits of the repeller have quantum amplitudes weighted by the stability eigenvalues, and the periodicorbit amplitudes interfere among each other as described by the zeta function. The more unstable the periodic orbit is, the less it contributes in (2.24). Therefore, only the least unstable periodic orbits play a dominant role. Let us consider here the case of a repeller that is a Smale horseshoe with three branches as described above. The periodic orbits are in one-to-one correspondence with the bi-infinite sequences constructed from the symbols {0,1,2} associated with the three fundamental periodic orbits. A complete list of periodic orbits can be established: { p} = {0,1,2,01,02,12, . ..}. Here,

558


the zeta function also factorizes into different zeta functions corresponding to the approximately good quantum numbers (i,e., associated with bending in our case), but each zeta function remains a product over all the periodic orbits

(4.14) where S,, Ap, pp, and pp are the action, the asymmetric stretching stability eigenvalue, the Maslov index, and the rotation number of bending, respectively, which are required for each period orbit. The product over the periodic orbits is expanded as in (2.24) into the so-called cycle expansion [lo, 141

P

= 1 - to - t l - t2 - (tol - t o t l ) - (rO2 - t 0 t 2 ) - (tI2 - t l t 2 ) - ... - \ * fundamental po

=O

(4.15)

The first three terms beyond 1 are the semiclassical quantum amplitudes of the three fundamental periodic orbits. The other terms appear as corrections that may be neglected in a first approximation. Indeed, each of these terms represents a difference between the amplitude of some periodic orbit o 1 w2 - - .w p minus the amplitudes of the shorter periodic orbits entering into its topological composition. In many cases, it turns out that the amplitude of the periodic orbit is very well approximated by the product of amplitudes of its topological parts so that an approximate cancellation holds within each of these terms [lo]. Of course, better approximations can be obtained by including more and more of these terms. In the case of symmetric molecules, we have that tl = t 2 so that our approximate quantization condition becomes [lo]


559

0 = 1 - to - 2tl

In the three-branch horseshoe, the periodic orbit 0 is hyperbolic with reflection and has a Maslov index equal to po = 3 while the off-diagonal orbits 1 and 2 are hyperbolic without reflection with the Maslov index p1 = 2 [ 101. Fitting of numerical actions, stability eigenvalues, and rotation numbers to polynomial functions in E can then be used to reproduce the analytical dependence on E. The resonance spectrum is obtained in terms of the zeros of (4.16) in the complex energy surface. We remark that an expression like (4.13) can no longer be derived in general because of the interference between the two amplitude terms in (4.16). This is the general feature of a nonseparable regime where the spectrum of resonances loses its regularity. When there exist two fundamental periodic orbits, we may expect that the spectrum of resonances still displays quasiperiodic regularities, as is the case for the three-disk scatterer [33]. However, for repellers with more than two fundamental periodic orbits, the regularities are even further destroyed and the spectrum becomes irregular. By inspection of (4.16) we can infer that the spectrum of the periods plays a crucial role in the distribution of the resonances. If the fundamental periodic orbits have very similar periods &So = (nearly degenerate period spectrum), we may expect that the resonance spectrum is still quite regular, as in the periodic case (4.12), but with an overall amplitude given by the sum of interfering amplitudes. On the other hand, the resonance spectrum will be irregular if the periods of the fundamental orbits are very different. A very important role is also played by the stability eigenvalues with respect to the unstable asymmetric stretching perturbations. If among the stability eigenvalues for two given periodic orbits one is much larger than the other, the corresponding amplitude may be neglected so that the condition (4.16) reduces to the periodic case (4.12) for the less unstable of both fundamental periodic orbits. For instance, if IAol >> (A, I > I, the resonance spectrum is determined by the off-diagonal periodic orbits 1 and 2 and should present a regular progression. Similar considerations apply to the eigenfunctions associated with the resonances. According to Eq.(2.26), the contribution of a periodic orbit to the eigenfunctions is also given in terms of the stability eigenvalue A,,. Therefore, the eigenfunction is localized essentially around the least unstable periodic orbits. This result shows that the

560


resonant states are localized around the off-diagonal periodic orbits 1 and 2 if the periodic orbit 0 is markedly more unstable than the off-diagonal ones. Since the off-diagonal orbits 1 and 2 are composing the periodic orbit 12, the resonant states would hence be localized around the hyperspherical periodic orbit 12. This coincides with an interpretation found in the literature according to which the orbit 12 represents the resonant periodic orbit (RF'O) [143]. Another remark is that the resonances exist only below the line defined by (2.18) and (2.19) so that there is a gap between the real energy axis and the resonance spectrum. This is the feature of a strongly open scattering system in which the decay process is ultrafast. This gap is given in terms of the topological pressure that is the leading zero, so(E) = P ( p ; E ) , of the inverse Ruelle zeta function, (4.17)

When p = {, the leading zero gives thus an approximate value of the gap (2.18) in the spectrum of scattering resonances. If the repeller has only one unstable periodic orbit, the topological entropy vanishes in Eq. (2.19) so that the gap is completely determined by the sum of positive Lyapunov exponents as described by (4.13). In comparison with the periodic case, the gap is reduced when the repeller becomes chaotic, which is a direct effect of emerging classical chaos on a quantum property. This lifetime lengthening which has been studied in detail by Gaspard and Rice [33] can be regarded as the precursor of the fluctuations of the quantum decay rates around the average RRKM rate, which appears as the degree of openness of the potential decreases from strongly open to weakly open. This lifetime lengthening studied by Gaspard and Rice also appears when the longest quantum lifetimes are compared with the average classical lifetime [33]. In Section 11, we made the distinction between quantal quantities such as survival amplitudes and quasiclassical quantities such as averages or correlation functions. The decay of the quantal quantities is determined by the scattering resonances while the decay of the quasiclassical quantities is determined by the classical escape rate. According to (2.46), the escape rate can be calculated as the leading zero of the classical zeta functions (2.44),which takes here the approximate form (4.17) with = 1. Gaspard and Rice have pointed out that the average classical lifetime is shorter than the longest quantum lifetimes for chaotic repellers while they coincide for periodic repellers [33]. A similar phenomenon exists in weakly open potentials where the quantum resonances are distributed around the average RRKM


561

quasiclassical rate, as described, in particular, by the Porter-Thomas distribution. Therefore, here also the quantum resonances may have a lifetime longer than the quasiclassical lifetime, which may also be considered as an effect of classical chaos. It should be emphasized that this argument is based on the distribution of the eigenenergies in their imaginary parts but not in their real parts. The periodic-orbit quantization can be used to calculate not only the resonances but also the full shape of the photoabsorption cross section using (2.26) and (2.27). This semiclassical formula for the cross section separates in a natural way the smooth background from the oscillating structures due to the periodic orbits. In this way, the observation of emerging periodic orbits by the Fourier transform of the vibrational structures on top of the continuous absorption bands can be explained. To conclude this section, let us add that the formulas (4.15H4.17) can be generalized in a straightforward way to repellers with more than three fundamental periodic orbits. In the following, these tools are applied to several dissociating molecules.

E. Ultrashort-Lived Resonances in Watomic Molecules 1. Hg12

Zewail and co-workers have reported femtosecond laser experiments on the photodissociation of mercury(I1) iodide (HgI,) [ 1511:

h~ + HgI,(X ' C i )+ [I * . Hg

1

*

I]'

HgI(X 2Ct)+ I(,P3/2)

---c

(4.18)

Motivated by this experiment, Burghardt and Gaspard have carried out a detailed analysis of the resonances of the transition complex on the semiempirical potential surface proposed by Zewail and co-workers [151], that is, a damped Morse potential for the two degrees of freedom of collinear motion. Bending motion is neglected in this model so that the bending rotation number is assumed to be zero. The results reported below must therefore be shifted by the bending zero-point energy, and also have to be corrected if the rotation number varies significantly with energy. The potential surface for the stretching dynamics is typical of strongly open systems, with a single saddle equilibrium point and two exit channels, i.e., we have a typical example to which the previous analysis applies. If the saddle is at the origin of energy, the total dissociation threshold is at an energy of 1800 cm-I, while the bottom of the exit channels lies at - 100 cm-' . The classical dynamics follows the first bifurcation scenario we discussed above, i.e., involving a supercritical antipitchfork bifurcation.

562


(a) Periodic Repeller for 0 cm-' c E < E, = 523 cm-' . In this regime, the repeller contains the single periodic orbit 0 of symmetric stretching motion that is unstable (hyperbolic without reflection). Equilibrium point and periodic-orbit quantization have been applied to this regime. Both semiclassical quantization schemes show that six resonances with m = 0, u = 0, and n = 0, ..., 5 exist in this regime, which are regularly spaced with a spacing corresponding to a period of 363 fs. The first resonance has a lifetime of 1 1 1 fs. The lifetimes become longer as energy increases. In Fig. 13, we observe that the resonances lie on the border of the spectral gap, which coincides with the curve corresponding to the escape rate, that is, to the Lyapunov exponent of the symmetric-stretch periodic orbit 0, because the repeller is here periodic: Im E = -(A/2)Xo(ReE) = hP(1/2; Re E) = (A/2)P(1;Re E ) . Differences between the lifetimes obtained from equilibrium point quantization and periodic-orbit quantization appear as the bifurcation is approached. The lifetimes are underestimated by equilibrium point quantization but overestimated by periodic-orbit quantization. The reason for the upward deviation in the case of periodic-orbit quantization is that the Lyapunov exponent vanishes as the bifurcation is approached. The quantum eigenfunctions,however, are not characterized by the local linearized dynamics but extend over larger distances that are of more unstable character.

0

500

lo00

1500

Re E [cm"]

Figure 13. Scattering resonances of a two-degree-of-freedom coilinear model of the dissociation of Hg12 [lo]. The filled dots are obtained by wavepacket propagation, the crosses by equilibrium point quantization with (2.8), the dotted circles from periodic-orbit quantization with (4.16). The solid lines ace the curves corresponding to the Lyapunov exponents, Im E = -(h/2)Xp(Re E ) , of the fundamental periodic orbits p = 0, 1, 2. The dashed line is the spectral gap, Im E = hP( 1/2; Re E). The long-short dashed line is the curve corresponding to the escape rate, Im E = (h/2)P(I;ReE).


563

The periodic repeller exists up to the antipitchfork bifurcation. (b) Transition Region for E, = 523 cm-' < E < Ehr = 575 cm-'. Above the antipitchfork bifurcation, the symmetric-stretch periodic orbit becomes elliptic in the interval 523 cm-' < E < 548 cm-' . It is embedded in a main elliptic island of phase-space area much smaller than 2&, which cannot sustain a long-lived resonant state. Actually a wavepacket calculation shows that a single short-lived resonance with m = u = 0 and n = 6 at E = 55 1 cm-I covers the transition region. The elliptic island is delimited by the two off-diagonal periodic orbits 1 and 2 which are born at the antipitchfork bifurcation. From there up to above 1500 cm-' ,the periodic orbits 1 and 2 are hyperbolic (without reflection). At E d = 548 cm-' , the elliptic island is destroyed by a period-doubling bifurcation at which the periodic orbit 0 becomes hyperbolic (with reflection) up to high energies. A last homoclinic tangency between the stable and unstable manifolds of the three periodic orbits 0, 1, and 2 occurs around Ehr = 575 f 5 cm-I. For E d = 548 cm-' < E c Ehr = 575 cm-', the main elliptic island no longer exists but small subsidiary elliptic islands of high periods still persist up to the last homoclinic tangency. At the antipitchfork bifurcation E,, the periods of the new periodic orbits 1 and 2 bifurcate supercritically from the period of 0, as seen in Fig. 14.

(c) Fully Chaotic Repeller for Eh, = 575 cm-' C E --t 1500 cm-'. In this region, the repeller is a Smale horseshoe with three branches (see Fig. 15). The orbits of the repeller are in one-to-one correspondence with sequences built from symbols (0, 1,2} associated with the three fundamental periodic orbits. All the orbits are unstable of saddle type. The periodic-orbit quantization based on the zeta function quantization

..

.. :: *.. ... ,...* .

I * .I

I .

0

*

0

500

I

E [cm-'1

lo00

.

.

.

.

1500

Figure 14. Period-energy diagram of the fundamental periodic orbits 0, 1, and 2 in HgI2 with the bifurcations of the transition region marked by dashed lines.

564


P

Q

Figure 15. Three-branch h a l e horseshoe in the 2F collinear model of Hg12 dissociation at the energy E = 600cm-' above the saddle in a planar Poincari surface of section transverse to the symmetric-stretch periodic orbit. The Smale horseshoe is here traced out in a density plot of the cumulated escape-time function (4.6).

condition (4.16) compares favorably with the resonances obtained from wavepacket propagation. In this region, 15 resonances with m = u = 0 and n = 7, 8, . .. have been identified numerically, The relative regularity of the spectrum is explained by the similarity of the fundamental periods, as seen in Fig. 14. The agreement between the periodic-orbit quantization and the wavepacket calculation is very good as far as the energies of the resonances are concerned. At low energies near the transition region, the lifetimes are again overestimated because the Lyapunov exponents approach zero, while the nonlocal features of the dynamics are not taken into account. At higher


565

energies, the lifetimes are well reproduced by the approximation (4.16). Their values range between 160 and 240 fs, in agreement with the results of the femtosecond laser experiments. Variations in the lifetimes are observed that we interpret as the effect of the interference between the three fundamental periodic orbits. This interference causes a lengthening of the lifetimes as compared with the lifetimes of the individual periodic orbits. In Fig. 13, we thus observe that the resonance spectrum extends below the spectral gap, Im E = trP(1/2;ReE). On the other hand, the curve corresponding to the escape rate, Im E = (A/2)P( 1; Re E), lies here below the spectral gap but above the curves corresponding to the Lyapunov exponents, Im E = -(R/2)Ap(ReE) with p = 0, 1, 2. This is evidence for the Gaspard-Rice lengthening of lifetimes due to classical chaos, which was previously observed in the three-disk scatterer [33]. The eigenfunctions associated with the resonances have been obtained via wavepacket propagation. They appear to be localized along the symmetricstretch periodic orbit 0, with a number of nodes equal to n and even under the exchange of iodine nuclei. Due to the relative stability of the symmetricstretch orbit, we have thus here a system where the hypothesis of the orbit 12 representing the W O ,that is, resonant periodic orbit, does not hold. 2. c02

Much work has been devoted to the photodissociation of C02 in its most intense electronic band between 85,000 and 90,000 cm-' [152-1551. This band, which corresponds to the C;: electronic state, shows irregular vibrational structures that have proved difficult to assign. The photodissociation on this electronic surface has been studied in the classical work by Kulander and Light, who used a collinear model based on a semiempirical LEPS (London-Eyring-Polanyi-Sato)surface [ 1521. Their calculation revealed irregular vibrational structures that have been more recently interpreted by Schinke and Engel [153], Kulander et al. [154] and Zobay and Alber [155] in terms of periodic orbits. The periodic orbits that have been identified in these works correspond to the orbits 0, 01, 02, 12, 011, 022, 012, and 0102, according to the symbolic dynamics we introduced. These results raise the question whether a Smale horseshoe indeed exists, by analogy with the case of HgI,. We have recently been able to show that the collinear periodic orbits in this model of C02 are in fact controlled by such a three-branch Smale horseshoe. Below we give a detailed description of the classical-dynamical regimes leading up to-and beyond-the regime characterized by the three-branch horseshoe and its symbolic dynamics. As the LEPS surface comprises all vibrational degrees of freedom, it is possible to include the effect of bending in the analysis. To this end, we have carried out the equilibrium point quantization for this system at energies just

'

566

P. GASPARD AND I. BURGHARDT Table I1 Resonance Constants of Several Dissociative Molecules

363 -103 108

TYf TY

7y 7,

111

h

Echannel Ediss.

47.8 -lo00 2800

AE*

-0.1

en

91.9 -30 -i49.1 -I .23

0 1

02 03 x1 I x12

+i1.7

XI3

x22 x23 x33 g22

-1.2

30 112 3.2 3 f l

35 244 6.2 6.4 590 -10,006 80,650 -23.4 952.4 136.9 - i 856.3 -2.8 21.7 +i53.9 1.23 -i2.3 -43.0

18 f 2

10f2

6 f 3

32.9 3863 -3193

21,300 1108 i 25 298 f 12 - i 1650 k i300 -9 f 3

1850 f 100 -80 -I 10

-0

-0

1.O ~

-

~~

~

~

Nore: Frequencies and energies in reciprocal centimeters and times in femtoseconds. aCollinear two-degree-of-freedom model by &wail et al. [15 I ] by quantization around the equilibrium point of linear geometry (bending ignored). bFour-degree-of-freedom model of I2CI602 with LEPS surface [ 1521 by quantization around the equilibrium point of linear geometry (bending and rotation included). ‘Dunham expansion fitted to experimental data of the Hartley band by Joens [158] (nonlinear equilibrium geometry). dThree-degree-of-freedommodel fitted to experimental data for the overall band structure by Schinke et al. [I591 (nonlinear equilibrium geometry). Tollinear two-degree-of-freedom model with the Karplus-Porter surface by Pechukas et al. [ 1431 (linear equilibrium geometry). f T! = 2h/wj with tc= 5308.84 fs/cm. g7:

= h/lw31.

hEnergy and lifetime of the first available resonance in the previous analyses.

above the unique saddle point of the LEPS surface [146]. Let us note that, for this surface, both the total dissociation threshold and the bottom of the exit channels are at a much larger separation in energy from the saddle than in the case of HgI,. Table I1 gives the relevant data. In the following, we will characterize the different dynamical regimes for COz on the LEPS surface. The scenario in a general way resembles the one for HgI,. However, the initial bifurcations are of the second type described in Section IV.C.2; that is, the symmetric-stretch orbit undergoes a subcritical

567


antipitchfork bifurcation, which is preceded by a pair of tangent bifurcations. We have the following scenario: (a) Periodic Repeller for 0 cm-’ < E < 2637cm-' . Below the pair of tangent bifurcations, the repeller is composed of the symmetric-stretch periodic orbit that is hyperbolic (without reflection). Equilibrium point quantization (including bending) shows that three resonances n = 0, 1, 2 with rn = u = 0 exist below the bifurcation (see Fig. 16). The period of the symmetric stretching motion at the saddle is equal to 35 fs. The lifetimes become longer as energy increases, as in HgI,. The lifetime extrapolated to the saddle turns out to be equal to 6.2 fs, which is very short. (b) Transition Region for Ett = 2637 cm-’ < E < E h t . The transition region is extremely narrow in this model. The pair of tangent bifurcations occur at Ett = 2637 cm-’, where the off-diagonal periodic orbits 1 and 2, of hyperbolic-without-reflection stability, as well as the elliptic periodic orbits 1’ and 2’ are born. The subcritical antipitchfork bifurcation occurs at E, = 2672 cm-’, where the orbits 1‘ and 2’ merge with the symmetric-stretch periodic orbit 0, which changes its stability from hyperbolic (without reflection) to elliptic. The orbit 0 remains elliptic up to a period doubling at E d 0

5

-200

E

-400

Y

w Y

-600 L 0

1

1000

I

I

3000 Re E [cm“]

2000

1

4000

I

5000

Figure 16. Scatteringresonances of the full rotational-vibrational Hamiltonian describing the dissociation of COz on a LEPS surface obtained by equilibrium point quantization with (2.8). The resonances with J = 0, ... 10 are given by dots. Their close vicinity explains the formation of “hyphens”, i.e., unresolved sequences of dots. Note that rotation is very slightly destabilizing in the present model. The successive hyphens are the bending progressions with uz = 0, . . . 5 . The solid line is given by the Lyapunov exponent of the symmetric-stretch periodic orbit 0 expressed as an imaginary energy.

.

568


= 2707 cm-', above which it is hyperbolic (with reflection) up to E = 54,000

cm-' (see below). The last homoclinic tangency is expected at a not much higher energy than E d . According to the equilibrium point quantization, there is no resonance in this tiny transition region, which remains much below the wave-mechanical phase-space resolution. We notice that the transition region is even narrower than in the HgI, system. However, the successive tangent and subcritical bifurcations have important consequences for the period of the orbit 0 relative to the one of 1 and 2. As shown in Fig. 17, the period of 0 has a slightly increasing monotonous behavior while the period of 1 or 2 rapidly decreases, which is in contrast with the Hgl, system (see Fig. 14). This behavior is due to the tangent bifurcation at which the curve of the period of 1 or 2 versus energy must have a vertical slope so that the branch of the curve corresponding to 1' or 2' joins the curve of 0 at the subcritical bifurcation from lower energies. This has the effect of forcing the period of 1 or 2 to be significantly different from the period of 0. (c) Fully Chaotic Repeller for Eh, c E -+ 54,000 cm-'. In this region, the repeller is a three-branch Smale horseshoe associated with the three fundamental periodic orbits 0, 1, and 2 (see Fig. 18). The Smale horseshoe has been constructed with the escape-time function as described above, as for the Hgl, system. All the orbits are unstable of saddle type. This result allows us to establish a complete list of periodic orbits in terms of the symbols {0,1,2}. A semiclassical quantization is thus possible over this large energy interval via the zeta function formalism. The shortest periodic orbits

60

40 20

0

0

5000

10000

E

15000

[cm-'1

Figure 17. Period+nergy diagram of the fundamental periodic orbits 0, 1, 2, 01, 02, 12 of the C02 system with the bifurcations of the transition region marked by dashed lines.


569

P

9

Figure 18. Three-branch Smale horseshoe in the collinear model of C02 dissociation at

the energy E = 10,759.4 cm-' above the saddle in a hyperbolic Poincari surface of section transverse to the symmetric-stretchperiodic orbit. The figure represents a density plot of the cumulated escape-time function (4.6), as in Fig. 15.

are thus (0,1,2,12,01,02,. . .}. Figure 17 shows that the composite orbits have periods close to the sum of periods of the fundamental orbits. As energy increases, the symmetric-stretch orbit 0 becomes very unstable while the offdiagonal orbits 1 and 2 have a Lyapunov exponent that decreases toward very small values. From this behavior, we may expect that the off-diagonal periodic orbits play the dominant role in the quantization and the resonant states will be localized around them, that is, around the hyperspherical periodic orbit 12, which is topologically composed of 1 and 2. This is what has been numerically observed by Kulander et al. [154]. On the other hand, the large

570


difference of periods between 0 and 1 or 2 provides an explanation for the irregularity of the resonance spectrum in this system. A detailed analysis in terms of a zeta function quantization according to (4.16) should give a clue as to what is the role of the different periodic orbits in the spectrum, especially taking into account their widely different stabilities. We will report on such an analysis in a future publication [146]. The semiclassical formula (2.26) and (2.27) for the photoabsorption cross section shows that the Franck-Condon region may be identified with the region around the symmetric-stretch periodic orbit 0. From this we may expect that all the periodic orbits with the symbol 0 contribute significantly to the photoabsorption cross section, which explains why the periods of 0, 0 1, or 02 appear in the Fourier transform of the band, while the periods of 1 or 2 and 12 do not. In accordance with this observation, Schinke and Engel [153] first suggested that the periodic orbits 0,012 and 0102, according to our symbolic dynamics, should be relevant to the analysis of the photoabsorption cross section. Zobay and Alber [ 1551 later carried out a semiclassical calculation of the absorption cross section, taking into account the periodic orbits 01,02,011 and 022 in addition to those mentioned above. Their analysis is based on the semiclassical expression for the absorption cross section given in Eq. (2.27) (see also Ref. [40]) and yields very good agreement with the exact quantum-mechanical result for the collinear model. (d) Transition to Another Fully Chaotic Repeller for 54,140cm-' < E. At E = 54,140 cm-', the symmetric-stretch periodic orbit 0 changes its stability from hyperbolic (with reflection) to elliptic in an inverse period-doubling bifurcation, which is followed over a very narrow energy interval by a pitchfork-type bifurcation. By the latter, the periodic orbit 0 becomes hyperbolic without reflection. This sequence of events occurs over less than 1 cm-'. It is preceded by bifurcations of other periodic orbits of the horseshoe, in particular of the orbit 01. During these bifurcations, two new periodic orbits are born in the system, and we conjecture that a new repeller with five fundamental unstable periodic orbits exists above 54,200 cm-' .Its trajectories should be in one-to-one correspondence with sequences built on the alphabet {0,1,2,3,4}. Its quantization is possible by a straightforward generalization of (4.14H4.16). Note, however, that this regime is well above the experimentally accessible region. Let us compare these results with the experimental data on C02. A photoabsorption spectrum of COz has been recorded in 1965 by Nakata et al. [156]. The vibrogram as well as direct Fourier transforms of the intense band under discussion reveal main time recurrences around 50 and 60 fs [14]. On the other hand, the above model shows periods corresponding to 0 and 01 or 02 around 40 and 60 fs, respectively. Although there is a qualitative agree-


57 1

ment, the differences suggest the need for a quantitative analysis based on ab initio surfaces. 3. H 3

This dissociative system, which represents the prototype system for chemical reaction dynamics, has been the object of many studies. Child et al. [143] have carried out a detailed analysis of the classical dynamics in a collinear model based on the Karplus-Porter surface. These authors have introduced the concept of PODS and first observed the subcritical antipitchfork bifurcation scenario in this system. The bottom of the exit channels is at -3 194 cm-' if the origin corresponds to the saddle of the Karplus-Porter surface. The pair of tangent bifurcations occur at E,, = 1670 cm-', which is followed by the subcritical antipitchfork bifurcation at E , = 2633 cm-' . The bifurcation scenario is thus similar to the CO2 system, and we may expect a three-branch Smale horseshoe in this system as well. The quanta1 character is even more prominent than in CO:!and the first reported resonances occur at 3863 and 7453 cm-', respectively, above the transition region (see Fig. 19). Here the zeta function semiclassical quantization with (4.16) should apply. The reported lifetimes are of 33 fs, which is

0

Q

-100

E

-200

u

w

U

-300

a

0

loo00

a

20000 Re E [cm-'1

3

m

Figure 19. Scattering resonances of 2F collinear models of the dissociation of H3 obtained by Manz et al. [ 1431 on the Karplus-Porter surface and by Sadeghi and Skodje on a DMBE surface [132]. The energy is defined with respect to the saddle point. The dashed lines mark the bifurcations of the transition to chaos: 1, 5, and 3 are the numbers of shortest periodic orbits, or PODS, in each region.

572


longer than the symmetric stretching period of about 10 fs, in contrast with both the HgI, and COz systems. In a recent work, Sadeghi and Skodje have calculated the resonances according to a different collinear model of H3, based on a doubly-manybody-expansion (DMBE) surface [132]. The energies of the two first resonances are very close to the ones on the Karplus-Porter surface, but the lifetimes are here given by 17 and 25 fs, respectively (see Fig. 19). The similarity with the COz system suggests that the periodic orbit 0 becomes more and more unstable as energy increases, while the periodic orbits 1 and 2 become less and less unstable. This behavior is in accordance with the localization of the resonant eigenfunctions around the off-diagonal periodic orbits 1 and 2, and with the lengthening of the lifetimes at high energies (see Fig. 19). The periodic-orbit quantization of the DMBE model has recently been carried out [157]. 4.

03

The Hartley band of ozone has been the object of systematic investigations. The vibrational structures on top of the band have motivated studies of the dissociation dynamics on the corresponding potential surface [ 1071. Johnson and Kinsey have interpreted the time recurrences in Fourier transforms of the band in terms of the periodic orbits of a two-degree-of-freedom model with a frozen bending angle [1071. The structure of the periodic orbits of this model turns out to be very similar to the periodic orbits of the collinear models described above. Drawing on this similarity, we may identify the periodic orbits observed by Johnson and Kinsey with the periodic orbits 0,Ol or 02,011 or 022, and 0111 or 0222 of a three-branch horseshoe, with periods of 43,70, 103, and 131 fs, respectively. Let us remark here that one may in principle carry out a periodic-orbit analysis that goes beyond the one for a two-degree-of-freedom model with frozen bending configuration. In particular, the periodic orbits of the three-degree-of-freedom model can be numerically obtained from the ones of a two-degree-of-freedom model by relaxing the constraint on the bend angle and letting it evolve according to the full dynamics. In a more recent work, Joens [158] has assigned the structures of the Hartley band using a Dunham expansion, that is equilibrium point quantization. The lifetime predicted by his analysis is extremely short, equal to 3.2 fs, while the symmetric stretching period is of 30 fs. Recall, however, that the interpretations in terms of equilibrium point expansions and in terms of periodic orbits are strictly complementary only for regular regimes. 5. H20

The first UV band of water has been analyzed by Schinke and co-workers,


573

who obtained a nice agreement between the experimental and theoretical shapes [159]. The vibrational structures on top of the band are very weak but regular. The corresponding period and lifetime of the time recurrences are respectively of 18 fs and about 6 fs. Recently, Hiipper et al. El601 have carried out a periodic-orbit quantization of the model of Ref. [159], as well as of another model [161] which predicts a shorter lifetime of about 3 fs instead of 6 fs.

6. Comparison of Lifetimes Table 11 shows a compilation of some of the data we collected for the above systems, which provide the lifetimes extrapolated to the saddle equilibrium point. As expected, a heavy molecule like HgI, has long period and lifetime as compared to the lighter species. In general, the period of the symmetric stretching oscillations behaves as expected, becoming shorter as the species becomes lighter. Concerning the lifetimes, Table I1 shows that they are usually three to four times shorter than the period such that the observable wavepacket recurrences are expected to be rather weak. For the comparison of periods and lifetimes, note here that the period is defined as the period of rotation of a trigonometric function like cos(2rf/T), while the lifetime is defined by the decay of an exponential function like exp(-t/.r), which shows that they differ by definition by a factor 2r. Apart from the general trend, a surprisingly short lifetime appears for 0 3 and a surprisingly long one for H3. This might be partially due to the different character of the potential surfaces for the different species, which will affect the dynamical instability. We hope that future works will shed more light on these observations.

V. CONCLUSIONS In this chapter we have summarized recent results obtained by applying semiclassical methods to the study of vibrational and dissociative dynamics on the femtosecond scale. In a first part, we described the methods of equilibrium point quantization and periodic-orbit quantization. The former method represents a perturbation treatment that is valid as long as an assignment in terms of good quantum numbers is possible. Beyond such regular regimes, we encounter irregular spectra related to nonlocal dynamics. Periodic-orbit theory provides an avenue to the understanding of such spectra on an intermediate energy scale. Periodic-orbit methods have been formulated on the one hand for classically integrable systems and on the other hand for classically hyperbolic systems: The former are described by the Berry-Tabor trace formula and the latter by the Gutzwiller trace formula. The Gutzwiller trace formula at its leading order is equivalent to a quantization condition expressed in terms

574


of a zeta function, which takes into account the interference between periodic orbits, that is characteristic of chaotic systems. Even though the periodic orbits are of infinite number, the analysis becomes manageable due to the fact that the shortest and least unstable periodic orbits provide the dominant contributions. For discrete spectra, we discussed the relationship between classical chaos and random-matrix properties, which concern the smallest spectral energy scale, below the mean level spacing. Note that dynamical instability, which is the signature of classically chaotic dynamics, directly affects the imaginary parts of the system’s eigenenergies, which is of immediate relevance to the lifetimes in scattering systems but does not play a direct role for bounded systems. Irregular spectra in bounded systems rather seem determined by the distribution of the periods of the periodic orbits or, more generally, of the interfering quantum paths. In the context of irregular spectra, we further summarized new results concerning their parametric properties. Turning back to the role of the periodic orbits emerging from the wavemechanical dynamics, we have shown that the new vibrogram technique may be used to provide evidence for such periodic orbits in the intramolecular vibrational dynamics. The vibrogram represents the period-energy diagram of the emergent periodic orbits. The periodic orbits that may thus be observed are necessarily limited to the shortest ones, since the accumulation of periodic orbits soon exhausts the limits of resolution of the technique. Nevertheless, several examples show clear evidence of the periodic-orbit signature. Of particular interest are changes in the stability of the periodic orbits, which can be observed, for example, in C2H2, due to a transition to chaos in the classical bending dynamics. This is in contrast to the isotopomer CzHD, whose dynamics can be modeled via a classically integrable effective Hamiltonian. Periodic orbits also play a key role in the semiclassical calculation of the transition-state resonances that characterize ultrafast dissociation processes. In this context, we have considered the dissociation of several triatomic molecules on potential-energy surfaces with a single saddle point. From a detailed analysis of the classical collinear dynamics for the symmetric species HgI, and C02, a general scheme may be inferred: Over a broad energy range, we have a classically chaotic regime where all periodic orbits are part of the global phase-space structure of a Smale horseshoe with three branches. This result paves the way for a periodic-orbit quantization based on the zeta function, which provides an understanding of the spectrum in terms of the periods and stabilities of a few fundamental periodic orbits. This theoretical scheme offers a unifying viewpoint with respect to a number of results obtained in previous research. On this background, we discussed possible dynamical scenarios and lifetimes for several molecules. Ultrafast dissociation, and reactive processes in general, can thus be


575

understood in terms of the dynamics of open systems and turn out to be markedly susceptible to the emerging regular or chaotic classical dynamics. In open systems, as opposed to bounded systems, a more direct connection exists between classical chaos and the properties of the resonance spectrum. We already mentioned above that the dynamical instability has direct consequences for the lifetimes of the metastable states. On the other hand, irregularities in the energy spectrum appear to be primarily related to the distribution of the periods. We thus believe that the explanations offered by the theoretical framework presented here will add to the interpretation of many experimental observations on elementary dynamical processes.

Acknowledgments We express our gratitude to G . Nicolis, I. Prigogine. and S. A. Rice for support in this research. We also thank M. Herman, R. Jost, J. Lievin, 1. Mills, J.-P. Pique, and R. Schinke for fruitful discussions as well as P. van Ede van der Pals for assistance in the preparation of this report. Financial support was granted by the National Fund for Scientific Research (FNRS Belgium) for P. G. and by an EEC doctoral fellowship, No.ERBCHBICFl30857 for I. B. The research is further financially supported by the Quantum Keys for Reactivity ARC Project of the CommunautC Franqaise de Belgique, and by the Banque Nationale de Belgique.

References 1. A. H. Zewail, Science 242, 1645 (1988); M. Gruebele and A. H. Zewail, Physics Today, May 1990, p. 24; L. R. Khundkar and A. H. Zewail, Annu. Rev. Phys. Chem. 41, 15 (1990); A. H. Zewail, J. Phys. Chem. 97, 12427 (1993). 2. R. M. Marcus, Faraday Discuss. Chem. Soc. 91,479 (1991). 3. E. Abramson, R. W. Field, D. Imre, K. K. Innes, and 3. L. Kinsey, J. Chem. Phys. 80, 2298 (1984); ibid.. 83,453 (1985); R. L. Sundberg, E. Abramson, J. L. Kinsey, and R. W. Field, J. Chem. Phys. 83, 466 (1985). 4. W. F. Polik, C. B. Moore, and W. H. Miller, J . Chem. Phys. 89,3584 (1988); W. F. Polik, D. R. Guyer, W. H.Miller, and C. B. Moore, J. Chem. Phys. 92,3471 (1990).

5. E. Haller. H. Koppel, and L. S. Cederbaum, Chem. Phys. Letr. 101,215 (1983): T. Zimmermann, H. Koppel, L. S. Cederbaum, G. Persch, and W. Demtriider, Phys. Rev. Letr. 61, 3 (1988).

6. A. Delon, R. Jost, and M. Lombardi, J. Chem. Phys. 95, 5701 (1991); R. Georges, A. Delon, and R. Jost, J. Chem. Phys. 103, 1732 (1995). 7. G. Sitja and J.-P. Pique, Phys. Rev. Lett. 73,232 (1994). 8. H. 0.Pritchard, The Quantum Theory of Unimolecular Reactions, Cambridge University Press, Cambridge, 1984; H. 0. Pritchard, J. Phys. Chem. 89, 3970 (1985). 9. D. C. Chatfield. R. S. Friedman, D. G. Truhlar, B. C. Garrett, and D. W. Schwenke, J. Am. Chem. SQC. 113, 486 (1991); D. C. Chatfield, R. S. Friedman, D. G. Truhlar, and D. W. Schwenke, Faraday Discuss. Chem. Soc. 91,289 (1991); R. S. Friedman and D. G. Truhlar, Chem. Phys. Len. 183,539 (1991); D. C. Chatfield, R. S. Friedman, D. W. Schwenke, and D. G. Truhlar, J. Phys. Chem. 96,2414 (1992).

576


10. I. Burghardt and P. Gaspard, J. Chem. Phys. 100,6395 (1993); I. Burghardt and P. Gaspard, J. Phys. Chem. 99, 2732 (1 995). 11. A. Holle, J. Main, G. Wiebusch, H. Rottke, and K. H. Welge, Phys. Rev. Lett. 61, 161 (1988); J.-M. Mao and J. B. Delos, Phys. Rev. A 45, 1746 (1992); T. van der Velt, W. Vassen, and W. Hogervorst, Europhys. Lett. 21, 903 (1993); J. Main, G. Wiebush, K. Welge, J. Shaw, and J. B. Delos, Phys. Rev. A 49, 847 (1994). 12. M. C. Gutzwiller, J. Math. Phys. 8, 1979 (1967); ibid., 10, 1004 (1969); ibid., 11, 1791 (1970); ibid., 12, 343 (1971); ibid., 18, 806 (1977); M. C . Gutzwiller, Phys. Rev. Lett. 45, 150 (1980); J. Phys. Chem. 92, 3154 (1988); M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, New York, 1990. 13. M. V. Berry and M.Tabor, Proc. Roy. SOC.Lo&. A 349, 101 (1976); J. Phys. A: Math. Gen. 10,371 (1977). 14. P. Gaspard, D. Alonso, and I. Burghardt, Adv. Chem. Phys. 90, 105 (1995). 15. W. Thining, Quantum Mechanics of Atoms and Molecules, Springer. New York, 1981. 16. I. M. Mills, in Molecular Spectroscopy: Modem Research, K. N. Rao and C. W. Mathews, Eds., Academic, New York, 1972, p. 115. 17. D. Papousek and M. R. Aliev, Molecular Vibrational-Rotational Spectra, Elsevier, Amsterdam, 1982. 18. W. H. Miller, R. Hernandez, N. C. Handy, D. Jayatilaka, and A. Willetts, Chem. Phys. Lett. 172, 62 (1990). 19. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978; A. J. Lieberman and A. I. Lichtenberg, Regular and Stochastic Motion, Springer, New York, 1983; J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. 20. M. J. Cohen, N. C. Handy, R. Hernandez, and W. H. Miller, Chem. Phys. Lett. 192,407

(1992). 21. L. E. Fried and G. S. Ezra, J. Chem. Phys. 86,6270 (1987). 22. G. D. Birkhoff, Dynamical Systems, Colloquium Publication 9, American Mathematical Society, New York, 1927; F. G. Gustavson, Astronom. J. 71,670 (1966); R. T. Swimm and J. B. Delos, J. Chem. Phys. 71, 1706 (1979); C. Jaff6 and W.P. Reinhardt, J. Chem. Phys. 71, 1862 (1979); ibid., 77, 5191 (1982); M. Robnik, J. Phys. A: Math. Gen. 17, 109 (1984); ibid., 25, 5311 (1992). 23. J.-A. Sepulchre and P. Gaspard, J. Chem. Phys. 102,6727 (1995). 24. T. Seideman and W. H. Miller, J. Chem. Phys. 95, 1768 (1991). 25. M. S. Child, Semiclassical Mechanics with Molecular Applications, Clarendon, Oxford, 1991. 26. T. Uzer and R.A. Marcus, J. Chem. Phys. 81,5013 (1984). 27. M. E. Kellman and E. D. Lynch, J . Chem. Phys. 85,7216 (1986); ibid., 88,2205 (1988); Z. Li, L. Xiao, and M. E. Kellman, J. Chem. Phys. 92, 2251 (1990); M. E. Kellman, J. Chem. Phys. 93,6630 (1990). 28. B. Grammaticos and A. Voros, Ann. Phys. 123,359 (1979). 29. W. H.Miller, J. Chem. Phys. 56, 38 (1972); ibid., 63,996 (1975); ibid., 83,960 (1979). 30. S. C. Creagh, J. M. Robbins, and R. G. Littlejohn. Phys. Rev. A 42, 1907 (1990). 31. P. Gaspard and D. Alonso, Phys. Rev. A 47, R3468 (1993); P. Gaspard, in Quantum Chaos. G. Casati and B. V. Chirikov, Eds., Cambridge, University Press, Cambridge, 1994; D. Alonso and P. Gaspard, Chaos 3,601 (1993).


577

32. A. M. Ozono de Almeida, Hamittonian Systems: Chaos and Quantization, Cambridge University Press, Cambridge, 1988. 33. P. Gaspard and S. A. Rice, J. Chem. Phys. 90,2225,2242,2255 (1989); ibid., 91, E3279 (1989). 34. B. Eckhardt and E. Aurell, Europhys. Lett. 9,509 (1989). 35. P. Gaspard and J. R. Dorfman, Phys. Rev. E 52, 3525 (1995). 36. A. Voros, J. Phys. A: Math. Gen. 21,685 (1988). 37. P. CvitanoviC and B. Eckhardt, Phys. Rev. Lett. 63, 823 (1989). 38. E. Bogomolny, Comm. At. Mol. Phys. 25, 63 (1990); M. V. Berry and J. P. Keating, J. Phys. A 23, 4839 (1990); Proc. Roy. SOC. Land. A 437, 151 (1992); 0. Agam and S . Fishman, J. Phys. A: Math. Gen. 26,2113 (1993). 39. S. Tomsovic and E. J. Heller, Phys. Rev. Lett. 67, 664 (1991); Phys. Rev. A 47, 282 (1993); M. A. Seplilveda, S . Tomsovic, and E. J. Heller, Phys. Rev. Left. 69,402 (1992). 40. B. Eckhardt, S. Fishman. K. Miiller, and D. Wintgen, Phys. Rev. A 45, 3531 (1992). 41. M. V. Berry, J. Phys. A 10, 2083 (1977). 42. A. Voros, Ann. Inst. Henri Poincare‘ A 24, 31 (1976); ibid., 26, 343 (1977). 43. M. Shapiro, J. Ronkin, and P. Brumer, Chem. Phys. Lert. 148, 177 (1988). 44. C. E. Porter and R. G. Thomas, Phys. Rev. 104,483 (1956); P. A. Moldauer, Phys. Rev. B 136, 947 (1964); T. A. Brody, J. mores, J. B. French, P. A. Mello, A. Pandey, and S . S. M. Wong, Rev. Mod. Phys. 53,385 (1981). 45. A. I. Shnirelman, Ups. Mat. Nauk 29, 181 ( 1974); Y.Colin de Verdi&re, Commun. Math. Phys. 102,497 (1985); S . Zelditch, Duke Math. J. 55,919 (1987). 46. M. Srednicki, Phys. Rev. E 50, 888 (1994). 47. E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984). 48. P. Gaspard, in Quantum Chaos: Quantum Measurement, P. CvitanoviC. I. Percival, and A. Winba, Eds., Kluwer, The Netherlands, 1992, p. 19. 49. A. M. Ozorio de Almeida and J. H. Hannay, J. Phys. A: Math. Gen. 20,5873 (1987). 50. M. Kub, F. Haake, and D. Delande, Phys. Rev. Lett. 71,2167 (1993). 51. J.-M. Mao, K. A. Rapelje, S. J. Bldgett-Ford, J. B. Delos, A. Konig, and H. Rinneberg, Phys. Rev. A 48,2117 (1993). 52. A. Bohm, Quantum Mechanics: Foundations and Applications, 3rd ed., Springer, New York, 1992. 53. H. Koppel, W. Domcke, and L. S . Cederbaum, Adv. Chem. Phys. 57, 59 (1984). 54. S. Weigert and R. G. Littlejohn. Phys. Rev. A 47, 3506 (1993); R. G. Littlejohn and S. Weigert, Phys. Rev. A 48,924 (1993). 55. W. H . Miller, Adv. Chem. Phys. 25, 69 (1974); ibid., 30,77 (1975). 56. P. Gaspard, in Quantum Chaos, G. Casati, I. Guameri, and U. Smilansky, Eds., Proceedings of the International School of Physics “Enrico Fermi,” Varenna, 1991, North-Holland, Amsterdam, 1993, p. 307. 57. R. Blumel, B. Dietz, C. Jung, and U.Smilansky, J. Phys. A: Math. Gen. 25, 1483 (1992); E. Doron, U. Smilanksy, and A. Frenkel, Phys. Rev. Lett. 65, 3072 (1990); Physica D 50,367 (1992). 58. M. V. Berry, in Chaos and Quantum Physics, M.-J. Giannoni, A. Voros, and J. ZinnJustin, Eds., North-Holland, Amsterdam, 1991, p. 251.

578


59. P. Camithers and F. Zachariasen, Rev. Mod. Phys. 55, 245 (1983). 60. P. Cvitanovif and B. Eckhardt, J. Phys. A: Math. Gen. 24, L237 (1991). 61. P. Gaspard and D. Alonso Ramirez, Phys. Rev. A 45, 8383 (1992). 62. W. R. Frensley, Rev. Mod. Phys. 62,745 (1990). 63. M. Pollicott, Invent. Math. 81, 413 (1985); D. Ruelle, Phys. Rev. Lett. 56, 405 (1986); D. Ruelle, J. Differ: Geom. 25,99, 117 (1987). 64. P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65, 1693 (1990); P. Gaspard, Chaos 3, 427 (1993); P. Gaspard and F. Baras, Phys. Rev. E 51,5332 (1995); P. Cvitanovi6. J.-P. Eckmann, and P. Gaspard, Chaos, Solitons, and Fractals 6, 113 (1995); P. Gaspard, Phys. Rev. E 53,4379 (1996). 65. M. A. Seplilveda and S. Mukamel, J. Chem. Phys. 102, 9327 (1995). 66. Z. Li, J.-Y. Fang, and C. C. Martens, J. Chem. Phys. 104,6919 (1996). 67. P.Gaspard, Prog. Theor: Phys. Suppl. 116, 369 (1994). 68. S. Mukamel, V. Khidekel, and V. Chernyak, Phys. Rev. E53, R1 (1996). 69. R. Jost, CNRS, Grenoble, private communication (1993). 70. P. Gaspard and X.-J. Wang, Phys. Rep. 235, 321 (1993). 71. K. A. Holbrook, M. J. Pilling, and S. H.Robertson, Unimolecular Reactions, 2nd ed., Wiley, Chichester, 1996. 72. W. Forst, Theory of Unimolecular Reactions, Academic, New York, 1973. 73. C. E. Porter, Statistical Theories of Spectra: Fluctuations, Academic, New York, 1965. 74. M. L. Mehta, Random Matrices and the Statistical Theory of Energy Levels, Academic, New York, 1967. 75. P. M. Bleher, Zh. Cheng, F. J. Dyson, and I. L. Lebowitz, Commun. Math. Phys. 154, 433, 469 (1993); P. M. Bleher, D. V. Kosygin, and Y. G. Sinai, Commun. Math. Phys. 170, 375 (1995). 76. 0.Bohigas, in Chaos and Quantum Physics, M.-J. Giannoni, A. Voros and J. Zinn-Justin, Eds., North-Holland, Amsterdam, 1991, p. 87. 77. 0. Bohigas, M. J. Giannoni, and C. Schmidt, Phys. Rev. Lett. 52, 1 (1984). 78. J. Bolte, G. Steil, and F. Steiner, Phys. Rev. Lett. 69, 2188 (1992). 79. P. Seba and K. Zyczkowski, Phys. Rev. A 44,3457 (1991); S. Albeverio and P. Seba, J. Stat. Phys. 64, 369 (1991). 80. G. Date, S . R. Jain, and M. V. N. Murthy, Phys. Rev. E 51, 198 (1995). 81. M. V. Beny and 1. P. Keating, J. Phys. A 23, 4839 (1990); C. Matthies and F. Steiner, Phys. Rev. A 44, R7877 (1991). 82. M. V. Berry, Proc. Roy. SOC. Lmd. A 413, 183 (1987). 83. M. Gutzwiller, Physica D 7 , 341 (1983). 84. M. V. Berry, Proc. Roy. SOC. b n d . A 400, 229 (1985). 85. J.-P. Pique, Y. Chen, R. W. Field, and J. L. Kinsey, Phys. Rev. Lett. 58, 475 (1987); J. P.Pique, J . Opt. SOC.Am. B 7, 1816 (1990). 86. J. Wilkie and P. Brumer, Phys. Rev. Lett. 69, 2018 (1992). 87. R. Aurich, J. Bolte, and F. Steiner, Phys. Rev. Lett. 73, 1356 (1994). 88. A. V. Andreev, 0. Agam, B. D. Simons, and B. L. Al’tshuler, Phys. Rev. Lett. 76,3947 ( 1996). 89. G. Casati and B. Chirikov, Physica D 86, 220 (1995).


579

90. P. Pechukas, Phys. Rev. Lett. 51,943 (1983); T. Yukawa, Phys. Rev. Lett. 54,1883 (1985); Phys. Lett. A 116, 227 (1986); K. Nakamura and M. Lakshmanan, Phys. Rev. Lett. 57, 1661 (1986). 91. J. Goldberg, U. Smilansky, M. V. Berry, W. Schweizer, G. Wunner, and G. Zeller, Nonlineariry 4, I (1991). 92. B. D. Simons and B. L. Al’tshuler, Phys. Rev. Lett. 70,4063 (1993);C. W. J. Beenakker, Phys. Rev. Lett. 70,4126 (1993). 93. I. C. Percival, J. Phys. B 6, L229 (1973); N. Pomphrey, J. Phys. 7, 1909 (1974). 94. D. W. Noid, M. L. Koszykowski, M. Tabor, and R. A. Marcus, J. Chem. Phys. 72,6169 (1980); J. Brickmann and R. D. Levine, Chem. Phys. Lett. 120, 252 (1985). 95. P. Gaspard, S. A. Rice, and K.Nakamura, Phys. Rev. Lett. 63,930, 2540(E) (1989); P. Gaspard, S. A. Rice, H.J. Mikeska, and K. Nakamura, Phys. Rev. A. 42,4015 (1990). 96. I. Guameri, K.Zyczkowski, J. Zaluzewski, L. Molinari, and G . Casati, Phys. Rev. E 52, 2220 (1995). 97. D. Saher, F. Haake, and P. Gaspard, Phys. Rev. A 44,7841 (1991): T. Takami and H. Hasegawa, Phys. Rev. Lett. 68,419 (1992). 98. J. Zakrzewski and D. Delande, Phys. Rev. E 47, 1650 (1993). 99. U. Stoffregen, J. Stein, H.-J. Stkkmann, M. Kui, and F. Haake, Phys. Rev. Lett. 74, 2666 (1995); H.-J. Stikkmann, J. Stein, and M. Kollman, in Quantum Chaos, G. Casati and B. Chirikov, Eds., Cambridge University Press, Cambridge, 1995, p. 661. 100. F. von Oppen, Phys. Rev. Lett. 73, 798 (1994). 101. Y. V. Fyodorov and H.-J. Sommers, Phys. Rev. E 51, R2719 (1995); Y. V. Fyodorov and H.-J. Sommers, Z Phys. B 99, 123 (1995). 102. P.Gaspard, K. Nakamura, and S. A. Rice, Comments At. Mol. Phys. 25, 321 (1991). 103. K. Nakamura, M. Lakshmanan, P. Gaspard, and S. A. Rice, Phys. Rev. A 46,6311 (1992); P. Gaspard and P. van Ede van der Pals, Chaos, Solitons and Fractals 5, 1183 (1995). 104. J. Nygiird, Quantum Chaos of the NO2 Molecule in High Magnetic Fields, Master Thesis, Copenhagen, 1996. 105. B. Jessen and H.Tomehave, Acra Math. 77, 137 (1945). 106. D. Wintgen, Phys. Rev. Lett. 58, 1589 (1987). 107. B. R. Johnson and J. L. Kinsey, J. Chem. Phys. 91, 7638 (1989); Phys. Rev. Lett. 62, 1607 (1989). 108. K. Hirai, E. J. Heller, and P. Gaspard, J. Chem. Phys. 103, 5970 (1995). 109. M.Baranger, M. R. Haggerty, B. Lauritzen, D. C. Meredith, and D. Provost, Chaos 5, 261 (1995). 110. D. C. Rouben and G. S. Ezra, J. Chem. Phys. 103, 1375 (1995). 111. R. Georges, A. Delon, F. Bylicki, R. Jost, A. Campargue, A. Charvat, M. Chenevier, and F. Stoeckel, Chem. Phys. 190, 207 (1995); A. Delon, R. Georges, B. Kirmse, and R. Jost, Faraday Discuss. Chem SOC. (1995). 112. J. Lihvin, M. Abbouti Temsamani, P. Gaspard, and M. Herman, Chem. Phys. 190,419 (1995). 113. L. Michaille, M. Joyeux, and J.-P. Pique, in preparation. 114. M. Abbouti Temsamani, P. Gaspard, M. Herman, R.Jost, and P. van Ede van der Pals, in preparation. 115. T. S. Rose, M. J. Rosker, and A. H. Zewail, J. Chem. Phys. 91, 7415 (1989).

