Advances in Atomic and Molecular Physics, Volume 9

Advances in ATOMIC A N D MOLECULAR PHYSICS VOLUME 9

Immanuel Estermann 1900-1973

ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited by

D. R. Bates DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND

Immanuel Estermann DEPARTMENT OF PHYSICS THE TECHNION ISRAEL INSTITUTE OF TECHNOLOGY HAIFA, ISRAEL

VOLUME 9

@

1973

ACADEMIC PRESS New York

London

A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT 0 1973,BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC. 111 Fifth Avenue, New

York,New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road,

LIBRARY OF

London N W l

CONGREsS CATALOG CARD

NUMBER:65-18423

PRINTED IN THE UNITED STATES OF AMERICA

Contents vii ix

LISTOF CONTRIBUTORS CONTENTS OF PREVIOUS VOLUMES

Correlation in Excited States of Atoms A . W . Weiss 1 4 9 19 43 44

I. Introduction 11. The Correlation Problem 111. Methods of Treating Correlation IV. Specific Correlation Effects V. Concluding Remarks References

The Calculation of Electron-Atom Excitation Cross Sections M . R . H . Rudge 1. 11. 111. IV. V. VI.

48 49

Introduction High Energy Theory High E,nergy Approximations Low Energy Theory Low Energy Approximations Concluding Remarks References

62 93 115 121 122

Collision-InducedTransitions Between Rotational Levels Takeshi Oka 127 134 160 202 204

I. Introduction 11. Theory 111. Experiment

IV. Concluding Remarks References V

vi

CONTENTS

The Differential Cross Section of Low Energy Electron-Atom Collisions D. Andrick 207 208 21 5 221 225 24 1

I. Introduction 11. Semitheoretical Background 111. Experimental Techniques IV. Evaluation Techniques V. Experimental Results References

Molecular Beam Electric Resonance Spectroscopy Jens C. Zorn and Thomas C. English I. Introduction 11. MBER Spectrometer Configurations 111. Observations of Molecular Spectra with MBER IV. State Selection with Electric Fields V. Experimental Methods VI. Energy Levels and Transitions VII. Molecular Properties of Molecules Molecules VIII. Quadrupole hfs in IX. Magnetic hfs in Molecules X. Stark-Zeeman Spectroscopy of Molecules XI. MBER Studies of Nan-% Diatomic Molecules XII. MBER Studies of Polyatomic Molecules XIII. MBER Studies of Vibrational State Populations XIV. Miscellaneous MBER Experiments and Applications References

'x 'x

'x

'x

244 241 252 255 264 280 285 289 294 296 301 304 307 312 314

Atomic and Molecular Processes in the Martian Atmosphere Michael B. McElroy I. Introduction 11. Atmospheric Composition 111. Photochemistry of C02 IV. Chemistry of the Ionosphere V. The Martian Dayglow VI. Evolution of the Martian Atmosphere VII. Concluding Remarks References

323 325 335 343 348 355 359 3 60

AUTHOR INDEX

365

SUBJECT INDEX

318

List of Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.

D. ANDRICK, Fachbereich Physik, Universitat Trier-Kaiserslautern, Kaiserslautern, Germany (207) THOMAS C. ENGLISH, Department of Physics and Astrophysics, University of Colorado, Boulder, Colorado (243) MICHAEL B. McELROY, Center for Earth and Planetary Physics, Harvard University, Cambridge, Massachusetts (323) TAKESHI OKA, Division of Physics, National Research Council of Canada, Ottawa, Ontario, Canada (127) M. R. H. RUDGE, Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast, Northern Ireland (47) A. W. WEISS, Institute for Basic Standards, National Bureau of Standards, Washington, D. C. (1) JENS C. ZORN, Department of Physics, University of Michigan, Ann Arbor, Michigan (243)

vii

This Page Intentionally Left Blank

Contents of Previous Volumes Volume 1

Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Ha// and A . T. Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H . Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K . Takayanagi The Study of Tntermolecular Potentials with Molecular Beams a t Thermal Energies, H . Pauly and J. P. Toennies High Intensity and High Energy Molecular Beams, J . B. Anderson, R . P. Andres, and J. B . Fenn AUTHOR INDEX-SUBJECTINDEX

Volume 2 The Calculation of van der Waals Interactions, A . Dalgarno and W . D. Dauison Thermal Diffusion in Gases, E. A. Mason, R . J. M u m , and Francis J. Smith

Spectroscopy in the Vacuum Ultraviolet, W. R. S. Carton The Measurement of the Photoionozation Cross Sections of the Atomic Gases, James A . R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V . Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer Mass Spectrometry of Free Radicals, S. N. Foner AUTHORINDEX-SUBJECTINDEX

Volume 3

The Quanta1 Calculation of Photoionization Cross Sections, A . L . Sfewart Radiofrequency Spectroscopy of Stored Ions. I : Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Worf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, F. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood AUTHOR INDEX-SUBJECTINDEX ix

X

CONTENTS OF PREVIOUS VOLUMES

Volume 4 H. S. W. Massey-A Sixtieth Birthday Tribute, E. H . S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H. C. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A . Buckingham and E. Gal Positrons and Positronium in Gases, P. A . Fraser Classical Theory of Atomic Scattering, A . Burgess and I. C . Perciual Born Expansions, A . R . Holt and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. C. Burke Relativistic Inner Shell Ionization, C. B. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W. 0. Heddle and R . G . W. Keesing Some New Experimental Methods in Collision Physics, R . F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J . Seaton Collisions in the Ionosophere, A . Dalgarno The Direct Study of Ionization in Space, R. L. F. Boyd AUTHOR INDEX-SUBJECT INDEX

Volume 5 Flowing Afterglow Measurements of Ion-Neutral Reactions, E. E. Ferguson, F. C . Fehsenfeld, and A . L. Schmeltekopf Experiments with Merging Beams, Roy H . Neytiaber Radiofrequency Spectroscopy of Stored Ions I1 : Spectroscopy, H . C. Dehtnelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical-Oscillator Analog, A. Ben- Reuven The Calculation of Atomic Transition Probabilities, R . J. S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s4’”pq, C. D. H. Chisholm, A . Dalgarno, arid F. R . Innes Relativistic Z Dependent Corrections to Atomic Energy Levels, Holly Tliomis Doyle AUTHOR INDEX-SUBJECT INDEX

Volume 6 Dissociative Recombination, J. N . Bardsley and M . A. Biondi Analysis of the Velocity Field in Plasma from the Doppler Broadening of Spectral Emission Lines, A . S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa

CONTENTS OF PREVIOUS VOLUMES

xi

The Diffusion of Atoms and Molecules, E. A . Mason and T. R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A . E. Kingston AUTHORINDEX-SUBJECTINDEX

Volume 7 Physics of the Hydrogen Maser, C. Audoin, J. P. Schertnann, and P. Grivet Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C . Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Parcncz, and Maurice Cohen General Theory o f Spin-Coupled Wave Functions for Atoms and Molecules, J . Gerratt Diabatic States o f Molecules-Quasistationary

Electronic States, Thomas F. O’Malley

Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A . J. Greenfield AUTHOR INDEX-SUBJECT

INDEX

Volume 8 Interstellar Molecules: Their Formation snd Destruction, D. McNaIIy Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y . Chen and Augustine C. Chen Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. I . Schoen The Auger Effect, E. H. S. Burhop and W. N. Asaad AUTHORINDEX-SUBJECTINDEX

This Page Intentionally Left Blank

CORRELATION IN EXCITED STATES OF ATOMS A . W . WEISS Institute for Basic Standards National Bureau of Standards Washington, D.C.

I. Introduction .................................................... 1 A. Scope and Limitations ...................................... 2 B. Aims ...................................................... 3 11. The Correlation Problem ........................................ 4 ................................ 111. Methods of Treating Correlation 9 A. SOC Methods .............................................. 9 B. The Multiconfiguration Self-consistent Field .................. 12 C. The Charge Expansion Method ................................ 13 D. Pair Theories ................................................ 16 IV. Specific Correlation Effects ...................................... 19 A. The Persistence of Asymptotic Degeneracy ...................... 20 B. Level Crossings ............................................ 23 C. Orbital Polarization ........................................ 25 30 D. Correlation Orbitals ........................................ 32 E. Series Pertubations .......................................... 36 F. Pair Correlations ............................................ G. Collapse of the 3d Shell.. ...................................... 40 43 V. Concluding Remarks ............................................ 44 References ......................................................

I. Introduction The importance of correlation for the accurate determination of atomic and molecular properties has been long recognized, and a large research effort has been devoted to this problem for some time now. A variety of methods have been developed and studied, and much has been learned about the correlation problem. Most of this work, however, has concentrated on the ground state and primarily on the ground state correlation energy.' While the potential importance of correlation for spectroscopy is clear, relatively little seems to have been said about the excited state problem as such. Actually, there is already a fairly substantial research literature dealing with correlation in excited states, and it is the intention of this article to 'For a comprehensive review of this work the reader is referred to the papers in Volume 14 of Advances in Chemical Physics. 1

2

A . W . Weiss

summarize the major results of this work, highlighting those features peculiar to the excited state problem. Much of what has been learned about the ground state is applicable to excited states as well, but often with a quite different emphasis. In addition, there appear to be some novel correlation effects peculiar to the excited state problem, and, of course, the net effect of correlation on the spectrum itself is often quite dramatic, if indeed not traumatic. A. SCOPEAND LIMITATIONS

As with any review, in order to maintain some internal coherence there must be restrictions and limitations, and this article will be no exception. In the first place, we will restrict ourselves to the way in which correlation affects only two properties, namely energies and oscillator strengths. The importance of the energy is obvious in view of the computational effort lavished on the ground state, although the viewpoint adopted here will be somewhat different. Usually these calculations have concentrated on getting an accurate value for the total correlation energy. Here we will emphasize more the way correlation affects the relative level positions, i.e., the overall structure of the spectrum. Oscillator strengths are well known as frequently sensitive indicators of correlation effects, and they will be discussed here primarily from this standpoint. The calculation off values as such is not our main concern as this topic has been reviewed quite well in recent years (Layzer and Garstang, 1968 ; Crossley, 1969). We will also concentrate on correlation effects in the lighter elements, up to about 2 = 18. In part this is because relatively little has been done on the heavier elements, although the promising work of Cowan certainly represents a welcome exception [see, e.g., Cowan (1968)l. In addition, for the heavier elements the correlation problem tends to be obscured by the more consistent and large departures from LS coupling. With respect to the lighter elements, we will also not discuss excited state correlation in the helium atom and its isoelectronic ions. In terms of the level of accuracy with which we will be concerned, the two-electron atom has been rendered essentially a solved problem by the definitive work of Pekeris (1958) and co-workers (Accad er al., 1971 ; Schiff and Pekeris, 1971). Our main concern will be with the large correlation effects associated with configurations of strongly interpenetrating electrons. This means that " small " correlation effects such as core polarizations fall outside the scope of this study, and they will not be discussed here. While they will play no role in the examples described here, it should be emphasized that these corrections are not unimportant and cannot always be neglected. For example, they are quite significant for transitions involving the higher Rydberg levels (Hameed

CORRELATION IN EXCITED STATES OF ATOMS

3

et al., 1968; Weisheit and Dalgarno, 1971), and the ordering of fine structure

levels of sodium-like ions (Phillips, 1933). The primary emphasis here will be on ab inifio correlation calculations for excited states. Semiempirical methods, except as they relate explicitly to the correlation problem, will not be discussed. Thus, we will not include the semiempirical parameterization methods commonly used in atomic spectroscopy [see, e.g., Edlen (1964) and Goldschmidt (1968)l; nor will any attempt be made to discuss the quantum defect theory (Seaton, 1966). Also, in view of their limited application to the excited state problem, no mention will be made of many-body perturbation theory (Kelly, 1963, 1969) or the atomic BetheGoldstone procedure (Nesbet, 1967, 1969; Moser and Nesbet, 1971).

B. AIMS The aim of this article is basically twofold. In the first place it is intended to give a general overview of current work directed at correlation in excited states, i.e., essentially a state-of-the-art summary. Secondly, we want to point out some of the specific correlation effects which have come to light, or been re-emphasized, in recent years, and which significantly affect the spectrum in terms of the energies orfvalues, or both. While some of these are properties of open shell systems generally, they are often much more critical for excited states. Also, it is hoped, in the coutse of discussion, to provide some indication of the level of accuracy one can reasonably expect from current methods. The basic outline, then, is the following: In order to establish the framework for the ensuing discussion, Section I1 will give a brief summary of the correlation problem as it is traditionally understood, setting out the necessary background equations and definitions. This will be followed by a review of the major methods currently being employed to investigate the correlation problem for excited states. For the tnost part they consist of one variant or another of the multiconfiguration expansion of the wave function, and it is largely in terms of this scheme that particular correlation effects will be discussed. Finally, we will describe examples of some particular correlation effects which are significant for, or peculiar to, excited states. Finally, a few words are in order about the very interesting dual nature of the excited state correlation problem. From the standpoint of a conventional correlation calculation there apear to be a variety of large, and novel, correlation effects. Frequently, however, these effects are readily interpreted in terms of the spectroscopy of the problem, e.g., level crossings in an isoelectronic sequence, series perturbations, etc. In other words, the dominant correlation effects often appear formally as the usual configuration interaction long familiar in the theory of atomic spectra (Condon and Shortley, 1935, Edlen, 1964). On the other hand, correlating the wave function often emphasizes

A . W . Weiss

4

configurations not usually considered in the conventional approach, such as terms representing orbital polarization. Furthermore, since the “ interacting configurations ” optimize the correlation corrections they often are not identifiable in terms of “real” excited states. While we will be discussing excited states primarily from the standpoint of correlation, we also want to emphasize this dual aspect of the problem. In short, one needs to respect both the spectrum as well as the correlation problem when setting out to make an excited state calculation, or to evaluate one already done.

II. The Correlation Problem Correlation energy is usually defined as the difference between the HartreeFock total energy and the exact energy (Lowdin, 1959), i.e., the eigenvalue of the many-electron Schrodinger equation. More generally the correlation error in any quantity-charge density, fvalue, etc.-is the difference betweeen the Hartree-Fock and the exact value of that quantity. Correlation thus refers to the residual error in the Hartree-Fock model. The many-electron Schrodinger equation is, for bound states, X Y = EY

(1)

with the Hamiltonian given by,

The summations here run over all the electrons of the atom; 2 is the nuclear charge; and atomic units have been used in writing Eq. (2), as they will be throughout this article unless otherwise specified. The unit of energy is twice the Rydberg constant (27.21 eV), and distances are measured in units of the Bohr radius. The problem is clearly a purely nonrelativistic one restricted to the case of LS coupling, the solutions of (I) being eigenfunctions of L2 and S2. “ Exact ” thus does not necessarily mean the experimental value, though in any reasonable application of the correlation concept the difference should not be significant, and for the lighter elements it is readily calculable. It should also be noted that the eigenvalue of (1) is the total energy of the atom; the observable excitation and ionization energies are differences of total energies. The Hartree-Fock approximation is the variational formulation of the independent paticle model. The fundamental assumption here is that the state of the atom is well described by a single conjguration, with each electron assigned to some one-electron function, or orbital. The total wave function

5

CORRELATION IN EXCITED STATES OF ATOMS

is then an antisymmetrized product of orbitals, and, in the traditional formulation of the Hartree-Fock, the orbitals are populated in accordance with the h f b a u principle. For a closed shell atom this means that the wave function is a single determinant, lP =((2n) 9-

I (Pla)q*P(2)(P2a(3). ..q. P(2n) I

(3)

cx and P refer to the m, = & 1/2 components of the one-electron spin functions,

and the 'pi, of course, are the orbitals. Application of the variational principle to a trial function of the form (3) then leads to the usual self-consistent field (SCF) equations for the orbitals (Hartree, 1955),

(4)

Fqi = ~~q~ where the Fock operator, F, has the form of a one-particle Hamiltonian,

F

=h

+ C ( 2 J i - Ki) i

(5)

Here, h is the hydrogenic Hamiltonian, the summation runs over all the orbitals of (3), and Ji and K iare, respectively, the Coulomb and exchange potentials defined by, Jiqj(1) = S d z 2 r 1 2 - 1 @ i ( 2 ) q i ( 2 ) ~ j ( l )

(64

The J ' s and K's depend on the solutions of (9,hence the self-consistency, and they represent an average field due to all the remaining electrons of the atom. In general, for excited states and open shells, the single configuration function is a linear combination of determinants such that the spin and orbital angular momenta of each electron are coupled to give an eigenfunction of L2 and S2,i.e., a pure LS state. This leads to a set of SCF equations with a Fock operator of the general form,

(7) The precise form of the Coulomb and exchange potentials, J and A', depends on the configuration and the particular LS state. This implies, of course, that the Hartree-Fock orbitals can be different for different terms of the same configuration. It should also be noted here that there are certain classes of excited states, such as 2p3p, where there is some ambiguity about defining the Hartree-Fock scheme. While such problems will not arise in our discussions, they should be kept in mind in thinking about the excited state correlation problem in general [see, e.g., Froese Fischer (1968a)l. F=h+J-X

A . W . Weiss

6

Before proceeding further it will be useful to indicate the precise formulation of the Hartree-Fock theory to which we refer. In keeping with the traditional ideas of atomic structure (Condon and Shortley, 1935) we assume that the orbitals are a product of radial and angular factors, Vnlm

=

&I( r ) yIm(e,4)

(8)

where Ylmis a spherical harmonic, and the radial factor, R,,,is obtained from the SCF equations (4). It is further assumed that the radial functions are the same for different values of m , and m,, and with the orbitals populated according to the Aufbau principle. Thus, e.g., a 2p orbital has the same radial dependence regardless of whether it is 2p0 or 2p,,, or with spin +$. These restrictions, of course, are not necessary, and the Hartree-Fock problem can be formulated without them, as has recently been done by Jucys (1967, 1969) in his extended method Hartree-Fock scheme. Doing so includes some of the correlation as defined here; indeed, as we will see, some important correlation effects can be related to an effective relaxation of some of these constraints. The averaging of the interelectronic interactions represented by the Coulomb and exchange potentials then defines the essential physical content of the idea of correlation. The Hartree-Fock model represents the effect of the electrons on each other in an average fashion, omitting the detailed way in which their motions are correlated. The residual error has thus come to be referred to as correlation. The most common method currently used to calculate correlation effects is the method of configuration interaction (CI), or superposition of configuration (SOC).The solution of (1) is approximated by a variational trial function of the expansion form, i

where 4o is a reference configuration and the 4i are “correction” terms in which one or more of the orbitals of q50 have been replaced by some “ virtual ” orbitals not occupied in 40.The variational principle then leads to the usual matrix eigenvalue equation for the energy and coefficients,

H i i a j = Eai i

with the matrix elements given by,

Hi j

=

(4i I 2 I 4j)

(1 1)

The eigenvalues of (10) are upper bounds to the energies of the corresponding states of the atom, and, with a judicious selection of configurations, the eigenvectors provide approximate wave functions for the appropriate state, or states.

CORRELATION IN EXCITED STATES OF ATOMS

7

If &, is the Hartree-Fock function then the remaining terms of (9) represent directly the correlation corrections. If not, then part of the remaining expansion serves to make up for the Hartree-Fock. While, for purposes of definition, we have followed the usual practice of referring correlation to the traditional Hartree-Fock model, it turns out that it is not necessary to actually carry out the calcualtions in this way. As we will see, many of the excited state calculations that have been done do not start with a HartreeFock reference state, and this does not appear to be a serious restriction. So long as the calculation is referred to a reasonable formulation of the independent particle model and the SOC trial function is sufficiently flexible, the major correlation effects seem to be correctly represented in the resulting wave function. One of the key questions that often comes up concerns the virtual orbitals of the SOC, or CI, expansion: this is probably best described by an example. In correlating the 2s2p6 ’S state of a fluorine-like system, the wave function might be of the form, 2s2p6 + 2s22p43d + 2s22p43s + . . . From a correlation standpoint, the virtual orbitals, 3d and 3s, are correlation orbitals and should be chosen so as to optimize the correlation content of their respective configurations, for the state under consideration. This usually results in the correlation orbitals being much more compact than the Rydberg orbitals o f t he excited states corresponding to the second and third configurations listed above. In terms of such spectroscopic orbitals this means that higher series members have been included. While one should probably not belabor the point, it will be convenient to distinguish betweeen correlation oriented or spectroscopically oriented calculations by labeling then SOC and CI, respectively. The expansion (9) should not be confused with the expansion method Hartree-Fock scheme (Roothaan and Bagus, 1963) which is essentially a matrix representation of self-consistent field theory. In this method the orbitals are expanded in some analytical basis leading to matrix SCF equations for the coefficients analogous to the integrodifferential equations of (4). For atoms this basis is usually a set of Slater-type orbitals which, apart from normalization, have the form rqe-cr, with c’s treated as additional variational parameters. The latter point is often of importance for defining the virtual orbitals of the SOC expansion. Since we will also be concerned with correlation effects on oscillator strengths, it will be useful here to set out the relevant formulas. The oscillator strength for a transition from state i to state.j is

f = :(AE/g)S

(12)

8

A . W . Weiss

where S is the multiplet strength, S=l E2 is the second-order energy, and the Yi are given by the usual formulas of perturbation theory. If there is a degeneracy in zeroth order, e.g., 2s' + 2pz 'S, the 'Po becomes a linear combination of the degenerate configurations, with coefficients given by the eigenvectors of the first-order perturbation matrix, Vp,

=

(YOPI VI y o q >

(23b)

The eigenvalues of (23b) then are the first-order energies. This question of asymptotic, i.e., large Z , degeneracy is important since it provides the basis for Layzer's idea of the complex. A complex is defined as the set of all configurations of the same parity and with the same principal quantum number occupancy, which are thus asymptotically degenerate. The theory, therefore, indicates that for large 2 the complex, rather than the configuration, provides the appropriate description of an atomic state. Since this asymptotic degeneracy occurs so often for excited states, the complex clearly has important implications for the excited state correlation problem in

CORRELATION I N EXCITED STATES OF ATOMS

15

highly ionized atomic species. As we will see. for neutral atoms as well one ignores the complex only at great risk. A brief consideration of the neutral spectrum along with the predicted complex structure is usually all that is needed to predict a good part of the important correlation effects. As the nuclear charge is turned up, a multitude of curve crossings often occur, with some states plunging deep into the energy level spectrum and with states regrouping themselves according to their complex. The situation is somewhat analogous t o the formation and dissociation of diatomic molecules, with the nuclear charge playing the role of internuclear distance; as in the case of molecules at equilibrium, the “memory” of the asymptotic structure often persists at the neutral end of the sequence. The behavior of oscillator strengths along an isoelectronic sequence can also be derived (Cohen, 1967; Wiese and Weiss, 1968) in a straightforward way by combining t h e f v a l u e and Z expansion formulas. It turns out that, for a transition where there is a change in principal quantum number, the f value is given by

where.fo, the large Z asymptote, is simply the oscillator strength for a manyelectron atom computed with purely hydrogenic orbitals, with due allowance for any asymptotic degeneracies. If there is n o change in principal quantum numbers, then (24a) becomes

f =f , / Z + f J Z 2 + . ’ ( showing that for such transitions the oscillator strength vanishes asymptotically. These formulas have provided the framework for an extensive study of regular trends in ,fvalue data along isoelectronic sequences (Smith and Wiese, 1971). Numerous calculations, taking the complex into account, have been made on energy levels [see, e.g., Godfredson (1966)] and f values (Cohen and Dalgarno, 1964; Crossley and Dalgarno, 1965). The theory has also been formulated in terms of a variational screening parameter formalism (Layzer, 1967) for calculating energy level trends. Complete second-order energy calculations have also been done (Layzer et al., 1964; Chisholm and Dalgarno, 1966) which involve a pairwise breakdown of E2 ;more will be said about these in the next section. In general, however, it is probably fair to say that the method has not been used extensively for very high precision correlation calculations on atoms with more than two or three electrons. One problem may stem from the slowness of the Z expansion for heavier atoms. At any rate, as indicated earlier, the greatest value of this theory for ouI purposes is in the theoretical framework it provides for understanding correlation effects in excited states.

16

A . W . Weiss

D. PAIRTHEORIES One of the main results of all the research on atomic and molecular correlation is that electrons correlate one pair at a time. Thus, the correlation energy, for closed shells, is predominantly a sum of pair energies, and the main correlation part of the wave function is a sum of pair correlation terms. Nonclosed shell systems introduce other effects, such as the complex and certain single-excitation configurations, which are in addition to the pair correlations. With the exception of Sinanoglu (1969a), little of this work has been directed toward excited states as such. As with the charge expansion theory, our main interest here with pair theories is in the extent to which theyprovideatheoretical framework for interpreting correlation calculations; we will, therefore, only require a sketch of the relevant main ideas. For a more detailed account of pair correlations the reader is referred to the articles in Adcances in Cherizical Physics, Volume 14. Probably the simplest entry into this subject is provided by second-order perturbation theory. The second-order energy is given by the well-known for mula,

where the summation runs of the complete set of zeroth-order excited states. If one takes H , as the sum of Fock operators, Eqs. (5) or (7), then the perturbation is the difference between the exact Hamiltonian and this H,, i.e., the residual interaction. In this case 4, is the Hartree-Fock function, and E, gives directly the second-order contribution to the correlation energy. If H o is taken as the sum of hydrogenic Hamiltonians, Eq. (25) is the secondorder energy of the charge expansion scheme, which includes a Hartree-Fock contribution. While our primary concern here is with correlation, the main conclusions about the structure of the second-order correlation energy are formally contained in the 2-expansion analysis of E, by Layzer and co-workers (Layzer et al., 1964), and we will follow these authors in the following discussion. They partition the second-order energy into a sum of two parts, E,

= E,' iE,"

where E,' includes all the contributions due to single principal quantum number excitations, and E," contains the double excitations. We will discuss E," first. It was pointed out by Bacher and Goudsmit (1934) that this part of E , can

CORRELATION IN EXCITED STATES OF ATOMS

17

be written as a weighted sum of pair energies, which, apart from the contributions of the complex, has the form

The electron pair with quantum numbers y,, S,, L, is coupled to the N - 2 remainingelectronswithquantum numbersy,, S,, L,.Thecoefficientsof the pair energies are the square of the two-particle fractional parentage coefficients (Chisholm and Dalgarno, 1966; Chisholm et al., 1969); the pair energies themselves are

Here u and p refer to a pair of occupied orbitals of 4, and the sum runs over the complete set of excited, virtual orbitals cr',P'. The number, Q , in (27) is N ( N - 1)/2 for a pair of equivalent electrons, where N is the number of such electrons in 4, , and it is N x M for a pair of nonequivalent electrons-N and M being the number of equivalent electrons in each nl shell. These pair energies (28), it should be noted, depend only on the pair and its S, and L, . All of the dependences on the number of electrons, configuration, final LS state, as well as the L, and S , of the N - 2 electron core are buried in the fractional parentage coefficients. Thus, formally at least, the E," part of the correlation for any state of any atomic system can be thought of as being made up of the same pair-energy building blocks. In thecaseof the charge expansion scheme, the pairs are rigorously transferable, since the zeroth-order orbitals are always hydrogenic, and one could conceive of building up a library of pair energies, as Layzer (1967) has suggested, for calculating E , for any atomic species. For the E," part of the correlation, this decoupling and transferability is not exact. The Hartree-Fock (zeroth-order) orbitals relax somewhat from species to species, and, in addition, we are only discussing the second-order perturbation form of the correlation energy. The exact form of the correlation energy will introduce some coupling among the pairs. Nevertheless, there is considerable evidence to support approximate pair transferability [see e.g., Sinanoglu (1969a)], and there has been a reasonably successful attempt to produce, semiempirically, a catalog of pair correlation energies (oksuz and Sinanoglu 1969b). For states with more than one configuration in the complex, the entire complex is probably the most logical choice for 40, i.e., a multiconfiguration reference state. However, it is more in the spirit of the conventional ideas of correlation to stay with a single configuration reference state and to treat these terms as a small number of additional, species dependent contributions to E, .

18

A . W. Weiss

The E,' term in (26) includes contributions from all those configurations with just one excitation of the principal quantum number, i.e., the single excitation terms as referred to the complex. Layzer and co-workers have distinguished three kinds of contributions to E2'. The first of these involve no change in the spin and orbital angular momenta. For example, for the ground state of boron, which is 2s22p ,P, such a term might be ( 2 ~ 3 s'S)2p. For a large class of cases, these terms in E, vanish identically by Brillouin's theorem (Brillouin, 1932), although there are many instances when they do not (Bauche and Klapisch, 1972). For closed shells it happens that not only do these terms vanish because of Brillouin's theorem, but they represent the only component of E2'.Thus, for closed shells the pair energies, (27) and (28), give the entire correlation energy through second order. The other components of E,' arise from terms where, (1) the spins are coupled differently than in &, and (2) there is a change and subsequent recoupling of the orbital angular momenta. For the ground state of boron again, an example of (1) would be ( 2 ~ 3 s3S)2p and two examples of (2), (2s3d 'D)2p and (2s3d 3D)2p. These kinds of corrections have been referred to as spin and orbital polarization correlations (Harris, 1970), and they are associated with effectively relaxing the spin and orbital symmetry constraints of the traditional Hartree-Fock theory. Experience indicates that the orbital polarizations often make a large contribution to excited state correlation, particularly when they include a spin recoupling as in the last example just cited. While this discussion has been based entirely on second-order perturbation theory, one can reach the same conclusions from more general arguments; this has been done by Sinanoglu and co-workers (see, e.g., Sinanoglu, 1964; 1969a, and references cited therein). They have analyzed the open shell correlation problem from the standpoint of perturbation theory (Silverstone and Sinanoglu, 1966a). and have obtained similar results from a variational approach (Silverstone and Sinanoglu, 1966b) by varying suitable portions of the energy. In the notation of Sinanoglu's many-electron theory (MET) the energy and wave function are written as E = EHF

+ Ei + E/ + Eu

(29)

+F Xi + X/ + x u

(30)

and =~

H

where the first term, of course, is the Hartree-Fock. The second term refers to the internal correlation and includes all those configurations involving excitations within the available Hartree-Fock sea, i.e., the set of all orbitals of the complex. While, in principle, this could include certain inner shell excitations (e.g., ls22s22p-+ 2s22p3), in practice this is not usually the case

CORRELATION IN EXCITED STATES OF ATOMS

19

and the " internal " correlation terms are identical to those of Layzer's complex. The third terms of (29) and (30) are called semi-internal correlations. They include all single excitations out of the Hartree-Fock sea the most important of which are the spin and orbital polarization terms and thus correspond to the E,' part of Eq. (26). Harris (1970) has referred to x i and xr collectively as the first-order wave function and the energy, the orbital correlation energy. The last terms in (29) and (30), of course, represent all the higher order effects, and they are expected to be dominated by the pair correlations. In terms of full-scale calculations based explicitly on such pair schemes little has been done on excited states as such. However, Sinanoglu and coworkers (oksiiz and Sinano$jlu, 1969a, b; Westhaus and Sinanoglu, 1969) have made extensive SOC calculations including the complex and all the semi-internal configurations for the valence excited states of the first row elements, 2s2p"+', 2p"+', as well as 2s22p". The all-external pair energies, Eqs. (27) and (28), have been estimated empirically by fitting the differences between these calculations and the observed energy levels, relying on the approximate decoupling and transferability of the closed shell-like pairs. The complex and orbital polarization terms appear to include the main effect of correlation on J' values. This is apparently because (1) they include those configurations with the largest coefficients, and (2) for the cases studied they also include the major spectroscopic configuration interaction effects. Pairwise calculations of the second-order Z-expansion contribution to the correlation energy have also been done on three-electron atoms (Chisholm and Dalgarno, 1966). An interesting calculation of three-electron atom oscillator strengths has also been done within the Z-expansion frame work, where the various pair contributions to the ,f value were calculated separately for two-electron systems and then assembled according to the prescriptions of the pair scheme (Dalgarno and Parkinson, 1968).

IV. Specific Correlation Effects We discuss here a number of calculations chosen to illustrate some of the particular correlation effects which are important for excited states. This is not, of course, intended as an exhaustive account of excited state calculations, and the selection is necessarily somewhat arbitrary. There is also considerable arbitrariness in classifying the correlation effects. For instance, some series perturbations are also examples of asymptotic degeneracies, and both may well represent an orbital polarization effect. As a result of this overlap there will be a certain amount of "cross-talk'' between the various subsections, which it is hoped will not prove too confusing.

A . W. Weiss

20

A. THEPERSISTENCE OF ASYMPTOTIC DEGENERACY Layzer has commented (Layzer et al., 1964) on the persistence of the asymptotic term interval ratios observed experimentally along isoelectronic sequences and has discussed the phenomenon in terms of the charge expansion series. Our interest here is in the way the large 2 degenerate mixings, i.e., the complexes, persist in variational calculations all along an isoelectronic sequence. Such configurations are often unbound for the neutral atom, and as Zincreases they come down into the spectrum, eventually clustering around the other members of the complex. One would thus expect substantial mixing in that region of the sequence where this clustering occurs. In fact, the effect of the complex is more far-reaching and often extends well along the neutral end of the sequence. The prototype example here is the well-known 2s2 + 2p2 mixing in the ground state of neutral beryllium, even though 2p2 'S is a distant state well up in the spectrum, perhaps even above the ionization limit. Table I shows an example of the effects of such a "plunging" term, The TABLE I MULTICONFIGURATION WAVEFUNCTIONS FOR 2s3p 3*'POF THE Be SEQUENCE" Z = co

Ne(VI1)

3P 2 ~ 2-0.062 ~ 2s3p 0.982 2 ~ 3 -0.028 ~ 2 ~ 4 -0.146 ~ 2p3d -0.074 a

'P 0.337 0.868 -0.160 -0.301 -0.069

3P

'P

-0.028 0.074 0.989 0.957 -0.059 -0.257 -0.082 -0.074 -0.099 0.057

3P

'P

3P

0.004 0.0 0.989 0.946 0.992 -0.086 -0.307 -0.064 -0.026 -0.022 0.0 0.094 -0.108 -0.115 -0.004

'P 0.0 0.936 -0.337 0.0 0.105

Odabasi (1969).

table gives the first few configurations in the wave functions for the 3P and 'P terms of 2s3p in the beryllium sequence, as given by SOC calculations based on the HFS procedure (Odabasi, 1969). Here, the degeneracy is with the corresponding terms of 2p3s. Experimentally the 3P term of 2p3s is already bound, and quite low, at B(II), while the 'P term first appears bound at C(1II). The asymptotic, Z-expansion calculation indicates that the mixingof 2p3s with 2s3p is quite small for 3'P but substantially larger for 'P, a feature which persists all along the sequence, except for the neutral atom. The strong mixing of 2s2p and 2s4p for neutral beryllium may be an artifact of the calculation due to the lack of term dependence of the orbitals in the HFS scheme. This term dependence is quite marked in Hartree-Fock calculations of 2s2p

CORRELATION IN EXCITED STATES OF ATOMS

21

(Hartree and Hartree, 1936), and a similar exchange potential effect may well occur for 2s3p. At any rate the preferential strong mixing of the 'P complex over the 3P is reflected throughout the sequence. This particular correlation effect is also manifested in the energy levels, as shown in Fig. 1 . Here the theoretical term energies are plotted in relation to the center of gravity of the original 2s3p. both before (HFS) and after mixing (SOC). The much stronger mixing in the 'P complex results in the 'P being pushed down below the 3P,except for the neutral atom. Experimentally the situation is essentially the same, although actually 'P is still slightly above 3P for B(I1). While correlation has here inverted the singlet and triplet, the main point is that the preferentially strong mixing in the 'P complex is already predicted asymptotically, and it carries through for most of the sequence.

0

0.1

0.2

0.3

l/Z+

FIG.I . Theoretical term energies for the 2s3p configuration in the beryllium isoelectronic sequence (Odabasi, 1969). The reduced energies are given relative to the center of gravity of the configuration without configuration mixing. a, without configuration mixing; 0, with configuration mixing.

22

A . W . Weiss

Another example of a calculation where this persistence effect appears is given in Table 11, which shows MCSCF wave functions (Froese Fischer, 1968c) for the ground state and lowest 2Dstate of several members of the A1 sequence. The asymptotic complex shows a substantial mixing of 3p3 with the ground state and of 3s23d with 3s3p2, and here the effect not only persists but TABLE I1 MULTICONFIGURATION WAVEFUNCTIONS FOR 3s23p 2P AND 3s3p2 2DOF THE A1 SEQUENCE'

3s23p 2P 3s23p 3P3 (3s3d 3D)3p 3s3pz 'D 3s3p2 3s23d (3p2 '93d (3pz V)3d 3s3d2 a

Z=

Si(I1)

S(IV)

Ca(VII1)

Fe(X1V)

0.982 0.189

0.985 0.171

0.988 0.152

0.990 0.140

0.990 0.110 0.093

0.716 0.619 0.119

0.881 0.467 0.077

0.919 0.390 0.055

0.932 0.358 0.046

0.947 0.304 0.033 0.087 0.039

00

Froese Fischer (1968~).

is enhanced toward the neutral end of the sequence. The asymptotic calculation also suggests several terms which were left out of the MCSCF calculations and which are of the orbital polarization, spin-flip type, in particular the third configuration listed for the ground state and the fourth of the excited state. Such correlation effects can have a profound influence on f values, as indicated by Fig. 2 which gives the f values for the 3s23p 2P-3s3p2 'D transition calculated for this sequence in both the H F and MCSCF approximation. One should note, in particular, the difference in scale of the two sets of oscillator strengths. The drastic reduction in the magnitude of thefvalues is due primarily to the interference of the contributions from 3s3p2 and 3s23d in the 2Dstate. The f values curve also appears to show a minimum near the neutral end, and the upper part of the figure, which plots (gf>'/2,shows the reason for it. The quantity(gf)'/2 is directly proportional to the transition matrix element itself, which changes sign between Si(I1) and S(II1) due to the change in the extent of the mixing, resulting in a minimum in the f value curve. Since the cancellation is so nearly complete, the additional smaller correlation effects omitted from this calculation may well alter this particular detail quite sub-

CORRELATION IN EXCITED STATES OF ATOMS

23

FIG.2. gf and (gf)"' values for the 3s23p 'P-3s3pZ 'D transition in the aluminum isoelectronic sequence (Froese Fischer, 1968~).The difference in scale should be noted for the Hartree-Fock (A)and configuration mixing (0) calculations.

stantially. Thus, thefvalue minimum may possibly not be real, although the basic trend is probably correct. The configuration mixing here in the 2D state is so severe that it probably does not make sense to describe it as 3s3p2, at least for the lower stages of ionization. A similar loss of identity occurs for the D states in the Mg sequence and has been commented on by Zare (1966). In such cases, a more elaborate calculation should probably include configurations designed to correlate each of the dominant components.

B. LEVELCROSSINGS In addition to the mixing with othermembersof thesamecomplexa" plunging" term can also have a strong effect on other configurations, not of the same complex but with the same symmetry, which it crosses on its way down.

A . W. Weiss

24

0.06

t

1

l/Z+

FIG.3. Oscillator strengths and reduced term energies for the boron isoelectronic sequence.fvalues are for the 2s22p-2s23stransition; Hartree-Fock values are indicated by and SOC values by x . The experimental f values are A (Bergstrom et al., 1969), 0 (Martinson and Bickel, 1970), and V (Heroux, 1967). The S-state experimental term energies are reduced by 1= Z-N-1, where N is the number of electrons in the ion.

+

An example of this is given in Fig. 3 which shows the ’ S levels and the 2s22p2s23s transition f values for the boron sequence. The SOC calculations here are based on the Hartree-Fock and use the pseudonatural orbital technique for the correlation orbitals (Weiss, 1969b, 1970). In neutral boron, 2s23s is the lowest lying ’ S state with 2s2p2 embedded in the S series well up in the spectrum. As the nuclear charge increases, 2s2p2 comes down, becoming the lowest ’ S term for the sequence, with the crossing occurring in the region near C(I1). Examination of the SOC wave functions shows 2s2p’ to be a relatively small correlation correction to 2s23sfor B(1) and Ne(VI), the dominant correlation term being the one correlating the 2s electrons, namely 2p23s. The 2s2p’ configuration represents a 2s-3s pair correlation, and, since this is an intershell pair, it is not surprising that the effect is relatively weak. In the vicinity of the crossing, however, 2s2p2 is a large correlation correction for 2s23s and introduces a large cancellation inthef value. The Hartree-Fock f value is quite regular for the entire sequence, but strong

25

CORRELATION IN EXCITED STATES OF ATOMS

mixing near the crossing puts a deep (near zero) and sharp minimum in the $value curve which has been confirmed experimentally (Martinson and Bickel, 1970). A similar situation occurs for the S states of the beryllium sequence (Weiss, 1970), althoughf-value data here are rather sparse.

C. ORBITAL POLARIZATION This correlation effect is inherent in open shell systems and has been calculated and discussed by many people (Layzer et al., 1964; McKoy and Sinanoglu, 1964; Weiss, 1967a; Bagus and Moser, 1968; Schaefer et al., 1969; Bunge, 1968). As discussed earlier this involves a single excitation from the complex where there is a change in orbital symmetry, and the most effective terms usually also involve a spin polarization with subsequent recoupling of the angular momenta. Thus, for example, two of the most important terms in the wave function for the ground state of carbon, 2s22p2 3P, are (2s3d 3D)(2pZ3P)

(2s3d 3D)(2p2'D)

and

As we noted above, such terms reflect, in effect, a relaxation of the HartreeFock restriction of individual orbitals to pure angular symmetry. In other words, the electrons are better able to keep out of each other's way when the orbitals are polarized. The effect shows up quite generally for excited states and is apparently responsible for much of the large semi-internal correlation found by Sinanoglu and co-workers (Oksiiz and Sinanoglu, 1969a; Sinanoglu, 1969a). Also, for excited states it often happens that one can find a straightforward spectroscopic interpretation for such large interactions. As a first example, we show in Table Ill some two-configuration, orbital TABLE 111 THESINGLE EXCITATION, ORBITAL POLARIZATION EFFECT IN THE GROUND STATE OF F-LIKEAND Cl-LIKE IONS -E

AE

AE

-E

Mg (1V)

2s22p5 2s22ps 3 (2s3d 3D)2ps

Ne(I1) 127.8177 127.8370 0.0193

195.9403 195.9638

3s23p5 3s23ps -t (3s3d 3D)3p5

Ar(I1) 526.2743 526.2995 0.0252

Ca(IV) 674.3197 674.3511 0.0314

0.0235

A . W. Weiss polarization calculations on 2s22p5 and 3s23p5 ions. The calculations augmented the analytical Hartree-Fock Slater type basis with three d functions, using a PSNO transformation to determine the final d orbitals, and varying the d exponents to minimize the energy. The correlation energy picked up by just the one extra configuration is substantial, ranging from 0.5 to 0.9 eV. This type of correlation term appears wherever the symmetry of the state permits the s + d polarization and usually affects the energy by about this magnitude. In the second row the effect is somewhat stronger because of the 3s-3d asymptotic degeneracy, and, in this case, it is actually part of the complex. Another very interesting example of orbital polarization correlation occurs in the sp6 excited states of these ions. It was noticed by Bagus (1965), in making Hartree-Fock calculations of the 2s ionization energy of neon (and the 3s of argon), that there seemed to bemorecorrelation energy associated with sp6thans2p6.This is illustrated in Table 1V which compares experimental and Hartree-Fock ionization energies for neon and argon, and some isoelectronic ions. The Hartree-Fock ionization energy, computed as the difference between two Hartree-Fock energies, is smaller than the experimental value forremoval of a2p(or 3p) electron, as expected. The 2s (or 3s) ionization energy, however, is larger than the experimental one. From a straightforward point of view, since more electrons means more pairs, oiie would expect the correlation energy to be larger for a state like s2p6than for sp6, where one electron has beenremoved. This anomaly has been discussed by Sinanoglu (1969b) and explained as being due to the large semi-internal correlation present in the sp6 open shell but absent in the closed shell s2p6.While this is true, it happens that here the main effect can be isolated in one particular interaction which has a simple, and classical, spectroscopic interpretation. One should also note that the s2p5state is also an open shell system and here the semi-internal correlation is not large enough to produce a similar anomaly. The precise nature of this correlation effect was actually indicated by Minnhagen (1963) in an analysis of the Ar(1I) spectrum. He noted that there is a strong configuration interaction between the 3s3p6 excited state and the 2S Rydberg term, 3s23p43d,with a large displacement of 3s3p6 down and 3p43d up. The sp6 term is the series perturber for the p4nd series, and this also appears to be the case for Ne(II), although the series perturbation is somewhat weaker. For the argon-like ions this interaction is actually part of the complex and so should persist along the sequence, as Table 1V seems to indicate. The ionization potential anomaly actually increases somewhat along the argon sequence, while dying out for the more highly ionized neon-like ions. To verify this strong series interaction effect a 2 x 2 SOC calculation was done on these states using the PSNO technique, with the results shown in Table V. It is clear that, with energy lowerings ranging from 2 to 7 eV, this

0 0

TABLE IV

IONIZATIONPOTENTIALS FOR

THE

E

is,

NEONAND ARGON SEQUENCES

P

I.P. (HF)"

I.P. EX^)^

AI

I.P. (HF)'

Ne(I) 2s22p6+ 2s22p5 + 2s2p6 3s23p6+ 3s23p5 + 3s3p6

0.7294 1.8123

0.7937 1.7815

0.5430 1.2198

ArW 0.5813 1.0745

I.P. EX^)^

AI

I.P. (HF)'

-0.0581 0.0202

6.0790 7.9846

MdIII) -0.0643 0.0308

2.8904 4.3848

2.9485 4.3646

1.8339 2.7316

1.8871 2.5769

2

z

Si(V)

Ca(II1) -0.0383 0.1453

1.P. EX^)^

AI

6.1366 7.9810

Y

-0.0576 0.0036

i!

Ti(V) -0.0532 0.1547

3.6100 4.7270

3.6791 4.5661

r A

-I

-0.0691 0.1609

5g -

a

Bagus (1965).

' Moore (1949).

Clementi (1965). Where the sp6 Hartree-Fock functions were not available, they were computed using essentially Clementi's basis set.

h

w

v

3!

5

A . W. Weiss

28

is probably the major qualitative mechanism for the correlation anomaly. Although not shown, the 2sz2p43s term was also investigated and found to have an order of magnitude smaller effect on the energy than 2p43d. The 3d orbital used in these calculations is a correlation orbital, in the sense that it was chosen solely with the aim of minimizing the energy of the lowest ’S state. This is the calculation labeled “ SOC.” For comparison, the same calculation (labeled “ C I ” ) was done using a Rydberg-like 3d orbital, which was taken from a Hartree-Fock calculation of the p4d configuration (c. of g.). TABLE V TWO-CONFIGURATION SOC AND CI CORRELATION CALCULATIONS ON spdO -E

2s2p6 2s2p6 2s2p6 3s3p6 3s3p6 3s3p6

A€

-E

A€

+ 2s22p43d(SOC) + 2s22p43d(CI)

Ne(l1) 126.7348 126.8316 0.0968 126.7402 0.0054

MNV) 194.4459 194.5328 0.0869 194.4583 0.0124

+ 3s23p43d(SOC) + 3s23p43d(CI)

Ar(1l) 525.5975 525.8262 0.2287 525.7871 0.1896

Ca(IV) 673.4220 673.6871 0.2651 673.6784 0.2564

-

This illustrates not only the large interaction with p4d but also the difference between using a variationally determined 3d(SOC) and one taken from an SCF on p4d c. of g. (CI).

The results show that the correlation orbital is much more effective in representing this correlation effect, the reason being that the correlation orbital includes, in an optimal way, the effect of the entire Rydberg series, a point to which we will return in the next section. Consistent with the charge expansion theory, the configuration mixing is much stronger for the second row ions, where it corresponds to the mixing of the complex, than for the first row, where it does not. In terms of correlation effects, this is an orbital polarization involving the excitation of two 2p electrons, 2p2 -+ 2s3d, with the obvious parallel for the 3p electrons. The relative lack of polarization represented by 2pz --* 2s3s, then, appears to be the reason that the term 2p43s (or 3p44s) is of so little significance. Such considerations would also suggest that d series should be more strongly perturbed than s series, as often appears to be the case. Not surprisingly the strong “ orbital polarization-series perturbation interaction has a large effect on the oscillator strengths as shown in Table VI.

”

CORRELATION IN EXCITED STATES OF ATOMS

29

The transition originates in the s2p5ground state and the two-configuration, orbital polarization approximation for this state (Table IV) has also been used in computing the f values. Correlation effects in this case interfere destructively, resulting in an f value much smaller than the Hartree-Fock. Whenever such cancellations occur, of course, the f value is very sensitive to small TABLE VI OSCILLATOR STRENGTHS FOR THE TRANSITION szp5+ sp6 SOC

CI

Ne(I1) 1 x 1 (HF)" 1 V

1x 2

I

2 x 2

I

V

V

0.160 0.130 0.088 0.015 0.060 0.064

0.160 0.130 0.147 0.115 0.107 0.226

Ar(1I) 1 x 1 (HF) 1 V

I V

2 x 2

1 V

Exp'

CI

MdW 0.131 0.102 0.120 0.082 0.094 0.179

0.131 0.102 0.096 0.021 0.074 0.079

0.035

Expb

1 x 2

SOC

0.227 0.227 0.263 0.263 0.010 0.012 0.000 0.001 0.002 0.001 0.014 0.015 0.009

Ca(1V) 0.207 0.225 0.025 0.003 0.010 0.023

0.207 0.225 0.022 0.002 0.008 0.020

This notation refers to the number of configurations used in each state; thus I x 2 is a one-configuration ground state and two-configuration sp6 state. Hinnov (1966). Lawrence (1969). (I

correlation corrections, and these results are not in good quantitative agreement with the available measurements, where available. However, the main qualitive effect is correctly reproduced by these simple wave functions; the fvalue is small compared to that of the Hartree-Fock. As with the energy, the spectroscopic configuration interaction (CI) is not as effective in representing the effect of the correlation on thefvalues, although it is much better for the second row ions [Ar(lI), Ca(IV)] than for those of the first row [Ne(II), Mg(1V)I.

A . W . Weiss

30

D. CORRELATION ORBITALS While the precise nature of the virtual orbitals needed to correlate an excited state is not a specific correlation effect, this topic is of sufficient importance to make a brief digression here worthwhile. It is well known, in making ground state correlation calculations, that the optimum correlation orbitals are usually pulled in to overlap the region of the atom where the charge density is greatest. The same thing happens with excited states as well [see e.g., Froese Fischer (1970)l even when the major correlation effect has a purely spectroscopic interpretation, as in the sp' example just cited. The difference in using a correlation and spectroscopic 3d orbital was most striking in 2s2p6 of Ne(II), for both the energy andfvalue, and the difference between these orbitals is indicated in Fig. 4. The correlation 3d for Ne(I1)

0

2

6

4

0

r-e

FIG.4. Radial functions for the sp6 states of Ne(1I) and Ar(1I). The two 3d orbitals are the correlation orbital appropriate for sp6 and (---) Hartree-Fock (c. of 8 . ) orbitals for 2p43d and 3p43d. (-)

31

CORRELATION IN EXCITED STATES OF ATOMS

overlaps very strongly with the 2p while the spectroscopic one, of course, is very diffuse and Rydberg-like. We can also see here the reason for the difference being so much smaller for Ar(I1). For Ar(I1) both orbitals were nearly as effective in correlating the 3s3p6 state for both the energy and osdillator strength, and indeed both 3d orbitals are quite similar, with the spectroscopic 3d overlapping quite strongly with the radial part of the 3p of the reference configuration. This is due to the collapse of the 3d shell which occurs near the end of the second row and is related to the wave functions rapidly approaching the situation predicted by the complex, as the nuclear charge increases. It should be noted that the Ca(IV) results are essentially insensitive to which orbitals are used. A similar example of this occurs in the 3s3p2 'D state of Si(I1) where certain fvalues are very sensitive to the correlation effects. Here there is a very large mixing with 3s23d and a correlation calculation (MCSCF) has been done (Froese Fischer, 1968b) as well as the more traditional configuration interaction one (Garstang and Shamey, 1967). Both calculations correctly predict the very small f value for the resonance line, which is due to configuration cancellation. In addition to this however, there are several two-electron transitions associated with this state, from 3s24p and 3s24f, which are made possible by the admixture of 3s23d. Froese Fischer's comparison of the two calculations is shown in Table VII, which gives both the gfvalues and the relevant one-electron transition integrals. The Garstang and Shamey calculation was done by semiempirical fitting of parameters for the configuration TABLE VII g f V A L u m AND u2 FOR

Si(I1)"

MCSCFb 3s3p2 2D-3s24f 2f 3s3p2 'D-3s24p *P

CI'

EXP -

02(3d4f)

1.43 0.127

4.94 0.63

-

gf u'(3d-4~)

0.406 0.233

1.82 1.30

0.311d,0.74' -

gf

These transitions become allowed primarily because of the mixing of 3s23d into the lowest 'D state, and both u2 and gfreflect the large difference between a variationally optimum 3d and spectroscopic one. The u2 in the CI column are actually obtained with the Hartree-Fock for 3s23d. Froese Fischer (1968b). Garstang and Sharney (1967). This calculation was done in several ways. The numbers quoted here are for a Slater parameter fitting of the energies and u2 from Hartree-Fock functions. Hey (1959). Schulz-Gulde (1969). These measurements were normalized to Froese Fischer's calculation of the 6350 8, multiplet.

32

A . W. Weiss

interaction and subsequent use of a 0’ calculated with the Hartree-Fock 3d. The difference in the resulting a2 is quite striking and is reflected in the calculated gf values. The Garstang and Shamey calculation correctly diagnoses the dominant correlation effect and includes the most important part of it. These particular transitions, however, are sensitive probes of this feature of the wave function, and they show that, in terms of a set of Hartree-Fock orbitals, it is necessary to take into account the full series interaction, i.e., 4d, 5d, etc. This appears to happen quite often. The importance of a particular correlation effect, and frequently the most important one, can often be readily understood in terms of the spectroscopy of the problem. For quantitative results, however, it is necessary to use functions dictated by the conventional ideas of a correlation approach.

E. SERIES PERTURBATIONS Another aspect of excited state correlation appears as the perturbation of Rydberg series by foreign terms. The perturbation here, of course, refers to deviations from the regular behavior of the series, e.g., the Ritz formula, due to configuration interaction between the series members and perturbing configuration. While these effects have been well known in atomic spectroscopy (Edltn, 1964), they often tend to complicate the correlation problem for such cases. Ordinarily, for a Rydberg state with one electron in a very diffuse orbital, one would expect core correlations to dominate and intershell effects to be quite weak, if not entirely negligible. The perturber, however, substantially alters this situation, introducing what may be alternatively interpreted as abnormally large intershell effects or as singularly emphasized semi-internal correlations. For a strongly perturbed series, these effects may well pervade the entire series, changing the character of the states to be strong mixtures of perturber and Rydberg terms; and these effects are best described in terms of a multiconfiguration reference state. In such cases, of course, it is necessary to correlate all configurations of the reference state. One example of a strongly perturbed series is the 3s2nd series of Al(I), where the perturber is 3s3p’ ’D. SOC calculations have been done on the five lowest members of this series using the PSNO technique (Weiss, 1973); some of these results are shown in Table VIII. The energies calculated with several truncations of the SOC functions are given here for the two lowest states. The first calculation includes, in addition to the series, only the important 3s’ correlation, that is, the 3s’ + p’ interactions. As expected the energy improvement is large and essentially the same as for the ground state of the ion, where the d electron has been completely removed. The perturber and other correlations effect only a relatively small further improvement in

n

s

EF

TABLE VIII

P

8

CORRELATION ENERGIES FOR THE Two LOWEST 'D STATES OF A1 (I)" --E,,

3s23d (3sz pp')'S Cnd (3s' pp')'S End SOC (50-conf.)

+ +

+ 3s3p2

241.7321 241.6792 241.7759 241.7847

AE 0.0371 0.0438 0.0526

3sZ4d (3s' pp')'S Cnd (3s2 pp')'S Cnd SOC (46-conf.)

+ +

+ 3s3p'

-EM

AE

AE(Al(II), 3s' 'S)

2

241.7067 241.7439 241.7484 241.7554

0.0372 0.0417 0.0487

-

?!

0.0368 -

8

The calculations referred to in the second row of this table include a number of terms of the form (npn'p 'S)nd rather than just the (3p2 'S)nd since the 3p of 3s3p2is somewhat different from that needed to correlate 3s'. The AEcolumns give the cumulative energy improvement over the Hartree-Fock, and the last column is the energy improvement for Al(I1) ground state. a

Q

?el !?

50

5

w w

34

A . W . Weiss

the energy. The implications with regard to the wave functions, however, are a bit deceptive, as can be seen in Table IX which gives the dominant composition of the wave functions for these calculations. In terms of the wave function composition, the 3s3p2 perturber, which might be interpreted as correlating the 3s3d (or 3s4d) pair, is more important than the 3s‘ correlation and is responsible for the quite substantial mixing in of the remainder of the series. This latter effect is, of course, represented by the 3s2d, term, where d, is a correlation orbital. TABLE IX WAVEFUNCTIONS FOR Configurations 3s23d 3s23d (3p2 %)3d 3s3p2 3S2d1 3s24d 3s24d (3p2 ‘S)4d 3s3p2 3S2d1‘ (3p2 ‘S)dl’

(3s2

THE

Two LOWEST ‘D STATES OF AI(1)”

+ pp’)’S Cnd 0.97 -0.21

-

0.97 -0.21

-

-

(3s2

+ pp’)’S Znd + 3s3p2

Full SOC

0.90 -0.19 0.32 -0.20

0.80 -0.16 0.46 -0.30

0.85 -0.19 -0.30 -0.34 -

0.60 -0.12 -0.46 -0.61 0.13

Only the dominant configurations are given. The orbitals, dl and dl’ are correlation orbitals representing the effect of the remainder of the series.

The full SOC calculation referred to here includes many more configurations, the most important of which appear to represent orbital polarization, or semi-internal correlations of the 3s3p2 component. These are terms such as d’(3p2 ’P), where the d’orbital overlaps the 3p quite strongly. The effect of such correlations is also quite marked, as shown in the last two columns of Table IX. Failing to correlate the perturber seriously underestimates the perturber composition as well as the degree of dilution of the Hartree-Fock Rydberg component. It is also of interest to note that no state is really identifiable as 3s3p2; which has been smeared out over the entire series. Oscillator strengths, of course, are also strongly affected by these correlations. In this case the f values, for transitions from the ground state, are strongly reduced, by a factor of 3 for the 3d and almost to zero for the 4d. Nor is this particular series an isolated example; strong series perturbations usually are faithfully reflected in the oscillator strength distribution. To further

35

CORRELATION IN EXCITED STATES OF ATOMS

, HARTREE-FOCK

0 1 -0.3

3r3d 3p2

384d

-0.2

-0.1

385d

I 0

E (o.u.) FIG.5. Theoretical term values and oscillator strengths for Al(I1) (Zare, 1967). The transitions originate in 3s3p 'P, and energies are given relative to the ionization limit.

illustrate this we show in Fig. 5 the results of HFS-SOC calculations (Zare, 1967) for another strongly perturbed series, the 3snd 'D series in Al(I1) perturbed by 3p2. The sequence of transitions here originates in 3s3p 'P. The perturber, in the Hartree-Fock Slater approximation, is initially near the bottom of the series, and the series perturbation correlation results in both a significant rearrangement of the energy levels and a redistribution of oscillator strength. Oscillator strength has been transferred up into the series, and the effect extends over the entire series. For instance, the very diffuse 3s5d state also feels the effect of the perturber, as reflected in the change infvalue. This transfer of oscillator strength from a low-lying perturber up into the series also does not seem to be an isolated phenomenon. One can argue, from simple model considerations, that if a perturber lies below a nonpenetrating series, with which it can interact, the result of the series perturbation will be

A . W . Weiss

36

TABLE X

RESONANCE LINEf VALUES ~

B(I) C(1) NO) Ne(l1) Si(I1) Ar(I1)

~

Transition

HF (len)

EXP

Series"

2s22p 'P-2s2p2 'D 2s22p2 'P-2s2p3 'D 2s22p3 4S-2s2p4 4P 2s22p5 2P-2s2p6 'S 3s23p 2P-3s3p2 'D 3s23p5 2P-3s3p6 'S

0.339 0.286 0.503 0.160 0.452 0.227

0.048b 0.076b 0.091 0.080b, 0.13d 0.035' 0.005f 0.009p

2s'nd 'D 2s22pnd 'D 2s22pznd 4P 2s22p4nd 2S 3s'nd 2D 3s23p4nd 2S

In each case, there is a higher-lying Rydberg series of the same symmetry as the upper state of these transitions, and for which there can be a strong series perturbation effect. * Lawrence and Savage (1966). Boldt (1963). Labuhn (1965). Hinnov (1966). 1 Savage and Lawrence (1966). Lawrence (1969).

to deplete the perturber's oscillator strength (Weiss, 1969a). In Table X we show, for a selection of resonance lines, a comparison of Hartree-Fock and experimentalfvalues. In each case, the upper state of thetransitionis avalence excited state, e.g., 2s2pn ', which lies below a d series for which it can act as a series perturber. In this case we are looking at the effect on the perturber of a series perturbation which appears as one particular kind of orbitalpolarization, or semi-internal correlation. Since thef'value is quite small, the precise numerical value is sensitive to all the other correlations. The reason for its smallness, however, is the cancellation introduced by the series perturbation phenomenon. We have already encountered several of these examples in the discussion of the sp6 states of Ne(I1) and Ar(II), and direct calculation of the series effects there did indeed give only qualitative accuracy. +

F. PAIRCORRELATIONS Any fully correlated excited state calculation, of course, will also include the pair correlations discussed earlier. While the more striking effects of correlation in excited states can be understood in terms of the complex or some form of semi-internal correlation, the residual pairs are necessary for the final structure of the spectrum. It is, therefore, worthwhile to consider not only the extent of their effect, but also how well this correlation can be calculated by currently available techniques.

CORRELATION IN EXCITED STATES OF ATOMS

37

The ab initio approach calculates all these effects directly, of course, by including all the supposedly important terms in the variational trial function. Each state is calculated by a separate variational calculation, with excitation energies determined as differences in total energies. The difficulties here are related to problems of convergence, such as completeness of the orbital basis and truncation of the SOC expansion. So far the accuracy attainable for relative energies seems to be, for the lighter atoms, in the 0.1-0.2 eV range. Sinanoglu and co-workers (Oksiiz and Sinanoglu, 1969a, b) have adopted a semiempirical approach, which starts from an ab initio calculation of the complex and semi-internal parts of the correlation. This usually is a relatively small SOC calculation, at least for the 2s"2pmtype of first row states considered so far by these authors. The pair correlation energies (Eq. 28) are then obtained by a least squares fitting of the remaining discrepancies between the experimental and calculated energies. In the terminology of our discussion of pair theories, they calculate E,' and fit E2", thus building up a catalog of pair energies for each nuclear charge. In addition to assumingapairenergyform of the residual correlation, the procedure assumes the transferability of the pair energies among different states and stages of ionization. While high precision calculations within this scheme have not been done tocheckthisassumption, the results and analysis of Oksiiz and Sinanoglu suggest an accuracy of 0.1-0.3 eV for first row excitation energies. Table XI illustrates the kind of results which can be obtained by these various methods for excitation energies of some three-electron atoms [B(I) and C(lI)]. It should be noted that, for the ground state as well astheseexcited states, the total energies are substantially depressed in all of the calculations, and this table indicates only the resulting relative positions. Both of the B(I) calculations are the ab initio type, one being a moderately exhaustive multiconfiguration SCF (Sibincic, 1972) and the other a somewhat larger SOC calculation based on PSNO's (Weiss, 1969a). In terms of the kind of accuracy we are considering (0.1-0.2 eV) there seems to be little choice between the two. The C(I1) results compare an ab initio, PSNO-SOC calculation with the MET semiempirical one, and once again there seems, in general, to be little significant difference. In addition, an MET type calculation was done using the pseudo natural orbitals to compute the internal and semi-internal correlation and then using tabulated MET pair energies. Here too the differences are not large, except for the 2p3 terms, which might incidate a limitation of transferability. The 2 D discrepancy is 0.4 eV. It should be pointed out that there may be a further small uncertainty, which is related to basis set limitations. The Oksiiz and Sinanoglu calculation used a quite restricted basis-essentially one Slater-type orbital of each symmetry, orthogonalized to the Hartree-Fock, and with fixed relative values of their [ parameters. The kinds of errors this entails are illustrated by Table XII,

w

00

TABLE XI

COMPUTED AND OBSERVED TERMENERGIES (RELATIVE TO 2s22p GROUND STATE)FOR B(1) AND C(I1) C(I1) Term 2s2p2 4P 2D

2s

ZP 2p3 4s 2D a

HF

SOC"

MCSCFb

Exp'

HF

SOCd

META'

METB'

0.0784 0.2172 0.3502 0.4010 -

0.1293 0.2202

0.1288 0.2271 0.3390 0.4424

0.1313 0.2180 0.3305 0.4421 -

0.1313 0.3285 0.4270 0.51 15 0.5953 0.6969

0.1941 0.3443

0.1929 0.3322

0.4446

0.4400

0.5102 0.6471 0.6889

0.5024 0.6503 0.6967

0.1896 0.3333 0.4425 0.5013 -

-

0.4420 -

-

-

Exp' 0.1959 0.3412 0.4395 0.5040

0.6469 0.6854

Weiss (1969b). A b inirio term energies. Sibincic (1972). Ab initio term energies. Moore (1949). Weiss (1967a). A b inifio term energies. Semi-internal correlation computed with PSNO's of d ; all external, semiempirical pair energies taken from Oksiiz and Sinanoglu (1969b). i)ksiiz and Sinanoglu (1969b).

+ F F

s. 2

TABLE XI1 THEORETICAL AND EXPERIMENTAL ENERGIES FOR C(I), 2s22p2 3P,AND 2s2p3 3Da METb

PSNO HF

I

+ SI

I + SI

+

“

pairs ”

SOC

I

+ SI

I + SI

+

“

pairs ”

Expc ~

2s22pz 3P 2s2p3 4D AE

-37.6886 - 31.3944 0.2943

-37.1471 - 31.4467 0.3004

- 31.191 1

-37.5143 0.2169

-37.1789 -37.4853 0.2936

-37.7434 -37.4388 0.3046

- 31.1874

-

-37.5063 0.2810

0.2919

-

+

The notation “ I SI” means internal plus semi-internal configuration mixing. Sinanoglu (1912). Moore (1949).

w

u3

40

A . W. Weiss

which compares both PSNO and limited basis calculations of the semiinternal correlation for the ground and 2s3p3 3D states of neutral carbon. The semi-internal energies differ by 0.1 and 0.2 eV for the two states, respectively, and the two calculated, semiempirical excitation energies are off by 0.4 and 0.3 eV. Similar differences in the semi-internal correlation also exist for the various states of C(II), and the better excitation energies there may indicate that the transferability assumption works better for the ion than the neutral atom. One problem here is the dependence of the semiempirical pair energies on the calculated values of the semi-internal correlation, and refitting the pair energies may well lead to better excitation energies. The computed f values for the transition between these states in C(1) are shown in Table XIII. The basis set dependence of the I + SI calculations appears to be about 20-30 %, which is also about the range of accuracy of the full SOC calculation. Also, the largest single effect on thefvalue, aside from the complex, is clearly the mixing of 2s22pnd into the 2s2p3 state, i.e., the series perturbation effect, which cancels out most of the Hartree-Fock contribution. The correlated wave functions without the pair terms do seem to give very good f values as found by Westhaus and Sinanoglu (1969) for a large number of transitions, although the reason does not seem to be related to the correlation sensitivity of the charge density, as they conjectured. At least, some recent scattering factor calculations (Brown, 1972) suggest that the Hartree-Fock charge density is already accurate to a few percent. In sum, the pair correlations are important for the energy but apparently not so much so for oscillator strengths, at least not for the 2s22p"-2s2p"+' type of transitions considered here. The semiempirical approach appears promising and seems comparable in accuracy to a fully correlated calculation, although the basis set question may require further clarification. In practice too, if one can adequately calculate the semi-internal effects, it often happens that a fully correlated, ab initio calculation is not substantially more difficult. Finally, we should note that the asymptotic degeneracy and semiinternal correlation problems become more complicated for other first row excited states, as e.g., 2s2pn3s, and for the 3-shell atoms of the second row. Extension of the semiempirical scheme here requires further study.

G . COLLAPSE OF THE 3d SHELL As we near the end of our discussion it seems appropriate to call attention to another correlation effect which has not yet received much attention, although it may well be important for certain excited states. This effect is related to the collapse of the 3d shell. As is well known in the first long period of the periodic table, the 3d electrons are buried in the 3s- and 3p-shell core, while in the second row the 3d orbital is Rydberg in character, at least for the

TABLE XI11 OSCILLATOR STRENGTHS FOR

2s2p3

I V

2s2p2

+ 2s22pnd I

V

SI

1 V

SOC

THE

2sz2p2 3P-2S2p3 3D TRANSITION I N C(1)

2s22p2

2 ~ 2 2 ~ 22p4

+

I + SI

0.286 0.332 0.096 0.009 0.117 0.026

0.203 0.432 0.054 0.031 0.076 0.077

0.190 0.495 0.057 0.042 0.078 0.097

1 V

SOC

MET"

EW

0.080 0.082 0.102 0.117

0.091b

0.076'

Nicolaides and Sinanoglu (1970). Boldt (1963). Lawrence and Savage (1966).

P

e

42

A . W . Weiss

low stages of ionization. In the intermediate region around argon and potassium, as Cowan (1970) has noted, the 3d shell teeters on the verge of collapse, and it is here that interesting differential correlation effects may occur. To get an idea of the magnitude of the correlation associated with the 3dcore penetration, we may consider the alkali-like systems, K(I) and Ca(I1). In neutral potassium the principal maxima for the Hartree-Fock 4s and 3d orbitals are at 4.3 and 8.3 a.u., respectively (Weiss, 1967b), and for the 3p core orbitals it is 1.2 a.u. The 3d is thus a diffuse Rydbergorbital wellremovedfrom the core. At the next member of the isoelectronic sequence, Ca(II), the 4s has shrunk somewhat to 3.3 a.u., but the 3d goes all the way in to 1.5 a.u., with 3p remaining practically unchanged. This sudden increase in core penetration should imply a substantial increase in the correlation energy, which is indeed the case. Table XIV shows the computed ionization energies for these states TABLE XIV HARTREE-FOCK AND EXPERIMENTAL ENERGIES FOR K(1) AND Ca(l1) Ed

E,,," HF

HF

Expb

Error

~~

K(1) 3P6 3p63d 3 ~

-599.01 74 -599.0755 ~ -599.1646 4 ~

Ca(I1) 3P6 -676.1542 3p63d -676.4981 3 ~ ~ -676.5698 4 ~ (I

0.0 -0.0581 -0.1472

0.0 -0.0614 -0.1595

0.0033 0.0123

0.0 -0.3439 -0.4156

0.0 -0.3739 -0.4363

0.0300 0.0207

Weiss (1967b). Moore (1949).

of K(1) and Ca(I1)-computed as the difference in Hartree-Fock total energies of the 3p6 and 3p6nl systems. While the difference in the correlation energies for 3p64s has increased somewhat, from 0.33 to 0.56 eV, the 3d error increases by an order of magnitude, going from 0.09 eV fo1 the Rydberg-like situation at K(I) to 0.82 eV at Ca(I1). One instance where such large core-correlation, or core-polarization, energies may play an important role is in the 3p53d states of argon and singly ionized potassium. The important point here is the possibly large diflerential correlation energies for the different states of this configuration. Cowan (1970) has commented on the large apparent series interactions associated with the

CORRELATION IN EXCITED STATES OF ATOMS

43

'P term. However, Cowan's discussion is based on HX calculations which use a single set of orbitals for all LS terms of the reference state, and, to a large extent, this is equivalent to allowing the Hartree-Fock 3d to be different for different terms of 3p53d (Hansen, 1972). What happens is apparently that the collapse of the 3d orbital of the 'P state lags behind that of all the others. Thus, for K(I1) the ' P 3d is still Rydberg-like while for all the other states it has collapsed into the core. There should, therefore, be large different correlation energies for these states associated with the core penetration, although they may possibly be masked somewhat by the different pair and semi-internal correlations of the different states. At any rate, the correlation aspect of this problem does not seem to have been discussed extensively as yet. We might also comment that such effects should also occur along second row isoelectronic sequences, since for increasing 2 the 3d orbital rapidly overlaps 3p, as we have seen in our discussion of the sp6 correlation in Ar(I1) and Ca(1V).