580


116. J.-P. Pique, M. Joyeux, J. Manners, and G. Sitja, J. Chem. Phys. 95, 8744 (1991). 117. J.-P. Pique, J. Manners, G. Sitja, and M. Joyeux, J. Chem. Phys. 96, 6495 (1992). 118. M. Joyeux, Chem. Phys. 185,263 (1994). 119. G. Chambaud, M. Joyeux, 1.-P. Pique, H. Ring, P. Rosmus, and G. Sitja, in preparation. 120. M. Joyeux, Chem. Phys. Lett. 247,454 (1995); J. Mol. Spectrosc. 175,262 (1996). 121. M. Joyeux, Chem. Phys. 203,281 (1996). 122. J. K. G. Watson, M. Herman, J. C. Van Craen, and R. Colin, J. Mol. Spectrosc. 95, 101 (1982); J. Vander Auwera, T. R. Huet, M. Herman, C. D. Hamilton, J. L. Kinsey, and R. W. Field, J. Mol. Spectrosc. 137, 381 (1989). 123. M. Abbouti Temsamani and M. Herman, J. Chem. Phys. 102,6371 (1995). 124. T.Carrington, Jr., L. M. Hubbard, H. F. Schaeffer 111, and W. H. Miller, J. Chem. Phys. 80, 4347 (1984). 125. S. A. B. Solina, J. P. O’Brien, R. W. Field, and W. F. Polik, Ber. Bunsenges. Phys. Chem. 99, 359 (1993). 126. M. Quack, Phil. Trans. Roy. Soc. Lond. A 332,203 (1990). 127. G. M. Lankhuijzen and L. D. Noordam, Phys. Rev. A 52,2016 (1995). 128. M. Bixon and J. Jortner, J. Chem. Phys. 48,715 (1968); ibid., 50,4061 (1969); J. Jortner, Pure Appl. Chem. 24, 165 (1970). 129. M. Lombardi, I? Labastie, M.C. Bordas, and M. Boyer, J. Chem. Phys. 89,3479 (1988); M. Lombardi and T. H. Seligman, Phys. Rev. A 47, 3571 (1993). 130. E. J. Heller, J. Chem. Phys. 68, 2066 (1978); S.-Y. Lee and E. Heller, J. Chem. Phys. 71,4777 (1979). 131. R. T. Pack, J. Chern. Phys. 65,4765 (1976). 132. R. Sadeghi and R. T.Skodje, J. Chem. Phys. 98, 9208 (1993); ibid., 99, 5126 (1993); R. Sadeghi and R. T. Skodje, Phys. Rev. A 52, 1996 (1995). 133. E. P. Wigner and L. Eisenbud, Phys. Rev. 72,29 (1947); A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30,257 (1958); E. P. Wigner, in Dispersion Relations and their Connection with Causality, E. P. Wigner, Eds., Academic, New York, 1964, p. 4 0 P. G. Burke, I. Mackey, and I. Shimamura, J. Phys. B: At. Mol. Phys. 10, 2497 (1977). 134. M. Quack and J. Troe, Them Chem.: Adv. Perspec. 6B, 199 (1981). 135. D. M. Wardlaw and R. A. Marcus, Adv. Chem. Phys. 70, 231 (1988). 136. W. H. Miller, Chem. Rev. 87, 19 (1987). 137. R. D. Levine, Adv. Chem. Phys. 70, 53 (1988). 138. W. F. Polik, D. R. Guyer, and C. B. Moore, J. Chem. Phys. 92, 3453 (1990). 139. W. H. Miller, R. Hernandez, C. B. Moore, and W. F. Polik, J. Chem. Phys. 93, 5657 (1990). 140. F. Remacle, M. Desouter-Lecomte, and J. C. Lorquet, Chem. Phys. 153, 201 (1991); M. Desouter-Lecomte and F. Culot, J. Chem. Phys. 98,7819 (1993); F. M. Izrailev, D. Saher, and V. V. Sokolov, Phys. Rev. E 49, 130 (1994). 141. R. Bliimel, in Quantum Chaos, G. Casati, 1. Guameri, and U. Smilansky. Eds., Proceedings of the International School of Physics “Enrico Fermi”, Varenna, 1991, North-Holland, Amsterdam, 1993, p. 385. 142. A. Buchleitner, D. Delande, J. Zakrzewski, R. N. Mantegna. M. Amdt, and H. Walther, Phys. Rev. Lett. 75,3818 (1995).


58 1

143. E. Pollak and P. Pechukas, J. Chem. Phys. 69, 1218 (1978); E. Pollak, M. Child, and P. Pechukas, J. Chem. Phys. 72, 1669 (1980); E. Pollak and M. S . Child, J. Chem. Phys. 73, 4373 (1980); E. Pollak and M. S . Child, Chem. Phys. 60, 23 (1981); E. Pollak, J. Chem. Phys. 74,5586 (1981); J. Manz, E. Pollak, and J. Romelt, Chem. fhys. Lett. 86,26 (1982); E. Pollak, J. Chem. Phys. 76,5843 (1982): E. Pollak, in The Theory of Chemical Reaction Dynamics, D. C. Clary, Ed., Reidel, Dordrecht, 1986, p. 135. 144. M. A. M. De Aguiar, C. P. Malta, M. Baranger, and K. T. R. Davies, Ann. fhys. 180, 167 (1987). 145. S. Wiggins, Global Bifurcarions and Chaos, Springer, New York, 1988; S. Wiggins, Chaotic Transport in Dynamical Systems, Springer, New York, 1992; E. Ott, Chaos in Dynamical Systems, Cambridge University Press, Cambridge, 1993. 146. I. Burghardt and P. Gaspard, in preparation. 147. N. K. Gavrilov and L. P. Shil'nikov, Math. USSR Sbornik 17,467 (1972); ibid., 19, 139 ( 1973). 148. S . Smale, The Mathematics of lime, Springer, New York, 1980. 149. P. Gaspard and S . A. Rice, Phys. Rev. A 48,54 (1993). 150. 0. Hahn, J. M. Gomez-Llorente, and H. S. Taylor, J. Chem. Phys. 94, 2608 (1991); R. T. Skodje and M. J. Davis, J. Chem. Phys. 88, 2429 (1988). 151. A. H. Zewail, Faraday Discuss. Chem. Soc. 91, 207 (1991); M. Dantus, R. M. Bowman, M. Gruebele, and A. H.Zewail, J. Chem. fhys. 91,7437 (1989); M. Gruebele, G. Roberts, and A. H. Zewail, Phil. Trans. Roy. SOC.Lond. A 332, 223 (1990). 152. K.C. Kulander and J. C. Light, J. Chem. Phys. 73,4337 (1980). 153. R. Schinke and V. Engel, J. Chem. Phys. 93,3252 (1990). 154. K. C. Kulander, C. Cerjan, and A. E. Orel, J. Chem. Phys. 94,2571 (1991). 155. 0. Zobay and G. Alber, J. fhys. B; Ar. Mol. Opt. fhys. 26, L539 (1993). 156. R. S . Nakata, K. Watanabe, and F. M. Matsunaga, Sci. Light 14, 54 (1965). 157. B. Hupper, Ph.D. Thesis, Carl von Ossietzky Universitat Oldenburg (1997). 158. J. A. Joens, J. Chem. Phys. 100,3407 (1994). 159. V. Engel, R. Schinke, and V. Staemmler, J. Chem. fhys. 88, 129 (1988); M. von Dirke and R. Schinke, Chem. Phys. Lett. 1%, 51 (1992). 160. B. Hupper, B. Eckhardt, and V. Engel, Semiclassical photodissociation cross section for H20, preprint (1996). 161. M. L. Doublet, G. J. Kroes, and E. J. Baerends, J. Chem. fhys. 103,2538 (1995).


GENERAL DISCUSSION ON REGULAR AND IRREGULAR FEATURES IN

UNIMOLECULAR SPECTRA AND DYNAMICS Chairman: M. Herman G. Casati: The change in some of our basic ideas about classical mechanics, enforced by dynamical chaos, is so deep that quantum mechanics may hardly ignore it. On the other hand, the investigation of the properties of quantum systems that become chaotic in the classical limit is bringing to light a qualitative picture of their quantum motion that could hardly be suspected at times when the qualitative understanding of quantum dynamics was modeled just after integrable cases and perturbation theory. For example, in the problem of excitation of Rydberg atoms under microwave fields a somewhat unexpected result is that the most efficient ionizing process is not a single-photon but a multiphoton one [l]. A remarkable fact is that, in spite of all fluctuations and fractal properties exhibited by quantum motion, strong empirical evidence has been obtained that the quantum evolution is very stable, in sharp contrast to the extreme sensitivity to initial conditions that is the very essence of classical chaos [2]. Some 30 years ago, a breakthrough in the understanding of the qualitative behavior of classical systems undergoing the so-called stochastic transition was made possible by the analysis of the “standard map” or “kicked-rotator” model. The main consequence of this transition is that a statistical picture of the Fokker-Planck type applies, leading to an energy absorption linear in time, in a diffusive-like fashion. One of the main problems is to what extent quantum mechanics mimics this behavior. Surprisingly enough, it turned out [3] that, typically, the quantum excitation follows the classical pattern only up to a finite time ts, after which the quantum rotator appears to enter a stationary regime and the energy absorption comes to an end. This quantum-mechanical suppression of the classical diffusive behavior is of a quite general nature and it has been confirmed, theoretically and experimentally, in the problem of excitation and ionization of hydrogen atoms under 583

584

GENERAL DISCUSSION

microwave fields [4,5]. The Hamiltonian of this model is given by (in atomic units)

where e and w are the intensity and frequency of the linearly polarized microwave field. In the classical case, when the field intensity exceeds a threshold value, the electron’s motion becomes chaotic and strong excitation and ionization takes place. In the quantum case instead, interference effects may lead to the suppression of the classical chaotic diffusion, the socalled quantum dynamical localization, and in order to ionize the atom, a larger field intensity is required (see Fig. 1). Another interesting example is the chaotic autoionization of molecular Rydberg states caused by the interaction of the electron with the degrees of freedom of the core. We consider the model in which the core consists of a positive Coulomb charge plus a rotating dipole that lies in the same plane of the electron orbit (m = 1). The Hamiltonian reads (atomic units)

L2 1 xcos++ycos+ H = !j(p: + py’)+ - - - + d 21 r r3 where d , L, I are the dipole moment, the orbital momentum, and the moment of inertia of the core, respectively. The analysis of the classical dynamics shows a transition to chaotic motion leading to diffusion and ionization [63. In the quantum case, interference effects lead to Iocaiization and the quantum distribution reaches a steady state that is exponentially localized (in the number of photons) around the initially excited state. As a consequence, ionization will take place only when the localization length is large enough to exceed the number of photons necessary to reach the continuum. In conclusion, quantum dynamical localization plays an important role in the excitation and ionization process of atoms and molecules. A question that remains open in connection with the previous talk is whether quantization via periodic orbits can account for this phenomenon.

1 . G. Casati, B. V. Chirikov, I. Guameri, and D. L. Shepelyansky, Phys. Rev. h f f . 57, 823 (1986); Phys. Rep. 154,77 (1987). 2. G. Casati, B. V.Chirikov, I. Guameri, and D. L. Shepelyansky, Phys. Rev. Leir. 56, 2437 (1986).

585

REGULAR AND IRREGULAR FEATURES

0.5

1.O

1.5

2.0

oo=on:

2.5

3.0

Figure 1. Comparison at identical parameter values of experimental and quantummechanical values for the microwave field strength for 10% ionization probability as a function of microwave frequency. The field and frequency are classically scaled, wo = niw and €0= rite, where no is the initially excited state. Ionization includes excitation to states with n above n,. The theoretical points are shown as solid triangles. The dashed curve is drawn through the entire experimental data set. Values of no, n, are 64, 114 (filled circles); 68, 114 (crosses); 76, 114 (filled squares); 80, 120 (open squares); 86, 130'(triangles); 94, 130 (pluses); and 98, 130 (diamonds). Multiple theoretical values at the same w o are for different compensating experimental choices of no and a.The dotted curve is the classical chaos border. The solid line is the quantum 10% threshold according to localization theory for the present experimental conditions. 3. G. Casati, B. V. Chirikov, J. Ford, and F. M.Izrailev, Lecrures Notes in Physics (Springer) 93,334 (1979);G. Casati and B. V. Chirikov, Quantum Chaos, Cambridge University Press, Cambridge, 1995. 4. G . Casati, I. Guameri, and D. L. Shepelyansky,IEEE J. Quantum Electron. 24,1420 (1988).

5. I. Bayfield. G. Casati, I. Guameri, and D. W.Sokol, Phys. Rev. Lett. 63,364 (1989).

6. F. Benvenuto, G. Casati, and D.

L. S. Cederbaum

L. Shepelyansky, Phys. Rev. Lett. 72, 1818 (1994).

1. Dr. Gaspard, you discussed periodic orbits of a number of triatomic molecules. 1 would like to know how many degrees of freedom have been included in the analysis? 2. To make the periodic orbit approach a competitive method of analysis, it would be relevant to extend it to more than two dimen-

586

GENERAL DISCUSSION

sions. As is well known, it is nontrivial to generalize the periodic-orbit approach to more dimensions. Could you discuss the advances made in this direction?

P. Gaspard

1. The theoretical analysis of HgI, by Burghardt and Gaspard has been carried out for the two-degree-of-freedom collinear dynamics. 2. The bending degrees of freedom can be taken into account as perturbations with respect to the collinear motion, thereby obtaining a complete description of the vibrational dynamics, as we discussed (Gaspard and Burghardt, “Emergence of Classical Periodic Orbits and Chaos in Intramolecular and Dissociation Dynamics,” in this volume). One of the possibilities provided by the ti-expansion theory is to include also nonlinear perturbations transverse to each periodic orbit. We think that periodic orbits for the full 3F dynamics can also be numerically obtained, for instance, in bent molecules. Approximate periodic orbits pertaining to a restricted 2F dynamics can be calculated in a first stage by freezing the bending degree of freedom. This constraint can thereafter be progressively removed while numerically tracking the periodic orbits in order to obtain the 3F periodic orbits.

A. H. Zewail: Dr. Gaspard has shown that for HgI, the period increases with energy. I assume that this reflects the “opening” of the potential in the coordinate perpendicular to the reaction coordinate, above the saddle point. P. Gaspard Yes. This is due to the damped Morse potential of your model of HgI,. There is thus a lengthening of the periods as the threebody dissociation is approached.

M. Quack I agree with most of what Prof. Field said. In fact, I have agreed with

this type of analysis for a long time. However, I have two remarks and a question. 1. The first remark concerns the abbreviation IVR. Prof. Field said that people even did not agree with what the three letters stand for, and in his lecture he actually never spelled out the words. He is, indeed, right that there are a variety of usages in reading words into IVR, the two most frequent ones being (a) “intramolecular vibrational relaxation” and (b) “intramolecular vibrational redistribution.” (There are also others, such as “intramolecular vibrationai randomization” and further, quite unrelated ones, such as “initial-value representation”). Personally, I strongly prefer the usage of “redistribution,” because this lan-


587

guage can be used both for the time-independent and the time-dependent pictures of the underlying physical phenomenon. It also allows for very general types of time dependence, whereas “relaxation” has an overly strong connotation of irreversibility, which may not always apply. There is also IVRR (“intramolecular vibrational-rotational redistribution”; see also Refs. 1 and 2). The basic physics of IVR(R) corresponds to a Hamiltonian being nonseparable (with a strictly separable vibrational molecular Hamiltonian there is no IVR), and the consequences of this can be seen both in stationary states and in time-dependent states. 2. The second remark concerns the question of the relation between the effective (algebraic) spectroscopic Hamiltonian and the “true” molecular Hamiltonian in coordinate and momentum space. The spectroscopic analysis Prof. Field has used gives an effective Hamiltonian, which is a matrix representation of the molecular Hamiltonian in some basis, which is not known and which in general cannot be written as the simple set of product functions in some set of coordinates as it was written down by Prof. Field. In the traditional spectroscopic treatment of such Hamiltonians going back to the work of Nielsen, Amat, Tarrago, Mills, and others decades ago, one had assumed that adequate basis functions and potential constants can be derived from such an analysis by perturbation theory. This was certainly the generally accepted wisdom until the 1980s (very widely assumed by many even today). If this were true to a good approximation, Prof. Field’s analysis would be fully justified. I do not want to exclude here the possibility that this works accidentally for the special case of the acetylene spectra discussed by him. However, when we introduced the polyad concept into the treatment of IR multiphoton excitation and IVR [l], we found that there were possible ambiguities and failures of the accepted treatment by perturbation theory. In the years following 1984 we have shown beyond any doubt that, indeed, the standard spectroscopic perturbation theory fails badly for a large class of Fermi resonance and Darling-Dennison resonance-type systems in the alkylic CH stretching-bending dynamics [3, 41. We have also less complete evidence on other systems. To illustrate the failure more quantitatively, I might quote the errors of anharmonic potential constants derived from the effective Hamiltonian analysis, which can be wrong by factors of 2-5 in individual cases, and the nature and structure of the basis states of H,ff, which can be quite different in reality from the simple analysis based on writing product functions CP I (Ql )@p2(Q2) * * .

588

GENERAL DISCUSSION

Thus, in order to derive quantitatively meaningful wavepacket dynamics of molecular motion (and also quantitative stationary-state wave functions) from the effective Hamiltonian, it is necessary to cany out afirther step in the analysis, which is highly nontrivial. We have canied out such analyses for a number of systems and I think an adequate understanding of the problem does exist nowadays and is accepted by the subcommunity most interested in this question. I like to illustrate this in the scheme from “molecular spectra” to “molecular motion” [5] (see Scheme 1).

High-resolutionmolecular spectroscopy Fourier transform spectroscopy Laser spectroscopy MOLECULAR SPECTRA

L t

1

Effective Hamiltonians Electronic Schriidin er equation L ? Rovibrational Schrodinger equation Molecular Hamiltonian +- Ab initio potential hypersurfaces

L

Time evolution operator (matrix) L MOLECULAR MOTION Molecular rate processes and statistical mechanics Scheme 1. Molecular spectra and molecular motion.

While the derivation of Heff is an important and useful step in this scheme (and I fully agree with Prof. Field that it is a most important exercise i,n spectroscopic “pattern recognition” [6]), the further step from H,ff to Hmol,tme must not be omitted in quantitative work, in general. The nature of the failure in the dynamics derived from the simplified Heft. analysis can be seen from the highly oversimplified sketch in two coordinates Ql, Q2 in Fig. 1. The error on the effective Hamiltonian basis (initial) state may be sufficiently large to introduce very large errors on the time propagated state. For quantitative results on some systems I can refer to Ref. 7, for example, where one can find also some interesting analytical considerations. Over the years we have assembled definitive experimental evidence for several molecular examples that the wavepacket evolution using incorrect (perturbation theory) initial states differs significantly from the evolution using the proper initial states. There remains no reasonable doubt about this, although this fact may not yet be widely recognized.


I

589

Q2

Figure 1. Sketch of probability densities as function of time in two coordinates: (1) correct initial state (from He# with transformation); (2) initial (basis) state of Heft (by perturbation theory); (3) true state at time t ; (4) state derived at time t (incorrect) if (2) is taken as initial state. 1. M.Quack, J. Chem.Phys. 69, 1282 (1978); Faraday Discuss Chem. SOC. 71, 309, 325,359 (1981); H.R. Dubal and M. Quack, J. Chem. Phys. 81,3379 (1984); Mol. Phys. 53,257 (1984). 2. M.Quack and W. Kutzelnigg, Be,: Bunsenges. Physik. Chem. 99, 231 (1995). 3. M.Lewerenz and M. Quack, J. Chem. Phys. 88,5408 (1988). 4. M. Quack, Ann. Rev. Phys. Chem. 41,839 (1990). 5 . M.Quack, Jerusalem Symp. 24,47 (1991); Mode Selective Chemistry, J. Jortner, R.

D. Levine. and B. Pullman, Eds., Reidel, Dordrecht (1991). 6. M. Quack, J. Chem SOC. Faraday Trans. 2,84, 1577 (1988). 7. R. Marquardt and M. Quack, J. Chem. Phys. 95,4854 (1991).

R. W. Field: I never claimed that molecular vibrations are even remotely close to being described by products of harmonic oscillators. I said that spectra are mysteriously more regular than they have any right to be. We can only adopt a simple representation for simple patterns. Harmonic oscillators are convenient because of their Au selection rules and matrix element u-scaling rules. Many of the inaccuracies of harmonic oscillators are irrelevant to dynamics, because the weak Au selection-rule-violating matrix elements between energetically well-separated basis states are much less important at early time than stronger Au-obeying interactionsbetween near-degenerate basis states. Of course it is possible to refine the model for the individual oscillators. This is especially important if the parameters determined from the analysis of a spectrum are going to be used to build an accurate and molecule-tomolecule transferablemodel for generic anharmonic couplings.I am sure you agree that one should start simple and build in complexity only as the specific class of problem at hand demands it. I continue to believe that one can construct reasonably accurate pictures of dynamics in configuration space (for the Franck-Condon

590

GENERAL DISCUSSION

active normal coordinates) and in state space (for the other modes strongly anharmonically coupled to the Franck-Condon active modes, restricted by approximate constants of motion) directly from intensity and frequency spacing information derived from feature-state progressions and from the width and complexity of each assigned feature state (polyad). The picture you drew on the board (Fig. 1 of the preceding comment by M. Quack), suggesting that a normal-mode picture would predict wavepacket motion in the opposite direction from reality, is ar best hyperbole. For large displacements, harmonic oscillators give an excellent description of the gradient of the potential in the Franck-Condon region.

M. E. Kellman: I believe the relations described by Prof. Quack among all the aspects of the effective spectroscopic Hamiltonian-its zero-order modes and the nature of these in coordinate space; the coupling terms in the spectroscopic Hamiltonian; and its connection to the potential-energy surface in coordinate space-are not completely understood, but there has been recent progress. Fried and Ezra [L. E. Fried and G. S . Ezra, J. Chem. Phys. 86,6270 (1987)] and McCoy and Sibert [A. B. McCoy and E. L. Sibert 111,J. Chem. Phys. 95,3476 (1991)] have shown how to obtain a spectroscopic Hamiltonian from a potential-energy surface in coordinate space. I have discussed [M. E. Kellman, Annu. Rev. Phys. Chem. 46,395 (1995)] the raising and lowering operators in the spectroscopic Hamiltonian in terms of actions defined among anharmonic modes, rather than as matrix elements of position and momentum operators for harmonic modes. So Prof. Field’s interpretation of the Hamiltonian makes a good deal of sense to me, though I agree with Prof. Quack that there are still difficulties in understanding just what all of us are doing when we use these Hamiltonians. In particular, the nature of the zero-order modes in this Hamiltonian, and what they look like in the ordinary molecular position and momenta coordinates, is something that, to my knowledge, is not yet understood.

M. Quack: In answer to the question by Prof. Kellman on exact analytical treatments of anharmonic resonance Hamiltonians, I might point out that to the best of my knowledge no fully satisfactory result beyond perturbation theory is known. Interesting efforts concern very recent perturbation theories by Sibert and co-workers and by Duncan and co-workers as well as by ourselves using as starting point internal coordinate Hamiltonians, normal coordinate Hamiltonians, and perhaps best, “Fermi modes” [I]. Of course, Michael Kellman himself has contributed substantial work on this question. Although all the available analytical results are still rather rough approximations, one can always


591

solve the problem “exactly” by numerical means, and we have done so in numerous cases. In that sense there is not a real problem left. We understand the theoretical essence of the problem and can carry out quantitatively valid calculations, even though there are no simple formulas around, which would be exactly valid. 1.

R. Marquardt and M.Quack, J. Chem. Phys. 95,4854 (1991).

M. E. Kellman: Prof. Field showed that there is a lot of order in the chaotic region of the C2H2 spectrum, by “unzipping” the spectrum into polyads defined by approximate quantum numbers. This gives a partial assignment of the spectrum, but not yet a complete assignment, because there are fewer polyad numbers than the N degrees of freedom of the system. This raises the difficult question, is there any way in these molecules to give a complete assignment with N quantum numbers? Another way of putting it is to ask if the spectrum can be further unzipped into sequences and progressions defined by a complete set of N approximate quantum numbers. We have been working at this problem and have recently made progress. We have used bifurcation theory to determine the anharmonic normal modes of the H20 spectroscopic Hamiltonian in the chaotic region, including the modes “born” from the original normal modes in bifurcations. This was used in a Husimi phase-space analysis [l] of the wave functions to assign the spectrum with quantum numbers appropriate for the bifurcated normal modes. These were used to “reconstruct” sequences and progressions. An example is shown in Fig. 1 for polyad P = 8 of the spectroscopic Hamiltonian. The sequences resemble those of the normal-mode region, but there are marked deviations as well. The figure shows as lines the levels calculated with both the 1 : 1 Darling-Dennison stretch-stretch resonance and the 2 : 1 stretch-bend Fenni resonance and compares these with levels shown as dots calculated by including just the 1 : 1 resonance. Particularly in the chaotic region, interaction of the F e d and Darling-Dennison resonances causes pronounced reorganization of the spectrum. In an attempt to avoid having to perform the difficult bifurcation analysis that leads to this assignment, John Rose and I devised a simpler diabatic correlation diagram approach that leads to the same results. We have applied this very recently to CzHz, where the spectral reorganization, due to the many couplings spoken of by Prof. Field, is much more pronounced than in H20. Nonetheless, the correlation diagram approach works very well. Figure 2 shows the correlation diagram for the polyad with 16 quanta of bend, “unzipped” into subpolyads defined by the effective Z-resonance quantum number obtained

592

GENERAL DISCUSSION

E

(cm

-'

27500

-[lAO -[26]0

-

[OW [16]2

-.

(52)'2mm -

27000

P =n,+n,+n,/2=8 -[06]4

-72414

26500

A

(71)+-0

-

26000

L

(60)'-4M'" -

25500

(7O)'2"Om -c

- (50)+'6 0

25000

(41)'s

A

-

11318 (31)+8

(80)+-0

I

I

I

1

I

C

2

4

6

8

Figure 1.

by our procedure. Figure 3 shows the energy levels unzipped into two types of sequences: 2-resonance subpolyads and subpolyads of the Darling-Dennison bend-bend coupling.

593


C2H2 P = 16 bend polyad I - resonance subpolyads 7200

cm-' 7100

7000

6900

6800

6700

6800

6500

6400

6300

I

I

I

full coupling Ho+ V Figure 2.

I

I

GENERAL DISCUSSION

594

C2H, P = 16 subpolyads Darling-Dennison"

l-resonance" 9800

Energy (cm-')

i

-

"

_

j

9600

9200

9000

-

In regard to control of energy flow, one can think of defining and trying to prepare time-dependent states closely related to these subpolyads (which contain time-independent levels of the spectroscopic Hamiltonian). If the proper time-dependent states could be prepared, energy could be made to flow with confinement to the chosen subpolyad. Now, 1 would like to pose a question to Prof. Field and Quack and to the entire conference. How do we use spectroscopic information to devise effective schemes to control chemical reactions? Should we try to force the molecule to follow our will? Or should we try to make use of what we learn from spectroscopy about what the molecule wants to do and use this knowledge to get the molecule to do what we actually want? 1. 2. Lu and M.E. Kellman, Chem. Phys. Leit. 247, 195 (1995).

R. W.Field: It is an extremely good idea to ask what the molecule "wants to do." Control schemes based on such knowledge are likely to be both more effective and easier to implement, because less frequent and forceful outside interventions will be required. A simplified picture of molecular dynamics might be very helpful to an experimentalist in designing a control strategy. It is very difficult to visualize the motion of an N-atom molecule in a full (3N - 6)-dimensional configuration space. A reduced dimension picture would serve as a stepping stone to insight.


595

The polyad model for acetylene is an example of a hybrid scheme, combining ball-and-spring motion in a two-dimensional configuration space [the two Franck-Condon active modes, the C-C stretch ( Q 2 ) and the trans-bend (Q4)] with abstract motion in a state space defined by the three approximate constants of motion (the polyad quantum numbers). This state space is four dimensional; the three polyad quantum numbers reduce the accessible dimensionality of state space from the seven internal vibrational degrees of freedom of a linear four-atom molecule to 7 - 3 = 4. M. Quack: Prof. Kellman has very nicely phrased the questions concerning reaction control: (1) Should we rather impose our will on molecules to have them do what we wish them to do? Or (2) should we not rather let the molecules do what they would like to do (but using our understanding in generating a good initial state)? He has addressed this question to everybody but did not get an answer. My answer would be, using an analogy between molecules and human beings, that it is neither nice nor possibly easy to use brute force on the molecules. However, often the molecules may be in a state where they do not really know what they want to do. Then we might use some very mild means to seduce them to do what we would wish them to do. As an early example for such mild seduction I might quote the theoretical scheme for potentially mode selective infrared laser chemistry of ozone [ l , 23, which predates some of the more widely publicized subsequent schemes using excited electronic states. 1. M.Quack and E. Sutcliffe, Chem. Phys. Lett. 99, 167 (1983). 2. M. Quack and E. Sutcliffe, Chem. Phys. Lett. 105, 147 (1984)

R A. Marcus: Concerning the report by Dr. Gaspard, I wonder if it might be useful to extract from the data on vibrational resonances the “natural motions” of the system, namely by looking at the typical periodic orbits (or the short-time wavepacket analogues), together with the relevant potential-energy contour that bounds the orbit. The resulting picture would be topological, that is, basis set free and coordinate system free. It could be a useful visual representation of the results and could be extended from pictures in two to then in three dimensions. To be sure, other than codifying the spectra, which the present algebraic scheme already does, it may not have other benefits unless some regularities appear for various classes of molecules. P. Gaspard: Concerning the issue raised by Prof. Marcus, I should remark that information on the shape of the periodic orbits in the original coordinates is lost at the level of a spectroscopic effective Hamil-

596

GENERAL DISCUSSION

tonian as given by a Dunham expansion. Such an information could be partially obtained if the electric dipole moment operator is also known. Further information can be extracted from observables depending directly on the position coordinates through the Stark and Zeeman effects. On the other hand, we should mention that, at the level of classical mechanics, periodic-orbit analysis provides a topological characterization of the system in terms of a symbolic dynamics, which appears as a common feature for a given class of systems. R. A. Marcus: I have a question for Dr. Gaspard concerning the following point. For some systems (e.g., H6non-Heiles), high-order perturbation theory provided a reasonable picture of the Poincar6 surface of section, if not performed to too high an order. It did not contain the irregularities of the actual motion, but nevertheless it provided a relatively good semiclassical quantization. Are there any analogous results in the systems you are studying, systems of more relevance to chemists, than Hinon-Heiles? P. Gaspard: To answer the question by Prof. Marcus, let me say that we have observed, in particular in HgI,, that higher order perturbation theory around the saddle equilibrium point of the transition state may indeed be used to predict with a good accuracy the resonances just above the saddle. However, deviations appear for higher resonances and periodic-orbit quantization then turns out to be in better agreement than equilibrium point quantization. Concerning the Poincar6 surface of section, it should be noticed that a sort of quantum surface of section can be constructed by intersection of the Wigner or Husimi transform of the eigenfunctions expressed in the quantum action-angle variables of the effective Hamiltonian, which can provide a comparison with the classical Poincark surfaces of section (e.g., in acetylene). M. S. Child: Prof. Marcus asks whether there is any way to identify “natural motions” of different species. We have a partial answer for the stretching motions of symmetrical AB2 species [l], based on the simple observation that the bonds of real molecules can break and that the dominant mechanism for interbond coupling in nonhydrides is via the kinetic energy. The old model of Wilson and Thiele [2] embodies just these characteristics by using Morse bond potentials and a kineticenergy coupling term dependent on atomic masses and the interbond angle. Furthermore the classical Hamiltonian may be scaled to a twoparameter form, in which energies are expressed as fractions of the dissociation energy and different molecules are identified by a coupling


597

parameter X = MB/(MA + MB)COS 8 , or equivalently by the ratio of small-amplitude antisymmetric and symmetric vibrational frequencies (w,/w,) = [(I - X)/(1 + X)]'I2.

0.5

0.4

$0.1

0

2

4

::0.2

0.1

frequency ratio

Figure 1. Symmetric-stretchclassical stability diagram. The energy is scaled by twice the bond dissociation constant.

The study that we have made was stimulated by the results of Pique et al. [3] on the dispersed fluorescence spectrum of CSz, which shows an assignable progression of Fermi resonance polyads between symmetric-stretch and bending motions, up to roughly half the dissociation energy, followed by a sharp transition to a softly chaotic regime. Moreover, the number of observed states rises as E 2 in the lower energy region and as E3 above the transition energy, indicating a transition from two-mode to three-mode coupling. Our suggestion is that this transition is associated with a change in character of the symmetricstretch periodic orbit from stable to unstable, which leads naturally to excitation of the antisymmetric stretch. Our model deals of course only with coupling between the two stretching motions. The method of analysis is an extension of Hill's method [4], which

GENERAL DISCUSSION

598

reduces to periodically forced motion of linearized displacements perpendicular to the periodic orbit. The extension is to allow for nonharmonic motion along the orbit, followed by a uniform asymptotic analysis rather than reduction to the Mathieu equation [2]. The resulting symmetric-stretch stability diagram is shown in Fig. 1, which shows a sequence of 1 : 1, 3 :2, 2 : 1, 5 :2, ... resonances on passing from the left to the right of the figure. The locations of different molecules in this phase plane are also indicated. Thus, for example, the facile transition from normal-mode to local-mode behavior in H 2 0 is seen to arise from the transition from 1 : 1 stability to instability. CS2, on the other hand, has a much larger q+.db ratio of approximately 5 :2 at low energies, rising to 2 : 1 when the symmetric stretch becomes unstable. It is also noticeable that this instability arises at roughly E/D = 0.5, which coincides well with the transition point in the experimental spectrum

PI.

As a crude two-mode simulation of the three-mode quantummechanical spectrum we have performed a spectral decomposition of two wavepackets initiated at the inner turning point of the symmetricstretch periodic orbit, in other words, with only zero-point excitation of the antisymmetric stretch. Figure 2a shows that a low-energy wavepacket contains only a simple progression in the symmetric stretch. However, increasing the mean energy of the wavepacket so that it spans the symmetric-stretch instability leads to the more complicated spectrum in Fig. 2b. Moreover, a similar break is found to occur at the same energy regardless of the magnitude of Planck’s constant, or equivalently of the actual masses of the atoms involved. The classical mechanics scales according to the relative atomic masses and fractional energy toward dissociation, and the qualitative appearance of the spectrum follows this scaling. Only the quantitative density of lines depends on the actual masses. To answer Prof. Marcus’s question, we may therefore conclude that the natural motions of the system are the short-time periodic orbits. Those that arise from the symmetric-stretch bifurcations depend on the frequency ratio: local modes in the 1 : 1 case, y-shaped orbits at the 3 :2 instability, horseshoes at the 2 : 1 resonances, and so on. 1. T. Weston and M. S. Child, in preparation. 2. E. Thiele and D. J. Wilson, J . Chem. Phys. 35, 1256 (1961). 3. G. Sitja and J.-P. Pique, Phys. Rev. Lett. 73, 232 (1994). 4. W. Magnus and S. Winkler, Hills Equation, Wiley-Interscience, New York, 1966.

M.S. Child: I have moreover a question for Prof. Field. Given that the Darling-Dennison resonance between the symmetric and antisym-

I-

599

REGULAR AND IRREGULAR FEATURES Packet energy = 0.143 I

0.04

0.03

0

0.05

0.1

0.15

0.2

0.25

0.3

Scaled energy

0.35

0.4

0.45

0.5

s (a)

Packet energy = 0.286

0'03 0.025

-

0.02

-

0.015

-

0.01

-

0.005

-

-

0

II

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Scaled energy (b) Figure 2. Spectral decomposition of two symmetric-stretch wavepackets at (ua/us)= 2.5, which is appropriate for CSz; ( a ) at reduced energy 0.143 and (b) at 0.286.

600

GENERAL DISCUSSION

metric stretch is a local stretch, is the Darling-Dennison resonance of the acetylene bends a local bend? If so, is it a step toward the vinylidene transition state?

R. W. Field: Each acetylene polyad contains zero-order states that are easily accessible via plausible direct or multiple-resonance 'A, -2 ' C i Franck-Condon pumping schemes. Each Evib 2 16,000 cm-' polyad also contains zero-order states that have good overlap with the half-linear acetylene t) vinylidene saddle-point structure. However, the eigenstutes that have detectable intensity in a Franck-Condon spectrum (-1 96 the intensity of the strongest transition within that polyad) typically have negligible (c fractional character of the best saddle-point overlapping state, and vice versa. It seems plausible that the bend Darling-Dennison resonance is capable of creating local-bender states by strongly mixing the cisand trans-bending normal-mode states. Such local-bender states would have excellent overlap with the acetylene t)vinylidene transition state. However, I do not know whether the increase in strength of the bend Darling-Dennison resonance as h n d E u4 + u5 increases will be overwhelmed by a Franck-Condon dilution effect owing to an even more rapid increase in the dimension of the polyad. However, I remain optimistic that something about the change in the w I :w I :w2 :- . - :w5 resonance structure that occurs at the energy of the acetylene c)vinylidene saddle point will be detectable in the dispersed fluorescence or stimulated emission pumping spectrum. E. Pollak: There is a unifying theme in the talks of Profs. Field and Gaspard. A huge body of work by people such as Child, Taylor, Farantos, Schinke, Tennyson, Schlier, and others has demonstrated the utility of the concept of periodic-orbit normal modes. At the bottom of the potential-energy surface, a periodic orbit corresponds to each normal mode. As one goes up in energy, bifurcations take place, and Child has beautifully analyzed the normal to local transition in terms of localmode periodic orbits. We have seen in H3+ how periodic orbits can be used to assign coarse-grained spectra in correspondence to short-time localizations. Taylor, Gomez Llorente, and co-workers have demonstrated the same for light-atom transfer systems, Na3, and more. Farantos has demonstrated how one can follow up in energy periodic-orbit bifurcation diagrams in three dimensions. The bottom line: Periodicorbit normal modes should become the language of high-energy molecular spectroscopy. Finally a question for Prof. Gaspard. You have shown the beautiful structure of periodic orbits in H3.Why, beyond a preliminary work of


601

Stine and Marcus, has no one given a good semiclassical analysis of the reaction probability in collinear H3? P. Gaspard: To answer the question by Prof. Pollak, we expect from our present knowledge that the periodic-orbit quantization of the H + H2 dissociative dynamics on the Karplus-Porter surface can be performed with the same theory as applied to HgI,. E. Pollak: The computation of Stine and Marcus for H3 was preliminary. I do not think that ghost orbits need be invoked but rather it is high time for a detailed and careful computation. B. A. Hess: Dr. Gaspard has introduced the vibrogram as a tool to extract periodic orbits from the spectrum by means of a windowed Fourier transform. This raises the question whether other recent techniques of signal analysis like multiresolution analysis or wavelet transforms of the spectrum could be used to separate the time scales and thus to disentangle the information pertinent to the quasiclassical, semiclassical, and long-time regimes. Has this ever been tried? P. Gaspard: As far as I know, the wavelet analysis of spectra has not yet been done and would be very interesting to develop. A remark is that the vibrogram also depends on the width E of the Gaussian window, which may be varied to construct another kind of plot. V. Engel: You showed the plot of the Nal molecule, which is a system with a curve crossing. The treatment of classical mechanics in such a system is not well defined. What can be done to treat the nonadiabatic effects if you calculate periodic orbits that are then used for interpretation? P. Gaspard To answer the question by Prof. Engel about curvecrossing problems, we should mention the work by Littlejohn and Weigert on matrix Hamiltonians [R. G. Littlejohn and S.Weigert, Phys. Rev. A 48,924 (1993)l. These authors have shown how to derive systematically the adiabatic Hamiltonians describing the classical motion on each potential surface. It turns out that surface hopping between surfaces requires one to use complex periodic orbits. J. Manz: The theoretical method of Prof. Field (See Field et al., “Intramolecular Dynamics in the Frequency Domain,” this volume.) evaluates the fluorescence dispersion spectra of HCCH in terms of the Fourier transform of the autocorrelation function,

where

602

GENERAL DISCUSSION

-

denotes the normalized (factor N ) initial (i) vibrational (v) eigenstate lClevi(q)prepared in the electronic excited state (e), multiplied by the g transition transition dipole functio! pge(q)for the electronic e and propagate9 [exp(- iHet/h)] on the electronic ground state ( 8 ) with Hamjltonian H e , as suggested by E. J. Heller [l]. I would like to ask Prof. Field about his model for pge(q).In principle, the transition dipole function pge(q) will affect all the subsequent quantitative results, in particular if pse(q)varies strongly along the vibrational coordinates q in the domain of +evi(q).In fact, one should expect substantial variations of pge(q)as the linear equilibrium structure of acetylene is transformed into vinylidene. 1. E. J. Heller, J. Chem. Phys. 68, 3891 (1978); Acc. Chem. Res. 14, 368 (1981).

R.W. Field: I must apologize for not being sufficiently clear about

the excitation scheme we use for our acetylene experiments. Although the initial and final states are both on the acetylene 2 'EB surface, the final state we prepare is the result of two electronic transitions +-i followed by --t i )rather than one vibrational-rotational infrared or Raman transition. There is a profound difference between the knowledge of the excitation function needed to describe electronic versus vibrational processes. The acetylene tj 2 electronic transition is a bent tj linear transition that would be electronically forbidden ('E;-'Ei) at the linear structure. The usual approximation is to ignore the possibility that the electronic part of the transition moment depends on nuclear configuration and to calculate the relative strengths of vibrational bands as the square of the vibrational overlap between the initial and final vibrational states (Franck-Condon factor). A slightly more accurate picture would be to express the electronic transition moment as a linear function of Qy (the trans-bending normal coordinate on the linear 2 'Xi state); in such a treatment, the transition moment would be zero at the linear structure and the vibrational overlap factors would be replaced by matrix elements of @(. Nevertheless, as long as one makes use of low vibrational levels of the state, neglect of the nuclear coordinate dependence of the electronic excitation function is unlikely to affect the predicted dynamics or to complicate any proposed control scheme. R. Jost: Prof. Field has shown three dispersed fluorescence spectra

A

(A


603

that contain information on ground-state vibrational energy levels of C2H2 up to 27,000cm-' ,but your analysis in terms of feature states and polyad quantum numbers breaks down at about 13,000 cm-'. In order to see if new feature states (i.e., new periodic orbits) appear at higher energy, I suggest to perform a windowed Fourier transform analysis (i.e., to get vibrograms as explained by P. Gaspard). It is tempting to think that the presence of the vinylidene minimum is responsible for the disappearance of some of the polyad quantum numbers, but new feature state@)[periodic orbit(s)] may appear above the vinylidene threshold. Prof. Field, did you try such a Fourier transform analysis? R. W.Field: No, I have not done it so far.

MOLECULAR RYDBERG STATES AND ZEKE SPECTROSCOPY

ZEKE SPECTROSCOPY E. W. SCHLAG Znstitut f i r Physikalische und Theoretische Chemie Technische Universitat Miinchen Garching, Germany CONTENTS I. Introduction 11. ZEKE Spectroscopy 111. Conclusion References

I. INTRODUCTION Zero electron kinetic energy (ZEKE) spectroscopy is a new form of spectroscopy with photoelectrons that, however, differs from photoelectron spectroscopy (PES) in some substantive ways. Basically it employs scanning photon sources much as in the photoionization efficiency (PIE) technique of Watanabe [ l ] (Fig. 1). The photoelectron spectroscopy of Siegbahn, Turner, and Terenin, on the other hand, employs a fixed wavelength in the vacuum ultraviolet (VUV)region of the spectrum and then analyzes the kinetic energy of the various photoelectrons emitted. This difference is more than one of procedure. Albeit it has proven to be nearly impossible to obtain electron energy peaks at a resolution of much better than some 100 cm-' in a standard experiment, ZEKE peaks are performed today at a resolution of some 0.1 cm-' and better. Such an increase in spectral resolution alone has made ZEKE spectroscopy a full-fledged spectroscopic technique. In addition, however, ZEKE spectroscopy is characterized by new selection rules that enable access to large number of levels, even normally forbidden levels. Hence, whereas in ZEKE spectroscopy nearly all levels are accessible

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, Xxth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

607

E. W. SCHLAG

608

(1954)

PIE

Photo Ionization Efficiency Watanabe

CambridgeMass.

PES

Photo Electron Spectroscopy Siegbahn Turner Terenin Kirnura

Uppsala Oxford St. Petersburg Okazaki

TPES

Threshold Photoelectron Spectroscopy Peatman

Zero Electron Kinetic Energy

-

(1984)

Munich

Anion ZEKE Neumark Drechsler

(1969)

Evanston

ZEKE

Muller-Dethlefs

(1961)

(1989) Berkeley Munich

Figure 1. Historical development leading to ZEKE spectroscopy.

if the energy is right, PES is a vertical spectroscopy that totally depends on the Franck-Condon principle in direct transitions. This is most clearly illustrated in Fig. 2a, where all vibronic states of nitric oxide are seen, including states in the Franck-Condon gap, whereas PES observes only the first five vibronic bands (Fig. 2). The scanning of a laser across the ionization threshold produces an almost continuous increase in the photocurrent with excess energy, as seen in Fig. 3a. Putting this same signal through a ZEKE “filter,” one produces the signal in Fig. 3b. In other words, the ZEKE signal that represents the ionic states of Nz+is seen here as a sharp signal that can be analyzed in terms of the species N2+ alone. This signal is buried but contained in Fig. 3a. It can be extracted by separating the direct ions produced above in the ionization energy, waiting some time until only high-lying Rydberg states survive, and now ioniz-

609

ZEKE SPECTROSCOPY

90,000 95.000

85,000

80.000

75,000

CM-I

105;OOO . 115;OOO CM-I

'

125,000

-P 10,000 2 -

m OD c

._

2

50,000

I

I

I

'

1

'

I

' '

I

7

I

' '

I

'

I

l~I'"'I""I""I""1"~'1"''~"

75,000 80,000 85,000 90,000 95,000 105,000 115,000 125,000 CH-I

I

10

CM-1

11 12 Binding energy (eV)

I

13

Figure 2. Comparison of the NO+ spectra with equal energy scales of (a) ZEKE spectrum,showing ion states up to d = 26; (b) total ion spectrum;and (c) photoelectron spectrum from Turner's book [2].

ing these with an extraction pulse. Hence, simply waiting produces the sharp ZEKE spectrum. This is due to the fact that it has been discovered that just below each possible state of an ion there reside high-n Rydberg states with an enormous lifetime of some 20-100 ps. This band of high Rydbergs is very narrow and only just below each ion state. Hence, separating these ZEKE

610

E. W. SCHLAG

a) photdonlzatlon

ZEKE filter b) ZEKE

W.vcnumk1 (an.')

Figure 3. Comparison of PIE spectrum of N2+ with the corresponding ZEKE spectrum: (a) photoionization spectrum in the region of the first IP [3]; (b) ZEKE photoelectron spectrum [41.

states out of the sea of ions gives a signature of all possible ionic states. This is totally different, in principle, from PES. Another example is the silver dimer Ag,, as shown in Fig. 4 [ 5 ] . Here the signature of the total spectrum of direct ions is seen in the top of Figure 4. In the same experiment destroying these with a spoiling field and analyzing the remaining ZEKE states with a delayed pulsed field give the correct vibronic spectrum of Ag2+, as shown in the bottom of Fig. 4. In the top spectrum there are clearly many excited ionic states produced directly, but these are autoionization resonances at energies other than the vibronic threshold of the E K E delayed pulsed-field ionization (PFI)spectrum. This technique has now been applied to many molecular systems, as shown in Fig. 5 . It can also been applied to van der Waals (vdW) complexes, where often all six intermolecular vibrational modes are observed. Another important application is to anions, where here the electron detachment produces ground-state neutral systems (Fig. 6). The anion state can be

61 1

ZEKE SPECTROSCOPY

Ag2 : Delayed PFI versus Direct Ions

-

X2Zi

B'II,(V-4)

+

X'X; ( v " - 0 )

Delayed PFI

v'-0

I

I

61800

'

i

'

l

"

62000

'

l

"

-'

'

l

62200

'

62400

M

Figure 4. Comparison of PIE spectrum of Ag2+ with the corresponding ZEKE spectrum

PI.

mass selected, and thus the neutral produced from this anion is also mass selected. Hence this provides a means for mass-selected neutral ZEKE spectroscopy. Since anions can be made of various metastable species, so can the neutral spectrum be measured for this metastable species. One example is the high-resolution neutral ground-state spectrum of a radical, here OH (Fig. 7). The various types of archetypal classes of systems that have been studied is shown in Fig. 8.

II. ZEKE SPECTROSCOPY Threshold electron spectroscopy, or its newest variant ZEKE spectroscopy, represents a new approach to these problems that has already afforded a broad set of new applications, particularly for soft bonds and metastable species, such as is characteristic for metastable reactive intermediates.

612

E. W. SCHLAG

Small Molecules

nitric oxide ammonia hydrogen sulfide carbon disulfide methyl iodide, ethyl iodide

Aromatic Molecules

benzene para-difluorobenzene phenol toluene

Van der Waals Complexes

benzene-argon benzene-krypton para-difluorobenzene-argon

H-Bonded Complexes

phenol-water phenol-methanol phenol-ethanol phenol-dimethylether phenol dimer

Mixed Complexes

phenol-water-argon

Metal Complexes

silver dimer

Molecular Complexes

nitric oxide dimer

Anions

hydroxyl radical iron oxide iron dicarbide

Figure 5. Molecular systems studied with ZEKE spectroscopy.

In the late 1960s steradiancy techniques were initiated that measured highly accurate drift-free thresholds (TPS), albeit with a modest increase in resolution. This work was started by Peatman et al. [6-81. This abandons all measurements of electron energies but instead measures the resonance wavelength via the steradiancy of the emitted electrons, a unique property at threshold. Only at threshold is it possible to produce electrons with no kinetic energy, and hence constitutes a source that lacks any angular divergence. A primitive optics, in fact a straight pipe, will, in the limit of extreme length, only transmit these threshold electrons. Coincident timing of these electrons also suppresses the remaining straight-through component 191. The threshold is ips0 fact0 exact (as exact as the light source) and cannot be affected by surface effects, for example (except in intensity, not energy). This became a new state selector of molecular ions since coincident with these threshold electrons are ions in a defined molecular state. The kinetics

613

ZEKE SPECTROSCOPY

;

f

=

Figure 6. Spectroscopy of neutral ground state via anion ZEKE spectroscopy. Rotationally resolved anion-ZEKE of OH

-m .u) op. W

Y N W

-

U(0) EA

EE Ii 1.5 em-1

14695 14700 14705

Figure 7. High-resolution spectroscopy of the ground state of the neutral OH radical.

E. W. SCHLAG

614

Molecular Catlons Fragment Cations

CH;

,C=C-C+

Catlon Complexes

van der Wsds Energies F-

Molecular Anions Fragment Anions

SH

I-H-l -

Transient Anions sholt lived

Au 2-

-

Fe C-C

Neutrals in Ground State IR. Ramen. MC.

6

Fragments in Ground State

SH

Transients in Ground State

I-H-l Fe-C-C

Reactive Intermediates

Au

Slate md Mess selected Spectra h Ground State

Surfaces - Clusters Figure 8. Types of systems studied with ZEKE spectroscopy.

of such state-selected ions has made important contributions to the understanding of the theory of unimolecular reactions [101 employing coincidence techniques (PIPECO). This TPS technique gives the exact location of states of molecular ions at a reasonable resolution of some 10-20 cm-' . The differences are demonstrated in Fig. 9, where the left-hand side shows threshold measurements. At the threshold, and only here, is the energy of the emitted electron near zero. Employing a detector for such zero kinetic energy electrons will show a peak only at the exact position of the molecular energy level (Fig. 10). The undifferentiated total electron current would, however, in contrast have kept increasing, since the excitation energy merely leads to an increasing velocity of the two charged particles produced. A very simple example (atomic argon) demonstrates this principle (Fig. 11). The total current simply reflects the onset of the ionization potential (IP) at the 2P3p state and the Rydberg series leading to the next level of the ionized atom, the 2 P , p state, this, however, not being directly visible. In contrast, the threshold spectrum (Fig. 11) clearly gives a peak for both states as well as suppresses the Rydberg spectrum, which generally leads to hot electrons.

615

ZEKE SPECTROSCOPY

t h r es ho Id spec t ros c opy

photo e l ec t ron spec f r o scop y

Figure 9. Comparison of threshold spectroscopy and photoelectron spectroscopy.

High Rydberg states were found, however, to have a very important property [Ill. Rydberg states with quantum numbers n = 100, ..., 200, just 1-2

'.' ;Y;. [k-

cm-' below the respective level of the free ions, have recently been discovered to be surprisingly long-lived, typically some 50 ps [11, 121, whereas some 10 ns would have been anticipated [13]. If now the laser excitation

... ..._.... .......[... I

..,. . ... . , , , .

_j.

.

.

M ("+,Nil

. . . .. . ... ... ., , , , . M Iv+, N I;

_......._............................ M (v: NTl

total Ion Signal

Threshold

w tuned

Signal

M(v.NI

Figure 10. Comparison of total ion spectrum with threshold spectrum.

E. W. SCHLAG

616 12 s

13 s

us

-

Ar : ION YIELD

+ E

U

ZEKE

127 000

128 000

L Icni’l

Figure 11. Argon spectrum.

occurs in a field-free environment, such Rydberg states will persist for some 50 ps. If an electric pulse is now applied after some 1-5 ps, only these highlying states will be observed and all other states will have disappeared. Since these high-lying states directly reflect the onset of each ionic level to which they converge within some 1-2 cm-’ ,this is a direct measure of the molecular ion spectrum displaced by some 1-2 cm-I , a number that can be calibrated or extrapolated away. This is the basis of modem ZEKEi spectroscopy of molecular cations. The resolution of ion spectroscopy now becomes astonishing in that it increases to near laser resolution and thus ion spectroscopy emerges as being on an equal footing with traditional high-resolution conventional spectroscopies. Figure 12 demonstrates the marked increase in resolution; we compare the historical first ZEKE spectrum (below) to the corresponding photoelectron spectrum [2, 14-16]. In fact, such a ZEKE spectrum will continue to display a rich set of energy levels in the ionic ladder up to u = 26 (top of Fig. 2), whereas the trace of the total electron current (middle photoionization efficiency trace) as a function of energy displays intially the traditional staircase function, which then becomes obscured by

617

ZEKE SPECTROSCOPY

Ebind lev1 9

11

10

12

-0 C

.-(31

m

W Y

W N I

I

0

I

10.0 E int Icm-’l

I

20.0

Figure 12. Comparison of the photoelectron spectrum of NO with the original ZEKE spectrum of the first band, with demonswation of rotational resolution of the ion state.

many autoionization and other resonances. Interestingly the photoelectron spectrum breaks off at u = 4 or u = 5 (bottom of Fig. 2). More recent experiments on nitric oxide are displayed in Fig. 13 (left side) [17], showing typical alternations due to parity but also showing large changes in the orbital quantum number of the outgoing electron. This was quite puzzling, until detailed theoretical calculations by McKoy et al. (right side Fig. 13) [ 181 confirmed this to be a new property of this spectroscopy in conformity to large-scale theoretical calculations. Figure 14 demonstrates a photoelectron spectrum (top) for benzene [19] compared with a current ZEKE spectrum (bottom) of some 0.2 cm-’ resolution [20], even demonstrating the expected splitting of the vibration due

618

E. W. SCHLAG Cdculstd

N, = O ).(

3

1

3

Ekprimantal

NA='

(b)

Exprimeold

5 Figure 13. Comparison of experimental ZEKE spectra of NO' [17] with ab initio calculated spectra [I81 N A reflects the initial rotational state of the transitions in the SIintermediate state. 0

1

200

100 I

I

E i ntl m e V 1

0

P

y1

W

a

0

P

y1

W

Y

W N

36288

-

36308

E [ cm"]

36328

Figure 14. Benzene spectra: ( a ) photoelectron spectrum; (b) ZEKE spectrum of a single band in (a).