V. Concluding Remarks All the important features of the ground state correlation problem exist for excited states as well and must, of course, be included in any complete correlation calculation. In addition, there are aspects of correlation which are either new or given a different emphasis, and this is probably best summed up by the dual need to respect both the demands of the spectrum and the implications of the charge expansion theory. Any configuration interactions indicated by a consideration of the spectrum will be present and often account for anomalously strong particular correlation effects. While the complex has always been recognized as an important part of ground state correlation, it is essential for an understanding of excited states, where the asymptotic degeneracies so often profoundly influence both the level structure and the f values. As to the accuracy attainable by present day correlation calculations, one can apparently expect something like 0.1-0.2 eV for excitation energies and f values in the range 20-30%. This, of course, is a general assessment of the situation, and individual cases may vary somewhat more. For instance, the ,/value for a transition that is weak because of cancellation may be considerably less reliable, due to the enhanced importance of otherwise minor correlation effects. It often happens too that the most outstanding effect of correlation can be understood in terms of just one or two correlation effects, in which case good results may be achieved by a relatively simple calculation. Finally, it should be clear that this article does not represent the last word on the subject. Many people are currently pursuing a variety of problems along the lines described here, and it would be surprising indeed if nothing new were to be learned. It would also be interesting to see the results of applying some of the other approaches to correlation, such as many-body perturbation theory (Kelly, 1969) or the atomic Bethe-Goldstone scheme (Nesbet, 1969).

44

A . W . Weiss

REFERENCES Accad, Y., Pekeris, C. L., and Schiff, B. (1971). Phys. Rev. A 4, 516. Bacher, R. F., and Goudsmit, S. (1934). Phys. Reu. 46,948. Bagus, P. S. (1965). Phys. Rev. A 139, 617. Bagus, P. S., and Moser, C. (1968). Phys. Rev. 167, 13. Bates, D. R., and Damgaard, A. (1949). Phil. Trans. Roy. Soc., London, Ser. A 242, 101. Bauche, J., and Klapisch, M. (1972). Proc. Phys. Soc., London (At. M o f . Phys.) 5, 29. Bergstrom, I., Bromander, J., Buchta, R., Lundin, L., and Martinson, 1. (1969). Phys. Lett. A 28, 721. Boldt, G. (1963). Z. Naturforsch. A 18, 1107. Brillouin, L. (1 932). J . Phys. (Paris) 3, 373. Brown, R. (1972). Phys. Rev. A 5,2141. Bunge, A., and Bunge, C. F. (1 970). Phys. Rev. A 1, 1599. Bunge, C. F. (1968). Phys. Rev. 168, 92. Burke, P. G., Hibbert, A., and Robb, W. D. (1972). Proc. Phys. Soc., London (At. Mol. Phys.) 5, 37. Chandrasekhar, S. (1945). Astrophys. J . 102, 223. Chisholm, C. D. H., and Dalgarno, A. (1966). Proc. Roy. Soc., Ser. A 290, 264. Chisholm, C. D. H., Dalgarno, A., and Innes, F. R. (1968). Phys. Rev. 167, 60. Chisholm, C. D. H., Dalgarno, A., and Innes, F. R. (1969). Advan. At. Mol. Phys. 5, 297. Clementi, E. (1965). IBM J . Rex Develop. Suppl. 9, 2. Clernenti, E., and Veillard, A. (1966). J . Chem. Phys. 44,3050. Cohen, M . (1967). Can. J . Phys. 45,2009. Cohen, M., and Dalgarno, A . (1964). Proc. Roy. Soc., Ser. A 280, 258. Condon, E. U.,and Shortley, G. H. (1935). “Theory of Atomic Spectra.” Cambridge Univ. Press, London and New York. Cowan, R. D. (1967). Phys. Rev. 163, 54. Cowan, R. D. (1968). J . Opt. Soc. Amer. 58, 924. Cowan, R. D. (1970). J . Phys. (Paris), Colloq. 31,4-191. Crossley, R. J. S. (1969). Advan. At. Mol. Phys. 5, 237. Crossley, R. J. S., and Dalgarno, A. (1965). Proc. Roy. SOC.,Ser. A 286, 510. Dalgarno, A., and Parkinson, E. M. (1968). Phys. Rev. 176, 73. Das, G., and Wahl, A. C. (1966). J. Chem. Phys. 44,87. Edltn, B. (1964). In “Handbuch der Physik” (S. Fliigg, ed.), Vol. 27, p. 80. SpringerVerlag, Berlin and New York. Edmiston, C., and Krauss, M. (1966). J. Chem. Phys. 45, 1833. Eissner, W., and Nussbaumer, H. (1969). Proc. Phys. Soc., London (At. M o f .Phys.) 2,1028. Friedrich, H., and Trefftz, E. (1969). J . Quant. Spectrosc. Radiat. Transfer 9, 333. Froese, C. (1 963). Can. J . Phys. 41, 1895. Froese, C. (1966). Can. J . Phys. 45, 7. Froese Fischer, C. (1968a). Can. J . Phys. 46,2435. Froese Fischer, C. (1968b). Astrophys. J . 151, 759. Froese Fischer, C. (1968~).J . Quant. Spectrosc. Radiat. Transfer 8, 755. Froese Fischer, C. (1969). Comput. Phys. Commun. 1, 151. Froese Fischer, C. (1970). Proc. Phys. Soc., London (At. Mol. Phys.) 3, 779. Garstang, R. H., and Shamey, L. J. (1967). In “The Magnetic and Related Stars” (R. C. Cameron, ed.), p. 387. Mono Book Corp., Baltimore, Maryland. Godfredson, E. (1966). Astrophys. J . 145, 308.

CORRELATION IN EXCITED STATES OF ATOMS

45

Goldschmidt, 2.B. (1968). In “Spectroscopic and Group Theoretical Methods in Physics” (F. Bloch, S. C. Cohen, A. de-Shalit, S. Sambursky, and I. Talrni, eds.), p. 41 1 , Wiley, New York. Gombiis, P. (1956). In “Handbuch der Physik” ( S . Fliigge, ed,). Vol. 36, p. 109. SpringerVerlag, Berlin and New York. Hameed, S., Herzenberg, A., and James, M. G. (1968). Proc. Phys. SOC.,London (At. Mol. Phys.) 1, 822. Hansen, J. E. (1972). Proc. Phys. SOC.,London (At. Mol. Phys). 5,1083. Harris, F. E. (1 970). J. Phys. (Paris), Colloq. 31, 4-1 1 I . Hartree, D. R. (1955). “The Calculation of Atomic Structures.” Wiley, New York. Hartree, D. R., and Hartree, W. (1936). Proc. Roy. SOC.,Ser. A 154, 588. Hartree, D. R., Hartree, W., and Swirles, B. (1939). Phil. Trans. Roy. SOC.London, Ser. A 238,229. Herman, F., and Skillman, S . (1963). “Atomic Structure Calculations.” Prentice Hall, Englewood Cliffs, New Jersey. Heroux, L. (1967). Phys. Reu. 153, 156. Hey, P. (1959). Z. Phys. 157, 79. Hinnov, E. (1966). J. Opt. SOC. Amer. 56, 1179. Hinze, J., and Roothaan, C. C. J. (1967). Progr. Theor. Phys. 40, 37. Hylleraas, E. (1930). Z. Phys. 65, 209. Jucys, A. P. ( I 952). Zh. Eksp. Teor. Fiz. 23, 129. Jucys, A. P. (1967). Int. J . Quantum Chem. 1, 31 1 . Jucys, A. P. (1969). Aduan. Chem. Phys. 14, 191. Kelly, H. P. (1963). Phys. Rev. 131. 684. Kelly, H. P. (1969). Aduan. Chem. Phys. 14, 129. Labuhn, F. (1965). Z. Naturforsch. A 20, 998. Lawrence, G. M. (1969). Phys. Rev. 179, 134. Lawrence, G. M., and Savage, B. D. (1966). Phys. Rev. 141,67. Layzer, D. (1959). Ann. Phys. (New York) 8,271. Layzer, D. (1967). Int. J . Quantum Chem. 1,45. Layzer, D., and Garstang, R. H. (1968). Annu. Rev. Astron. Asfrophys. 6,449. Layzer, D., Horiik, Z., Lewis, M. N., and Thompson, D. P. (1964). Ann. Phys. (New York) 29. 101.

Lowdin, P. 0. (1955). Phys. Rev. 97, 1509. Lowdin, P. 0. (1959). Aduan. At. Mol. Phys. 2, 207. McKoy, V., and Sinanoglu, 0. (1964). J. Chem. Phys. 41, 2689. Martinson, I., and Bickel, W. S . (1970). Phys. Lett. A 31, 251. Minnhagen, L. (1963). Ark. Fys. 25,203. Moore, C. E. (1949). Nut. Bur. Stan. (U.S.), Circ. 467. Moser, C . M., and Nesbet, R. K. (1971). Phys. Rev. A 4, 1336. Nesbet, R. K. (1967). Phys. Reu. 155, 56. Nesbet, R. K. (1969). Aduan. Chem. Phys. 14, 1. Nicolaides, C., and Sinanoglu, 0. (1970). Phys. Lett. A 33, 178. Nussbaumer, H. (1969). Mon. Notic. Roy. Astron. SOC.145, 141. Nussbaumer, H. (1971). Astrophys. J . 166,411. Odabasi, H. (1969). J. Opt. SOC. Amer. 59, 583. Oksiiz, I., and Sinanoglu, 0. (1969a). Phys. Rev. 181,42. Oksiiz, I., and Sinanoglu, 0. (1969b). Phys. Reu. 181, 54. Pekeris, C. L. (1958). Phys. Rev. 112, 1649. Phillips, M. (1933). Phys. Rev. 44, 644.

46

A . W . Weiss

Roothaan, C. C. J., and Bagus, P. S. (1963). Methods. Comp. Phys. 2, 47. Savage, B. D., and Lawrence, G. M. (1966). Astrophys. J. 146,940. Schaefer, H. F., Klemm, R. A., and Harris, F. E. (1969). J. Chem. Phys. 51,4143. Schiff, B., and Pekeris, C. L. (1971). Phys. Rev. A 4, 885. Schulz-Gulde, E. (1969). J. Quant. Spectrosc. Radiat. Transfer 9, 13. Seaton, M. J. (1966). Proc. Phys. Soc., London 88, 801. Sibincic, Z. (1972). Phys. Rev. 5, 1150. Silverstone, H. J., and Sinanoglu, 0. (1966a). J. Chem. Phys. 44,1899. Silverstone, H. J., and Sinanoglu, 0. (1966b). J. Chem. Phys. 44, 3608. Sinanoglu, 0. (1961). Proc. Roy. SOC.,Ser. A 260, 379. Sinanoglu, 0. (1964). Advan. Chem. Phys. 6, 315. Sinanoglu, 0. (1 969a). Advan. Chem. Phys. 14,237. Sinanoglu, 0. (1969b). Comments. At Mol. Phys. 1, 116. Sinanoglu, 0. (1970). J. Phys. (Paris), Colloq. 31,4-83. Sinanoglu, 0. (1972). In “Topics in Modern Physics” (W. Brittin and H. Odabasi, eds.), p. 287. Colorado Univ. Press, Boulder, Colorado. Slater, J. C. (1951). Phys. Rev. 81, 385. Slater, J. C. (1960). “Quantum Theory of Atomic Structure,” Vol. 2. McGraw-Hill, New York. Smith, M. W., and Wiese, W. L. (1971). Astrophys. J.,Suppl. Ser. 23, (196) 103. Steele, R., and Trefftz, E. (1966). J. Quant. Spectrosc. Radiat. Transfer 6,833. Trefftz, E. (1950). Z. Astrophys. 28, 67. Vizbaraite, I., Shironas, V., Kavetskis, V., and Jucys, A. P. (1956). Opt. Spekrrosk. 1,277. Warner, B. (1968). Mon. Notic. Roy. Astron. SOC.139, I . Weisheit, J., and Dalgarno, A. (1971). Chem. Phys. Lett. 9, 517. Weiss, A. W. (1967a). Phys. Rev. 162, 71. Weiss, A. W. (1967b). J. Res. Nut. Bur. Stand., Sect. A 71, 157. Weiss, A. W. (1969a). Phys. Rev. 178, 82. Weiss, A. W. (1969b). Phys. Rev. 188, 119. Weiss, A. W. (1970). Nucl. Instrum. Methods 90. 121. Weiss, A. W. (1973). (to be published). Westhaus, P., and Sinanoglu, 0. (1969). Phys. Rev. 183, 56. Wiese, W. L., and Weiss, A. W. (1968). Phys. Rev. 175, 50. Zare, R. N. (1966). J. Chem. Phys. 45, 1966. Zare, R. N. (1967). J. Chem. Phys. 47, 3561.

THE CALCULATION OF ELECTRON-A TOM EXCITATION CROSS SECTIONS M . R . H . RUDGE Department of Applied Mathematics and Theoretical Physics The Queen’s University of Belfast Belfast. Northern Ireland

I . Introduction .................................................... I1 . High Energy Theory .............................................. A . Coulomb Functions .......................................... B. Boundary Conditions and Cross-Section Expressions for the e-H Problem .................................................... C . The Conservation Theorem . . . . . . . . . . ...................... D . Variational Principles for the Scattering E . Integral Identities and Reciprocity Relat ............ F. Eigenfunction Expansions . . . . . . . . . . G . Excitation of Helium . . . . . . . . . . . . . . . . ................. H . Reliability Criteria . . . ...................................... 111. High Energy Approximations .................................... A . The First Born Approximation ................................ B. The Born-Oppenheimer Approximation .......................... C. The BOMC Approximation . .............................. D . Orthogonalized Born-Oppenh Approximations . . . . . . . . . . . . . . E . The Approximations of Bonham, Ochkur. and Rudge ............ F. First-Order Distortion Approximations .......................... G . Second-Order Approximations . . . . . . . .................... .................... H . The Method of Regional Trial Functio IV Low Energy Theory . . . . . . . . . . . . . . . . . . . .................... A . Partial Wave Boundary Conditions and ection Expressions .... B. Variational Principles .......................................... C . Resonance Theory . . . . .................................. D . The Close-Coupling Me .................................. E . A Minimum Principle .................................. F. An Alternative Expansion .................................... G . Polarization Effects .......................................... H . Threshold Behavior . .................................. V . Low Energy Approximatio .................................. A . Unitarized Approximations . . . . . . . . . . . .. ............. B. The Close-Coupling Method . . . . . . . . . ..................... C . The Correlation Method ...................................... D . Pseudostate Expansions . . . . . ............................. E Variational Procedures ........................................ VI . Concluding Remarks ...............................

.

.

................................... 47

48 49 49 50

53 56 59 60 62 63 67 69 69 72 80 83 89 93 93 97 99 104 105 108 109 111 115 115 117 117 118 120

48

M . R . H . Rudge

I. Introduction The calculation of electron-atom excitation cross sections has been the subject of a number of excellent reviews (Bates et ul., 1950; Massey, 1956; Seaton, 1962; Burke and Smith, 1962; Peterkop and Veldre, 1966; Moiseiwitsch and Smith, 1968; Burke, 1969). The present article includes more recent work but is also designed to provide a survey that is of value to new research workers in the field of theoretical atomic physics. Inevitably, therefore, some topics are included that appear also in the references cited. Atomic units (e = m = h = 1) will be used throughout, but cross section values will be expressed in units of nuO2 (= 8.797 x cm2). It is often convenient to express energies in terms of a dimensionless variable ER

= EllET

where E, is the energy of the incident electron and ET is the threshold energy for the particular transition under discussion. We shall be little concerned with elastic scattering events. No electron-atom or electron-ion scattering problem can be solved exactly; even the simplest is already a complicated three-body problem. One must therefore develop techniques that provide a close approximation to the exact solution of the problem. There are two main approaches. In the first of these, perturbation theory, a solution is developed in terms of the exact eigenfunctions which belong to an approximate Hamiltonian. In the second approach, the variational method, trial functions are treated as approximate eigenfunctions of the exact Hamiltonian. The two methods are to some extent equivalent but the variational method can offer a rather clearer picture of the physics of the collision process and will therefore be the one most closely adhered to in this article. The technique that can be used to calculate an excitation cross section depends very much on the energy at which the cross section is required. We can think of a high energy regime wherein ER 2 20, a low energy regime wherein ER 5 1.5, and an intermediate energy range between these two. The precise definition of what constitutes a high energy regime and what constitutes a low energy regime varies from transition to transition. Broadly speaking, however, two such regimes do exist in which the appropriate theoretical techniques for calculating the cross section differ. Our development of the theory will therefore take cognizance of this. It would be a pleasant surprise for the theoretician to find that, in the intermediate energy range, he thus had at his disposal two techniques both of which were adequate. Unfortunately, it is more commonly found to be the case that neither of the two techniques is adequate at these energies; this, of course, poses a considerable problem. A significant part of the theory will be concerned with the excitation of

ELECTRON-ATOM EXCITATION CROSS SECTIONS

49

atomic hydrogen, partly because this is a problem of fundamental importance and partly because the discussion of the basic ideas is then less likely to be obscured by algebraic complexity. Perversely, however, the degeneracy of the atomic energy levels for hydrogenic systems gives rise to some complications that do not occur for other atomic systems.

11. High Energy Theory Let us consider the theory of scattering of an electron by a hydrogenic system at high energies. Relativistic effects will be neglected and the nucleus will be taken to be infinitely massive and at the origin of coordinates. It is convenient to first list some properties of Coulomb functions which we shall use in the subsequent discussion. A. COULOMB FUNCTIONS

The equation [V2 + k 2 + 2z/r]x = 0 has a solution

-

~ ( z-, klr) = [2nq/(l - e-2"q)]"2exp i(ao(q)- k r)lFl(iq, 1, i(kr - k * r))

(2)

where q = z/k

(3)

and uf(q) = arg

r(l+ I -iq)

(4)

From the form of Eq. ( I ) it follows that the functions x*(z, - klr), ~ ( klr), z , and x*(z,kl r) are also solutions of the equation. The various solutions differ in their asymptotic forms. It is shown by Gordon (1928), that x ( z , - klr) = k - 1 ' 2 r - 1~ i - f e i u ' ( q ) ( 21)P,(k /+ * P)F,(z,klr) I

(5)

where

I'K

x

k-"' sin(kr - +fn

( I + 1 - iq,2/ + 2, - 2ikr)

+ q In(2kr) + o f(q))

(6) (7)

M . R . H . Rudge

50

Using Eq. (7) and the completeness relation for the Legendre polynomials P l ( k * P), it follows that ~(z, k I r)

-

2n/ikr {a(&,P) exp i(kr

r+m

+ q In(2kr)) - S(&

- P)

exp - i(kr

+ q In(2kr)))

(8)

where

@(k, P) = S(cr + P) + (ik/2n)f,

(-

cr, P)

(9)

andf,( -&, P) is the Rutherford scattering amplitude defined by the equation

fR(-&,p)

= qr(i - iq) exp

-

2iq In(sin e/2)/2kr( I +iq) sin2(e/2)

(10)

where cos 0 = -r( P. These formulas simplify greatly in the case where z = 0; in particular there is then no distinction between the functions ~ ( 0 , k I r) and x*(O, klr). Equation (8), given for plane waves by Percival and Seaton (1958), is a very useful one which will find application in the ensuing sections. B. BOUNDARY CONDITIONS AND CROSS-SECTION EXPRESSIONS FOR PROBLEM

THE

e-H

The stationary states of the hydrogen atom are characterized by three quantum numbers (n, I, m). It is convenient to parametrize these through an index i so that to each value of i there corresponds a set of quantum numbers ( n i , li , m i ) . Let the corresponding eigenstates be $(il r) which satisfy the equation [-

+V2 - l/r]$(i]r)

= Ei$(ilr)

(1 1)

Let E be the total energy of the system and let ki2 = 2 ( E - Ei) The Schrodinger equation which describes the e-H system is

where

and r12

= Ir, - r2l

(15)

rl and r2 being the position vectors of the two electrons relative to the proton.

ELECTRON-ATOM EXCITATION CROSS SECTIONS

51

The interelectron potential r;; can be expanded in a series of Legendre polynomials according to the formula

and so in the region r2 -+ co the Hamiltonian 2 is given by

2 = HAS+ o(rT2)

(17)

where

Solutions of the equation

are given by

It follows that if the atom is in state i = p initially, the solution as r2 -+ co must have the form Y p ( k p l r 1 , '2)

-

rdrl-m

$(plrl)exp(ikp 'r2) + r;' Cfpj(kp9 f 2 )

x $(jlrl) ex~(ikjr2)

j

(21)

Clearly the asymptotic form (21) takes proper account of terms of order

r i l but not those of order r;", with n > 1. The form of the wave function in the region r l / r z -+ co is given in a similar way by

and corresponds to the situation in which the incident particle is captured while the initially bound electron recedes from the atom. It should be noted that asymptotic region specified in Eq. (21) and (22) excludes the possibility of ionization for which process a different asymptotic form is required. The coefficients.fpj(kp, P2) and g p j(kp, PI) are respectively the direct and exchange scattering amplitudes. If the electrons were in some way distinguishable, these two amplitudes would describe distinct physical processes. In fact the electrons are indistinguishable and therefore the description of the excitation process will involve a combination of the two amplitudes the form of which is dictated by the Pauli principle.

M . R. H. Rudge

52

We define a spin coordinate CJ = & $, a spin quantum number p = It +, and a spin function 8(p I CT)which is unity if p = CJ and zero otherwise. x denotes collectively the spin and space coordinates. A spin function for two electrons is given by X(S,M,ICJ,,CJ,) =

c

(23)

~ ~ ~ l t ~ 2 I ~ ~ s ) ~ ~ ~ ~ I l C J I ) ~ 1 ~ 2 1 ~ 2 )

PIP2

where ( t p 1+p2 IS, M,) is a Clebsch-Gordan coefficient [EC(++S;p,, M , p z ) in the notation of Rose (1957)l. It follows that the totally antisymmetric function is given by

W x , ,xz>= 2-”2~iS,M,l~,,a2){Yir,,r2) + (-I)’Y(r2,

r,)}

(24)

The total spin S, on neglecting relativistic effects, is a good quantum number and if S is specified the scattering amplitude is diagonal in S and independent of its component M , . The corresponding amplitude can be seen from Eq. (24) to be fpj(Slkp’f2>=.fpj(kp,P)

+ (-l)’gpj(kp?f)

(25)

The differential cross section for given spin S is then given by r p j (Sl k p

3

3)

= kjikplfpj (SIk p ,

2) I

(26)

In general the spin orientation is not known and the corresponding differential cross section is obtained by averaging over the possible spin orientations. Using the notation

[I] = 21 + 1

(27)

the differential cross section for a system of unpolarized spin is given by ZpjWp,

P) =

(1[SI)-’ 1

[SIIpj(SIkp,

(28)

f>

S

S

=kj/kp(Ifpj(kp,P)12

+ Igpj(kp,P)12

-Re(fpj(kp,

P)g,*j(kp$f)))

(29)

The cross section in this case depends therefore upon the magnitudes of both of the scattering amplitudes and upon their relative phase. If energy differences due to fine or hyperfine interactions are ignored, the scattering amplitude in an angular momentum representation r can be seen to be related to that in a different representation y through the equation fpj

(r,r’I kp

9

f >=

C (Y I

vv’

r)fpj

(7, Y‘ I k p

3

P)(r‘I Y’>

(30)

where (y I r) is the appropriate scalar product. As a particular example of this formula we can consider transitions between the hyperfine levels of two

53

ELECTRON-ATOM EXCITATION CROSS SECTIONS

F the total atomic spin, the components of which are hf, and M,, we can take

s states. If I is the nuclear spin and

y = ( S ,M , 1, MI)

(31)

r = ( F , M,,

(32)

1

trp2)

and it follows that

where W is a Racah coefficient. The differential cross section for a transition from total atomic spin F t o total atomic spin F‘, if the incident electron beam is unpolarized, is given by

On using orthogonality properties of the recoupling coefficients this reduces to the expression Ipj

( F , F‘I kp 2) 7

I

I

- f p j ( 1 kp 3 P>

= (kj [F’1/8kp [I]) I f p j (0 kp 9 = ( k j [F’]/2kP[Ill lgpj (kp.f )

I

I

(35) (36)

The spin change cross section, as measured by a change in the hyperfine state of the atom, is therefore seen to depend only upon the exchange scattering amplitude. Equations (29) and (36) show, therefore, that differential cross section and spin change differential cross section measurements can be used to check different aspects of the theoretical amplitudes. More general expressions can be derived for scattering of polarized atoms by polarized beams of electrons (Bederson, 1970; Farago, 1971 ; Kleinpoppen, 1971). C. THECONSERVATION THEOREM Let Y j( k , , Sl r l , r2) be an exact solution of the Schrodinger equation which satisfies the boundary conditions

M . R . H . Rudge

54

The left-hand side of Eq. (38) can be explicitly evaluated by the use of Green's theorem and Eq. (8). There results the identity that

A particular case of Eq. (39) is given by p can be written

= q, k, = k,

. The result in this case (40)

4nrlkp I m ( f p p (SI kp kp)) = Qtot ( S ) where Ptot(S)= kil

1i J" Ifpj(Slkp, ')I2

(41)

d?

kj

is the total cross section. Equation (40) simply expresses the conservation of charge in the collision process. Equations (39) and (40) show that the exact scattering amplitudes are complex. D. VARIATIONAL PRINCIPLES FOR

THE

SCATTERING AMPLITUDES

Variational principles serve the important function of providing for the theoretician a framework within which he can construct approximations which yield scattering amplitudes which are at least stationary. They also serve the useful purpose of allowing exact theoretical relationships to be deduced. The first variational principles for scattering amplitudes were given by Kohn (1948) and can be used, provided that the trial functions have the correct asymptotic form and are continuous. There is clearly, however, advantage to be gained from allowing the greatest possible flexibility in the choice of trial function. It can happen for example that one has knowledge of the form that the wave function must have in certain regions though this form may be quite inadequate or possibly gravely incorrect in other regions. In order to make use of such knowledge a variational principle is required that admits different trial functions in different regions and does not necessarily demand that the trial function within one region should match those in contiguous regions. Such variational procedures can be developed and we follow that given by Rudge (1970a) for the case in which there are two regions defined as Region TI, r2 > r ,

Region T,,

r l > r2

Let Y,' and Y,' be trial functions defined such that Y,' in the region T i , where Yr'

-

12-cc

$ ( p I r l > exp(ik, . r J

+ r;'

n

(k,,

f2)$(n

= Y:),

Yq' =

I r l >exp(ik,, r d

"6" (42)

55

ELECTRON-ATOM EXCITATION CROSS SECTIONS

Yfl:2)

-

r;' C g & , . ( - k , , P1)IC/*(nlr2)exp(iknr,) n

rl+m

Let

and let

L,,

=Ll,

4-Ll,

(47)

Define the operator D

= [(ajarl) - (ajar2)]

(48)

and let m

K,,

=

-(4n)-'s 0 -

r4 dr /dP, df, {(Yb') + YF))D(Yf)- YL'))

+

( Y y - Y';'))D(Y;') Y y ) } r , = r * = r

(49)

It was shown by Rudge (1970a) that a variational principle for the direct scattering amplitude is given by

w,,, + K,, -f,, (kp k,)) 3

It can also be shown that if a trial function

and if that

z,,and R,

9,(rl , r,)

= Y,

=0

(50)

9 ,is defined by the equation

(r,, rl)

(51)

are defined by replacing Y, by

q, in Eqs. (46) and (49),

w,, + R,,

- gP4(kp'k4))= 0

(52)

For the particular case in which the trial functions are the sam'e in the two regions, Eqs. (50) and (52) become

w,,-f,, (k,

k,))

=0

(53)

d( t,,- g p y (k, k,))

=0

(54)

7

9

which are the results given by Mott and Massey (1965). For the more general case of scattering by a hydrogenic positive ion with nuclear change Z , the functions exp(ik,, . r) and exp( - ik, . r) are replaced respectively by the func-

M . R . H . Rudge

56

tions x(Z,k, Ir), ~ ( 2-k, , I r). A variational principle for the scattering amplitudef,,(S( k,, f ) of Eq. (25) can be written

Wnf,,(SIk,, kq) - J ” y q ( - k q l x , , x 2 ) ( z - E ) y p ( k , l x , , x 2 ) d ~ , d 2 )= 0 (55) where integration over x implies integration over the spatial coordinates and summation over the spin coordinates.

E.

RECIPROCITY RELATIONS

INTEGRAL IDENTITIES AND

It is possible to deduce exact integral expressions for the scattering amplitudes by making particular choices of trial functions in the above variational expressions. If, for example, we choose

Y p r ( k p l r l r2) , = $(pIrd exp(ik, r2) ~,‘(-kqbl

9

r2)

= YJ-kqIr1, r2)

(56)

(57)

then Eq. (53) gives the integral identity that Y ( p l r l ) exp (ik, . r2)dr1dr2 (58)

In a similar way it can be seen that

s

2 w , , (k, , k,) = ‘l’,( - k, I r2 , r l >

$ ( p I r , ) exp(ik, * r2) dr, dr2 (59)

Equation (58) can be rewritten

s

2nfpq(k, , kq) = {Yi(&‘ - E ) Y , - Y q ( 2- € ) V i ) dr, dr,

(60)

The right-hand side of Eq. (60) can be evaluated by use of Green’s theorem to give the reciprocity relation that

f,, (k,

9

kq) =f q * p( - kq > - k,)

(61)

A similar result can be obtained for the exchange scattering amplitude and thus fpq

(Sl k, k,) 9

=&,*

(Sl - kq - k,) 9

(62)

57

ELECTRON-ATOM EXCITATION CROSS SECTIONS

Making use of the reciprocity relations (61) and (62), it can be seen that the integral expressions (58) and (59) can be written in the alternative form 2nf,,(k,,

k4) = J'f',(k,Ir,,rJ

2wp,(k,, k,,)

=I

$*(qIrA exp(-ik, . r z ) d r l d r z

(63) $*(q[r,) exp(-ik,

'f',,(kplr2, r l )

*

rJdrldr2 (64)

Many other such exact integral expressions can be deduced corresponding to one of the trial functions being an exact solution while the other is an approximate solution having the correct boundary conditions. The point of interest is that they can all be derived as special cases of the variational principles and that, naturally enough, the " interaction potentials" in the integrals are dependent upon the choice of wave functions. It is of some interest to develop integral expressions which involve Coulomb waves rather than plane waves. The exact solution which satisfies boundary conditions of the form (21) and (22) satisfies the Schrodinger equation rl

rz

(65)

r12

while IC/*(qlr,)X(Z, -kqlr,)

=0

(66)

It follows from these two equations that, with 0,= $*(q I rl)X(2, - k, I r,),

-I@,

2€ j',['(?

- 1

r12

dr, dr,

Making use of Eq. (8) then gives the result that

1-z

R

2nf,,(k,,

k4) = lim exp(iq In(2kR)) [dr, d3, Jo rzzYp(R-rm

r2

-

-)r121

@q

drz (68)

It can be shown in a similar way that R

2ngpq(k,, kq) = lim exp(iq In(2kR)) Jdr, di, R-'m

where

6q(rl,r2) = @,

Jo

r,'Y,(-

1-2

r2

-

-)1

6qdr,

112

(69) (r, , rl).

M . R. H . Rudge

58

The entire expressions (68) and (69) are convergent because the divergent phase factor outside the integration cancels with its complex conjugate which arises from the integration itself. Correspondingly, if the external factor is omitted then the integration gives rise to a divergent phase factor (Rudge, 1968). Some authors (Day eta/., 1961 ; Kang and Sucher, 1966; Kang and Foland, 1967a,b) have pointed out that if Z is chosen equal to unity and the Coulomb wave approximated by a plane wave then Eqs. (63) and (64) are recovered without the electron-nucleus term in the interaction potential the removal of which they therefore advocate. It is clearly, however, an unsound theoretical procedure to develop an integral identity and then approximate the functions which appear in it since this will not, in general, be in harmony with the more basic variational principles.

F. EIGENFUNCTION EXPANSIONS The set of functions {$(il r)} is complete with a continuum. It follows therefore that the total wave function can be expanded in the form

where the functions F(ilr) can be determined by appeal to the Schrodinger equation and where denotes a summation over the discrete states plus an integration over the continuum. It is of interest to examine the structure of the expansion (70). The essential features of the expansion were exhibited by Castillejo et a/. (1960) who simplified by replacing the Coulomb potentials with Yukawa potentials. By making use of Eq. (69) their method can be readily applied to the case of Coulomb potentials. From a physical point of view the completeness of the set means that the expansion must describe all possible distinct physical events. The direct scattering is clearly represented in Eq. (70) but it must also represent exchange scattering. In order to see that this is so, Eq. (70) can be written more fully as

si

Following Castillejo et al. (1960) the function F(kIr,) can be represented in the basis { $ ( i \ rz)} through the equations F(klr) =

s A j ( k ) ( k j 2 k’)-’$(jIr) -

(72)

j

where A j ( k ) = 2-1/27c-3/2~ Y ( r ~ , ~ , ) ~ ~ ~ $ * ( j -klrl) l r z ) ~dr,dr, (1,

(73)

ELECTRON-ATOM EXCITATION CROSS SECTIONS

59

Hence

There exist poles in the integrand of Eq. (74) and a way of treating these must be specified. The appropriate procedure is to let the energy E become complex

E=E+iE and let the positive constant E vanish after the integration has been performed. Following this procedure and making use of Eq. (8) gives the result that Y(rl,r2)

-

rllr2-

-(n/2)1’*r;1 r.

$(jlr2)Aj(kj) exp(i(kjrl

+ qj ln(2kjr,))

(75)

j

Examination of Eqs. (69) and (73) shows that Eq. (75) can be written y

p (kpI rl, r2)

-

r;’

r ~ l r ~ - m

2 g p j ( k p I Pl)Ic/(jI rl>exp(ikjrl)

(76)

j

It can thus be seen that an expansion in a set of eigenstates for just one of the electrons does indeed include exchange events which might at first sight be thought to need an expansion containing bound states for either electron. G. EXCITATION OF HELIUM The theory for helium differs insofar as the bound state wave functions cannot be exactly determined. The effect this has on the variational principles for the scattering amplitudes has not been altogether satisfactorily dealt with though the matter is discussed by Demkov (1963). However, provided that it is supposed that the atomic wave functions are exact, the variational principle for the scattering amplitude can be deduced to give a result analogous to that of Eq. ( 5 5 ) , viz.

where X is the appropriate Hamiltonian. In Eq. (77) S and Ms refer to the spin quantum number of the atom, while p refers to that of the free electron. The totally antisymmetric functions V(kl x 1x2 x3) are related to functions Y(kl x1x,; xj) which are antisymmetric in particles one and two only, through the equation

M . R . H . Rudge

60

The differential cross section is given by

Making use of Eq. (78) in Eq. (77) and using orthonormality properties of Clebsch-Gordan coefficients this reduces to the expression I,, (k, k,) 9

= k,/k,@ss,

(If,, (kp

?

k4) I - 2 Re(f,, (k, k4)g;, (kp k,)) + [SIlg,, (kp k4)I 9

3

9

where

’>(80)

6j2afpq(k,, k,) - jY’,(-k4lrI,r2; r3) x

( 2- E)Yp(kp~r,,rz; r3) dr, dr, dr,

and

a( 2w,,

(k, k,) 3

J- ‘y, ( - k, I

r l >r3 ;

1

=0

(81)

rz>

x ( X - E)Yp(kplrl,r,; r3) dr,,dr, ,dr,

It can be seen, from Eq. (80), that singlet-triplet or triplet-singlet transitions only involve the exchange amplitude gp4.

H. RELIABILITY CRITERIA It is convenient to conclude this section on high energy theory by seeing what criteria have emerged whereby theoretical techniques can be assessed. There are in the first place what might be called theoretical criteria, which constitute those tests which can be applied on the basis of general theoretical principles, before detailed calculations are performed and compared with experiment. These tests can be listed as follows (T.C. denoting “ theoretical criterion ”).

T.C.(I) Consercation Conditions Conservation conditions are given by Eqs. (39) and (41). As will be seen later, these can be expressed more simply if a partial wave analysis is performed, but in the absence of such analysis these equations are obeyed by the exact amplitudes and it is some guide to see whether or not they are obeyed by approximate amplitudes. In particular the conservation calculations imply complex scattering amplitudes and so if a theoretical amplitude is real, that is evidence that its phase, at least, is in error.

ELECTRON-ATOM EXCITATION CROSS SECTIONS

61

T.C.(2) Reciprocity Conditions Reciprocity conditions are given by Eqs. (61) and (62) and it is certainly desirable that a satisfactory theory should yield amplitudes which obey these relations. If T.C.(2) is satisfied then the cross sections will obey detailed balance.

T.C.(3) Vuriational Principles

If an approximation results from a variational principle, it can be said to be the best approximation that can be achieved with the wave functions used in a variational sense. This means that the errors are second order though if the principleis only a stationary one, this does not necessarilyimply that the errors are numerically smaller than could be achieved by some other, nonvariational, expression. If the variational principle can be shown to give upper or lower bounds, and those discussed heretofore do not, the importance of a variational formulation is greatly enhanced. From a practical point of view a formulation in terms of a variational principle gives some guide as to what are good or bad features of the wave functions used, this in turn can make way for systematic improvements. T.C.(4)Threshold Behauior The behavior of cross sections at an energy close to a threshold energy can be deduced from general theoretical considerations and is discussed in Section IV,H. It is of importance that a low energy theory should yield the correct threshold dependence of the cross section. For some cases, high energy theories also give the correct threshold behavior.

T.C.(5) Resonance Structure In the vicinity of threshold energies cross sections may exhibit a resonance structure and a low energy theory that fails in appropriate cases to predict such structure can be judged to be unsatisfactory. In the second place there are those tests which can be applied by comparison of theory with experiment and these are of varying degrees of refinement. We list these empirical criteria (E.C.) as follows: E.C.( I ) Total Cross Sections

For many practical purposes a fairly reliable estimate of the total cross section is all that is needed and a theory can be judged to be partially satisfactory if it provides this information. Most theories are less than completely

62

M:R. H . Rudge

successful even in this regard in that they can provide adequate information only in certain energy ranges and perhaps for certain types of transition. Provided, however, that its limitations are clearly understood, such a theory is a useful one. The structure of Eq. (29) shows, however, that this test is of low resolution, since it involves three distinct terms any one or two of which may be wrong yet lead to reasonable results by virtue of cancellation of error. Equally, it can be seen that it is possible to generate a theory which gives rise to reasonable magnitudes of the scattering amplitudes but can lead to poor results if their relative phase is in error; the latter quantity is the more difficult to calculate accurately. E.C.(Z) Exchange Cross Sections

Equation (36) shows that for e-H scattering a comparison between experimental and theoretical spin change cross sections provides a check on 1gI2 while Eq. (80) shows that singlet-triplet transitions in helium similarly depend upon 1gI2.This therefore provides an additional check on the theory. E.C.(3) Differential Cross Sections

It is possible for a theory to satisfy E.C.(l) and E.C.(2) but predict an incorrect angular distribution of scattered electrons. E.C. (3) is therefore a more stringent test and it is of importance to know for what range of energies and angles a theory yields correct differential cross sections. E.C.(4) Polarization of’Impact Radiation

If the atom is excited to a state which decays radiatively, the emitted radiation will be polarized, the degree of polarization depending upon the relative population of the magnetic sublevels. E.C.(4) therefore provides a further, more severe test of theoretical predictions. Clearly the theoretician would like to produce a technique which is satisfactory in all of these and other respects; a universal tool as it were. In the event what has been produced more closely resembles a set of spanners, each of which is useful provided that its limitations are understood.

111. High Energy Approximations Let us now turn to an examination of procedures whereby the scattering amplitudes can be calculated. We begin with the simplest and best known approximation, that of Born (1 926).

63

ELECTRON-ATOM EXCITATION CROSS SECTIONS

A. THEFIRSTBORNAPPROXIMATION The approximation consists of choosing, in the variational principle, Eq. (58), the trial functions

~ , ( k p l r 1 J z )= $(PI',)exP(ik,.rz) yq(-kqIr1,r2)

= $*(q1 TI)exp(

-ik,

(83) *

(84)

rz)

The direct scattering amplitude is then given by the equation .f;q

1

(k, k4) = (2~)-' $(P I rl)$*(q I rl>

exp(iq * r2) dr, dr,

(85)

where q = k, - k,

The Born differential cross section for excitation is given by

where ET is the excitation energy of the transition and

is the generalized oscillator strength (Bethe, 1930); $,,(O) oscillator strength. The total cross section is given by

is the optical

The fact that, for high incident energies, the main contribution to the integral of Eq. (90) comes increasingly from the region of small q, allows the determination of the high energy form of Qpq(Bethe approximation). Thus with El = $kp2,

for optically allowed transitions, where A is proportional to $, (0), while for nonoptically allowed transitions

where B is proportional to the quadrupole moment of the transition.

M . R . H . Rudge

64

Clearly the trial functions (83) and (84) are extremely crude and open to criticism. They take no account of exchange effects nor do they take account of the polarization of the atom by the impinging electron and the effect which this has in distorting the plane wave. In Section II,C a conservation theorem was derived. It can be seen that in the first Born approximation fpp(kp,kp) is real and so the conservation theorem is not satisfied. For transitions between strongly coupled levels this can give rise to serious errors. This said, however, it is to be expected that these defects will become decreasingly important as the energy of the incident electron increases. Therein lies the importance of the first Born approximation in that it can provide a means of normalizing relative experimental measurements at energies which are high enough for it to be accurate. There, is however, no a priori means of establishing which energy is high enough. A cross section for which absolute experimental measurements are available is the I'S-2'P cross section in helium (Jobe and St. John, 1967; Moustafa Moussa et al., 1969). It is of interest therefore to compare theory and experiment for this cross section. From the theoretical standpoint it is clearly a very important matter in trying to assess the accuracy of the first Born approximation for excitation of helium that the atomic wave functions used should be of high accuracy. A good deal of careful study of this matter has been carried out using different formulations of the matrix elements which, while identical when exact atomic functions are used, are not so when approximate functions are used and thus provide a useful test of their accuracy (Kennedy and Kingston, 1968; Bell et al., 1968a,b; Bell and Kingston, 1968; Kim and Inokuti, 1968). Calculation (1) of Kennedy and Kingston (1968) shows good consistency between two different formulations of the matrix element and in Fig. (1) this calculation of the 1 'S-2lP cross section in the first Born approximation (FBA) is compared with the experimental values. The figure shows, as is typical for optically allowed transitions, that the FBA overestimates the cross section at low energies; for this case experiment and the FBA do not merge within the range of ER depicted. Turning to predictions of the differential cross section it can be seen that Eq. (88) provides a useful means of assessing the approximation. We can define an apparent generalized oscillator strength as 6 A ( d

= ET

kpq2

I p q (kp

7

kq)/2kq

(93)

where Z, (kp,k4) is the experimentally measured differential cross section. According to the Born approximation this should be independent of k , . Conversely, any observed variation in the apparent generalized oscillator strength serves as an indication of the breakdown of the first Born approximation. It is convenient, following Miller and Platzman (1957), to plot $A (9) against q 2 . Such a plot, for the I'S-2lS cross section in helium, obtained by Vriens et al. (1968), is shown in Fig. (2).

ELECTRON-ATOM EXCITATION CROSS SECTIONS

65

FIG.I . The helium I'S-2'P cross section comparing the first Born approximation (Kennedy and Kingston, 1968) with two experimental measurements (Jobe and St. John, 1967; Moustafa Moussa et al., 1969).

.,

q'

FIG.2. The apparent generalized oscillator strength as a function of momentum transfer squared for the 1 'S-21Scrosssectionin helium (Vriens ef u l . , 1968). 0 , 4 0 0 eV; 0 , 3 0 0 eV; A,225 eV; A , 200 eV; 0 150 eV; I00 eV.

It can be seen from this figure that the +A(q) do not tend to lie on a common curve as the energy increases. What is apparent however is that they tend to lie on a common curve for small values of q 2 . Truhlar et a / . (1971) have also compared Born predictions and experimental values of the differential cross

66

M . R. H. Rudge

section for excitation of the 2'P state and similarly find that the Born predictions are too small at large scattering angles though the total cross section is too large. As remarked by Truhlar et al. (1971) it can therefore be seen that the FBA can more accurately be described as a small momentum transfer approximation than as a high energy approximation, although as the energy increases the relative error in the total cross section arising from this behavior diminishes. It is a more difficult matter to determine the region of validity of the Born approximation where there are no measurements of the differential cross section. A case in point is the Is-2s and Is-2p transitions in atomic hydrogen. Is-2s-2p close coupling calculations (Burke et al., 1963; Damburg and Peterkop, 1962) apparently lead to the conclusion that the Born approximation is valid at quite low values ER x 5-10. Burke et al. (1963), however, noticed that in a partial wave expansion the largest contributions to the cross section arose from different angular momentum values in the two approximations. This suggests that the agreement is fortuitous. Burke et al. (1967b) suggested that in view of the doubt concerning normalization to Born cross sections, an alternative procedure might lie in normalization to low energy theoretical calculations. This view was reiterated by Geltman and Burke (1970) who considered it doubtful that the Born approximation is valid for these transitions at E, = 200 eV (ERx 20), this being the energy at which Long et al. (1968) normalized their experimental data. The problem has been further studied by Damburg and Propin (1972). They consider a model of the e-H collision in which the close-coupling equations are approximated by retaining only the dominant terms in the potentials. They show for this model, in the energy range 200-1000 eV, that coupling effects neglected in the FBA can account for a 20% change in the Is-2p cross section. The results they obtain from this model, however, lie below the Born results which is in the opposite sense from that required to establish agreement between experiment and the low energy theoretical results. It therefore appears that the FBA is quite a good approximation for small angle scattering but very poor for large angle scattering. Considerable doubt exists as to the energy at which it predicts accurate total cross sections. Its predictions concerning the polarization of impact radiation emitted from the 2p state of hydrogen are compared with experiment by Ott et al. (1970) and prove to be quite good for ER 2 3 . One futher theoretical point can be made concerning the evaluation of the generalized oscillator strengths for nonhydrogenic systems. The usual way of calculating bound state wave functions is to minimize the energy. This leads to bound state wave functions which should be accurate within the region which contributes most strongly to the energy integral, but may not be

ELECTRON-ATOM EXCITATION CROSS SECTIONS

67

elsewhere. In calculating generalized oscillator strengths an alternative procedure is to develop bound state wave functions that give rise to their best value in a variational sense. Khare and Rudge (1965) adopted this approach in calculating oscillator strengths in the H- continuum. More recently Schneider (1970) has adopted a similar approach to the evaluation of generalized oscillator strengths in helium with considerable success. B. THEBORN-OPPENHEIMER APPROXIMATION The most salient defect of the first Born approximation lies in its neglect of exchange effects. This omission was remedied by Oppenheimer (1928) whose approximation for the exchange scattering amplitude consists in making the choice y p

(kp I r l ?rz) = $(P I rl) exp(ik, . r2)

(94)

~ q ( - k q l r 2 , r l=) $*(qIrdexp(-ik,.r,)

(95)

in the variational principle of Eq. (54). The corresponding scattering amplitude is given by 1 2 v 0 , , ( k p , ky)= J$(Plr1)$*(qlrz~ - - - exp i(k,. (r: r J

r2

- k; rA dr, dr2 (96)

Similar expressions can be deduced for atoms other than hydrogen. The term I/r, can equally well be replaced by I/r, in Eq. (96) though for other atomic systems, for which exact wave functions are not available, similar changes do lead to different results. The combination of the Oppenheimer exchange scattering amplitude and the Born direct scattering amplitude yields the Born-Oppenheimer approximation. At first sight the Oppenheimer expression appears to be no less correct than the Born expression and it might therefore be expected that the combined result, accounting as it does for the possibility of the incident electron being captured, would improve the first Born approximation at low energies. The facts are far otherwise, as can be judged by Table I where the results of experiment and calculation for a few, not atypical. cases are listed. Given in the table are values of the maximum cross section and the reduced energy at which this maximum is reached as given by experiment. by the first Born approximation, and by the BornOppenheimer approximation. The poorness of the Oppenheimer approximation is very striking especially for the singlet-triplet transitions in helium where, as can be seen from Eq. (80), it is only the exchange amplitude which contributes to the cross section. Before discussing the many alternative approximations which have been proposed to calculate the exchange scattering

M . R . H . Rudge

68

TABLE I A COMPARISON BETWEEN THE POSITIONS A N D MAGNITUDES OF T H E MAXIMA OF SOME CROSS SECTIONS Experiment

Cross section

QH( 1s-2~) QH( 1s - 2 ~ )

Q , , ~ ( 'I~ - 2 3 s )

Q,te( I 'S-23P)

ER

1.14 4.4 1.3 1.08

Qmm

0.16" 0.82' 0.06d 0.025"

FBA

ER

Qmax

1.33 2.4

0.25 1.3

BOA

ER 1.66b 1.2Id >1.29d

Kauppila ef a/. (1970). Bell and Moiseiwitsch (1963). McGowan e t a / . (1969). Bell ef a / . (1966). Jobe and St. John (1967).

amplitude it is worthwhile t o consider why the Oppenheimer exchange scattering amplitude should be so much worse than the Born direct scattering amplitude, when the same basic approximation, that the wave functions are products of a bound state and an undistorted plane wave, is a common feature of both approximations. One way of looking at this problem is to notice that the approximate wave function (83) is a reasonable one if rz % r, because in this region it is clear that the nuclear charge is screened by the atomic electron and polarization effects are small. In the region r2 < r l , on the other hand, this is no longer the case and the wave function in this region is very poor. The same considerations apply t o the trial function in Eq. (84). The Born scattering amplitude involves the product of two such wave functions which are plausible at least in the region rz $ r l , but poor in the region rz < r , . For small values of q, especially, the integration in Eq. (85) is dominated by the region r2 > r l . The errors in the wave functions d o not, thus, greatly influence the scattering amplitude, this being particularly the case for small q and correspondingly large E , . In the Oppenheimer approximation the situation is very different. The trial function (94) is reasonable if r2 9 rl while the trial function (95) is reasonable only if r2 @ r l . Whatever, therefore, the relative values of r2 and r, at least one of the wave functions which appears in the exchange scattering amplitude (96) is poor. This argument suggests an alternative approach t o the problem of calculating the exchange scattering amplitude. First, however, let us consider some other approximations designed to improve upon the Oppenheimer result.

69

ELECTRON-ATOM EXCITATION CROSS SECTIONS

It can be seen that for a n excitation process the term I/r, in the Born approximation, Eq. (85), makes no contribution because of the orthogonality of the atomic states. In the Oppenheimer approximation, Eq. (96), this is not so. This observation has lead some authors to advocate the removal of this term. Day et ul. (1961) have argued that alternative integral expressions can be derived for the scattering amplitude which d o not involve such a “core term’’ and that if these integral expressions are then approximated by replacing the wave functions appearing in them by plane waves, then a n expression is obtained which is identical with that of Oppenheimer but with no core term. This argument has been criticized in Section II,E. Nevertheless it can be asked whether, despite theoretical objections, simply dropping the 1 “rzterm in the Oppenheiiner amplitude proves to be a pragmatic way of improving it. Truhlar et a/. (1968) have carried out a careful study of ;i wide variety of approximations, including this one, which they term the Born-Oppenheimer-minus-core” (BOMC) approximation. Their results show it to be extremely poor and worse than the BO approximation in all the cases they have investigated. Some of their results for elastic scattering are displayed in Table 11. Further evidence of the effect of dropping core terms is provided by the work of Steelhammer and Lipsky (1970) who show that, for electron helium scattering, the retention of the core term is necessary in order to obtain a differential ctoss section of the correct shape. “

D. ORTI-IOGONALIZED BORN-OPPENIIEIMER APPROXIMATIONS An alternative expression for the exchange scattering amplitude was developed by Feenberg (1932) and found little application for many years. His scattering amplitude can be expressed variationally (cf., Heddle and Seaton, 1964; Abiodun and Seaton, 1966). On correcting a misprint in the definition of V , ( r l . r,) in the paper of Feenberg (1932), it can be seen that his scattering amplitude can be regarded as arising from the choice of trial functions, T,,(k,,l r , , r 2 ) =

$(PI

r , ) exp(ik,, . r2)

‘I),, ( - k , I rz , rl ) = $*cc/ I rd(exp - (ik, . r , ) -

.$(PI

(97) r, ))

(98)

where ci =

s $ * ( p l r) exp( - ik, r) dr

(99)

The choice of r in Eq. (99) makes t h e two trial functions orthogonal, as they are in the first Born approximation, and thereby reduces the effect of the “core term” in the BO approximation in which a = 0. From a theoretical point of

M . R. H . Rudge

70

view the OBO approximation is certainly preferable t o the BOMC approximation. There is, as remarked by Abiodun and Seaton (1966), no compelling theoretical reason to suppose that it is better than the BO approximation; however it has already been noted that this latter approximation gives results which are far too large and so it might be expected that the orthogonalized method will give some improvement. In Table 11, taken from the work of TABLE 11 THE EXCHANGESCATTERING AMPLITUDE FOR Is-1s EXCHANGE SCATTERING AT AN ANGLE OF 18"

0.005 0.05 0.15 0.25 0.35 0.5 0.72 1.125 2.0 a

-5.761 -4.011 -1.782 -0.699 -0.137 0.261 0.455 0.476 0.340

9.769 8.010 5.501 4.402 3.119 2.261 1.557 0.942 0.468

-2.861 -1.600 -0.107 0.522 0.787 0.901 0.861 0.682 0.412

2.185 0.982 0.276 0.629 0.583 0.550 0.641 0.588 0.369

Truhlar et a / . (1968). Burke and Schey (1962).

Truhlar ef al. (l968), is shown a comparison, for the elastic scattering of an electron by a hydrogen atom in its ground state at an angle of 18", between the magnitude of the exchange scattering amplitude as given by the BO, BOMC, and OBO approximations and as given by the close-coupling calculations of Burke and Schey (1962). It can be seen from the table that there is only slight improvement in the goBovalues. Elastic scattering, however, is a situation in which simple theories might be expected to be inadequate. A very similar approach to that of Feenberg (1932) was adopted by Mittleman (1962) and by Bell and Moiseiwitsch (1963). Bell and Moiseiwitsch (1963), like Feenberg (1932), develop their argument from the point of view of perturbation theory. From the variational standpoint their method is equivalent to replacing Feenberg's choice of trial function in equation (98) by the choice YJ(-kqIr2, r,) = $*(41r2){exp(-ikq

* r , )- M P l r l ) - P$I4lrl,>

(100)

where

/ = /$*(qJr)exp(-ik;r)dr

(101)

71

LLECTRON-ATOM EXCITATION CROSS SECTIONS

The trial function thus contains an additional orthogonalization term. The approximation, which has some resemblance to those of Bates (1958) and of Bassel and Gerjuoy (1960), is named the FOE, first-order exchange, approximation by Bell and Moiseiwitsch (1963), and is identical with that of Feenberg (1932) in the particular case of elastic scattering for which results have already been discussed. For excitation the comparison is more favorable. In order to judge unambiguously an approximation for the exchange scattering amplitude, it is necessary to consider cross sections, such as spin change cross sections in atomic hydrogen, or singlet-triplet transitions in helium which depend only upon the exchange scattering amplitude. In Fig. 3 the results of

7

FIG.3. The helium 1 'S-2'P cross section comparing the FOE approximation (Bell et al., 1966) with experimental data (Jobe and St. John, 1967).

Bell e / a/. (1966) for the l'S-23P cross section in helium are compared with the experimental data of Jobe and St. John (1967). The results at maximum can can be seen to be about a factor of ten too large, though this is an improvement upon the Born-Oppenheimer values. The first-order exchange approximation results for transitions between s states differ more markedly from those of the €30approximation. Bell e / a/. (1966) calculate, by the FOE method, a maximum cross section for the l'S-Z3S transition in helium of 0.14naO2as compared with the value 1.21 given by the BO approximation. Experimental results of Schulz and Fox (1957) indicate a peak value of about 0.05, those of Fleming and Higginson (1964) a value of 0.03. By using scaling relations, a value of 0.07 may be inferred from the resultsof Zapesochnyi (1967). For this cross section the FOE method is therefore much better than the BO

M. R. H. Rudge

72

approximation though, at maximum, the FOE results appear to be about a factor of three too large. It might be thought preferable to introduce additional bound-state terms into both trial functions, rather than just one of them, and to allow the coefficients of these terms to be determined variationally rather than by fixing them through orthogonality. This approach has been little pursued though Abiodun and Seaton (1966) have done so for the s-wave elastic scattering by atomic TABLE 111 VALUESOF

THE

EXCHANGE SCATTERING AMPLITUDE FOR S-WAVEELASTIC SCATTERING BY ATOMIC HYDROGEN“

Wave number k 1.o 2.0 3.0 4.0 5.0

BO 0.5000 -0.0320 -0.0360 -0.0212 -0.0125

FOE -0.1400 -0.1759 -0.0716 -0.0322 -0.0166

Variational

Accurate

-0.0461 -0.1399 -0.0661 -0.0312 -0.0164

-2.4492 -0.2431 -0.0736 -0.0311 -0.0160

Abiodun and Seaton (1966).

hydrogen. Their results are shown in Table 111, where the column headed “variational” gives the results of the method outlined, while the column headed “ accurate denotes theoretical solutions of single-state close-coupling calculations [see Abiodun and Seaton (1966)l. Though theoretically more appealing, the results show that for this case at least the variational determination of the parameters does not greatly improve matters. ”

E. THEAPPROXIMATIONS OF BONHAM, OCHKUR,

AND

RUDGE

Bonham (1962) used a Fourier transform technique to develop an approximate expression for the Oppenheimer amplitude which is valid at high energies. His result can be derived more easily as follows. Let

Then

V 1 2 F= -4n$*(qIr,) exp(ik, r l )

(103)

F = 4n exp(ik, * r,)G(rl)

(104)

Writing

ELECTRON-ATOM EXCITATION CROSS SECTIONS

73

gives the result that

kP2G = $*(q 1 r l )

+ 2ikp . V, G + V I 2G

( 105)

For large kP2 Eq. (105) can be solved iteratively through the equations

If G(r,) is approximated by GI (r,) and the I/r, term is ignored, it can be seen that the Oppenheimer amplitude is replaced by the much more simple expression

Bonham (1962) used the expression (108) to take account of exchange in computing inelastic cross sections in the energy range 10-80 keV. For such energies the procedure is a very reasonable one on two counts. First the Oppenheimer amplitude should be reliable, and second the approximation to it, made in solving Eq. (103, should also be valid. Ochkur (1964) rederived the result (108) and made the much more drastic approximation of adopting this expression for the exchange scattering amplitude at all energies. He argued that the Oppenheimer amplitude is only correct at high energies and therefore the leading term in its expansion in powers of kp’ is more correct than the full amplitude. The omission of the core term is justified on the grounds that it is of higher order in powers of k p l than the term which is retained. Clearly the expression can be developed in powers of either k,’ or k4’ and Ochkur’s prescription is that

where k , is the greater of k, and k,. Results obtained by Morrison and Rudge (1967) for the Ochkur approximation to the 11S-23P cross section in helium are shown in Fig. 4. It can be seen, by comparison with Fig. 3, that the method constitutes a remarkable improvement over the BO and FOE results. However, it is not obvious why this should be so. Tt is by no means clear theoretically why the leading term in the expansion of the Oppenheimer amplitude should be a better approximation than the full amplitude, and while the iterative scheme of Eq. (105) is a valid one for large kP2 this is certainly not the case for small kp2. Again for transitions between states with a small energy difference the result (109) predicts a very large, and, for elastic scattering, an infinite amplitude at threshold. Thus, on the one hand,

M . R . H. Rudge

74 0 08

I

I

I

I

I

I

FIG,4. The helium I 1S-23Pcross section comparing the Ochkur-Bonham and OchkurBonham-Rudge approximation (Morrison and Rudge, 1967) with experimental data (Jobe and St. John, 1967).

the approximation lacks theoretical justification but on the other, for some transitions at least, leads to quite good results. In an attempt to resolve this dichotomy Rudge (1965a,b) sought to reinterpret Ochkur's result by determining what pair of trial functions in the variational principle would give rise to the expression (109) with the view that if the approximation is a good one then this should be reflected in the trial functions. It is readily shown that the result (109) can be obtained by choosing ~ p ( k p l r l J 2=) 44Plrl) exP(ikp *

r2)

(1 10)

Equations (1 10) and (1 11) represent a pair, though not necessarily a unique pair [cf. Green (1967)], of trial functions which generate the Bonham-Ochkur result exactly. The trial functions reveal only one obvious feature, namely that they are orthogonal while the Oppenheimer functions are not. The trial function (1 11) contains two defects, the first of these being its singularity at the nodes of $(plr), the second being that it does not have, except at high energies, the proper asymptotic form. The latter feature can be remedied (Rudge 1965a,b) by replacing kp2 in Eq. (111) by [k, - i(2Zp)'/2]2 where I p

ELECTRON-ATOM EXCITATION CROSS SECTIONS

75

is the ionization energy of the initial state, the corresponding amplitude then being g;, (k, > k,) =

df;,(kp

3

kJk,

- ~(2~p~”21z

(1 12)

If the same trial functions, given by Eqs. ( I 10) and (1 11) with the appropriate change in the denominator, are used to calculate g,.,,, (- k,, -kJ, it is readily shown that reciprocity is satisfied. On the other hand if, as would be reasonable, the different trial functions

and Yq(-kqI~1Jz)=

$*(qIrJ exP(-ik, *rz)

(1 14)

were used to calculate g,,,,. ( - k,, - k,,), reciprocity would not be satisfied and the corresponding amplitude would be g4*,,*( - k, , -kp) = q’f,q (k,, , k,)/[k,

-

W,)”21z

(1 15)

This observation has led Bely (1966% 1967) to advocate replacing the denominator in Eq. (1 12) by its modulus. However, the fact that two different pairs of trial functions give results which differ in their phase is not a sufficient reason to indicate that the Bely result is necessarily better than the result (1 12). It is, however, an unattractive feature of the method that the form of the trial function for the two states is different. The result ( I 12) is in harmony with the variational principle and removes the possibility of a vanishing denominator. The effect this has on elastic scattering can be seen from Table IV, taken from Truhlar et al. (1968), where a comparison is made between the Ochkur result, the Rudge result, and accurate values of the exchange scattering amplitude. Conservation limits are exceeded by the Ochkur result, Eq. (109), but the modified values of Eq. (1 12) remain quite reasonable. Examples of inelastic transitions where the modification is important occur in calculating exchange transitions between excited states of helium. Ochkur and Bratsev (1966) have calculated transitions between the 23S state and various singlet states: the authors remark that conservation limits are exceeded in calculating, for example, 23S-2’P or 23S-31P cross sections. In Table V the effect of the modification for these cross sections is indicated. Larger changes still occur for the 2’S-23S deexcitation cross section which was calculated by Morrison and Rudge (1967) along with cross sections for excitation of the Z3S, 23P, 33S, 33P, and 33D states of helium from the ground state. Their results for the 11S-23P cross section are indicated in Fig. 4 for which transition the OB and OBR results do not differ greatly from each other but are very different

M . R. H . Rudge

76

TABLE IV A COMPARISON OF EXCHANGE SCATTERING FOR 1s-Is SCATTERING AT AN AMPLITUDES ANGLEOF 18"" Energy

goc"

lSRI

0.005 0.05 0.1 5 0.25 0.35 0.5 0.72 1.125 2.0

199.902 19.902 6.570 3.904 2.764 1.906 1.296 0.799 0.41 5

1.979 1.809 1.516 1.301 1.137 0.953 0.765 0.553 0.050

IgRSIb

2.185 0.982 0.276 0.629 0.583 0.550 0.641 0.588 0.369

See Truhlar e t a / . (1968). BS denotes Burke and Schey (1962). TABLE V 23S-n1P CROSS SECTIONS I N HELIUM AS CALCULATED IN THE OCHKUR-BONHAM (OB)" AND OCHKUR-RUDGE (OR) APPROXIMATIONS

0.0735 0.1 10 0.147 0.184 0.221 0.294 0.368 0.551 0.735 1.103 1.470 1.838 2.573 3.676

99 44 20 I1 6.3 2.6 I .4 0.40 0.17 0.050 0.021 0.01 I 0.0040 0.0014

14 9.7 5.9 3.9 2.6 1.3 0.78 0.27 0.12 0.040 0.018 0.0088 0.0036 0.0013

-

2.1 1.5 0.94 0.41 0.21 0.063 0.027 0.0080 0.0033 0.001 7 0.00063 0.00022

1.1 0.89 0.60 0.29 0.16 0.052 0.023 0.0073 0.003 1 0.00 16 0.00060 0.00021

Ochkur and Bratsev (1966).

from the BO and FOE results. As far as total exchange cross sections are concerned, therefore, the Bonham-Ochkur approximation is a great improvement over the other methods which have been discussed for transitions between states with a large energy difference. The procedure of Rudge im-

77

ELECTRON-ATOM EXCITATION CROSS SECTIONS

proves this agreement, in particular for transitions between states with a small energy difference. The cross sections which have been discussed involve the magnitude of gpq(k, , k,) only. In general the cross section depends also upon the magnitude off,,(k,, k,) and upon its phase relative to that of gpq(kp,k4). This relative phase for nonexchange transitions is of considerable importance but the phase of the scattering amplitude is given much more poorly in general by these approximations than are their magnitudes. It is for this reason that the exchange scattering amplitudes have been judged by considering exchange cross sections only. It can be argued that while a good relative phase cannot be guaranteed, it is a t least a consistent procedure to use the same form of trial function in calculating both the direct and exchange scattering amplitudes. This approach was used by Morrison and Rudge (1966) in evaluating excitation cross sections for atomic hydrogen. They chose one of their trial functions to be that of Eq. (110) while the other was taken to be

in calculating the direct scattering amplitude, the exchange amplitude being calculated with Y,( - k q l r 2 , r,). In Fig. 5 their results are compared with 0 2(

0.15

0 OIC

0 0:

I

1

1

,

1

,

20

30

40

50

60

70

,

,

80

1

90 100

,

110

I

I

I

120 130 140

3

ER

FIG.5. The hydrogen Is-2s cross section comparing the Born, Ochkur-Born, and Morrison-Rudge theoretical values (Morrison and Rudge, 1967; Morrison, 1968) with experimental data (Kauppila et al., 1970).

78

M . R . H . Rudge

experimental data for the Is-2s transition in atomic hydrogen. These experimental data are taken from the work of Kauppila et af. (1970) from which a cascade contribution, as calculated theoretically by Morrison and Rudge (1966), has been subtracted. Also displayed in the figure are Born and BornOchkur results (Morrison, 1968). In Fig. 6 a similar comparison is shown for

FIG.6. The hydrogen Is-2p cross section comparing the Born, Born-Ochkur, and Morrison-Rudge theoretical values (Morrison and Rudge, 1966) with experimental data (Long et al., 1968; McGowan et al., 1969).

the 1s-2p excitation cross section in hydrogen. In this case the experimental data are those of McGowan et al. (1969) and of Long et al. ( I 968). The latter data were normalized to Born’s approximation at 200 eV; as has been mentioned, there is some doubt concerning this procedure. Again a cascade contribution has been subtracted. In Fig. 7 the polarization of the impact radiation as given theoretically by the Born-Ochkur amplitudes and by the procedure of Morrison and Rudge (1966) is compared with the experimental data of Ott et al. (1970). It can be seen that the total cross section predictions are given extremely well by the Morrison-Rudge method ; the Born-Ochkur method is little different from that of the Born approximation. As far as the relative population of the magnetic sublevels is concerned, as indicated by the polarization P in Fig. 7, the situation is reversed, the Born approximation giving a good estimate while the Morrison-Rudge technique is very poor in this regard. If $ ( p I r) is an excited state of atomic hydrogen then the trial function ( 1 16) suffers the further drawback of being infinite at the nodes of ~)(plr) though

79

00-

02-

\

I

I

I

I

40

50

I

60

\\ \p

MORR -RUOGE

FIG.7. Polarization of H(I s-2p) impact radiation comparing Born-Ochkur and Morrison-Rudge values (Morrison, 1968) with experimental data (Ott et al., 1970).

the scattering amplitude, if treated as a principal value, remains finite. This feature suggests that Eq. ( 1 16) should not be used in this case. Calculations by Van Blerkom (1968) confirm this. Turning to the calculation of differential cross sections it can be seen that for nonexchange transitions the Born-Ochkur method predicts spurious zeros in the differential cross section where (2/q2 - I/kP2)= 0 (Truhlar er al., 1971). Similar spurious zeros occur if the method of Bely (1966a, 1967) is used. They d o not occur if Eq. ( 1 12) is used because of the complex denominator. Turning to pure exchange transitions, Vriens et al. (1968) have measured differential cross sections for the 1'S-23S cross section. They found that the differential cross section peaks in the forward direction and disagrees in this respect with the Ochkur calculations of Miller and Krauss (1968) who used Hartree-Fock wave functions. Calculations of Morrison (l968), using more accurate wave functions, confirm the conclusion that while the experimental results peak within 5" of the forward direction, the theoretical results peak typically near 30'. The matter was considered further by Steelhammer and Lipsky (1970). They showed that, for this case, better agreement with the differential cross section could be obtained by retaining terms other than the first in the approximation which Bonham made to the Oppenheimer amplitude. It can be seen therefore that, while the Bonham-Ochkur-Rudge approximations predict much better total exchange cross sections than the BO, BOMC, or FOE methods, they do so at the expense of removing too much flux from the forward direction.

80

M . R.H . Rudge

F. FIRST-ORDER DISTORTION APPROXIMATIONS In comparing theory with experiment for the excitation of the 2lP state in helium, Truhlar et al. (1971) do not find that the inclusion of exchange, by any of the methods which have been discussed, remedies the defects in the first Born approximation at large scattering angles. The experimental evidence of Truhlar et al. (1971) has been confirmed by Opal and Beaty (1972). It is reasonable to suppose that the distortion of the plane wave plays a more important role in describing the large angle scattering than in describing the small angle scattering. A simple modification of the Born approximation is the distorted wave (DW) approximation, sometimes referred to as the distorted wave Born approximation (DWBA). For e-H scattering the method is

The physical basis of the method therefore is that the motion of the free electron is influenced by only a partially screened nuclear field whereas in the FBA the net effect of the nucleus-electron and electron-electron potential is taken to be zero, i.e., total screening. The method takes no account of the effect of the distortion of the atomic states which in turn produces additional polarization potentials that influence the motion of the free electron. This effect is accounted for, in principle at least, by second-order treatments of the problem. The DW method can be extended to scattering by other systems in an obvious manner. In particular Madison and Shelton (1971) have used this technique to compute differential cross sections for the excitation of the 2'P state in helium. One of their sets of results (which they label FG) is depicted in Fig. 8 and compared with experimental data. It can be seen that there is good agreement at angles near 90". Total cross sections cal-

ELECTRON-ATOM EXCITATION CROSS SECTIONS

I

I

I

I

81

I

FIG. 8. Differential

helium

(1 'S-2'P) cross sections as given

theoretically (Madison and Shelton, 1971) and experimentally (Truhlar et a / . , 1971; Opal and Beaty, 1972). E = 82 eV.

I

1

I

30

60

90

I

120

I

I50

0

Scattering angle (degrees1

culated by the DW method are typically too large, indicating a deficiency in the method at small scattering angles. A different approach has been advocated by Geltman and Hidalgo (1971 ; Hidalgo and Geltman, 1972). Referring to Eq. (68) they argue that the FBA results if the exact wave function, ~ , , ( k p l r l , r 2 )is replaced by $(plr,) exp(ik, . r2) and if Z is chosen equal to zero. They then advance a n alternative, the " Coulomb projected Born approximation," which results from Eq. (68) by discarding the phase function, by making the same replacement of the exact wave function but by now choosing 2 equal to unity. The scattering amplitude is therefore given by

fp4(

k

,. kq) = -(2n)-'

1

/ $ ( p l r l ) $ * ( d r l ) - exp(ik,, . r2)x(l, -kqlr2)dr, dr, rI 2 (123)

The development of the method is not, therefore, a variational one and the two states appearing in the approximation are treated asymmetrically; in theinitial state thenuclearfield is treated as though it were completely screened

M . R . H . Rudge

82

while in the final state it is treated as though it were completely unscreened. Differential cross sections computed by Hidalgo and Geltman (1972) for the l'S-2lP transition in helium are shown in Fig. 9 from which it is apparent that the method does, however, yield better differential cross sections at large angles than does the Born method. It is not clear whether or not this advantage is gained at the expense of a worse differential cross section at small scattering angles. Calculations of the total cross section are needed to ascertain this. 0.0

-1.0

2

-20

N

FIG. 9. Differential cross sections for the 1's-2'P transition in helium as given theoretically by Hidalgo and Geltman (1972), by the Kang-Foland method (Truhlar et al., 1971), and experimentally (Truhlar et al., 1971; Opal and Beaty, 1972). E = 82 eV.