619

%EKESPECTROSCOPY

U

6l

62

u -

O0

7L000

3

"\,

4:

75000

w,

0

w2

2l

lcrn-ll

76000

Figure 15. Spectrum of para-ditluorobenzene: ( a ) high-resolution photoelectron spectrum [19]; (b) ZEKE spectrum in the same energy range [21].

to the Jahn-Teller effect showing rotational resolution. Figure 15 shows a high-resolution photoelectron spectrum for para-difluorobenzene compared to the ZEKE spectrum below [21]. Figure 16 demonstrates the results for a cluster phenol-water [22]. The top of Fig. 16 is the total current [23], repeated in our laboratory (middle), and the ZEKE spectrum below demonstrates the direct observation of the new soft modes derived from intennolecular vibrations and their many overtones and combinations for the soft-modes phenol-methanol clusters [24]. Such normal vibrations are pictorially displayed in Fig. 18. An interesting demonstration for a chemical application to neutral ground states is in a mixture that represents a study of Fe complexed to a hydrocarbon, in this case the complex FeC4H2. This complex can be uniquely identified in the anion mass spectrum of an iron target in a hydrocarbon stream (top of Fig. 19). Threshold photodetaching this mass-selected species in a very high resolution ZEKE experiment then directly produces the unequivocal spectrum of the ground state of this interesting reactive intermediate, demonstrating that the structure must have linear C-C-Fe bonding. This can have many interesting application to the chemistry of such intermediates. Figure 8 gives a short summary of the types of systems that are amenable to this new form of high-resolution photoelectron spectroscopy for extreme

620

E. W. SCHLAG

phenol water

65 000

6L 500

Icm-’l

6 L 000

total energy / cm-1

Figure 16. Spectrum of the phenol-water complex: (a) total ion signal [23]; (b) total ion signal [22]; ( c ) ZEKE spectrum [22].

metastables (even transition states [25]) and for reactive intermediates, thus opening up a new spectroscopic look at metastable states and soft modes. III. CONCLUSION

Zero electron kinetic energy spectroscopy provides a new tool in the study of chemical systems. In particular, it is applicable to molecules that are not “in a bottle” (metastable species) which nevertheless are of fundamental

62 1

ZEKE SPECTROSCOPY

27600

28ooo

Ionising Laser Energy / cm-'

28500

Figure 17. Phenol-methanol ZEKE spectrum via the vibrationless state of S1. n

(J

(24Ocrn.')

Z (2%:257 cm-')

V

1 y ' (328 cm")

p ' (84 cm-')

y

(261 cm")

p"(67cm-')

Figure 18. Phenol-water complex: intermolecular normal modes.

622

E. W. SCHLAG

I

1

I

‘98

i

1

I

100

I

1

l

I

1

t

102

l

I

I

1

1

l

I

104

I

.

106

I

I

I

I

I

I

amu

I

108

I

I

I

I

I

110

I

I

I

I

I

I

112

.

I

I

*

I

I

I

I

I

114

Figure 19. Neutral spectrum as mass-selected anion spectrum.

importance to chemistry as intermediates in chemical reactions. The struc-

ture of these systems is important for an understanding of the mechanism of chemical reactions. This applies not only to cations and anions as objects of study but also to free radicals, complexes, hydrogen bonded or just vdW complexes, and other unstable species. In spite of their substantial instability, these can be mass selected as anions and then studied as a spectrum of the mass-selected neutral ground state of these species. Considering the great interest in such species, this should become one of the major areas of application of this new spectroscopy.

ZEKE SPECTROSCOPY

623

References I . K. Watanabe, (a) J. Chern. Phys. 22, 1564 (1954); (b) ibid., 26, 542 (1957). 2. D. W. Turner, C. Baker, A. D. Baker, and C. R. Brundle, Molecular Photoelectron Spectroscopy, Wiley, London, 1970. 3. 3. Berkovitz and B. Ruscic, J. Chem. Phys. 93, 1741 (1990). 4. F. Merkt and T. P. Softley, Phys. Rev. A 46, 302 (1992). 5. Ch. Yeretzian, R. H. Hermann, H. Ungar, H. L. Selzle, E. W. Schlag, and S. H. Lin, Chem. Phys. Lett. 239, 61 (1995). 6. W. B. Peatman, Ph.D. Thesis, Northwestern University, Evanston, 1969. 7. W. B. Peatman, T. B. Borne, and E. W. Schlag, Chem. Phys. Lett. 3,492 (1969). 8. T. Baer, W. B. Peatman, and E. W. Schlag, Chem. Phys. Lett. 4, 243 (1969). 9. P. M. Guyon, T. Baer, L. F. A. Ferreira, 1. Nenner, A. Tabche-Fouhaile, R. Botter, and T. R. Govers, J. Phys. B 11, L141 (1978). 10. T. Baer, in Gas Phase Ion Chemistry, M. T. Bowers, Ed., Academic New York. 1979, p. 153. 11. G. Reiser, W. Habenicht, K. Muller-Dethlefs, and E. W. Schlag, Chem. Phys. Left. 152, 119 (1988). 12. W. G. Scherzer, H. L. Selzle, and E. W. Schlag, Z. Naturforsch 48% 1256 (1993). 13. E. W. Schlag and R. D. Levine, J. Phys. Chem. 96, 10608 (1992). 14. K. Muller-Dethlefs, M. Sander, and E. W. Schlag, Chem. Phys. Lett. 112, 291 (1984). 15. K. Muller-Dethlefs and E. W. Schlag, Ann. Rev. Phys. Chem. 42, 109 (1991). 16. K. Muller-Dethlefs. E. R. Grant, K. Wang, V. B. McKoy, and E. W. Schlag, Adv. Chem. Phys., in press. 17. M. Sander, L. A. Chewter, K. Miiller-Dethlefs, and E. W. Schlag, Phys. Rev. A 36,4543 (1987). 18. H. Rudolph, V. McKoy, and S. N.Dixit, J. Chem. Phys. 90,2570 (1989). 19. E. Sekreta, K. S. Viswanathan, and J. P. Reilly, J. Chem. Phys. 90,5349 (1989). 20. R. Lindner, H. Sekyia, and K. Muller-Dethlefs, Angew. Chem. 105, 1384 (1993). 21. D. Rieger, G. Reiser, K. Muller-Dethlefs, and E. W. Schlag, J. Phys. Chem. 96, 12 (1992). 22. G. Reiser, 0. Dopfer, R. Lindner, G . Henri, K. Muller-Dethlefs, E. W. Schlag, and S. D. Colson, Chem. Phys. Lett. 181, 1 (1991). 23. R. J. Lipert and S. D. Colson, J. Chem. Phys. 89,4579 (1988). 24. T. G. Wright, E. Cordes, 0. Dopfer, and K. Muller-Dethlefs, J. Chem. Soc. Faraday Trans. 89, 1609 (1993). 25. G . Drechsler, C. Bksrnann, U. Boesl, and E. W. Schlag, J. Mof. Srruct. 348,337 (1995). Note: cf ZEKE homepage: http://www.chemie.tu-muenchen.de/zeke

DISCUSSION ON THE REPORT BY E. W.SCHLAG Chairman: G. Casati

G. Gerber: Prof. Schlag, could you shortly explain what is an inverse Born-Oppenheimer (BO) situation?

624

E. W. SCHLAG

E. W. Schlag: It simply means that the electron is the slowest particle compared to the more rapid nuclear motions. For this reason, each rotation has its own Rydberg series. This is referred to typically as an inverse BO representation, similar to a Hund’s case d. S. A. Rice: Have you studied any excited ion states with degenerate vibrations to ascertain the coupling of the vibrational angular momentum to the electron angular momentum? E. W. Schlag: Yes, we do see splitting of the ion states. In particular, we see the Jahn-Teller splitting for the first time for benzene ions and in great detail. A. H. Zewail: Are we supposed to think, according to the discussion by Prof. Schlag, that for high-n Rydberg orbitals we have an electronic rnanifoEd within each vibrational state, the inverse of conventional molecular description? E. W. Schlag: Each rotational or vibrational level has its own private Rydberg series that are strongly coupled to other series and are driven by entropy to the highest n state of all accessible manifolds. M. Shapiro: The cuckoo effect: How do we know where the ZEKE electron comes from? If A is present among B, the ZEKE electron can come not from B, influenced by A, but from the minority species A itself. E. W. Schlag: The pickup is from a sea of ions of one species that picks up the E K E electron of the other species. Thus it shows up at the wrong energy. When the sea of ions is shut off, the signal disappears since it is at the wrong mass. This demonstrates that ZEKE states have an existence of their own and need not be formed necessarily from optically excited low4 Rydberg states.

SEPARATION OF TIME SCALES IN THE DYNAMICS OF HIGH MOLECULAR RYDBERG STATES R. D.LEVINE The Fritz Haber Research Center for Molecular Dynamics The Hebrew Universily Jerusalem, Israel and Department of Chemistry and Biochemistry University of California Los Angeles Los Angeles, California

CONTENTS I. Background

11. Preliminaries

111. Dynamics A. Effective Hamiltonian B. Trapping Versus Dilution Iv. Concluding Remarks References

1. BACKGROUND There are many motivations for the study of the unusual dynamics of high Rydberg states of molecules. The two that most capture my imagination are the exceptionally wide range of time scales involved (Fig. 1) and the unusual limiting situation of a very slow electron being perturbed by the faster motion of the nuclei in the core about which it revolves. What this means is that, as one varies the hydrogenic principal quantum number n, it is possible Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scafe. XXfh Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0471-18048-3 0 1997 John Wiley & Sons, Inc.

625

R. D.LEVINE

626

/-.time resolved

'

-I

Stark modulatio

1

10 100 principal quantum number/n

Figure 1. Orbital period of an electron moving in a Coulomb field, the time scales of some internal and external perturbations [3a-3d], and the observed (shorter, see below) lifetime for the polyatomic molecule known as BBC [4]. Note that at the highermost values of n the decay lifetime begins to shorten; cf. Fig. 4.

to cover the entire range from the Born-Oppenheimer limit to its inverse (Fig. 2). Zero electron kinetic energy (ZEKE) spectroscopy [6a, 71 has opened up the study of high Rydberg states and, in particular, has drawn attention to the many time scales that are involved. Extremely long living (tens of microsecond) states (which can be detected by their ionization by a pulsed weak electrical field) have been observed [ l , 6b, 7-17]. However, shorter living (submicrosecond scale) states can also be seen [4, 18, 191, and the preliminary experimental evidence is that the shorter living states are more intense whereas the extremely stable states constitute about 5-20% of the ZEKE signal [19]. There is also some experimental indication and much theoretical evidence for a prompt (i.e., comparable to an orbital period) decay. Our interest in the study of the lifetime of high Rydberg states was motivated by the phenomena of delayed ionization [20]. Multiphoton (and also single-photon excitation of large molecules [21,221 and of clusters [24-321) has revealed that such molecules do not necessarily promptly ionize, even though they contain enough energy to do so without the assistance of an electrical field or any other external perturbation. There is also no experimental doubt that they decay also by electron emission (since one can detect the electrons and even measure their kinetic energy distribution [33]). There can, however, be other decay mechanisms, in particular dissociation of the

627

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

-1"

-20

1 " hverse '

"

"

0

'

"

'

Born-Oppenheimer (~=200) "

0.2

"

'

"

"1

"

0.4

0.6

0.8

U nz

I

Figure 2. Effect of the frequency w of the perturbation by the core on an electron moving in a Bohrdommerfeld orbit of high eccentricity (low angular momentum). Plotted vs. the angle u, which varies by 2r over one orbit. Note that the perturbation is localized near the core. In the inverse Born-Oppenheimer limit o( >> 1) the perturbation oscillates many times during one orbit of the electron. (For further details and the formalism that describes the motion at high x as diffusive-like (see Refs. 3c and 5.) For higher angular momentum 1 the effective adiabaticity parameter is x( I - e) = x12/2, where e is the eccentricity of the Bohr-Sommerfeld orbit. States of high 1 are thus effectively decoupled from the core.

energy-rich molecule (or cluster) into neutral fragments. We have indeed proposed [21] that the energy-rich molecule stores the energy in the very many degrees of freedom of the core and that it is the slow, diffusivelike transfer of the energy back to the outgoing electron that is rate determining. This point of view does account for the size dependence [21] and also provides definite predictions for the dependence on the total energy (Fig. 3).

,

0

0.5

I

1.5

2

Energy I Ionization potential

Figure 3. Transition from the small- to the large-molecule limit. (The large molecule is twice as large as the middle one, and both are far larger than the small one, which undergoes prompt ionization.) A schematic plot of the yield of ionization vs. the available energy scaled by the ionization potential.

R. D. LEVINE

628

The study [ 181 of time-resolved ZEKE spectroscopy was undertaken to obtain a better and detailed understanding of the phenomena of delayed ionization. The earlier work by Even and co-workers at TeI Aviv University had several limitations. First, there was a delay time between excitation and the time at which detection could begin of about 100 ns. In the range of states of interest (say, n > 100) the experiment was therefore blind to what happens during the first 100 orbits or more (cf. Fig. 1). In the decade of 100-1OOO ns the signal decayed roughly exponentially but with a long-time component of roughly 5-20% of the total intensity (depending on the molecule, the wavelength, etc.). These results were obtained for a number of (jet-cold) aromatic molecules, all of which exhibited a “turnover” in that the (shorter) lifetime increased with increasing principal quantum number, as does the orbital period (Fig. 1) but then started to decrease (Fig. 4). Another limitation of the early Tel Aviv experiments was that the noise level was such that one could not examine the long-time (microsecond-scale) component at a good signal-to-noise level. Furthermore, experiments on Hg atoms suggested that for times longer than 1 ps the atoms were subjected to external perturbations. A third limitation was that at the time one did not have a precise enough value for the ionization potential, and therefore one did not know whether the optically prepared state was above or below the threshold for ionization. This has since been rectified [4] by observation of many members of Rydberg series that allowed an independent determination of the ionization potential. Thereby an assignment of n can be made and hence provided the lifetime as a function of n, as shown in Fig. 1. In the experimental arrangement used by Schlag and co-workers at the

DCA

.

* -800 0

.i700 s“ 6 0 0 c

500”

-8

I

I

I

I

-6 -4 -2 0 Energy below threshold /cm*’

Figure 4. The measured (shorter) decay lifetime (in nanoseconds) of dichloroanthracene (DCA) vs. the energy below the threshold to ionization. The curve is a fit for a diffusional motion of the electron about an anisotropic core in the presence of a weak (stray) DC field. For the derivation of the kinetic description from a Hamiltonian see Ref. 3c.


629

Technical University Munich, the delay time before detection can begin is about 10 ps (or down to 1 ps in the recent experiments by Muller-Dethlefs and co-workers [6a]). Most of the time-resolved experiments were, so far, carried out for benzene. The question of whether the longer decay observed (but not well characterized) in the Tel Aviv experiment is the same as the long-time (microsecond scale) decay observed for benzene in a variety of Munich experiments [6b, 7, 9, 10, 13, 35, 361 is still open. It is suggestive that it is, but the definite experiments are not yet done. At the same time, interest in ZEKE spectroscopy focused attention on a rather different aspect, namely the extreme sensitivity of the high Rydberg states to external perturbations [2, 3a, 6, 8, 11, 12, 15, 17, 37401. What is clear is that the presence of an electrical and/or magnetic field or external charges can elongate the lifetime of states into the microsecond range. From a practical point of view this is very beneficial since it gives rise to a larger ZEKE signal that is useful in both the spectroscopic and the analytical applications. On the other hand, it clouds the issue of what effects are inherent to the isolated molecule and what is due to external perturbations. Of course, there is the other side of the coin. The external perturbations provide probes with the right time scales for the study of high Rydberg states; cf. Fig. 1. Detailed considerations (Fig. 5 ) show that the effect of external perturbations is not invariably in the direction of elongating the lifetime. Indeed, the dynamical computations also shown in Fig. 5 support the interpretation that the decrease in the lifetime with increasing electrical field is due to the lowering of the threshold for ionization [41]. This, we think is the same effect

v)

0.7 0.6 -

3. 0.5

\

-

0.4 -

0.3 0.2 -

-Computed

.\.

0.1 01 ” ” 0 0.2 0.4 0.6 0.8

1

o

.j ’

’

1.2 1.4 1.6

DC field I (V/cm)

Figure 5. Experimental (shorter) lifetimes of DABCO in the presence of a DC field (in addition to a stray field of 0.1 V/cm) compared to a computation using classical mechanics [ I ] for an electron revolving about a quadrupolar anisotropic core. The smooth line is a fit to (see Ref. 1 for more details). an exponential dependence on

630

R. D. LEVINE

that is due to the shortening of the lifetime as one approaches threshold, as seen in Fig. 4. In terns of the picture of delayed ionization, this effect corresponds to reducing the value of the ionization potential and so, at a given energy, is equivalent to moving to the left on the abscissa of Fig. 3. 11. PRELIMINARIES

We consider that the essential physical nature of the decay of high Rydberg states (and of delayed ionization in general) is that there is a very congested set of bound states coupled to a sparse continuum. In the language of transition-state theory, the number of states of the transition state is far smaller than the number of states of the energy-rich molecule to which they are coupled. This is also the case for an ordinary unimolecular dissociation, and there are indeed many useful analogies between the two types of time evolution. This makes the study of Rydberg states of relevance to reaction dynamics in general. Indeed, the two very actively debated questions of mode-selective chemistry and of control of chemical reactions can be usefully studied through their close analogies in the Rydberg case. There are two sources of congestion of states. One is orbital. There is the n2 degeneracy of a hydrogenic state of principal quantum number n. Moreover, since the spacing of adjacent hydrogenic states is (in atomic units) K 3 , it follows that there are n5 distinct quantum states per atomic energy unit or n5/2 Ry states per wavenumber, where Ry = 105cm-' is the Rydberg constant. For the high n's of interest, this orbital degeneracy is by itself quite high. Of course, these states will remain bunched unless the spherical degeneracy of the Coulombic field is lifted, for example, by an anisotropic molecular core about which the Rydberg electron revolves and/or by an external DC electrical field. To completely resolve the degeneracy, one needs to also break the cylindrical symmetry, and this can be done by a variety of ways including external perturbations (a magnetic field, other molecules, etc.). Moreover, as has been well documented for atoms [42], when the central field is not strictly Coulombic, additional perturbations such as a DC field can mix states of different n. The other source of congestion is due to the molecular core. It is most readily discussed using the inverse Born-Oppenheimer point of view to define the zero-order quantum numbers. Here each state of the ionic core has its very own series of high Rydberg states. The physical reality of this approximation is the observation [36,43] of the long-time stable ZEKE states not just below the lowest ionization threshold but also just below the threshold of ionization processes that leave an excited ionic core. Indeed, it is for this very reason that ZEKE spectroscopy is useful for the spectroscopy of ions (or for such neutrals that are produced by ionization of negative ions


63 1

[44]). The high value of the total energy (comparable or larger than the ionization potential) means that many states of the ionic core are available, up to and including neutral fragmentation channels. (Of course, the higher the energy of the core, the less energy is in the electronic excitation.) As in the case of the orbital degeneracy, the availability of isoenergetic states does not necessarily mean that they are coupled to one another. Indeed, the very use of the inverse Bom-oppenheimer approximation implies a weak coupling between these zero-order states. Here, too, the coupling is often limited by selection rules and one role of external perturbers is to relax these rules. Under the circumstances that the number of bound states is high and ionization requires the most of the available energy is localized in the electron, the dynamics can exhibit two or more time scales [45] (Fig. 6). The prompt decay is a minority route. In this mode, the optically excited electron departs without having sampled much (or even any) of the bound phase space that is available. All other decay channels are delayed. Figure 6 is a schematic drawn for an intramolecular coupling of uniform strength between all the bound states. The actual phase space has bottlenecks (cf. Fig. 7), and these cause a spread in the magnitude of the decay rates. Even so, the typical result of computational studies in which the Hamiltonian is diagonalized is that the rates tend to cluster into groups of quite different magnitudes. A quantum mechanical proof of the bifurcation of the decay modes is

prompt

1o-'q- . -

I

2

,

0

,

1

2

, \ 4

1

6

log (density of qbound states) Figure 6. The average decay width for a given uniform coupling strength V to the continuum (as indicated) vs. the density of (quasi)bound states (logarithmic scale). When the bound phase space is congested, the individual state rates bifurcate into a few (equal in number to the number of dissociation channels) promptly decaying states and many "trapped states whose decay is far slower. (See Section 1II.B.) In the real molecule there are considerable variations in the magnitude of the intramolecular couplings. Therefore, the magnitudes of the rates are scattered but fall within two (or more) groups.

632

R. D.LEVINE

simple and will be given shortly. It must however be emphasized that this is not an inherently quanta1 result. The delay due to trapping in a congested bound phase space is essentially the point made by Lindemann in interpreting the delayed dissociation of energy-rich polyatomic molecules. It was further elaborated and quantified by RRKh4 (Rice, Ramsperger, Kassel, and Marcus). Their initial (RRK) result for the dissociation rate can be cast in the present terms as

where v is the frequency of motion along the bottleneck, K is the number of dissociative continua (the number of states with enough energy along the reaction coordinate for dissociation to take place), and N is the number of (quasi) bound states. When K r ) and 8 is the angle between the vectors to the electron and the ion, respectively. Now (r) = n2 and (r-*) = n-3 so that the relative importance of the core vs. the external dipolar contributions scales as (c1/n)(R2/(r)*)and it is possible for the ion to overwhelm the importance of the anisotropy of the core. Both the terms in (3.3) and the external fields affect the electron when it is primarily far away from the core. In terms of Fig. 2, this is the midrange where u = ?r. The influence of the core is primarily near the inner turning point where u = 0, 2a. Even at high n's one needs to follow the system for many orbital periods if one is to mimic the experimental results. The difficulty is compounded if one measures the time in units of periods of the core motion. This suggests that the time evolution be characterized using the stationary states of the Hamiltonian rather than propagating the initial state. We have done so, but our experience is that in the presence of DC fields of experimental magnitude (which means that Stark manifolds of adjacent n values overlap), and certainly so in the presence of other ions that break the cylindrical symmetry and hence mix the m1 values, the size of the basis required for convergence is near the limit of current computers. In our experience, truncating the quan-

636

R. D. LEVINE

tum mechanical basis cannot be trusted without extensive convergence tests. (There is a theoretical reason for this, which is discussed below.) Hence we tend to regard classical mechanics as providing a viable alternative (particularly so in view of the already mentioned) very high density of states. Indeed, for the times in the microsecond range the mapping [3c, 51 is the most realistic tool. To discuss the separation of time scales, we begin with the argument that a system that reaches the continuum via a narrow bottleneck can exhibit more than one time scale [45a,b,f, 511. Particular attention will be given to the question of when this will be the case. The argument begins by considering the time evolution in the bound subspace. As is well known [52,53,54], one can confine attention to the bound levels by the introduction of an effective Hamiltonian H in which the coupling to the continuum is accounted for by a rate operator I’: In general, I’ contains both a Hermitian and an anti-Hermitian part. The latter causes a “level shift,” and if it contributes, we include it in H so that r is Hermitian. The usual discussion of dynamics in a congested bound-level structure begins with the dynamics induced by H. In other words, H is diagonalized first and only then is the role of I’ examined. Here we shall first diagonalize r and only then discuss how off-diagonal terms in H modify the results. The essential point is that if there are K independent decay channels, then the N x N matrix r is necessarily a matrix of rank K. If K < N,then I’ has N - K (or more) zero eigenvalues. The N - K associated eigenvectors have a zero width. If the time evolution is determined by r, then these states are trapped forever. The limiting situation when this is the case is the “trapping limit.” In the genera1 case H has nondiagonal elements between the different eigenvectors of I?. The time evolution will mix the different eigenvectors of I’, and the coupling strength to the continuum will be spread over all those states that are coupled by H. The reason for the emphasis is that this is where the selection rules on which states can be coupled by H are of key importance. The more states are coupled by H, the more is the given coupling to the continuum spread over more states. The opposite limit to trapping is when the mixing induced by H is dominant. We proceed to a detailed derivation of these results.

A. Effective Hamiltonian

To describe the dynamics in a congested bound phase space above the threshold for dissociation, it is convenient to use an effective Hamiltonian, which


637

is an operator confined to the subspace of bound states: !Jf = QHoQ + Q(V + U)Q + Q(V + U)P(E+- PHp)-'P(V + U ) Q

(3.5)

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V + U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Ho + V+ U,where HOis the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V + U)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. The coupling to the continuum is implicitly contained in the last term in (3.5). The decay is due to the imaginary part of the Green's function P(E+ - PHP)-'P. In what follows we assume: (i) The spectrum of PHP is in principle known. Typically, what is readily known is the spectrum of PHoP and the additional coupling terms can give rise to so-called final-state interactions (discussed, e.g., in Ref. 55). (ii) The contribution of the real part, the so-called level shift [53], due to the coupling to the continuum can be neglected. This is justified if the coupling terms QHP are only weakly energy dependent [561. If this is not the case, then the effective Hamiltonian contains an additional Hermitian term. Otherwise, the last term in (3.5) is a purely anti-Hermitian operator and hence can be written as iQrQ, where r is Hermitian:

In (3.6) the matrices are in the bound (Q) subspace and I' is the Hermitian rate operator. The essential point of this chapter is that the physics of the problem we wish to address dictates that the rank of the matrix I' is small compared to the dimension N of the bound subspace, where N is the rank of H or of Q. The proof that I' is of a low rank begins with the explicit form

r = ?~QHPG(E - PNPIPHQ or

(3.7)

R. D. LEVINE

638

where the states ) k ) are states of the continuum that are isoenergetic with the zero-order bound states In). The summation over the index k in (3.7) is over the number of linearly independent isoenergetic states of PHP. Physically these are dissociative states where the intramolecular coupling is still “on”, that is, they are what one loosely refers to as transition states. In the RRK terminology these are the states with enough energy along the reaction coordinate. The range K of the index k is therefore the number of such states. In the notation of transition-state theory, K = N f ( E- Eo), where EO is the threshold energy for dissociation. For any but a diatomic molecule, most ways of partitioning the energy E among the internal degrees of freedom results in that there is not enough energy in the reaction coordinate so that K is small compared to the range N of values that the index n can assume. To prove that the rank of the matrix I’ is K, (3.7) is written as

k= I

The K vectors Vk are linearly independent and have N components, where if N is the number of bound states,

n= I

Some, or many, matrix elements in (3.9) may vanish because of selection rules. It is convenient to take the vectors Vk as eigenvectors of the N x N Hermitian matrix r, N n= 1

N

K

n=1 k=l

The first K such eigenvectors can be normalized by


639

Note that since the matrix r is positive (semi)definite, the first K eigenvalues rk, k = 1, ... , K, are all nonnegative. The other N - K eigenvectors of r all have zero as an eigenvalue. (Note that the nonvanishin eigenvalues of r must equal those of the K x K Hermitian matrix 42 a V V.) The time evolution is determined by the full effective Hamiltonian H and not by the rate matrix F alone. One cannot therefore discuss the time evolution without reference to the matrix H.Say, however, H and r commute, [H, I"] = 0. A simple condition that ensures this result is that the bound states are strictly degenerate. If H and r commute, the eigenvectors of r evolve in time independently of one another. In the basis of states defined by the N eigenvectors of r there will be K states that will decay by direct coupling to the continuum and N - K states that are trapped forever. An arbitrary initial state is a linear combination of the N eigenvectors of I' and hence can have a trapped component. The K eigenstates that are directly coupled to the continuum decay promptly because they carry the entire coupling strength

f

K

TrI'=x

rk

(3.12)

k= I

so that, when K > N-'Tr r. The Rydberg state which is optically prepared in a typical ZEKE experiment is usually directly coupled to the continuum [45c, 571. Other considerations being absent, it should decay promptly, possibly with a stable, trapped component. The point is that the initially prepared state is also directly coupled to many other states, due both to external perturbations 1371 and to intramolecular coupling [3b]. The conclusion that the initial state has two components, one that decays promptly and one that is trapped, is thus only valid in zero order (so-called golden rule limit). One needs to allow for the coupling terms represented by V and U.

B. Trapping Versus Dilution

In the golden rule approximation, the decay rates are the eigenvalues of the matrix I'. The result that when the bound-state subspace is dense, I' is of a rank smaller than the number of bound states means that, in the golden rule approximation, some of these bound states will remain bound. That is, they are trapped states and do not decay. Of course, the golden rule is just

640

'

R. D.

LEVINE

the first-order approximation. But in first order it follows that states can be strictly classified as either decaying or as stable states. This is just as in the simple RRK picture, where states either have enough energy along the reaction coordinate or they do not and only the former do dissociate. Analytical considerations applied to the full dynamics (where it is rather than r that determines the time evolution) and computational results [45Q suggest that even when the coupling is allowed for, remnants of the zero-order dichotomy between promptly decaying and stable states do survive, that K states can either decay promptly, without much sampling of the bound space of N states, and that N - K states have a delayed decay. The larger is the density of bound states, the slower is the decay of the states that, in first order, are trapped. Figure 6 is the schematic case while Fig. 8 shows explicit computational results.

1 0.8

0.6 0.4

0.2

s

0.8

0.6 0.4

0.2

n., -3

-2

-1

0

1

log (t /ns)

2

3

Figure 8. Time evolution, obtained via a numerical diagonalization of the effective Hamiltonian H (adapted from Ref. 45) when N = K and N > K for two different initial states. Shown is the survival probability of the initial state (dashed line) and the probability to remain bound (full line) vs. time. The reason for the difference between these two is due to the system sampling the rest of the bound phase space. At higher number N of bound states, for a given value of K, the delayed decay would be shifted to even longer times while the survival probability will remain essentially unchanged, showing that the delay is due to sampling of the bound phase space.


To discuss the role of are uncoupled

641

H,consider first the limit where the bound states

31 = I&,- ir

[&,r]= 0

(trapping limit)

(3.13)

where I&,is diagonal in the basis of states that diagonalize r. Then the eigenvectors of H are the eigenvalues of r and the eigenvalues have a real part, which is the energy (i.e., the eigenvalue of €I,-,), and an imaginary part, which is the eigenvalue of:'l

H

* I = (El - rl)q

I = I , . ..,N

rr= 0

for I > K

(3.14)

An arbitrary initial state cp can be expanded using the eigenvectors of I' as a basis where, since these states are orthogonal [cf. Eq. (3.1 l)], N

(3.15) I=l

It follows that in the trapping limit an arbitrary initial state will have a component that is trapped and does not decay:

(3.16)

The trapped component is the component of the initial state in the subspace of the N - K trapped eigenvectors of I".

A special case of these results, discussed in Ref. 51, is when all the states are degenerate, €I,-, = El, where I is the identity matrix. This special form is sufficient to show the trapping phenomena but it is not necessary. Strict trapping is possible even when the states are not degenerate. The necessary condition is that the Hamiltonian H in the bound space commutes with the

642

R.D.LEVINE

matrix I?. Here we discussed a more general case where the two matrices commute, namely H = &, where 6 is diagonal. The general case corresponds to allowing for H being nondiagonal, so that (3.13) is replaced by

H=&+H1-ir

(3.17)

As in (3.13) we take the matrices in this equation to be defined in the basis that diagonalizes I‘ so that, in the absence of HI, the effective Hamiltonian is diagonal and the dynamics manifests trapping. Computing the time evolution is equivalent to diagonalizing the effective Hamiltonian. One can proceed by using perturbation theory and thereby eliminate the effect of the intramolecular coupling HI in successive orders. This route is particularly useful if the perturbation HI itself has terms of different orders, as is often the case for real molecules. In this way one can discuss the sequential sampling of phase space [45b, 58-61] with the modification that dissociation is possible. Another route is to exactly diagonalize, which can be done numerically (as, e.g., in Fig. 8). Either way, the coupling mixes the trapped and the promptly decaying states. If the mixing is so complete that one cannot uniquely correlate the eigenstates of H with the eigenstates of r (or of &),then the distinction between trapped and promptly decaying states is lost upon diagonaIization of H . The eigenstates cannot be said to be of one kind or the other. Since the intramolecular coupling causes a “repulsion” of the energy levels of H, the region of dominant coupling is when the spacings of the levels of H exceeds the magnitude of the eigenvalues of r. In the language of resonance scattering theory [52,53,54,62] the “resonances” are far apart in energy and so are isolated. In such a coupling regime it is more reasonable to first diagonalize H and then to regard r as a perturbation. In the language used below, in this range the coupling to the continuum has been effectively democratically diluted over all the states. This is the limit of a low density of states, and the merging of the two branches is clearly seen in Fig. 6. Also seen therein is that when the coupling is to the continuum is weaker, a higher density of states before the bifurcation into two types of decay is evident. For a high density of states it will typically be the case that dilution is incomplete. What this means is that when an eigenstate w, of H is expanded in the basis that diagonlize r, K

N

I= I

l=K+l

(3.18)


643

one or the other of the two terms dominate. As an eigenstate of 31 the state w, decays independently of all others and its rate of decay is

Those eigenstates of 31 that are made up predominately of prompt states will decay promptly and vice versa, the extent of dilution being measured by the expansion coefficients lc;tI2, I 5 K . The dramatic effect of the dilution of the coupling strength to the continuum is to endow the trapped states with a finite (but long) lifetime. We consider that this is the origin of the extremely long decay times observed in ZEKE spectroscopy. Dilution also tends to extend the lifetime for the prompt decay. For a discussion of dilution from a time-dependent point of view (the so-called time stretch), see Refs. 3a and 3d. The overall effect of dilution is to cause a more uniform distribution of the lifetimes, so that the shorter lifetimes are longer and the very long lifetimes are shorter. In other words, dilution and trapping, which are due to different terms in the effective Hamiltonian, have an essentially opposite effect. When dilution is the dominant effect, the distribution of lifetimes is unimodal and narrower. (Increasing dilution means moving to the left in Fig. 6.) When trapping dominates, the distribution of lifetimes is at least bimodal. The long lifetimes corresponding to states that would be fully trapped in the absence of coupling and the short lifetimes are the prompt decays. It is possible for the distribution of lifetimes to have more than two modes because the coupling need not be a simple process but can manifest a sequential character [45b, 58, 60,611. A structure in the matrix H1 can give rise to additional bifurcations in the distribution of lifetimes. The distinction between prompt decay and trapped states has been made here for the basis that diagonalizes the r matrix. When the Hamiltonian H that governs the evolution in the bound space is not diagonal in that basis, the distinction between the two types of states is valid only to lowest order and the actual dynamics mixes the two types of states. It is possible to define trapped states even in the general case [45d] by diagonalizing the Hermitian operator €I-'I', and this is equivalent to a simultaneous diagonalization of H and r. The resulting generalized eigenvectors can be strictly classified according to whether their eigenvalues does or does not have an imaginary part and there are K of the former. However, the required transformation is not unitary and so the generalized eigenvectors do not diagonalize functions

644

R. D. LEVINE

of H so that they are mixed by the time evolution (except, as in the above, to first order). Finally we explicitly discuss external perturbations as a source of dilution. When the molecule is coupled to its environment, strictly speaking, one should use a Hamiltonian description for the combined system. Just as we eliminated the explicit coupling to the continuum by the use of an effective Hamiltonian, one can do the same for the perturbation due to the surroundings. In the lowest order, when such perturbations are weak and essentially static, the effect is equivalent to an additional coupling mechanism, denoted as U in Eq.(3.5). States that otherwise would be trapped can acquire a finite width when U is included in H. Since such a description is valid only when U is a weak perturbation, one might think that the effect will be minor. The reason that this is not necessarily so is the high degeneracies typical of highly excited Rydberg states. The optical excitation of the Rydberg state typically accesses only very low angular momentum states. The breaking of the spatial symmetry of the isolated atom by an external anisotropy (as in Ref. 12) or by a magnetic field (Ref. 1) couples in a very large number of states, many more states than are coupled when only an external electric field is present since such a DC field retains the cylindrical symmetry of the problem. One point about the perturbation of the Rydberg electron is therefore the number of states that are coupled by H. The other is the dispersion in the eigenvalues of H. The extent of dilution depends on the failure of H and r to commute. Larger energy spacings favor dilution, where larger is measured in respect to W ' T r r. Since the typical long decay time is orders of magnitude longer than an orbital period of a Rydberg electron, it follows that N-lTr r is very much smaller than the spacing of (unperturbed) Rydberg states (which at high n's are fractions of a wavenumber). The long time evolution governed by the small spacings in the eigenstates of H is therefore sensitive to unusually small spacings, of the order of Ry/n5, in reciprocal centimeters. In computational studies of quantum dynamics this implies that truncating a basis set can lead to severe truncation errors in the long-term time evolution. In experimental studies it means that slight disturbances can have observable effects, particularly so at longer times.

IV. CONCLUDING REMARKS Prompt and delayed ionization is familiar for very energy rich molecules. The special feature of high Rydberg states is the initial state that is optically prepared, a state directly coupled to the continuum on the one hand and to a very dense bound manifold on the other. The dynamical theory necessary to describe such states has been reviewed, with special reference to the extremely long-time decay. It is suggested that this resilience to decay is due


645

to admixture of states that are, in zero order, trapped. These can be mixed in due to both intramolecular and external perturbations.

Acknowledgments This work would not have been possible without the close collaboration with the experimental groups of E. W. Schlag (Munich) and of U. Even (Tel Aviv). I am very grateful to them and to their co-workers, H. L. Selzle and K. Muller-Dethlefs in particular. It is also a pleasure to thank my co-workers Eran Rabani, FranGoise Remade, and Leonid Baranov, whose contribution is reflected through the references to their papers. The work was supported by the Volkswagen Foundation and by SFB 377.

References I . A. Muhlpfordt, U. Even, et al.. Phys. Rev. A 51, 3922 (1995). 2. E. Rabani and R. D. Levine, J. Chem. Phys. 104, 1937 (1995).

3. (a) E. Rabani, L. Y. Baranov, et al., Chem. Phys. Lett. 221, 473 (1994); (b) E. Rabani, R. D. Levine, et al., J. Phys. Chem. 98, 8834 (1994); (c) E. Rabani, R. D. Levine, et al., Ber. Bunsenges. Phys. Chem. 99,310 (1995); (d) E. Rabani, R. D. Levine, et al., J. Chem. Phys. 102, 1619 (1995). 4. U. Even, R. D. Levine, et al., 1.Phys. Chem. 98, 3472 (1994). 5. E. Rabani and R. D. Levine, J. Chem. Phys. (1996). 6. (a) K. Muller-Dethlefs and E. W. Schlag, Ann. Rev. Phys. Chem. 42, 109 (1991); (b) G. 1. Nemeth, H.L. Selzle, et al., Chem. Phys. Lett. 215, 151 (1993). 7. E. W. Schlag, W. B. Peatman, et at., J. Elect. Spectry. 66, 139 (1993). 8. X.Zhang, J. M. Smith, et al.. J. Chem. Phys. 99, 3133 (1993). 9. C. E. Alt, W. G. Scherzer, et al., Chem. Phys. Len. 224, 366 (1994). 10. I. Fischer, R. Lindner, et al.. J. Chem. Soc. Faraday Trans. 90, 2425 (1994). I 1. F. Merkt, Chem. Phys. 100, 2623 (I994). 12. F. Merkt and R. N. Zare, J. Chem. Phys. 101, 3495 (1994). 13. G. I. Nemeth, H. Ungar, et al., Chem. Phys. Left. 228, (1994). 14. W. G. Sherzer, H. L. Selzle, et al., Phys. Rev. Lett. 72, 1435 (1994). 15. C. E. Alt, W. G. Scherzer, et al., J. Phys. Chem. 99, 1660 (1995). 16. A. Muhlpfordt and U. Even, J. Chem. Phys. 103,4427 (1995). 17. (a) M. J. J. Vrakking and Y.T. Lee, Phys. Rev. A 51, 894 (1995); (b) M. J. J. Vrakking and Y. T. Lee, J. Chem. Phys. 102, 8818 (1995). 18. D. Bahatt, U. Even, et al.. J. Chem. Phys. 98, 1744 (1993). 19. U. Even, M. Ben-Nun, et al., Chem. Phys. Lett. 210,416 (1993). 20. E. W. Schlag and R. D. Levine, J. Phys. Chem. 96, 10668 (1992). 21. E. W. Schlag, J. Grotemeyer, et al., Chem. Phys. Left. 190, 521 (1992). 22. R. Weinkauf. P. Aicher, et al., J. Phys. Chem. 98, 8381 (1994). 23. E. E. B. Campbell, G. Ulmer, et al.. Z. Phys. D 24,81 (1992). 24. P. Wurz and K. R. Lykke, J. Phys. Chem. 96, 10129 (1992). 25. B. A. Collings, A. H. Amrein, et al., J. Chem. Phys. 99, 4174 (1993). 26. M. Foltin, M. Lezius, et al., J. Chem. Phys. 98, 9624 (1993).

646

R. D. LEVINE

27. T. Leisner, K. Athanassenas, et al.. J. Chem. Phys. 99,9670 (1993). 28. K.Toglhofer, F. Aumayr, et al., Europhys. Len. 22, 597 (1993). 29. C. Yeretzian, K. Hansen, et al., Science 260,652 (1993). 30. H. Hohmann, C. Callegari, et al., Phys. Rev. Len. 73, 1919 (1994). 31. H. Hohmann, R. Ehlich, et al., Z. Phys. D 33, 143 (1995). 32. B. D. May, S. F. Cartier, et al., Chem. Phys. Lett. 242, 265 (1995). 33. H. Weidele, D. Kreisle, et al., Chem. Phys. Len. 237, 425 (1995). 34. D. Bahatt, 0. Cheshnovsky, et al., 2.f: Phys. Chem. 184,253 (1994). 35. H.-J. Dietrich, R. Lindner, et al., 1.Chem. Phys. 101, 3399 (1994). 36. W. G. Schemer, H. L. Selzle, et al., Phys. Rev. Lerr. 72, 1435 (1994). 37. W. A. Chupka, J. Chem. Phys. 98,4520 (1993). 38. S. T.Pratt, J. Chem. Phys. 98, 9241 (1993). 39. M. Bixon and J. Jortner, J. Chem. Phys. 103,4431 (1995). 40. F. Merkt, S. R. Mackenzie, et al., J. Chem. Phys. 103, 4509 (1995). 41. L. Y. Baranov, Y. Kris, et al., J. Chem. Phys. 100, 186 (1994). 42. M. Born, Mechanics of the A r m , Blackie, London, 1951. 43. H. Krause and H. J. Neusser, J. Chem. Phys. 99, 6278 (1993). 44. D. M. Neumark, Ann. Rev. Phys. Chem. 43, 153 (1992). 45. (a) F. Remacle and R. D. Levine, Phys. Leu. A 173, 284 (1993); (b) F. Remacle and R. D. Levine, J. Chem. Phys. 98,2144 (1993); (c) F. Remacle and R. D. Levine, J. Chem. Phys. 104, 1399 (1996); (d) F. Remacle and R. D. Levine, Mol. Phys. 87,899 (1995); (e) F. Remacle and R. D. Levine, J. Phys. Chem. 100, 7962 (1995); (9 F. Rernacle and R. D. Levine, J. Chem. Phys. 105,4949 (1996). 46. G. Herzberg and C. Jungen, J. Mul. Specrrosc. 41, 425 (1972). 47. R. S. Berry, J. Chem. Phys. 45, 1228 (1966). 48. A. Russek, M. R. Patterson, et al., Phys. Rev. 167, 17 (1968). 49. E. E. Eyler, Phys. Rev. A 34, 2881 (1986). 50. R. D. Gilbert and M. S. Child, Chem. Phys. Leu. 287, 153 (1991). 51. F. Remacle, M. Munster, et al., Phys. Len. A. 145, 265 (1990). 52. H. Feshbach, Ann. Rev. 19, 287 (1962). 53. R. D. Levine, Quantum Mechanics of Molecular Rate Processes, Oxford, Clarendon, 1969. 54. S. Nordholm and S . A. Rice, J. Chem. Phys. 62, 157 (1975). 55. J. A. Beswick and J. Jortner, J. Chem. Phys. 68, 2277 (1978). 56. M. Desouter-Lecomte, J. Lievin, et al., J. Chem. Phys. 103,4524 (1995). 57. Merkt and Softley, lnr’l. Rev. Phys. Chem. 12, 503 (1993). 58. J. P. Pique, Y.Chen, et al., Phys. Rev. Lert. 58,475 (1987). 59. D. E. Logan and P. G. Wolynes, J. Chem. Phys. 93,4994 (1990). 60. K. Yamanouchi, N. Ikeda, et al., J. Chem. Phys. 95,6330 (1991). 61. M. J. Davis, J. Chem. Phys. 98,2614 (1993). 62. M. Bixon, J. Jortner, et al., Mol. Phys. 17, 109 (1969).


GENERAL DISCUSSION ON MOLECULAR RYDBERG STATES AND ZEKE SPECTROSCOPY: PART I Chairman: G. Casati

J.4. Lorquet: The energy resolution of Prof. Schlag’s spectra is indeed impressive. But would there be an advantage in recording the spectra at a low resolution, just good enough to resolve the vibrational structure, in order to get a better appreciation of the intensity pattern? Could then that pattern be interpreted as a superposition of several progressions? Also, can one rationalize any observed dependence of the intensities on experimental parameters of the ion source? E. W. Schlag: The intensities can in many cases be calculated from the Franck-Condon principle, but in some cases there are important exceptions that are most clearly interpreted in the inverse Born-Oppenheimer (BO) framework. Here one clearly sees the superposition of Rydberg stacks from individual rotations or vibrations. R. D.Levine: To answer the question of Prof. Lorquet, let me say that the peaks in the ZEKE spectra correspond to the different energy states of the ion. From the beginning one was able to resolve vibrational states, and nowadays individual rotational states of polyatomics have also been resolved. The ZEKE spectrum is obtained by a (weak) electrical-field-induced ionization of a high Rydberg electron moving about the ion. The very structure of the spectrum appears to me to point to the appropriate zero-order description of the states before ionization as definite rovibrational states of the ionic core, each of which has its own Rydberg series. Such a zero-order description is inverse to the one we use at far lower energies where each electronic state has its own set of distinct rovibrational states, known as the Born-Oppenheimer limit. D. Gauyacq: I have a short comment on Prof. Schlag’s remark on the multichannel quantum defect theory (MQDT) approach to ZEKE spectra: The first ZEKE spectrum of NO was actually interpreted by using MQDT as early as 1987 [ 13. In this work, a full calculation of the ZEIE peak intensities is carried out by the MQDT approach, which 647

648

GENERAL DISCUSSION

takes into account the relative population of the intermediate levels (see p. 859 and Fig. 14 of Ref. 1). 1. S. Fredin, D.Gauyacq, M. Horani, Ch. Jungen, G . Lefevre, and F. Masnou-Seeuws, Mol. Phys. 60, 825 (1987).

E. W. Schlag: Thank you for pointing out this reference to me which totally agrees with our originally measured spectrum. R. Schinke: Prof. Levine, I have two questions: 1. Did you and your co-workers compare the results of classical and quantum mechanical calculations? 2. For an electronic system, how do you define a transition state necessary for calculating an RRKM rate constant?

R. D. Levine: Our classical and quantal simulations use the same Hamiltonian. However, we have not, so far, performed the dynamical computations on the same system by both classical and quantum mechanical methods. We have not done so because, as I discussed, one needs a rather large basis set if the quantal computation is to converge. In particular, computation on the role of the electrical field using a limited basis set (using not enough 1 values) give completely spunous results, such as a lengthening of the lifetime. These results change markedly when a basis that properly spans the Stark manifold is used. So we have focused the quantal effort on a few topics. Encouraged by your interest we will certainly try a direct quantalclassical comparison. Of course, we have a clear idea that classical mechanics will be at some fault if the spacings of the states of the ionic core are wider than the strength of the coupling of such states due to the electron. (And for this reason we prefer to treat the vibrations by semiclassical or quantal means.) I agree with you that it will be useful to have a direct quantitative comparison and we will work on providing it. J.-C. Lorquet: Leaving Rydberg states aside for a moment and concentrating on unimolecular reactions, I wonder whether a relationship exists between the trapping mechanism described in your present communication and the healing process you introduced many years ago? As a second question, how does the efficiency of both of these processes vary as one increases the internal energy? I imagine that the numbers N and K increase with energy in a different way. One might imagine that the higher the internal energy, the greater the tendency of exploding rapidly. R. D. Levine: An important motivation for the detailed understand-

RYDBERG STATES AND ZEKE SPECTROSCOPY I

649

ing of the dynamics of high molecular Rydberg states is indeed that they provide a wonderful laboratory for examining many of the issues we want to explore for unimolecular reactions of energy-rich polyatomic molecules. Two recent papers [F. Remacle and R. D. Levine, J. Chem. Phys. 104,1399 (1996) and J. Phys. Chem. 100,7962 (1996)l. A particular illustration of this cross-cultural influence is the question of whether one can pump states directly coupled to the transition state. In the Rydberg problem the answer is very much “yes” because these are often the states that are optically excited. They are directly coupled to the continuum but are also coupled to other Rydberg series, where n is lower and the core is more excited. Healing is said to occur when a bond that is already broken, so that the products begin to recede from one another, is “healed”; the products approach again and recross the transition state back to the bound region of phase space. When there are many (quasi)bound states, healing can occur in an off-diagonal manner in that the (quasibound) state that they come back to need not be the same as the (quasibound) state that they dissociated from. Healing thus provides for an enhanced mixing of the bound phase space. Healing thus provides for a longer lifetime. In formal theory healing is due to the rate operator part of the effective Hamiltonian being nondiagonal in the zero-order basis. The rate operator is a nonnegative definite operator and can be diagonalized. You and your co-workers have argued that under the circumstances that we are discussing (a dense bound phase space coupled to a sparse continuum) there will necessarily be many “trapped” eigenvectors of zero eigenvalue for the rate operator. However, the dynamics is described by the full effective Hamiltonian, and this will mix in the eigenvectors of the rate operator. A trapped eigenstate will thus not be stable but may have a very long decay time. In other words, healing can provide for a bottleneck to dissociation. J.-C. Lorquet: That’s a very good point. In order words, resurrection is an extreme case of healing, isn’t it? R. D. Levine: I am coming from Jerusalem. Ch. Jungen: I would like to make a comment on the concept of “inverse Born-Oppenheimer approximation,” which has been invoked by both speakers this morning. Here is a quotation from a paper published by R. S . Mulliken [J. Am. Chem. Soc. 86, 3183 (1964)l more than 30 years ago: “In most discussions on molecular wave functions,

650

GENERAL DISCUSSION

the validity of the Born-Oppenheimer approximation is assumed. This approximation is most nearly accurate when the frequencies of motion, which can be gauged by energy level spacings, are much larger for the electronic than for the nuclear motions. In a Rydberg state series, as n increases, the frequencies of the Rydberg electron become smaller and smaller relative to those of nuclear vibration and rotation. This leads to more or less radical changes in coupling relations.” Mulliken then goes on to denote the two limiting cases as Rydberg-coupled and Rydberguncoupled wave functions. I do not think we need a new word here for something that has been in use for a long time.

R. D. Levine: As I am sure you will agree, it is not possible to

categorically state that one zero-order basis set is “better” than another. At best one can state that a particular basis is more convenient. If one intends to actually diagonalize the Hamiltonian, then the convenience is a matter of technicalities: Which basis offers a better computational route? If, however, one wants to use the zero-order basis to discuss physics, then, as we all agree, one would like the basis to diagonalize as much of the problem as is possible. My point is that the Rydberg states span an enormous coupling range and what can be a useful zero-order basis in one range of n need not necessarily be so in a quite different range. This is in part due to the n3 dependence of the orbital period of the electron (cf. Fig. 1 of the chapter by R. D. Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg states,” this volume). There is also the question of which zero-order basis is more “natural” for the experiment. Of course, here, too, simplicity is in the eye of the beholder, but as I already argued, to my mind, ZEKE spectroscopy points out the advantage of the inverse Born-Oppenheimer zero-order basis. I recognize that MQDT is based on the fact that near the core the separation inherent to the inverse Born-Oppenheimer basis is no longer valid and the MQDT procedure switches basis. It does so because it seeks to diagonalize the Hamiltonian in an efficient manner. We recognize the coupling by labeling the inverse Born-Oppenheimer states as providing a zero-order basis. It is precisely the nature and role of this coupling that we wish to elucidate. For a derivation of the Herzberg-Jungen formula starting with our point of view see the appendix of E. Rabani and R. D. Levine, J. Chem. Phys. 104, 1937 (1996).

From a purely operational point of view, please note that the individual peaks in what one calls the ZEKE spectrum (E. W. Schlag,


65 1

“ZEKE Spectroscopy,” this volume) are obtained by ionizing the Rydberg series which is built on a particular state of the ionic core. The inverse Bom-Oppenheimer picture is thus natural when one begins the discussion with what is actually the raw observation. M. Shapiro: Concerning the inverse Bom-oppenheimer regime, I wonder if we are not simply in a decoupling regime. The relevant parameters for the Bom-Oppenheimer approximation are not so much the relative frequencies of the electronic and nuclear motions (were this the case in the continuum, when the spacing between the electronic levels is zero, we would always be in the “inverse Bom-Oppenheimer” regime, which is clearly not the case) as the magnitude of the “nonBom-Oppenheimer” coupling terms and the level separations. There are systems where, in fact, as we increase the size of the system (which is what we do when we go to a high Rydberg state), the non-Bom-Oppenheimer terms go to zero faster than the separation between levels, and the Bom-Oppenheimer approximation stays valid. Therefore I wonder if you actually checked the magnitude of these terms? R. D. Levine: The case of the Rydberg electron orbiting a rotating ionic core is formally analogous to that of a very low energy rare-gas atom bound by van der Waals forces to a rotating diatomic. In both cases the system can dissociate by rotational energy of the core being made available to the translational motion. What is new in the Rydberg problem is that because of the great depth of the Coulomb well, the already bound electron can also lose energy by going down in that well while the core is further rotationally excited. In both cases, the same zero-order basis set, in which the faster motion (namely the rotation) adapts itself to the slower one, is useful [e.g., M. Shapiro, R. D. Levine, and B. R. Johnson, J. Chem. Phys. 52, 1755 (1970)l.In the Rydberg case this basis set is the one in which the electron is solved for in the field of the core. This is the zero-order basis that we (and others) use. In both cases what the basis set provides is only a zero-order description because in both cases one knows that the system can exit to the continuum. A. H.Zewail: I have two questions for Profs. Schlag and Levine: 1. For lower n we observe femtosecond dynamics [l]. Is it reasonable to think that because of the low n (low angular momentum I ) the probability for penetrating the core is much larger, leading to femtosecond instead of microsecond dynamics?