0

1

-0

Lo s Y)

-3.0

V

+.

r

n 2-4.0

I

J 0

-5.0

-60

I

30

I

I

I

I

60

90

120

150

I

Scattering angle (degrees)

A symmetrical version of the treatment of Geltman and Hidlago was advanced by Kang and Foland (1967a,b), who chose to approximate &he integral expression (68) by discarding the phase factor, choosing Z = 1, and by replacing Y,(k, I r l , r2) by $ ( p I r,)X*( 1, - k, 1 rz). This method has been used by Truhlar et al. (1971) to compute differential cross sections for the l'S-2lP transition the results again being shown in Fig. 9. It can be seen that this more symmetrical Coulomb projected Born approximation yields very poor results at large scattering angles.

ELECTRON-ATOM EXCITATION CROSS SECTIONS

83

It is clear, therefore, from the work of Madison and Shelton (1971) that allowance for distortion effects plays an important role in the description of large angle scattering. The work of Geltman and Hidalgo shows that the use of Coulomb waves in the description of one of the states also improves the description of the large angle scattering. It is not clear, however, why one state should be described differently from the other or why, when they are treated symmetrically, as in the treatment of Kang and Foland (1967a,b), this improvement is lost.

G. SECOND-ORDER APPROXIMATIONS The first Born approximation neglects not only exchange effects but also those effects due to the polarization of the atom by the incoming electron and the effect that this has in distorting the plane wave. The latter effects can be studied by means of the second or higher order Born approximations. Even the second Born approximation, however, has not proved to be capable of being evaluated exactly and so further approximation is required. A variety of treatments have been recently proposed (Glauber, 1958; Holt and Moiseiwitsch, 1968; Birman and Rosendorff, 1969; Bonham, 1971a,b; Bransden and Coleman, 1972; Bransden et af.,1972). They share the common feature of using the closure approximation introduced by Massey and Mohr (1934). On writing y & b I J 2 )

=

1 $(ilrI)Fi(r2)

( 124)

i

the functions Fi(r)can be seen to satisfy the coupled equations

(V2 + ki2)Fi(r) = 2 C .i

vij(r)Fj (r)

(125)

where F,(r)

N

6,,exp(ik,. r) +fpi(k,,P)r-' exp(ik,r)

( 126)

r + ru

An iterative solution of equations (125) is given by the scheme (V2 + ki2)FI")(r)= 2 C Vij(r)Fjfl-')(r)

(127)

j

where F;')(r) = Ji, exp(ik, . r)

( 128)

These iterations constitute the various orders of Born approximations. In particular the second Born approximation corresponds to the equations (Vz

+ k , z ) F : 2 ' ( r )= 4

Vij(r) jG(kj21r,r')Vj,(r')exp(ik, j

r')dr' (129)

M . R. H . Rudge

84

where the Green function is given by G(kj2I r, r’) = (27~)=

lim

,,+o+

- (4n)-’

.r

exp ik . (r - r‘) dk (k; + iq - k 2 )

exp ikjlr - r‘I Ir - r‘I

No technique has been advanced whereby the infinite sum in Eq. (129) can be evaluated exactly. It is therefore convenient to rewrite it in the form N

(V2

+ ki2)F‘:’’(r) = 4 1 Vij(r)~C(kj21r,r’)Vj,,(r’)exp(ikp.r’)dr’ +R(r) j= 1

(132) where R(r) comprises the remaining terms. This remainder can be approximated in the manner of Massey and Mohr (1934) by replacing the Green functions G(kj2 [r, r’) by a single function C( 6’ I r, r’). The summation can then be performed by using the closure property of the bound states, to give the result that

The equations (132) then become N

(V2

+ ki2)F12’(r) = 4 1 Vij(r) sG(kj2 Ir, r’)Vjp(r’) exp(ik,.

r’) dr’

j = 1

+ 4 J K i p(r, r’) exp(ik,

*

r’) dr’

(135)

where

The treatment of elastic scattering presented by Massey and Mohr (1934) corresponds to using Eq. (135) with N = 1 and k 2 = k 1 2 .This can be shown to give rise to a logarithmic divergence in the imaginary part of the forward scattering amplitude (Moiseiwitsch and Williams, 1959). One procedure which avoids this difficulty is the truncated second Born approximation wherein R(r) is set equal to zero; it has been used by Kingston et al. ( I 960), Kingston and Skinner (1961), and Moiseiwitsch and Perrin (1965).

ELECTRON-ATOM EXCITATION CROSS SECTIONS

85

A second procedure, termed the simplified second Born approximation by Holt and Moiseiwitsch (1968), uses Eq. (135) with the choice

E2 =ki+,

(137) A third and more elaborate scheme has been advanced by Bransden and Coleman (1972). They make no use of the Born iterations but do make use of the closure technique. They write the equations (125) in the approximate form

In this method the equations for states 1 to N are treated exactly and the remaining states are included only insofar as the coupling between these and the first N states is retained. Bransden e f al. (1972) discuss various possible choices of the mean energy E2 which enters into the kernel of Eq. (138). A fourth method is that due to Glauber (1958) an adaptation of whose approach is also used by Bransden and Coleman (1972) in solving Eq. (138). It emerges as a type of impact parameter approximation but bears some similarity to the other second-order treatments. We follow the development of Birman and Rosendorff ( I 969). Starting from the exact integral equation for the wave function given by Y P ( k P h J 2 ) = $(plr,) exP(ikp *

r2)

+ 2 i1

J$cil~l)$*cilr,~~

x G(kjz I r2 , r,’) V(r,’, r2’)Y(kplr1’,r2’) dr,’dr,’

(139)

the following substitution is made: y p (kp

I r1, r2> = M

I 9 r2)

exP(ikp * rz)

(140)

It follows, from Eq. (139), that

4(rl,r2) = +(Plr,)

+ 2 c J $ ( j l r l ~ ~ * ( j l rexp l ~ )- ikp. i

(rz -rz‘)

x V(rl’,r2’)G(kj2Ir, , r2’)rj5(r,’,rz‘)dr,’ dr,’

(141)

If, once more, the Green functions are approximated by setting kj2 = kP2 and the closure relation then used, on defining, R

= r2 - r,’

( 142)

Eq. (136) reduces to

m, f*) = 9

1

$ ( ~ ~ ~ ~ ) - j - - ~ R - ’ V (-R)d)(r,,r2 r,,r~ -R)expi(k,R-

k;R)dR

(143)

M . R. H . Rudge

86

For large k , the integral in Eq. (143) can be estimated by using the plane wave version of formula (8) and omitting, as is justified for large k,, the term involving a(%, + 8).The resulting equation is i

Jm

4(ri9r2)= $klri) - V r i 3 r 2 - RkP)4(r1,r2 - RG,) dR (144) k, 0 On choosing kp to be the z axis, a solution of Eq. (144) can then be written

where ( p 2 , & , z2) are the cylindrical polar coordinates of the vector r 2 . The Glauber approximation now results from using Eqs. (140) and (145) in the integral expression (63). The result thus obtained is,

Equation (146) is a complicated six-dimensional integral and can be further approximated. On writing

r2 = b +z2&, (147) the integrand in Eq. (146) can be seen to contain the function exp(iz2q . k,). Now q . k,

= k,

2

- k, * k,

(148)

which is approximately zero for large k,(k, N" k,) and small scattering angles. Glauber therefore replaces the function exp(iz2q &), by unity, in which case a considerable simplification of Eq. (146) ensues, since the integration over z2 can be performed explicitly to give the result that

f,, (k, kq) = (2x1- lik, J- $kJIr1)$*(q I r1)l-(r13 3

P2

9

42)

exp(iq * b) dr1 d2b (149)

where

r = 1 - exp(i;y)

( 150)

with X=

1 " --/ k,

Ur1,p94,z)dz

(151)

--OD

It can be seen that the Glauber approximation involves a number of subsidiary approximations, the overall effect of which can only be judged by a

87

ELECTRON-ATOM EXCITATION CROSS SECTIONS

comparison of detailed calculations with experimental data. A few general remarks can be made, however. First, it can be seen that the method of approximating the Green functions leads to the same difficulty for elastic scattering as that found by Moiseiwitsch and Williams (1959), namely a logarithmic divergence in the imaginary part of the forward scattering amplitude. This difficulty can be circumvented by making an alternative choice of E2 as has been discussed by Birman and Rosendorff ( 1969) and by Joachain and Mittleman (1971). Second, it can be seen that the reduction of Eq. (146) to the form (149) involves the assumption that q * kp = 0. If the quantization axis is taken along k, the result (149) can be shown to lead to a zero cross section for excitation of the m ,= 0 sublevel of the 2p state in hydrogen. The predicted polarization radiation is in consequence quite wrong, ( P = -3/11). Byron (1971) has evaluated the more accurate form of the method given by Eq. (146) and finds that the predicted polarization is in that case in quite good agreement with experimental values. The evaluation of Eq. (146) is difficult, however, being a six-dimensional integral, which Byron (1971) evaluated by a Monte Carlo technique, while Eq. (149) can be reduced to a very much more simple twodimensional integral. A reasonable prediction of the impact radiation polarization can be achieved within the framework of the approximation (149), if rather than choosing k, to be quantization axis and assuming that q * k, = 0, instead i is chosen to be the quantization axis where t q = 0 (Gerjuoy et ul., 1972). The vector Z lies symmetrically between k, and k,, and its designation as quantization axis has also been advocated by Chen er al. (1972) in an " eikonalized " treatment of the distorted wave method. We turn now to the results of numerical calculations ; Table VI taken from TABLE VI CROSSSECTIONS FOR

THE

Is-2s EXCITATION CROSSSECTION OF ATOMIC HYDROGEN" TSBA'

SSBA'

Ei

FBA

N=3

N=4

N=5

N=3

N=4

N=5

2.0 4.5 8.0 12.5

0.1019 0.0476 0.0272 0.0175

0.0876 0.0438 0.0259 0.0170

0.0884 0.0442 0.0261 0.0171

0.0906 0.0447 0.0262 0.0171

0.0837 0.0447 0.0265 0.0173

0.0822 0.0444 0.0264 0.0173

0.0814 0.0443 0.0264 0.0173

Holt and Moiseiwitsch (1968). Truncated second Born approximation. Simplified second Born approximation.

M . R . H . Rudge

88

the work of Holt and Moiseiwitsch (1968), shows a comparison for various values of N between the first Born and the truncated and the simplified second Born approximations for the ls-2s cross section in atomic hydrogen. The difference between the various second Born methods is slight but it can be seen from Table VI, and from Fig. 10, that for this case the second Born 0 20

I

1

1

l

l

l

l

1

I

I

I

I

I

\ 0 15

P 0 10

0 05

I

20

1

30

I

40

I

50

I

60

I

70

I

80

90 100

I

I

110

120

0

ER

FIG.10. The H(ls-2s) cross section as given experimentally (Kauppila et al., 1970), by the FBA, SSBA (Holt and Moiseiwitsch, 1968), by the Glauber approximation (Tai etal., 1970), and by Bransden et al. (1972).

methods are an improvement over the FBA. For the ls-2p transition in hydrogen the results of Holt and Moiseiwitsch (1968) are closer to the FBA. For the l'S-2lP cross section in helium, Holt er al. (1971) find that the second Born and first Born results are in close agreement for E R 2 10 but that the second Born results exceed the first Born for E R < 8. The authors conclude that the method is not valid below E R w 8 because of the slow convergence of the Born series and the neglect of exchange effects. Figure 10 also illustrates the results of the Glauber calculations of Tai et al. (1970) and the results of Bransden et al. (1972) for the ls-2s cross section in atomic hydrogen. It can be seen that the three " second-order" theories depicted are in good agreement with the experimental data of Kauppila et al. (1970) for ER 2 6 . Below this energy there are significant departures; the results of Bransden er al. (1972) show the closest agreement with the shape of the experimental curve. Figure 11 shows a similar comparison of theory with experiment for the ls-2p cross section. Glauber calculations have been carried out by both Tai

ELECTRON-ATOM EXCITATION CROSS SECTIONS

89

20

45

05

FIG.1 1 . The H(1s-2p) cross section as given theoretically by the FBA, SSBA (Holt and Moiseiwitsch, 1968), Glauber approximation (Tai et al., 1970), by Bransden et al., (1972), and experimentally (Long ef al., 1968, McGowan et al., 1969) with correction for polarization of the radiation (Ott et al., 1970) and cascade (Morrison and Rudge, 1966).

et al. (1970) and Ghosh et al. (1970) for this cross section. Here it can be seen

that the Glauber approximation gives better agreement with the experimental data than do the other methods. Tai et al. (1970) also find, despite the fact that the Glauber technique would seem to be a small angle approximation, that its predictions of large angle differential cross sections are in satisfactory agreement with experimental data. The Glauber approximation therefore enjoys some degree of success though it makes no allowance for exchange effects and does not admit of obvious systematic improvements. In the latter regard, the method of Bransden and Coleman (1972) is preferable.

H . THEMETHOD OF REGIONAL TRIALFUNCTIONS The methods which have been discussed previously have for the most part been based upon the Kohn variational principles and have all used wave functions which are continuous and have continuous derivatives. A disadvantage of these procedures lies in the fact that it is often possible to write down a functional form of the wave function which, while adequate in certain regions, is wholly inadequate in others. Thus, as has been noted in Section III,B, the trial function (83) is a reasonable one if r2 9 r , but is entirely unjustified if r2 < r l . It is consequently to be expected that the FBA will be

90

M . R . H. Rudge

poor at large scattering angles and that the BOA will be unreliable. The obvious remedy lies in the attempt to ensure that the chosen form of trial function is only used throughout the region in which it is expected to be a reasonable approximation to the true wave function. Since it is not the wave function itself which is required, but only those aspects of it which relate to the calculation of cross sections, the trial functions could, in general, be be chosen without regard to the continuity requirements satisfied by the exact wave function. Little use has been made of more general functions of this type in scattering theory, though their application has been considered in calculations of bound state energies (see McCavert and Rudge, 1972a, and references therein) and in solid state calculations (Kohn, 1952; Kohn and Rostoker, 1954; Leigh, 1956; Bevensee, 1961; Schlosser and Marcus, 1963). A difficulty which is encountered in using discontinuous trial functions can be illustrated if a trial function which contains step functions is introduced into the Kohn variational principle (53). It can be seen that the integrand in the variational expression then contains products of a step function with the derivative of a delta function and that these products give rise to divergent integrals. The first attempt to use discontinuous functions in scattering theory was that of Layzer (1951), and his method, though not a variational one, suffers from this difficulty. The Layzer method has been studied by opik (1955). Temkin (1962) considered the s-wave e-H elastic scattering problem by using a trial function which had the form

wr,, rz) =

(I1 1 2 ) -

l4(r,

9

rMr2

- rl)

(1 52)

where q is the unit step function and the function 4 contains linear parameters. Temkin (1962) chose these parameters by minimizing the integrals I(S ) , where

d/an being the normal derivative to the line rl = rz in the ( r l , r z ) plane. This method encounters no difficulty from the step functions and the condition H(S)=0

(155)

generates solutions which satisfy the boundary conditions. The condition (155) does not of itself, however, guarantee that these solutions are solutions of the Schrodinger equation and so is incomplete as a variational principle. This difficulty was circumvented by Temkin (1962) through a careful choice of 4 but if, for example in the triplet case, 4 were chosen to be of the form 4(rI,r 2)

=

mdmJ

(156)

ELECTRON-ATOM EXCITATION CROSS SECTIONS

91

where P(r,) is the ground state radial function of the hydrogen atom, then the condition (1 55) would yield only the trivial solution for the function F(rz). In order to be able to use discontinuous regional trial functions a variational principle is therefore needed which does not give rise to divergent integrals and for which an inverse principle can be established which guarantees that the solutions of the variational principle are solutions of the Schrodinger equation which satisfy the boundary conditions. Such a principle for the scattering amplitudes was given by Rudge (1970a) and is described in Section II,D. In order to give some guide as to the possible utility of the method, McCavert and Rudge (1970)considered the scattering of an electron by atomic hydrogen or by the alkalis Li, Na, and K, using as trial functions

y p(kpI rl r2) = $CP I r*)exp(ikp . rz), = 0,

r2 > r1 rz < rl

f

y q ( - k q l r , , r z ) = $*14lrl)exp(-jk;r2),

YZ > Y I

(157) (158)

rz < rl

= 0,

Clearly, setting the trial functions equal to zero in one of the regions is a crude approximation which could be readily improved upon. However, Eqs. (157) and (158) together with the variational principles (50) and (52) yield very simple expressions for the scattering amplitudes and give some guidance concerning the merit of the approach. Figure 12 depicts the results obtained by McCavert and Rudge (1970) for the 1s-2p cross section in atomic hydrogen, a comparison being shown with 20

15

First Born 7

la 10

05

1

1

1

1

1

1

1

20

30

40

50

60

70

80

1

1

9 0 100

1

410

1

I20

ER

FIG.12. The H(I s-2p) cross section as given experimentally (see Fig. 1 I), by the FBA, and by the regional approximation of McCavert and Rudge (1970).

M . R . H . Rudge

92

first Born and experimental results. At high enough energies the results of the regional plane wave and the first Born approximation agree for this cross section though it is apparent from the figure that at E R = 10 there is still a substantial difference. Figure 13 shows a comparison between computed and experimentally observed values of the polarization of the Lyman radiation ; above E R = 2.5 there is satisfactory agreement.

04

I

1

I

1

p

03 -

I

011 el a1

(1969)

0 2 i

OI

P-

-

O, 0

I

I

I

I

I

20

30

40

50

60

70

ER

FIG.13. Polarization of H(ls-2p) impact radiation as given experimentally (Ott et al., 1970) and in the regional approximation (McCavert and Rudge, 1970).

Figure 14 illustrates results obtained from the Na(3s-3p) cross section. Values computed by McCavert and Rudge (1970) are compared with FBA values (Vainshtein et al., 1965) and with experimental data (Enemark and Gallagher, 1972). In this case the regional trial function method again gives better results than the FBA but there remains a very substantial disagreement with experimental data. This disparity reflects the need in computing cross sections for a strong transition to use a partial wave analysis. The general picture, however, which emerges from these and other calculations of McCavert and Rudge (1970) indicates that the basic idea is not an unreasonable one and that the trial functions (157) and (158) can serve as a basis for further improvements. One systematic scheme whereby such improvements can be effected is described in Section IV,F, while results of a unitarized scheme that makes partial allowance for coupling effects is described in Section V,A.

93

ELECTRON-ATOM EXCITATION CROSS SECTIONS

Experiment

-

1

I

1

I

I

I

IV. Low Energy Theory In the low energy regime the effects of exchange, polarization, and distortion are all of importance. In order to account for these effects adequately it is necessary to seek more refined trial functions than those considered in trying to describe the high energy regime. In order to do this it is convenient to decompose the scattering amplitude into an expansion in spherical harmonics and to then find approximations for these components rather than for the full amplitude. The theory has been developed by Blatt and Biedenham (1952) and by Percival and Seaton (1957). A. PARTIAL WAVEBOUNDARYCONDITIONS A N D CROSS-SECTION EXPRESSIONS Let us again restrict ourselves to the e-H problem since the extension of the discussion to more general problems is straightforward. Let y stand for the set of quantum numbers (SM,nl, m,1, mz)and let j be a channel index in terms of which the quantum numbers which comprise y are parametrized. For ease of notation the quantum numbers which refer to the entrance channel denoted by p , will be primed while those which constitute an arbitrary exit

94

M . R . H . Rudge

channel j will be unprimed. Let ,j,(kr) and n,(kr) be functions which are related to the spherical Bessel functions and are defined as solutions of the equations

and are such that lim (kr)-('+')j,(kr) = constant r-10

j , (kr)

N

k-

1 / 2 sin(kr

- ih)

r-+m

lim (kr)'n,(kr) = constant r-rO

n,(kr)

-

k-'I2 cos(kr - fln)

r-tm

Referring to Section II,B we expand the various quantities which appear in the asymptotic form of the wave function thus

fpj(SIk,, P2) = 2ni(k,kj)-'l2

C

~ ' z ' ~ ~ z Y ~ , m z ( P ~ ) Y(164) ~~~z~(kp)~

12mz lz'mz'

The external factor and the internal phase factor in Eq. (164) are introduced for subsequent convenience and Tpj is the representative of the scattering amplitude in the basis of spherical harmonics. TPiis diagonal in the quantum numbers (S, Ms), and is an element of the transmission matrix T. It follows from the above equations and from the boundary conditions of Section II,B that in the representation y,

where $ 3 r 1 , r 2 ) = (rl r2)t-I CP(nl1 l r l ) Y , ~ m ~ ( ~ ~ ) Y i , m ~ ( ~ 2 ) i

x ( ( 2 ~ 5, ~T$;)jI2(kjr2) +iT$; nf2(kjr2)}(166) The T matrix is diagonal in the total angular momentum L and so it is convenient to introduce an alternative representation r = {SM, LML nl,12}. Then $3r1,r2) =

1 (11m112mzl ~ M L M ~ ) ( r l , r 2 )

mum

( 167)

95

ELECTRON-ATOM EXCITATION CROSS SECTIONS

and conversely

Defining the coupled spherical harmonics I2 LML

I

P2)

=

c (l1m

I LM,) ~1

1 1 2 ~ 2

mimi

Iml

(PI) yitrn, (P2)

(169)

it follows from Eqs. (166) and (167) that $ 3 1 J 2 )

0 - 1 r21-I

c P ( 4 Irl)Y(ll LLMLIP,,P2) j

rzlrt+m

(170) where the channel index now labels the quantum numbers which constitute r and where x {(2hPj- T$))j,z(kjr2)+iT';;)n,,(kjr2)}

Conversely

Ti:) =

c (l1m1l2m 2~ L M L ) T ~ ) ( l l ' m l ' 1 2 ' m 2 ' ~ L ~ ~(172) )

LML

The total cross section for excitation from state (n'l,') to (nl,) is given by

On substituting Eq. (172) into Eq. (164) the summations over the ClebschGordan coefficients can be carried out using their orthogonality properties to give the result that

independent of kp . Cross sections for excitation of the individual magnetic sublevels, however, are dependent upon k, . Further cross-section expressions are derived in the papers of Percival and Seaton (1957) and of Blatt and Biedernharn (1952). We shall henceforth use the representation r and therefore omit the superscripts referring to this. The representation I' can also be taken to include a further conserved quantity, the parity Il = (- I)"+'*. It follows from the properties of the scattering amplitudes which have been derived that the elements of the T matrix are not independent. Thus substituting Eq. (164) into the result (62) shows that

T-T=O

(175)

A similar substitution into the result (39) shows further that

T + T+ = TT+

(1 76)

96

M. R. H . Rudge

In these matrix equations the indices run only over the open channels (kj2> 0). Defining the scattering, or S matrix through the equation

T=I-S

(1 77)

where I is the unit matrix, it follows from (175) and (176), that

S=S and that

ss+ = I

In general one might seek solutions $p(rl,rz) which, instead of having the form (170), are chosen to satisfy the boundary conditions

where the matrix elements apj and bpi are real. The T matrix can then be related to the matrices a and b through the equation

T = -2i(a-'b)(I - i(a-'b))-'

(181) If a is chosen to be the unit matrix then b is written R (or sometimes K) and called the reaction matrix. From the properties of T it then follows that

R=R* R=R It follows conversely that any approximation which generates a real and symmetric R matrix gives a unitary and symmetric S matrix which implies that reciprocity is obeyed and the conservation theorem is satisfied in the sense that the incoming flux is balanced by the outgoing flux in those channels which have been included in the calculation. If these do not include all the open channels, it follows that the flux which, in an exact calculation would go into the channels which have been omitted, is being transferred to those channels which have been included. Since the R matrix is real and symmetric it can be diagonalized by an orthogonal matrix X in the form

R

= %(tan q)X

(183) where the matrix tan q is diagonal with entries tan qj which are the eigenvalues of the R matrix. The notation corresponds to the form which the matrix takes in the single channel case and the qj are termed the eigenphase shifts. A second normalization which is useful in discussing threshold behavior was introduced by Ross and Shaw (1961) and corresponds to writing b = k'+'/z (184)

a = Mk-(l+1/2)

(185)

ELECTRON-ATOM EXCITATION CROSS SECTIONS

97

where k'"'2 is the diagonal matrix with entries

(k'+1/2)ij= 6 i j k j 1 2 ( j ) + ' / 2 The T matrix is then given in terms of the M matrix by the equation T = -2ik'+'/2 (M - jk21+1)-1kl+1/2

(1 86)

(187)

B. VARIATIONAL PRINCIPLES The determination of the cross sections depends on calculating the T matrix which in turn is expressed through Eq. (181) in terms of the matrices a and b. We choose the normalization a = 2-1/21

in Eq. (180) and seek a variational principle for the R matrix. As discussed in Section (11, D) greater generality is achieved by using regional trial functions and we therefore follow the treatment of Rudge (1970a,b, 1972). With the regions defined as in Section (II,D) let

Using Green's theorem it follows that JP4 (i)

-

46Rp, k + jOmr4drSdP, d?2{$!(i)Dd$:)

- 81);

D$:(i)}r,=r2=r(193)

where the minus sign occurs if i = 1, and the plus sign if i = 2. If the trial functions are continuous and have continuous first derivatives, the integral in Eq. (193) vanishes. The integrals I,, also vanish and so the variational principle in this case is given by 6(Lp,- +Rp4)= 0

(194)

M . R . H . Rudge

98 Conversely, if t,hp and

$q

are solutions of the variational principle (194) then

Z, must vanish; this implies that the solutions of the variational principle are solutions of the Schrodinger equation. If the trial solutions are not continuous the integral of Eq. (193) is of first order and an additional term is therefore needed to compensate. Define

+ 1 r4 dr SdP, dP2{($',2)- $r))D($:(') + m

Kp4=

0

- ($:(')

I),*('))

+ $:('))N$',2)- $r))}r,=rz=r (195)

Then 2

a,

+ Kp4- tR,) = i = 1(1:; + 1 ~ ~+a' ) )0 r4 dr idPl dP2{($',2)- $y)) x D(6$;(') + s $ y ) ) + ($y - p ) D ( 8 $ p + S$',2)) + ( 6 $ y + Sip)

6(L,

x D($y -

$62)) + (S$b" + d $ : ) ) D ( $ y

- $;(2))}r,=rz=r

(196)

Since the exact solution satisfies the conditions

it follows that a variational principle in this case is given by a(Lpq+ Kpq- f R P J = 0

(198)

The converse follows, namely that the solution for the variational principle (198) satisfies the Schrodinger equation and the equations (197). The variational principle (198) is not unique. It can be seen that integrals can be defined similar to those of Eqs. (153) and (154) which have second-order variations. For example if a,

1, = Jo r4dr JdP, dP,(($F' - $f))($:(2) Z2 = JOmr4drJdP, d P , ( D ( $ ~ ) $b")D($:")

then 61, = 61, = 0

- $q*(l)))rl=rz=r

- $q*(')))rl=rz=r

(199) (200)

ELECTRON-ATOM EXCITATION CROSS SECTIONS

99

Similarly the integrals

and

have second-order variations 61, = 61, = 0

More generally therefore

+

6(Lp, Kp4- t R p ,

+ 1 CjZj) = 0

(205)

j

where the Cj are arbitrary constants and the Zj are any functionals, of which the above are examples, that have a zero first variation. It is of considerable theoretical interest to decide whether or not the variational principles are extremum principles. If so, the usefulness of the principle is greatly enhanced especially if the trial functions contain nonlinear parameters. This question is considered in Section IV,E.

C. RESONANCE THEORY

A feature of low energy electron-atom collisions is the occurrence of resonances in both elastic scattering and excitation cross sections. It is appropriate therefore to briefly consider the theory of resonance scattering. More detailed reviews have been given by Burke (1968) and by Smith (1966). It is instructive to first consider the theory of single channel or shape resonances. Let V be a potential such that V(r) = 0 ,

r >R

(206)

and consider s-wave scattering by this potential. The radial wave function satisfies the equation

+

[(d2/dr2) k2 - 2V(r)]$(r)

=0

(207)

The phase shift 9 can be written in the form 9 = ?o

+ ?1

where 90 =

-kR

(208)

M . R . H . Rudge

100

is the phase shift which arises from scattering by a hard core potential and k cot

' 11

= $'(R)/$(R)

(210)

is the phase shift which represents the effect of the potential for r < R. In the region r < R Eq. (207) can be solved subject to either of the two sets of boundary conditions

or

These boundary conditions are not those appropriate to the scattering problem but generate eigenfunctions 4 1 ( n l r ) and eigenvalues E,(n) where 1. = 1 refers to the boundary conditions (21 1) and 1, = 2 to those of Eq. (212). The wave function can be written

It follows from Eq. (210) that if

k 2 = 2E1(n) (214) corresponding to an energy equal to one of the pseudobound states then I ] ~is a multiple of n while if k'

= 2E2(n)

(215)

then I ] ~= n/2. The solution of Eq. (207) subject to the boundary conditions (212) thus generates a set of energies at which there exists a resonance for the q1 part of the phase shift. Near one of these energies, denoted by E,, we can write Eq. (210) as cot

I]1 = ( E - E M

r,/2)

(2 16)

where 2 / r , = d/dE(cot Ill)& (217) The partial cross section described by the phase shift I ] ~then takes the BreitWigner form

The true partial cross section will exhibit a resonance at an energy E,' for which I],, + I], = n/2.

ELECTRON-ATOM EXCITATION CROSS SECTIONS

101

If (E,’ - E,) is small it is given by (E,’ - E,)

= E,r,R/[(2E,)1’2

+ T1R/2]

(219)

Writing this energy shift as A, gives the form of the partial cross section as

It can be seen that the definition of the resonant energy E, is in some measure arbitrary since it depends upon the arbitrary, though convenient, division of the phase shift given by Eq. (2081. Er could be defined otherwise; if so the expression for the shift would also change but the true resonance energy E,’

= E,

+ A,

is a feature of the potential and independent of the choice of ‘lo and q,. The resonance width is also independent of the definition of the resonance energy. For narrow resonances E,‘ z E, . The S matrix boundary conditions for scattering by the potential V are e - i k r - S(k)eikr

$s (r)r;m

(221)

where S ( k ) = exp 2iq =

[(cot VI + i)/(cot q - i ) ]

(223)

If the solution (221) is analytically continued it follows that at a bound state, k = ix, S(ix) must have a pole. Equations (216) and (223) show that a pole in the S matrix exists at E

=

E,

+irp

(224)

A resonance can therefore be regarded as belonging to the complex eigenvalue (224) and the resonance width can then be interpreted as the lifetime of a decaying state. Let us now turn to the e-H scattering problem. The total wave function can be expanded as in Eq. (70). Suppose that

i = 1, ... N

ki2> O and

(225)

ki2 < O

i> N

Following Feshbach (1958) let P be a projection operator which selects these N open channels. Then explicitly N

M . R. H . Rudge

102

where the Dirac notation in this equation implies integration over the coordinates r,. In subsequent equations the brackets will denote integration over all coordinates. Let (227)

Q=l-P

The relations P2=P,

Q2=Q,

(228)

PQ=O

follow from the definitions. It can be seen that the Schrodinger equation can be written in the form P(%' - E)(P + Q ) Y

=

Q(%' - E ) ( P + Q)W

=0

(229)

giving the equations

+ HpQ Q Y = 0 (HQQ- E)QY + HQpPY = 0 (Hpp

- E)PY

(230) (231)

where for example

H p p = P%'P

(232)

The study of Eqs. (230) and (231) depends on splitting them into a resonant and a nonresonant part and this choice can be made in a number of ways. One convenient method is to consider the nonresonant part to be given by the equations (HPP

- J w Y ) o= 0

(233)

Equation (230) can be solved formally as PY

=(PY)o

+ ( E + i& - H p P ) - l H p QQ Y

(234)

and Eq. (231) then becomes (Hpg

+ HQp(E+ iE - H p p ) - l H p Q- E)QY + HQp(PY)o= 0

(235)

and substituting in Eq. (230) gives (Hpp

- E)PY

= HpQ(H' - E ) - l H Q p ( P Y ) O

(236)

where H'

+ HQp(E+ iE - H p P ) - ' H p Q

= HQQ

(237)

Let [ P Y ( -k,)l0 be a solution of Eq. (233) which has the asymptotic boundary conditions given by Eqs. (44) and (45) while [PW(k,)] has the boundary condi-

ELECTRON-ATOM EXCITATION CROSS SECTIONS

103

tions (42) and (43). By using methods similar to those used in deriving variational principles for scattering amplitudes it follows that (fp4

( k p k 4 ) -f;,( k p k,)) 7

9

= (2nJ-I

(243)

and the resonance width Ts,is given by TS = -2n

(244)

The Feshbach method thus shows that the scattering amplitude can be broken into two parts. One part gives rise to a scattering amplitude which varies slowly with energy though it may exhibit shape resonances. The other part varies rapidly in the neighborhood of one of the eigenvalues of IfQQ and then exhibits a resonance. Physically this corresponds to a quasi-stationary state of the system wherein both electrons are excited. If the resonances are narrow, their positions can be found by solving the bound state problem defined by Eq. (239). Providing the trial functions are chosen to be orthogonal to the open channel subspace, the Rayleigh-Ritz procedure applied to Eq. (239) yields upper bounds on the energies gS(O'Malley and Geltman, 1965). Other calculations of resonance positions, for example those of Lipsky and Russek (1966), have sought to determine the resonance positions without orthogonalizing the trial functions to the open channel space. This procedure is an admissible one since in effect it corresponds to making a different choice

104

M . R . H . Rudge

of projection operators. However, an injudicious choice of trial function can, in this case, give rise to very different values of 4. As in the discussion of + A, emerges from a suffishape resonances the true resonance position ciently accurate calculation independently of the choice of projection operators. However, the shift and the width can be determined only if the calculation includes the part of the wave function which arises from the open channels. The eigenfunctions QS represent a reasonably good approximation to the true wave function at a resonance energy in the vicinity of the nucleus. They can therefore be used to compute the radiative decay rates of narrow resonant states (Dickinson and Rudge, 1970; McCavert and Rudge, 1972b). D. THECLOSE-COUPLING METHOD In Section (I1,F) it was seen that in principle the scattering can be described by means of the expansion (70) in which the functions F(ilr2) are uniquely defined as ~ ( i ~ r=, )/ $ * ( i I r l ) ~ ( r l ,r~ dr,

(245)

However, the approach is of little practical value and so it is more convenient to write Y(Slrl,r2) = 2-l”

S (1 + ( i

l ) s ~ , 2 ) $ ( j I r , ) ~ ( ~ I r 2 ) (246)

where PI, permutes the coordinates rl and r 2 . The expansion (246) is now overcomplete and constraints must be imposed to ensure that the exchange events that are described by the second part of the summation are not in the continuum part of the first summation and equally to ensure that the direct scattering events do not emerge twice. This can be achieved by imposing the orthogonality conditions

($(ilr)IF(Nlr))= 0,

i = 1 ... N - 1

(247) The constraints (247) are a necessary feature of the exact solution. In practice the summation (246) must be truncated, and the constraints (247), while no longer necessary, are nonetheless still found to be convenient (Burke and Seaton, 1971). Thus taking N

Yf(SJrl,r2) = 2-l”

1 ( I + (i= 1

l ) s ~ 1 2 ) ~ ( ~ ~ r l ) ~ ( i ~ r(248) 2)

equations for the unknown functions F(ilr2) can be obtained by imposing the conditions that

ELECTRON-ATOM EXCITATION CROSS SECTIONS

105

Equation (249) is equivalent to Eq. (233) of the projection operator formalism and conditions of the form (247) can be added. In practice a partial wave analysis is needed. If now u denotes the quantum numbers { S M , L M , n } and i parametrizes the quantum numbers {nl, 12}, a trial function can be written 21’2$p(a]rI,r2> = (rl r2)-’

c (1 + ( - ~ ) ~ ~ 1 2 ) ~ ( i l ~ l ) ~ ( i ~ ~(250) l,P~)~(il~ i

A procedure which is equivalent to that of Eq. (249) consists of inserting trial functions of the form (253) in the variational principle (194) which then gives rise to a set of coupled integrodifferential equations for the functions F ( i l r ) .These are of the form

where V i j is a direct potential and Wij is the exchange kernel. The detailed forms of V i j and W i j are given by Percival and Seaton (1957). It can be readily shown that the equations (251) generate a real and symmetric R matrix. A recent survey (Burke and Seaton, 1971) discusses the close-coupling equations more generally with particular emphasis on the numerical techniques that can be used to solve them.

E. A MINIMUM PRINCIPLE Minimum principles have been the subject of a good deal of theoretical effort. Prominent in this are the papers of Hahn et al. (1964), Gailitis (1965a,b), Sugar and Blankenbecler (1964), and Hahn and Spruch (1967). These papers give a full account of the theory; this section will be restricted to one important aspect which is contained in the work of Hahn et al. (1964) and of Gailitis (1965a,b) and which is suggested by Eq. (240). In Eq. (240) it can be seen that the denominator, for a range of E at least, is negative but that the form of the numerator allows little to be said concerning its sign. More can be said if a similar analysis is applied to elements of the R matrix rather than to the scattering amplitude. Let (PY),be a solution of Eq. (233) which has the asymptotic form (180). Then PY = (P‘f‘),

+ ( E - HpP)-’HpQ QY

(252)

where the Green function is a standing wave Green function, and so the term ie is omitted, and (Hpp

- E ) ( P Y )= H p Q ( H ‘ - E ) - ’ H Q p ( P Y ) o

(253)

M . R. H . Rudge

106 where

Let (PY) satisfy the same boundary conditions (180), (more generally a different choice of entrance channel could be made) and let upj= d P j . From Eqs. (233) and (253) it follows that

( ( P Y ) ( H P P - J%PY)o) - ((P'y)O(HPP - E)(PY')) = ( ( p y ) O H P Q ( E - H ' ) - l H Q P( P y ) O )

(255)

On evaluating the left-hand side by use of Green's theorem it follows that Rpp

- (Rpp)O

= 2((p'y)0

HPQ

(H'-

'H Q P ( P y ) O )

(256)

where (R,Jo is a diagonal element of the R matrix as given by the closecoupling equations. Introducing the eigenfunctions mS where

( H ' - &J@,

=0

(257)

Eq. (256) becomes

The right-hand side of Eq. (258) is positive if g S >E ,

Vs

(259)

If the condition (259) is satisfied, Eq. (258) shows that ( R p p )> (R,,)o

(260)

which shows that the true diagonal elements of the R matrix are greater than the values given by the close-coupling equations where the open channels only are included. If H' does not support a discrete spectrum the inequality (259) is certainly satisfied because the continuous spectrum of H' runs to infinity from a value greater than E. If H' does support a discrete spectrum then, following Gailitis (1965a,b), a more general result can be obtained by extending the choice of trial function from the form (250) to the form

where the functions $ j ( r l , r 2 )are zero at infinity and orthogonal to the functions (P(i1 r l ) , P ( i l r 2 )It. can then be shown that if, in the basis of the functions $ j , the Hamiltonian supports as many eigenvalues as there are eigenvalues of H ' lower than E, the inequality (260) still holds where (R,J0 is replaced by its value given through the expansion (261).

107

ELECTRON-ATOM EXCITATION CROSS SECTIONS

In the neighborhood of resonances it is more convenient to work in terms of the matrix R-' and the relation (260) becomes

Equations (260) and (262) therefore imply that ?p

'

hp)0

The existence of the minimum principle implies that the addition of any further term in the second expansion in Eq. (261) must improve the calculation. In particular if there is only one open channel, the phase shift will approach the true phase shift from below. An illustration of the theorem, taken from the work of Gailitis (1965a) is given in Table VII. In this table the TABLE VII

CALCULATIONS OF e-H ELASTIC SCATTERING ( L = 0, S = 0 ) PHASESHIFTS ILLUSTRATING THE MINIMUM PRINCIPLE ~~

k2

Is"

0.01 0.04 0.09 0.16 0.25 0.36 0.49

2.396 1.870 1.508 1.240 1.031 0.869 0.744

a

II

= 1, = Ob

2.417 1.897 1.536 1.269 1.065 0.910 0.797

II = I ,

=

2.542 2.053 1.683 1.402 1.188 1.028 0.917

lb

II

= I , = 2b

2.547 2.060 1.690 1.409 1.194 1.034 0.923

Correlation' 2.553 2.067 1.696 1.415 1.202 1.041 0.930

Burke and Schey ( I 962). Gailitis (1965a). Schwartz (1961).

column labeled Is are the close-coupling values obtained by retaining just the ground state in the expansion. The next columns indicate the results of including various closed channel terms of the form given by Eq. (261) while the last column indicates the very accurate results of Schwartz (1961) obtained by a variational calculation which explicitly included correlation terms. The numerical calculations of Gailitis can be seen to confirm the theoretical expectations provided by the minimum principle. The results of Schwartz (1961) to which they appear to converge, while accurate, do not represent a bound. The results obtained by Gailitis (1965a) using the expansion (261) with only the Is state in the first summation are an improvement over closecoupling results wherein the first summation contains the Is, 2s, and 2p states and the second summation is omitted.

M . R . H. Rudge

108

F. AN ALTERNATIVE EXPANSION A systematic way in which the method of regional trial functions described in Section III,H can be improved has been advanced by Rudge (1970b, 1972). As an alternative to the expression (250) he writes

where r2 > rl, the trial solution for r2 < rl being given by =(-l)s$,(4rl,~z) (265) The expansion (264) is not, like that of Eq. (246), overcomplete and takes some explicit account of correlation effects. The boundary conditions at rl - r2 = 0 are not automatically satisfied by the expansion (264), however, while they are by the expansion (246). The functions F,(ilr) can be chosen by appeal to the variational principle (198) which is appropriate for such a trial function. On writing the scattering functions F(ilr) as a column vector F and if (LF), is the column vector with elements $,(4~2JI)

it is found (Rudge, 1972) that ALF +A’F’

+ (V + +A‘‘ + A - A -IT( - l)s-LK”)F = 0

(267) where A,& V, and A are matrices which depend upon r and are readily evaluated. For large r the matrix A is the unit matrix and the equations (267) reduce to the same form as the close-coupling equations (251). The equations (267) are only coupled differential, rather than integrodifferential, equations and thus represent a considerable simplification of the scattering problem. The set of equations (267) are not, however, a unique way of determining the functions F(il r ) because any functional which has a zero first variation can be added to a functional which has been made stationary. This is a feature which arises from the use of discontinuous trial functions [cf., Leigh (1956)l. In certain cases, for example with L = 0 and S = 1, the functions F(il r ) given by the equations (267) oscillate for small r and the R matrix correspondingly depends upon the starting values in the outward integration of the equations. Rudge (1972) therefore solved the equations (267) with the subsidiary condition that the stationary value be such that the integral I , of Eq. (199) should be a minimum. Othei conditions could be chosen. In the one state calculations of Rudge (1970b) a set of equations which, in the notation of Section IV, B, satisfied the condition &I

were solved.

+K l ) =o

(268)

ELECTRON-ATOM EXCITATION CROSS SECTIONS

109

The main advantage of the alternative expansion (264) therefore lies in the fact that it leads to a relatively simple set of equations while its main disadvantage is that there exist more than one possible set of equations which may differ in their convergence properties though not in their final result. One other attractive feature of the equations (267) lies in the fact that the matrices are energy independent and do not therefore need recalculating for each energy value. G . POLARIZATION EFFECTS If an electron impinges upon an atom in an s state, the atomic orbital is distorted and gives rise to a potential which behaves asymptotically as r-4. It was shown by Massey and Mohr (1934) that this behavior was implicit in the scattering equations and investigated in greater detail by Castillejo et a/. ( 1960). In Eq. (251) the exchange terms vanish exponentially and the direct potentials at least as fast as r - 2 for large r. It therefore follows that F(ilr)

-

r+m

2 C VijFj/(ki2- k j 2 )

(269)

j

since V;j vanishes more rapidly than V i j. If i = I represents the channel with n = I , II = 0, and 1,(1) E 1 2 , it follows, again retaining only the leading terms for large r, that

where VPOL =

C V?j/<Ej

-

El)

(271)

j

The values o f j for which I, = 1 are the dominant terms in Eq. (271) and retaining just these gives the result of Castillejo et a/. (1960) that VPOL

where the dipole polarizability

tl

- u/2r4

(272)

is given by

If the sum in this equation is truncated at n = 2, then 65.8 % of the polarizability is obtained while if all bound states are included this rises to only 81.4%. At low energies it is to be expected that the polarization potential will strongly influence the scattering and it would therefore seem appropriate to include this effect as fully as possible. Burke and Schey (1962) suggested that the close-coupling expansion might be augmented by further states, which

110

M . R . H.Rudqe

are not eigenstates of the atom Hamiltonian, chosen in such a way as to fully account for the polarizability. They did not, however, deduce the form of these states. The problem was considered further by Damburg and Karule (1967) whose technique is similar to that of Temkin (1957, 1959). If (274) then on writing

4 = S an p(n, 11 r)

(275)

a = (+rPU,Olr))

(276)

n

it can be seen that

Equation (274) can be solved exactly and with E l = - 1/2

9 = $r’(

1

+ +r) exp( - r )

(277)

It follows that if the close-coupling expansion contains the Is state and the 2F state where $25

= Nr -

yt,

(QW)

(278)

with N a properly chosen normalization constant, the 1s scattering function will contain a polarization potential governed by the exact dipole polarizability in contrast to the situation wherein the hydrogen 2p state is included. This treatment has been extended to account for the exact long range potentials in the excited states by Damburg and Geltman (1968). Further refinements and detailed calculations have been performed using an expansion which contains these “pseudostates” by Burke et af. (1969). There are two main drawbacks to the pseudostate expansion approach. First, the pseudostates introduce nonphysical thresholds and at energies close to these the method can give rise to difficulties; second, a larger set of coupled integrodifferential equations must be solved to obtain an R matrix the dimension of which is the number of physical channels. An alternative approach has therefore been advanced by Vo Ky Lan (1971, 1972) and by Feautrier et al. (1971) which is much closer to the polarized orbital method introduced by Temkin (1957, 1959) and by Temkin and Lamkin (1961). The basis of the method of Vo Ky Lan (1971, 1972) and to Feautrier et al. (1971) is to replace the atomic orbitals which appear in the close-coupling method by perturbed atomic orbitals. Their approach is developed quite generally for complex atoms and can be illustrated by again considering polarized 1s state of atomic hydrogen. In this case the function $(1 Ir,) is replaced by $(I Ir,) + +poL(rl>~z)

(279)

ELECTRON-ATOM EXCITATION CROSS SECTIONS

111

where where and

4 is the function (277). If P(r2) is chosen to be

P(r2) = -(37~-’/~/4)r;~q(r~ -rl) (282) where q is the unit step function, Eq. (279) is identical with that derived by Temkin (1959) and Temkin and Lamkin (1961). Vo Ky Lan (1972) avoids the discontinuity of Eq. (282) by determining P ( r 2 ) by the method of Stone and Reitz (1963) and of Stone (1966). The method of Feautrier et al. (1971) further differs from that of Temkin and Lamkin (1961) in that the scattering functions are determined within the orbital expansion by appeal to a variational principle. The polarized orbital method therefore has the attractive feature that a single polarized basis orbital gives the exact asymptotic potential and is therefore more economical than the pseudostate expansion method.

H. THRESHOLD BEHAVIOR Let t,bp(r1,r2)be an exact solution of the Schrodinger equation which satisfies R matrix boundary conditions. If t,bp(r1,r2)is represented by the satisfy Eq. (250) wherein the summaexpansion(250) then the functions F(’(ilr2) tion runs over all states. It follows from Eqs. (159) and (251) that an exact expression for the matrix element R,, is given by .a

R,,

=

-2

CJ i

where 1,

=

+

j l z ( k q r )VqjF(jIr) (

0

I, ( 4 ) . It follows therefore that P4

w+‘i2 (k,

+

0)

provided that

1

lim ~j J r ” * ’ ( V q j F ( j l r-JWqj(r,r‘)F(J1r’)dr‘ ) dr#O *a

k,+O

(285)

The exchange terms vanish exponentially for large r and the closed channel functions vanish more rapidly than the open channel functions. The convergence of the expansion (285) depends therefore upon the behavior of the Vqjwherej is an open channel. Consider the situation

k j # k,,

anyj

(286)

M . R . H . Rudge

112

and where I, = 0. The integral

is then bounded provided that

and that

n a. This approach was developed by Burke et al. (1971) and calculations reported by Burke and Robb (1 972) show it to be extremely promising. C. THECORRELATION METHOD Burke and Taylor (1966; Taylor and Burke, 1967) extended the work of Gailitis (1965a) by adding to a truncated close-coupling expansion that included the Is, 2s. and 2p states of hydrogen a series of terms that involved powers of the interelectron distance r I 2 . This has the effect of explicitly including correlation effects in the wave function and should therefore improve

118

M . R. H . Rudge

it in the vicinity of small r , 2 .They found that in the energy range between that required to excite the n = 2 states and that to excite the n = 3 states, the addition of these terms improves the calculation as evidenced by the behavior of the eigenphases. However, on comparing results of the correlation method with the calculations of Burke et al. (1967a) wherein the n = 3 states were explicitly included, it was found that the resonances which arise from the n = 3 states and are due to long range dipole effects are not well represented by the correlation terms. Above the n = 3 threshold the method was found to give rise to spurious resonances. It thus appears that the correlation method is of value only if all the open channels are included in the close-coupling expansion and the energy is below that at which resonances which arise from the first set of closed channels begin.

D. PSEUDOSTATE EXPANSIONS It was seen in Section IV, G that the sum which represents the dipole polarizability of atomic hydrogen is only slowly convergent. It is not unreasonable to suppose therefore that the slow convergence of the close-coupling expansion is related to this and that if the expansion were modified to include the polarization potentials exactly, faster convergence would be achieved. Even so, as remarked by Damburg and Karule (1967), it may still be necessary to take better account of short range correlation effects. If the above argument is correct, it is to be expected that in calculating excitation cross sections for the alkalis, good results could be achieved by retaining just a few states which include the lowest s and p states, since in this case the polarizability sum is dominated by the term which arises from the resonance transition. Evidence that this is so is provided by the recent close-coupling calculations of the 3s-3p cross section in sodium by Moores and Norcross (1972) whose results agree closely with the experimental values of Enemark and Gallagher (1972). As has been seen, however, the 2p state in hydrogen contributes only 66% of the polarizability and this ratio is still lower in the case of the inert gases. Burke et a/. (1969) have therefore carried out calculations of elastic scattering cross sections for hydrogen by adding suitably chosen pseudostates to the truncated close-coupling expansion and Geltman and Burke (1970) have extended this work to thecalculation of the Is-2s and 1s-2pcross sections near threshold. Table VIII displays these results and compares them with results of the correlation method (Taylor and Burke, 1967) with six-state close-coupling calculations (Burke et al., 1967a) and with the “best” results of Geltman and Burke (1970). The latter results were obtained by taking the best result of the preceding columns for each partial wave as judged by the smallest eigenphase sum. It can be seen from the table that the pseudostate calculation is in close agreement with these “best” data.

I19

ELECTRON-ATOM EXCITATION CROSS SECTIONS

TABLE VIII CALCULATIONS OF THE 1 s-2p CROSSSECTION I N ATOMICHYDROGEN SHOWING THE EFFECT OF PSEUDOSTATES

kZ

1s-2s-2p"

0.76 0.78 0.81 0.83

0.342 0.294 0.353 0.41 1

Ir

ls-2s-2p-3s-3p-3d"

ls-2s-2p-3s-3p-3db Correlation' 0.238 0.246 0.301 0.346

0.264 0.265 0.322 0.381

0.239 0.233 0.273 0.308

"

Best " b

0.237 0.244 0.296 0.351

Burke et a/. (1967a). Geltman and Burke (1970). Taylor and Burke (1967).

Burke and Webb (1970) extended the pseudostate calculations to higher energies but used a different choice of pseudostates which were not chosen to give the exact polarizabilities. Their results for the H( 1s-2s) cross section are depicted in Fig. 16 and compared with experimental data, from which it can be seen that good agreement is obtained. It can be argued in this case that at higher energies it is less important to obtain the exact polarizabilities but that the addition of pseudostates is still beneficial since they partially account for loss of flux to channels, includingcontinuum channels, which have not been explicitly included in the calculation.

0051

1

I

20

1

I

40

30

I

50

El7

FIG. 16. The H(ls-2s) cross section as given experimentally (Kauppila el al., 1970) and by a pseudostate expansion (Burke and Webb, 1970).

M . R . H . Rudge

120

The alternative expansion described in Section IV, F is not a pseudostate expansion in the sense described above but nevertheless is a different type of expansion which we therefore consider under this heading. It is more closely related to correlation effects than to polarization effects. Results obtained by Rudge (1972) for the ' S phase shifts are shown in Table 1X and TABLE IX

e k

1s"

1s-2s-2p'

0.1 0.2 0.3 0.4 0.5 0.6 0.7

2.276 1.770 1.448 1.208 1.016 0.855 0.716

2.529 2.034 1.659 1.376 1.160 0.999 0.888

+Wls)