652

GENERAL DISCUSSION

2. When the level structure is that dense for high n, you have a distribution of lifetimes, and in the language of intramolecular dynamics you would expect a fast component and a much slower average decay. 1. M. H. M.Janssen et al., Chem. Phys. Lett. 214, 281 (1993); H. Guo and A. H. Zewail, Can. J. Chem. 72,947 (1994).

E. W. Schlag 1. Yes, indeed, coupling to the core is stronger for lower n, leading to shortened lifetimes for low n. 2. For complex lifetimes at high n, there are expected to be a series of couplings leading to short and long lifetimes.

R. D. Levine: The latest results for benzene as reported by Prof. Neusser suggest that benzene is the molecule we should study at all time scales. In other aromatic molecules (and also in other molecules [U.Even et al., J. Phys. Chem. 98, 3472 (1994)l) you have reported that there are at least two times scales for the decay of high Rydberg states. A faster (hundreds-of-nanosecond) scale and a slower one (microsecond range). Even the faster time scale is two to three orders of magnitude slower decay than one would expect from an extrapolation of the decay measured at low or intermediate n Rydberg states. Theory [F.Remacle and R. D. Levine, Phys. Lett. A 173,284 (1993)] and simulations suggest that there should also be a prompt decay corresponding to a yet shorter scale. Your statement that you can now begin the time-resolved ZEKE spectroscopy at much earlier times than before makes me suggest that benzene be one of the first molecules that you choose for experimental study. A. H. Zewail: I wanted to emphasize in my comment that the dynamics is rich and the definition of a “lifetime” is only possible if the entire temporal behavior is unraveled. E. W. Schlag: It is very hard to do the ideal experiment of perfect state selection in a field-free environment. The perfect experiment here may well not be the ideal ZEKE experiment since some field appears to be needed for state mixing. D. M. Newark: I would like to make a comment to Prof. Schlag. One expects an anion ZEKE spectrum to have the same overall intensity profile as the corresponding photoelectron spectrum only if direct detachment is the only process that occurs. However, FeO- has several dipole-bound and valence-excited states near the detachment threshold.

RYDBERG STATES AND ZEKE SPECTROSCOPY 1

653

So it is not too surprising that the FeO- ZEKE spectrum is very different from the photoelectron spectrum. E. W.Schlag: That is a good point, although there are some other very profound intensity changes in anion-ZEKE spectra. D. M.Neumark: We have recently investigated resonant multiphoton detachment in C4, C,, and C,. Photoelectron spectroscopy followed by multiphoton excitation strongly suggests that “thennionic emission” occurs in these small clusters, and this is supported by our observation of delayed electron emission from C, and C,. The latter results yield the emission rate as a function of total energy. We can compare these results to RRKM calculations and find the calculated rates to be an order of magnitude too high. Do you have any insights into possible nonstatistical effects in this electron emission process that might explain thk discrepancy? (See Figs. 1-5 .)

1

21200

.

t

22200

.

n

23200

.

m

24200

.

I

25200

P

Q m

Figure 1. One-color resonant multiphoton detachment spectra of C,, C 6 . and C,.

654

GENERAL DISCUSSION

One-Photon

hu4.66eV

-z

.a v)

c

e

B

Multi-Photon

0.m

0.20

0.40

0.60

0.80

1.m I.P

Electron Kinetic Energy (eV)

Figure 2. Time-of-flight photoelectron spectra of C, . Top: one-photon direct detachment with photon energy 4.66 eV. Bottom: resonant multiphoton detachment with photon energy 2.04 ev. The electron affinity of linear c6 is 4.18 ev.

Time (nanoseconds)

Figure 3. Electron emission time profiles from resonant multiphoton detachment of C; at various photon energies. The time width at full width at half maximum (FWHM)of the laser pulse is 30 ns.


655

c; x%,

c; x*n,

-

Figure 4. Schematic drawing of C, resonant multiphoton detachment mechanism X2n, transition (IC, internal confor excitation at the band origin of the C2n, version; TE, thermionic emission).

5.0 47wo

48000

51m

53OOo

Total Photon Energy (m')

Figure 5. Comparison of experimental and calculated electron emission rate constants for C, and C,. Experimental results are shown in solid circles connected by a dashed line. Rates are calculated using microcanonical statistical model for thermionic emission. Solid lines are results when ab initio vibratiohal frequencies are used. Dashed line (C6 only) is calculation in which the value of the lowest vibrational frequency in C, was reduced by 30%.

656

GENERAL DISCUSSION

R. D. Levine: Prof. Neumark, your very detailed results on the excess energy dependence of the rate of delayed detachment of the electron from small carbon clusters should help test an ongoing discussion. One possibility is to view the process as that of thermionic emission. Now thermionic emission is the oldest version of transition-state theory known to me. What one assumes is that the system is in thermal equilibrium and the rate is given by the rate at which thermal electrons will cross a (hypothetical) surface surrounding the cluster, where the barrier height is the work function. In other words, the theory takes the rate of crossing of the transition state to be rate determining. It is not possible for me to fit your interesting data “by eye,” but very superficially I would say that your measured rate is too slow as compared to the predictions of the theory based on thermionic emission. Sometime ago we [E. W. Schlag and R. D. Levine, J. Phys. Chem. 96, 10608 (1992)l discussed the possibility that crossing the barrier is not rate determining. Rather, the slow process is the exchange of energy between the electron and the nuclear degrees of freedom (electron-phonon coupling is the technical term). This gives rise to a diffusive-like motion of the electron. Typical results are shown in Fig. 3 of the chapter by myself (R. D. Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume) where your clusters qualitatively correspond to the rightmost curve shown therein. The situation is analogous to the so-called energy diffusion regime in the Kramers picture of reactions in solution except that here “the molecule acts as its own solvent” [E. W. Schlag, J. Grotemeyer, and R. D. Levine, Chem. Phys. Lett. 190, 52 1 (1992)l.

D. M. Neumark: I talked to Prof. Marcus, and he mentioned some-

thing similar.

V. Engel: Let me come back to the distribution of lifetimes of the ZEKE Rydberg states. I wonder if there is a simple picture behind. Con-

sider a much simpler molecule, namely the NaI molecule Prof. Zewail told us about. There you have a bound state coupled to a continuum. It can be shown that in such a system the lifetimes of the quasibound states oscillate as a function of energy. In fact, Prof. Child showed with the help of semiclassical methods that there are lifetimes ranging from almost infinity to zero [l]. That can be understood by the two series (neglecting rotation) of vibrational levels obtained from the adiabatic and diabatic picture. If two energy levels of different series are degen-

RYDBERG STATES AND ZEKE SPECTROSCOPY 1

657

erate, an infinite lifetime is obtained; in the case the excitation energy is between two levels, a shorter lifetime occurs. For a Rydberg electron in a molecule there are many series, but the picture behind might be the same. 1 . S. Chapman and M. S . Child, J. Phys. Chern. 95,578 (1991).

J. Manz 1. The classical trajectory simulations of Rydberg molecular states carried out by Levine (“Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume) remind me of the related question asked yesterday by Prof. Woste (see Berry et al., “SizeDependent Ultrafast Relaxation Phenomena in Metal Clusters,” this volume). Here I wish to add that similar classical trajectory studies of ionic model clusters of the type A; ’B; have been carried out by 0. Knospe and R. Schmidt [I]. Here the two charged clusters A; and B; rotate around each other, similar to the rotation of the Rydberg electron e- around its cationic center M+ in the Rydberg state M+ . e- . Several properties of the trajectories for A; - B; and M+ - e- are found to be analogous, except for the different masses and effects of internal motions in the fragments A; and B; [l]. 1. 0. Knospe and R. Schmidt, Z. Phys. D 37, 85 (1996); 0. Knospe and R. Schmidt, Phys. Rev. A, 54, 1154 (1 996).

2. Prof. Schlag (“ZEKE Spectroscopy,” this volume) has introduced a new sequential technique of ZEKE spectroscopy: In the first step, a negative ion M- is photoionized, yielding the neutral core M of the excited Rydberg state of the anion M-*. In the second step, M is further photoionized, yielding the cationic core M+ of the excited Rydberg state of the neutral molecule M*. The overall sequence is thus M- -M-*

-M

+ e- -M* + e- -+M++ 2e-

and this sequential scheme is (at least formally) similar to the new NeNePo method which has been introduced into femtosecond chemistry by L. Woste and co-workers (see Ref. 1 and Berry et al., “SizeDependent Ultrafast Relaxation Phenomena in Metal Clusters,” this volume). I would therefore like to ask Profs. Schlag and Woste to comment on the relations between their new techniques. I . S . Wolf, G. Sommerer, S. RUU, E. Schreiber, T. Leisner, L. Woste. and R. S. Berry, Phys. Rev. Lett. 74,4177 (1995).

3. Prof. Levine (“Separation of Time Scales in the Dynamics of

658

GENERAL DISCUSSION

High Molecular Rydberg States,” this volume) has pointed to the fascinating situation that exists in Rydberg molecules in electric fields for energies E close to the ionization threshold Ethr,that is, the number of may be even larger (near-) degenerate (quasi)bound states &,,,nd(E) than the number of near-degenerate dissociative states Ndiss(E). I would like to ask Prof. Levine about the energy dependence of the ratio

I would expect that R ( E ) increases as E approaches Ethr, due to the enormous increase in N b o u n d ( E ) relative to Ndi,,(E). This conjecture may have important consequences for the stability of Rydberg molecules; cf. the experimental results of Prof. Schlag, which show that highly excited molecular Rydberg states may have exceedingly long lifetimes (see Schlag, ‘‘ZEKEi Spectroscopy,” this volume). Usually the stability of Rydberg molecules is attributed to dynamical effects; that is, any ionization processes are limited to the rare situations where the Rydberg electron interacts with the molecular core (see Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume). On the other hand, the stability of Rydberg molecules could also be interpreted in terms of an “entropic effect”; that is, the formation of bound states rather than ionization states could be due to the increasing ratio R ( E ) as defined above.

-

E. W. Schlag: On the way up from M- -+M M+ it would be often helpful to state select M via ZEKE spectroscopy. It would also be useful for its overtones. L. Woste: In stationary spectroscopy ZEKE certainly provides spectroscopic results at an impressive resolution. Using femtosecond pulses one can certainly not excite specific states as compared to ZEKE. The Fourier transform of the wavepacket evolution, however, exhibits also spectral resolution that easily reaches and even exceeds what we see in ZEKE spectra. For this reason, I do not see any disadvantage in using “femtosecond NeNePo” to probe states of a prepared molecule. E. W. Schlag: I would suggest that an interesting experiment is ZEKE selection on the way to further decomposition. This would directly answer the question. R. D. Levine: The suggestion of Prof. Manz about the dynamics of extremely high Rydberg states is a very interesting one to discuss.


659

Unfortunately it is not going to be easy to test experimentally or even to simulate on the computer. The reason is the extreme sensitivity of states of high n to external perturbations. In the laboratory, stray electrical fields, which cannot be completely avoided (or black-body radiation) will cause ionization of these states. Even on the computer, numerical roundoff errors will act as external noise. M.S. Child: Perhaps I can add a comment about time scales. For small nonpolar species such as H2 and N2 the dominant interaction between the Rydberg electron and the nuclear vibrational and rotational motion occurs within a small radius around the ionic core, which is traversed in a fraction of a femtosecond. This short encounter justifies the “sudden” treatment of vibration and rotation in MQDT theory, while also permitting Born-Oppenheimer estimates of the necessary quantum defect functions. It is also central to the n-3 scaling law because the core transit time is almost energy independent, while the Rydberg orbit time increases as n3. In dipolar situations the electron will continue to rotate with the core out to a radius such that the ion-dipole anisotropy is small compared with the relevant rotational energy separation for the ion core. Again since the switch-off distance is insensitive to energy, the dipole transit time will also be roughly the same for all n and one again expects an n-3 scaling law, but with a different coefficient. If this reasoning is correct, it is hard to see how the presence of a dipole can substantially enhance the lifetimes of ZEKE states. E. W. Schlag: The paper by Akulin et al. [l] explicitly calculates this model. 1.

V. M. Akulin, G . Reiser, and E. W. Schlag, Chem. Phys. Lett. 195, 383 (1992).

R. D. Levine: I completely agree with the picture of Prof. Child, and indeed this is how we calculate the effect of coupling to the vibrations [E. Rabani and R. D. Levine, J. Chern. Phys. 104, 1937 (1996)]. As to the special case of dipolar anisotropy, see L. Ya. Baranov, F. Remacle and R. D. Levine, Phys. Rev. A 54,4789 (1996). U. Even: In a recent series of papers [M. Bixon and J. Jortner], using a model Hamiltonian quantum treatment, it is shown that all multipole contributions to 1 mixing are negligible when compared with 1 mixing by low external fields. Thus the long lifetimes associated with ZEKE states are attributed (in atoms and in molecules) to the external fields alone. M. Herman: There is a striking, obvious and fascinating comparison between overtones and Rydberg states. Both cases present a high

GENERAL DISCUSSION

660

density of levels and exhibit scaling factors (referring to the acetylene case, to be extended to smaller and possibly larger species). How far can one pursue the comparison? What are the chances that a model developed in one case could help elucidate the other case?

V. S. Letokhov: In his exciting and dynamical report Prof. Schlag mentioned the historical development leading to ZEKE spectroscopy. Let me comment on this point. Laser-induced WMPI and ZEKE spectroscopies belong to the rich family of laser ionization spectroscopy techniques (Fig. 1) [l]. In resonance photoionization of a neutral particle (atom or molecule), the absorption by the particle of a few laser photons gives rise to an easily detectable pair of charged particles-a photoion and a photoelectron. In most experiments, the number of charged particles

r Mt

/

Photoion mass spectra

M*

+ eK

'Photoelectron spectrum

k

Photoions Photons

Je

Photoelectrons 'Resonant ionization optical spectrum

(b)

(a1 Figure 1. Versions of photoionization spectroscopy wherein not only the dependence of the multiphoton ionization efficiency on the laser wavelength is subject to measurement, but also the mass spectrum of photons and energy spectrum of photoelectrons: (a) energy-level diagram; (6)collision of a neutral particle with laser photons.


66 1

(i.e., the photoionization efficiency) as a function of the laser wavelength at one or, in principle, several resonance excitation steps is subject to measurement. This simplest version of photoionization spectroscopy is referred to as resonance photoionization spectroscopy (Fig. 1). The technique provides fairly accurate information on the structure of the quantum transitions of atoms and molecules with extremely high sensitivity. An excited molecule can be ionized by several pathways, charged fragments of various masses being formed in the process. A mixture of charged particles of various masses is also formed in the course of photoionization of a mixture of isotope atoms or molecules or else of molecules with close absorption lines. To identify more accurately the charged particles formed, we can in addition analyze the mass spectrum, that is, change over to photoionization muss spectrometry, as is mentioned in Ref. 2. This spectroscopic technique is being widely used in experiments with molecules and mixtures of isotopic atoms [I, 31. In my papers in 1975-1976 I considered various applications of laser resonance photoionization for the detection of single atoms and molecules, particularly by means of laser optical mass spectrometry [3, 41. The first steps in the realization of this idea were the successful experiments on two-color photoionization of polyatomic molecules in the gas cell [3, 51 and in the mass spectrometer chamber [6, 71. It seems to me that these works are a first demonstration of the REMPI technique. Independent experiments of Prof. Schlag and Prof. Bemstem have been cited by Prof. Schlag. In this respect I should mention also Prof. Mamyrin’s work at the Ioffe Institute, who realized the mass reflectron in 1966. Now this is a most useful type of TOF mass spectrometer for experiments with laser pulses. Finally, the photoelectron formed as a result of the decay of the ionic state of a particle acquires some excess energy, equal to the difference between the initial energy of the particle and the ion energy. This energy excess is transferred to electron translation; that is, it is converted to kinetic energy. If it populates different energy levels, there occurs a whole spectrum of energies of the photoelectrons produced in the course of ionization. Measurement of the photoelectron energy spectrum provides additional information about the particle being ionized and lies at the root of one or more photoionization spectroscopy versions-photoelectron spectroscopy of excited states with resonance ionization. Zero electron kinetic energy spectroscopy is based on the detection of low-kinetic-energy electrons and tuning of the laser wavelength XZ

662

GENERAL DISCUSSION

in Fig. 1. This technique allows to obtain a high spectral resolution from the optical channel for the ionizing state (0.1 cm-' and better), as compared with 100 cm-' in standard photoelectron spectroscopy, exploiting the spectral resolution of the electron channel. As concerning the development of ZEKE spectroscopy, I should say that we are using this technique for experiments with atoms routinely since perhaps 1978 (see relevant references in Ref. 1). In the case of multistep ionization of atoms we excite high-lying states below (Rydberg states) and above (autoionization states) the ionization limit. In both cases we exploit pulsed electric time-delayed ionization and (or) detection of ions or electrons. This technique allows us to detect charged particles with low kinetic energy. In the case of photoion detection we can denote this technique as ZIKE (zero ion kinetic energy) spectroscopy. This technique is very powerful for the study of narrow metastable autoionization (AI) states (above the ionization limit). In our experiments we have discovered the very narrow (-0.01 cm-') autoionization state in gadolinium above the ionization limit with nanosecond lifetime IS]. Such A1 states are very important for the practical realization of laser isotope separation of uranium by the resonant multistep ionization technique. Many other AI states of rare-earth elements have been studied by the ZIKE technique (see Ref. 1). The nature of the narrow autoionization long-lived atomic and molecular states may be quite different. (See also Ref. 9.) Finally I am very impressed how efficiently Prof. Schlag has used the ZEKE technique, which was already well known for atoms, in the study of molecular states. 1. V. S. Letokhov, Laser Photoionization Spectroscopy, Academic, Orlando, 1987. 2. R. V. Ambartsumian and V. S. Letokhov, Appl. Opt. 11(2), 354 (1972). 3. V. S. Letokhov, In Tunable Lasers and Their Applications, A. Mooradian, T. Jaeger, and P. Stokseth, Eds., Springer-Verlag.Berlin, 1976, p. 122. 4. V. S. Letokhov, Physics Today 30(5), 23 (1977). 5. S. V. Andreev, V. S. Antonov, I. N. Knyazev, and V. S. Letokhov, Chem. Phys. 45(1), 166 (1977). 6. V. S. Antonov, I. N. Knyazev, V. S. Letokhov, V. M. Matyuk, V. G. Movshev, and V. K. Potapov, Pis'ma ul. Tekhn. Fiz. 3(23), 1287 (1977) (in Russian). 7. V. S. Antonov, I. N. Knyazev, V. S. Letokhov, V. M. Matyuk, V. G. Movshev, and V. K. Potapov, Opt. Lett. 3(2), 37 (1978). 8. G . I. Bekov, V. S. Letokhov, 0. I. Matveyev, and V. I. Mishin, Pis'ma 2.Eksp. Teor. Fiz. 28, 308 (1978). 9. G. I. Bekov, E. P. Vidolova-Angelova, L. N. Ivanov, V. S. Letokhov. and V. I. Mishin, Optics Commun. 35, 194 (1988); Zh. Eksp. Teor. Fiz. 80, 866 (1981).

E. W. Schlag: Prof. Letokhov, the interesting development here is


663

that there are many molecular states above the IP that have anomalous lifetimes of some 100 ps. This is the required signature for this spectroscopy. This was not expected and is the basis of ZEKE since 1984. This gives a new rich spectroscopy of ions. Some atoms have a few states also above the IP but due to a different mechanism. Technically pulsed-field ionization detection has been used for both atoms and molecules, but generally speaking for atoms below the IP, whereas for molecules these are detecting imbedded long-lived states that are ZEKE states in the ionization continuum.

V. S. Letokhov: As concerning the autoionization states of atoms, for example Gd, the lifetime is in the nanosecond range. But according to our calculations some atoms, for example Yb, have very long-lived (microsecond range) lifetimes (see Refs. 1 and 2). You have observed long-lived Rydberg autoionization states for molecules. It seems to me that such states have already been well known in atomic physics. I also have a comment about the historical development of laser resonance ionization spectroscopy with mass spectroscopy (REMPI + MS) leading to ZEKE. Zero electron kinetic energy spectroscopy is just one version of this technique, which is quite known in atomic physics. As concerning the observed long-lived Rydberg autoionization states of molecules I should say that by the same technique we observed long-lived autoionization states for atoms about 20 years ago. Such states can live even on the microsecond time scale [2]. Moreover, other researchers have observed long-lived autoionization Rydberg states in atoms, which have the same origin as in molecules. The long lifetime of these states is explained by weak coupling with the continuum. I . V. S. Letokhov, Laser Photoionization Spectroscopy, Academic, Orlando, 1987. 2. G. I. Bekov, E. P. Vidolova-Angelova, L. N. Ivanov, V. S. Letokhov, and V. I. Mishin, Opt. Commun. 35, 194 (1988); Zh.Eksp. Teor. Fiz. 80, 866 (1981).

E. W. Schlag: The term zero-kinetic-energy electrons was used already by us in Chem. Phys. Lett. 4, 243 (1969). The new mechanism of redistribution of low-l into high-l molecular states is the basis of ZEKE states. T. P.Softley: The aims of ZEKE spectroscopy are conceptually different from the atomic pulsed-field ionization experiments that predate ZEKE. In the latter, the aim is always to observe and study the individual Rydberg states. In ZEKE spectroscopy the aim is to detect small batches of Rydberg states lying below successive vibration-rotation thresholds of the ion without specific interest in the individual Ryd-

664

GENERAL DISCUSSION

berg spectra. The experimental methodology is the same, but the aims differ.

M. Chergui: Let me invoke a rather exotic behavior of Rydberg states that may or may not relate to the lifetime lengthenings discussed in the previous talk. From a nonexpert point of view, my impression is that the lifetime lengthenings here reported are due to external perturbations, and Prof. Levine has shown us the cases of the electric and magnetic fields. But there exists another type of external perturbation, and this is the case of a condensed-phase environment. We have measured the lifetime of the low-n (actually n = 3) Rydberg states of NO in Ne, Ar, Kr, and Xe matrices [l]. We have observed a lifetime lengthening of up to two orders of magnitude (see Fig. l), as compared to the gas phase value, in going from low-polarizability matrices (Ne) to high-polarizability ones (Xe). In order to lift any ambiguities and to stress that the environment specifically affects the Rydberg electron, I would like to add the following points: (a) The effect cannot be due to a modification of the ground-state wave function as we have checked this point against valence transitions, which show no lifetime modification.

1.4 10'

-

1.2 10'

h

I 'A4

2000 0

1

2

3

, A

4

atomic polarizability (lO%d) Figure 1. Lifetime of the n = 3 Rydberg states of NO in Ne, Ar,Kr, and Xe matrices vs atomic polarizability.


665

(b) The effect is not caused by mixing of the Rydberg state with near-resonant quadruplet states of infinite lifetime.

(c) For a given matrix, the Rydberg lifetime changes dramatically with the trapping site of the molecule. 1. F. Vigliotti, G. Zerza,and M. Chergui, in Femtochemistry, Ultrafmt Chemical and Physical Processes in Molecular Systems, M.Chergui, Ed., World Scientific, Singapore, 1996.

njl

-3

’

nj2

.,

- 4 - 3 - 2 - 1

nj4

0

nj8

1

log 1o(dmo)

nil6

2

732

3

11164

4

I 5

Figure 1. Stability borders for the system Hamiltonian at fixed w = 0.001 and m = 0.4. The lower curve is the usual chaos border € 0 = 1/50Wk’3 = ( m ~ / w ~ ) ’ / 4 / 5 0 ( ~ m 3 )For L/L2. small 00 this border approaches the usual static border eo = 0.13. The “magic mountain” of stability is delimited from below by the stabilization border € 0 1200 mg and from above by the destabilization border € 0 = (l6L/r)(wo/rn0)~ with L = In[ 2eo/(er)/(womo)].The dashed lines co = (em/w)(wo/mg) are drawn at constant e: ( a ) e = 0.0025; ( b ) e = 0.05;( c ) e = 1. The border (3) below which the Kepler map description is valid is given, in the present case, with fixed m and w , by the line €0 = O.Z(oo/mo) (not drawn in the figure). The present picture is drawn at fixed w and m. If instead we keep no fixed, then the system will always be stable in the region to the right of the dotted vertical line given by worn: = 3.

$-1

666

GENERAL DISCUSSION

G. Casati: I have the following general comment. When a Rydberg atom, prepared in some initial state, interacts for a certain time with a radiation field of a given intensity and frequency, one would naively expect the ionization probability to be a monotonically increasing function of the field intensity. This is however not the case. Instead, it may happen that on increasing the field strength above a critical value the atom becomes increasingly stable against the field-induced ionization (see Fig. 1). This phenomenon is a property of the classical motion and qualitatively can be understood, as mentioned by Levine in his talk, by the fact that by increasing the field strength, the electron is kept far from the nucleus, and this prevents ionization.

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS USING ZEKE SPECTROSCOPY T.P. SOmLEY,* S. R. MACKENZIE, F. MERKT, and D. ROLLAND Physical and Theoretical Chemistry Laboratory Oxford, United Kingdom

CONTENTS I. Introduction 11. State-Selected Ion-Molecule Reactions A. Principles of State Selection B. Experimental 111. Examples of Preparation of State-Selected Ions A. Hydrogen, H; B. Carbon monoxide, CO+ C. Nitrogen, N$ D. Nitric oxide, NO+ IV. Studies of Ion-Molecule Reactions A. H; + H2 H; + H B. Collision Energy Resolution C. Transmission Effects D. Rydberg State Perturbation by Collision V. Rydberg State Lifetimes VI. Experimental Measurements of Rydberg Lifetimes VII. MQDT Calculations of Spectra of Autoionizing Rydberg States A. Method Employed in the Calculations B. Calculations for Argon C. Calculations for Nitrogen VIII. Conclusions References

-

*Report presented by I: l? Sofley Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on rhe Femtosecond lime Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard,

Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

667

668

T. P. SOFTLEY, S. R, MACKENZIE, F.MERKT, AND D. ROLLAND

1. INTRODUCTION

The delayed pulsed-field ionization (PFI) of Rydberg states now lies at the heart of an expanding range of experiments in gas-phase spectroscopy and dynamics. Foremost are the techniques of zero-kinetic-energy(ZEKE) photoelectron spectroscopy [l-31 and mass-analyzed threshold ionization (MATI) spectroscopy (41. Over 100 molecules have been studied using these techniques ranging from H2 to large aromatics, resulting in a wealth of spectroscopic data on molecular cations and an improved understanding of photoionization dynamics [l] and Rydberg channel couplings [2]. The ZEKE spectra of anions [5] and clusters [6] have also been recorded, and ZEKE or MATI has been used as a state-specific probe in studies of intramolecular dynamics in the femtosecond (see Ref. 7 and Baumert et al., “Coherent Control with Femtosecond Laser Pulses,” this volume) and picosecond [8] time domains. In a separate development Rydberg tagging time-of-flight spectroscopy has proved to be a valuable means for determining product velocity distributions in photofragmentation [9] and bimolecular collisions [lo]. Recent research has concentrated both on broadening the range of applications and on improving the understanding of the physical mechanisms underlying these PFI techniques. In the latter category, there has been considerable interest in explaining the unexpected long lifetimes of high Rydberg states of a wide range of molecules [ll-181. The study of the dynamics of molecular Rydberg states has now become a topic of importance in its own right, with particular interest in the significance of intramolecular couplings and external electric and magnetic field effects. This work is a natural follow-up to two decades of measurements of properties of atomic Rydberg states [191. In this chapter, results of importance to both categories are reported. In the first part a new application of PFI to the study of state-selected ion-molecule reactions is described, and in the second part some novel experiments and calculations relating to Rydberg state dynamics are presented. The goal of the ion-molecule reaction studies is to select ions in unique vibration-rotation quantum states and to study the effects of selectivity on reaction cross sections. The additional possibility of controlling the spatial alignment of the ionic angular momenta using the polarization properties of lasers is also of interest and is under investigation. Rotational effects on reaction cross sections act as a probe of anisotropy in the reaction potential-energy surface, which might arise from long-range forces (see Ref. 20 and Troe, “Recent Advances in Statistical Adiabatic Channel Calculations of State-SpecificDissociation Dynamics,” this volume) or shorter range valence interactions. The effects of ionic rotation have received very little attention either theoretically or experimentally.

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS 669

11. STATE-SELECTED ION-MOLECULE REACTIONS A number of techniques have been used previously for the study of stateselected ion-molecule reactions. In particular, the use of resonance-enhanced multiphoton ionization (REMPI) [21] and threshold photoelectron photoion coincidence (TPEPICO) [22] has allowed the detailed study of effects of vibrational state selection of ions on reaction cross sections. Neither of these methods, however, are intrinsically capable of complete selection of the rotutional states of the molecular ions. The TPEPKO technique or related methods do not have sufficient electron energy resolution to achieve this, while REMPI methods are dependent on the selection rules for angular momentum transfer when a well-selected intermediate rotational state is ionized; in the most favorable cases only a partial selection of a few ionic rotational states is achieved [23]. There can also be problems in REMPI state-selective experiments with vibrational contamination, because the vibrational selectivity is dependent on a combination of energetic restrictions and Franck-Condon factors. In this chapter, the design and instigation of a new experimental method aimed at achieving an improved level of ionic state selectivity is described, which allows, in small molecules, the selection of individual rotational states. The technique, developed from ZEKE spectroscopy, makes use of the existence of a pseudocontinuum of high-n Rydberg stakes of the neutral molecule located energetically below each ionic threshold (corresponding to each vibration-rotation state of the ion), which in the range of principal quantum number n 2 100 show metastability with respect to decay processes such as fluorescence, predissociation, and autoionization. Experiments have shown that even the Rydberg states of complex molecules with many internal degrees of freedom, or Rydberg molecules with electronically excited core states, can live for tens of microseconds [17,24-263 (see also Section V). In order to exhibit such stability, the Rydberg electron must have virtually no interaction with the ionic core of the molecule, which itself may be considered to exist in a well-defined internal quantum state. The Rydberg electron is easily ionized by the application of a small pulsed electric field. The ion core is unaffected in this process (but see next paragraph), and therefore the resultant ion is left in the same well-defined state. Thus, the P H of Rydberg states provides the basis for a method of ionic state selection. However, the formation of “prompt ions” by direct photoionization at the same time as Rydberg state excitation is generally unavoidable, and therefore such nonselected ions must be separated from those arising from field ionization. The possibility of making such a separation was first demonstrated by Zhu and Johnson [4] in the MAT1 experiment. An important consideration for the purposes of ionic state selection is the

670

T. P. SOlTLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

possibility that the ionizing pulsed electric field actually induces a change of rotational state of the ion core by enhancing the interchannel couplings 127, 281. In ZEKE or MATI spectroscopy this question is disregarded because one only collects all the ions or electrons produced by delayed PFI in a given energy range, and the actual states of the ions formed do not matter. Nevertheless, a Rydberg state (v+N+nl)could be coupled by the pulsed field to the (u+N+- x d ’ ) continuum in a forced rotational autoionization, leaving the ion in rotational state N + - x instead of the selected N+.Two arguments offer strong evidence that such processes do not occur. First, there is no reason why forced rotational autoionization, if it is happening, should be confined to occur within the field ionization range; it could also occur below this range and would give rise to long red tails on every ZEKE or MAT1 peak. The general observance of well-defined field ionization behavior in ZEKE spectroscopy points to the nonoccurrence of forced autoionization. Second, in order to survive the several-microseconddelay before field ionization, the Rydberg electron must almost certainly have found its way into a high-m,, high-Z state (see Section V) in which the electron is noninteracting with the core. In this state the forced autoionization, which requires core penetration, cannot occur because the pulsed electric field will preserve the high-mr value of the Rydberg electron and hence the nonpenetrating character.

B. Experimental The state-selected ion-molecule reaction experiment is illustrated schematically in Fig. 1 and has been described in more detail elsewhere [29].

I EXCLTATION AND DISCRIMINATfoN 2 FIELD 1ONlZATlON AND ACCELERATION

AND PRODUCT DETECTION

STATESELECTED /////#

BEAM

ALS

Figure 1. Schematic diagram of the state-selected ion-molecule experiment.

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS

67 1

Molecules are excited in a doubly skimmed pulsed supersonic beam using a two-color resonance-enhanced multiphoton excitation process, to Rydberg states with the desired ionic core quantum state. Prompt ions are deflected out of the beam by applying a small perpendicular field (typically Fd = 0.1 V/cm) or, alternatively, are retarded using a field parallel to the beam axis. The surviving Rydberg molecules continue unaffected by the field into the extraction region (with the exception of the very highest states, which are field ionized) and then, after a suitable time delay, they are field ionized by a pulsed field (-4 V/cm) to produce state-selected ions. The metastability of Rydberg states is crucial for the selectivity because it is generally necessary to allow a period of a few microseconds between excitation and field ionization to give adequate separation between Rydberg molecules and prompt ions. The extraction pulse serves a second important purpose in that it accelerates (or decelerates if the polarity is reversed) the ions in the neutral beam direction, and the degree of acceleration is determined by the duration and amplitude of the pulse. This causes a slippage in the beam between the ions and the neutrals and hence a much enhanced ion-neutral collision rate. Reactive collisions can therefore occur over a length, which in the present setup is variable in the range 5-15 cm. A quadrupole mass filter with resolution below 1 amu is placed at the end of this reaction zone and is tuned to transmit either the parent ions or the product ions, which are detected by a multichannel plate (MCP) detector. We choose to vary the collision energy by changing the duration (20 ns-20 ps) rather than the amplitude of the extraction pulse, because changing the duration does not affect the field ionization probability (and hence the spectroscopic discrimination between Rydberg states belonging to different series). In preliminary experiments using the new apparatus we have produced rovibrationally state-selected Ht, CO+, Nt, NH;, and NO+ and have begun Hi + H to study simple bimolecular reactions, for example, H$ + H2 at low collision energy ( 4 . 5 eV) [29]. For studying the H;/H2 reaction, where the molecular ions react with the species that itself is the precursor neutral molecule, the beam consists of neat H2 only, backing pressure of 2 bars. However, for other reactions such as CO+ + H2, the ion precursor and reactant gases are coexpanded from the same nozzle. The translational temperature has been measured (see Section 1V.B) to be of order 2 K, and the rotational temperatures, determined from REMPI spectra, are typically in the range 5-20 K (except H2). The laser system consists of two dye lasers (Spectra Physics PDL 3) pumped by a single Nd-YAG laser (Quanta Ray GCR 270). Dye Laser 2, pumped at 355 nm, operates in these experiments in the range 370-500 nm (5-15 mllpulse), whereas Dye Laser 1, pumped at 532 nm, operates at 660-600 nm and is frequency tripled using KD*P and BBO crystals

-

672

T. P. SOFTLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

in an INRAD autotracking system to produce light at 220-200 nm (0.3-1 mJ/pulse). The two beams, which are temporally overlapped, are combined on a dichroic mirror and are then focused into the vacuum chamber using a 15-cm lens to intersect the molecular beam perpendicularly. The beam waste at the focal point is approximately 0.4 mm.

III. EXAMPLES OF PREPARATION OF STATE-SELECTEDIONS In all cases a two-color multiphoton process is used to excite the molecules to the high-n Rydberg states. In the remainder of this text we use a primed notation to refer to the transitions from the ground state to the intermediate state (e.g., Q‘ branch, 0’ branch, etc.), whereas an unprimed notation is used for transitions from the intermediate state J’ to the Rydberg states with core rotation quantum number N + ;for example, S(2) implies the transition J‘ = 2 - + N + = 4, whereas S’(2) implies J” = 2 J‘ = 4.

-.

A. Hydrogen, H i

The hydrogen molecular ion, H,+ is prepared by 2 + 1’ multiphoton excitation via the E, F ’ C i , u’ = 0 state, with the frequency tripled Laser 1 tuned to around 202 nm (0.5 &/pulse) and Laser 2 to ~ 3 9 6nm (5-10 mJ/pulse). The transitions Q’(O), Q’(l),Q‘(2), and Q’(3) are well resolved in the transition to the intermediate state (rotational temperature =180 K), and Q-type transitions to the high Rydberg pseudocontinuum are used to produce ions in u+ = 0, Nt= 0, 1,2,3. Approximately 1000 state-selected ions are detected per pulse after transmission through the mass filter. The S-type transitions, which in principle could allow population of ionic states up to N + = 5, are too weak to be observed. Using a DC discrimination field of 0.1 V/cm and a pulsed extraction field of 3.7 V/cm, a resolution of 5 cm-’ is obtained in the PFI peaks (however, see Section IV.D and Fig. 9). Full details of the state-selective preparation of Hi are given elsewhere [29]. An alternative procedure demonstrated earlier [30] was to use a tunable extreme ultraviolet source, based on four-wave mixing of pulsed ultraviolet laser beams, to excite directly from the ground state to the high Rydberg states. On balance it appears that the multistep excitation is a more efficient use of the dye laser energy and is less demanding experimentally. Moreover, for all ions except H;, the enhanced selectivity gained by exciting via an intermediate well-defined rotational level is a major advantage.

B. Carbon Monoxide, CO+ The CO+ ions are prepared by 2 + 1’ excitation via the E ’n,u’ = 0 state; Laser 1 is frequency tripled to produce light of 215 nm (0.5 ml/pulse) and


.. .

673

. . .. .. . . . . ... .. . .*

' p , , ,

z.-

M

e v)

a

-

.

CI

S'(3) &

Figure 2. The 2 + 1 REMPI spectrum of the E 'H-X 'CC transition of CO (0-0 vibrational band).

Laser 2 is at 4 9 6 nrn (5 ml). The 2 + 1 REMPI spectrum of the E 'II- X 'C+transition is shown in Fig. 2, from which a rotational temperature of 6 K may be deduced. From the point of view of optimizing selectivity, this transition, which shows well-resolved P,R,and S' branches, is most likely to be a preferable starting point compared to the B-X or C-X bands of this molecule; the two-photon transition intensity for the latter bands would be completely concentrated in the Q'-branch lines, which would not be well resolved. Figure 3a shows the PFI spectrum obtained by selecting the intermediate state J' = 2 with Laser 1 tuned to the S'(0) transition. Six peaks are observed corresponding to N + = 0 to 5. The DC/pulsed fields used were 0.2 and 3

113010.0

113030.0

113050.0

~ o t atmn l value / cm-'

113070.0

113090.0

Figure 3. Pulsed-field ionization spectra from the E 'II (u' = 0) state of CO with the initial level J' = 2 pumped via (a)S'(0) and (b) R'(1).

674


V/cm, respectively, giving a resolution of 4cm-’ , sufficient to observe the individual rotational states. The peaks correspond to 10-50 ions per pulse. Figure 3b shows the PFI spectrum probed via the same intermediate state J‘ = 2 but with the R’(1) transition as the initial excitation process. The two spectra show quite different rotational line intensities; the Q-type transition to N + = 2 is the strongest line when the R’(1) transition is pumped (Fig. 3b), but in the S’(0) spectrum (Fig. 3a) it is weaker than the P- and R-type transitions. Also the T(2) peak is observed in the spectrum via S’(0) but not via R’(1). The observed differences are explained in part by parity selection rules. The S’(0) transition populates only the e component (II+) of the J’ = 2 state According whereas the R‘( 1) transition populates only thef component (W). to Xie and Zare [31], the selection rule applying is N + - J’ + p+ + I = odd where 1 is the angular momentum of the departing electron and the Kronig symmetry indices are p+ = 0 for the ionic 2C+ state and p’ = 1, 0 for the intermediate II- and II+ states, respectively. For the present case this implies N + - J ’ = f l , + - 3 ,... forleven,n+(e) or N + - J’ = 0, f 2 , k4,. . . for 1 even, n-(f) or

forZodd,W(f) for 1 odd, II+(e)

Fujii et al. observed a strong propensity to the even-l selection rule [32] when using laser-induced fluorescence to determine the ionic state distribution following direct ionization from the same intermediate state, observing no Q-type transitions following S‘-branch excitation. The even4 propensity is to be expected because the electron is ionized from a nearly pure p orbital and hence I should equal 0 or 2. In the PF’I spectrum via S‘(O), however (Fig. 3a), the Q(2) line gains some intensity by rotational channel coupling with Rydberg states converging to the N ++ 1, limit; that is, the Q-type transition is gaining intensity from an R-type transition. This is shown more clearly in the Q(3) line following S’(1) excitation in Fig. 4, where the apparently noisy structure observed on the Q-type transition is resolved into individual Rydberg states converging to the N + = 4 limit. The intensity reduces to zero between each Rydberg state, suggesting that there is no intrinsic intensity for direct excitation to the N + = 3 pseudocontinuum from the J’ = 3(e) level: All the intensity is borrowed. Nevertheless, the position and overall width of this feature are determined by the field ionization of the pseudocontinuum. The coupling that gives rise to the intensity borrowing must occur between states of the same overall parity. Therefore, the even-l Rydberg states populated via the R’-type transition must couple to Rydberg states with odd-2 character associated with the N + = J’ limit, mediated by the core dipole. The PFI spectrum of CO illustrates nicely the interplay between pure photoionization dynamics and Rydberg channel couplings, as is often observed

il


1

1 13040.0

nd4

68

113042.0

76

70

113044.0

113046.0

Total term value I cm-'

78

80

113048.0

113050.0

Figure 4. The Q(3)transition in the PFI spectrum of CO following S'(1) excitation showing Rydberg states converging to N + = 4. The total term value is given with respect to CO X 'E+,v" = 0, J" = 0.

in ZEKE spectra [2]. This interplay is important in controlling our ability to produce specifically chosen states effectively. In many cases it is possible to produce a wider range of ionic quantum states using PFI than in direct ionization. Vibrational intensities do not necessarily follow Franck-Condon factors, although most of the known examples of non-Franck-Condon behavior occur in small molecules.

C. Nitrogen, NZ

The lowest accessible two-photon transition of N2 is the a" 'E; - X 'c;'g system, for which the 0-0 vibrational band is at =202 nm. The intensity is concentrated in the strong Q' branch, with S' and 0' branches weaker by two orders of magnitude. The S' branch transitions in the REMPI spectrum allow an estimate of the rotational temperature of 7 K. implying a significant population of J" = 0 , . . .,4. The rotational lines of the Q' branch are not resolved in the present study, and in order to obtain rotational selectivity, it is preferable to use the well-resolved S' and 0' branch transitions in the two-photon step under high-power conditions [33]. Figure 5 shows the PFI spectrum from J' = 0,2,3,4 leading to population of the N; ions in N + = 0,. .., 6 . Again the spectroscopic resolution is =4 cm-' . Approximately 100 selected ions per laser pulse are produced in this way, compared to the 600 ions per pulse at lower laser power, but with only partial state selection, when exciting via the Q/Q'-type sequence. The 0 branch line profiles

676

T. P. SOFTLEY, s. R. MACKENZIE, F. MERKT, AND D. ROLLAND

N+=

0' ! 'z

m

I

3

I

4

I

5

I

6

J'=4

125640.0

125660.0

125680.0

125700.0

125720.0

125740.0

125760.0

TOUIterm value / cm-' Figure 5. The P H spectra from J' = 0,2,3,4 of the a" 'Z; state of N2 leading to population of the Nt ions in N + = 0,. ..,6. The total term values are given relative to N2 X 'C;, V" = 0,J" = 0.

in the PFI spectrum show a superimposed structure associated with the coupling of Rydberg states = n = 60,. . .,80, converging on a higher rotational state of the ion, with the field-ionized pseudocontinuum in question. This gives an overall factor-of-2 enhancement to the intensities of the 0 branch lines compared to the S branch. The Q branch transitions are in any case the strongest.

D. Nitric Oxide, NO+ The optimum method for production of NO+ is to use the 1 + 1' excitation via theA 2C+ state (226 + 337 nm). The ZEKE spectrum is well documented [34] and the resolution achieved to date in our experiments is sufficient to obtain good state selectivity. We have also investigated the 1 + 1' ionization via the B *IIstate, primarily because of the convenience of wavelengths used (202 + 387 nm,very close to those used for N2 and H2).The B 211 state of NO has a potential minimum at larger internucleardistance compared to the ground state of either NO or NO', and therefore good Franck-Condon overlap can only be obtained via 'v > 0. The most convenient level to access in this work was u' = 4, and the 1 + 1REMPI spectrum is shown in Fig. 6. The PFI spectrumobtained via the P;,(5.5) transition (J' = 4.5) is shown in Fig. 7.The spectrum is weak and corresponds to the production of only a few ions per pulse but shows that a very wide range of rotational states can be populated, apparently rather more than in the excitation from the A 2C+ state [34].

-


-m . .-8 :

R

:

6.5

-4;(

v)

I + -

t- i

5!

9:. 5;.

-

,

i4.5

55

-

5

3.5 2 5 15

-

-

zs 1.50.5

35

1

I

45

3.5

I

-

L

d

4

Figure 6. The 1 + 1 REMPI spectrum of the B 2111p - X 2111/2 (4-0) transition of NO.

The reason for the weakness of the PFI spectrum is of interest in itself, in that the one-photon ionization from the B 'II state to the ground state of the ion is formally forbidden as it involves a nominal two-electron change:

-

N 0 + ( 3 u ) * ( l ~ ) ~ ;'E+ X

N 0 ( 3 0 ) ~ ( 1 ~ ) ~ ( 2Ba2) n~ ;

It is therefore surprising that this process can be observed at all, and the intensity can be ascribed to a configuration mixing of the B 'II state with ]. the rotational intenthe C 211 state [configuration ( 3 ~ )1~~() ~ ( 3 p ) 'Indeed, sity distribution is similar to that shown in the ZEKE spectrum via the C state

-

N+=b

'1 5

I

4

I

I

6

5

I

7

-

d

el

.+ % v1

*

E74700.0

-

74750.0

74800.0

TOM~term value / cm-I

Figure 7. The PFl spectrum of NO from the J' = 4.5 level of the B leading to population of the NO+ ions in N + = 0 , . .. ,7.

74850.0

(d= 4) state

678

T. P. SOFIZEY, S. R. MACKENZIE, E MERKT, AND D. ROLLAND

[35]. The configuration mixing is much stronger above u' = 7 of the B state

[36], and one would expect a much stronger spectrum in excitation from higher levels. The use of the u' = 4 level of the €3 state as a suitable intermediate for ion preparation is not to be recommended, although it is potentially a valuable starting point for the one-electron excitation to Rydberg states converging to the u 3C+ state of the ion [configuration ( 3 0 ) * ( 1 ~ ) ~ ( 2 ~ ) 'which ], would however require the second laser frequency at =78,000 cm-' .

-

IV. STUDIES OF ION-MOLECULE REACTIONS A. HZ

+ H2

HS

+H

The Hi + H2 reaction is studied using pure normal hydrogen in the molecular beam with the Hi prepared as described in Section IILA. The 1000-shot averaged Hi and H; signals are recorded by tuning the quadrupole mass fiiter to masses 2.0 and 3.0, respectively. Although the Hi parent ion peak is -100 times greater in intensity, there is no remnant signal at mass 3.0 due to H;. The H; signal corresponds to a few ions per pulse. Figure 8 shows the variation of the measured H;-H,+ signal ratio as a function of center-of-mass collision energy E,, for Hi selected in u+ = 0, N + = 1. The data are uncorrected for various transmission losses and dynamical factors (see below), but the figure illustrates the scope and accuracy of the measurements. After applying instrumental corrections, the collision energy dependence of the signal follows that predicted using the Langevin model. We have measured

I

0.0

100.0 200.0 Center of mass collision energy / meV

300.0

Figure 8. Variation of the measured H;-H$ signal ratio as a function of center-of-mass collision energy for Hi selected in u+ = 0, N + = 1.


similar profiles for other rotational states N + = 0,. . .3, but any differences from the curve shown in Fig. 8 were within the error bars of the measurements. Hence, at the present time we are unable to make any authoritative statements about the magnitude of the rotational effects on the reaction cross section, except to say that the variation between N + = 0 and N + = 3 does not appear to be any greater than the value of 10% suggested by Chupka et al. [37]. Unfortunately the largest effects of rotation would be expected at the lowest collision energies, but this is currently where the error bars are largest. Nevertheless, the results shown demonstrate the ability of this apparatus to study reactions, not just state selection, and work is in progress to improve the sensitivity at low collision energy. In evaluating the viability of a new experimental technique for studying ion-molecule reactions, a number of factors must be considered. Ultimately our aims are to measure relative cross sections for reactions as a function of both the internal energy of the ion and the collision energy. It is important that the collision energy can be varied down to =10 meV where the rotational energy may be comparable with the translational energy.

B. Collision Energy Resolution The velocity distribution of the H2 reactants in the direction parallel to the molecular beam has been measured by exciting the molecules to stable Rydberg states around n = 80 (v+ = 0, N + = 0) and then allowing them to continue from the excitation region all the way to the MCP detector, a distance of about 20 cm, under field-free conditions. The width of the time-offlight distribution shows that the translational temperature is =2 K.The ions produced by field ionization in the extraction region are all given an equal acceleration by the pulsed field, assuming the field is perfectly homogeneous, and therefore the laboratory velocity spread for the ions will be identical to that of the neutral molecules. If these velocity-spreading effects dominate the collision energy distribution, then we expect a f 3 meV collision energy spread at E,, = 10 meV and +I0 meV spread at E,, = 100 meV for the H2 + Hl reaction. The angular divergence of the molecular beam is difficult to determine but is likely to be a less significant factor. Ideally we would like all reactions to occur after the field ionization pulse is switched off and prior to the reactants entering the quadrupok, that is, at a nominal constant collision energy. It is important to ensure therefore that the ionization pulse is as short as possible to minimize collisions during acceleration. Given that the acceleration is controlled by both the magnitude and duration of the pulse, there is some scope to test whether this problem is significant and to limit its effect. For a pulse of length 100 ns and amplitude 3.7 V/cm, the probability of collision is approximately 2% of that along the remainder of the flight. Concerning the collisions within the quadruple

680

T. P. SOFIZEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

filter, it is unlikely that the resultant product ions will have the correct trajectory to be transmitted to the detector, and moreover, the parent HZ ions will penetrate the quadrupole very little when it is tuned to mass 3. In any case the lower beam density in this region further downstream reduces the probability of collision. It may therefore be concluded that the collision energy is well controlled for >95% of the collisions.

C. Transmission Effects

+ H2 reaction is that it is highly exothermic and the product H; ions are light. The average center-of-mass frame product ion velocity is of order 3500 ms-I ,compared to the center-of-mass velocity of, for example, 4850 ms- at a collision energy of 0.1 eV. This restricts the detection angle in the center-of-mass frame (i.e., the products scattered in the forward direction are preferentially detected) with the limits depending on exactly where the collision takes place and on the center-of-mass velocity. The HS + H2 reaction shows a strong backward-forward peaking, and hence the collection efficiency is still high; nevertheless, the integral cross section cannot be determined in this case without detailed knowledge of the product angular distribution. This transmission factor will be less of a problem for reactions with heavier product ions and lower energy release. An improvement to the apparatus might be to carry out the PFI within a guiding octupole. The transmission probability is also affected by the angular divergence of the neutral beam and by divergence of the ions due to space charge effects. The excitation of the neutral molecules takes place at the focal points of two pulsed lasers, and therefore a small bunch of Rydberg states are formed in a volume of approximately 0.1 mm3. There is some natural spreading of the group of high Rydberg molecules due to the molecular beam velocity distribution, as the neutrals drift into the extraction region, where we estimate that the volume occupied by Rydberg states is approximately 1 mm3.Great care has to be taken to ensure that we do not produce too many ions, so as to avoid space charge effects. One method to reduce the ion density without a corresponding reduction in the total number of ions is to use a cylindrical focusing arrangement producing the initial bunch of Rydberg molecules in a line focus 4 x 0.1 mm. Such an arrangement allows one to take full advantage of the laser energy available, especially in the case where the two-photon transition is saturated, while avoiding space charge effects.

A significantproblem with the Hi

D. Rydberg State Perturbation by Collision The presence of a large number of ions in the excitation volume, which are mainly caused by one-color direct ionization at the same time as Rydberg state excitation, can lead to additional problems as illustrated in Fig. 9. This


124000.0

124100.0

124200.0

124300.0

68 1

124400.0

TO^ term value / cm-'

Figure 9. The Q(1) rotational line in the PFI spectrum of H2 recorded with increasingly tight laser focus from (a) to ( d ) .

shows the Q(1) rotational line in the H2 PFI spectrum using a field ionization pulse of just 10 V/cm with Figs. 9a-d resulting from increasingly tight focusing of Laser 1 and Laser 2. The spectra in Figs. 9b-d show that ions are produced by field ionization, not only when the neutrals are excited to Rydberg states within the calculated field ionization range (n > 90), but also when they are excited initially to levels well below this range, even to n = 20. Our interpretation is that the presence of high densities of ions leads to n-changing collisions, or charge exchange, in which the low-n states are upconverted to high-n states that are then field ionized. The observed spectra reflect the initial excitation rather than the final states ionized. In practice, it is possible to suppress this effect by lowering the laser power or defocusing the beams, as shown in Fig. 9a. Again the use of a cylindrical lens is helpful in this respect.

V. RYDBERG STATE LIFETIMES A factor of key importance in these studies is the metastability of the longlived Rydberg states, that are field ionized to produce state-selected ions. This metastability, in which the states exist for tens of microseconds, has been the subject of much discussionrecently [ 11-18] and is not expected a priori. A simple extrapolation of the known decay behavior of low-n Rydberg states using the conventional n3 scaling law leads to predictions of lifetimes for the highn states that are too short by several orders of magnitude. For example, in N:! the u+ = 0, N + = 2 Rydberg states with n 5 200 are predicted to decay by rotational autoionizationin less than 20 ns, whereas the experiments show that such Rydberg molecules survive for many microseconds (see Section VI).

682


Chupka [Ill, in a landmark paper, pointed out the importance of electric fields in enhancing the stability of the Rydberg molecules. The transition intensity for the initial excitation process canies the molecules into low4 states, which have core-penetrating orbits. These states are unstable with respect to decay processes, because of the energy exchange when the electron is close to the core. However, the effect of the electric field (stray fields of order 20-50 mV/cm are almost unavoidable in most ZEKE experiments) is to mix the low4 states with the degenerate sets of nonpenetrating high-I states (hydrogenic manifolds), and these latter states have no Rydberg electron+ore interaction and hence are long lived. The sharing out of lifetime and transition intensity results in a set of populated states whose lifetime is greater than the zero-order low4 states by a factor of approximately n. The existence of inhomogeneous electric fields, which could for example be created by the presence of high densities of ions near the Rydberg molecules, leads to rnr mixing (or more strictly M J mixing in most systems) and this would lead to further lifetime dilution, by a factor approaching n. Thus in the limit of complete Z and ml mixing, the lifetimes would scale as n5 rather than n3. Recent experiments have demonstrated that the presence of background ions can lead to enhancement of ZEKE signals [7], presumably through lifetime lengthening effects, although other experiments appear to demonstrate the lack of such effects [38]. Levine and co-workers, in their theoretical studies of high Rydberg lifetimes [14], have concentrated more on the importance of intramolecular processes, utilizing a classical trajectory approach. They point to the importance of rotational-electronic energy exchange between the ion core and the Rydberg electron. In this picture the Rydberg electron drifts from one orbital angular momentum state to another, exchanging angular momentum and energy with the rotating core each time it comes into the core region. It is probable that such effects are more important in larger molecules when the rotational energy spacing and Rydberg energy spacing become comparable in magnitude, with the additional requirement for strong coupling (and hence a highly anisotropic ion core). Recently they have shown that the presence of an external field has important effects in modifying this “diffusive motion,” leading to less frequent collisions with the core [14]. They have also adopted a quantum mechanical approach based on diagonalization of a complex Hamiltonian (similar to the method of Bixon and Jortner [13], as described in Section VI1.B) to demonstrate a “trapping effect,” which occurs when the density of coupled bound states exceeds the density of continuum states (see Ref. 18 and Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume). Such an effect is more likely to be prominent in molecules than atoms, because of the large number


683

of interacting Rydberg series. It has been shown that this situation leads to a bifurcation of lifetimes, particularly when the mean level spacing is less than the decay width. The ideas of Chupka and those of Levine are not incompatible. A general framework for discussing Rydberg state lifetimes must embrace both intramolecularand external effects. Two limiting cases may be envisaged for any atom or molecule; the low-external-field regime, in which intramolecular effects dominate the dynamics, and the high-field limit, where the nature of the molecule becomes almost irrelevant and the external effects dominate. In the intermediate regime the intramolecular and external-field effects may show a complex interaction, with the field modifying and perhaps facilitating the intramolecular couplings. Ultimately, as the field is increased, intramolecular couplings can become effectively switched off (see Section VII.C), and lifetimes are determined by the dilution of the zero-field bound-continuum couplings. In this sense the general framework we refer to is analogous to the well-known transition in atoms from the Zeeman effect at low magnetic field to the Paschen-Back effect at high magnetic field. The matter that is not clearly established is whether the transition to the high-field regime occurs under rather similar external conditions for all molecules or whether it is highly dependent on the rotational constant, ion-core anisotropy, or possibly the density of vibrational states. In the context of designing experiments such as the one described in Sections 11-IV, three fundamental questions should be addressed: Are the weak homogeneous fields present in most experiments sufficient to cause the observed lifetime lengthening, and if not, what is the relative importance of homogenous and inhomogeneous field effects? Is there a universal stabilization mechanism valid for the complete range of systems from simple atoms to complex molecules or are intramolecular effects more dominant in some cases than others? Is the stabilization highly variable from one experimental setup to another, and what can be done to control it? In the following sections some new experimental and theoretical results are presented that shed further light upon the answers to these questions.

VI. EXPERIMENTAL MEASUREMENTS OF RYDBERG LIFETIMES Three basic methods for determining Rydberg lifetimes may be envisaged. 1. High-resolution measurement of linewidths of transitions to individual

eigenstates.

684

T. P. SOFILEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

2. Measurement of the survival probabiiity of Rydberg molecules as a function of delay time for pulsed-field ionization. 3. Direct observation of the appearance of decay products in real time.

It should be noted with respect to method 1 that in most Rydberg excitation experiments to date pu1sed lasers have been used even to obtain the highest resolution spectra, and this naturally precludes the measurement of lifetimes longer than -10 ns. We have carried out careful measurements, in the apparatus described in Section 11, of lifetimes as a function of principal quantum number for Rydberg states of nitrogen converging to the N + = 2 , 3 ionic thresholds [24] using methods 2 and 3. These states can in principle undergo rotational autoionization to the continua associated with N + = 0, 1, respectively. The molecules are excited from the J’ = 2, 3 intermediate levels of the a 1qstate (see 3 parallel to the Section 1II.C) in the presence of a small DC field ~ 0 . V/cm, molecular beam direction, which is used to retard the prompt ions relative to the neutral Rydberg states. A pulsed field is applied after a time delay of typically 7 ps to ionize the Rydberg molecules, and all ions are accelerated by this pulse to the detector. The ion time-of-flight profile shows two peaks in general, as shown in Fig. 10, the early peak due to field-ionized Rydbergs and the later peak due to the retarded prompt ions formed immediately on

0.10

.g

0.09

A

0.08

g om

h

--2

3

a!

0.06

0.05

.gJ 0.04

.g .$

0.03

0

d

0.02

4

0.01 0.00 -0.01 27.0

27.5

28.0

Ion timc of flight / ps

28.5

29.0

Figure 10. The Nt ion time-of-flight profile following Rydberg state excitation around n = 150 (N+= 2) in the presence of a 0.3-V/cm retarding field. The dotted line shows the one-color direct ionization signal, which is subtracted from the total signal in the analysis of the results.


excitation. A signal also appears at times intermediate between the field ionization and prompt ionization peaks, that can be ascribed to ions formed by autoionization after a short time delay with respect to excitation. An exponential decay curve due to autoionization is obtained as a shoulder on the prompt ionization peak, and as shown in Fig. 10, the time of flight can be converted into a real time measurement of the time of ionization (method 3). A series of measurements of the time-of-flight profiles at different excitation energies has been carried out [24], from which the ionization dynamics as a function of n can be extracted. Experiments of type 2 have also been carried out in which the pulse delay is varied over the range 1-26 ps and the remnant population detected by PFI. For the np Rydberg states converging to the N + = 2 limit in N2 there are three J levels ( J = 3,2, l), arising from the coupling of N+ and I , and these are optically prepared in approximately equal proportions. Only the J = 1 set can couple to the N + = Onp(J = 1) continuum in the absence of an electric field. For n = 80,. . . ,100 it is found experimentally that roughly one-third of the states decay by fast rotational autoionization with 7 c 500 ns (presumably the J = 1 set), while two-thirds are not observed either by delayed-field ionization or by fast autoionization at all. Note that the total excitation probability is assumed to be constant across the threshold region and is determined from the magnitude of the direct ionization signal above the N + = 2 threshold. For the nonautoionizing J = 2, 3 states there must be some other decay process, presumed to be predissociation, competing on a submicrosecond time scale. Hence virtually none of the populated n = 80,. . . ,100 states survive the 7-ps delay before field ionization. The 0.3V/cm field is apparently having very little stabilizing effect in this range of n. For n = 100,. ..,140 the lifetimes with respect to autoionization gradually increase to = 2ps at n = 140 (a longer tail is observed on the prompt-ion timeof-flight peak) while the fraction of molecules undergoing predissociation on the experimental time scale decreases, thereby increasing the fraction of states that survive to be field ionized. The increased lifetimes are in part due to the DC field inducing the mixing of the short-lived np, J = 1, states with the longer lived high4 states and with the J = 2,3 states. The effects of 1 mixing and J mixing should only be observed when the np Rydberg states are submerged into the hydrogenic high-2 manifolds, predicted to occur at n > 100, and this explains why the increased lifetimes are observed in the range n = 100,. ..,140. The decreased importance of predissociation is also caused by I mixing. Surprisingly the fraction of states undergoing autoionization within 7 p s does not increase, despite the fact that when 1 and J mixing occur, all states should have comparable autoionization lifetimes. It is probable that there is a subtle interplay between predissociation, autoionization, and rnr mixing in this range of n.

686


For n = 160 a decay time of approximately 3 ps is measured for 25% of the Rydberg states and a very long lifetime >30 ps for the remaining 75%. The lifetimes >30 ps cannot be explained purely in terms of a democratic I and J mixing of all states in this range, caused by the homogeneous field. The multichannel quantum defect theory (MQDT) calculations discussed in the next section suggest an average lifetime of -5 ps in a homogeneous field, which is quite close to that observed for the fast-decay component. It therefore appears that other processes make an important contribution to the stabilization for a large fraction of the Rydberg states with n > 150. The inhomogeneous field created by ions produced within the laser excitation volume is likely to be a significant factor [12] in causing m mixing and hence further stabilization.

VII. MQDT CALCULATIONS OF SPECTRA OF AUTOIONIZING RYDBERG STATES The MQDT picture [39, 401 of a Rydberg molecule in an electric field [41, 421 reveals much about the esentid physics of the Rydberg stabilization problem. The space of the electron is divided primarily into three regions. The effects of the external field are only felt in the outermost region III, where the total potential for the electron is a sum of Coulombic and external field terms (in ax.) 1 r

V ( r )= --

+ Fz + - -

where Fz is the homogeneous field term, but further terms may be added for an inhomogeneous field. The effects of ion-core anisotropy are assumed to be negligible in this region. If the field is purely homogeneous, the hydrogenic Stark wave functions (the solutions of the Schrodinger equation in region 111) are best described in a parabolic coordinate system. In the intermediate region 11 the electron experiences forces identical to those in a field-free hydrogen atom; that is, the ion core may be considered as a point charge and the field effects are negligible in comparison to the Coulombic force. The wave function must be a linear combination of field-free hydrogenic solutions to the Schrodinger equation, most conveniently expressed in spherical polar coordinates. In the short-range region, molecule-specific interactions of the Rydberg electron with the ion core occur, and the hydrogenic functions are no longer true eigenfunctions of the system; there is interaction between the long-range channels, including mixing of channels with different ioncore states. It should be stressed that although the Schriidinger equation is


687

molecule independent in the external regions I1 and Ill, the need to make a smooth join to the wave function in region I leads to a molecule-specific combination of the hydrogenic functions as a valid solution. This may be generally written as

where & is the wave function for the core state with quantum numbers y and is a hydrogenic function for the Rydberg (or free) electron characterized by quantum numbers i. However, as the field and/or effective principal quantum number increases, the importance of couplings in region I decreases, the electron spends a greater fraction of its time in regions I1 and 111, and the total solutions converge toward a molecule-independent hydrogenic solution. In the execution of the MQDT calculations described below, we take a handful of input parameters-the energy-independent quantum defects and transition moments for the molecule or atom of interest-derived from a spectroscopic analysis of a limited energy region, or ab initio calculation, and these can then be used to calculate spectroscopic and dynamical properties over wide energy ranges without further adjustment of parameters. A photoionization spectrum for a given field is simulated by determining the linear combination of hydrogenic functions, as in Eq. (2), as a function of energy and then using it to calculate the transition probability. In order to extract dynamical information regarding the lifetimes 7 of the autoionization features in the spectrum, the simplest procedure is to use the formula +iy

_T -S

5 . 0 1 4 ~lo-'* r/cm-'

(3)

where r is the calculated linewidth [full width at half maximum (FWHM)]. More correctly the lines should be fitted to a Fano line shape to extract the true width. However, both procedures cause difficulties when overlapping complex resonances occur and then a time-dependentcalculation of the decay becomes desirable.

A. Method Employed in the Calculations The method used in these calculations is based on the theoretical developments of Sakimoto [42] and is also an extension of our previous work on the Stark spectrum of spin-orbit autoionized Rydberg states of argon [43] and of the vibrationally autoionized states of hydrogen [44, 451. Only the homoge-

688

T. P. SOFTLEY,S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

neow component of the electric field is included in these calculations; hence MJ-mixing processes are not described. A fundamental assertion is that the open-channel components of the wave function are unchanged by the electric field, and therefore these can still be represented in region 111by the quantum numbers E, ml, and y [ I , m l = i in Eq. (Z)]. The closed channels, however, are mixed by the field and are defined by the quantum numbers y, mi and the parameter p, which is the separation constant (value, 0 .. . , 1) in the solution of the Schriidinger equation in parabolic coordinates; takes INT( Y - m) values at a given effective principal quantum number v and corresponds to the INT(v - m) pairs of values for the parabolic quantum numbers nl and n2. Tunneling effects are ignored in the calculations. The /3 values need to be determined for each field and v value [43].The total photoionization cross section, proportional to the oscillator strength, is given by a summation of partial photoionization cross sections over all the open channels (yo,I , m); only the open channels with non-zero quantum defects need be included.

where dr;s..m,, and dkom are transition moments to closed-channel final states y~plrm”and open-channel final states yJm, respectively:

with A;:,,,,,, representing the admixture of bound-channel $p“rn“ into the continuum wave function y,lm. In Eq. ( 5 ) the Stark scattering matrix x F is related to the K matrix (reactance matrix) via

x F = (1 + iKF)(l - iKF)-’

(6)

The KF matrix is related to the conventional K matrix of MQDT by a frame transformation from parabolic to spherical co-ordinates: the K matrix is then related by a further frame transformation to the quantum defects [43, 451. The first term in Eq. (4)gives a contribution to the photoionization intensity borrowed from the “bound-state” spectrum. The dkomterm represents direct photoionization, and the overall expression allows Fano-type interference between these terms. In Eq. ( 5 ) A is a phase shift in the parabolic rep-


689

resentation of the closed-channel components and is calculated as described in Ref. [43]. The theory is common to both system (N2and Ar) for which calculations are presented here and the differences occur in the detailed determination of the KF and dF matrices in the two cases.

B. Calculations for Argon

The first system chosen is the autoionizing Rydberg states of argon converging to the 2P1/2 ionization limit. In a previous paper [43] the results of similar calculations for the Rydberg states in the range n = 15, ... ,20 were reported and compared with the single-photon excitation spectrum obtained by the same authors. At low field the spectrum is dominated by transitions to the ns’ and nd’ states, but as the field increases, the np’ states and the high-2 manifolds appear. The agreement obtained between experiment and theory was generally very good in terms of linewidths and positions as a function of applied field. In the present case the aim is to extend the calculations to the high-n Rydberg states n = 80,. ..,100; the extension has been made possible primarily by improvements in efficiency of the computer code. Of particular interest is a comparison with the experimental results of Merkt [46], who obtained the ZEKE-PFI spectrum of Ar in the presence of three different DC fields ( 4 . 1 , 1 and 2 V/cm) with a pulsed-field delay time of 200 ns. At 0.1 V/cm the PFI spectrum shows a constant signal over the range n = 90,. ..,200 and then drops off at lower n due to autoionization. This suggests that the states with 90 < n < 200 have lifetimes 2 200 ns, implying a stability more than 50 times greater than predicted using an n3 scaling law for the optically accessible ns’ series and lo00 times greater than for the more intense nd‘ series. The effect of increasing the electric field was found experimentally to lead to significant shortening in the lifetime of the high-n Rydberg states and hence loss of signal in the delayed PFI. The method used in the calculations follows that explained in detail in Ref. [43]. The input quantum defects pa defined with respect to the Russell Saunders coupling scheme, which is the appropriate short-range basis, are given in Ref. 43. At energies corresponding to u = 100 the total number of open and closed channels in the final KF matrix is 414. Figure 11 shows the calculated photoionization spectra in the region of n = 90 for various fields in the range 0.001-2 V/cm. A step size of cm-’ was used in the calculations. At 0.001 V/cm (Fig. 1la) the 92s (sharp) and 90d (broad) Rydberg states appear strongly, and from the linewidths we ~ At 0.1 infer that the lifetimes are approximately 3.8 and 0 . 2 respectively. V/cm the n = 90 manifold appears superimposed on the tail of the 90d resonance. The manifold levels are sharper than the s and d resonances and

690

T. P. SOnLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

I

128528.10

I

128528.15

I

128528.20

128528.25

Wavenumber /cm -’ Figure 11. Multichannel quantum defect theory simulations of the photoionization cross section of Ar versus excitation wavenumber, in the presence of a DC field of magnitude ( a ) 0.001 V/cm, (b) 0.1 V/cm, (c ) 0.2 V/cm, ( d ) 0.3 V/cm and (e) 2.0 V/cm.

have an average width corresponding to approximately 25 ns. These lifetimes are much shorter than the experimental observations of Merkt, who found lifetimes > 200 ns [46]. It is noticeablethat as the field increases to 0.2 V/cm (Fig. 1Ic), the 90d resonance is strongly shifted in position and narrows quite dramatically, whereas the 92s resonance is barely changed in intensity and actually becomes slightly broader. It is only at around 0.3 V/cm (Fig. 1Id) that the 92s resonance starts to become absorbed into the high-l manifold. The main reason for this delayed mixing is that the s states can only interact with the manifold via a chain of A1 = 1 coupling through the np and nd Rydberg states. It is only at 0.3 V/cm that the np states become immersed in the manifold. At 2 V/cm (Fig. Ile) the spectrum is dominated by hydrogenic manifold structure and there is strong overlap between many manifolds. Other calculations (not shown) indicate that there is no major qualitative change in average lifetimes between 1 and 5 V/cm, although the resonances become


less isolated at increased field, due to greater overlap of manifolds. A notable feature is that there is quite a wide range of widths predicted at 2 V/cm, if anything more so than at 0.3 V/cm. The main intensity occurs in lines with widths corresponding to lifetimes in the range 3-50 ns. This raises an important question as to whether ZEKE experiments measure the entire population initially excited in the pseudocontinuum or whether only a small fraction is observed that survives the initial decay. These calculations may also illustrate the bifurcation of lifetimes suggested in the model of Remacle and Levine [18] (see also Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume). In recent work [13], Bixon and Jortner have used a matrix diagonalization approach to calculate lifetimes for autoionizing Rydberg states in argon for individual manifolds of a given n (where n = 100,. ..,280) in the presence of an electric field. Their method involved the use of a complex matrix of dimension =2n of the effective Hamiltonian H = Ho + eFz - $’, although the off-diagonal matrix elements of ir were omitted, making this method equivalent to earlier calculations on Xe [47]. They concluded that there were two distinguishable field ranges, as defined by the value of the reduced field,

F=

’

(F/V cm- )a5 3.4 x 109[p(mod l)]

(7)

where p is the quantum defect for the initially populated state. In range A (0.7 1 P 1 2), the onset of coupling between the low-l and high4 manifolds occurs, and the states show a bimodal distribution of lifetimes: There is a long-lived component whose transition intensity increases and whose lifetime decreases with increasing F, while there is a short-lived component showing the opposite trends. This range corresponds to Figs. l l b and l l c (F = 0.96, 1.92, respectively). In range B (F 2 3 corresponding to Figs. lld and 1le), there is a democratic sharing of lifetime between all coupled states with the resultant mean lifetime

The sum in the denominator represents the total coupling width of all the low4 states to the continuum (or continua). The predictions of Bixon and Jortner are in semiquantitative agreement with the MQDT results presented here; for example they would predict a mean lifetime of approximately 18 ns for the long-lived states in the region of n = 90 in the high-field limit.

692

T. P. SOFIZEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

The matrix diagonalization method suffers two disadvantages that are not intrinsic to the MQDT calculations presented in this chapter. First, in order to account for the effects of overlap between adjacent manifolds, a very large matrix would need to be diagonalized, particularly in the high-n range, and convergence may be difficult. The second disadvantage comes in the requirement for a knowledge of the matrix elements of I?; the diagonal elements can only be obtained from measurements of experimental linewidths for low-n autoionizing states, while there is no straightforward method to obtain the off-diagonal elements other than by calculation [17]. The zero-field quantum defects are also required to calculate (Ho).For some of the basis states of interest in argon, the diagonal matrix elements of r are unknown experimentally, and educated guesses need to be made. In the MQDT treatment, the overlapping manifolds are automatically included in the calculation because the principal quantum number n is no longer a parameter in the formalism and is effectively replaced by the energy as a continuous variable. A channel wave function can describe the properties of an infinite range of n values for given angular momentum quantum numbers. The contributions to the wave function at a given energy from states of all n values are automatically included. Second, the quantum defects are used in MQDT to characterize not only the line positions but also the bound-continuum interactions, and there is no need to determine interaction widths independently. The interactions corresponding to the off-diagonal matrix elements of I' are included automatically. The disadvantage of the MQDT method for the Stark effect is that fairly large complex matrices ( 4 0 0 x 500) must still be handled and inverted at each point on a very fine energy grid. Although these operations may be more feasible than diagonalizing complex matrices with dimensions of thousands, the method is still very computer intensive. In this work we have demonstrated that the MQDT method can be applied to obtain meaningful results in the high-n region. Although it is particularly interesting to compare our results with those of Bixon and Jortner for argon, and satisfying to find agreement, we are also interested in determining whether there are molecule-specific effects that cannot be determined by simply calculating properties of atomic systems. Of particular interest is to determine the effects of interactions between the many Rydberg series converging on different vibration-rotation states of the ion, which is equivalent to the rotational electronic couplings in the Levine model [14].

C. Calculations for Nitrogen For nitrogen we would ideally like to calculate the excitation spectra of the high-n N+ = 2 , 3 Rydberg states discussed in Section VI in the presence of a 0.3-V/cm field, including as many as possible of the interacting Rydberg series


693

converging on different ionic rotational thresholds. However, there is a need to compromise between including sufficient rotational channels and calculating for high enough n. We are restricted by the dimensionsof the matrices involved (which on the particular computer system being used means 400 x 500). Two types of calculationshave been perfomed in the first type only N + = 0 (open) and N + = 2 (closed) channels are included, which can be carried out for up to n = 120. In the second type the N + = 4 , 6 channels are also included but the calculationsare carried out at lower n (516).The complicationin the latter case is that the N* = 0 channel only becomes open at an energy corresponding to -n = 80 of the N + = 2 channel, and therefore rotational autoionization cannot occur for n c 80. However, in this work the N + = 0 limit is falsely lowered to below n = 16 of N + = 2. Although this is a physically artificial calculation, it does have validity within the general framework of MQDT, which treats bound states and continua uniformly and assumes an energy independence of the quantum defects. Furthermore, it is expected that the results obtained at a given value of P will mimic those obtained at a corresponding value of P for higher n Rydberg states. The u+ > 0 channels are ignored in these calculations. We assume R-independent quantum defects using the values given in Ref. 48 for thep channels, and in the absence of further information,the calculated values for s and d Rydberg channels belonging to the A 211uand B 2qion-core states from Ref. 49. Redissociation is not included in the calculations,although in principle it could be. The appropriateframe transformationsfor the diatomic molecule case are discussed in detail in Ref. 45, although the integration over vibrational wave functions need not be included for R-independent quantum defects. Figure 12a shows the calculated spectra (0.3 V/cm field) in the region of the 70p Rydberg states with the three J components showing exactly the behavior described in Section VI; that is, two long-lived components and one short-lived component are predicted, with linewidths corresponding to lifetimes of 200 ns for J = 2, 3 and 1 ns for J = 1. In zero field the autoionization lifetimes of the J = 2 , 3 components should be infinite, and therefore the calculated lifetimes are an indication of the extent of J mixing induced by the field. However, as shown in Fig. 126, at n = 80 the effect of the field is apparently already sufficient to mix completely these states such that they have a comparable lifetime. In the experiments, however, there is no evidence for such J mixing occurring in the range n = 80,. ..,100; that is, only one-third of the states autoionize rapidly, suggesting that predissociation is competing on a time scale of a few nanoseconds and is still causing decay of two-thirds of the populated states. At n = 95 in the calculations, only high4 manifold structure is apparent with a mean lifetime of 689 ns. An extrapolation of this lifetime to n = 160 using an n4 scaling law would give a maximum lifetime of 5.0 ps, which is reasonably consistent with the fast

694


r

1

I

I ' J=2

I

I J=3

k (b)n=80

125661.10

125661.12

125661.14

1 25661.16

125661.18

Wavenumber /cm-'

Figure 12. Multichannel quantum defect theory simulations of the photoionization cross section of N2, field = 0.3 V/cm (a) near n = 70 and (b) near n = 80, including N + = 0, 2, M J = 1 channels only, with excitation from J' = 2, M; = 1.

decay observed experimentally for ~ 2 5 % of the states but totally incompatible with the long-lived component (>30 ~ s )It. must be stressed once again that only the effects of homogeneous fields are included in these calculations at the present time. Figures 13a and 13b show a comparison of spectra calculated at n 16 with and without the N + = 4, 6 channels included. At 100 V/cm (F = 0.5) there is considerable mixing between the N + = 2 Rydberg states and the N + = 4 hydrogenic manifold, which is primarily induced by the electric field and does not exist at much lower field. This leads to significant differences from the case where rotational couplings with the higher N + channels are ignored. On the other hand, at 200 V/cm (F = 1.5), where the hydrogenic manifolds completely dominate the spectrum (Fig. 14), the effect of the rotational channel mixing is somewhat less, and the inclusion of the N + = 4, 6 channels leads only to a few additional lines that do not perturb the N + = 2 transitions very strongly, Figure 14 shows a small part of the N + = 2 n = 16 manifold structure, each group of lines being one Stark component split by fine-structure interactions into three or four components.

-


695

1000 V/cm

(a) N=0.2 only

L (b) N=0.2,4,6

125212.0

125

125216.0

0

12

18.0

Wavenumber /cml

Figure 13. Multichannel quantum defect theory simulations of the photoionization cross section of N2, near n = 16, field = lo00 V/cm; ( a ) including only N + = 0 . 2 and (b)including N + = 0. 2, 4, 6, with excitation from J' = 2, M; = 1 .

2000 V/cm

(a) N=0,2 only

1

d I

L 125225.0

(b)N=0.2,4,6

1

125 30.0

Wavenumber /cm"

Figure 14. Multichannel quantum defect theory simulations of the photoionization cross section of Nz,near n = 16, field = 2000 V/cm: ( a ) including only N + = 0, 2 and (b) including N + = 0, 2, 4, 6, with excitation from J' = 2, M; = 1.

696

T. P. SORLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

The most interesting conclusion at this point is that the mixing between 1 components belonging to a particular rotational channel is so strong at high fields that the coupling with other rotational channels is effectively switched off. The reason for this is that the high-Z states do not undergo rotational channel couplings, and therefore when every state has a large admixture of high4 components, there is very little interchannel coupling. This point reiterates the comments made in Section V concerning the transition to the high-externalfield limit. Analogous behavior has been observed in calculations for HZ[50].

VIII. CONCLUSIONS In Sections 11-IV it is demonstrated that rotationally state-selected ions can be produced in sufficient quantities to study reactions. Product ions for lowenergy collisions (Ecoll values of 0.04-0.24 eV) have only been detected to date in the reaction of H2+ + Ha, but it is likely that this method should be reasonably general. The importance of Rydberg state lifetime enhancements has been discussed in depth in Sections V-VII. The comparison of MQDT calculations with experimental results shows clearly that the inclusion of homogeneous field effects only (1 mixing, no m mixing) is insufficient to explain the observed lifetimes in the experiments, pointing to the importance of inhomogeneous fields. A question raised by the argon calculations is whether the majority of PFI experiments measure only a small fraction of the total excitation cross section to the pseudocontinuum of high Rydbergs, because of the wide range of lifetimes of these states, even in the high-field limit. The N2 calculations show that internal rotational-electronic couplings can be initially enhanced as a homogeneous electric field is increased, but the couplings are effectively switched off at sufficiently high field. In the future further investigation of the effects of electronic-rotational coupling must be carried out to determine under what conditions the transition between the regime dominated by internal couplings and that dominated by external-field effects occurs in various molecular systems.

Acknowledgments We are grateful to the EPSRC for an equipment grant and to the EU Human Capital and Mobility Program (no. CHRX-CT93-0150) and the British Council for their additional support. S. R. M. gratefully acknowledges the EPSRC for his Studentship and F. M. thanks St John’s College, Oxford, for his Research Fellowship.

References 1. K. Miiller-Dethlefs. E. W. Schlag, E. Grant, K. Wang and V. McKoy, Adv. Chem. f h y s . 90, 1 (1995).

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS 697 2. F. Merkt and T.P. Softley, Inf. Rev. Phys. Chem. 12, 205 (1993). 3. 1. Powis, T.Baer, and C. Y. Ng, Eds.. High Resolution Laser Photoionization and Photoelectron Studies, Wiley, Chichester, 1995. 4. L. Zhu and P. Johnson, J. Chem. Phys. 94, 5769 (1991). 5 . C. C. Arnold, Y. Zhao, T.N. Kitsopoulos, and D. M. Neumark, J. Chem. Phys. 97,6121 ( 1992). 6. 0. Dopfer, G. Reiser, K. Muller-Dethlefs, E. W. Schlag, and S. D. Colson, J. Chem. Phys. 101,974 (1994). 7. M. J. J. Vrakking. I. Fischer, D. M. Villeneuve, and A. Stolow, J. Chem. Phys. 103,4538 (1995). 8. X. Zhang, J. M. Smith, and J. L. Knee, J. Chem. Phys. 100, 2429 (1994). 9. M.N. R. Ashfold, 1. R. Lambert, D. H. Mordaunt, G. P. Morley, and C. M. Western, J. Phys. Chem. 96,2938 (1992). 10. F. Merkt, H. Xu and R. N. Zare, J. Chem. Phys. 104,950 (1996). 11. W. A. Chupka, J. Chem. Phys. 98,4520 (1 993). 12. F. Merkt and R. N. Zare, J. Chem. Phys. 101, 3495 (1994). 13. M. Bixon and J. Jortner, J. Chem. Phys. 103,4431 (1995). 14. E. Rabani, R. D. Levine, A. Muhlpfordt, and U. Even, J. Chem. Phys. 102, 1619 (1995). 15. M.J. J. Vrakking and Y.T. Lee, J. Chem. Phys. 102,8818 (1995). 16. S. T. Pratt, J. Chem. Phys. 98, 9241 (1993). 17. X.Zhang, J . M. Smith, and J. L. Knee, J. Chem. Phys. 99,3133 (1993). 18. F. Remacle and R. D. Levine, J. Chem. Phys., 104, 1399 (1996). 19. T. F. Gallagher, Rydberg Aroms, Cambridge University Press, Cambridge 1994. 20. D. C. Clary, Annu. Rev. Phys. Chem. 41,61 (1990). 21. S. L. Anderson, Adv. Chem. Phys. 82, 177 (1992). 22. P. M. Guyon and C. Alcaraz, Proc. SPIE 1858, 398 (1993). 23. H. K. Park and R. N. Zare, J. Chem. Phys. 99,6537 (1993). 24. F. Merkt, S. R. Mackenzie, and T. P. Softley, J. Chem. Phys. 103,4509 (1995). 25. H. Krause and H.J. Neusser, J. Chem. Phys. 99,6278 (1993). 26. W. Kong, D. Rodgers, and J. W. Hepbum, J. Chern. Phys. 99,8571 (1993). 27. C. R. Mahon, G. R. Janik, and T. F. Gallagher Phys. Rev. A 41, 3746 (1990). 28. F. Merkt, H.H. Fielding, and T. P. Softley, Chem. Phys. Left. 202, 153 (1993). 29. S. R. Mackenzie and T. P. Softley, J. Chem. Phys. 101, 10609 (1994). 30. F. Merkt, S. R. Mackenzie, and T. P. Softley, J. Chem. Phys. 99,4213 (1993). 31. J. Xie and R. N. Zare, J. Chem. Phys. 93,3033 (1991). 32. A. Fujii, T. Ebata, and M. Ito, Chem. Phys. Lett 161, 93 (1989). 33. S. R. Mackenzie, F. Merkt, E. J. Halse, and T. P.Softley, Molec. Phys. 86, 1283 (1995). 34. G. Reiser and K. Muller-Dethlefs, J. Phys. Chem. 96,9 (1992). 35. K.Muller-Dethlefs, M. Sander, and E. W. Schlag, Chem. Phys. Left. 112, 291 (1984). 36. M. Raoult, J. Chem. Phys. 87,4756 (1987). 37. W. A. Chupka, M. E. Russell, and K.Refaey, J. Chem. Phys. 48, 1518 (1968). 38. C. Alt, W. G. Schener, H. L. Selzle, and E.W. Schlag. Chem. Phys. Left.240,457 (1995).

698

T. P.SOFTLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

39. M. I. Seaton, Rep. Prog. Phys. 46, 167 (1983). 40. U. Fano, Phys. Rev. A 2, 353 (1970). 41. D. A, Harmin, Phys. Rev. A 24,2491 (1981). 42. K. Sakimoto, J. Phys. B 22,2727 (1989). 43. H. H. Fielding and T. P. Softley, J. Phys. B 25, 4125 (1992). 44. H. H. Fielding and T. P. Softley, Chem. Phys. Lett. 185, 199 (1991).

45. H. H. Fielding and T. P.Softley, Phys. Rev. A 49, 969 (1994). 46. F. Merkt, J. Chem. Phys. 100, 2623 (1994). 47. W. E. Emst, T. P. Softley, and R. N. Zare, Phys. Rev. A. 37,4172, (1988). 48. K. P. Huber and Ch. Jungen, J. Chem Phys. 92, 850 (1990).

49. W. M. Kosman and S. Wallace, J. Chem. Phys. 82, 1385 (1985). 50. T. P. Softley, A. J. Hudson and R. Watson, J. Chem. Phys. 106, 1041, (1997).

DISCUSSION ON THE REPORT BY T. P. SOFTLEY Chairman: M. S. Child D. M. Neumark: Prof. Softley, what is the kinetic-energy resolution of your ion-molecule experiment? I am particularly concerned about collisions that occur while the ions are being accelerated by the pulsed field. T.P.Softley: At 10 meV collision energy for the Hi + H2 reaction the collision energy resolution is approximately +3 meV. This is due to the translational temperature of the beam and angular divergence primarily. The acceleration pulses are short, confining collisions during acceleration to less than 5%. R. D. Levine: The rate of an ion-molecule reaction can be governed either by the (physical) polarization potential in the reactant's region or by a possible barrier of chemical origin along the reaction coordinate, Similarly, the stereochemistry of ion-molecule reactions is of particular interest because it can be due to either the anisotropy of the potential in the entrance valley [R. D. Levine and R. B. Bernstein, J. Phys. Chem. 92,6954 (1988)J or to the angle dependence of the chemical barrier [E. Rabani, D. M. Charutz, and R. D. Levine, J. Phys. Chern. 95, 10551 (1991)l. Rotational state selection is one way to probe the steric requirement €32 of reactions. Your preliminary result that the reaction rate for Hi iis only weakly dependent on the rotational state will thus attract much attention. This is particularly so since the system is simple enough from


699

a quantum chemistry point of view that ab initio potential-energy surfaces of realistic accuracy can be computed. The study of the dynamics of diatom4iatom ion-molecule reactions is only beginning, and there are already a number of puzzling issues. To my mind, not the least of which is the rather high rate of such reactions. In view of much recent work on the N; + H2 reaction and the possible role of the charge transfer channel, N2 + H;, it might be of interest to look at this reaction directly. T. P. Softley: At the present time, we do not say that there are no rotational effects on the cross section of the H; + H2 reaction but that our signal-to-noise levels at the lowest collision energies (where these effects might occur) are not sufficiently good to draw any conclusions (see the current chapter). D. M. Neumark: Didn’t Chupka investigate rotational effects in the H; + H2 reaction many years ago? T. P. Softley: Chupka concluded that the cross section for H; + H2 did not vary by more than 10% over the range of rotational levels studied, although in common with our studies at present, they were not able to increase the accuracy beyond this level [W. A. Chupka, M.E. Russell, and K.Refaey, J. Chem. Phys. 48, 1518 (1968)l. B. Kohler: I have a question for T. Softley: Experimentally, how do you distinguish predissociation of the initially populated high Rydberg states from other decay mechanisms? T. P. Softley: The predissociation yield is determined indirectly. The total number of molecules excited is known, by comparison with above-threshold ionization yields. The number of autoionizing Rydbergs and the number of field-ionized Rydbergs is also known. The fraction of dissociating Rydbergs is determined by subtracting the total ionization signal from the total excitation signal (presumed to be equal to that above threshold).

QUANTUM DEFECT THEORY OF THE DYNAMICS OF

MOLECULAR RYDBERG STATES CH.JUNGEN Laboratoire Aimk Cotton du CNRS Universitk de Paris-Sud Orsay, France

CONTENTS I. Introduction

11. Frame Transformations and Bound States

[II. High Orbital Angular Momentum States IV. States in the Electronic Continuum V. Determination of Quantum Defects from Experiment VI. Conclusion References

I. INTRODUCTION With the advent of the technique of zero-kinetic-energy (ZEKE) photoelectron spectroscopy there has been renewed interest in the physics of highly excited atomic and molecular Rydberg states and, in particular, the breakdown of the Born-Oppenheimer approximation in those states. The recent discussion of the long-lived so-called ZEKE states [ 11, that is, Rydberg states with principal quantum numbers n L 100, has focused on the mixing of Rydberg channels characterized by different values of the core quantum numbers u+, N + (u+ vibration, N + rotation) as well as different values of the electron orbital angular momentum 1. Rydberg states with n 1 100 correspond to Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femfosecond Time Scale, XXth Solvny Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt. I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

701

CH.JUNGEN

702

orbits of almost macroscopic dimension (of the order of for n = 100) so that weak external perturbations such as electric stray fields become effective in mixing the unperturbed Rydberg channels [2]. This mixing “dilutes” the short-lived low4 states among the manifold of long-lived high-Z states and hence leads to a lifetime lengthening. On the other hand, a molecular Rydberg electron is also strongly attracted by the positively charged core, and collisions with the latter also lead to mixing of Rydberg channels. It has been surmised [3] that this intramolecular Rydberg dynamics also plays a role in the lifetime lengthening of the ZEKE states. This mechanism is related to the breakdown of the Born-Oppenheimer approximation and as such has been studied for a number of molecules, mainly diatomic systems, in the framework of multichannel quantum defect theory. The purpose of this contribution is to provide a brief review of the work that has been carried out along these lines in the author’s group and where the role of the vibrational and rotational degrees of freedom has specifically been considered. The reader is also referred to a recent review by Lefebvre-Brion [4] on rotationally resolved autoionization of molecular Rydberg states. 11. FRAME TRANSFORMATIONS AND BOUND STATES

A central feature of molecular quantum defect theory is the use of frame transformations. These provide an elegant way of treating the breakdown of the Born-Oppenheimer approximation that occurs systematically once an electron is excited into a high Rydberg state. The realization that this breakdown occurs is quite old. Mulliken [ 5 ] , in the first of his papers on molecular Rydberg states, wrote in 1964 (p. 3 189). “In most discussions on molecular wave functions, the validity of the Born-Oppenheimer approximation is assumed. This approximation is most nearly accurate when the frequencies of motion, which can be gauged by energy level spacings, are much larger for the electronic than for the nuclear motions. In a Rydberg state series, as n increases, the frequencies for the Rydberg electron become smaller and smaller relative to those of nuclear vibration and rotation. This leads to more or less radical changes in coupling relations. . . . In the B.O. approximation, the wave function for the lowestn Rydberg states of a diatomic molecula with a closed-shell core takes the

form

where the operator A in Eq. (1) makes % antisymmetric in all the electrons. As n increases and the inner loops of the Rydberg molecular orbital become

QUANTUM DEFECT THEORY OF MOLECULAR RYDBERG STATES

703

less and less important relative to the outer loops, the Rydberg electron withdraws more and more from influencing the rotational and vibrational motions of the nuclei and for large n the wavefunction takes the form

Here J / R , , ~ is no longer included within the electronic factor of the B.O. approximation." Mulliken referred to Eqs. (1) and (2) as Rydberg-coupled and Rydberg-uncoupled wave functions, respectively. As far as rotational motion is concerned, the transition between Eqs. (1) and (2) is equivalent to transitions between Hund's coupling cases (a) and (b) (uncoupling of the electron spin) or between cases (b) and (d) (uncoupling of the electron orbital angular momentum I). The corresponding recoupling transformation is implied in Van Vleck's work [6] published in 195 1. The connection with multichannel quantum defect theory was established by Fano [7] in 1970. By combining the concepts of Seaton's atomic quantum defect theory [8] with the frame transformation approach used in electron-molecule scattering [9], he showed how multichannel quantum defect theory (MQDT) can account for rotation4ectron coupling (1 uncoupling) in H 2 . A comprehensive account of the theory of rotation-electron coupling as well as of vibration-electron coupling in diatomic molecules was given by Jungen and Atabek [lo] in 1977. These authors based their application to the ungerude Rydberg states of H2 on ab initio theory, and they showed that MQDT couple with frame transformation approach accounts for nonadiabatic level shifts quite accurately without requiring the knowledge of the usual momentum coupling functions, which are necessary in the customary coupled equations calculations. The key quantity in their approach is a rovibronic reaction matrix of the form

Kv+N+, ,+"+'

=

where K is the body-fixed reaction matrix K"(R) = tan 7rp"(R) that depends on molecular symmetry (electronic angular momentum component A) and nuclear configuration (internuclear distance R). Here, p is the quantum defect in the usual sense. The terms J and M are the total angular momentum of the molecule and its space-fixed component, which are conserved for an isolated molecule in field-free space. The terms )u+N+)denote the uncoupled

704

CH. JUNGEN

or asymptotic Rydberg channels whereas IRA) denote the coupled or shortrange channels. The transformation between the two limits is embedded in the bra-kets J ’ ( ~ + N +IRA)JMfor which explicit expressions can be found in the relevant papers. Equation (3) is written for a given electronic state of the core and a given value 1 of the electron orbital angular momentum. The paper by Jungen and Atabek [lo] also contains a sketch of the extension of the theory to include additional purely electronic interactions, such as the mixing of different partial waves 1 by the nonspherical field of the molecular core and/or interactions with core-excited Rydberg channels. This was not implemented at the time, however. This gap has been filled only recently in a series of papers by Ross et al. dealing with the gerade Rydberg states of H2 [ll]. Equation (3) is now generalized to take the form

where K$(R) = tan 7rp$(R) is a nondiagonal quantum defect matrix with i or j specifying a given electronic core state and associated Rydberg orbital angular momentum 1. In the application to gerade H2 the manifold of Born-Oppenheimer channels was represented by a 3 x 3 electronic matrix including sog and dug electrons associated with the * ground state of H$ and a po, electron associated with the excited core. All other molecular symmetries were represented by a single matrix element. The connection with ab initio theory here and in Eq. (3) above is the following: The clamped-nuclei reaction matrix K$Z) is determined such that when entered into a multichannel quantum defect calculation for fixed R and A, it yields precisely the corresponding known ab initio clamped-nuclei electronic energies U k ( R ) of the molecule. This approach has been used to calculate 382 observed gerude singlet and triplet excited-state levels of HZ with total angular momentum values (exclusive spin) between N = 0 and N = 5, spanning a range of 23,000 cm-I and associated with 12 electronic states. The mean deviation observed (calculated) is only 5.8 cm-’ . By comparison, the errors of the ab initio input potential-energy curves [12] depend on the electronic state and vary with R, but a reasonable mean value is probably about 2 cm-’. In turn, the nonadiabatic level shifts themselves amount up to several hundreds of wavenumber units.

’

*

xi


705

111. HIGH ORBITAL ANGULAR MOMENTUM STATES As the orbital angular momentum 1 of the Rydberg electron increases, the centrifugal potential term +&I+ 1)/2r2 becomes more and more effective in keeping the Rydberg electron outside the core region. For a threshold electron (E = 0) the inner turning point for classical motion in a Coulomb field is ro = !1(1+ 1). For 1 = 3 one thus has ro = 6 a.u., which is larger than the core size of a small molecule. On the other hand, any molecule possesses electric multipoles as well as polarization fields that extend beyond the core region. These contribute to the body-frame quantum defects p*(R) of all Rydberg channels, but for 1 1 3 their contribution dominates. This means that the body-frame quantum defects can be evaluated in terms of the ioncore multipole moments and polarizabilities. For example, for a symmetrical molecule (no core dipole moment) the body-frame quantum defect for an f electron interacting with a C diatomic core is given by [13] A 1 p 3 , 3 ( ~ , R=) -(1 + 4 ~ ) a ( R-) 630

Here E is the electron energy in Rydbergs; ;[2all(R) + a l ( R ) ] = a(R) and i[all(R)- a l ( R ) ] are, respectively, the isotropic and anisotropic dipole polarizabilities of the core that are nuclear-coordinate dependent, and Qzz is its quadrupole moment. Expressions analogous to Eq. ( 5 ) for I 2 3 are readily derived. Equation ( 5 ) takes account of the long-range field components proportional to r-k with k up to 4 . For higher accuracy higher multipole moments and hyperpolarizabilities must also be included. In the case of HZ all these quantities are known from ab initio theory and are available for a wide range of R values. High4 Rydberg states have been observed for this molecule for 1 = 3 , 4 , 5 by Fourier transform spectroscopy [14-161 and for 1 = 5,6 by microwave spectroscopy [17-201, and they have also been calculated [14-161 from first principles by multichannel quantum defect theory by using formulas of the type of Eq. ( 5 ) in conjunction with the frame transformation approach of EQ. (3) (i.e., neglecting I mixing). In the calculations terms up to r-6 were taken into account. It has been found that the calculations yield the observed rovibronic Rydberg levels to within about 0.5 cm-' for 1 = 3 and to within about 0.04 cm-' for I = 5, showing that quantum defect theory provides an adequate description of high-1 molecular states just as for the more familiar low4 states. A problem that apparently has not

706

CH. JUNGEN

yet been tackled and may be of interest for studying the physics of high-Z ZEKE states is the calculation of channel couplings off-diagonal in I based on the long-range field model, followed by use of Eq. (4).

IV. STATES XN THE ELECTRONIC CONTINUUM Some of the earliest applications of MQDT dealt with vibrational and rotational autoionization in H2 [21-251. One concept that emerged from these studies is that of complex resonances [26), which are characterized by a broad resonant distribution of photoionization intensity with an associated rather sharp fine structure. These complex resonances cannot be characterized by a single decay width; they are the typical result of a multichannel situation where several closed and open channels are mutually coupled. The photoionization spectrum of H2 affords a considerable number of such complex resonances. The H2 molecule is a system for which quite recently it has been possible to measure in unprecedented detail state-selected vibrationally and rotationally resolved photoionization cross sections in the presence of autoionization [27-291. The technique employed has been resonantly enhanced multiphoton ionization. The theoretical approach sketched above has been used to calculate these experiments from first principles [30], and it has thus been possible to give a purely theoretical account of a process involving a chemical transformation in a situation where a considerable number of bound levels is embedded in an ensemble of continua that are also coupled to one another. The agreement between experiment and theory is quite good, with regard to both the relative magnitudes of the partial cross sections and the spectral profiles, which are quite different depending on the final vibrational rotational state of the ion. V. DETERMINATION OF QUANTUM DEFECTS FROM EXPERIMENT

The key quantities in the traditional Born-Oppenheimer theory of molecules are the coordinate-dependent electronic energies. They supply the potentials for nuclear motion from which the level fine structure can be predicted. These curves or surfaces need not necessarily be obtained from ab initio theory. The inverse approach is followed in most spectroscopic work in that the potentialenergy surfaces or sections thereof are extracted from experiment. Indeed, the structural information contained in the electronic energies provides the most commonly used interface for the comparison between ab initio theory and experiment. Without this key feature of the theory, molecular physics could never have progressed as it has in the past decades.


707

An analogous situation occurs in multichannel quantum defect theory where the coordinate-dependent quantum defect matrices of Eiq. (4) above play a similar role. In a number of instances the inverse procedure has indeed been followed here too. Thus, for Liz, a 2 x 2 quantum defect matrix involving su and du channels has been determined [3 11 from mixed Rydberg series at the equilibrium internuclear distance Re as well as the derivative of each of its elements with respect to R. Similarly, for N2,2 x 2 matrices representing the p andf series and their interactions have been determined for R = Re from the highly resolved vacuum ultraviolet (VUV) absorption spectrum [131. The NO molecule is another example where quantum defect matrices have been extracted from experiment and have been compared directly to corresponding quantities calculated from first principles [32, 331. Work on the Rydberg states near n* = 14 is in progress for the very strongly dipolar molecule CaF. Here it has been possible to extract from the observed fine structure of the super-1-complex the full nondiagonal quantum defect matrices including the partial-wave components 1 = 0,. .., 3 for A = 0,. . . ,3, that is, in all 20 matrix elements [34].

VI. CONCLUSION It appears in conclusion that molecular multichannel quantum defect theory is a method of choice for treating channel couplings in high Rydberg states of molecules. This theory is based on coordinate- and symmetry-dependent quantum defect matrices, which in tum are related to the clamped-nuclei Bom-Oppenheimer potential-energy surfaces or equivalent electron-core scattering phase shifts. Nonadiabatic effects are taken into account through geometric frame transformations,while the customary radial momentum coupling or Coriolis coupling functions are not required. Interference effects between various channel interaction paths are fully taken into account, as is exemplified by the calculations of complex resonances discussed above. We expect that such complex resonances are also important in the physics of very high Rydberg states.This approach can also be extended to account for external electric fields [35] and Rydberg electron wavepacket dynamics [36]. References 1. G . Reiser, W. Habenicht, K. Muller-Dethlefs, and E. W. Schlag, Chem. Phys. Lett. 152, 119 (1988). 2. W. A. Chupka, J. Chem. Phys. 98,4520 (1993); 99,5800 (1993). 3. D. Bahatt, U. Even, and R. D. Levine, J. Chem. Phys. 98, 1744 (1993). 4. H. Lefebvre-Brion, in High Resolution Laser Photoionization and Photoelectron Studies, I. Powis, T. Baer, and C. Y. Ng, F A , Wiley Series in Ion Chemistry and Physics, Wiley, Chichester, 1995. p. 171.

708

CH. JUNGEN

5. R. S. Mulliken, J. Am. Chem. Soc. 86,3183 (1964); 88, 1849 (1966); 91,4615 (1969). 6. J. H. Van Vleck, Rev. Mod. Phys. 23,213 (1951). 7. U. Fano, Phys. Rev. A 2, 353 (1970). 8. M. J. Seaton, Rep. Prog. Phys. 46, 167 (1983). 9. N. F. Lane, Rev. Mod. Phys. S2,29 (1980). 10. Ch. Jungen and 0. Atabek, J. Chem. Phys. 66,5584 (1977). 11. S. C. Ross and Ch. Jungen, Phys. Rev. A 49, 4353 (1994); 49, 4364 (1994); 50, 4618 (1994). 12. S.Yu and K. Dressler, J. Chem. Phys. 101,7692 (1994). 13. K. P. Huber, Ch. Jungen, K. Yoshino, K. Ito, and G. Stark, J. Chem. Phys. 100, 7957 (1994). 14. Ch. Jungen, I. Dabrowski, G. Herzberg, and D. J. W. Kendall, J. Chem. Phys. 91, 3926 (1989). 15. Ch. Jungen, I. Dabrowski, G. Herzberg, and M. Vervloet, J. Chem. Phys. 93,2289 (1990). 16. Ch. Jungen, I. Dabrowski, G. Herzberg, and M. Vervloet, J. Mol. Spectrosc. 153,ll (1992). 17. W. G. Sturms, P. E. Sobol, and S. R. Lundeen, Phys. Rev. Lerr. 54,792 (1985). 18. W.G. Sturms, E. A. Hessels, and S. R. Lundeen, Phys. Rev. Lett. 57, 1863 (1986). 19. W. G. Sturms, E. A. Hessels, P.W. Arcuni, and S. R. Lundeen, Phys. Rev. Lett. 61,2320 (1988).

20. W. G. Sturms, E. A. Hessels, P. W. Arcuni, and S. R. Lundeen. Phys. Rev. A 38, 135 (1988). 21. G. Herzberg and Ch. Jungen, J. Mol. Specrmsc. 41,425 (1972). 22. Ch. Jungen and Dan Dill. J. Chem. Phys. 73, 3338 (1980). 23. M. Raoult and Ch. Jungen, J. Chem. Phys. 74, 3388 (1981). 24. N. Y. Du and C. H. Greene, J. Chem. Phys. 85,5430 (1986). 25. J. A. Stephens and C. H. Greene, J. Chem. Phys. 100,7135 (1994). 26. Ch. Jungen and M. Raoult, Faraday Discuss. Chem. Soc. 71,253 (1981). 27. M.A. OHalloran, F! M. Dehmer, F. S. Tomkins, S. T. Pratt, and J. L. Dehmer, J. Chem. Phys. 89, 75 (1988). 28. M. A. OHalloran, S. T. Pratt, E S. Tomkins, J. L. Dehmer, and P. M. Dehmer, Chem. Phys. Lerr. 146,291 (1988). 29. J. L. Dehmer, P. M. Dehmer, S. T. Pratt, F. S. Tomkins. and M.A. O’Halloran, J. Chem. Phys. 90,6243 (1989). 30. Ch. Jungen, S. T. Pratt, and S. C. Ross, J. Phys. Chem. 99, 1700 (1995). 31. A. L. Roche and Ch. Jungen, J . Chem. Phys. 98,3637 (1993). 32. S. Fredin, D. Gauyacq, M. Horani, Ch. Jungen, G. Lefivre, and F. Masnou-Seeuws, Mol. Phys. 60, 825 (1987). 33. M. Raoult, J. Chem. Phys. 87,4736 (1987). 34. Ch. Jungen, N. A. Harris, and R. W. Field. (in press). 35. H. H. Fielding and T. P. Softley, Phys. Rev. A 49, 969 (1994). 36. H. H. Fielding, J. Phys. B: At. Mol. Opt. Phys. 27, 5883 (1994).


709

DISCUSSION ON THE REPORT BY CH. JUNGEN Chairman: M. S. Child U. Even 1. Prof. Jungen, at what distances did you calculate these quantum defect matrix elements? 2. How sensitive is the calculation to the division of space into the three regions?

Ch. Jungen 1. I should have mentioned that in this example only one vibrational channel is included in the calculations, that is, only rotation-electron coupling is taken into account. Vibration-electron coupling will be discussed in the second part of my talk. 2. The switch-over point does not have to be specified since this is a “sudden approximation.’’All that matters is that a value r, exists for which (i) the field due to the core has vanished except for the Coulomb component and (ii) the Born-Oppenheimer approximation is still valid.