+

e+ H(ls)'S PHASESHIFTS

-

1s-2s-2p-3p" Variationalb ls-2s-2p-2pC 2.542 2.051 1.679 1.397 1.181 1.022 0.912

2.553 2.067 1.696 1.415 1.202 1.041 0.930

2.532 -

1.663 -

1.162 -

0.881

~

~~~~~~

ls-2s-2pd 1s' 2.491 2.396 1.974 1.871 1.596 1.508 1.32 1.239 1.092 1.031 0.93 0.869 0.817 0.744

Rudge (1 972).

* Schwartz (1961). Burke et al. (1969). Burke and Schey (1962). John (1960).

compared with values that have been obtained using close-coupling expansions both with and without pseudostates and with the very accurate variational results of Schwartz (1961). The most elaborate results stem from the comparatively simple problem of solving a set of four coupled differential equations while the least complex calculation only involves the solutions of a single differential equation. These results compare very favorably with the other methods for which the computational problem is greater. We conclude this section by remarking that the polarized orbital closecoupling method of Feautrier et d . (1971) has recently been applied to electron-oxygen scattering (Vo Ky Lan et a/., 1972). The results of this method, which is an economical way of including polarization effects, agree closely with experimental data in the near threshold region.

E. VARIATIONAL PROCEDURES It follows from the variational principle (194) that if is the value of the integral L,, calculated using trial functions the asymptotic form of which can be expressed in terms of a matrix RFj then the corrected R matrix is

ELECTRON-ATOM EXCITATION CROSS SECTIONS

121

The unitarized Born approximation for example uses this result wherein RFj = 0 while in the close-coupling method Lb',' = 0. In some circumstances it is convenient to solve the close-coupling equations approximately, corresponding to both LFi # 0, R!; # 0, and then to determine a corrected R matrix through Eq. (305). This procedure, however, is unsatisfactory if the correction is large, as it will be in the vicinity of resonances. The approach of Saraph et al. (1966, 1969) is to choose a different normalization. Referring to Eq. (180) they write a = cos T - sin zp

b

= sin T

(306)

+ cos ~p

(307) where T is a diagonal matrix. By suitably choosing z it can be ensured that the correction to the p matrix is small. A case where this procedure is extremely valuable is the excitation of terms of the configuration 2p4. The energy splitting between the terms is small and if they are assumed to be zero, then the set of coupled integrodifferential equations can be uncoupled and cross sections determined in this relatively simple exact resonance approximation (Seaton, 1953a,b). An improved approximation can then be found by variationally correcting the p matrix. Other variational treatments have been advocated with a view to speeding up the computational process. The variational principles can be used to derive differential equations for unknown functions as in the close-coupling method, or to derive differential equations for the unknown functions and equations for the coefficients of correlation terms which are represented analytically as in the treatments of Gailitis (1965a) and Taylor and Burke (1967). If the continuum functions are also represented analytically, the problem reduces entirely to the algebraic problem of determining coefficients. Harris and Michels (1969) and Nesbet (1968, 1969) have used this approach to extend earlier treatments of elastic scattering (Harris, 1967) to the inelastic problem. Nesbet (1969) has shown that use of the variational principle for the R matrix may, at certain energies, give rise to spurious singularities but that where this is so the variational principle for the matrix R - ' can be used, and conversely. The method has been used by Seiler et al. (1971) who show that this algebraic close-coupling method can give results as accurate as those obtained by solving the integrodifferential equations, with considerable saving in computer time.

VI. Concluding Remarks In the introduction it was remarked that there exist two energy regimes in which theoretical treatments are adequate but that in the intermediate energy range a problem persists. It is clear, however, that as a result of using

122

M . R . H . Rudge

different types of eigenfunction expansions and of using improved numerical techniques, the range of applicability of the “ low energy” treatments is being extended. At the same time developments have occurred which have led to new “ high energy ” approximations which are more reliable at lower energies than previous methods. The intermediate energy problem therefore remains, but is being steadily eroded.

ACKNOWLEDGMENT It is a pleasure to thank Mrs. P. Sterling for her careful and patient typing of a difficult manuscript.

REFERENCES Abiodun, R. A., and Seaton, M. J. (1966). Proc. Phys. Soc., London 87, 145. Bassel, R. H., and Gerjuoy, E. (1960). Phys. Rev. 117, 749. Bates, D. R.(1958). Proc. Roy. Soc., Ser. A 247, 294. Bates, D. R., Fundaminsky, A., Leech, J. W., and Massey, H. S. W. (1950). Phil. Trans. Roy. SOC.London, Ser. A 243,93. Baz, A. I. (1958). Sou. Phys.-JETP, 6, 709. Baz, A. I. (1959). Sou. Phys.-JETP, 9, 1256. Bederson, B. (1970). Comments A t . Mol. Phys. 2, 160. Bell, K. L., and Kingston, A. E. (1968). Proc. Phys. Soc., London ( A t . Mol. Phys.) 1, 322. Bell, K. L., and Moiseiwitsch, B. L. (1963). Proc. Roy. Soc., Ser. A 88, 57. Bell, K. L., Eissa H., and Moiseiwitsch, B. L. (1966). Proc. Phys. Soc., London 88, 57. Bell, K. L., Kennedy, D. J., and Kingston, A. E., (1968a). Proc. Phys. Soc., London ( A t . Mol. Phys.) 1, 204. Bell, K. L., Kennedy, D. J., and Kingston, A. E. (1968b). Proc. Phys. SOC.,London (At. Mol. Phys.) 1, 322. Bely, 0. (1966a). Proc. Phys. Soc., London 87, 1010. Bely, 0. (1966b). Proc. Phys. Soc., London 88, 833. Bely, 0. (1967). Nuouo Cimento B 49, 66. Bely, O., Moores, D. L., and Seaton, M. J. (1963). I n “Atomic Collision Processes” (M. R. C. McDowell, ed.), p. 304., North-Holland Pub]., Amsterdam. Bethe, H. A. (1930). Ann. Phys. 5, 325. Bevensee, R. M. (1961). Ann. Phys. (New York) 12, 222. Birman, A., and Rosendorff, S. (1969). Nuovo Cimento B 63, 89. Blatt, J. M., and Biedenharn, L. C. (1952). Rev. Mod. Phys. 24, 258. Bonham, R. A. (1962). J. Chem. Phys. 36, 3260. Bonham, R. A. (1971a). Phys. Rev. A 3, 298. Bonham, R. A. (1971b). Phys. Rev. A 3, 1958. Born, M. (1926). Z . Phys. 38, 803. Bransden, B. H., and Coleman, J. P. (1972). Proc. Phys. Soc., London ( A t . Mol. Phys.) 5,537. Bransden, B. H., Coleman, J. P., and Sullivan, J. (1972). Proc. Phys. Soc., London (At. Mol. Phys.) 5, 546.

ELECTRON-ATOM EXCITATION CROSS SECTIONS

123

Burke, P. G. (1963). Proc. Phys. Soc., London 82,443. Burke, P. G. (1965). Aduun. Phys. 14, 521. Burke, P. G. (1968). Aduun. At. Mol. Phys. 4, 173. Burke, P. G. (1969). Lec. Theor. Phys. l l c , 1. Burke, P. G., and Robb, W. D. (1972). Proc. Phys. Soc., London (At. Mol. Phys.) 5, 44. Burke, P. G., and Schey, H. M. (1962). Phys. Reu. 126, 147. Burke, P. G., and Seaton, M. J. (1971). Methods Comp. Phys. 10, 1. Burke, P. G., and Smith, K. (1962). Rev. Mod. Phys. 34, 458. Burke, P. G., and Taylor, A. J. (1966). Proc. Phys. Soc., London 88, 549. Burke, P. G., and Webb, T. G. (1970). Proc. Phys. SOC.,London (At. Mol. Phys.) L131. Burke, P. G., Schey, H. M., and Smith, K. (1963). Phys. Reu. 129, 1258. Burke, P. G., Ormonde, S., and Whitaker, W. (1967a). Proc. Phys. Soc., London 92,319. Burke, P. G., Taylor, A. J., and Ormonde, S. (1967b). Proc. PhyA. Soc., London 92, 345. Burke, P. G., Gallagher, D. F., and Geltman, S. (1969). Proc. Phys. Soc., London (At. Mol. Phys.) 2, 1142. Burke, P. G., Hibbert, A., and Robb, W. D. (1971). Proc. Phys. Soc., London (At. Mol. Phys.) 4, 153. Byron, F. W. (1971). Phys. Reu. A 4, 1907. Castillejo, L., Percival, I. C., and Seaton, M. J. (1960). Proc. Roy. Soc., Ser. A 254, 259. Chen, J. C. Y., Joachain, C. J., and Watson, K. M. (1972). Phys. Rev. A 5, 2460. Damburg, R., and Geltman, S . (1968). Phys. Rev. Lett. 20, 485. Damburg, R., and Karule, E. (1967). Proc. Phys. SOC.,London 90, 637. Damburg, R., and Peterkop, R. (1962). Proc. Phys. Soc., London 80, 563. Damburg, R., and Propin, R. (1972). Proc. Phys. Soc., London (At. Mol. Phys.) 5,533. Day, R. B., Rodberg, L. S., Snow, G. A., and Sucher, J. (1961). Phys. Reu. 123, 1051. Demkov, Yu. N., 1963, “Variational Principles in the Theory of Collisions.” Pergamon, Oxford. Dickinson, H. O., and Rudge, M. R. H. (1970). Proc. Phys. Soc., London (Af. Mol. Phys.) 3, 1284. Doughty, N. A., Seaton, M. J., and Sheorey, V. B. (1968). Proc. Phys. Soc., London (At.Mol. Phys.) 1, 802. Enemark, E. A., and Gallagher, A. (1972). Phys. Rev. A 6, 192. Farago, P. S. (1971). Rep. Progr. Phys. 34, 1055. Feautrier, N., Van Regemorter, H., and Vo Ky Lan (1971). Proc. Phys. Soc., London (At. Mol. Phys.) 4, 670. Feenberg, E. (1932). Phys. Rev. 40,40. Feshbach, H. (1958). Ann. Phys. (New York) 5, 357. Fleming, R. J., and Higginson, G. C. (1964). Proc. Phys. SOC.,London 84, 531. Gailitis, M. (1963). Sou. Phys. JETP 17, 1328. Gailitis, M. (1965a) Sou. Phys. JETP 20, 107. Gailitis, M., (1965b).In “Atomic Collisions 111” (V. Ya Veldre, ed.) translated in J.I.L.A. Information Center Report No. 3. Univ. of Colorado, Boulder. Gailitis, M., and Damburg, R. (1963). Proc. PhyA. SOC.,London 82, 192. Geltman, S., and Burke, P. G. (1970). Proc. Phys. SOC.,London (At. Mol. Phys.) 3, 1062. Geltman, S., and Hidalgo, M. B. (1971). Proc. Phys. SOC.London (At. Mol. Phys.)4, 1299. Gerjuoy, E., Thomas, B. K., and Sheorey, V. B. (1972). Proc. Phys. Soc., London (At. Mol. Phys.) 5 , 321. Ghosh, A. S . , Sinha, P., and Sil, N. C. (1970). Proc. Phys. SOC., London (At. Mol. Phys.) 3, L58. Glauber, R. I. (1958). Lect. Theor. Phys. 1, 315.

124

M . R.H . Rudge

Gordon, W. (1928). Z. Phys. 48, 180. Green, T. A. (1967). Proc. Phys. Soc., London 92, 1144. Hahn, Y., and Spruch, L. (1967). Phys. Rev. 153, 1159. Hahn, Y., O’Malley, T. F., and Spruch, L. (1964). Phys. Rev. B 134,397. Harris, F. E. (1967). Phys. Rev. Lett. 19, 173. Harris, F. E., and Michels, H. H. (1969). Phys. Rev. Lett. 22, 1036. Heddle, D. W. O., and Seaton, M. J. (1964). I n “Atomic Collision Processes” (M. R. C. McDowell, ed.), p. 137. North-Holland Publ., Amsterdam. Hidalgo, M. B., and Geltman, S. (1972). Proc. Phys. SOC.,London ( A t . Mot. Phys.) 5 , 617. Holt, A. R., and Moiseiwitsch, B. L. (1968). Proc. Phys. Soc. London, ( A t . Mot. Phys.) 1, 36. Holt, A. R., Hunt, J., and Moiseiwitsch, B. L. (1971). Proc. Phys. Soc.,Londorz ( A t . Mol. Phys.) 4, 1318. Joachain, C. J., and Mittleman, M. H. (1971). Phys. Reo. A 4, 1492. Jobe, J. D., and St. John, R. M. (1967). Phys. Rea. 164, 117. John, T. L. (1960). Proc. Phys. Soc., London 76, 532. Kang, I. J., and Foland, W. D. (1967a). Proc. Phys. SOC.,London 92, 263. Kang, I. J., and Foland, W. D. (1967b). Phys. Rev. 164, 122. Kang, I. J., and Sucher, J. (1966). Phys. Lett. 20, 22. Kauppila, W. E., Ott, W. R., and Fite, W. L. (1970). Phys. Rev. A 1, 1099. Kennedy, D. J., and Kingston, A. E. (1968). Proc. Phys. SOC.,London ( A t . Mot. Phys.) 1, 195. Khare, S. P., and Rudge, M. R. H. (1965). Proc. Phys. Soc., London 86,355. Kim, Y . K., and Inokuti, M. (1968). Phys. Rev. 175, 176. Kingston, A. E., and Skinner, B. G. (1961). Proc. Phys. Soc., London 77, 724. Kingston, A. E., Moiseiwitsch, B. L., and Skinner, B. G. (1960). Proc. Roy. SOC.,Ser. A 258, 245. Kleinpoppen, H. (1971). Phys. Rev. A 3, 2015. Kohn, W. (1948). Phys. Rev. 74, 1763. Kohn, W. (1952). Phys. Ree. 87, 472. Kohn, W., and Rostoker, N. (1954). Phys. Rev. 94, 1111. Layzer, D. (1951). Phys. Rev. 84, 1221. Leigh, R. S. (1956). Proc. Phys. SOC.,London 69, 388. Lipsky, L., and Russek, A. (1966). Phys. Rev. 142, 59. Long, R. L., Cox, D. M., and Smith, S. J. (1968). J . Res. Nut. Bur. Stand., Sect. A 72, 521. McCavert, P., and Rudge, M. R. H. (1970). Proc. Phys. Soc., London ( A t . Mol. Phys.) 3, 1286. McCavert, P., and Rudge, M. R. H. (1972a). Proc. Roy. Soc., Ser. A 328, 429. McCavert, P.,and Rudge, M. R. H. (1972b). Proc. Phys. Soc., London ( A t . Mol. Phys.) 5,832. McCavert, P., and Rudge, M. R. H. (1972~).Proc. Phys. Soc., London (Ar. Mot. Phys.) 5, 508.

Macek, J. H., and Burke, P. G. (1967). Proc. Phys. Soc., London 92, 351. McGowan, J. W., Williams, J. F., and Curley, E. K. (1969). Phys. Rev. A 180, 132. Madison, D. U., and Shelton, W. N. (1971). Abstr. I.C.P.E.A.C. Conf. 7th, p. 768. Martins, P. de A. P., and Seaton, M. J. (1969). Proc. Phys. Soc., London ( A t . Mol. Phys.) 2, 333. Massey, H. S. W. (1956). In “Handbuch der Physik” (S. Flugge, ed.), p. 36. Springer, Verlag, Berlin and New York. Massey, H. S. W., and Mohr, C. B. 0. (1932). Proc. Roy. Soc., Ser. A 136,289. Massey, H. S. W., and Mohr, C. B. 0. (1934). Proc. Roy. Soc., Ser. A 146, 880. Miller, K. J., and Krauss, M. (1968). J . Chem. Phys. 48, 2611.

ELECTRON-ATOM EXCITATION CROSS SECTIONS

125

Miller, W. F., and Platzman, R. L. (1957). Proc. Phys. Soc., London 70, 299. Mittlernan, M. H . (1962). Phys. Rev. Lett. 9, 495. Moiseiwitsch, B. L., and Perrin, R. (1965). Proc. Phys. Soc., London 85, 51. Moiseiwitsch, B. L., and Smith, S. J. (1968). Rev. Mod. Phys. 40, 238. Moiseiwitsch, B. L., and Williams, A, (1959). Proc. Roy. Soc., Ser. A 250, 337. Moores, D. L. (1966). Proc. Phys. Soc., London 88, 843. Moores, D. L. ( I 967). Proc. Phys. Soc., London 91, 830. Moores, D. L., and Norcross, D. W. (1972). Proc. Phys. Soc., London ( At : Mol. Phys.) 5, 1482. Morrison, D. J. T. (1968). Ph.D. Thesis, Queen’s University of Belfast. Morrison, D. J. T., and Rudge, M. R. H . (1966). Proc. Phys. Soc., London 89,45. Morrison, D . J. T., and Rudge, M. R. H . (1967). Proc. Phys. Soc., London 91, 565. Mott, N.F., and Massey, H. S. W. (1965). “The Theory of Atomic Collisions,” 3rd ed. Oxford Univ. Press (Clarendon), London and New York. Moustafa Moussa, H. R., De Heer, F. J., and Schutten, J. (1969). Physica (Utrecht) 40, 517. Nesbet, R. K. (1968). Phys. Rev. 175, 134. Nesbet, R. K . (1969). Phys. Rev. 179, 60. Ochkur, V . I . (1964). Sou. Phys. JETP. 18,503. Ochkur, V. I., and Bratsev, V. I. (1966). Sou. Asfron.-AJ9, 797. O’Malley, T. F., and Geltrnan, S. (1965). Phys. Rev. A 137, 1344. Opal, C. B., and Beaty, E. C. (1972). Proc. Phys. SOC.,London ( At . Mol. Phys.) 5, 627. Opik, U. (1955). Proc. Phys. Soc., London, Sect. A 68,377. Oppenheinier, J. R. (1928). Phys. Rev. 32, 361. Ornionde, S., and Smith, K. (1964). In “Atomic Collision Processes” (M. R.C. McDowell, ed.), p. 114. North-Holland Publ., Amsterdam. Ott, W. R.,Kauppila, W. E., and Fite, W. L. (1970). Phys. Rev. A 1, 1089. Percival, 1. C., and Seaton, M. J. (1957). Proc. Cambridqe Phil. Soc. 53, 654. Percival, I . C., and Seaton, M. J . (1958). Phil. Trans. Roy. Soc., London, Ser. A 251, 113. Peterkop, R., and Veldre, V. (1966). Advan. At. Mol. Phys. 2, 263. Robertson, H. H . (1956). Proc. Cambridge Phil. Soc., 52, 538. Rose, M. E. (1957). “Elementary Theory of Angular Momentum.” Wiley. New York. Ross, M., and Shaw, G. (1961). Ann. Phys. (New York) 13, 147. Rudge, M . R. H . (1965a). Proc. Phys. SOC.,London 85, 607. Rudge, M. R. H . (1965b). Proc. Phys. Soc., London, 86, 763. Rudge, M. R. H. (1968). Proc. Phys. Soc., London (At. Mol. Phys.) 1, 130. Rudge, M. R . H. (1970a). Proc. Phys. Soc., London ( A t . Mol. Phys.) 3, 173. Rudge, M. R. H. (1970b). Proc. Roy. Soc., Ser. A 318, 497. Rudge, M. R. H. (1972). Proc. Roy. Soc., Ser. A 327, 425. Salmona, A,, and Seaton, M. J. (1961). Proc. Phys. Soc., London 77, 617. Saraph, H. E., and Seaton, M. J . (1971). Phil. Trans. Roy. Soc., London, Ser. A 271, 1. Saraph, H. E., Seaton, M. J., and Sheniming, J . (1966). Proc. Phys. Soc., London 89, 27. Saraph, H. E., Seaton, M. J., and Shemming, J. (1969). Phil. Trans. Roy. Soc., London,Ser. A 264, 77. Schlosser, H., and Marcus, P. M. (1963). Phys. Rev. 131, 2529. Schneider, B. (1970). Phys. Rev. A 2, 1873. Schulz, G . J., and Fox, R. E. (1957). Phys, Rev. 106, 1179. Schwartz, C . (1961). Phys. Rev. 124, 1468. Seaton, M. J . (1953a). Phil. Trans. Roy. Soc., London, Ser. A 245,469. Seaton, M. J. (1953b). Proc. Roy. SOC.,Ser. A 218, 400. Seaton, M. J. (1955). C. R. Acud. Sci. 240, 1317.

126

M. R. H . Rudge

Seaton, M. J. (1958). Mon. Notic. Roy. Astron. Soc., 118, 504. Seaton, M. J. (1961a). Proc. Phys. SOC., London 71, 174. Seaton, M. J. (1961b). Proc. Phys. SOC., London 71, 184. Seaton, M. J. (1962). I n “Atomic and Molecular Processes” (D. R. Bates, ed.), p. 374 Academic, Press, New York. Seaton, M. J. (1966a). Proc. Phys. SOC.,London 88, 801. Seaton, M. J. (1966b). Proc. Phys. SOC.,London 88, 815. Seaton, M. J. (1969). Proc. Phys. SOC., London ( A t . Mol. Phys.) 2, 5 . Seaton, M. J. (1970). In “Computational Physics,” p. 19. Inst. Phys. Phys. SOC.,London. Seiler, G. J., Uberoi, R. S., and Callaway, J. (1971). Phys. Rev. A 3, 2006. Smith, K. (1966). Rep. Progr. Phys. 29, 373. Steelhammer, J. C., and Lipsky, S. (1970). J. Cheni. Phys. 53, 1445. Stone, P. M. (1966). Phys. Rev. 141, 137. Stone, P. M., and Reitz, J. R. (1963). Phys. Rev. 131, 2101. Sugar, R., and Blankenbecler, R. (1964). Phys. Rev. B 136, 472. Tai, H., Bassel, R. H., and Gerjuoy, E. (1970). Phys. Rev. A 1, 1819. Taylor, A. J., and Burke, P. G, (1967). Proc. Phys. SOC., London 92, 336. Ternkin, A. (1957). Phys. Rev. 107, 1004. Ternkin, A. (1959). Phys. Rev. 116, 358. Ternkin, A. (1962). Phys. Rev. 126, 130. Temkin, A., and Lamkin, J. C. (1961). Phys. Rev. 121, 788. Ser. A 294, 160. Thompson, D. G. (1966). Proc. Roy. SOC., Truhlar, D. G., Cartwright, D. C., and Kuppermann, A. (1968). Phys. Rev. 175, 113. Truhlar, D. G., Rice, J. K., Kuppermann, A., Trajmar, S., and Cartwright, D. C. (1971). Phys. Rev. A 1, 778. Vainshtein, L., Presynyakov, L., and Sobel’man, I. (1961). Sou. Phys.-JETP. 18, 1383. Vainshtein, L. A., Opykhtin, V., and Presnyakov, L. (1965). Sou. Phys.-JETP 20, 1542. Van Blerkom, J. (1968). Proc. Phys. SOC.,London ( A f . Mol. Phys.) 1, 423. Vo Ky Lan (1971). Proc. Phys. SOC.,London (At. Mol. Phys.) 4, 658. Vo Ky Lan (1972). Proc. Phys. SOC.,London ( A f .Mol. Phys.) 50, 245. Vo Ky Lan, Feautrier, N., Le Dorneuf, M., and Van Regemorter, H. (1972). Proc. Phys. SOC.,London. ( A f .Mol. Phys.) 5 , 1506. Vriens, L., Simpson, J. A., and Mielczarek, S. R. (1968). Phys. Rev. 165, 7. Wigner, E. P. (1948). Phys. Rev. 73, 1002. Wigner, E. P., and Eisenbud, L. (1947). Phys. Rev. 72, 29. Zapesochnyi, I. P. (1967). Sou. Astron.-AJ 10, 766.

COLLISION-INDUCED TRANSITIONS BETWEEN ROTATIONAL LEVELS TAKESHI OKA Division of Physics National Research Council of Canada Ottawa, Ontario, Canada

I. Introduction .................................................... A. Elementary Process ......... ................ B. Experimental Method ...................................... 11. Theory ........................... ...................... A. Hamiltonian ............................. .............. C. Overall Symmetr D. Time-Dependent E. Intermolecular P F. Local Symmetry 111. Experiment

127 129 134 134

................. ................

.........

183

1V. Concluding Remarks . . . . . . . . . . . . . . . . . . 202 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

I. Introduction A. ELEMENTARY PROCESS The elementary process which will be discussed in this paper is shown in Fig. 1. A molecule in a quantum state $ “collides” with another molecule (or atom) and changes its quantumstate t o $’.I Contrarytomanyother studies of collisions in which the main interest is in scattering, that is, the change of translational energy and the deflection of the path, the interest in this paper is in the change of the internal molecular state from $ to $’. We will consider only changes in the rotational states of molecules where the energy change is Not bothering about the philosophical interpretation of quantum mechanics, we assume that $ and 4‘ are stationary states of isolated molecules. 127

T. Oka

128

POTENTIAL REG I ON

FIG.1. Collisional process.

smaller than, or at most of the order of, kT at room temperature. The change of vibrational or electronic state of a molecule which usually requires a much higher energy change and thus occurs with a much lower rate is neglected in this paper. Very often for polyatomic molecules the change of rotational energy in a collision is much smaller than kT and a rather weak intermolecular interaction is sufficient to cause the transition $ --t I)’.Therefore many of the “collisions” discussed in this paper are not of the head-on type but are subtle encounters in which one molecule simply passes another with a rather large distance (typically 10 20 A) between them. The ultimate goal of the study is to understand the elementary process to the extent that we can calculate the probabilities of the transitions I) -+ t,b’ for all possible $’ for a given initial condition of the collision (that is, the wave function of the collision partner, the relative velocity, and the impact parameter). We are far from the goal. Our lack of knowledge about the collision process is obvious if we compare it with our knowledge of another process which also changes the rotational state of a molecule, namely, the radiative process shown in Fig. 2. For a radiative rotational transition, the selection rules, i.e., the relation between $ and $‘, are well established, and the probability of a transition can be accurately calculated if molecular quantities such as the permanent electric dipole moment, transition moments, and energy levels are known. Also, by using spectroscopy, it is relatively easy to N

FIG.2. Radiative processes.

COLLISION-INDUCED ROTATIONAL TRANSITIONS

129

single out one transition and measure its probability. For the collision process, however, it is only recently that we have acquired techniques that enable us to study it in detail. We are just beginning to understand the " selection rules " governing collision-induced transitions and to measure some transition probabilities. The theory has been worked out only for simple cases like H,-H, collisions or for the limiting case of a long range dipole-dipole interaction when perturbation theory can be used. Therefore it is not easy to describe the theory in a logically deductive way; the theoretical arguments given in Section I1 are heavily dependent on the intuition obtained from the experimental studies described in Section 111. The information obtained from the study of collision-induced rotational transitions can be useful in two ways: (1) to determine the intermolecular potential and to obtain molecular quantities such as multipole moments, polarizabilities, etc; (2) to understand the molecular relaxation process and apply it to explain observed results in other fields of study. The application (1) has not been very successful so far, mainly because of a lack of reliable theory. For example, the pressure broadening parameters have been used to determine the quadrupole moment of molecules. However, a much more thorough understanding of the collision process is needed before we can hope to obtain accurate values of molecular multipole moments from experiments. I believe that we are at the stage of understanding collisions in terms of various molecular quantities measured by other methods. As for the application (2), a detailed understanding of the elementary process is rarely needed to explain the resulti of normal laboratory experiments. Since, except for equilibration between different nuclear spin modifications of molecules such as ortho and para hydrogen, rotational relaxation occurs fast enough to give a Boltzmann distribution, a detailed analysis is not needed. However, a great incentive to the study of the collision process has been given by the recent discovery of interstellar molecules, which are observed to have anomalous rotational distribution. Because of the low density of molecules in interstellar space, the collision process and certain radiative processes have similar rates. Moreover, since the molecular cloud is not in thermal equilibrium, the radiative and kinetic temperatures are different. In order to discuss rotational distribution of molecules in such conditions, a detailed knowledge of the elementary process is needed. B. EXPERIMENTAL METHODS Until recently the experimental methods used to study the collision process were limited mostly to the measurement of ultrasonic dispersion [for a summary see Herzfeld and Litovitz (1959) and Cottrell and McCoubrey (1961)l and to the measurement of pressure broadening of spectral lines [for a summary see Birnbaum (1967)]. Measurements of ultrasonic dispersion give

130

T. Oka

the probabilities, Pi,, of collision-induced transitions averaged over all the pi (Pir)/N where the sign ) designates the initial and final levels, i.e., average over various collision parameters such as relative velocity, impact parameters, mutual orientation of the molecule, pi denotes the density of a quantum level i , and N is a normalization factor. In a pressure broadening experiment two quantum levels (1 and 2) are specified as initial levels for the collision-induced transitions and the broadening gives the sum of the probabilities of transition from the two levels to all possible final levels, i.e., {(Plr) ( P 2 , ) } / n ,where n is a normalization factor. These results give the average time that a molecule stays in the particular rotational levels 1 and 2, but do not give direct information on the selection rules, i.e., the relation between the initial and final levels. In order to obtain such information we have to monitor the initial and thefinal rotational levels separately. In the last decade three such methods have been developed using, in turn, molecular beams, optical fluorescence, and double resonance. In these methods a known nonBoltzmann distribution is established in some molecular levels (initial levels) of a gas at low pressure, and the consequence of collisions is obtained by monitoring populations of molecules in other levels (final levels).

1


=

I ( m I T(t) I t7> I *

(23)

where the unitary property of T, T-' = Tt, has been used. Substituting Eq. (22) into Eq. (19) and using T-' = Tt, we obtain

ihT(t) = exp[-H,t/ih]V(t) exp[H,t/ih]T(t) The equation of motion for the matrix element is therefoie ih(nI$(t)(m)

=c(nl

V(t)Ik) as (klT(t)lm)eiwnkr

k

(24)

(25)

where w,k = (En - Ek)/h. For a weak intermolecular interaction, T(t) is not very different from the unitary matrix 1; we can therefore expand T as

T = 1 + T,

+ T,

a * *

(26)

and use the iteration procedure for the calculation. The first approximation gives

Those readers who are interested only in qualitative discussions on collision-induced transitions can skip this section and proceed to Section II,E.

COLLISION-INDUCED ROTATIONAL TRANSITIONS

145

An application of this formula to the collision problem gives the first-order approximation of the transition probability as

where R designates rotational levels and AEr was given in Eq. (13). Equation (28) indicates that for weak collisions the probability of a transition is proportional to the square of intermolecular potential; for example, if only dipole-dipole interaction is considered in a pure gas, the probability is proportional to p4. Since the rotational wave function R , contains only the Eulerian angles ( a , ,/I1, y , ) as variables, only the angle-dependent part of V ( t ) is important in the calculation. The angle-independent part of V ( t )affects the probability indirectly through the deflection of the path. In this respect the results discussed in this paper are complementary to those from molecular beam scattering experiments in which the angle-independent part of V(r) is important. The explicit form of V(f)for electric multipole interaction is summarized in the next section. Anderson (1949) treated the pressure broadening of NH, using Eq. (28) based on the assumption of AE, = 0, linear path, and a dipole-dipole interaction. For a dipole-dipole interaction and a linear path, the time dependence of Y(f)is

-

~ ( t=) [PI P2 - 3 ( P l - R U ~ ~ 1 ~ ~ 1 1 ~ ~ = ~ L ~ / R ~ 8, [ Ccos O S0,( 1 - 3 C O S ~$) sin 0 , sin O ~ ( C O S-( Cp2) ~ , - 3 sin2 i+b cos 4, cos Cp2} - 3 sin $ cos i+b(cosCp, sin 0, cos 0, cos Cp2 cos 8, sin O,)]

+

+

(29)

where 0 and Cp are the polar angles of the dipole moment and R, cos $, and sin $ (see Fig. 8 ) are expressed as R2 = b2 f v 2 t 2 ,

cos i+b = b / R ,

sin $ = vtlR

FIG.8. Geometry of molecular collision.

(30)

T.Oka

146

Performing first the time integral s’g V(t)dr,which is elementary for AE, = 0, and then taking the matrix element using known matrix elements of direction cosines, Anderson calculated the probabilities of collision-induced rotational transitions in NH, . The probability is proportional to p4/b4u2h2.The implication of this formula is obvious if it is written in the form ( b / ~ ) ~ ( p ~ / b;, h ) ~ p2/b3his the energy of interaction and b/u is the duration of collision. The probability is a sharp rising function of the impact parameter for small b. Since the probability cannot be greater than one, Anderson introduced the following three approximations which are still used for b < b*(P(b*)= 1)

PI#(b)= 1 - cos(2P(b))”2

P2#(b)= 1 They are shown in Fig. 9. The total cross section CT is CT

=

JOm2xbP(b)db

It is interesting to see that if approximation 2 [Eq. (31b)l is used for the dipole-dipole interaction,

b-

FIG.9. The transition probability as a function of the impact parameter b (Anderson, 1949). ---, Approximation no. 1 ; -, approximation no. 2; ---, approximation no. 3 ; ---, probable shape of curve: a and b (two possibilities).

COLLISION-INDUCED ROTATIONAL TRANSITIONS

147

In other words, nb*' gives exactly half of the total cross section. Since the perturbation method is a good approximation only when the probability is much less than 1, Eq. (33) indicates that for appreciably more than half of the inelastic collisions, the perturbation treatment is not a good approximation. The situation will be worse for higher order multipole interactions which the probability is an even more sharply rising function of the impact parameter. This limitation of the accuracy of the calculation may not be very serious for the calculation of pressure broadening which depends on the average time a molecule spends in a rotational level. Once a molecule is disturbed strongly enough it will change rotational state or phase and therefore the curves shown in Fig. 9 give a good approximation of the probability that a molecule will change its rotational state. However, if a probability of a particular transition Ic/ + Ic/' is to be calculated, this limitation is certainly much more serious. Rabitz and Gordon (1970) have used a method in which the region 0 < b < b* is divided into many equally spaced intervals, various transition probabilities are calculated, and each of them is normalized to the total transition probabilities. Some such consideration will be needed also for those regions of b > b* where the probability is not much smaller than one. Tsao and Curnutte (1962) extended Anderson's treatment to the cases in which AE, is not negligible. For such cases Eq. (28) indicates that the probability is given by the Fourier component of the time variation of the intermolecular potential V ( t ) at the frequency AEJh. It will be shown in the next section that if a multipole expansion is used, V ( t ) is given as a sum of the products of rotation matrices Dfn,,m,(~lP1yl),Dfn2,m2(~2P2yZ) and a spherical harmonic Y,:(O, $)/R'+', where 0 and 4 are polar angles of the intermolecular vector with respect to a specified axis system. The time integral in Eq. (28) operates only on Yl:(8, $)/I?+'. The choice of the space fixed axis is arbitrary; we can take

4 =0

and

0 = I(/

4 =$

and

9=n/2

(see Fig. 8)

or

(34)

The former has been customary but the latter may save calculation because for odd m values Ylz (n/2, 4) = 0. If the linear path approximation is used the time dependence of t,h and R is given by Eq. (30). Therefore the integral is of the form

T. Oka

148

1.4

1.2

1c.

0.E

4

0.6

0.'

0.:

FIG.10. The resonance factors for dipole-dipole(d-d) and quadrupole-quadrupole(q-q) interaction as functions of x = wb/v, where fiw is the change in the total rotational energy of the colliding molecule (Van Kranendonk, 1963),

where x = ob/u. For a first-order multipole interaction, n in Eq. (35) is always an odd number and the integral is given by a modified Bessel function K,,,(x) (Foley, 1946). Tsao and Curnutte calculated various time integrals in terms of modified Bessel functions, and Gray and Van Kranendonk (1966) gave a general expression for the integral. For the second-order interaction (induction or dispersion) n may be an even number; the time integral is elementary. The time integral squared and summed over orientations is called the resonance factor. The dependence of the resonance factor go(x) on x = o b / u is shown in Fig. 10 (Van Kranendonk, 1963). It is seen that for a dipole-dipole interaction the resonance factor is maximum at x 1, that is, when

-

v/2nb = 1 / 2 n ~ , AEJh

-

(36)

where T, = b/v represents the time of duration of the collision. It is to be noticed that the resonance factor is not a very rapidly varying function of x in the region of x smaller than about 1.5. This means that if T, is sufficiently small (impact theory), the consideration of resonance does not affect the

COLLISION-INDUCED ROTATIONAL TRANSITIONS

149

result very much; Anderson (1949) used this as a justification when he neglected AE, in his calculation. Using typical values of u = 5 x lo4 cmjsec and b = 10 A, we find that T~ is of the order of 2 psec; therefore AE, smaller than 5 cm-' may be neglected. Figure 10 indicates that the quadrupole-quadrupole interaction has a resonance at much larger x, essentially because of the faster time variation of interaction for the same T ~ . The formalism of Tsao and Curnutte, and Gray and Van Kranendonk has been used extensively by Murphy and Boggs (1967a,b, 1968, 1969a,b, 1971) for the calculation of pressure broadening. For other calculations of pressure broadening a reference should be made to Birnbaum's review paper (1967). The iteration using the T matrix in Eq. (26) can be extended further and we obtain in general

Rabitz and Gordon (1970) extended the approximation to the second order and calculated ratios of various rates of collision-induced transitions in HCN which agreed well with the experimental results (see Section 111,D). Similar calculations have been done by Prakash and Boggs (1972)for H,CO, H,CCO, and HCN. For stronger collisions, the deflection of the path should also be considered. Some such effort has been made by Dillon et al. (1970) and by Dillon and Godfrey (1972) and applied to the pressure broadening of 0,. Much more effort in this direction will be needed before we understand quantitatively the results of stronger collisions such as those discussed later in this article. In some collisions it happens that the transition probability from an initial level n to a particular level m is much larger than those to any other levels. This is realized in rotational resonance (Anderson, 1949) in which molecule 1 in the J , = J level collides with molecule 2 in the 5, = J- 1 level (Fig. 7). During the interaction molecules exchange a virtual quantum back and forth without changing the total rotational energy of the pair (AE, = 0 ) ;the density matrix oscillates between n = (J,J- 1) and m = ( J - 1, J ) . The typical case of rotational resonance is realized for light molecules for which all m,,k's except on,,are much larger than the time variation of V ( t ) .For such a case Eq. (25) indicates that < n I T ( t )I m ) is a sum of terms inversely proportional to w n k .Therefore we retain only the resonant term and neglect the others to obtain iWn I

*I

m> = (0

I v(t)I m>(mI TO)I m>

(38)

T. Oka

150

Also from Eq. (24) we have

ih(ml$lm)

= (ml

v(t)In)(nlT(t)(m)

(39)

The solution of these equations is

and

Therefore the transition probability is

A different derivation of this formula will be found in Landau and Lifshitz (1963, Section 39, problem 3). Equation (41) indicates that the interacting molecules complete the exchange of a virtual quantum back and forth at times t,, t , , . . . etc. which satisfies

1: ( J , J -

1 I V(t)IJ- 1,J)dt = h/4

f : ( J , J - 1 I V(t)(J- 1,J) dr = h/4,

etc.

(42)

(see Fig. 11). The time needed for an exchange is t

-

14(J,J- 11 V(t)lJ- l , J ) / h J -'

(43)

For example, for two polar molecules with a dipole moment of 1 D at the distance of 10 A, t is estimated from r3h/4p2 to be of the order of 0.5 psec; therefore for favorable molecular orientations the exchange will occur four times during the duration of a collision which lasts for about 2 psec. E. INTERMOLECULAR POTENTIAL The intermolecular potential given in Eq. (3) can be written in terms of spherical harmonics as

COLLISION-INDUCED ROTATIONAL TRANSITIONS

151

\

I

2

J

J-I

FIG.1 1 . Exchanges of a virtual quantum during one collision.

where 8, Pi, and Pk represent angular orientations of the vectors (see Fig. 6), Ylm(P) are spherical harmonics corresponding to the angular orientation P, (i, !& k j with I = I , I , and m = -(ml+ m,) is a 3j symbol, and C l l l ris a constant C1112= ( - 1)”{4~(21+lj!/(211 + 1)!(212 + 1)!}’/2/(2Z+ (45)

+

The history and derivation of Eq. (44) can be found in Gray’s paper (1968). It should be noted that Eq. (44) is rigorous for long-range intermolecular interaction. The spherical harmonics Y,,(P), in the space-fixed coordinate system can be expressed in terms of those in a molecule-fixed system. Ylm,(?o), as I

Ylm

(3)

=

C- I

m’=

0k.m

( ~ P YY )I m , (Po)

(46)

where Dk.,(c&> is a matrix element of the rotation matrix with Eulerian angles (aPy) that relates the orientation of the molecule-fixed axes to that of the space-fixed axes [see, for example Edmonds (1957)l. Introducing molecular multipole moments M I , defined as

where the summation goes over all electrons and nuclei. Eq. (44) can be rewritten

T. Oka

152

Thus we have obtained a general formula for an intermolecular interaction expressed in terms of electric multipole moment operators. For neutral molecules the summation starts from I , = 1 and I, = 1 (dipole-dipole interaction) which has R dependence as R - 3 . The rotational wave functions of molecules 1 and 2 operate only on the rotation matrices D ~ l , m l ( ~ l Pand l ~ D!,&, l) ( a 2 P 2 y 2 ) respectively, , the time integral only on Yz(R)/R''', and the electronic and vibrational wave functions only on the multipole moment operators MIlm1. and M I ,m z l . The intermolecular potential averaged over the ground vibronic state is obtained from Eqs. (7) and (48) to be

v, = (0,O I V (0,O) =

cA,

(49)

~ m * ( R > ~ ~ ~ ~ m ~ ( ~ l P 1 ~ 1 (a2 ) ~ fPn2 2~ Z2 ' )m - 2 ~ ~ 1 m m1Z , ,/ R~' + '~ z

A

and

vi + v, = -ec' C' l, e2.m

I(0,Ol

= -

x

1 cK A

i;'Ielu11 e 2 u 2 > I 2

Eel",+ E e l u2

ui

c

A K B l B2 rAiC yl:

ni.nz

D:1*+iii1,, ml + t i i 1 ( ~ l P l ~ l > D $ z , + E z m , .z + i i i z

(R>yIKi

( ~ P22

1+1+2

(50)

Y~)/R

where A indicates a group of subscripts 1,,1,, m , , m 2 ,m1',m2', MI,,,= (0,O I Mfm I0,O) is the multipole moment in the ground vibronic state, and

and

B, = (- l)mi+tiii+mi'+ifi' ( 2 n i

li

li

+ 1)

The summation extends over the region li 2 mi 2 - I i , etc., and I li + liI 2 ni 2 I li - Ii I. In deriving Eq. (50) a contraction formula for the rotation matrix elements has been used (Brink and Satchler, 1968). The coefficient rAK is given by -AA

-

c,1,(01 el

I

I

lO>(Ol

M t 1 m 1 , el >(el Mil

ez

Eel

Ml2mz,le2>(e2

I M I ,w l )lo>

+ Eel (51)

for the dispersion potential V, , and by

rAK =

~

~

~

~

~

~

~

+

f ~ ,&2 mq 2 1 , ~ 1 ~2 i ~ i i 1z1~.m~i 'lC l E (l . ~

~

( 5~ 2 ) ~

:

COLLISION-INDUCED ROTATIONAL TRANSITIONS

153

for the induction potential V i ,where a component of the polarizability tensor a;: is given by a;; =

1 (01 M l m I k ) ( k I k

10)

Ek

(53)

with the summation k covering all the excited electronic and excited vibrational states. Equations (49) and (50) give the general form of the intermolecular potential. Using the expressions for the rotational matrices (Edmonds, 1957) and spherical harmonics, a straightforward calculation yields all terms of the multipole interaction. Some explicit forms of such terms have been given [see, for example, Buckingham (1967)l. For multipole interaction of low order the direct calculation such as that given by Eq. (29) is simpler, but Eqs. (49) and (50) are particularly useful in the discussion of selection rules because they give the angular dependence of the intermolecular potential explicitly as products of rotation matrices. Remembering the fact that a matrix element of the rotation matrix Dip,,, (apy) is an eigenfunction of the symmetric rotor Hamiltonian [see, for example, Edmonds (1957) or Brink and Satchler (1968)l and using the product rules for it, we obtain

The transition J k M + J'k'M' occurs when the matrix element is nonvanishing, that is

Ak = k - k ' AM

= M-M'

= m'

I 1

=m

I 1

(55)

( A J J= IJ-J'I I 1 and

J+J'rI For the second-order interaction given in Eq. (50), a transition J k M + J'k'M' occurs when the matrix element

(56) is nonzero, that is

Ak

+ iiiJI n I I+ 1 m + Fii I n I 1 + I

= m'

AM=

IAJI s n I 1 + I

(57)

154

T. Oka

and J + J ' 2 n 2 I!-

I1

Equations (55) and (57) give the selection rules for collision-induced transitions caused by an electric multipole interaction. It may be noted from Eq. (49) that, for an interaction involving permanent electric moments, the selection rules for the transition in molecule 1 is dependent only on the angular dependence D:,.,, (alPlyl) of the multipole moment All,,,of the same molecule and is independent of that of the collision partner Alzm 2 1 as long as the latter is nonzero. For example if the dipole moment of an axially symmetric molecule is used (for which I = 1 and rn = 0 ) , the selection rules given by the first-order theory [Eq. (28)] are AJ = 0, & 1, A M = 0, k 1, and Ak = 0 regardless of which electronic multipole moment of the second molecule is used. It is for this reason that the collision-induced transitions in polar molecules observed by the microwave double resonance method are dominated by the dipole-type transitions for such a wide variety of collision partners (see Section 111,C).

F. LOCALSYMMETRY AND " SELECTION RULES" The selection rules for individual molecules can be obtained from Eqs. (55) and (57) if we know from symmetry considerations which of the permanent multipole moments and second order moments are nonvanishing. Let us start with a discussion of the permanent multipole moments Alm = (OIM,,,,IO). For normal molecules, the wave functions 10) of the ground vibronic state belong to the totally symmetric species of the point group of molecules, and thus the multipole moment Alm is nonvanishing only when M I , in Eq. (47) is also totally symmetric. The transformation properties of M I , with respect to five point-group operations C,,(z) (2nln rotation around the z axis), Z (inversion), and o h ( x y ) ,ov(xz), o , ( y z ) (plane reflections) are considered below. For the operation C, ( z ) which corresponds to 4i+ 4i 2n/n we obtain

+

C,,(z)M,, = eznmiinMlm C,,( z )

(58)

This indicates that for a molecule with an n-fold axis the permanent multipole moment Aimexists only when m is a multiple of n. For the inversion I(ri + - r i ) and b h ( x y ) (Oi + n-Oi) we have similarly

ZM,,= (- I)lM,, I

(59)

and ah(xy)Mlm = ( -

l)l+"M1rnoh(xy)

(0)

COLLISION-INDUCED ROTATIONAL TRANSITIONS

155

When the point group of a molecule contains the symmetry operations ~ , ( x z )(4i -4i) and o,(yz) (4i n -4d, then MI: = (MIrnf MI-,)/2 rather than the individual MI,, belongs to an irreducible representation which transforms6 as +

6 , (xz)M:, =

f (- l)"MIf, 6, ( X I )

(61)

and 6"(Yz)M,: =

f MI:

6"(yz>

(62)

The moment M , , transforms similarly to M I m. Using Eqs. (58)-(62) we find the nonvanishing permanent electric moments as listed in Table 111. The number of nonvanishing moments has been given previously (Bhagavantam and Suryanarayana, 1949; Jahn, 1949; Buckingham, 1967). In deriving the result, the z axis is chosen to be along the axis of maximum rotation symmetry except for the molecules with C, symmetry, in which the axis is taken perpendicular to the 6 plane and for molecules with Td symmetry in which the axis is taken along the axis of the C , and the S4 operation. The derivation of Table 111 is quite straightforward except for spherical top molecules Td and Oh where the spherical harmonics do not necessarily belong to irreducible representations. For these molecules the spherical harmonics were expressed in terms of Cartesian coordinates x, y , z and the C, transformation (x + y , y + z, z + x) was used to find the totally symmetric expression. Using Table 111 and Eq. (55) we can work out the selection rules for the rotational quantum numbers, J , k , M . For example, we see that for a molecule with an ti-fold rotation axis, Ak =jn

( j , integer)

(63)

because of Eq. (58). Care should be taken in applying Eq. (63) to spherical top rotors that the z axis of the rotational wave functions is along the z axis taken to classify multipole moments. For example in CH, the wave functions of D,, type should be used instead of those of C3, type. The nonvanishing moments for the second order interaction can also be derived from symmetry considerations. For the second-order interaction, It is possible to show that Eq. (48) can be expressed in terms of M,, and M k in a symmetric form by using the following properties of rotation matrix, spherical harmonic, and 3j symbol

D;&py) = ( - 1 y - m o ' - m , - m (OrPy) Y,-rn(&+) = (-l)mY,x8,+)

TABLE 111 PERMANENT ELECTRIC

MULTIPOLE MOMENTS OF MOLECULES

(Mlo, Mil*) x (Mia, Mil*) indicates any combination of (Mlo, M i l + ,and M i l - ) and(Mio, Mlo+, Mil-). (MilMll)* =MiIM1l

iMi-iM1-i.

T. Oka

158

M I , does not have to be totally symmetric; however, since (0 I M I , I eo) and (ev I MI,[ 0) are constants [see Eqs. (51) and (53)], the symmetry of the excited vibronic wave functions has to be such that M I , I ev) and (ev I M I , are simultaneously totally symmetric. By applying Eqs. (58)-(62) we can find which interactions are nonvanishing for individual molecules. For example, application of Eq. (58) indicates that Thus, in order for M , , 1 ev) to be totally symmetric, I ev) has to transform

C, (z) I eu)

I

= e - l n m i I ne v )

that is, ( e v I C, (z)-' = e2nmi/n(evI Therefore in order for (eo I M I , to be totally symmetric rn + i?i has to be a multiple of n. From such considerations the second-order interaction terms are derived as shown in Table IV. For the dispersion potential V, in Eq. (7)all the terms given in Table IV contribute, whereas for the induction potential, only those terms which include the permanent moments given in Table 111 contribute. The extension of Tables 111 and IV for higher moments and for molecules with other symmetries is straightforward. The selection rules for rotational quantum numbers J, k , M for individual molecules and interaction can be given by using Tables I11 and IV and Eqs. (55) and (57). In addition to these quantum numbers, the rotational wave functions of molecules with a plane of symmetry have anotherrelevant quantum number, parity (+), which corresponds to the symmetry of the wave function with respect to space inversion [see Chapter VIII in Landau and Lifshitz (1963)l. The parity selection rule is