SUBPICOSECOND STUDY OF BUBBLE FORMATION UPON RYDBERG STATE EXCITATION IN CONDENSED RARE GASES M.-T.PORTELLA-OBERLI, C.JEANNIN, and M. CHERGUI* Znstitut de Physique Expirimentale Universite‘ de Lausanne Lausanne-Dorigny, Switzerland

The fate of Rydberg states in the condensed phase is still a matter of debate from both a spectroscopic and a dynamical point of view. As far as dynamics is concerned, it is now well established that following absorption of a photon by low-n Rydberg states, a “bubble” is formed around the excited atom or molecule due to the repulsion of the surrounding closed-shell atoms by the Rydberg electron [ 1, 21. In conventional spectroscopy, this is inferred from the strong blue shifts of absorption bands and the strong absorption-emission Stokes shifts of several hundreds of millielectron-volts, which have been observed for pure and doped rare-gas solids or liquids [l-31. Recently, molecular dynamics (MD)simulations have confirmed this picture [4, 51. The aim of this work is (a) to determine the time scales for bubble fonnation and (b) to investigate the dynamics of the cage relaxation process and if any, in particular, look for coherences of the cage vibrations and their disappearance. It has been suggested that bubble formation upon Rydberg state excitation corresponds to an increase of the cage radius by up to 20% following photoexcitation [3-51, and it would be of interest to investigate the dynamics response to the crystal driven out of equilibrium by an ultrashort laser pulse. *Communication presented by M. Chergui Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femrosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

71 1

712

M.-T. PORTELLA-OBERLI, C. JEANNIN, AND M. CHERGUI

The system we have singled out for this study is NO in Ar matrices. The reasons are as follows: (a) The NO molecule has low-lying Rydberg states due to its low ionization potential. This is an advantage over rare-gas atoms, which need vacuum ultraviolet (VUV) photons for excitation [11. (b) The spectroscopy of its Rydberg states in condensed rare gases is now well established [3,6]. This is an advantage over metal atoms (in particular alkali atoms), whose spectroscopy in the condensed phase is still a matter of debate [7]. (c) It is, to our knowledge, the sole molecule to exhibit a fluorescencethat has also been well characterized by time- and energy-resolved studies [3, 81. This fluorescence stems from the lowest lying A (u = 0) state (vibrationless), and it turns out to be of great use for probing the time evolution of the Rydberg state, as we will see later.

* c'

In order to carry out the study some prerequisites are necessary: (i) Observables need to be established that would allow us to probe the cage relaxation process in real time. For that matter a preliminary nanosecond experiment was carried out that consisted in pumping the A(v = 0) level by a first laser and then probing transitions from this level to higher lying Rydberg states by means of a second tunable VIS-XR laser after a time delay of -20 ns. The detected signal is the depletion of the A(u = 0) fluorescence. The results of this study are published elsewhere [9] and show several Rydberg-Rydberg transitions from the A(u = 0) level around 1.1,0.8, and 0.6 pm,which can be used to probe the dynamics going on the A(u = 0) state by transient absorption or fluorescence depletion. Of particular interest are the bands near 0.8 pm which fall close to the fundamental of our Ti-Sa laser. It should be stressed that these transitions correspond to the relaxed configuration of the cage around the A(u = 0) state. (ii) A scheme had to be found to pump the A(v = 0) level that has its maximum absorption at 6.33 eV in Ar matrices (FWHM = 100 meV) and probe its evolution in time on one of the Rydberg-Rydberg transition bands. This was done by taking the fundamental (around 800 nm) of the Nd-Y1F pumped Ti-Sa regenerative amplifier, tripling it and mixing the fundamental and the frequency tripled pulses in a &barium borate (BBO) crystal (100 pm thick) to generate the hard UV pump beam around 200 nm. Part of the fundamental beam was used as probe beam after passing through a delay line. The zero delay between pump and probe pulses and their cross-correlation were done by crossing the pump and probe beams in a second BBO crystal

BUBBLE FORMATION IN CONDENSED RARE GASES

713

in place of the sample and detecting the difference frequency signal around 266 nm. The samples were prepared as described in Ref. 9. The experiment therefore consists in probing depletion of A(u = 0) fluorescence as a function of delay time between the pump and the probe pulses. The difference between the undepleted fluorescence minus the depleted fluorescence is plotted in Fig. 1 as a function of pumpprobe delays for excitation of the A(u = 0) absorption band at 195 nm. The cross-correlation of the pump and probe beams is also shown in Fig. 1 and shows a UV pulse width of -200 fs. It is found that the signal rises with the cross-correlation of pumpprobe pulses and then shows a decay on a time scale of a few picoseconds followed by stabilization at a level some 4040% lower than the maximum. Pursuing the pumpprobe scan up to -120 ps does not yield any additional structures, and the level reached after -2-5 ps (depending on hpump) remains constant throughout. It is tempting to interpret the trace in Fig. 1 as a loss of population in the first few picoseconds following excitation, followed by stabilization of the population. However, in steady-state measurements with synchrotron radiation, it was clearly established that excitation of the A(u = 0) Rydberg state of NO leads first to a cage relaxation followed by a nonradiative relaxation to near-resonant valence levels on a nanosecond time scale, which can compete with the nanosecond time scale of radiative decay to the ground state [lo, 111. Rather, we believe that our signals reflect the evolution in time of an absorption coefficient between the A state and the upper state (or states) to which the probe wavelength is tuned. The point that remains to be clarified is: Does the evolution of the absorption coefficient for the probe transition reflect that of the cage? Recently, Jortner and co-workers [4, 51 carried out simulations on the bubble formation in Arn (n = 12,. ..,200) clusters following Rydberg excitation of a Xe impurity to the 6s state. The interesting feature in relation with our system is that NO-Ar and Xe-Ar have virtually the same difference potential between their ground and first excited Rydberg state [12], and therefore the forces involved therein should be very similar. According to these MD simulations for large clusters (n = 146) at 10 K, following excitation at the band maximum of the Xe (6s) absorption, there is a sudden expansion of the cage with an increase by some 10% of the cage radius, in about 200 fs after the pump pulse. This is followed by an oscillation of the cage for some 2-2.5 ps at a period of 400-500 fs around the radius of the expanded cage. A further expansion of the cage takes place on a long time scale between 2 and 5 ps where a more complex oscillatory pattern is exhibited. For times larger than 5 ps, the cage has reached its final equilibrium configuration, which corresponds to an increase by -20% of the ground-state cage radius. The time

714

--

M.-T. PORTELLA-OBERLI, C. JEANNIN, AND M. CHERGUI

1.0 : 0.8

.-ro

-

6P

0.6 0.4

g 0.2 v1

u

0.0

-:-0.2 -0.4 h

v

-0.6

B 8u1 -0.8

5

-1.0

-1.2 -10000

-5000

0

5000

lo000

15000

20000

25000

Time delay (fs)

Figure 1. Pumpprobe signal for the A(u = 0) Rydberg level of NO in Ar matrices at 4 K. The plotted signal is the fluorescence in the presence of the pump only minus the fluorescence in the presence of pump and probe pulses, as a function of the time delay between them: pump 195 nm, probe 784 nm.The upper curve is the cross-correlation of the pump and probe pulses.

scales for complete cage relaxation inferred from these simulations are very close to the ones obtained from our pumpprobe experiments, and the latter reflect the structural modifications of the cage. However, the trajectories obtained by Jortner and co-workers [4,51 could not explain our pump-probe scans as it would mean that the signal stabilizes after the first 200-400 fs and changes again only after 2 to 5 ps. Our results would be more in line with the adiabatic bubble expansion suggested for free electrons in liquid He [13]. A complete characterization of the bubble expansion mechanism is underway, including other media such as solid Ne and solid hydrogen, and molecular dynamics simulations for the case of NO in Ar matrices are being carried out. Finally, a more extensive account of the methodology of this experiment can be found in ref. 14.

References E. E. Koch, Electronic Excitations in Condensed Rare Gases, Springer Tracts in Modem Physics, Springer, Berlin, 1985. 2. I. Ya. Fugol, Adv. Phys. 27, 1 (1978). 3. M. Chergui, N. Schwentner, and V. Chandrasekharan, J. Chem. Phys. 89, 1277 (1988). I. N. Schwentner, J. Jortner, and


715

4. D. Scharf, J. Jortner, and U. Landman, J. Chem. Phys. 88,4273 (1988). 5. A. Goldberg and J. Jortner, in Femtochemistry, Physics and Chemistry of Ultrafast Processes in Molecular Systems, M. Chergui, Ed., World Scientific, Singapore, 1996, p. 15. 6. M. Chergui, N. Schwentner, and W. Bohmer, J. Chem. Phys. 85, 2472 (1986). 7. N. Schwentner and M. Chergui, J. Chem. Phys. 85, 3458 (1986).

8. F. Vigliotti, G . Zerza, and M. Chergui, in Femtochemistry, Physics and Chemistry of Ultrafast Processes in Molecular Systems, M. Chergui, Ed.,World Scientific, Singapore, 1996, p. 654. 9. G. Zerza, F. Vigliotti, A. Sassara, M. Chergui, and V. Stepanenko, Chem. Phys. Lett., 256, 63 (1996). 10. M. Chergui, R. Schriever, and N. Schwentner, J. Chem. Phys. 89, 7083 (1988). 11. M. Chergui and N. Schwentner, J. Chem. Phys. 91, 5993 (1989). 12. R. Reinginger, private communication; 1996 A. Kohler, Dissertation, Hamburg, 1986. 13. M. Rosenblit and J. Jortner, Phys. Rev. Lett. 75,4079 (1995). 14. M.-T. Portella-Oberli. C. Jeannin and M. Chergui, Chem. Phys. Lett. 259,475 (1996).

DISCUSSION ON THE COMMUNICATION BY M. CHERGUI Chairmun: M. S. Child

L. Woste: Prof. Chergui, the bubble created by your Rydberg electron is very exciting. It should have a remarkable dimension. Do you think that your NO molecule can rotate inside the bubble, just to give more evidence to that bubble? M. Chergui: Indeed, we estimated the increase of the cage radius to be of the order of 5-10% [see Chergui et al., J. Chem. Phys. 89, 1277 (1988)], as compared to the ground state. These are dramatic local modifications comparable to those of large-scale character that could be induced by a hydrostatic pressure. To answer your question, we have carried out preliminary polarization measurements in which the polarization of the Rydberg fluorescence was seen to remain unchanged as compared to that of the pump laser. This would suggest that the molecule does not rotate inside the bubble. K. Yamanouchi: Recently, we investigated the interatomic potential V R , ~ ( Rof ) the Rydberg states of a HgNe van der Waals dimer by optical-optical double-resonance spectroscopy. It was demonstrated that VRyd(R) sensitively varies as a function of the principal quantum number n [ J . Chem. Phys., 98, 2675 (1993); ibid., 101, 7290 (1995); ibid., 102, 1129 (1995)], and in the lowest Rydberg states of Hg(7 3S~)Neand Hg(7 'So)Ne, the interatomic potentials exhibit a distinct barrier at around R 4 A. The existence of the barrier was interpreted in terms of a repulsive interaction caused by the 7s Rydberg

-

716

M.-T.PORTELLA-OBERLI, C. JEANNIN, AND M. CHERGUI

electron and the attached rare-gas atom on the basis of the quantum defect orbital approach. As n increases, it was found that the barrier disappears in the Franck-Condon region and VRyd(R) converges rapidly to the interatomic potential of the ion core, Vion(R), of HgNe+. We also determined the ion-core potential Vion(R) as a convergence limit of VRyd(R). By assuming VRyd(R) to be represented by a simple sum of the ioncore potential and the repulsive exchange interaction, Vex(K), that is, VR~~(R = )Vion(R) + Vex(R), the repulsive contribution Vex(R) for the lowest Rydberg state (n = 7) was extracted. The shape of Vex(R), which has a barrier at K - 3 A,was very similar to that of the probability distribution of the Rydberg electron for n = 7 evaluated using its quantum defect orbital wave function. Therefore, it was concluded that the barriers in the interatomic potentials for Hg(7 3S1)Ne and Hg(7 'So)Ne are caused by the exchange repulsive interaction between the 7s Rydberg electron and the rare-gas atom. The bubble of the Rydberg electron of NO in the rare-gas matrix observed by Dr. Chergui can be interpreted by a similar model, which we proposed for the Rydberg states of a free HgNe van der Waals molecule. The driving force for the structural relaxation in the cage of rare-gas atoms could originate from the repulsive exchange interaction between the Rydberg electron of the NO moiety and the surrounding rare-gas atoms. M. Chergui: This goes along with our interpretation. The A *C+ state has n = 3 and a quantum defect of -1.1; that is, it shields the ionic core quite efficiently and, in the ground-state configuration of the molecule-matrix system, the Vion(R) you mentioned should be negligible. This does not mean that at much larger NO-Ar distances a weak van der Waals attraction does not appear for the A state and indeed it has been observed in molecular beam studies [see Quaid et al., Chem. Phys. Lerf. 227,54 (1994); Tsuji et al., 1.Chern. Phys. 100,5441 (199411. Concerning higher Rydberg states of NO in rare-gas matrices which we probe from the A(v = 0) level [Zerza et al., Chem. Phys. Left., 256, 63 (1996)], we also invoke a balance between Vion(R) and Vex(R) for increasing n in the interpretation of our results. B. A. Hess: My naive view of Rydberg states in the condensed phase was that they are simply quenched. Do I have to change this picture on account of your experiments? M. Chergui: Yes, in the past the first experiments on the absorption of Rydberg states in condensed rare gases showed that either they disappeared from the spectrum or only the lowest ones, bearing


717

a parentage with the atomic or molecular states, could be observed while the higher ones were interpreted as Wannier excitons (see, e.g., Schwentner, Koch, and Jortner, in Electronic Excitations in Condensed Rare Gases, Springer Tracts in Modem Physics, Berlin, 1985). More recently, we showed that higher states of NO in rare-gas matrices still bear a molecular character [Chergui et al., J. Chem. Phys. 85, 2472 (1986); Zerza et al., Chem. Phys. Lett., 256, 63 (1996)l. Now concerning the fluorescence, while rare-gas atoms in rare-gas matrices do exhibit fluorescence (see the above reference of Schwentner et a].), NO is, to our knowledge, the sole molecule to exhibit Rydberg fluorescence in the condensed phase. That and the fact that NO has low-lying Rydberg states are the reasons we singled it out as our model system in this study.

T. Kobayashi 1. Prof. Chergui, how about doing experiments on the polarization dependence? 2. If the delay time is made very long, an oscillatory structure may be observed in case the sensitivity is high enough. M. Chergui 1. As mentioned in my reply to Prof. Woste (see above), such experiments are underway. 2. We have scanned up to -120 ps and seen no oscillatory structure whatsoever.

M. Herman: Do you see any evidence of NO dimers? Could they possibly interfere? M. Chergui: Dimers absorb at -207 nm in rare-gas matrices and excitation of this band does not yield any fluorescence [Chergui et al., Chem. Phys. Lett. 201,187 (1993)l. Furthermore, our detection is based on the fact that we record the depletion of the fluorescence of one of the A(0, u”) bands due to NO monomers. There is therefore no possibility that NO dimers could interfere with our measurements. A. H. Zewail: Prof. Chergui’s observations are very interesting. As discussed in this conference, coherent wavepacket motion has been observed in condensed phases in many laboratories. In Prof. Chergui’s experiment it is now possible to study the time scale for “bubble” formation. Molecular dynamics should tell us the nature of forces which maintain any coherence in such systems.

718

M.-T. PORTELLA-OBERLI, C. JEANNIN, AND M.CHERGUI

L. Woste: A speculative remark Could the Rydberg molecule even form a bubble in a gas cell (scattering environment)? M. Chergui: “Bubble” formation is a consequence of the change in molecule-environment interactions. If the density of the gas is such that you probe, from the ground state, a distribution of distances for which this interaction is repulsive in the excited state, then you will form a “bubble.” The relevant regime of densities need not be very high since Rydberg states are extremely sensitive to the environment [for the case of NO, see Miladi et al., J. MoZ. Spect. 55,81 (1975)and 69,260 (1978); Morikawa et al., J. Chem. Phys. 89,2729 (1988); and Chergui et al., Chem. Phys. Let?. 216, 34 (1993)l. W. E. Schlag: In the future we will find these long lived states in other media as well, I think. Bound states on surfaces may become one such example.


GENERAL DISCUSSION ON MOLECULAR RYDBERG STATES

AND ZEKE SPECTROSCOPY PART 11 Chairman: M. S. Child

R. D. Levine: Prof. Jungen, your results for the low Rydberg states of molecules with exceptionally high dipole moment prompt me to ask if you can extend such multichannel quantum defect theory (MQDT) computations to the region of high n’s. The motivation is that the coupling of the zero-order inverse Bom-oppenheimer states is due to the electrical anisotropy of the core. With a large dipole moment this coupling will be quite strong and, because of the rather long range of the electron-dipole potential, can be quite effective. Ch. Jungen 1. What I would do is the following: (a) For low 1 take the quantum defect matrix elements as they result from the fittings. (b) For high I evaluate them in elliptic coordinates assuming no penetration (the effects of the dipole field are then fully included); in this way a full calculation with an arbitrary number of 1 components can be carried out. 2. Notice that in our calculation we have both closed and open ionization channels. This means that the quantum defect/frame transformation approach appears to work very well both below and above the “critical region” for which n - 100,. . . ,1OOO.I would find it surprising if the approach failed in that region.

W. H. Miller: I believe that the reason the multichannel quantum defect theory (MCQDT) works well is that it assumes the ordinary Bom-oppenheimer approximation (i.e., that the electron follows the molecular vibrational and rotational motion adiabatically) in the region close to the molecule, but not so in the region far from the molecule (where the electron moves more slowly than molecular vibration and rotation). The “frame transformation” provides the transition between 719

720

GENERAL DISCUSSION

the body-fixed basis function in the interior region and the space-fixed basis functions in the exterior region.

Ch. Jungen: All I can say at this point is that the quantum d$fect/frame transformation approach appears to work for CaF around n = 14. We have chosen CaF, which is so highly polar, in order to ascertain this, and this is also the reason why we have made ab initio calculations in order to compare experiment and theory. I suppose that the dipole field is averaged out by the rotational motion, and thus one can get away with the customary frame transformation approach. L. S. Cederbaum: Prof. Jungen, you mentioned that MCQDT takes into account nonadiabatic effects. I would like to point out that this approach only considers those nonadiabatic effects that arise due to the motion of the Rydberg electron. The nonadiabatic effects in the ion core are not considered. These effects can often be substantial. The MCQDT could probably be extended to include long-range potentials other than Coulomb, for instance, dipole potentials.

Ch. Jungen: You are quite right. The Renner-Teller effect in the water ion is an example of this kind. The strong vibronic coupling in the core leads to nonadiabatic effects in the Rydberg states of the neutral species, which are of course not accounted for by the coordinate dependence of the quantum defect and have to be taken into account separately. With regard to your second remark I would like to say this: The calculations I showed for CaF and BaF Rydberg states are examples where the core field is a Coulomb field but where a very strong dipole field is superimposed. Here we have used the generalized version of quantum defect theory, which takes account of this modified long-range field.

B. A. Hess: Prof. Jungen, in your talk you emphasized that you don’t have to calculate matrix elements of a/aQ or Coriolis coupling. My impression is that this is due to your most appropriate choice of a diabatic basis, which is generally what ab initio quantum chemists do when they want to avoid singularities in the adiabatic basis. On the other hand, the absence of explicit Coriolis coupling matrix elements is due to the transformation to a space-fixed coordinate system.

RYDBERG STATES AND ZEKE SPECTROSCOPY I1

72 1

Ch. Jungen: The vibronic coupling is included through the R dependence of the diagonal and off-diagonal quantum defect matrix elements. The effective principal quantum number, or more precisely the quantum defect, gives a handle on the electronic wave function. The variation with R then contains the information concerning the derivative with respect to R of the electronic wave function.

T.P. Softley: I would like to ask Prof. Jungen if there is any experimental evidence for the need to include singlet-triplet interaction in a quantum defect description of H2? Ch. Jungen: Oka has recently obtained Fourier transform infrared (FTIR) emission spectra of the 5g4f transition in H2. These spectra are currently being analyzed. They show singlet-triplet splittings of the order of 0.04 cm-', which thus are comparable in magnitude to the hyperfine splittings of Hi. (This had actually already been observed by Jungen, Dabrowski, Herzberg and Vervloet [J. Chem. Phys. 93,2289 (1990)l.) The new aspect in Oka's work is that satellite transitions are observed that demonstrate that a recoupling of angular momenta occurs when the singlet-triplet splitting becomes very small.

R. W. Field: I'd like to make a comment and ask two questions. My comment is the following. The internuclear distance (R) dependence of the quantum defect ( p ) expresses one of the most important mechanisms for the exchange of energy between the Rydberg electron and the vibrating nuclei. In a sense, the quantum defect function p ( R ) is a generalization of the Born-oppenheimer potential-energy curves. I am concerned with convenient experimental methods for determining d p / d R . There appear to be at least three ways to determine this crucial coupling constant, d p / d R . The first involves analysis of Au = -An* = 1 perturbations that occur at relatively low n* when the Rydberg orbit period is equal to the vibrational period. For example, for we = 500 * * cm-', the n , u = 0 - n - 1, u = 1 perturbation occurs at n* = 7.6. If the quantum defect is expanded about R:, the equilibrium internuclear distance of the ion core,

722

GENERAL DISCUSSION

the nonzero part of the perturbation matrix element (eu)p(R)le’u’)= (e(dp/dRle’)(ul(R - R:)(u f 1) provides an experimental value of d p / d R . A second way of sampling d p / d R is through the rate of vibrational autoionization. The In*, u+ = 1) state is autoionized by the ionization continuum of u+ = 0.For 0: = 500 cm-‘, all u+ = 1 levels with n* 2 14.8 lie above the u+ = 0 ionization limit. Once again, the (ul(R - R:)lu f 1) vibrational matrix element picks out the d p / d R coupling constant. A measurement of the vibrational autoionization rate gives an independent measure of d p / d R . The third method seems very different from the first two. However, it most elegantly illustrates how an entire Rydberg series forms a single structural unit. Suppose the potential curves for the ion, V+(R),and for a low-n* member of a Rydberg series, V,,(R), are both known. Then the R dependence of the vertical ionization energy determines d p / d R at R:,

p ( R ) = n - 4 1 f 2 [ V + ( R) Vn*(R)]-’/’ where d p / d R at R: can be evaluated directly or expressed as a power series in w t -we,,* and R t - Re,,*. This last method is ideally suited for quantum chemical calculations, because the problem of diffuse orbitals is absent for the ion core and minimized for a low-n* Rydberg state. (It also implies useful n * scaling laws for u: - w ~ , and ~ + - Re,n* at the low-n* end of a series where the Born-oppenheimer approximation is sufficiently well obeyed to permit construction of potential-energy curves.) At last, here are the two questions: 1. Have there been examples where the intrachannel d p / d R has been independently determined by the perturbation matrix element, autoionization rate, and V + ( R )- V,,(R) methods? 2. Do the values of d p / d R determined by the various methods typically agree? If not, which method do you believe is most reliable?

Ch. Jungen: Yes, I would agree with Prof. Field that autoionization widths in the continuum, spectral perturbations at high n, and the


723

distortion of the vibrational frequency at low n with respect to that of the ion in principle provide the same information on vibration-electron coupling. This has been demonstrated in the case of H2.

B. Kohler: My question to T. Softley ties in to one of the major themes of the meeting, namely coherence. In your presentation you

briefly mentioned that it may be important to consider an initially coherent superposition of states in the preparation step of experiments on highly excited Rydberg states. Several groups have now prepared coherent electronic wavepackets using picosecond (and shorter) pulses. Would this kind of an initial state be useful for any of the classes of pulsed-field ionization experiments that you have described?

T.P. Softley: There is little doubt that in most ZEKE experiments using nanosecond lasers the Rydberg level structure is so dense that a coherent superposition of levels is populated initially, and the correct description of the dynamics should be a time-dependent one. It is possible that some control over the dynamics could be achieved using some of the methods described earlier in the conference, for example, simultaneous excitation through three-photon and one-photon transitions, using third-harmonic generation. The MQDT calculations we have performed yield a solution to the time-independent Schriidinger equation; in principle the time-dependent information can be obtained by Fourier transformation, possibly using the vibrogram method described by Gaspard. M. S. Child: I would like to ask to T. Softley what limitations apply to vibrational states of H2+ that can be prepared by the mass-analyzed threshold ionization (MATI) technique. In particular it would be interesting to known whether the states observed in Carrington’s H3+predissociation excitation experiment [ 11 could be more selectively prepared by the reaction H;(u)

+ H2 -+H$(u’) + H

with known initial u values. 1. A. Carrington and R. A. Kennedy, J. Chern. Phys. 81, 91 (1984).

T. P. Softley: It has been demonstrated by Hepburn et al. that H; can be formed in u = 13 by a pulsed-field ionization technique from a

724

GENERAL DISCUSSION

low vibrational level of the E, F state of Hz.It would be feasible for us to study reactions of Hi (v = 13) in our experiments.

S. R. Jain: I have two questions for Prof. Levine:

1. With respect to the debate on Born-Oppenheimer vs. inverse Born-Oppenheimer regimes, we know that adiabaticity leading to Born-Oppenheimer description is associated with a geometric phase. Similarly, it is not difficult to envisage a geometric phase with an inverse Born-Oppenheimer description. As we have seen in the talk of Prof. Levine, in situations where there are time-varying fields, does it not give us an opportunity to settle (to some extent) this debate by observing (or not observing) this phase? 2. What guarantees the area-preserving nature of your classical map PI?

1. E. Rabani, R. D. Levine, and U. Even, Bet Bunsenges. Phys. Chern. 99, 310 (1995).

R. D. Levine

1. The discussion of the geometric phase requires that the zero-order basis be a good approximation. If there is mixing, then the concept of a phase factor is replaced by a unitary matrix. Indeed, one can think of the frame transformation of MQDT as just such a matrix. If there is no coupling, then the geometric phase is, in essence, what gives rise to the quantum defect of Rydberg states. 2. As to your other question, a generating function for our map is given by

where k is the index of the orbit and E is the energy of the Rydberg state, E = - 1/2n2. The map is given by

W. H. Miller: In treating electronically nonadiabatic processes one often introduces (usually on physical grounds) a diabatic model, which has a nondiagonal electronic potential matrix, and then neglects any remaining derivative coupling. The total (vibronic) wave function is


725

then a multi-(electronic) state expansion, and no “geometric phase” needs to ever be introduced (for it is implicitly there!). The geometric phase could arise if one diagonalized the diabatic Hamiltonian, that is, going to the adiabatic representation.

B. A. Hess: The notion of a geometric phase generally requires an adiabatic situation, where an adiabatic connection and adiabatic transport can be defined. In the presence of many nearby avoided crossings, in a highly nonadiabatic situation (as in the case of the inverse Bom-oppenheimer regime), the notion of a geometric phase is ill defined. S. R. Jain: After the initial work of Berry, it is now well-known that cyclicity or adiabaticity is not necessary to associate geometric phase from the work of J. Anandan and Y.Aharonov [Phys. Rev. Left. 58, 1863 (1989)l.

M.Lombardi: What is not needed is the validity of the adiabatic approximation, that is, that there is no transition between adiabatic states. But the geometric phase is defined by following states along a path in parameter space (here nuclear coordinates) with some continuity condition. In the diabatic representation, there is no change of basis at all and thus the geometric phase is identically zero. Do not confuse adiabatic basis (which is required) and adiabatic approximation (which may not be valid). Consider the example of the Jahn-Teller interaction at a conical intersection of two electronic surfaces of energy, the pre-Berry example of geometric phase studied by Longuet-Higgins. A doubly degenerate electronic state at a highly symmetric reference configuration (e.g., an E state of an X3 molecule at equilateral configuration) is split by a doubly degenerate vibration [an e vibration described by a radial ( r )and an angular (4) parameter]: One of the two degenerate states is lowered in energy such that the minimum occurs for a lower symmetry (isosceles instead of equilateral) for a given value of the distortion r = ro and any phase 4. The vibrational energy levels of that molecule can be (and usually are) computed by diagonalizing a Hamiltonian matrix set up in a diabatic basis, that is, a fixed basis independent of r and 4. Then if the vibronic interaction is strong, the lower energy levels computed that way follow a law El = BZ(Z + 1) with half-odd integer I , in a system with absolutely no spin. In the diabatic basis, this is a nonunderstandable peculiarity that pops out of the black box that effects the diagonalization of the matrix. However, as pointed out by

726

GENERAL DISCUSSION

Longuet-Higgins,one can also set up the problem in an adiabatic basis, obtained by rediagonalizing the electronic potential at every value of r and 4. One obtains in that way the two adiabatic electronic surfaces, and the lower energy levels are confined to the lower surface, avoiding the center of the cone, that is, the reference equilateral configuration. Longuet-Higgins rationalized this half-odd integer value of I by noticing that, when one follows by continuity the adiabatic basis along a full turn around the center, the two electronic states change sign, exactly as does a spin after a full turn, and this change is the geometric phase. (But the total wave function, composed of electronic and nuclear parts, is unchanged, the change of sign of the electronic state being compensated for by a change of sign of the nuclear wave function, so that there is no contradiction with unicity of the total wave function.) So this halfodd integer value of I is an observable consequence of geometric phase, connected with properties of adiabatically following a basis in parameter space. This property is evidenced only if the vibronic interaction is strong, causing enough levels in the lower adiabatic surface to recognize this Ef law; otherwise there are only weak perturbations between pairs of degenerate E states. Nevertheless, nothing forbids making all computations in the diabatic basis, which has no geometric phase. The preceding discussion pertained to dynamical situations in intermediate cases, in which it is necessary to take into account interactions between adiabatic states, enabling transitions between them.

M. S. Child: With regard to the question about Berry’s geometric phase, the most common molecular manifestation is the half-odd angular momentum associated with the Jahn-Teller effect. Two cases may be distinguished in the context of Rydberg spectroscopy, according to whether the electronic degeneracy involves the excited Rydberg orbitals or states of the positive ion core. Examples of the former type are well established for NH3 [l] and H3 121 and a quantum defect treatment has been given for the latter species by Stephens and Greene [3]. The theoretical situation is more complicated if the ion core is Jahn-Teller active. Mueller-Dethlefs [4] has given a very complete picture of the Jahn-Teller dynamics of C&j+, but the consequences for interaction with benzene Rydberg electrons remain to be worked out. 1. 2. 3. 4.

M. N. R. Ashfold, R. N. Dixon, and R. J. Stickland, Chem. Phys. 88 463 (1984). G. Herzberg, Faraday Discuss. Chem. Soc. 71, 165 (1981). J. A. Stephens and C. H.Greene, Phys. Rev. Lett. 72, 1624 (1994). I. Fischer, R. Linder, and K. Muelier-Dethlefs. J . Chem. SOC. Faruday Trans. 90, 2425 (1994).

TRANSITION-STATE SPECTROSCOPY AND PHOTODISSOCIATION

PHOTODISSOCIATION SPECTROSCOPY AND DYNAMICS OF THE VINOXY (CH2CHO) RADICAL D. L. OSBORN, H. CHOI, and D. M. MUMARK* Department of Chemistry University of California Berkeley, California and Chemical Sciences Division Lawrence Berkeley Laboratory Berkeley, California

CONTENTS I. Introduction 11. Experimental 111. Results IV. Discussion A. CH3 + CO Channel B. D + CD2CO Channel V. Conclusions References

I. INTRODUCTION Photodissociation experiments have become one of the most valuable tools in chemical physics for the purpose of understanding how excited electronic states couple to the dissociation continuum. These experiments, and the the*Report presented by D. M.Neumark Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Eme Scale, Mth Solvay Conference on Chemistry. Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

729

730

D. L. OSBORN, H. CHOI, AND D. M. NEUMARK

ory developed to explain them, have yielded considerable insight into the variety of dynamical processes that occur subsequent to electronic excitation [l]. From these studies, one hopes to obtain bond dissociation energies, characterize the symmetry of the excited state, measure the product branching ratios, and determine if the excited state undergoes direct dissociation on an excited-state surface, predissociation via another excited state, or internal conversion to the ground state followed by “statistical” decay to products. There are two clear recent trends in photodissociation experiments. One is to perform extremely detailed mesurements on systems where the basic photodissociation dynamics are reasonably well understood. As an example, experiments have been reported on CD31 in which the parent molecule is state selected and oriented prior to photodissociation [2]; multiphoton ionization of the CD3 fragment combined with imaging detection yields angular and kinetic-energy distributions for each product rotational state. An alternate direction is to apply well-developed photodissociation techniques to larger and/or more complex species, with the goal of seeing what new phenomena appear that are absent in the more commonly studied small-model systems. The work of Butler [3] on acetyl halides is a nice example of this; these studies have probed subtle nonadiabatic effects that dramatically affect the product branching ratio. The approach taken in our laboratory combinesboth of these trends. Specifically, we have developed a new experiment that allows us to study, for the first time, the photodissociationspectroscopyand dynamics of an importantclass of molecules: reactive free radicals. This work is motivated in part by the desire to obtain accurate bond dissociation energies for radicals, in order to better determine their possible role in complex chemical mechanisms such as typically occur in combustion or atmospheric chemistry. Moreover, since radicals are open-shell species, one expects many more low-lying electronic states than in closed-shell molecules of similar size and composition. Thus, the spectroscopy and dissociation dynamics of these excited states should, in many cases, be qualitatively different from that of closed-shell species. While one might expect that the techniques developed for photodissociation studies of closed-shell molecules would be readily adaptable to free radicals, this is not the case. A successfulphotodissociation experiment often requires a very clean source for the radical of interest in order to minimize background problems associated with photodissociating other species in the experiment. Thus, molecular beam photofragment translation spectroscopy, which has been applied to innumerable closed-shell species, has been used successfully on only a handful of free radicals. With this problem in mind, we have developed a conceptuallydifferent experiment [4] in which a fast beam of radicals is generated by laser photodetachment of mass-selected negative ions. The radicals are photodissociated with a second laser, and the fragments are detected in coinci-

PHOTODISSOCIATION OF THE VINOXY RADICAL

73 1

dence with a position- and time-sensing detector. This detection scheme yields high-resolution photofragment energy and angular distributions for each product channel. The negative ion production scheme ensures that only the desired radicals are produced. We have studied several radicals with this experiment during the last few years. In this chapter we report new results on the photodissociation spectroscopy and dynamics of the vinoxy radical, CH2CHO. The vinoxy radical is an important intermediate in hydrocarbon combustion; it is one of the primary products of ethylene oxidation. The electronic absorption spectrum was obtained by Hunziker [5],who found two bands with origins at 8000 and 28,760 cm-' . On the basis of ab initio calculations by Dupuis [6], these were assigned to transitions from the X 2A" ground state to the A 2A' and B *A" states, respectively. The higher energy B 6 X band extends at least to 35,700 cm-I . In Hunziker's experiment, this band shows some partially resolved features near the origin, but little structure remains above 3 1,200cm-I .Inoue [7] and Miller [8] have measured laser-induced fluorescence (LIF) spectra of the B +- X band, and the appearance is very different. These spectra consist of only a few sharp features between the origin and 30,200 cm-I ;no fluorescence signal is observed for excitation to the blue of this. The comparison of the absorption and LIF spectra implies that rapid predissociation occurs beyond 30,200 cm- ,thereby quenching any fluorescence. This is consistent with a recent hold-burning study by Gejo et al. [9]. Hence, this is an attractive speciesfrom the perspective of our photodissociation experiment, since we can examine the competition between predissociation and fluorescence in more detail. The primary photochemistry of the vinoxy radical is also of interest. Several relatively low-lying dissociation channels are available [ 10, 111:

'

--

CH2CHO CH3 + CO CH2CHO H + CHzCO CH2CHO ---t CH2 + HCO

A E = 0.05 eV AE = 1.45 eV AE = 4.1 eV

(1) (2) (3)

Channel (l), the lowest energy channel, requires a hydrogen shift, while (2) and (3) are simple band fission channels. Channels (1) and (2) are accessible from all levels of the B state, whereas channel (3) can be accessed only at the blue edge of the B c X band. Our experiment reveals that all levels of the B state in CH2CHO predissociate. We observe dissociation to channels (1) and (2). Kinetic energy distributions at several dissociation energies indicate that CH3 + CO production takes place following internal conversion to the ground-state potential-energy surface. These distributions are indicative of a barrier of 1-2 eV along the reaction coordinate to dissociation on the ground-state surface. A comparison of our results to photodissociation studies of the acetyl (CH3CO) radical

732

D. L. OSBORN, H.CHOI, AND D. M. NEUMARK

[12, 131 suggests that this barrier corresponds to the isomerization barrier between vinoxy and acetyl on the ground-state surface.

II. EXPERIMENTAL The fast-beam photodissociation instrument has been described in detail previously [4].Briefly, a beam of vinoxide (CH2CHO-) anions is generated by bubbling O2 at 3 atm through acetaldehyde at -78°C. The deuterated species (CD2CDO-) is made in a similar fashion. The mixture is introduced to the spectrometer through a pulsed molecular beam valve. Ions are generated by means of a pulsed discharge assembly attached to the faceplate of the valve through which the gas pulse passes. By firing the discharge just after the valve is open, one forms a variety of ions that cool significantly (-50 K) in the resulting free jet expansion. The pulsed beam passes through a skimmer, and negative ions in the beam are accelerated up to 8 keV and mass separated by time of flight. The vinoxide ions are then photodetected with an excimer-pumped dye laser. The CH2CHO- and CD2CDO- were photodetached at 663 nm (1.870 eV) and 667 nm (1.859 eV), respectively. These energies are just above the electron affinities [14] of CH2CHO (1.824 eV) and CD2CDO (1.818 eV). Remaining ions are deflected out of the beam, leaving a fast pulse of mass-selected vinoxy radicals. These are photodissociated with a second excimer-pumped dye laser. The photofragments are detected with a microchannel plate detector that lies on a beam axis 1 m downstream from the dissociation region. A blocking strip across the center of the detector prevents parent radicals from reaching the detector, whereas photofragments with sufficient recoil energy miss the beam block and strike the detector. These fragments are generally detected with high efficiency (up to 50%) due to their high laboratory kinetic energy. Three types of measurements were performed in this study. First, photodissociation cross sections were measured, in which the total photofragment yield was measured as a function of dissociation photon energy. In these experiments, the electron signal generated by the microchannel plates is collected with a flat metal anode, so that only the total charge per laser pulse is measured. The beam block is 3 mm wide for these measurements. To perform photodissociation dynamics experiments, the dissociation laser is tuned to a transition where dissociation occurs. Most of the dynamics studies described here employ a time-and-position sensing photofragment coincidence detection scheme [ 151 in which a dual wedge-and-strip anode [4] is used to collect the electron signal from the microchannel plates. For each dissociation event, we measure the distance R between the two fragments on the detector, the time delay T between their arrival, and the individual displacements of the two fragments, r1 and r2, from the detector center. From this we obtain the center-of-mass recoil energy ET, the scattering angle with


733

respect to laser polarization, 8, and the two photofragment masses ml and m2 via

e =tan-'

($)

(4)

Here EO and uo are the ion beam energy and velocity, respectively and 1 is the flight length from the photodissociation region to the detector. An 8-mmwide beam block is used for these measurements; a narrower block results in crosstalk between the two halves of the anode. The coincidence scheme works very well so long as m1/m2 I -5 and is thus quite suitable for channel (1). However, for channel (2), the fragment mass discrepancy is too large to perform this type of measurement. The recoil velocity of the heavy fragment is too small to clear the beam block, and the laboratory energy of the H (or D) atom is so low that their detection efficiency drops considerably. Since either of these effects makes coincidence measurements difficult, if not impossible, we performed a somewhat less complex experiment to detect and analyze channel (2). The flat-anode configuration of the detector (with the 3-mm beam block) was used, and we simply measure the time-of-flight distribution of all fragments at the detector. This was used previously to measure the kinetic-energy release in NCO photodissociation [161. For the present study, we only investigated CD2CDO since the fragment mass ratio for channel (2) is smaller than for CH2CHO. As will be seen below, our ability to measure the kinetic-energy distribution for channel (1) via the coincidence scheme allows us to subtract the contribution of this Channel from the time-of-flight measurement. This enables us to isolate the contribution of channel (2).

HI. RESULTS Photodissociation cross sections for CH2CHO and CD2CDO are shown in Fig. 1. Both spectra consist of sharp, extended vibrational progressions indicative of predissociation. A comparison of Fig. 1 with the LIF and absorption spectra [5, 7, 81 shows that we observe predissociation all the way to the origin of the B X band; this is labeled peak 1 in Fig. 1. We observe photodissociation over the entire range of the absorption band. However, peaks 1-7 are the only peaks seen in the LIF spectrum. Peaks 1,2,5, and 6 are particularly prominent in the LIF spectrum; the latter three peaks are assigned

.-

734

29000

D. L. OSBORN. H.CHOI, AND D. M. NEUMARK

3oooo

31000

32000

33000

Photon Energy (cm-') Figure 1. Photodissociation cross section of CHzCHO (top) and CDzCDO (bottom). Peaks 1-7 are the only features seen in the laser-induced fluorescence spectrum of CH2CHO (Ref. 8).

by Yamaguchi [17]to the 9;, 8;, and 7h transitions involving the C-C-0 bend, C-C stretch, and CH2 rock, respectively. While peak 1 is the most intense peak in the LIF spectrum, it is the weakest of the four in Fig. 1 . This shows that the competition between LIF and predissociation tilts sharply in favor of the latter even over the small energy range spanned by peaks 1 4 . Photofragment coincidence data were taken at several of the peaks in Fig. 1. Mass analysis of the fragments showed that only coincidences corresponding to channel (l), CH3 + CO (or CD3 + CO), were seen at all dissociation wavelengths examined. As discussed previously, the time and position data yield a coupled translational energy and angular distribution P(ET,e), which can be written as

Here P(ET) is the angle-independent translational energy distribution, and ~ ( E Tis) the (energy-dependent) anisotropy parameter, with - 1 5 P I 2. Fig. 2 shows the P(ET) distributions for CH2CHO at the four disso-


735

1

Translational Energy (eV) Figure 2. Translational energy distributionsP ( E T ) for CH2CHO four dissociation energies corresponding to peaks A-D in Fig. I .

+

CH3

+ CO taken at

ciation energies indicated by peaks A-D in Fig. 1. The distributions all peak at nonzero translational energy. The most striking feature of Fig. 2 is the insensitivity of the P(&) distributions to photon energy; only a very small shift toward higher ET is seen over the entire energy range that was probed. The P ( E T ) distributions for CD2CDO photodissociation are similar and again show little variation with photon energy. Figure 3 shows the average anisotropy parameter at each photon energy for CH2CHO photodissociation. This shows that the angular distributions are isotropic (fl 0) at B state excitation energies below 1000 cm-’, but for higher energies we find fl in the range of 0.4-0.5. Note that the angular distribution becomes anisotropic in the energy range where fluorescence is quenched. Finally, the photofragment time-of-flight distribution for CD2CDO photodissociation at 31,980 cm-I is shown in Fig. 4. This will be analyzed in detail in the next section, but for now it suffices to point out that the wings in the distribution are from D atoms, indicating that photodissociation channel (2) to D + CD2C0 is indeed occurring.

736


Figure 3. Anisotropy parameter 0 for CH2CHO energies.

3.4

3.5

3.6

3.7

-.

CH3

3.8

+ CO at several dissociation

3.9

4.0

Time of Flight (ps) Figure 4. Photofragment time-of-flight spectrum for CD2CDO excited at 3 1,980 cm- I , including experimental results, contributions from D + CD2CO and CD3 + CO channels, and total simulated spectrum. The CD3 + CO contribution is obtained from an independent measurement of the energy and angular distribution at this energy using the photofragment coincidence detection scheme. The inset shows the translational energy distribution for the D + CD2CO channel (with 0 = 1.2) used to simulate the contribution of this channel to the timeof-flight spectrum.


737

IV. DISCUSSION

A. CH3 + CO Channel The shape of the P(Er) distributions for channel (1) and their insensitivity to excitation energy is characteristic of statistical decomposition over a barrier. Distributions of this type are often seen in infrared multiphoton dissociation of molecules in which there is a barrier to product formation on the groundstate potential-energy surface [ 181 and also for electronic excitation in which internal conversion to the ground state occurs prior to dissociation [19]. The rationale for these distributions is that dissociation is statistical up until the top of the barrier. At this point, the most likely trajectories have nearly zero translational energy since this maximizes the number of vibrational states perpendicular to the reaction coordinate that can be populated. This is true regardless of the total excitation energy, provided dissociation is sufficiently slow so that energy randomization can occur. However, once the barrier is traversed, the photofragments move apart too quickly for the newly available energy to be fully randomized, so that the translational energy distribution peaks at some fraction of the barrier height, typically 4040%. It therefore appears that channel (1) occurs via internal conversion from the initially excited B (2A”) state to the X (2A”) state and that the peaking of the P(&) distributions around 1 eV translational energy is caused by a barrier between 1.2 and 2.4 eV on the ground-state potential-energy surface. We now consider the location of this barrier along the reaction coordinate. Internal conversion from the B state will result in highly vibrationally excited CH2CHO. In order to dissociate to CH3 + CO, this species must first isomerize to the acetyl radical, CH3C0, and then break the C-C bond in this radical. One certainly expects a barrier to be associated with isomerization, and the photodissociation experiments by North et al. [121 indicate that there is a barrier associated with breaking the C-C bond in the acetyl radical. Hence, there should be two barriers along the reaction coordinate, as shown in Fig. 5 . Given this reaction coordinate, we now want to consider which barrier is responsible for the maximum in the P(&) distributions. In North’s experiment, acetyl chloride (CH3COCl) is photodissociated at 248 nm to yield C1 + CH3CO. Time-of-flight analysis of the photofragments shows that the CH3C0 radical undergoes secondary dissociation when it contains more than 17 f 1 kcal/mol of internal energy, indicating that this is the barrier height for the reaction CH3CO + CH3 + CO. This reaction is endothermic by only 11 kcal/mol, however, so the barrier is 6 kcal/mol (0.26 eV) above CH3 + CO products. These energetics are consistent with recent ab initio calculations by Deshmukh et al. [13]. Thus, according to our model for the dissociation, this barrier cannot be responsible for the peak

738


Figure 5. Energetics for CH2CHO dissociation, including qualitative picture of the reaction coordinate for CH3 + CO production on ground-state potential-energy surface. See text for discussion of barrier heights.

at 1 eV in the P(&) distributions, since one generally expects only a fraction of the barrier height to manifest itself in this manner. This suggests that the peak in the translational energy distributions reflects the isomerization barrier in Fig. 5. If this is so, the barrier should lie 1-2 eV above the products, and preliminary ab initio calculations in our group indicate that this is a reasonable range of values. Our interpretation implies that once isomerization occurs, the resulting energized CH3CO falls apart very rapidly, before energy randomization can happen. Otherwise, the smaller barrier would determine the product translational energy distributions. This is what happens in the infrared multiphoton dissociation (IRMPD) of CH3N02, where the CH30 + NO product channel is formed by isomerization to CH30NO followed by dissociation with no exit barrier [18]. The translational energy distribution peaks at zero, consistent with the dynamics being determined by the absence of an exit barrier for dissociation rather than the isomerization barrier, which lies about 0.6 eV above the products. However, the CH30N0 well lies 1.8 eV below CH30 + NO, whereas the CH3CO well is only 0.7 eV deep relative to the exit barrier to dissociation. In addition, the ratio of the excitation energy to the well depth is much higher in our experiment than in the IRMPD study (>5 vs. 4).Thus, once isomerization to CH3CO occurs, dissociation may be so rapid that the well does not noticeably affect the final state dynamics. Alternatively, concerted elimination to CH3 + CO may occur at the top of the isomerization barrier, in


739

which case the resulting reaction path would not pass through the CH3CO well (and over the second barrier) at all. Clearly, the detailed dissociation mechanism on the ground-state surface will have to be examined in more detail once a reasonable potential-energy surface is developed. At excitation energies close to the band origin, fluorescence still competes effectively with dissociation. We interpret this to mean that the initial internal conversion to the ground state is sufficiently slow at these energies that fluorescence is competitive. However, it appears that as the vibrational energy in the B state increases, internal conversion becomes so rapid that fluorescence is completely quenched. This is consistent with the angular distributions. These are isotropic (p 0) for those transitions that exhibit fluorescence, but anisotropic distributions occur at higher excitation energies where the fluorescence is quenched, indicating the lifetime with respect to dissociation has dropped substantially. Thus, it is the early-time dynamics following excitation that determine the outcome of the competition between fluorescence and dissociation. The rather abrupt change in the relative quantum yields for fluorescence vs. dissociation is of considerable interest. A possible mechanism has been proposed by Yamaguchi [20] based on ab initio calculations of the various CH2CHO excited-state energies as a function of the C-C torsional angle. He finds that energy of the B(2A”) state rises only slightly (3300 cm-’) over the entire torsion angular range. In contrast, the A(’A’) state, which is well separated in the planar geometry, approaches to within 2200 cm-’ of the B state at a geometry where the planes CH2 and CHO groups are perpendicular. The implication here is that, with minor adjustments, one can imagine an intersection between the two states at relatively low excitation energy of the B state, and this is what promotes the abrupt increase in the internal conversion rate with energy. Our results are certainly consistent with this if one views internal conversion to the A state as the rate-limiting component in a two-step internal conversion process that finishes on the ground electronic state.

B. D + CDzCO Channel

The time-of-flight distribution in Fig. 4 has contributions from channels (1) and (2). However, we know the detailed form of the energy and angular distribution for channel (1) from our photofragment coincidence measurements. The contribution from the CO3 and CO fragments to Fig. 4 can then be determined by using a Monte Car10 simulation to calculate the fragment time-of-flight distribution from channel (1) based on the known energy and angular distribution; this procedure, which averages over all relevant experimental parameters, is described in more detail in Ref. 16. The result is shown as the solid line in Fig. 4. The comparison of the simulation with the data shows that channel (2) is responsible for the far wings of the distribution (from the D atoms) and the sharp peak at the center.

740


This latter feature must come from CD2C0, which barely recoils out of the beam and therefore misses the smaller beam block used in this detector configuration. The inset to Fig. 4 shows a center-of-mass translational energy distribution for channel (2) that, when iun through our Monte Carlo program, generates the time-of-flight distribution shown by the dashed lines in Fig. 4. This distribution, when added to that for C03 + CO, adequately reproduces the total experimental time-of-flight distribution. We can therefore learn about the dynamics of channel (2), although not in as much detail as channel (1). The translational energy distribution for channel (2) also peaks away from zero, implying that there is an exit channel barrier to hydrogen atom loss. This is not unusual for the ground-state dissociation of a radical to a radical (D) + closed-shell species (ketene). Although we do not have data for channel (2) at the whole series of excitation energies as we do for channel (l), it is reasonable to expect that channel (2) also proceeds by internal conversion to the ground state followed by statistical decomposition over a barrier. It is clearly of interest to know the branching ratios between the two channels. Unfortunately, this is complicated by two factors. First, we only observe CDzCO fragments near the high-energy limit of the translational energy distribution; slower fragments strike the beam block and are not detected. This effect is taken into account in the Monte Carlo generation of a time-of-flight spectrum, but it does mean that the low-energy portion of the translational energy distribution is poorly determined. Second, the detection efficiency of the D atoms is considerably lower than that of the heavy fragments, but we do not know exactly how low. We estimate the detection efficiency to be between 1 and 10%;the true branching ratio clearly depends on this value. It does appear that channel (2) is a major channel, contributing at least 50% to the total fragmentation. A more detailed analysis is currently underway to better quantify this channel.

V. CONCLUSIONS This work represents the first study of the photodissociation dynamics of the vinoxy radical. We observe predissociation over the entire B 2A'' +-X *A" band of CHzCHO, including the origin at 28,760 cm-' . Two dissociation channels are observed: CH3 + CO and H + CH2CO. Translational energy distributions for the CH3 + CO channel are largely independent of excitation energy, indicating that this channel is likely due to statistical decomposition on the vinoxy ground-state potential-energy surface. The translational energy distributions all peak near ET = 1 eV, and we believe this is indicative of the isomerization barrier for conversion of vinoxy (CHzCHO) to the acetyl radical (CH3CO) prior to dissociation. A comparison of our results


74 1

with previous laser-induced fluorescence work on CH2CHO shows that predissociation dominates over fluorescence at excitation energies > 1000 cm-' above the band origin. This indicates a greatly increased internal conversion rate above this energy, possibly due to another excited state of vinoxy intersecting the B 2Att state. The results presented here show that our instrument can also be used to investigate dissociation channels in which the mass disparity of the two fragments is very large, namely the H + CH2CO channel; the study of this channel was facilitated by using CDzCDO. Although the dynamics of this channel cannot be elucidated at the same level of detail as the CH3 + CO channel, our ability to study it at all represents an important extension of the capabilities of the instrument.

Acknowledgments This research is supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

References I . R. Schinke, Photodissociation Dynamics, Cambridge University Press, Cambridge, 1993. 2. J. W. G. Mastenbroek, C. A. Taatjes. K. Mauta, M. H. M. Janssen. and S. Stoke, J. Phys. Chem. 99,4360 ( I 995). 3. M. D. Person, P. W. Kash, and L. J. Butler, J. Chem. Phys. 97, 355 (1992). 4. R. E. Continetti, D. R. Cyr, D. L. Osborn, D. J. Leahy, and D. M. Neumark, J. Chem.

Phys. 99, 2616 (1993); D. J. Leahy, D. L. Osborn,D. R. Cyr, and D. M. Neumark, J. Chem. Phys. 103,2495 (1995). 5. H. E. Hunziker. H. Kneppe, and H. R. Wendt, J. Photochem. 17,377 (1981). 6. M. Dupuis, J. J. Wendoloski, and W. A. Lester, Jr., J. Chem. Phys. 76, 488 (1982). 7. G. Inoue and H. Akimoto, J. Chem. Phys. 74, 425 (1981). 8. L. F. DiMauro, M. Heaven, and T. A. Miller, J. Chem. Phys. 81,2339 (1984). 9. T. Gejo, M. Takayanagi, T. Kono, and I. Hanazaki, Chem. Lett. 2065 (1993). 10. S. G. Lias, J. E. Bartmess, J. F.Liebman, J. L. Holmes, R. D. Levin, and W. G. Mallard, J. Phys. Chem. Ref- Data 17 (Suppl. 1) (1988). 11. C. W. Bauschlicher, Jr., J. Phys. Chem. 98, 2564 (1994). 12. S. North, D. A. Blank, and Y. T. Lee, Chem. Phys. Lett. 224, 38 (1994). 13. S. Deshmukh, J. D. Myers, S. S. Xantheas, and W. P. Hess, J. Phys. Chem. 98, 12535 ( 1994).

14. R. D. Mead, K. R. Lykke, W. C. Lineberger, J. Marks, and J. I. Brauman. J. Chem. Phys. 81,4883 (1984). 15. D. P. DeBruijn and J. Los, Rev. Sci. Instrum. 53, 1020 (1982). 16. D. R. Cyr, R. E. Continetti, R. B. Metz, D. L. Osborn, and D. M. Neumark, J. Chem. Phys. 97,4937 (1992). 17. M. Yamaguchi, T. Momose, and T. Shida, J. Chem. Phys. 93,4211 (1990). 18. A. M. Wodtke, E. J. Hintsa, and Y.T. Lee, J. Phys. Chem. 90, 3549 (1986).

742


19. X. Zhao, R. E. Continetti, A. Yokoyama, E. J. Hintsa, and Y. T. Lee, J. Chem. Phys. 91, 4118 (1989).

20. M. Yamaguchi, Chem. Phys. Lett. 221,531 (1994).

DISCUSSION ON THE REPORT BY D. M. NEUMARK Chairman: J. Manz

M. Shapiro: Prof. Neumark, in general, I was struck by the similarity between your data on the CH3O radical and the photodissociation of CH31, which we analyzed many years ago. In particular, it is interesting to note that the degree of excitation of the umbrella mode is similar and appears to increase with increasing excitation of the parent molecule. Obviously the exit channel dynamics dominate here. The other comment I would like to make is that the positive value of the P parameter you observe is due to a quantum interference effect. A simple mixing of the ground state with the excited state in the final continuum state hardly affects the directional properties of the dipole matrix elements per se because the transition dipole matrix elements between different states within the ground state are very small. Namely, if Pex=

+

then

because

which represents a high overtone transition.

D. M. Neumark: We indeed take the translational energy distribution from CH30 dissociation to be evidence for exit channel interactions on a repulsive potential-energy surface. This is in contrast to photodissociation of the vinoxy radical, for which very little variation of the CH3 + CO translational energy distribution occurs over a 0.5-eV range of excitation energy.


743

Regarding your comment on the photofragment angular distributions, the 3;66:, transition in the 2A1-* 2E band of methoxy radical (involving single excitation of the degenerate CH3 rocking mode) is nominally allowed because of Jahn-Teller coupling in the 2E ground state. However, you are correct in pointing out that this does not explain the positive value of 0 observed for this perpendicular electronic transition. The most likely explanation is that the upper level is vibronically mixed with an a1 vibrational level of a nearby electronic state of E symmetry; this argument has also been proposed by Terry Miller (Ohio State University) to explain new spectroscopic features seen in the electronic spectrum of CH3O. J. The: Prof. Neumark, how well under control do you have energy and angular momentum of the dissociating excited species? D. M. Neumark: In cases where we can measure it (02, for example), the rotational temperature is 30-50 K. T.P. Softley: Would you comment on the vibrational temperature of the photodissociated parent molecules in the beam. For 0 2 you clearly observe u” = 5 6 , but is this also the case for methoxy? D. M. Neumark We make no effort to produce vibrationally cold 02, since the B +-X transitions to predissociating upper state levels are rotationally resolved and completely understood. In the case of CH30, we detach the CH3O- just above the detachment threshold so that we do not produce vibrationally excited CH3O. K. Yamanouchi: There are many kinds of small polyatomic molecules that emit detectable fluorescence in the energy region where they predissociate. Acetylene, S02, and CS2 are examples of such molecules. The two processes, fluorescence emission and dissociation, are in general competing processes. In order to discuss predissociation dynamics, it is also important to derive a state-specific rate constant based on the measurements of absorption and dissociation cross sections. D. M. Neumark: The competition between fluorescence and predissociation is an interesting and complex problem, as evidenced by the observation that it is quite different in the vinoxy and methoxy radicals. Unfortunately, high-resolution absorption spectra are not available for these radicals so it is not so straightforward to compare absorption spectra with LIF or predissociation spectra. R. A. Marcus: The large amount of energy liberated as translational energy in the CH2CHO -+ CH3CO -+ CH3 + CO reaction was very interesting. Do you have information on the energy distribution among the other coordinates of the products: If the first step, the transfer of

744


an H from the CHO group to the CH:! group is vibrationally adiabatic, that is, no significant change in this H-stretching vibrational quantum number, and if the C=O and other CH fragments are largely spectators, then all of the excess energy of the highly exothermic process will go into the C-C vibration and hence into translational energy of separation of the CH3 and CO fragments. The actual distribution will depend on how short-lived the hot CH30 is. D. M. Neumark: Once the vinoxy radical surmounts the isomerization barrier, we don’t really know if it passes through the acetyl minimum before dissociating to CH3 + CO, or if instead CO elimination is concerted. This would clearly be an interesting theoretical problem to pursue. K. Yamashita: Prof, Neumark, you talked about the experiment in which anions are photodetached to the bound states of neutral molecules, and these bound states are excited to dissociate by a second photon. However, you can also excite anions to the transition state of neutral reactions instead of bound states and can reach the resonance states, as you do for several systems. My question is then, is it possible to excite these resonance states directly by your second photon? If it is possible, you would be able to control photodissociation product channels, because there should be cases where some resonance states localize on one side of the transition state and other resonance states localize on another side. D. M. Newnark: In our current experiment we use pulsed nanosecond lasers, and the lifetimes of even the longest lived transition-state resonances are too short to excite with a second photon. A related and very interesting experiment would be to measure the product energy distributions resulting from decay of these resonances. For example, one could photodetach IHI- and measure the I + HI kinetic-energy distributions as a function of electron kinetic energy. This requires a “triple-coincidence” experiment that cannot be done easily in our laboratory. However, experiments of this type are currently being carried out by Prof. Robert Continetti at the University of California at San

Diego.

RESONANCES IN UNIMOLECULAR DISSOCIATION: FROM MODE-SPECIFIC TO STATISTICAL BEHAVIOR R. SCHINKE,* H.-M. KELLER, H. FLOTHMANN,

M.STUMPF, C. BECK, D. H. MORDAIRVT, and A. J. DOBBYN Max-Planck-Institut jiir Stromungsforschung Gottingen, Germany

CONTENTS 1. 11. 111. IV. V. VI. VII.

Introduction Potential-Energy Surfaces Quantum Mechanical Calculations HCO: A Textbook Example of Regular Dynamics K O : A Spectroscopic Challenge HNO: A Mixed Reguhr-Irreguhr System HO2: Classical Chaos Reflected in Dissociation Rates and Product-State Distributions VIII. Resume and Outlook References

I. INTRODUCTION Resonances in open systems, that is, systems whose total energy is higher than the first dissociation threshold, are an old theme of scattering dynamics. For very elementary introductions the reader is referred to, for example, Messiah (Ref. 1, Chapter III), Cohen-Tannoudji et al. (Ref. 2, Chapter XIII), Satchler (Ref. 3, Chapter 4), and Schinke (Ref. 4, Chapter 7). They *Report presented by R. Schinke Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale. XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

745

746

R. SCHINKE et al.

are induced by the temporary trapping of some part of the available energy in internal modes (collectively represented by r in what follows) that are “orthogonal” to the fragmentation coordinate R. The storage of energy in these internal degrees of freedom can significantly delay the rupture of the bond between the two (or more) entities because first sufficient energy must be transferred to the dissociation coordinate. Resonances are well-known features in all kinds of scattering [5-71, for example, nuclear reactions, electron collisions, scattering of atoms and molecules from solid surfaces, photodissociation, and molecular spectroscopy. The latter topic is particularly enlightening, because with modem experimental and theoretical tools all facets of resonances can be studied in great detail [8]. Recent progress in the minute investigation of resonance states in polyatomic molecules will be highlighted in the present contribution for the Twentienth Solvay conference in chemistry. A typical example of a molecule ABC in its electronic ground state 2 is depicted in Fig. 1, showing a one-dimensional cut through a multidimensional potential-energy surface (PES). The quantum states below the AB(X, 0) + C threshold are true bound states with discrete energies. When the total energy is increased above the first threshold (E = 0). it becomes a continuous variable. However, this does not necessarily imply that the progression of bound states immediately dies out when E > 0. On the contrary, it can persist into the continuum, even to high energies above the threshold, with one important distinction: Unlike true bound states, resonance states have a finite energetic width. In view of the time-energy uncertainty relation, resonances have a finite lifetime whereas true bound states live forever (if we ignore spontaneous emission). The quantities that specify a resonance are the energy E$\, the width Ak“), where k(’) is the decay rate of the resonance, and last but not least the final distribution of the particular quantum states /3 of the products, P(i)(/3). The energy of a resonance is essentially determined by the PES in the inner (i.e., bound) region. Its width (or lifetime #) = l/k(i) if the resonance has a Lorentzian line shape) depends on the coupling between the inner region and the exit channel, that is, the efficiency of internal vibrational energy redistribution (IVR) between the dissociation coordinate R on one hand and the internal coordinates r on the other and therefore on the shape of the potential near the transition state (TS). The lifetime of the temporary excited complex can range from a few tens of a femtosecond for a rather fast process to microseconds, or even longer, for a weakly bound van der Waals complex. Finally, the product-state distributions reflect both the wave function at the TS and the dynamics (i.e., the energy exchange) in the exit channel [4, 9, lo]. In order to fully understand the fragmentation dynamics of a system, it is, in principle, necessary to consider all three of these observables for as

747

RESONANCES IN UNIMOLECULAR DISSOCIATION

I

+

dissociation coordinate R

Figure 1. Schematic representation of resonances in the continuum of a polyatomic molecule ABC(X) dissociating into products AB(X, 0) and C. The left-hand side shows an absorption-type cross section O&&) with a rich resonance pattern. The term p ( E ) is the density of states at the energy E and N & ( E ) is the number of states at the TS, orthogonal to the dissociation path, that are accessible at energy E. Several experimental schemes for a spectroscopic analysis of resonances are also indicated. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

many resonances as possible. The possibility of measuring and calculating resonance widths and final-state distributions makes photodissociation, from the dynamical point of view, more interesting than bound-state spectroscopy; in the latter case only the excitation energies can be analyzed. Experimentally one can investigate resonances by various spectroscopic schemes, as indicated in Fig. 1: by direct overtone pumping [ l l ] from the ground vibrational state, by vibrationally mediated photodissociation [121 using an excited vibrational level as an intermediate, or by stimulated emission pumping (SEP)[13-151 from an excited electronic state. In all cases it is possible to scan over a resonance and thereby determine its position E Z and its width hkc').A schematic illustration of an absorption or emission spectrum is depicted on the left-hand side of Fig. 1; all of the more or less sharp structures at energies above threshold are resonances. Figure 2 shows an overview SEP spectrum measured for DCO [16]. It consists of

8PL ODEEL

OOBEL

WtEL

OOZEL

OOOEL

OOSZl

OOBZL

OOttL

OOZEL

00021

Excitation Energy / (cm-') Figure 2. Overview SEP spectrum for DCO(2) as measured by Stiick et al. [16]. The energy is measured with respect to the vibrational ground state (0, 0.0). Each vibrational band consists of four different rotational lines. For a detailed discussion of the assignment the reader is referred to the theoretical analysis of Keller et al. [17]. (Reprinted with permission of the American Institute of Physics, from Ref. 16).

749


a long series of vibrational bands each being composed of four different rotational lines. The bands below 5500 cm-' are true bound states while all bands above this energy are resonances. (The highest bound state has an energy of 5471 cm-' above the ground vibrational state [17].) A finer scan of this spectrum bears out that the resonance widths do not depend much on the particular rotational state, within the same vibrational band, but depend sensitively on the vibrational excitation. Finally, the distribution of quantum states of the product molecule, f i i ) ( n , j ) ,can be monitored with a third laser [18]. (In what follows the quantum numbers n a n d j specify the vibrational and rotational state of the fragment molecule AB.) As examples we show in Fig. 3 two measured rotational distributions of CO in the fragmentation of a particular vibrational resonance of HCO [19], in comparison with the results

0.2

n

;3

a

-

303

-

oexperiment

n

a

;3

a.

i Figure 3. Comparison of measyed [19] and calculated [20] rotational state distributions following the dissociation of HCO(X) for two different initial total angular momentum states.

750

R. SCHINKE et al.

of an ab initio calculation [20); the total angular momentum states are 303 and 312.The particular shapes of these distributions reflect, in a qualitative sense, the angular dependence of the wave function at the TS, as we will elucidate below. Although each resonance has its own individual width and final-state distribution (quantum state speciJicify), one can broadly distinguish between two limits: mode selectivity and statistical behavior [21, 221. In the first case, both the widths and the product-state distributions depend in a predictable way on the kind and the degree of excitation of the molecule, that is, on the quantum numbers with which the resonance can be labeled; mode specificity obviously requires some kind of unique assignment. In the second case, a systematic dependence on the particular resonance is not possible, simply because a meaningful assignment of the resonance states does not exist. Because of the lack of a clear-cut relationship between the resonance on the one hand and k(’)and P(’)(P)on the other, a statistical analysis might be more meaningful. The widths and the final-state distributions can be thought of as reflections of the nodal pattern of the underlying resonance wave function and therefore, we argue that mode specificity is confined to systems with mainly regular motion whereas statistical behavior is normally found in irregular, that is, classically chaotic systems. Mode-specific behavior is expected when the density of states is relatively small and/or when the coupling between the modes is weak. In contrast, statistical behavior is believed to occur when the density of states is large and/or the internal coupling is strong. In the first case exact quantum mechanical (or possibly semiclassical) calculations on a complete and accurate PES are indispensable if one wants to compare theoretical predictions with detailed experimental data. In the statistical case it is hoped that much simpler calculations based on statistical assumptions, for example, the Rampsberger-Rice-Kassel-Marcus (RRKM)[23] theory or the statistical adiabatic channel model (SACM) [24,25], can be used to describe the average values of decay rates and final-state distributions. These theories do not require detailed knowledge of the complete PES, but only a small portion of it in the region of the TS. The fragmentation of a molecule in its ground electronic state is commonly known as unimolecular dissociation [26-281. [For a recent review see Ref. 29 and the Faraday Discussion of the Chemical Society, vol. 102 (1995).1 Because of its importance in several areas of physical chemistry, such as combustion or atmospheric kinetics, there is a high demand of accurate unimolecular dissociation rates. On the other hand, however, the calculation of reliable dissociation rates by dynamical methods (i-e., the solution of the classical or quantum mechanical equations of motion) is, for obvious technical problems, prohibited for all but a few simple molecules. For


75 1

this reason one is forced to use very simple models for predicting dissociation rates, the accuracy of which is not a priori known. The most widely employed method is based on the assumption that prior to dissociation all (zero-order) states, or the classical phase space in classical mechanics, are statistically populated [23, 30, 311. The dissociation rate as a function of energy E is then given by

where N l ( E ) is the number of accessible states at the transition state “orthogonal” to the fragmentation path, p ( E ) is the density of states, and h is Planck’s constant. The various versions such as RRKM, variational RRKM, or SACM differ by how the numerator is calculated. By definition, Eq. (1) can only be expected to yield a reasonable approximation for the average quantum mechanical rate, while fluctuations, which normally are caused by quantum interference, cannot be reproduced. In the last two years or so our research group has analyzed in detail the dissociations of HCO [32, 331 DCO [16, 171, HNO [34], and H02 [35-371 on their ground-state PESs using quantum mechanical methods. In the case of HCO we concluded that it is an essentially regular system with mostly assignable wave functions and that it illustrates mode specificity. On the other extreme, analyses of the bound-state wave functions as well as the energy spectrum of the bound states showed that the motion of H02, at energies near the H + 02 threshold, is mainly irregular, that is, classically chaotic [36]. The dissociation rates and intensities in an absorption-type spectrum can be well modeled by statistical theories [37]. Most interestingly, the dissociation of HNO embraces both regular and irregular motion; it thus represents an intermediate case between HCO and HO2. The purpose of this chapter is a detailed comparison of these systems and the elucidation of the transition from regular to irregular dynamics or from mode-specific to statistical behavior. The main focus will be the intimate relationship between the multidimensional PES on one hand and observables like dissociation rate and final-state distributions on the other. Another important question is the rigorous test of statistical methods for these systems, in comparison to quantum mechanical as well as classical calculations. The chapter is organized in the following way: The three potential-energy surfaces and the quantum mechanical dynamics calculations are briefly described in Sections I1 and 111, respectively. The results for HCO, DCO, HNO, and H02 are discussed in Sections IV-VII, and the overview ends with a short summary in Section VIII.

752

R. SCHINKE et al

II. POTENTIAL-ENERGY SURFACES The calculations presented here include all three internal degrees of freedom. Since we consider the decay of an unstable complex into two (or in principle more) fragments (“half collision”), it is necessary to use coordinates appropriate for scattering calculations rather than normal coordinates, for example [4]. Furthermore, since the fragmentation into a single product channel is investigated, the usual Jacobi coordinates can be used: R, the distance from the atom C to the center of mass of the diatomic fragment AB; r, the vibrational coordinate of the AB entity; and y , the angle between R and r. The HCO PES has been determined by extensive ab initio calculations at the multi-reference configuration interaction (MRCI) level using the MOLPRO program package [32]; the almost one thousand original energy points were then fit to an elaborate analytical expression [33]. In order to compensate for a small underestimation of the CO stretching frequency by the ab initio calculations, the PES was slightly modified. The PES for HNO has also been constructed by ab initio calculations, again making use of MOLPRO [38]. About 1200 points have been calculated and a three-dimensional spline interpolation scheme was employed to calculate the potential between the grid points. In this case, too, slight modifications have been applied in order to make the theoretical term energies agree better with the experimental ones. The PES for HO2 is the double many-body expansion (DMBE) IV potential of Pastrana et al., which has been constructed by using both ab initio and experimental data [39]. Contour plots of all three PESs are depicted in Fig. 4. The left-hand side shows the potentials as functions of R and r for fixed y and the right-hand side demonstrates the (R, y) dependence for fixed r. In order to present in the clearest way the similarities and differences, the scales of the axes are the same for each molecule. The HCO PES has the shallowest well (D, = 0.834 eV) and a small barrier separates the inner region from the exit channel. The values of 2.14 potential wells for HNO and HO2 are considerably deeper (0, and 2.37 eV, respectively) and no potential barriers hinder the bond rupture. Let us first discuss the (R,r ) behavior. In all three cases the bonds of the diatomic entities, CO, NO, and 0 2 , significantly change when the hydrogen atom is attached. In the well the anharmonicity along r is much larger than in the free molecule. This effect is least pronounced for HCO and most dramatic for HO2; in the latter case the opening of the 0 + OH reaction channel is clearly seen at large 02 separations. Since the fundamental frequency w r associated with r is smaller in the case of H02 than for HNO, and since the anharmonicity is so strong in this coordinate for H02, the density of states is substantially larger for HO2, despite the fact that the dissociation


753

Figure 4. Contour plots of the potential-energy surfaces for HCO, HNO. and H02.The left-hand side shows the (R, r ) dependence, with the angle y being fixed at the equilibrium in the well. The right-hand side highlights the (R, y ) dependence, with r fixed at the equilibrium. The spacing of the contours is 0.25 eV and the lowest contour is 0.25 eV above the minimum. Energy normalization is so that E = 0 corresponds to H + XO(r,). The Jacobi coordinates R, r, and y are as described in the text, with y = 180” corresponding to H-X-0. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

energies are similar. The number of bound states is merely 15 for HCO, 215 for HNO, and 361 (365) for the odd (even) parity of H02. In all three cases the coupling between R and r, qualitatively indicated by the angle between the two “reaction channels,” is relatively weak, and therefore high excitation in r is, roughly speaking, the “carrier” for the many resonances above the threshold observed for all three molecules [32]. In each case more or less “pure” vibrational states with excitation only in r are found up to very high energies. The decoupling of r from the other two modes seems to be most effective for HCO, less so for HNO and H02. The coupling between the angle y and the dissociation coordinate R is always large if Jacobi coordinates are used. At low energies deep inside the well, this coupling is linear and normal coordinates are usually better suited for interpretation and assignment than are Jacobi coordinates. However, if the molecular dynamics above the dissociation threshold is studied, the normalmode picture breaks down and scattering coordinates have to be employed.

754

R. SCHINKE et al.

The coupling between R and y is reflected by the curvature of the minimum energy path inside the well, which seems to be roughly equal in all three cases. The PES for HCO has a very shallow second minimum at small angles that, however, is not seen in the figure because it is too high in energy. For HNO this second minimum occurs at lower energies and is clearly seen in Fig. 4. The PES for HO2 is symmetric with respect to y = 90” and the two wells are obviously equally deep. The PES of HCO has a barrier and therefore the TS is clearly defined. When the molecule leaves the well region, it has to “squeeze” through this bottleneck, and that is clearly shown by the stationary wave functions [32, 401. HCO is an example of a “tight” transition state. In the case of HO2 there is no potential barrier, but nevertheless a distinct bottleneck is still seen where the wide HO2 well (wide with respect to r ) quite abruptly narrows and turns into the relatively cramped exit channel. Here, the 0 2 force constant changes rapidly with R. Thus, as we will discuss below, H 0 2 can still be considered as an example of a tight transition state. The situation is different, however, for HNO: No potential barrier hinders the dissociation and the NO force constant changes very smoothly with the distance from the H atom. If we look from the free-product side, the narrow exit channel gradually opens into the wide HNO well. HNO exemplifies a system with a “loose” transition state. The consequences will become apparent below. The general behavior of the potentials in the asymptotic channel can be qualitatively understood in simple chemical terms [341. It is not difficult to surmise the existence of a long series of resonances in each case. They are basically the result of high overtone excitation in the vibrational coordinate r. the mode that for all three molecules is rather weakly coupled to the dissociation coordinate. In order to enable the system to dissociate, first a sufficient amount of energy has to be transferred from motion in r to motion along R, which, depending on the coupling strength, can be very time consuming. Simultaneous excitation in the angular degree of freedom generally accelerates the fragmentation because y and R are usually rather strongly coupled. Finally, direct excitation of the dissociation mode normally leads to rather short-lived complexes. This is, of course, only a qualitative picture. Pronounced differences among the systems considered will be emphasized below. At the end of this section, it is worthwhile to point out that resonances in quantum mechanics are intimately related to the existence of trapped classical trajectories. The smaller the classical forces between r and y on one hand and R on the other, the longer is the lifetime and vice versa. In this sense it might be helpful for understanding the complex quantum dynamics by imagining the trajectories of a classical billiard ball moving on multidimensional potential-energy surfaces (see, e.g., Chapter 5 of Ref. 4).


755

111. QUANTUM MECHANICAL CALCULATIONS

Continuum wave functions are required to fulfill specific boundary conditions in the exit channel, that is, in the limit R-+-, and therefore they are much more difficult to Calculate than bound-state wave functions. In our applications we use a modification of the log-derivative version of Kohn’s variational principle [37]. The essential steps can be summarized as follows: (i) The coordinate space is divided in an inner (R < R,) and an outer (I7 > R,) region, where R, is chosen so that the coupling between the vibrational-rotational channels of the diatom is negligibly small. (ii) In the inner region an appropriate set of energy-independent basis functions is constructed from a large primitive basis set by a general contraction-truncation scheme, and the corresponding Hamiltonian matrix is diagonalized; this yields, as a by-product, all the bound-state energies and wave functions. (iii) In the outer region the appropriate coupled-channel equations are solved approximately for each value of the energy E. (iv) Matching the solutions in the inner and the outer regions at the boundary R, leads to algebraic equations from which the wave functions for each energy and all open channels P of the diatom, *$), are obtained. The diagonalization of the Hamiltonian matrix is time consuming, but as this step is independent of E, it has to be done only once. Varying the energy is very efficient and normally several thousand energies are included in one scan of the spectrum. Having the partial wave functions *$’, we usually calculate the overlap with an “initial” wave function xo and determine an absorption-type cross section according to [4].

where the sum runs over all open product channels /3. If we were to calculate an absorption spectrum, we would take xo as a particular vibrational state in the ground electronic state; if, on the other hand, our goal would be the calculation of a SEP spectrum, the initial wave function must represent a vibrational state in an upper electronic manifold. In any case, because we calculate directly the wave functions (rather than discrete energies in the complex plane), we can determine any observable and compare it to its experimental counterpart. Plotting the stationary wave functions of the system facilitates the assignment of the resonances in terms of vibrational quantum numbers (if there is any) and illustrates the overall dissociation path. For this reason we usually consider the so-called total wavefunction * E (Chapter 2 of Ref. 4), that is, a

756

R. SCHINKE et al.

superposition of all partial wave functions Of’for a given energy, weighted by the overlap factor (xo IOf’).Figure 5 depicts several examples of resonance wave functions for HCO. The left-hand side shows the (R, r) dependence for fixed angle y while the right-hand side features the (R, y) dependence for fixed r. Although these wave functions look like bound-state wave functions, they are, nevertheless, real continuum wave functions; except for state (3, 0, 0), the asymptotic tails are too small, relative to the amplitudes in the inner region, to be seen on this scale. The assignment in terms of the three quantum numbers u1 (R, H-C stretch), u2 (r, C-O stretch), and u3 (7, H-C-O bending) is rather obvious, except for the wave function displayed in the lowest panel. In the next section we will elucidate the relationship between the shape of these wave functions on one hand and the dissociation rates and the final rotational state distributions of CO on the other. Resonances are features inherent in the Hamiltonian and show up as more or less sharp structures in all quantities that contain the wave functions, for example, the absorption cross section aab,(E).The absorption spectrum obviously depends on the initial-state wave function x o , and therefore the various resonances are weighted with different Franck-Condon factors, which complicates the analysis of the spectrum. A numerically more convenient quantity is

where the norm 11.11 is calculated only in the inner region R < R,. Division by the cross section compensates for the weighting with the overlap matrix element in the definition of the total wave function. For an isolated resonance with hrentzian shape it can be shown that 7(E,,) evaluated at the center of the resonance is the lifetime of the resonance, and therefore we will call ; ( E ) the lifetime function in what follows. The lifetime function is largely independent of the particular initial state, and in our applications we found that in general it gives a better resolution of the resonance structures, especially in the overlapping regime. Figure 6 shows ?(E) for all three systems as a function of the total energy available for the fragments, E-Eo, on a logarithmic scale. The energy regime is chosen to be the same in all three cases in order to demonstrate the different densities of (resonance) states. The spectrum of HCO has been calculated to much higher energies, about 1.7 eV above threshold, and even at these high energies there still exist many very narrow resonances [32]. The spectrum for HNO and H02 have been determined up to 0.7 and 0.4 eV, respectively. Most of the resonances for HCO can be unambiguously


3.0

1801

2.5

*

757 I

150 120 -

2.0

3.0

La

2.5

i-

120 -

2.0

4 180 - 1

3.0 L

150 -

2.5

r-

150 120 -

2.0 3.0

La

2.5 2.0

lZO

&

2.5

i-

150 120.-

2.0

2.0

t

i180 -I

3.0

2.5 3.0

R

3.5

4.0

4.5

2.0 2.5

3.0 3.5

R

4.0 4.5

Figure 5. Selected resonance wave functions for HCO. The left-hand panel shows (R,r) cuts for fixed angle y and the right-hand panel illustrates the (R, y ) behavior for fixed value of r. The distances are given in a0 and the angle is given in degrees. Plotted is the modulus square of the wave function. Except for the lowest panel, all wave functions are easily assigned by the quantum numbers U I (H-CO stretch), u2 (C-O stretch), and u3 (H-C-Obend).

758

R. SCHINKE et al.

0.1

0.2

E-Eo / e V

0.3

0.4

Figure 6. Lifetime function ?(E), defined in F q (3), is shown on a logarithmic scale as a function of the available energy E - Eo above threshold. Atomic time units are used (1 a.u. = 2.42 lo-” s). The numbers on the vertical axis indicate powers of 10. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

assigned to the three quantum numbers (UI,u2, ug) mentioned above. Only in some cases the accidentally strong coupling between two energetically close states may disturb the clear picture. In the case of HNO the majority of states have an irregular nodal structure, although a fair number of resonances still can be assigned; the assignable states are mainly associated with excitation in the NO mode with none, one, or two quanta in the bending mode. Finally, for HOz an assignment is hopeless in view of the fact that the bound states close to threshold are already irregular [36]; going into the continuum does not make the dynamics more regular [37]. Figure 6 clearly bears out that for each of the three molecules the lifetimes fluctuate a great deal across the entire


759

energy regime shown. The “spectra” for HNO and HO:! distinctly illustrate how the resonances gradually begin to overlap as the energy increases. Extracting decay rates k(’)from the spectra is simple if the resonances are isolated. In that case the lifetime function, for example, is to a good approximation given by

where hk is the full width at half maximum of a resonance. With increasing energy the average spacing between the resonances decreases, and at the same time the average width increases so that the resonances begin to overlap. This general behavior makes the extraction of rates gradually more tedious and at some point it is even impossible [41]. This problem, together with the fact that the larger rates are obscured in the background, means that the rates that we can extract from the spectra with reasonable confidence are limited to values smaller than about 1013 s-’. The dissociation rates extracted from the three spectra in Fig. 6 are depicted as a function of the energy above the threshold, E - Eo, in Fig. 7. Before we discuss details of the three systems, we mention only that in all three cases the rates fluctuate a great deal about an average and that this average gradually increases with energy, as generally predicted by statistical theories. Whether or not the statistical models correctly predict this average is a question at the heart of unimolecular dissociation. Although the fluctuations of the rates appear to be similar, we will stress in the following sections that their origins are rather different for, for example, HCO and HOz: Modespecific variation of the wave functions for HCO and irregular behavior in the latter case.

IV. HCO. A TEXTBOOK EXAMPLE OF REGULAR DYNAMICS The dissociation of the formyl radical HCO into H and CO is an illuminating example for studying in great detail the breaking of a bond in a triatomic molecule. It is a textbook example of resonances in polyatornic systems. Because of its importance in combustion, the fragmentation of HCO has been intensively studied both experimentally and theoretically. For a recent overview covering the literature up to 1994 see the review by Neyer and Houston [ 181. The most detailed spectroscopic investigation was published less than half a year ago by Tobiason et al. [42], who used stimulated emission pumping to resolve about 60 resonances. On the theoretical side, many research groups have exploited the dissociation of HCO as

760

R. SCHINKE et al.

0

0.1

0.2

E-E,

IeV

0.3

0.4

Figure 7. Dissociation rates k as extracted from the quantum mechanical calculations (open circles). The statistical rates are represented by the step functions and the filled circles represent the classical rate constants as obtained from elaborate classical trajectory calculations. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

a test case of developing time-independent [43-46] as well as time-dependent [40, 471 quantum mechanical methods. Most of these studies used the Bowman-Bitman-Harding (BBH) PES, which was constructed by-at that time extensive-ab initio electronic structure calculations about 10 years ago [48]. Although this PES provides a generally realistic representation of the dissociation process, the number of details, which became available with the


76 1

most recent experimental data for HCO [42] and DCO [16, 491, demanded a new, hopefully more accurate PES. Simultaneous with the latest round of SEP experiments, a new global ab initio PES has been constructed on the MRCI level [32] that is the basis for all our dynamics calculations. After a slight adjustment of the potential in order to correct for an underestimation of the CO stretching frequency by about 25 cm-' , the measured term energies of the 15 bound states and the about 60 resonances are reproduced with a root-mean-square (rms) deviation of about 20 cm-I, compared to 50 cm-l obtained with the BBH potential [42]. The deviation of our calculations is more or less uniform over the entire energy range considered while the deviation obtained with the BBH PES increases rather drastically with E. There are many interesting facets of this elementary fragmentation process. In this presentation we emphasize primarily the mode specificity, that is, the more or less sytematic dependence of the resonance widths and the final-state distributions on the excited mode and the number of quanta in each mode. (For more complete analyses the reader is referred to Refs. 32 and 33 and some future publications.) Due to the low density of (resonance) states and the weakness of the intramolecular coupling, especially between the dissociation coordinate R and the CO stretching mode r, the vibrational dynamics of HCO is mainly regular, even at energies high above the dissociation threshold. The wave functions in the upper four panels of Fig. 5 intriguingly reflect this regularity: They show a clear-cut nodal pattern that in most cases allows a straightforward assignment (u,, u2, u j ) (the three modes have been defined above). Only if two resonances are accidentally very close in energy can they mix with the result that the wave functions do not have a simple behavior, thus complicating the assignment. The lower panel of Fig. 5 shows an example; in this case it is more sensible to make the assignment in view of the energy of the state rather than the wave function pattern. The rate constants k extracted from the resonance widths are depicted as a function of the access energy E - EO in the upper part of Fig. 7. In order to emphasize the comparison with the other two systems, only a small portion above the threshold is shown. The rates show the generic behavior typical for a system with a barrier in the exit channel: An exponential increase reflecting tunneling through the barrier is followed by a leveling off as the energy exceeds the adiabatic potential barrier, that is, the potential barrier plus the zero-point energies corresponding to the modes orthogonal to the dissociation coordinate [50]. Superimposed on this secular behavior are pronounced fluctuations that encompass about three orders (!) of magnitude. These fluctuations are typical for a mode-specific system [21, 51-53] and basically reflect the different types of wave functions of the various resonance states.

762

R. SCHINKE et al.

Resonances with pure excitation of the CO stretching mode (0, u2, 0) (an example with u2 = 8 is shown in Fig. 5) have the smallest rate and therefore the longest lifetime; energy transfer from r to R is rather inefficient, and therefore the system needs a long time before enough energy is accumulated in the dissociation coordinate to permit dissociation. On the other extreme, direct excitation in R allows a rather rapid bond rupture, and therefore the resonances (u1, 0, 0) have the shortest lifetimes. Excitation of the bending mode (0, 0, u3) leads to lifetimes that are between C-O excitation as the lower limit and H-C excitation as the upper extreme. This mode specificity is further elucidated in Fig. 8, where we show the widths for several

2

6

I0

12

"2

Figure 8. Resonance widths (in cm-l) for the dissociation of HCO as a function of the CO stretching quantum number y for several series as indicated. The terms vl and u3 are the H-CO stretch and H-C-0 bending quantum numbers, respectively. (Reprinted, with permission of IOP Publishing, from Ref. 8.)

763


resonance series as a function of u2. Of course, mode specificity requires an assignment, whether it be in normal modes, local modes, or whatever coordinates. Without any classification, representations like the one in Fig. 8 would not be possible. The calculated resonance widths agree in most cases very favorably with those measured by Tobiason et al. in the SEP experiment [42]. Some examples are displayed in Fig. 9. The calculations using the new PES reproduce the general trends much better than the previous calculations employing the BBH potential, especially at high energies [42]. Nevertheless, there are still a few resonances for which substantial disagreement is found; which particular features of the PES are responsible for these failures is not known to us. In accord with common experience, we found that resonance widths are much more difficult to improve by slight modifications of the PES than the resonance energies. The relatively strong dependence of the widths on the excited modes can easily be read off the scales in Fig. 8.

j1

II \ \

1

:jr"" (1 .vz

2)

40

20

0 1

"2

,' 2

3

4

5

6

7

8

"2

Figure 9. Comparison of measured [42] and calculated [33] resonance widths (in cm-') for selected HCO resonances. The terms UI. u2, and u3 are the H-CO stretch, C-0 stretch, and H-C-O bending quantum numbers, respectively. (Redrawn from Ref. 42.)

764

R. SCHINKE et al.

In view of the strong mode specificity of the dissociation of HCO, it is not at all surprising to find that the statistical rate does not reproduce well the quantum mechanical average, as seen in Fig. 7. The statistical prediction is significantly larger than the quantum mechanical average rate (note that the logarithmic scale is quite extended in the figure). However, one must keep in mind that the particular way, in which we determine dissociation rates, namely the extraction of resonance widths from a spectrum, is “blind” to very large rates, because the corresponding resonances are hidden in the background. Including the very large rates would certainly push up the quantum mechanical average and thus improve the agreement with the statistical rate. Tunneling corrections [50] are not incorporated in the statistical rate so that it starts abruptly when the first channel opens at the TS. Next we turn to a brief discussion of the final-state distributions, Z’(”(n,j), of the CO fragment, where n and j are the vibrational and rotational quantum numbers, respectively. Let us emphasize again that each resonance has a distinct final-state distribution and that the dissociation process can be claimed to be completely understood only when all P(i)(n,j)are known. This is a formidable task, not only experimentally. In the case of HCO the final-state distributions for approximately 140 resonance states have been calculated but not fully analyzed at present time. A systematic investigation is currently in progress. Figure 10 depicts the vibrational distributions for the (0, u2, 0) series of resonances for u2 = 8,. ..,11, and a very systematic variation with the CO stretching quantum number is observed: With increasing u2 the distributions shift in a gradual manner to larger and larger n values and always peak near the highest accessible state. This trend can be qualitatively explained by an adiabatic picture in which the CO degree of freedom is adiabatically separated from the other two modes [32]. Originally the energy is mainly stored in the r coordinate and the amount of energy contained in R is insufficient to overcome the barrier, A vibrationally nonadiabatic transition with a Auz value of 4-5 (internu1 vibrational energy redistribution, IVR) is required to pump enough energy into the dissociation mode in order to allow the H-CO bond to break. This scenario, which is quite similar to the An = 1 propensity rule often found in the fragmentation of many weakly bound van der Waals molecules (see Ref. 54 or Chapter 12 of Ref. 4) explains the n = u2 - 5 propensity seen in the final vibrational state distributions. Rather different vibrational distributions will be presented for HO2 in Section VII. Rotational state distributions in fragmentation processes often reflect, in a quite direct manner, the angular dependence of the wave function at the transition state, that is, the y -dependent distribution of dissociating molecules before they enter the exit channel [9, 10, 55, 561. In a semiclassical picture, the modulus square of the TS wave function determines the initial conditions

765


= 11

“co Figure 10. Final vibrational state distributions of CO following the decay of the (0, u2,O) resonances with y = 8,. ..,1 1 . The arrows mark the highest accessible state for the respective

resonance. (Reprinted, with permission of the American Institute of Physics, from Ref. 32.)

of a swarm of classical trajectories that are started at the “point of no return” with positive momentum PR, that is, that immediately lead to fragmentation. HCO nicely exemplifies this “mapping” of the transition-state wave function. Figure 11 shows the rotational distributions following the decay of some of the (0, u2, 0) resonances. They all look rather similar, having a bimodal shape: a main maximum at low values of j, a node near j = 10 or so, and a less intense maximum at higher rotational states. The main maximum is superimposed by a secondary oscillation that can be explained in a Franck-Condon-type picture [20, 321. This bimodality reflects the bimodal shape of the TS wave functions, an example of which is shown in Fig. 12. Since the wave functions have the same qualitative shape for all the (0, u2,O) resonances shown in Fig. 11, it is not surprising that the resulting rotational state distributions are also qualitatively similar. Note that at the TS the wave function has a node in the angular coordinate although inside the well the bending degree of freedom is not excited. This indicates that in this particu-

Figure 11. Final rotational state distributionsof CO following the decay of the (0,u2, 0) resonances with u2 = 4, . . . , 8. The vibrational state of CO is the state with the largest probability. (Reprinted, with permison of the American Institute of Physics, from Ref. 32.)

rotationol state j

i

767


180,

2.0

1

,

I

1

1

3.0

I

1

i

1

1

LO

R Iq,l

1

i

i

I

5.0

i

i

i

i

1 1

6.0

Figure 12. Angular dependence of the (0, 7, 0) resonance wave function of HCO. (Reprinted, with permission of the American Institute of Physics, from Ref. 32.)

lar case the bending degree of freedom is also involved in the IVR process described above. In view of the statistical behavior of the fragmentation of H 0 2 (discussed below), we underline that for HCO, in an adiabatic sense, a single bending level is excited at the TS and the angular dependence of this single level is mapped into the rotational distribution of CO. Other examples are the photodissociations of CINO(Tl) [56, 571 and FNO(S1) [58, 591. Rotational distributions have been measured for only very few vibrational resonances [191 and excited rotational states of HCO; two examples in comparison to our ab initio results are depicted in Fig. 3. They show the same overall behavior as the distributions for J = 0 depicted in Fig. 11, a main peak at low values of j and a broader but less intense maximum at larger rotational states. As discussed in Ref. 32, the first peak originates from an angular region where the rotational-translational coupling in the exit channel is very weak, and therefore it can be explained by a Franck-Condon mapping mechanism (see Chapter 10 in Ref. 4). In such cases it has been demonstrated that the shape of the initial state wave function (the TS wave function in the fragmentation of a resonance state) is more or less directly reflected in the final-product-state distribution. This mapping is mediated by a Fourier transformation between the y a n d j representations [60]. Since the angular parts of the wave functions for the overall rotational states 503 and

768

R. SCHINKE et al.

J13are different, it follows that also the resulting CO rotational distributions differ. This explains the different oscillatory patterns overlaying the first peaks in the distributions shown in Fig. 3. The first system for which this J state dependence has been demonstrated was the dissociation of water in the first absorption band [61]. The second peak in the CO rotational state distributions stems from an angular region where the coupling is not negligible. In such cases details of the angular dependence of the initial-state wavefunction are more or less erased during the final fragmentation [62], with the consequence that t h e j distribution does not depend much on the total angular momentum state. In conclusion, the dissociation of HCO in its ground electronic state is an intriguing example of mode specificity. The dynamics is mainly regular and almost all states can be easily assigned either by a Dunham expansion of the term energies or by considering the nodal structures of the stationary wave functions. Not unexpectedly, the statistical rate is larger than the quantum average by at least a factor of 5. The final-state distributions depend sensitively on the modes that are excited and on the number of quanta in each mode, and they can be explained, at least qualitatively, by simple models. The calculated energies and widths agree quite well with the best resolved measurements available today. It was certainly worth calculating a new ab initio potential-energy surface for this system.

V. DCO: A SPECTROSCOPIC CHALLENGE When hydrogen is substituted by deuterium, the spectroscopy changes dramatically. While almost all bound and resonance states in HCO are readily assignable, only few states in DCO can be unambiguously classified. The reason is a l :l :2 “resonance” (i.e., near degeneracy) of the D-C stretch, C - 0 stretch, and D-C-0 bending frequencies: w1 = 1900.6 cm-’, w2 = 1804.8 cm-’, and 2w3 = 1675 cm-’ (according to the calculations in Ref. 17). Especially the resonance between the two stretching modes destroys the assignment, which is so straightforward in the case of HCO. The nearly 100% mixing between the corresponding wave functions is best seen in an “experiment” that only theorists can perform. Let us consider the system XCO with the mass of the fictitious atom X, mx,changing continously from 1 (X = H) to 3 (X = T). The resulting term energies of the three relevant excited vibrational states are depicted as functions of mx in Fig. 13. For mx = 1 the two stretching states (1, 0, 0) and (0, 1, 0) are well separated in energy and the assignment provides no problem. The variation of mx in principle does not affect the C - 0 stretching frequency and 0 2 would stay constant (in a diabatic sense). On the other and therefore hand, the X-CO frequency scales approximately as 1/&


769

decreases with increasing mass (again, in a diabatic sense). However, as a consequence of Wigner's noncrossing rule, the two adiabatic curves are not allowed to cross but avoid each other, and this avoided crossing occurs just in the vicinity of mx = 2 (i.e., DCO). The state that for hydrogen has the clear character of excitation in the CO mode finally turns into excitation in the X-C mode for tritium and vice versa. This situation is completely analogous to avoided crossings of two potential-energy curves as functions of the internuclear distance. The mixing between the two stretching states and the analogy with mixing of electronic states become even more apparent in the inset of Fig. 13 where we show, for one particular state, how the expectation values of the kinetic energies in R and T , (TR)and ( T J , change with mx. These curves are very reminiscent of the diabatic mixing angle near an avoided crossing of two potential curves that belong to electronic states with the same symmetry. As a consequence of the strong mixing of the two stretching states, the corresponding wave functions are rotated in the (R, r ) plane by about 45" away from the axes. This is clearly seen in the top panel of Fig. 14, where we show the wave functions of DCO as functions of R and r for all those states that are unexcited in the angular coordinate. The states that are assigned as (0,1,O) and (1,O. 0) actually are better described in terms of n o m l modes, which account for the mixing, rather than by local-mode quantum numbers referring to the Jacobi coordinates (R, r). However, the degree of mixing changes with the total energy; that is, the avoided crossing seen in Fig. 13 (on page 770) shifts in a complicated manner to other masses mx when the energy is increased. Thus, while some states are reasonably well described in terms of normal modes, an assignment by local modes is more appropriate for other states (see, e.g., the states in the most left column of Fig. 14). Furthermore, to make things worse, the state mixing also changes when the bending degree of freedom is also excited by one or several quanta. In conclusion, a meaningful assignment that is valid over the entire energy regime is not possible. More information about this fundamental question of spectroscopy can be found in the original publications [16, 171. Mixing of states when they are nearly degenerate is, of course, an old theme of spectroscopy (Fermi resonances) and is observed in many polyatomic molecules. Incidentally we note that the same general effect has been found by us in the ground state of HNO. However, there the mixing occurs between the NO stretching and the bending states [38]. In conclusion, although the PESs for HCO and DCO are the same, the spectroscopies of the two isotope variations are very different. While almost all states in HCO can be rigorously assigned, the opposite is true for DCO. Nevertheless, a detailed comparison between the more than one hundred resonance energies measured by SEP [16] and the ab initio calculations [17]

770

R. SCHINKE et al.

HCO

1400

1

DCO

1.5

2

TCO

2.5

3

Figure 13. Excitation energies (with respect to the vibrational ground state) of the second, third, and fourth excited vibrational states of XCO as functions of the mass m x . The dashed lines schematically indicate diabatic energy curves. The inset shows the expectation values of the kinetic energies (measured in terms of the corresponding values in the ground vibrational state) of the fourth excited state [(I, 0, 0) for HCO]. (Reprinted, with permission of the American Institute of Physics, from Ref. 17).

is possible and yields a similarly good agreement as for HCO. The deviations for the widths are slightly larger than for HCO, though. Because of the lack of a clear-cut assignment, the mode-specific behavior of the dissociation rates as illustrated in Fig. 8 for HCO does not exist. Yet, DCO cannot be described as an irregular or “chaotic” system; the wave functions still appear too regular to be considered as chaotic (in contrast to the HO2


77 1

Figure 14. Contour plots of the wave functions for DCO in the ground bending states

v3 = 0. The numbers on the right-hand side denote the polyad number N = U I + ~2 + u3/2. The quantum numbers above each column, (UI, v2. u3), indicate the assignment if the substantial

mixing between the modes were not present. For more details see the text and the original publication (171. For ease of the visualization the relevant potential cut is shown in the upper left comer. (Reprinted, with permission of IOP Publishing, from Ref. 8.)

772

R. SCHINKE et al.

wave functions discussed below). On the other hand, severe mixing between zero-order states is the prerequisite for chaos, and in this qualitative sense DCO might be interpreted as a “precursor” for truly irregular systems like HOz. It will be interesting to investigate in the future whether an assignment based on periodic classical orbits is better suited for DCO than traditional methods [63, 641.

VI. I h O : A MIXED REGULAR-IRREGULAR SYSTEM The potential well for HNO is roughly a factor of 3 deeper than for HCO, which leads to a considerably higher density of states p ( E ) ; while the well for HCO supports only 15 bound states, there exist 215 bound states for HNO. The consequence of this increase of p ( E ) is an increase of accidental (near) degeneracies and thus of the mixing between zero-order states. In other words, the quantum dynamics gradually becomes more complex and irregular with rising energy. Nevertheless, series of quite regular states exist even at high energies where the majority of states are already irregular. Examples of wave functions for a regular and an irregular state, whose energies are both above the H + NO continuum, are depicted in Fig. 15. While the regular wave function has a quite clear nodal structure and can be assigned to 11 quanta of excitation in a mode in the (r, y) plane, the irregular wave function is obviously unassignable. It should be clear that the dissociations of these two states will give quite different rates as well as product-state distributions. A detailed analysis of all bound and resonance states is in progress and will be published at a later date. The coexistence of substantial regular and irregular parts of the classical phase space is also reflected by the statistics of the bound-state energy levels shown in Fig. 16. The nearest-neighbor level statistics on the left-hand panel is between the Poisson distribution, predicted for a regular system, and the Wigner distribution, which according to random-matrix theory describes irregular systems [65,66]. The Brody parameter [67]is 0.55 f 0.1 and clearly bears out the mixed character of HNO. The right-hand panel shows the & statistics and again the numerical results for HNO lie between the predictions for the regular and the fully irregular case. The PES for HNO does not have a barrier between the well region and the exit channel; furthermore, the TS is quite loose so that also the potential curves for the lowest adiabatic vibrational-rotational states are purely attractive at large and intermediate H-NO distances (see Fig. 8 of Ref, 34). Therefore it does not come as a surprise that the dissociation starts right at the threshold with rates that are large compared to HCO (Fig. 7). As for HCO

\

\ %/

H.

1

w

Figure 15. Three-dimensional plots of a regular (a) and an irregular (b)continuum wave function in the dissociation of HNO. The energies are in the same range.

(a)

z

h

G

a'

u)

T

0

.5

1

.5

\

\

5

L \

0

1

d<S>

2 3 4

0

1

2

I

I I

1 I

Figure 16. Left-hand panel: The nearest-neighbor energy spacing distributions P(sl(s));the quantum mechanical results are represented by the histograms (full lines) while the prediction for a regular system (Poisson distribution) and that for an inegular system (Wigner distribution) are represented by the short dashes and the long dashes, respectively. Right-hand panel: The z3(iiL)} distributions; the quantum mechanical results are represented by the full lines while the prediction for a regular system and that for an irregular system are represented by the short dashes and the long dashes, respectively. (For more details see Ref. 36. Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

1

Id0

J

I C

Y

h.

3, I I

1


775

the dissociation rates fluctuate a great deal about some average. However, the fluctuations are much less pronounced than in the former case, encompassing only one order of magnitude rather than three. While for HCO the fluctuations reflect mainly the mode specificity; that is, excitations of the various modes lead to substantially different lifetimes. In the case of HNO they are predominantly the consequence of the irregular behavior of the wave functions, as the one shown on the right-hand side of Fig. 15. Because of the generally larger degree of irregularity (or mode mixing), it is not unexpected that the statistical assumptions are on the average better fulfilled. Thus, although the rates predicted by RRKM seem to be, as for HCO, an upper limit, the deviation from the quantum mechanical average is “only” a factor of 2 or so. In our analysis of absorption-type spectra in order to extract the dissociation rates, the very broad resonances, which make up the “background” of the spectrum, are not taken into account. Therefore, the true quantum average is actually higher than the average of the points shown in Fig. 7, which means that the deviation between the statistical rates and the quantum average is probably smaller than a factor of 2. Although the analysis of HNO is not yet at all complete, we can summarize the presently available data in the following way: HNO is a mixed regular and irregular system, and therefore the requirements for a statistical analysis to be trustworthy are better fulfilled than for HCO. As a consequence the RRKM dissociation rate is on the whole in better agreement with the quantum mechanical average. Nevertheless, regular states exist up to very high energies, even above the fragmentation threshold, and this regularity shows up in the final-state distributions.

VII. HO2: CLASSICAL CHAOS REFLECTED IN DISSOCIATION RATES AND PRODUCT-STATE DISTRIBUTIONS The HO;! PES has roughly the same well depth as the HNO potential. However, since the anharmonicity along the 0 2 coordinate r is significantly larger than for the NO mode in HNO (see Fig. 4), the density of states is also significantly higher. There are altogether 726 bound states (361 with odd and 365 with even parity [36]) compared to “only” 215 for HNO. Since the intramolecular coupling is certainly not weaker than for HNO, it is not difficult to surmise that H02 is more irregular than the former system, and this is clearly born out by the statistics of the bound states shown in Fig. 16. Both the level spacing statistics and the 3 statistics are very close to the predictions for irregular systems; the Brody parameter is 0.92 0.1, compared to 0.55 & 0.1 for HNO. The high degree of irregularity is also illustrated by the wave functions. Figure 17 (on page 777) shows examples of continuum wave functions for neighboring resonances in two energy regimes, the first

*

776

R. SCHINKE et al.

one being just above threshold, which is roughly 0.1 eV. In all cases the wave functions show an irregular nodal pattern that clearly rules out any assignment, at least in the usual spectroscopic sense. Also, there is no systematic trend as one goes from one resonance to the next one; the nodal pattern changes in an unpredictable way. The same kind of fluctuating behavior is, not unexpectedly, also shown by the dissociation rates and the final-state distributions of 0 2 , as we will elucidate below. Extracting the rates from the spectrum shown in the lower part of Fig. 7 is certainly not staightforward. At energies only slightly above threshold the resonances are very narrow and do not overlap, so that in most cases a representation by Lorentzians is unambiguous. However, with increasing excess energy the resonances start to overlap, which makes the extraction of rates cumbersome and finally impossible. For a more detailed discussion see Ref. 37. This inherent problem must be kept in mind when we consider the rates for the dissociation of H02 into H and 0 2 shown, for both parities, in the lower part of Fig. 7. As for HNO, the minimum energy path of the PES does not have a barrier, and therefore the fragmentation starts right at threshold with relatively large rates. Again, the rates fluctuate a great deal about an average; the fluctuations encompass about two and a half orders of magnitude near threshold and are still one order of magnitude at the highest energies considered. Of all the systems discussed in the present contribution, HOz is the most irregular one, and therefore it is not surprising to find that the requirements of the statistical models are fulfilled the best: the RRKM rate goes right through the quantum mechanical points and satisfactorily reproduces the quantum average [37]. In addition, the rates obtained from classical trajectories (the result for one energy is included in Fig. 7) also agree quite well with both the quantum average and the RRKh4 prediction. Without showing results we mention that the distribution of (classical) lifetimes can be well represented by a single exponential over the entire energy range, which also suggests that the fragmentation is random [30]. The unimolecular dissociation of HO2 has recently received much interest; Mandelshtam et al. [68],for example, performed a similar study and obtained similar results for the eigenenergies and the dissociation rates. Let us now turn the discussion to the final-state distributions. If the wave functions show an irregular, unpredictable behavior, it is not difficult to surmise that the final-state distributions behave similarly. Figure 18 shows the ratio of the probabilities with which the n = 0 and the n = 1 vibrational states of 0 2 are populated. While the vibrational state distributions for HCO change in a systematic way as one goes from one resonance to the other, the n = O/n = 1 ratio for H02 fluctuates a great deal as a function of (resonance) energy. There is no apparent systematic behavior, except that with increasing energy the average decreases and approaches 1, as predicted by statis-

777


-

-

E 0.15eV

E 0.25 eV

a

t

o

20

LO

60

[degl

80

o

10

20

i

30

o

20

LO

60

T Ideg1

mo

20

10

30

j

Figure 17. Plots of resonant wave functions for H02 as functions of R and y for two energy regimes, E - 0.15 eV and E - 0.25 eV. The 02 coordinate r is fixed. Shown is the logarithm of the modulus square of the total wave function q ~The . right-hand side in each column depicts the corresponding rotational state distributions at these energies. The precise energies (in eV) are from top to bottom: 0.151, 0.152, 0.154, 0.156, 0.158, and 0.252,0.254, 0.257, 0.259, 0.263, respectively. The coordinate R ranges from la0 to 5ao. (Reprinted, with permission of the American Institute of Physics, from Ref. 37.)

778

R. SCHINKE et al. 103

la*

F:--

.

7

100

quanta1

:

.

-

.-. . . .. . : :=-> -.-- - ____ .. . . . . . .

: ’# I **

‘\

Y)z)

3

- classical - -- PST

- ..

10’

I

I

:

la-’

63

*.

-

=

9

-------a

*

0135

6

-

0

--L

.

.

1

0.L5

a5

Figure 18. Vibrational product state distributionof 0 2 following the dissociation of HO2, both quantum and classical, together with the predictions of PST. (Reprinted, with permission of the American Institute of Physics, from Ref. 37.)

tical arguments. The prediction due to phase-space theory (PST [69]) yields a ratio that is on average too small; that is, the vibrational distribution is too “hot” compared to the quantum results. The ratio obtained from classical trajectories is in better agreement with the quantum calculations, especially at the higher energies. At low energies, close to the threshold for the n = 1 channel, violation of zero-point energy overestimates the n = 1 probability and therefore leads to a significant underestimation of the n = O/n = 1 ratio. The final rotational state distributions of the products in the fragmentation of a polyatomic molecule contain additional clues about the intra- and intermolecular dynamics, especially about the coupling in the exit channel. In realistic as well as model studies it has been observed that the rotational state distributions of the photodissociation products “reflect” the angular dependence of the wave function at the transition state and the anisotropy of the PES in the exit channel [4, 9, 101. HO2 is no exception. Examples of final rotational state distributions are shown, together with the corresponding wave functions, in Fig. 17. Several general features are immediately apparent: (i) The number of occupied rotational channels increases steadily with energy, which is simply a result of energy conservation. (ii) While at low energies all states accessible at this energy are populated, at higher energies the levels close to the energetic cutoff remain unfilled; the gap to the highest accessible state increases gradually with E.


779

(iii) The distributions show, except for the lowest energies, a very complicated oscillatory behavior, with the number of oscillations generally increasing with energy; the latter observation is in accord with the increase of the number of minima/maxima in the angular dependence of the TS wave function as E increases. (iv) There is no apparent correlation in the oscillatory structures as one goes from one resonance to the next one; the fluctuations seem to be random and unpredictable, just like the nodal structures in the wave functions. The results of the quantum calculations are compared, for four energies, to the results of classical trajectory calculations as well as two types of statistical models in Fig. 19. First of all, except for the pronounced oscillations, which are certainly caused by quantum interferences, the classical rotational state distributions agree on average extremely well with the quantum distributions; especially their extent is accurately reproduced. Phase-space theory, which assumes that for total angular momentum J = 0 all asymptotically accessible states are equally filled, yields poor agreement with the exact results. The PST distributions extend all the way to the highest possible state, while the quantum as well as classical distributions, which incorporate the full dynamics from the inner region to the asymptotic product channel, die off at lower rotational states. The gap increases with the total available energy. On the other hand, the statistical adiabatic channel model (SACM [70, 7 11). which assumes that all energetically accessible adiabatic bending/rotational states are equally populated at the transition state and that there is no further coupling between the adiabatic states from the TS to the products, yields too “cold” rotational state distributions. The vertical bars in Fig. 19 indicate the highest state,jSAcM, that would be populated according to SACM. The true distributions extend to significantly higher rotational states, although in each case ~ S A C Mmore or less marks the beginning of the roughly exponential decay toward the high-j tail. While PST assumes strong coupling among the rotational states all the way to the products, SACM, on the other hand, completely neglects any coupling between the adiabatic states beyond the TS. The truth, at least in the present case, lies inbetween. If Ref. 37 we applied a very simple model that, apart from the oscillations, yields reasonable agreement with the observed rotational distributions. It is based on two assumptions: first, that the wave function at the TS is a linear superposition of all energetically accessible (bending) states with equal weighting and, second, that the final rotational state distribution, on average, is obtained by expanding the TS wave function in terms of the final rotor states (Franck-Condon mapping [4]; see also Refs. 10,72, and 73). Dynamics in the exit channel, which is small but not negligible in the case of HOz, is not taken into account in this model. In conclusion, the dynamics of HOz at high energies, below as well as

R. SCHINKE et al.

780

0

10

20

-

Figure 19. Comparison of the quantum (filled circles, long dashes) and the classical (solid lines) rotational product distributions of 02(n = 0) following the dissociation of HOz for four energies as indicated; the precise energies of the correspondingquantum resonances are 0.1513, 0.2517, 0.3507, and 0.4471 eV, respectively. Also shown are the distributions obtained from PST (short dashes). All distributions are normalized so that the areas under the curves are equal. The arrows on the j axis indicate the highest accessible state at the respective energy and the vertical bars on the classical curves indicate jSACM. the highest populated state according to the SACM. (Reprinted, with permission of the American Institute of Physics, from Ref. 37.)

above the dissociation threshold, is mainly irregular, and therefore the fragmentation rate is on average satisfactorily described by statistical models. The strong fluctuations of the state-specific rate constants are caused by the irregular, randomlike behavior of the corresponding wave functions, which is in accord with the fact that the distribution of the fluctuations seem to


78 1

follow the predictions of random-matrix theory [37]. On the other hand, statistical methods, which make unrealistic assumptions about the dynamics in the exit channel beyond the TS,at least for H02, do not yield satisfactory final vibrational and rotational state distributions for the 0 2 fragments.

VIII. RESUME AND OUTLOOK The series of molecules HCO, DCO, HNO, and HO2 is well suited for a demonstration of the transition from regular to irregular dynamics in unimolecular dissociation processes on ground-state PESs with relatively deep wells. Whether a molecule shows regular or chaotic motion depends intimately on the density of states p(E) and/or the coupling between the various degrees of freedom. In the cases discussed in the present overview both p ( E ) and the degree of internal mixing of zero-order states increase gradually from the mainly regular system HCO to the essentially irregular dynamics of H02, and the dissociation rates as well as the final-state distributions show this. The wave functions for HCO have mostly a regular nodal pattern, and this simple behavior is reflected in the observables. The decay rates and the final-state distributions for H02, too, reflect the corresponding wave functions. But in this case the wave functions have complicated, irregular nodal structures, and the same general behavior is found in the rates and the fragment distributions. They fluctuate a great deal about some average, having no systematic behavior as one proceeds from one resonance to the next one. The wave functions are the central quantities, and therefore we showed examples for all the four systems considered. Inspection of wave functions reveals more or less directly how the observables generally behave. This is, of course, an obvious statement. Knowing the wave functions is the great asset of theory compared to experimental spectroscopy, and it is not difficult to surmise that many spectroscopic misinterpretations could be avoided if the underlying wave functions were known. HCO is a relatively simple system. However, that does not mean that it is uninteresting. On the contrary, the availability of high-resolution experimental data for about 60 resonance states (and about 120 for DCO) calls for high-quality quantum dynamics calculations. The level of detailed comparison between experiment and theory possible for HCO (and likewise for DCO) is unprecedented for a dissociating molecule, which makes it a real challenge for dynamical theory. Because of the large amount of experimental data (energies, widths, rotational constants, and final-state distributions), an accurate PES is required. A less accurate potential might reveal the general dissociation dynamics but certainly not all the fine details. In this respect HCO is actually a rather difficult system; both the PES and the dynamics calculations are required to be highly accurate.

782

R. SCHINKE et al.

In the case of H02 the observables show erratic looking fluctuations that are certainly the result of complicated quantum mechanical interferences. Interference structures always depend sensitively on details of the dynamics and ultimately on details of the PES, and therefore, one might be tempted to believe that the potential can be determined by an “inversion” of experimental data. That is very likely not the case, simply because the sensitivity of the observables with respect to the PES is too strong, that is, the variations of rates, for example, with changes of the potential are nonlinear. In other words, no calculated PES will be sufficiently accurate to allow a state-bystate comparison with experimental data (provided they exist). For example, fluctuating dissociation rates have been measured for H2CO [74-761 and CH3O [77,781 and fluctuating rotational state distributions have been observed in the dissociation of, for example, NO2 [79,801. Can one ever hope to reproduce these data on a state-by-state level? Certainly not! One should be content with reproducing the average dissociation rate and the average rotational distribution. However, that can be achieved with less accurate potentials and more approximate dynamics calculations. For example, for the dissociation of H02 we found that the average rate is well described by statistical theories; classical trajectories even reproduce both the average rate and the average final-state distributions of the 0 2 fragments reasonably well. Average values, however, are less sensitive to details of the PES, at least in the case of HO2, and are determined by relatively small regions of the coordinate space. For example, the statistical rate is mainly determined by the potential near the transition state, and the average final-state distribution depends mainly on the forces in the exit channel from the transition state to the asymptote. In this sense HO:! is actually the simpler of the two systems, HCO and H02. In the future we will attempt to extend the calculations to systems with still higher densities of states. In all cases studied up to now, the atomic fragment is hydrogen, leading to a small reduced mass associated with the dissociation coordinate. We are currently calculating PESs for OClO and NO2 in their ground electronic states. For the latter system measured rates as well as rotational and vibrational state distributions are available, and it will be interesting to compare them with the results of classical trajectory calculations. Another interesting question concerns the applicability of statistical models for predicting final-state distributions, depending on the nature of the transition state, that is, whether it is “tight” or “loose.” Furthermore, we will consider the construction of an average quantum mechanical rate directly from time-dependent wavepacket calculations in order to avoid the problems associated with overlapping resonances. This seems to be necessary if one wants to assess the reliability of statistical methods in a more unambigious way.


783

Acknowledgments R. S. and H. M. K. gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich357 Molekulare Mechanismen Unimolekularer Prozesse. A. J. D. is grateful for a fellowship from the Royal Society/Deutsche Forschungsgemeinschaft under the European Science Exchange Programme. D. H. M. is grateful for a fellowship from the European Union under the Human Capability and Mobility Programme. R. S. and H. M. K. gratefully acknowledge their ongoing collaboration with H.-J. Werner, P. Rosmus, F.Temps, and E. A. Rohlfing on the dissociations of HCO and DCO. Many fruitful discussions with W. L. Hase, J. Troe,C. B. Moore,W. H.Miller, H. Reisler, and many other colleagues have greatly influenced the authors’ general understanding of unimolecular dissociation.

References 1. A. Messiah, Quantum Mechanics, Vols. I and 11. North-Holland, Amsterdam, 1972. 2. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Vols. I and 11, Wiley, New York, 1977. 3. G. R. Satchler, Introduction to Nuclear Reactions, Oxford University Press, New York, 1990. 4. R. Schinke, Photodissociation Dynamics, Cambridge University Press, Cambridge, 1993. 5 . D. G. Truhlar, Ed., Resonances in Electron-Molecule Scattering, van der Waals Complexes, and Reactive Chemical Dynamics, American Chemical Society, Washington, DC 1984. 6. V. I. Kukulin, K. M. Krasnopolsky, and J. HorEek, Theory ofResonances, Principles and Applicafions, Kluwer, Dordrecht, 1989. 7. M. Garcia-Sucre, G. Raseev, and G. C. Ross, Eds., Half Collision Resonances in Molecules: Experimental and Theoretical Approaches, American Institute of Physics, New York, 1991. 8. R. Schinke, H.-M. Keller, M. Stumpf, and A. J. Dobbyn, J. Phys. B 28, 3081 (1995). 9. R. Schinke, A. Untch, H. U. Suter, and J. R. Huber, J. Chem. Phys. 94, 7929 (1991). 10. H. Reisler, H.-M. Keller, and R. Schinke, Commun. At. Mol. Phys. 30, 191 (1994). 11. F. F. Cnm, in Molecular Photodissociation Dynamics, M. N. R. Ashfold and J. E. Baggott, Eds., The Royal Society of Chemistry, London, 1987. 12. F. F. Crim, Ann. Rev. Phys. Chem. 44, 397 (1993). 13. C. E. Hamilton, J. L. Kinsey, and R. W. Field, Annu. Rev. Phys. Chem. 37, 493 (1986). 14. F. J. Northrup and T. 3. Sears, Annu. Rev. Phys. Chem. 43, 127 (1992). 15. H.-L. Dai and R. W. Field, Eds., Molecular Dynamics and Spectroscopy by Stimulated Emission Pumping, World Scientific, Singapore, 1994. 16. C. Stiick, X. Li, H.-M. Keller, R. Schinke, and F. Temps, J. Chem. Phys., 106,5333 (1997). 17. H.-M. Keller, T. Schroder, M. Stumpf, C. Stock, F. Temps, R. Schinke, C. Bauer, P. Rosmus, and H.4. Werner, J. Chem. Phys., 106, 5359 (1997). 18. D. W. Neyer and P. L. Houston, in The Chemical Dynamics and Kinetics of Small Radicals, K. Liu and A. Wagner, Eds., World Scientific, Singapore, 1994. 19. D. W. Neyer, X. Luo, I. Burak and P. L. Houston, J. Chem. Phys. 102, 1645 (1995). 20. H.-M. Kellcr and R. Schinke.

784

R. SCHINKE et al.

21. B. Hartke, J. Manz, and J. Mathis, Chem. Phys. 139, 123 (1989).

22. W. L. Hase, S.-W. Cho, D.-H. Lu, and K. N. Swamy, Chem. Phys. 139, 1 (1989). 23. D. M. Wardlaw and R. A. Marcus, Adv. Chem. Phys. 70,231 (1988). 24. M. Quack and J. Troe, in Theoretical Chemistry, Vol. 6b: Theory ofScattering, D. Henderson, Ed., Academic, New York, 1981. 25. J. Troe, in State-Selected and State-to-State Ion-Molecule Reaction Dynamics, Part 2: Theory, M. Bear and C. Y. Ng, Eds., Wiley, New York, 1992. 26. P. J. Robinson and K. A. Holbrook, Vnimolecular Reactions, Wiley-Interscience, London, 1972. 27. W. Forst, Theory of Unimoiecular Reactions, Academic, New York, 1973. 28. R. G. Gilbert and S. C. Smith, Theory of Vnimolecular and Recombination Reactions, Blackwell, Oxford, 1990. 29. W. H. Green, Jr., C. B. Moore, and W. F. Polik, Annu. Rev. Phys. Chem. 43, 591 (1992). 30. W. L. Hase, in Modern Theoretical Chemistry, Dynamics of Molecular Collisions, Part B, W. H. Miller, Ed., Plenum, New York, 1976. 31. W. L. Hase, Arc. Chem. Res. 16,258 (1983). 32. H.-J. Werner, C. Bauer, P. Rosmus, H.-M. Keller, M. Stumpf, and R. Schinke, J. Chem. Phys. 102, 3593 (1995). 33. H.-M. Keller, H. Flothmann, M.Stumpf, R. Schinke, H.-J. Werner, C. Bauer, and P. Rosmus, J. Chem. Phys. 105,6983 (1996).

34. M. Stumpf, A. J. Dobbyn, D. H. Mordaunt, H.-M. Keller, R. Schinke, H.-J. Werner, and K. Yamashita, Faraday Discuss. Chem. Soc. 102, 193 (1995). 35. A. I. Dobbyn, M. Stumpf, H.-M. Keller, W. L. Hase, and R. Schinke, J. Chem. Phys. 102, 5867 (1995). 36. A. J. Dobbyn, M. Stumpf, H.-M. Keller, and R. Schinke, J. Chem. Phys., 103,9947 (1995). 37. A. J. Dobbyn, M. Stumpf, H.-M. Keller, and R. Schinke, J. Chem. Phys. 104,8357 (1996). 38. H. D. Mordaunt, H. Flothmann, M. Stumpf, A. J. Dobbyn, H.-M. Keller, R. Schinke, and

K. Yamashita, to be published.

39. M. R. Pastrana, L. A. M. Quintales, J. Brandb, and A. J. C. Varandas, J. Phys. Chem. 94,8073 (1990). 40. R. N. Dixon, J. Chem. Soc. Faraday Trans. 88,2575 (1992). 41. U. Peskin, H. Reisler, and W. H. Miller, J. Chem. Phys. 101, 9672 (1994). 42. J. D. Tobiason, J. R. Dunlop, and E. A. Rohlfing, J. Chem. Phys. 103, 1448 (1995). 43. D. Wang and J. M. Bowman, J. Chem. Phys. 100, 1021 (1994). 44. D. Wang and J. M. Bowmann, Chem. Phys. Lett. 235,277 (1995). 45. 46. 47. 48. 49. 50.

V. Ryaboy and N. Moiseyev, J. Chem. Phys. 103,4061 (1995).

T. P. Grozdanov, V. A. Mandelshtam, and H. S. Taylor, J. Chem. Phys. 103,7990 (1995). S. K. Gray, J. Chem. Phys. %, 6543 (1992). J. M. Bowman, J. S. Bittman, and L. B. Harding, J. Chern. Phys. 85, 911 (1986). J. D. Tobiason, J. R. Dunlop, and E. A. Rohlfing, Chem. Phys. Lett. 235, 268 (1995). W. H. Miller, Chem. Rev. 87, 19 (1987). 51. R. H. Bisseling, R. Kosloff, and J. Manz, J. Chem. Phys. 83, 993 (1985). 52. B. Hartke and J. Manz, J. Chem. Phys. 92,220 (1990).


785

53. B. Hartke, A. E. Janza, W. Karrlein, J. Manz, V. Mohan, and H.-J. Schreier, J. Chem. Phys. 96,3569 (1992). 54. J. A. Beswick and J. Jortner, Adv. Chem. Phys. 47,363 (1981).

55. A. Vegiri, A. Untch, and R. Schinke, J. Chem. Phys. 96,3688 (1992). 56. D. Solter, H. J. Werner, M. von Dirke, A. Untch, A. Vegiri, and R. Schinke, J. Chem. Phys. 97, 3357 (1992). 57. C. X.W. Qian, A. Ogai, L. Iwata, and H. Reisler, J. Chem. Phys. 92,4296 (1990).

58. A. Ogai, J. T. Brandon, H. Reisler, H. U. Suter, J. R. Huber, M. von Dirke, and R. Schinke, J. Chem. Phys. 96,6643 (1992). 59. A. J. Dobbyn, M. von Dirke, R. Schinke, and R. Fink, J. Chem. Phys. 102,7070 (1995). 60.R. Schinke, R. L. Vander Wal, J. L. Scott, and F. F. Cnm, J. Chem. Phys. 94,283 (1991). 61. R. Schinke, V. Engel, P. Andmen, D. Hausler. and G. G. Balint-Kurti, Phys. Rev. Leu. 55, 1180 (1985).

62. 63. 64. 65.

A. Untch, K. Weide, and R. Schinke, J. Chem. Phys. 95,6496 (1991). J. M. Gomez Llorente and E. Pollak, Annu. Rev. Phys. Chem. 43, 91 (1992). P. Gaspard, D. Alonso, and I. Burghardt, Adv. Chem. Phys. XC, 105 (1995). T. Zimmermann, L. S. Cederbaum. H.-D. Meyer, and H. Koppel, J. Phys. Chem. 91,4446

(1987). 66. M. C. Gutzwiller, Chaos in Classicafand Quantum Mechanics, Springer, New York, 1990. 67. T. A. Brody, J. Flores, J. B. French, P. A. Mello. A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. 53, 385 (1981). 68. V. A. Mandelshtam, T. P. Grozdanov. and H. S. Taylor, J. Chem. Phys. 103, 10079 (1995). 69. P. Pechukas, J. C. Light, and C. Rankin, J. Chem. Phys. 44,794 (1966). 70. M. Quack and J. Troe, Ber. Bunsenges. Phys. Chem. 78,240 (1974). 71. M. Quack and J. Troe, Ber. Bunsenges. Phys. Chem. 79, 170 (1975). 72. S. A. Reid and H. Reisler, J. Chem. Phys. 101, 5683 (1994). 73. U. Peskin, W. H. Miller, and H. Reisler, J. Chem. Phys. 102, 8874 (1995). 74. W.F. Polik, C. B. Moore, and W. H. Miller, J. Chem. Phys. 89, 3584 (1988). 75. W. F. Polik. D. R. Guyer, W. H. Miller, and C. B. Moore. J. Chem. Phys. 92,3471 (1989). 76. W. H. Miller, R. Hernandez, C. B. Moore, and W. F. Polik, J. Chem. Phys. 93, 5657 (1990). 77. A. Geers, J. Kappert. F. Temps, and J. W. Wiebrecht, J. Chem Phys. 99,2271 (1993). 78. F. Temps, in Molecular Dynamics and Spectroscopy by Stimulated Emission Pumping, H.-L. Dai and R. W. Field, Eds., World Scientific, Singapore, 1994. 79. M. Hunter, S. A. Reid, D. C. Robie, and H. Reisler, J. Chem. Phys. 99, 1093 (1993). 80. S. A. Reid, D. C. Robie, and H. Reisler. J. Chem. Phys. 100, 4256 (1994).

DISCUSSION ON THE REPORT BY R. SCHINKE Chairman: J. Manz

S. A. Rice: Prof. Schinke, can you correlate parts of the distribution of resonances you have calculated with the “scars” of the wave function

786

R. SCHINKE et al.

that are associated with unstable classical trajectories? In particular, can the widths of the resonances be grouped so as to show whether or not intramolecular energy transfer competes with the reaction? Using the language of kinetics, I believe that it should be possible to associate the scars with barriers to intramolecular energy exchange. If my assertion is correct, the presence of scars can be taken as a signal, in the quantum mechanical case, of deviation from the statistical limit description of unimolecular reaction rate.

R. Schinke: We did not analyze all resonance wave functions in the case of HO2. But from the limited investigation performed we note that we did not see pronounced scars. This is in accord with your assertion that scars are a signal of deviation from the statistical limit. HO2 is mainly irregular at energies around the dissociation threshold, and this is nicely documented by the classical calculations, which show a single exponential for the lifetime distribution function (Fig. 9 in Ref. 34 of the current chapter). On the other extreme, HCO is mainly regular, and the classical distribution function consists of (at least) three exponentials. The corresponding quantum mechanical wave functions show in most cases clear nodal structures that could be interpreted as scars. Thus, in general, I agree with your assertion that scars are associated with barriers to intramolecular energy transfer whereas the lack of scars indicates very fast internal vibrational energy transfer and therefore statistical decay. R. A. Marcus: Prof. Schinke has certainly described an array of exciting results. In the case where your wave functions showed a complicated pattern, it would be useful (for the case of an isolated internal resonance) to seek out the relevant vibrational periodic trajectories to sort out the series of such states and relate (directly or indirectly) to Kellman’s algebraic analysis of bound states. Another way of calculating the distribution of product states would be to apply an extension of RRKM that Wardlaw, Klippenstein, and I developed. However, judging from your observations, the reaction is highly vibrationally nonadiabatic, considering, for example, the considerable difference in vibrational quantum number uco in HCO and CO and the major change in bending -+ rotational state. In that case a Franck-Condon approach would seem to be much more appropriate than any adiabatic or near-adiabatic or statistically adiabatic model. R. Schinke: Concerning the rotational state distribution of 02, we applied actually a Franck-Condon mapping model that reproduces the rotational state distribution at the transition state. A full description is given in the original publication. However, the coupling is not negli-


787

gible in the exit channel, and therefore one must go beyond this model and incorporate the final channel coupling. Classical mechanics is certainly a very good approximation. J. ’ h e : Let me point out that the ab initio calculations of the H02 surface still must be in error in their long-range part. High-pressure H + O2 recombination experiments clearly show that the potential is completely loose and not “semirigid” like the potential you used for illustration. For loose potentials SACM and classical trajectory calculations of product distributions (in the classical range) nicely agree. R. Schinke: The calculations for H02 are certainly model calculations as I underlined in my talk. First, as you pointed out, the potential might not be the most accurate one, especially at large H + 02 separation. Second, there might be other electronic states that are involved within the considered energy range. D. M. Newark: In Lineberger’s photoelectron spectrum of the HCO- ion, resonances above the H + CO dissociation threshold were observed. Were any resonances seen in this experiment that were not seen (at higher resolution) in the stimulated emission pumping work? R. Schinke: The earlier spectroscopicexperiments of the Lineberger group certainly probe the same parts of the potential-energy surface as the experiments of the Rollfing group, with which I compared our theoretical calculations. The Rollfing data are the most complete and the best resolved. M. E. Kellman: Following up on Stuart Rice’s point regarding scars of periodic orbits, are there abrupt changes in reaction rates as nodal patterns of the vibrational wave functions change in the regular regime, for example in DCO, where the local-mode nodal pattern breaks down? R. Schinke: No, we did not clearly see abrupt changes in the dissociation rate as nodal patterns of the underlying wave functions change. The rates fluctuate strongly with energy, and since there is no clear-cut assignment, it is difficult to recognize any schematic trends.

PHOTODISSOCIATING SMALL POLYATOMIC MOLECULES IN THE VUV REGION: RESONANCES IN THE 'E+-fC' BAND OF OCS K. YAMANOUCHI,* K.OHDE, and A. HISHIKAWA Department of Pure and Applied Sciences College of Arts and Sciences The University of Tokyo Tokyo, Japan

CONTENTS 1. Spectra of Dissociating Molecules 11. Absorption Spectrum of O C S in the VUV Region 111. PHOFEX Spectrum of the 'c+-'c' Band of OCS

IV. Fano Profile in the VUV-PHOFEX Spectrum of O C S V. Concluding Remarks References

I. SPECTRA OF DISSOCIATING MOLECULES Recent advances in ultrashort laser technology has enabled us to investigate dynamics of molecules in a time domain, and furthermore, the success of a theoretical interpretation of the results of time-domain experiments by a moving wavepacket on a potential energy surface (PES)impressively demonstrated the importance of time-domain experiments [l]. On the other hand, it is well-known that a spectrum in a frequency domain and an autocorrelation function in the time domain can be transferred with each other via a Fourier transformation [2]. Therefore, it can be said that the spectrum *Communication presented by K. Yamanouchi Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond lime Scale, XXrh Sofvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

789

790

K. YAMANOUCHI, K. OHDE, AND A. HISHIKAWA

represented in a frequency domain has essential information regarding the dynamics of a molecular system after it is excited to an electronically excited PES. The interpretation of a spectrum from a dynamical point of view can also be applied to a spectrum containing a broad feature associated with direct and/or indirect dissociation reactions. From such spectra dynamics of a dissociating molecule can also be extracted via the Fourier transform of a spectrum. An application of the Fourier transform to the Hartley band of ozone by Johnson and Kinsey [3] demonstrated that a small oscillatory modulation built on a broad absorption feature contains information of the classical trajectories of the vibrational motion on PES, so-called unstable periodic orbits, at the transition state of a unimolecular dissociation. In this decade dissociation dynamics of a variety of molecules has been investigated both experimentally and theoretically 141. On the experimental side, rich information regarding photodissociation dynamics has been derived by measuring a product-state distribution of photofragments, a photofragment excitation (PHOFEX) spectrum, and a Doppler profile of photofragments. From these photodissociation experiments as well as from the measurements of absorption spectra exhibiting broadened features, it has been known that most polyatomic molecules dissociate very rapidly after absorbing VUV light. However, as far as photodissociation studies in the vacuum ultraviolet (VUV) wavelength region are concerned, laser spectroscopy has afforded only fragmental information at fixed wavelength such as 193 and 157 nm. Until very recently, almost no elaborate effort has been made in order to extract the dissociation dynamics directly from an observed absorption spectrum in the VUV region. In absorption spectra measured under the bulb conditions, it is in general difficult to identify homogeneous linewidths associated with a lifetime shortening separately from inhomogeneous contribution, such as overlapping vibrational hot bands and rotational broadening. Even when free jet expansion is employed to cool down molecules, it has been difficult to sample molecules only in the coldest central region of the free jet. In the present study, we demonstrate that PHOFEX spectrum of jet-cooled molecules is most suitable to derive an absorption profile in the VUV region representing only the homogeneous broadening. 11. ABSORPTION SPECTRUM OF OCS IN THE VUV REGION Among the absorption spectrum of relatively simple polyatomic molecules, a spectral structure of the strong 'C+-'C' band of OCS in the 160-140-nm region [5] is noteworthy. Though there is a broad background-like structure in the entire absorption band, there are seven distinct features with a sep-

PHOTODISSOCIATING SMALL POLYATOMIC MOLECULES

791

aration of -800 cm-' . The wavelength positions for these seven peaks are 156.1,154.5,152.6,150.6,149.0,147.3,and 145.7nm. The absorption cross cm2. From the Fourier sections of these absorption peaks are as high as transformation of the absorption spectrum recorded by McCarthy and Vaida 161 under jet-cooled conditions, we identified recently a period of 41 fs, corresponding with intervals of -800 cm-', as a clear recurrence time in the autocorrelation function [7].This period should represent some vibrational dynamics on the PES of the excited 'C+ state. Previously, based on an absorption spectrum, two vibrational progressions were identified and were ascribed to two stretching modes; that is, a progression of the C-S stretching mode carrying strong intensity and a weaker progression of the C-S stretching mode built on one quantum of the C-O stretching mode. It is known that the excitation to the upper 'E+ electronic state leads to only one dominant dissociation channel of CO(X C+)+ S( S), and a quantum yield of S('S) was derived to be >80% in the entire 160-140 nm of the absorption band [S]. If the PES for the excited 'E+state is mostly repulsive along the dissociation coordinate and is correlated exclusively to CO(X IC+) + S('S), an intense progression should be assigned to the C-O stretch rather than the C-S stretch, since a spectral structure of a dissociating state reflects vibrational modes orthogonal to a reaction coordinate.

'

111.

'

PHOFEX SPECTRUM OF THE 'E+-lC+ BAND OF OCS

In order to clarify the vibrational dynamics on the PES of the C+ electronic state of OCS, we attempted to record a high-resolution ' F - ' E + absorption profile of OCS free from the inhomogeneous broadening by introducing a technique of PHOFEX spectroscopy with a tunable high-resolution V W laser light source [7,9, 101. Advances in nonlinear laser spectroscopy have made it possible to generate an intense, coherent, and tunable light in the VUV wavelength region by four-wave sum and difference-frequency mixing of tunable visible and UV laser beams in a nonlinear medium like rare-gas and metal vapor [ll]. It has been shown that the VUV light in the wavelength region between 200 and 115 nm can be continuously covered by using Sr, Mg,Hg, Kr, and Xe as nonlinear media. One advantage of this VUV light is its high resolving power as high as (x/6X) - lo6 originating from the high-resolution nature of visible and UV laser light adopted in the nonlinear mixing. In the measurements of the PHOFEX spectrum, we scanned the VUV wavelength while the photofragment of S( IS)was monitored by exciting the S(3DI)-S('S) transition by UV laser light. Since only the fluorescence emitted from the S(3D1)fragments in the central region of the free-jet expansion was collected, the photoabsorption of ultracold (-5 K) OCS was selectively

792


detected. Since the quantum yield of S('S) is S O % , the PHOFEX spectrum thus measured is almost identical with an absorption spectrum measured under the ultracold conditions. Therefore, the spectrum should be free from rotational broadening and overlapping of vibrational hot bands, and the spectral feature in the PHOFEX spectrum represents only a homogeneous contribution, reflecting directly the photodissociation dynamics. As described later, the recent ab initio molecular orbital (MO) calculation of the excited-state PES of O C S by Hishikawa et al. [lo] shows that the 'C+ state is mostly repulsive along the dissociation coordinate r(0C- - -S). When we measured the PHOFEX spectrum of the lowest energy peak located at 154.5 nm, we found that the spectral peak exhibits significantly narrower width (-42 cm-I) [9] than that observed previously by McCarthy and Vaida under the jet-cooled conditions. By subtracting (i) the instrumental resolution and (ii) the rotational contribution estimated by using the rotational constant of the electronic ground 'E+state from the observed peak width, the homogeneous width was derived, and then the lifetime of the broadened feature was estimated to be 133 fs. The rotational temperature was estimated from the rotational structure of the A-X transition of CO contained in the sample gas by a trace amount. If the vibrational period of 41 fs derived from the Fourier transform of the absorption spectrum of the 'C+-'C+band [6] represents that of the vibrational motion near the transition-state region of the unimolecular dissociation, the lifetime associated with the 154.5-nm peak indicates that OCS vibrates more than three times along the direction perpendicular to the dissociation coordinate prior to the complete dissociation. There are two vibrational modes, the in-phase stretch and the O X - S bend, which are orthogonal to the dissociation coordinate, corresponding with the C-S stretch. Since both of the two C+ electronic states involved in the transition are known to be linear at their equilibrium position, a long progression may not be expected for the bending excited states. Therefore, the distinct progression with an interval of -800 cm-' observed in the 'C+-'C+ band is ascribed to the vibrational progression for the in-phase stretch. The interesting findings are that the PHOFEX peak at 152.6 nm is significantly broader than the peak at 154.5 nm and the peak at 150.6 nm is much broader. The lifetimes for the peaks at 152.6 and 150.6 nm were derived from the broadened peak widths to be 44 and 27 fs, respectively. It can be said that OCS vibrates only once along the in-phase stretching mode when it is excited to the 152.6-tun peak and less than once when excited to the 150.6-nm peak. In Fig. 1, the PHOFEX spectrum of the three most intense vibronic bands of OCS are compared with the corresponding absorption spectrum measured under the jet-cooled conditions by McCarthy and Vaida [63. It is clearly seen in this figure that the PHOFEX peaks were significantly narrower than the absorption peaks. This difference in the peak width demonstrated that (i)


MOO0

65Ooo

660v

WAVENUMBER / cm-

793

67000

Figure 1. Comparison of the VUV-PHOFEX spectrum of a part of the I C+-' I? band of the jet-cooled OCS and the corresponding absorption spectrum measured by McCarthy and Vaida [6].

the broadening due to the rotational structure and vibrational hot bands is largely decreased in the PHOFEX measurements, since only the coldest spatial region of the free-jet expansion is sampled in the PHOFEX experiments, and (ii) the spectral resolution in the PHOFEX measurements, determined by the resolution of a tunable VUV laser, is very high.

IV. FAN0 PROFILE IN THE WV-PHOFEX SPECTRUM OF OCS It can also be noticed in Fig. 1 that spectral features for these three peaks are not symmetrical; that is, their spectral shape deviates considerably from a simple Lorentzian line shape. Since the rotational contribution in the peak width in the PHOFEX spectrum is -I cm-', which is significantly smaller than the observed peak width, these asymmetrical spectral features are regarded as Fano-type profiles, which can appear in a spectrum for quasibound states. A Fano profile was originally derived to interpret an asymmetrical spectral feature of autoionizing atoms [ 121, but it can also be identified in the electric spectrum of some simple molecules, which indirectiy or directly dissociate. It has been known that a transition from an electronic ground state to a resonance state in the excited-state PES,formed through a mixing between zero-

794


order bound and continuum states, can exhibit an asymmetrical line shape, when there are nonzero transition moments to both of the bound and continuum zero-order states. It has been reported that H2 [13], Cs2 [14], and 0 2 [15] predissociate via a coupling with a dissociation continuum and Fano profiles were identified in their spectra. In the case of Cs2, Kim and Yoshihara identified a clear q-reversal in the progression of vibronic bands, each of which exhibits an asymmetrical line shape. In the optical-optical double resonance (OODR) spectrum of the 7 3 S ~Rydberg state of HgNe, Okunishi et al. [ 161 observed a characteristic asymmetrical line shape caused by a potential barrier. The resonances resulting in the asymmetrical spectral profiles for HgNe can be classified as the shape resonance, since the dissociation occurs on a single electronic potential energy curve, while those found in H2, Csz, and 0 2 can be classified as the Feshbach resonance [ 181. In the present measurements of the PHOFEX spectrum of OCS, the observed Fano profile is categorized into the Freshbach resonance, in which a zero-order bound state associated with the vibrational motion perpendicular to the dissociation coordinate couples with the zero-order continuum state associated with the motion along the dissociation coordinate. This type of resonances was also identified of the photodissociation of FNO by Reisler and co-workers [18]. By using a Fano line shape formula, the observed PHOFEX peaks were fitted, and an asymmetry parameter q and a width are determined for these three peaks. It is clearly seen in Fig. 1 that the peak width increases substantially as energy increases and that these three peaks exhibit asymmetric line profiles, though a degree of asymmetry decreases as energy increases. We performed the fitting of the observed peak profile to a Fano line formula, expressed as

where e = (E--Er)/(I'/2), with E, and r representing the energy and the full width at half maximum (FWHM) of the resonance state, respectively, and an asymmetric parameter q reflecing the interference between the discrete and dissociative states. From the least-squares fitting, r = 105(10) cm-' and q = -8.1(7) were derived for the 152.6-nm peak and I' = 186(20) and q = -20(3) for the 150.6-nm peak. Previously, we determined the corresponding values for the 154.5-nm peak as r = 41.6(2) cm-' and q = - 3 3 9 ) [7]. The observation indicates that the dissociation rate increases sensitively as energy increases in the energy range covering these three peaks. It is noted that the q value becomes negatively large as energy increases. In other words, the


795

observed asymmetrical line shape approaches a symmetrical Lorentzian line shape. Recently, Hishikawa et al. [lo] calculated ab initio PESs for the electronically excited states of O C S and showed that the excited state in the 160-140-nm region is rather isolated from the other electronic states and that the slope of the PES along the dissociation coordinate is extremely flat in the Franck-Condon region from the ground vibrational state of the electronic ground state. The PES for the '72 state has a shallow valley along the dissociation coordinate r(OC-- -S) from the Franck-Condon point, and a shallower slope extends along r(0-- -CS). From the wavepacket calculation on the ab initio PES,it was demonstrated that a wavepacket moves along the direction perpendicular to the dissociation coordinate while the dissociation proceeds rapidly. The autocorrelation function derived from the wavepacket dynamics exhibits a period of 48 fs as a recurrence time for the wavepacket associated with the transition-state vibrational motion, which is comparable with the experimental period of 41 fs [6]. This reasonable agreement of a vibrational period near the transition state confirmed that the overall feature the 'I7state PES is described well by the theoretical PES. An asymmetrical line shape is known to characterize a resonance state in a bimolecular reaction. Sadeghi and Skodje [ 191derived a line shape formula representing a spectral profile for a barrier resonance in D + H2. They showed that their formula can represent well a spectral profile for resonances associated with a reactive periodic orbit as well as that for barrier resonances associated with a periodic orbit dividing surface. We have attempted to apply their formula to the PHOFEX spectral peaks of OCS. The least-squares fit to the observed peaks using their formula for an even wavepacket results in a reasonable fitting as a simple Fano formula. It should be noted that a physical significance of determined parameters in Sadeghi and Skodje's formula may not be straightforwardwhen the formula is applied to resonances other than barrier resonances. However, their analysis of resonance wave functions near the transition state of the bimolecular reaction is informative in the sense that a spectral feature such as a width and an extent of asymmetry is dependent on a type of resonance. For example, the width of the resonance peaks need not increase (or decrease) monotonically as energy increases. Indeed, in the case of the reaction D + H2 barrier resonances in the lower energy region exhibit much broader profile than resonances for reactive periodic orbits located in the higher energy region. In the photodissociation of OCS, the resonance width can also be influenced by a shape of the excited-state PES,which governs a type of resonance at a particular energy. It is expected that the PES near the transition state could be further characterized from the observed characteristicvariation of the peak profiles by simulating the absorption spectrum on the basis of a theoretically derived PES.

796


V. CONCLUDING REMARKS

Through measurements of the PHOFEX spectrum of jet-cooled OCS, it has been demonstrated that detailed information of the dissociation dynamics on the mostly repulsive PES can be derived from the PHOFBX spectrum of jet-cooled molecules. The vibronic transitions at 154.5, 152.6, and 150.6 nm in the E+- C+ transition of OCS exhibit significantly narrower peak widths in the PHOFEX spectrum recorded by monitoring the S( S) fragments than those in the previously recorded absorption spectra, because (i) only the coldest region of a free jet is sampled in the PHOFEX measurements, resulting in a spectrum free from rotational and vibrational congestion, and (ii) the spectral resolution of the employed light source in the VUV region (i.e., a so-called VUV laser) is significantly high. The PHOFBX spectrum of OCS clarified for the first time that the peak shape of the three vibronic transitions are all asymmetric. The asymmetry of the peak profile was interpreted as a Fano line shape, which appears through the mixing between bound and continuum zero-order wave functions resulting in a dissociative eigen-wave function. The extent of the mixing depends sensitively on the shape of the PES in the transition-state region. It is certain that rich information is contained not only in the width and intensity of a vibronic transition peak but also in its profile. Such an asymmetric line profile can be regarded as a general feature for structured absorption profiles observed in the VUV region.

'

Acknowledgments This work is supported in part by a Grant-in-Aid for the Priority Area (No. 07240106) from the Ministry of Education, Science, Sports and Culture. The authors thank K. Yamashita (University of Tokyo) for his helpful discussion on the potential-energy surface. They also thank C. D. Pibel, S. Liu, and R. Itakura for their involvement in the present study.

References 1. A. H.Zewail, J. Phys. Chem. 97, 12427 (1993). 2. E. J. Heller, in Chaos and Quantum Physics, NATO Les Houches Lecture Notes, A. Voros, M. Gianonni, and 0. Bohigas, Eds., North-Holland, Amsterdam, 1990. 3. B. R. Johnson and J. L. Kinsey, Phys. Rev. Lett. 62,1607 (1989); J. Chem. Phys. 91,7638 ( 1989). 4. R. Schinke, Photodissociation Dynamics, Cambridge University Press, Cambridge, 1993. 5. J. W. Rabalais, J. M. McDonald, V. Scherr, and S. P. McGlynn, Chem. Rev. 71,73 (1971). 6. M. I. McCarthy and V. Vaida, J. Phys. Chem. 92,5875 (1988).

7. K. Yamanouchi, K. Ohde, A. Hishikawa, and C. D. Pibel, Bull. Chem. Soc. Jpn. 68,2459 (1995). 8. G. Black, R. L. Sharpless, T. G. Slanger, and D. C. Lorents, J. Chem. Phys. 62, 4274 (1975).


797

9. C. D. Pibel, K. Ohde, and K. Yamanouchi, J. Chem. Phys. 101, 836 (1994). 10. A. Hishikawa, K. Ohde, R. Itakura. S. Liu, K. Yamanouchi, and K. Yamashita, J. Phys. Chem., 101,694 (1997). 11. K. Yamanouchi and S. Tsuchiya, J. Phys. B: Opt. Ar. Mol. Phys. 28, 133 (1995). 12. (a) U. Fano and J. W.Cooper. Rev. Mod. Phys. 40,441 (1968); (b) U. Fano, Phys. Rev. 124, 1866 (1961). 13. M. Glass-Maujean, J. Breton, and P. M.Guyon, Chem. Phys. Lert. 63,591 (1979). 14. (a) B. Kim and K. Yoshihara, J. Chem. Phys. 99, 1433 (1993); (b) B. Kim, K. Yoshihara, and S. Lee, Phys. Rev. Lett. 73,424 (1994). 15. S. T. Gibson and B. R. Lewis, J. Elec. Spec. Rel. Phenom., 80,9 (1996). 16. M. Okunishi, K. Yamanouchi, K. Onda, and S . Tsuchiya, J. Chem. Phys. 98,2675 (1993). 17. M . S. Child, Molecular Collision Theory, Academic, London, 1974. 18. (a) A. Ogai, J. Brandon, H. Reisler, H. U. Suter, J. R. Huber, M. von Dirke, and R. Schinke. J. Chem. Phys. 96,6643 (1992); J. T. Brandon, S. A. Reid, D. C. Robie, and H. Reisler, J. Chem. Phys. 97, 5246 (1992). 19. R. Sadeghi and R. T. Skoje, J. Chem. Phys. 102, 193 (1995).

DISCUSSION ON THE COMMUNICATION BY K. YAMANOUCHI Chainnun: J. Manz D. M. Neumark: Prof. Yamanouchi, which photofragment are you looking at?

K. Yamanouchi: In the VUV-PHOFEiX measurements, the photofragment of S('S) was monitored by exciting it to the S(3D,)state by the UV laser light and by detecting the laser-induced fluorescence emitted from S(3Dl).Since only the fluorescence from the S fragments produced in the central region of the free-jet expansion was collected, the photoabsorption of ultracold (-5 K) OCS was selectively detected.

PHASE AND AMPLITUDE IMAGING OF EVOLVING WAVEPACKETS BY SPECTROSCOPIC MEANS MOSHE SHAPIRO Department of Chemical Physics The Weizmann Institute Rehovot 76100, Israel CONTENTS 1. Introduction 11. Theory of Wavefunction Imaging 111. Imaging of a Highly Rotating Na2 Molecule

Acknowledgments References

We show how one can image the amplitude and phase of bound, quasibound and continuum wavefunctions, using time-resolved and frequencyresolved fluorescence. The case of unpolarized rotating molecules is considered. Explicit formulae for the extraction of the angular and radial dependence of the excited-state wavepackets are developed. The procedure is demonstrated in Na2 for excited-state wavepackets created by ultra-short pulse excitations.

I. INTRODUCTION Since the development of ultra-fast pump-probe methods [l], in which one detects the temporal evolution of molecular absorption [l, 21, or emission

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femfosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley 8t Sons, Inc.

799

800

MOSHE SHAPIRO

[3]-[5], the general feeling [6]has been that such “real time” measurements contain enough information to yield the phase as well as the amplitude of the time-evolving wavefunction. In recent publications [7,81 we have presented a practical procedure for using spectroscopic experiments to derive complex wavefunctions. Our solution, which involves using both time-resolved and frequency-resolved data, makes no prior assumptions about the fluorescence frequency-internuclear distance correspondence [ 1, 51, the degree of localization of the wavepacket, or the degree of harmonicity. So far, our wavefunction imaging method was limited to non-rotating molecules. In the present paper we deal with rotating molecules. In particular, we show how to image unpolurized time-evolving mixed excited states. We explore the accuracy of our method by demonstrating the imaging of the density-matrix of a highly rotating Na2 molecule.

11. THEORY OF WAVEFUNCTION IMAGING

Consider the fluorescence from a molecular wavepacket excited from the ground electronic state by a short pulse of light. We assume that the initial energy of the molecule is E,,g,,s, where u, j denote, respectively, vibrational and rotational quantum numbers, with well defined magnetic quantum m,

where xuS,j,(R) is a ro-vibrational wavefunction which is a function of R(= (RI,R2, ...,RN)- the collection of internuclear distances specifying the shape of the molecule. The molecule is subjected to the action of a linearly-polarized pulse of the form, 0) on damping the oscillatory nature of the integral. One also sees the stationary phase region of the integrand for the case above the barrier, E = 0.6 eV, near the value of pi = 1.0. There is no stationary phase region for the energy below the barrier, E = 0.3 eV, and this is of course why the transmission probability is small in this case (i.e., in the tunneling regime). Figure 6 shows the results obtained for N ( E ) for several values of A. We do not obtain satisfactory results for A = 0, but for a wide range of A > 0 we obtain quite stable results that are relatively insensitive to the particular value of this smoothing parameter. This is precisely the behavior one wishes to see. It is also significant that the results in Fig. 6 are accurate for some ways into the “classically forbidden” tunneling regime, in this case for energies as much as 0.1 eV or so below the barrier, down to a transmission probability o f = 10-3.

IV. CONCLUDING REMARKS One thus has a theoretical methodology that allows one to compute the rate constant for a chemical reaction “directly,” without having to solve the com-

THEORIES OF REACTION RATES

867

plete state-to-state reactive scattering problem, yet also “correctly,” without inherent approximation. One can of course carry out the correct quantum calculation only for relatively small molecular systems, but is is possible to utilize a semiclassical approximation (which includes the effects of interference and tunneling) that should be applicable to larger systems.

Acknowledgment This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract NO. DE-AC03-76SF00098.

References 1. W. H. Miller, J. Chem. Phys. 62, 1899 (1975). 2. W. H. Miller, S. D. Schwartz, and J. W. Tromp, J. Chem. Phys. 79,4889 (1983). 3. (a) T. Seideman and W. H. Miller, J. Chem. Phys. 96,4412 (1992); (b) T. Seideman and W. H. Miller, J. Chem. Phys. 97, 2499 (1992). 4. U. Manthe and W. H. Miller, J. Chem. Phys. 99, 3411 (1993). 5. (a) An interesting set of papers by many of the founders of the theory-Wigner, M. Polanyi, Evans, Eyring-is in Trans. Faraduy Soc. 34, Reaction Kinetics-A General Discussion, 1938, pp. 1-127; (b) F? Pechukas, in Modern Theoretical Chemistry, Vol. 2: Dynamics of Molecular Collisions, Part B, W. H. Miller, Ed., Plenum, New York, 1976, Chapter 6. (c) D. G. Truhlar, W. L. Hase, and J. T. Hynes, J. Chem. Phys. 87, 2664 (1983). 6. W. H. Miller, Accrs. Chem. Res. 26, 174 (1993). 7. D. Thirumalai, B. C. Garrett, and B. J. Berne, J. Chem. Phys. 83, 2972 (1985). 8. (a) A. Goldberg and B. W. Shore, J. Phys. B 11, 3339 (1978); (b) C. Leforestier and R. E. Wyatt. J. Chem. Phys. 78, 2334 (1983); (c) C. Cerjan, D. Kosloff. and T. Teshef, Geophysics 50, 705 (1985); (d) R. Kosloff and D. Kosloff, J. Comput. Phys. 63, 363 (1986); (e) D. Neuhauser and M. Baer, J. Chem. Phys. 90,4351 (1989); (f)D. Neuhauser, M. Baer, and D. J. Kouri, J. Chem. Phys. 93, 2499 (1990); (g) D. Neuhauser, J. Chem. Phys. 93, 7836 (1990). 9. R. W.Freund, SIAM J. Sci. Star. Comput. 13, 425 (1992). 10. H. 0. Karlsson, J. Chem. Phys. 103,4914 (1995). 11. (a) U. Manthe, T. Seideman, and W.H.Miller, J. Chem. Phys. 99, 10078 (1993); (b) U. Manthe, T. Seideman, and W. H. Miller, J. Chem. Phys. 101,4759 (194). 12. J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys. 82, 1400 (1985). 13. S. Keshavamurthy and W. H. Miller, Chem. Phys. Leu. 218, (1994) 189. 14. J. H. Van Vleck, Proc. Natl. Acad. Sci. 14, 178 (1928). 15. (a) W. H. Miller, Adv. Chem. Phys. 25,69 (1974); ibid., 30,77 (1975); (b) V.P. Maslov and M.V. Fedoriuk. Semiclassical Approach in Quantum Mechanics, Reidel, Boston, 1981; (c) M. C. Gutzwiller. Chaos in Classical and Quantum Mechanics. Springer, New York, 1990. 16. (a) W. H. Miller, J. Chem. Phys. 53, 3578 (1970); (b) W. H. Miller and T. F. George, J. Chem. Phys. 56, 5668 (1972); (c) W. H. Miller, J. Chem. Phys. 95, 9428 (1991). 17. (a) M. F. Herman and E. Kluk, Chem. Phys. 91, 27 (1984); (b) E. Kluk, M. F. Herman,

868

W. H. MILLER

and H. L.Davis, J. Chem. Phys. 84,326 (1986); (c) M.F. Herman, J. Chem. Phys. 85, 2069 (1986).

18. (a) E. J. Heller, J. Chem. Phys. 94,2723 (1991); (b) M. A. Sepulveda and E. J. Heller, J. Chem. Phys. 101, 8004 (1994); (c) E. J. Heller, J. Chem. Phys. 95,9431 (1991). 19. (a) G. Camplieti and P. Brumer, J. Chem. Phys. 96,5969 (1992); (b) G. Camplieti and P. Brumer, Phys. Rev. A 50,997 (1994).

20. (a) K.G. Kay, J. Chem. Phys. 100,4377 (1994); (b) K. G. Kay, J. Chem. Phys. 100,4432 (1994); (c) K. G. Kay, J. Chem. Phys. 101, 2250 (1994). 21. (a) N. Makri and W. H. Miller, Chem. Phys. Lett. 139, 10 (1987); (b) N. Makri and W. H. Miller, J. Chem. Phys. 89, 2170 (1988). 22. B. W. Spath and W. H. Miller, J. Chem. Phys. 104,95 (1996).

DISCUSSION ON THE REPORT BY W.H. MILLER Chairman: R. Jost J. Maw: Prof. Miller has presented to us a quantum theory for rate coefficients, based on the expression [ 1-51

+

A

where Q is the partition function, and the symmetrized flux operator, = (P+ P ~ / Z In practice this integral has to be evaluated only over a short time t, and the trace (tr) involves only wavepackets propagated in small regions close to the transition state. Since this is a conference that centers much attention on femtosecond chemistry, I would like to ask Prof. Miller whether he could specify the decisive time scale that he needs to evaluate k ( T ) for some specific system. If this time scale is short enough (

Chemical Reactions and Their Control on the Femtosecond Time Scale: 20th Solvay Conference on Chemistry

Chemical Reactions and Their Control on the Femtosecond Time Scale

Chemical Reactions (Essential Chemistry)

Chemical Reactions (Essential Chemistry)

Chemical Reactions (Essential Chemistry)

Chemical Reactions Their Theory and Mechanism

Proceedings of the 20th International Conference on Fluidized Bed Combustion

Fullerenes: Chemistry and Reactions

Fullerenes: Chemistry and Reactions

Proceedings of the 20th International Conference on Fluidized Bed Combustion

Chemistry: Principles and Reactions

On Time and Being

On Time and Being

On Time and Being

Introduction to Dynamic Spin Chemistry.. Magnetic Field Effects On Chemical and Biochemical Reactions

The Engineering of Chemical Reactions

Equadiff 9: Conference on Differential Equations and their Applications : Proceedings

Green Chemical Reactions

Chemical Studies on Tuberculin

Oscillations in Chemical Reactions

43rd International Conference on Decision and Control (IEEE CDC 2004)

Chemistry and Chemical Reactivity

Conference on Commutative Algebra

Ravaged! Primer on Large-Scale and Destructive MiningRavaged! Primer on Large-Scale and Destructive Mining

Mosquitoes and Their Control

Mosquitoes and Their Control

Chemistry and Chemical Reactivity

Conference on Harmonic Analysis

Conference on Group Theory

Time Frequency and Time-Scale Methodes

Superelectrophiles and Their Chemistry

Chemical Reactions and Their Control on the Femtosecond Time Scale: 20th Solvay Conference on Chemistry