+*+,

-*-

foreven

I or 1 + 2

for odd

I or I + 1

and

+--

(64)

The selection rules for J , M , and parity of a specific molecule can be derived from Eqs. ( 5 9 , (57), and (64). In general, for the dipole interaction, AJ= +l,O,

A M = +1,0,

+ *-

(65)

for the quadrupole interaction (and for the dispersion interaction of r - 6 ) , A J = f2, +1,0,

A M = +2, * l , O ,

++++

-4+-

(66)

and for the octopole interactions (and for the dispersion interaction of r-'), A J = *3, + 2 , +1,0,

A M = +3, + 2 , +1,0,

+ *-

(67)

COLLISION-INDUCED ROTATIONAL TRANSITIONS

159

The rules together with the k selection rule in Eq. (63) give the full selection rules. Care should be taken in applying Eqs. (63)-(67) to individual molecules because sometimes the rules are mutually exclusive. For example, for a linear molecule with C,, symmetry (such as OCS) in the ground vibronic state, A J = 0, AM = 1 cannot be caused by the first-order dipole interaction because the parity relation + ++ - is not satisfied. As discussed in the previous sections, the accuracy of the first-order perturbation treatment is rather limited; a higher order treatment of the interaction is necessary for stronger collisions. Therefore even for an electric multipole interaction of relatively low order, the values for A J and AM may become large for strong collisions; furthermore, the parity rule breaks down. However, the Ak rule given by Eq. (63) still remains valid because this is the rule based on symmetries of permuting identical nuclei. These symmetries leave the Hamiltonian invariant even for strong collisions as discussed in Section I1,C. For example, for the (123) permutation in NH, , (123)V= V(123)

or

(123)V(123)-’= V

(68)

where V is any intermolecular interaction. On the other hand, the wave functions of a symmetric rotor transform as (123) IJ, k)

= e Z n i kIJ, / , k)

(69)

and ( J ’ , K‘1(123)-’

= e-2niki3(J’, K‘(

Thus, noting that (J’k’l VJJk) = (J’k’1(123)-’ . (123)V(123)-’ . (123)lJk)

- e 2 n i ( k - k ’ ) / J ( ~ t k r I VIJk)

we find the rule Ak

= k-k’ = 3n

(n; integer)

(70)

in order for the matrix element (J’k‘l VJJk) to be nonzero. This rule, which was first obtained experimentally (Oka, 1968b) holds even for strong collisions. This selection rule is essentially the same as the ortho -ortho, para ++para rule given in Eq. (18) but is more useful in quantitative understanding of the observed transition. Since higher energy of impact is necessary to cause transitions with higher Ak values [higher value of n in Eq. (70)], the transition probability decreases with increasing n. For example, transitions from K = I k I = 4 to K = I k I = 5 are extremely weak because, although these are para-para transitions, a change of Ak = 9 is needed. The selection rules for collision-induced transitions are composed of two kinds of rules, one corresponding to the permutation symmetry which holds

160

T. Oka

rigorously if the inhomogeneous magnetic interaction is neglected, and the other, such as those discussed in this section, which gives some quantitative preference of transitions. In order to distinguish the latter from the former we use quotation marks (i.e.,) “ selection rules ” for the latter.7 All symmetry operations discussed in Eqs. (58)-(62) except C,,(n # co) give “ selection rules.” The C , operation in a linear molecule is a special case for which Tables 111 and IV and Eqs. (55) and (57) give selection rules A1 = 0 for any higher order interaction. However, since it is not a permutation operation, the C , operation is not an invariant operation during a strong collision and therefore A l # 0 transitions are weakly allowed.

111. Experiment A. PRELUDE

In 1965, Cox, Flynn, and Wilson published a paper in which they describe, among other things, a double resonance experiment on the OCS molecule as shown in Fig. 12. A dc electric field is applied to the molecule so that the Approximate parity

FIG.12. Energy levels of OCS used by Cox, Flynn, and Wilson (1965). The effect of pumping vc was not observable on the signal vSz. From this result the authors concluded that the collisioninduced transition y is not very fast.

J= I

0 +I

J=O

0

+

rotational levels are split into M components. Then two microwave radiations (vp and v, in Fig. 12) are applied simultaneously to the molecule; both radiations have their electric field parallel to the dc field so that only A M = 0 transitions occur. A relatively strong radiation (v,) “ pumps ” molecules from the J = 0, M = 0 level to the J = 1, M = 0 level and a weak radiation (v,) monitors the resultant change in other rotational levels. Cox et al., monitored

’The words “propensity rules” have been used for the latter (Ottinger, Velasco, and Zare, 1970). “Preference rules” may also be used.

161

COLLISION-INDUCED ROTATIONAL TRANSITIONS

not only the transition v,, ( J = 2 + 1 for M = 0), which is directly connected with the pumping transition and showed expected increase in intensity due to the pumping, but also the transition v,, ( J = 2 +- 1for M = & 1). Since the latter transition is not directly connected to the pumping transition, any effect of the pumping on it must be due to transfer of molecules from the pumped levels to the signal levels by collision. They did not see any effect on the vS2 signal and concluded that " this argues strongly against a dominant, rapid AM = 1, AJ = 0 collision (shown as y in Fig. 12) for this molecule at least."' Although the result was negative, this experiment indicated a possibility that if a suitable rotational energy system was chosen, the microwave double resonance could be used for the study of rotational energy transfer. The first successful observation of such an effect was made a year later (Oka, 1966) in a rotational level system of ethylene oxide (CH,CH,O) as shown in Fig. 1 3. The transition 2, 2, (v,) at 34157.1 MHz was " pumped " and the transitions 3,, C321 (vsl) at 23135.5 MHz and 3,, +3,, (v,,) at 2361 1.5 MHz were monitored. In this system, since both of the signals v,, and v,, are separated from the pumping transition, any effect of the pumping on the signals must be due to collisional transfer. It was observed when vp was pumped that the signal vS1 decreased (by about 12%) and the signal vS2 increased (by about 7%). It was immediately clear that this shows that collision-induced transitions obeying dipole selection rules are preferred-a conclusion which was expected from Anderson's theory (1949) but had not been confirmed experimentally. Since this molecule has a dipolemoment along the b axis of the moment of intertia tensor, the b-type transitions e o t t o e shown in Fig. 13 with wavy lines are dipole allowed [see, for example, Townes and Schawlow (1955)l. The increase of molecular population in the 2,, level (6n > 0) is transferred selectively to the 3,, and 3,, levels and the decrease of molecular population in the 2 , , level (6n < 0) is transferred to the 32, level; therefore the population difference and hence the intensity of absorption for the transitions vSI and v,, decreases and increases, respectively. The other transition v,,(2,, +-2,,) was not affected at all by the pumping because, although energetically close to the pumped levels, these levels are not connected to the pumped levels with the dipole rules. This experiment has established that, contrary to the often used assumption that rotational energy transfer occurs randomly, there is a rather detailed preference for molecules with regard to which levels to make transitions by collision. Ronn and Wilson (1967) confirmed the results not only in

,

+-

* In retrospect this is not surprising because, as discussed in Section II,F, a A J = 0, A M = 1 transition does not satisfy the dipole parity rule This experiment was later repeated with a higher sensitivity and decrease of uS2 was detected which is caused by collision-induced transitions of dipole typeJ = 2 + 1, M = 1 c0,and J = 1 0,M = 1 + 0 (Oka, 1969, unpublished).

+*-.

f-

T. Oka

162 CH2-

CH,

\0/ 8 n>O

330

3 2I

312

212-

FIG.13. Rotational energy levels of ethylene oxide in which the four-level double resonance signal was first detected. When the 221c- 212 transition was “pumped”, the intensity of the 330-321(vIJ transition decreased, that of the 321- 312 (v12) increased, and that of the 21, - 202 (vS3) did not change. The results were explained qualitatively by assuming the dipole-type collisional transitions shown in the picture by wavy arrows (Oka 1966).

ethylene oxide but in several other molecules. More extensive studies of the collision process were difficult in the ethylene oxide molecule, partly because the signals were rather weak and partly because the rotational energy levels are complicated. After trying several molecules it was found that those molecules which have rotational energy levels composed of pairs of symmetrically split levels are more readily studied; thus the K-type doubling spectra of H,CO, H,CCO (Oka, 1967a; Roger, Freund, Evenson, and Oka, 1973), the 1-type doubling spectrum of HCN (Oka, 1967a; Gordon, Larson, Thomas, and Wilson, 1969), the inversion spectrum of NH, (Oka 1967b; 1968a, b; Roussy, Demaison, and Barriol, 1969; Daly and Oka, 1970; Fabris and Oka, 1972), and the rotation-torsion spectrum of CH,OH (Lees and Oka, 1968, 1969) have been used. In this review I would like to summarize these experimental results; more details will be found in the original papers. B. GENERAL PRINCIPLES OF OPERATION As described in the previous section, two microwave radiations are used in the experiment: a strong one called the pumping radiation and denoted by vp , and a weak one called the signal radiation and denoted by v, . In contrast to

COLLISION-INDUCED ROTATIONAL TRANSITIONS

163

normal three-level double resonance in which the pumped transition and the signal transition have one level in common, a four-level system (Fig. 5a) is used for collisional studies. In such a system, since the pumped transition and the signal transition are not directly connected, the effect of pumping appears on the signal only through collisional transfer of molecules from the pumped levels ( 1 , 2) to the signal levels (3, 4).9 The pumping radiation is powerful enough to saturate the pumped transition. The efficiency of saturation 4 is given by Karplus and Schwinger (1948) as

where n i ( i = 1, 2 ) denotes the molecular populations under the pumping, niO denotes the equilibrium (Boltzmann) population, pI2 is the transition dipole moment, E, is the microwave electric field, and z is the relaxation time for the transition 1-2. Equation (71) indicates that when resonant radiation is used for pumping (w = wo), efficient saturation is achieved if

The value of 1/2m can be estimated from the pressure broadening parameter; at the pressure of 10- 100 m Torr used in the experiment, 1/2nz is typically 0.2 2 MHz. Depending on the value of p 1 2 ,that is, a product of the value of permanent dipole moment and the matrix element of the direction cosine, proper pumping field has to be chosen.” For example, for AJ = AK = A M = 0 transitions which are mostly used in the experiment, the transition moment is expressed as pEKM/J(J 1). For a typical case with a dipole moment of I D and for transitions with K - J , the power density of 1 W/cm2 is sufficient to satisfy the condition given in Eq. (72) for all M values (except M = 0), whereas for transitions with K 4 J, either higher pumping power or lower sample pressure must be used. The fact that the M = 0 level is not pumped in the @branch transitions does not affect the result very much if J B I and the effect is monitored by other Q-branch transitions, because the intensity of the signal

-

+

The radiative relaxation is much slower than the collisional relaxation under normal laboratory conditions (except for the case of the three-level maser). The effect of wall collision can be discriminated from that of the intermolecular collision by pressure dependence as will be shown later. l o The microwave electric field E (V/cm) in a waveguide is given in terms of the power density P (W/cm’) by the following formula:

E = {4 ~ o P / [ 1 - ( X / h , ) 2 ] ” 2 } ” 2 where ‘lo= 1 2 0 ~(-377 0) and h and h, are wavelengths of the microwave radiation in the waveguide and the cutoff, respectively [see Southworth (1950)l. A useful rule of thumb is that if the power density is expressed in milliwatts per square centimeter, the square root of the value gives the approximate field in volts per centimeter.

164

T. Oka

is proportional to M Z and collision-induced transitions with A M > I are slower than those with A M = 0, f 1. However, for low J experiments this should be taken into account. If the saturation condition [Eq. (72)]is satisfied, the deviation of the molecular population n, from the normal Boltzmann distribution nio,6ni = n, -ni0, is given by"

an,

-6n, = [nzO- n10]/2N nP0hv,/2kT

=

(73)

where npo denotes the average of n10 and nzo. The signal radiation monitors the collision-transferred population deviation in levels 3 and 4 . The relative change in intensity of the signal absorption is expressed as 17 = AZ/Z = (an, - 6 n , > / ( t 1-~n30) ~

(74)

Although the population deviations 6ni caused by the pumping is small compared with ni because hv, < kT, the value of 17 can be large [for example 30.7% in H2C0, and 31.6% in HCN, Oka (1967a)],because the denominator qf Eq. (74) is also small; therefore the measurement can be done easily. The intensity of the signal radiation should be small because the radiation merely monitors the population difference without affecting it. Thus for the field of the signal radiation E , , the reverse of Eq. (72) should be satisfied, that is ~ ( 3EJh 4

< 1/27~5

(75)

Normally a signal radiation power of the order of 1 pW is used so that the signal power is about a million times smaller than the pumping power. Because of the smallness of ani, we can use linear rate equations for analysis of the results. Without pumping radiation we have

1 (k..n.' - k..n.') = Jl

I

1J

1

=0

(76)

j

Because of the principle of detailed balancing

k j i / k i j= nio/njo

(77)

the individual expressions in the brackets in Eq. (76) are zero. Under the continuous application of the pumping radiation, we have

C ( k j i n j- k i j n i )+ M ( 6 , , - 6,,)(nZ- n , ) = ri, = 0

(78)

j

where 6,, and diZ are Kronecker's delta and M is the rate constant of radiative transitions under strong pumping radiation. Because of Eq. (72), M is In this discussion a Q-branch transition is considered. For R-or P-branch transitions, components for each M-value should be considered separately.

COLLISION-INDUCED ROTATIONAL TRANSITIONS

165

much larger than the kij’s;therefore, from Eqs. (78) for i = 1 and 2, we obtain n , = n2 (or Eq. (73), Sn, = - S n 2 ) . For othervaluesof i weobtain,bysubtracting Eq. (76) from Eq. (78),

c (kj,6nj

- k i j S n i )= 6rii = 0

(i # 1, 2)

(79)

j

Therefore if we know all kij’s we can determine 6ni’s by substituting Eq. (73) into Eq. (78) and solving Eq. (79). Our procedure will be the reverse. From the observed value of q = AI/I, we obtain information of Sn, - Sn, by Eq. (74) and try to relate it to the kij’s. Equation (79), which has infinite number of kij’s is not applicable to practical problems ; we therefore need some approximation. The first approximation is to retain only those terms in Eq. (79) which include 6ni with i = 1, 2, 3, 4, i.e., k , , S n , + k2,Sn2 k4,6n4 - k,iSn, = 0

1

+

j

kI4Sn,

+ k2,Sn2 +k,,Sn,

-

C k,jSn4 = 0 i

This approximation is based on the idea that we perturb the population of levels 1 and 2 and consider the populational changes in levels 3 and 4, assuming other levels work as a thermal bath, i.e., Sn, for other levels are zero. This approximation is reasonable for most of the cases because the h i ’ s ( i # I , 2) are much smaller than Sn, and Sn, in magnitude [the largest so far observed is of the order of 1/ 5 (Oka, 1967a)l. Some cases where higher approximation is needed will be discussed later (Section III,D,5). Equations (80) are simplified considerably in rotational level systems where levels are composed of symmetrically split doublets. Examples of K doublets, I doublets, and inversion doublets are shown in Fig. 14. For example, for the case of K-type doubling (H2CO) or I-type doubling (HCN)

k,3

g

k24

= kaJ

k,, z k2, = kyL

ck3j i

k d j = kar

+ kyr + k , + k ,

j

where k , , etc., represent rate constants for the transitions specified in Fig. 14, CI and p are dipole type transitions with a change of parity, y’s are parity conserving transitions, and 5 represents all the other transitions to the thermal bath. For every constant with a superscript indicatinga transition from higher level to a lower level, there exists another constant with a superscript indicating a reverse process. These constants are related by Eq. (77), k,r/kaA= exp( - A E / k T ) .

T.Oka

166 K-doubling I-doubling

inversion doubling

parity

parity

J

J-l

FIG. 14. Rotational energy schemes of symmetrically split levels. (a) Indicates K-doubling or /-doubling levels where the order of parities of the levels alternates (H2CO or HCN). (b) Indicates inversion doubling levels where the order of parities does not alternate (NH,). When the dipole parity rule t+- is obeyed, the absorption signal decreases in the former but increases in the latter with pumping.

+

For the transition P,AE is sosmall that k,? is equal toksl toagood approximation. There has been no experimental evidence for the relation k,, z k24 and kI4 z k2, in Eqs. (81); these equations are assumed from theoretical considerations. For example k,, = k24 is assumed because both the transitions 1 -,3 and 2 --t 4 change parity, change J by 1, do not change K, and have nearly equal energy gap.12 If Eqs. (81) are assumed, we can derive from Eqs. (80) that k,$ - kYL an, - Sn4 = kUt k,? + 2k, + k , (6n1 - an21

+

This formula expresses the efficiency of transferring a population difference from the 1-2 pair to the 3-4 pair of levels. Using Eqs. (73), (74), and (82) and the principle of detailed balancing we have

q = r AI =-YP.

k,? - k ,

v, k,?

+ k,? + 2k, + k,

l 2 A recent proposal of Townes and Cheung (1969) for explaining anomalously low excitation temperature of interstellar H 2 C 0 is based on the breakdown of (81b). They argue that k I 4and k 2 , are different because the changes in K + in the asymmetric rotor expression J X - K + are different.

COLLISION-INDUCED ROTATIONAL TRANSITIONS

167

A n k+

> kX

%
k, ark,< k , .

The procedure for deri,ving Eq. (83) can be applied to any other system with a pair of symmetrically split doublets such as the one shown in Fig. 15. In general we have

AI

vp

v = - =I - v, ’ 2k,

k’,

- k,’

+ k,’ + k‘, + k ,

(84)

All we obtain from a steady state experiment is one value of AZ/Z for each four-level system ; therefore we cannot determine individual rate constants from this experiment alone. The pressure broadened linewidth Av gives an independent quantity

AV = ( k p + k’,

+ k,’ + kc)/2.

(85)

Various other methods to separate individual rate constants have been developed, but none has proved to be generally applicable to any system; these methods will be described in Section II1,E. In this article we use primarily Eqs. (83) and (84) for the discussion of collision-induced transitions. These equations indicate that the sign of q gives direct information about which of k, and k, or k, and k, is larger than the other. For example, if the observed q is negative in the system shown in Fig. 14a, we know that k, > k,,that is, the parity changing transition is preferred to the parity conserving transition. The preference rule thus obtained will be called a “selection rule” (see footnote 7). We cannot obtain quantitative

T. Oka

168

preference from the measurement alone but, by using the fact that the rate constants are positive, we can set an upper limit for k , / k , . For example, if vp/v, 1.5 and the observed q is 30% [H,CO (Oka, 1967a)], k , cannot be more than half of k, , even if we assume k , = 0, because for normal four-level systems k , is at least of the order of k, . I 3 Obviously the larger the value of q, the more rigorous the “selection rules,’’ i.e., the ratio of the rate constants of preferred transitions of those of unpreferred transitions is larger. By using various combinations of J, K and J’, K ‘ (Fig. 1 9 , we can obtain “selection rules” for rotational quantum numbers J and K. If the intensity of the (J’K’) absorption line is affected by the pumping of the ( J , K ) line, this is regarded as evidence that collision-induced transitions between the ( J , K ) levels and the ( J ’ , K ’ ) levels are “ a l l o ~ e d . ”On ~ ~the other hand, the reverse of the statement is not necessarily true, that is, the fact that the intensity of the (J’, K’) line is not affected by pumping the ( J . K ) line does not necessarily mean that the transitions between them are “forbidden,” because k , may be equal to k , . However, the latter case is not experienced frequently; in most cases there exists a parity “selection rule” which favors one of the transitions, 4 or x, over the other. Sometimes (as discussed in Section II1,C) the reason for the existence of such a parity rule is not clear, but its existence is very convenient. Therefore, if transitions of a particular kind are not observed in several systems, we can conclude with some certainty that that type of transition is “forbidden.” The selection rules obtained from such experiments are summarized in Section III,C. A block diagram of a double resonance apparatus is given in Fig. 16. It has been said “In every experiment there is one central feature or part, which really matters. This must be first rate, but it is not always necessary that everything else be of the highest quality obtainable ” [Thomson (1959)l. The central feature of the present experiment is the filter in the detection path. The filter should pass the signal radiation but stop the pumping radiation efficiently. Since the power of pumping radiation is lo6 times more than that of signal radiation, discrimination by a filter of at least 70 db is required. It appears likely that the inconsistent results by Roussy et al. (1969) resulted from difficulties with the filter. Three types of filters have been used (a) commercial low pass filters, (b) waveguide cutoff filters, and (c) cavity filters; they are useful when (a) vp > v, + 3 GHz, (b) vp < v, - 0.5 GHz, and (c) I vp - v, I > 30 MHz, respectively. The cavity filter has the advantage that the

-

l 3 It should be remembered that the transition 6 includes all the transitions to the thermal bath and therefore also includes the a-type transitions from the J - 1 to J - 2 levels. l4 Sometimes it happens that the non-Boltzmann distribution broughtinto the(J, K)levels by pumping is transferred to the(J’, K’)levels via some other levels, say ( J ” , K”).Such a case will be found by performing systematic experiments over many (J, K ) levels.

COLLISION-INDUCED ROTATIONAL TRANSITIONS

4

9 US

us

169

t FREOUENCY

FILTER

PUMPING KLYSTRON

METER

DUMMY

LOAD

discrimination is much sharper in frequency but has the disadvantage that the transmission loss is large (typically 90%); the appropriate filter has to be chosen for each experiment. For more details of experimental procedures readers are referred to the original papers (Oka, 1967a, 1968a, 1969). Two qualitative remarks can be made from Eqs. (83) and (84): (1) In order to have a large signal, it is advantageous to use the system in which vp > v,; ( 2 ) since all k’s are proportional to pressure, the value of q = AI/I is independent of pressure. Therefore, accurate pressure measurement is not needed.’ C.

“

SELECTION RULES

”

The microwave double resonance method can be applied for studies of collision-induced transitions only in polar molecules. It is convenient to classify such transitions into categories according to the collision partner. The three types of collision partners which will be discussed here are (1) polar molecules, (2) nonpolar molecules, and (3) rare-gas atoms. In case (I), (polar molecule)-(polar molecule) collisions, the signs of the observed signals are explained by assuming that the collision-induced transitions obeying dipole selection rules are preferred. A great many transitions have been observed in a variety of molecules such as (CH2),0 (Oka, 1966); F,CCH,, (CH,),S: HCN (Ronn and Wilson, 1967); H2C0, H,CCO (Oka, 1967a); NH, (Oka 1968a); CH,OH (Lees and Oka 1969); and C,H,OH (Seibt, 1972), with a variety of polar collision partners and there have been no exceptions. Whenever a collision-induced signal is observed in a four-level system in which at least a pair of levels are connected by a dipole-type transi-

’’

A pressure dependence of 7 occurs at high pressures where the pumping efficiency decreases and at low pressures where wall collisions (collisions without “selection rules ”) are not negligible (Oka, 1967a).

T. Oka

170

tion, the sign of the signal is explained by assuming that the dipole transition is preferred. The dipole-type transitions constitute transitions with AJ=O, +I,

A M = O , +1,

parity+--

with additional rules; for a linear molecule, A1 = 0, for a symmetric top molecule, A K = 0, and for an asymmetric top molecule, A K - = even, A K + = odd (pa transition), A K - = odd, A K + = even (pc transition), and A K - = odd, AK+ = odd (pb transition). In some cases, transitions with AJ > 1 have been observed in (polar molecule)-(polar molecule) collisions. For example AJ = 2 transitions were observed in H,CO and HDCO and AJ = 2, 3, 4 transitions were observed in HCN and H,CCO (Oka, 1967a). The analysis of rate equations (Oka, 1967a) and triple resonance experiments (Lees and Oka, 1968) has indicated that these AJ > 1 transitions are not of the stepwise nature that are caused by two or more successive AJ = 1 transitions, but are transitions in a single collision. This has been interpreted as a result of higher order dipole-dipole interaction rather than that of a quadrupole or higher multipole interaction (Rabitz and Gordon, 1970). In the case (2), (polar molecule)-(non,polar molecule) collisions, the qualitative features of the observed signals are generally similar to the case (1). Again the dipole-type transitions are always preferred to the corresponding transitions of other types. This has been observed for a variety of collisions such as NH,-H, (Lees and Oka, 1968);CH,OH-H,, O,, CH, (Leesand Oka, 1969); NH,-H, ,para-Hi, ,HD, D, ,0, ,N,, CH, , SF6 (Daly and Oka, 1970); and H,CO-H, , para-H, , D,, 0, , CH,, SF, (Roger et al., 1973). This probably indicates that most of the rotational transitions are induced by interaction of the dipole moment of the polar molecule and the multipole moment of the nonpolar molecule. It should be remembered (Section II,E) that as long as the dipole moment of the polar molecule is used, the selection rules given by the first-order perturbation is of dipole type regardless of which electronic multipole of the second molecule is used. One noticeable difference between the (polar molecule)-(polar molecule) collisions and the (polar molecule)-(nonpolar molecule) collisions is that in the latter the ratio of the rate of the AJ > 1 transition to that of the AJ = 1 transitions is higher than that of the AJ = 1 transition. For example AJ = 2 transition have been detected in NH,-H, , para-H, , HD, and D, collisions by using the microwave triple resonance method (Lees and Oka, 1968) and also by solving rate equations for normal double resonance results (Daly and Oka, 1970) but no such transitions have been observed for NH,-(polar molecule) collisions. In the CH,OH-H, collisions, the AJ = 1,2,3,4,5 transitions have been detected whereas in pure CH,OH only the AJ = 1 transition has been detected (Lees and Oka, 1969). This confirms the expectation that the interaction with

+

COLLISION-INDUCED ROTATIONAL TRANSITIONS

171

a nonpolar molecule is of shorter range than that with a polar molecule, and higher order transitions have to be considered. Violation of the AK = 0 rule in NH, was looked for (Daly and Oka, 1970) but was found only recently in NH,-H, collisions (Fabris and Oka, 1973). It has been indicated theoretically as well as experimentally that an H, molecule in the J = 0 level acts like a rare-gas atom as a collision partner since the average electric multipole moment for the J = 0 level is zero (Fabris and Oka, 1973). Thus it has been shown that in discussing rotational equilibrization of interstellar molecules the question of whether the interstellar H, molecules are para-H, or normal H, is crucial. In case (3), (polar molecule)-( rare-gas atom) collisions, the “ selection rules are markedly different from those for cases (1) and (2). Since the interaction is of much shorter range, the dipole rules are no longer obeyed; AJ > 1 transitions are observed with large intensity relative to the AJ = 0, L- 1 transitions, A K # 0 transitions are observed, and no uniform parity rules are noticed. The only remaining rule is that of the k quantum number in symmetric top molecules, i.e., Ak = 3n (Oka, 1968b). The transitions of case (3) have been studied in NH,-He, Ar, Xe (Oka, 1967b, 1968b; Fabris and Oka, 1973); (CH,OH, CH,NH,)-He, Ar, Xe (Lees and Oka, 1969); H,CO-He, Ne, Ar, Xe (Roger et al., 1973); C,H,OH-He (Seibt, 1972); and HCN-He, Ar, Xe (Cohen and Wilson, 1972). Experiments using NH, have provided the most extensive information since the richness of the inversion spectrum allowed a systematic study using many four-level systems. In Fig. 17 a comparisonof case ( I ) and case (3) is given. In bothcasesgivenin Fig. 17aandFig. 17b,respectively, a non-Boltzmann distribution is introduced into J = 5, K = 4 levels (double circled in the figure) by pumping and population change in other levels were monitored. In the NH,-NH, collisions (case I), the effect of pumping was observed only on two levels (circled in Fig. 17a) whereas in the NH,-(raregas atom) collisions, many other levels were affected by the pumping. Thus we see that more collision-induced transition (shown in Fig. 17 with wavy arrows) are “alloued” in the latter. This is not interpreted as an indication that these non-dipole-type transitions occur more efficiently in the (polar molecule)-(rare-gas atom) collisions but rather an indication that in the (polar molecule)-(polar molecule) collisions the dipole-type transitions are so predominant that other transitions are obscured.I6 Readers are referred to original papers cited above for more details of the results. In the following, some crucial experiments for various selection rules are summarized. ”

l 6 It should be remembered that we are measuring ratio of rate constants given in Eqs. (83) and (84); the denominator of these expressions always contains the rate of dipole-type transitions.

T.Oka

172 a

NH3 -NH,

NH,-He

E(cm-')

,A,Xe

E(crn-'l

1 - 1

2

I

,

I

I

I

3

4

5

6

7

I

FIG. 17. Comparison of four-level double resonance experiments in NH3 for weak collisions (NH3-NH3) and strong collisions (NH3-He). In the former only two transitions obeying the dipole selection rules ( A J = 0, i l , AK = 0) were observed whereas in the latter many other transitions were observed. The only remaining selection rule in the latter is Ak = 3n ( n ; integer) (Oka, 1968a, b ; Fabris and Oka, 1973).

I . Parity Rule The parity is a relevant quantum number if a molecule has a plane of symmetry. Especially in linear molecules (HCN, OCS), planar molecules (H,CO), and inverting molecules (NH,), each rotational level has a unique parity" (Herzberg, 1966). The experiment shown in Fig. 14 is a typical example of many other experiments which indicated a systematic parity rule. In the Kdoubling or /-doubling levels (Fig. 14a), the order of the parities of the split levels alternates as J changes, whereas in the inversion doubling levels (Fig. 14b) it does not.'* Therefore when the parity changing transitions, a, have larger probabilities than the parity conserving transition, y , the pumping decreases the absorption signal in the former (Fig. 14a) but increases it in l 7 This is because the permutation-inversion group of these molecules contains the space inversion operation E*. This is because the parity is proportional to (-l)J, (-1)" and (-l)K+ in a linear molecule, in a symmetric top molecule, and in a planar asymmetric top molecule, respectively. The relevant rotational quantum numbers in the discussion of parity are those which correspond to the angular momentum perpendicular to the plane of symmetry associated with the inversion operation E* [Oka (1973)l.

173

COLLISION-INDUCED ROTATIONAL TRANSITIONS Stork Effect

1 -doubling

FIG.18. Four-level resonance experiments in OCS. A four-level system caused by Stark effect ( M = 0, i 1) and that caused by /-doubling in the excited bending state are compared. Because of the difference in parity relation (indicated in the figure with and -), the pumping incremes the signal in the former but decreases in the latter (Oka, unpublished).

+

the latter (Fig. 14b). This has been observed for many collision partners as will be described later. An example of reversing the sign of the signal in the same molecule because of the presence of the systematic parity rule is shown in Fig. 18. In these experiments four-level systems of OCS caused by the Stark effect (Fig. 18a) and the I doubling (Fig. 18b) are compared; these two four-level systems are similar in that the splittings are much smaller than the spacing of rotational levels but are different in parity relations as shown in the figure. Because of the preference of the parity changing transitions shown in Fig. 18 with wavy arrows, the pumping increases the signal in the Stark system but decreases it in the /-doubling system. The experimental results so far obtained indicate that for collision-induced transitions with AJ = 0 or AJ = f 1 (and A K = 0 ) , parity changing transitions always have larger probabilities than parity conserving transitions; the only exception to this rule has been the case of NH,-He collisions. As mentioned earlier, the existence of such a general parity rule in the (polar molecule)-(polar molecule) collisions and in the (polar molecule)-(nonpolar molecule) collisions is understood as due to the dominance of the first order

T.Oka

174

dipole interaction, but the reason for the existence of the same rule for (polar molecule)-(rare-gas atom) collisions (except He) is not clear. The longest range interactions in such collisions are of the r - 6 type, that is, the dispersion and the induction interaction which involve off-diagonal dipole matrix elements of the rare-gas atom; the parity rule for both of these interactions is + c* + and - t* -. In this respect the preference of parity conserving transitions in NH,-He collisions is easier to understand. The experimental results shown in Fig. 19 indicate very definitely that, in NH,-Ar and NH,-Xe collisions, the parity changing transitions are preferred (Oka, 1967b). As shown in Fig. 19 even in NH,-He collisions, the parity changing transitions

-5

t

i +

FIG.19. Plot of the values AI/Z versus J observed in the [(J 1, K),-(J, K),] (J= K) four-level systems for NH3-NH3 (a),NH3-He (O), and NH,-Ar (0) collision. The positive values of Al/I for the NH3-NH3 and NH3-Ar collisions indicate that the parity changing +++-transitions are preferred. On the other hand, the negative values of AI/Zfor J = I , 2, 3 of the NH3-He collisions indicate that the parity conserving transitions are preferred. (Oka, 1967b, 1968b).

COLLISION-INDUCED ROTATIONAL TRANSITIONS

175

are preferred for large J , K values ( J z 5). No explanation has been given as to why this happens and why He gives different parity rules from Ar and Xe. It should be emphasized that NH,-He has been the only case in which reversing of the signal is observed in dipole-type transitions. Similar experiments have been performed in H,CO-He, HCN-He, CH,OH-He, and C,H, OH-He collisions, but no reversing has been observed in these systems. For collision-induced transitions with AJ = 2 (and A K = 0), parity conserving transitions have larger probabilities than parity changing transitions. Many transitions of this type have been observed and there have been no exceptions. In general, in the system where transitions with large AJ (AK = 0) are observed (Oka, 1967a), the parity rule is + ++ - for odd AJ and + c) +, and - tf - for even AJ (except 0). For strong collisions in which both AJ and A K change [which so far have been observed only in NH,-(rare-gas atom) and in NH,-H, collisions], no uniform rules have been observed. Frequently, it is observed that the same four-level system has different parity rules for NH,-He collisions and for NH,-Ar or NH,-Xe collisions, although the latter two seem to have the same parity rule for most of the systems studied. In the NH,-He collisions, the + tf - transitions are preferred in 29 of the total of 45 four-level systems which gave nonvanishing AZ/Z and the + f-.t + transitions are preferred in the other 16 systems. In the NH,-Ar collisions the numbers are 19 versus 17, and for the NH,-Xe collisions it is 12 versus 16 (Oka, 1968b). Although the parity relations are random, it has been noted that the values of AZ/Z are more frequently negative in the NH,-He collisions (31 negative versus 14 positive), whereas they are more frequently positive in the NH,-Ar collisions (31 positive versus 5 negative) and in the NH,-Xe collisions (21 positive versus 7 negative). It looks as if transitions between the same inversion levels are favored in the NH,-He collisions, while transitions between the different inversion levels are favored in the NH,-Ar or NH,-Xe collisions regardless of the symmetry of the rotational state. At present no explanation has beeen given for this. Although the reason for the preference of one transition over the other (compare the transitions 4 and x in Fig. 15) is not well understood, the existence of the preference is very convenient for studies of J and K “ selection rules.” 2 . J “Selection Rule”

For most of the systems studied, the transitions with AJ = k 1 have larger probabilities than the corresponding AJ > 1 transitions, and the probability decreases as AJ gets larger; a few exceptions to this rule are NH,-(rare-gas atom)(again!); CH,NH,-He(Leesand Oka, 1969);and HD(Akins,Fink,and Moore, 1970). The dipole rule AJ = & 1, 0 is obeyed most strictly in collisions

T. Oka

176

(a 1

(b)

FIG. 20. Comparison of (a) the A J = 2 experiment and (b) the A J = I experiment. The signs of 8n denote the expected variation of molecular populations in each level by cascading dipole-type transitions. The arrows in the bottom figures of absorption show the observed increase or decrease of the signal (Oka, 1968a).

when both molecules are polar and light. Examples of this case are the NH,NH, collisions and CH,OH-CH,OH collisions in which only AJ = 0, f 1 transitions are observed. Figure 20 shows a n experiment which indicates the dominance of the AJ = 1 transition. When the inversion spectrum of NH, in the (5, I ) rotational levels ( J = 5 and K = 1 ) is pumped, the absorption signal at the (4,1) levels increases its intensity, whereas the signal at the (3, I ) levels decreases. The opposite direction in the changes of the signals is understood as a result of the parity rule discussed earlier. Moreover, it is observed that the magnitude of the decrease of the signal at the ( 3 , I ) levels agrees within experimental uncertainty to that calculated on the assumption that the non-Boltzmann distribution produced in the (5, I ) levels is transferred to the (3, 1) levels by cascading. In the NH,-H, collisions, however, the change of the signal observed at the ( 3 , 1) level is so large that it clearly indicates the existence of the single AJ = + 2 transitions. Even for a (polar molecule)-(polar molecule) collision, the existence of the AJ > 1 transitions are clearly observed when the molecules are heavier and the levels are more closely packed. Figure 21 indicates a series of observed signals in I-doubling system of H C N ; the direct I-doubling transition with J = 13 was

+

177

COLLISION-INDUCED ROTATIONAL TRANSITIONS

HCN L = I

30

DCN I = I

Ar/i 20 %

\

10%

\.

\\

,

1

J=lI

J:IO

.--T-

I I

J=9 J=8

(dipole)

5.12

J:ll

JzlO

J=9

FIG.21. Observed values of Al/I for I-doubling rotational energy levels of HCN and DCN; the /-doubling line of J = 12 and J = 13 were pumped, respectively. The sign 0 represents the observed values of A l / I . The sign represents the on the basis of dipole-type cascading model, and A represents the of dipole- quadrupole-type cascading model (Oka, 1967a).

pumped and the changes of signals at other [-doubling lines with J = 12, I I , 10, and 9 were monitored. A large variation of the signal was observed even for J = 9 which cannot be explained by cascading transitions. I n Fig. 21 the results expected from the cascading transitions are drawn with broken lines. The large difference between the observed values and the values calculated on the assumption of cascading indicates the existence of the AJ > 1 transitions. An experimental method to discriminate the populational transfer due to single transitions from that due to cascading transitions was developed by using a triple resonance technique, the energy scheme of which is shown in Fig. 22. In this experiment, in addition t o the pumping and the signal radiation, we use one further strong radiation, the “clamping” radiation. The role of this radiation is to artificially increase the transition probability between the components of the J - 1 levels and thus to interrupt the cascading process which transfers the population difference from the J doublet to the J - 2 doublet. The experiment proceeds a s follows. We first observetheJ - 2 transition and adjust the frequency of the clamping radiation to the J - 1 resonance; this results in a change of the signal which is denoted by AI, . Then, keeping the clamping at resonance, the pumping radiation is tuned to the J resonance. This produces a furthet change of the signal, AZ, which is caused by single AJ = 2 transitions. The results obtained from such experiments established the existence of the AJ > I transitions and gave relative values of the rate of ( A J = I 2 transitions to those of (AJI = 1 transitions. Collision-induced transitions with largest values of AJ have been observed

T. Oka

178

A J=l

J=Il

Pumping

J = 10

Clamping

J=9

Signal

Normal

AJ.2

Clamping

Clamping +Pumping

FIG.22. A six-level system for a triple resonance experiment and the observed results for /-doubling of HCN. The J = I 1 doublet levels were “ pumped,” the J = 10 levels were “clamped,” and the J = 9 levels were observed. The oscilloscope traces show the intensity changes of the J = 9 /-doubling line of HCN for each process (Lees and Oka, 1968).

in collisions with He atoms because for such collisions the denominator in Eq. (84) is small. Figure 23 indicates the results of an experiment of CH, OH-He collisions in which Q-branch series of CH,OH have been used; a transition of Q-branch series with J = J , was pumped and transition with J = J, = J , AJ were observed. The experiment was performed for J , = 12, 11, 10, and 9 and the observed change of the signal extended to levels with AJ = 6, 6, 5, and 4, respectively. Readers are referred to the original paper for more detailed discussions. In this case there was a rather systematic tiend in the dependence of intensity on AJ. In the NH,-(rare-gas atom) collisions shown in Fig. 24, no such trend has been observed. In this case no symmetry restriction exists and the decrease of the signal with increasing AJ is simply a result of the decrease of transition probabilities due to the increase of AEr (see Eq. 13) which has to be converted to translational energy. It is surprising that the collision-induced signal has been observed with AEr as large as 400 cm-’ (Oka, 1968b).

+

t

l3

10

0

CH, O H

!Y T

1

- He

Js = 12

v

II 10 9

0

J,

2

3

AJ

=

r

J5- A J

1

4

5

6

FIG.23. Variation of AI/I with AJ for the Q-branch in CH,OH-He collisions (Lees and Oka, 1969).

. .

-

-2 -3 --4 -5 -

9

:

A

.

.

"

~

u -5

-4

-3

-2

-I

1

2

3

4

5

AJ

FIG.24. Observed values of AZ/I versus AJ for NH,-rare-gas collisions. Each point represents some hundred measurements on a four-level system. The zero values of A I / l in the NH,-Ar and the NH,-Xe collisions are omitted for the sake of simplicity. No obvious selection rules are noticed. The decrease of A l / l for high J values is due to conservation of energy and angular momentum during a collision process. He, A Ar, U Xe (Oka, 1968b).

180

T . Oka

3. k “Selection Rule” A systematic study of the k “selection rule” has been done only for NH, for which inversion splitting provides many absorption lines in the microwave region. For other splittings such as k doubling or I doubling, the magnitude of the splitting is highly dependent on the value of k or I and systematic study of the k rule has not been possible. For most collision partners only AK = 0 tIansitions have been observed in NH, . The only exceptions to this are the NH,-H, collisions in which Ak = 3 transitions were observed weakly (Fabris and Oka, 1972) and the NH,-(rare-gas atom) collisions in which Ak = f 3 and Ak = + 6 transitions were observed (Oka, 1968b). The selection rules Ak = 3n ( n : integer) was obtained empirically from these observations, As an example let us consider the experiments shown in Fig. 17b which led us to the k rule. In these experiments of NH,-(rare-gas atom) collisions, non-Boltzmann distribution is introduced to the ($4) inversion levels by the pumping radiation. The effect of punipingis observed ink = 7 and k = 1 levels which are separated from the pumped levels by Ak = + 3 . The effect was also observed in the levels with k = 2 but the magnitude of the signal is smaller; these transitions are interpreted as due to k = 4 c* -2 and k = - 4 - 2 which correspond to Ak = + 6 transitions. The only reason for interpreting these transitions as the ones with Ak = + 6 rather than the ones with Ak = + 2 is that the former interpretation explains the observed results in some 127 four-level systems more systematically. The effect of pumping the k = 4 levels was not observed on k = 5 levels, although energetically these levels are very close and both levels correspond t o para-NH, . This is because the k = 4 levels and the k = 5 levels are connected by Ak = + 9 transitions rather than by Ak = f l transitions; in order for the former transitions to occur, 9 units of angular momentum have to be either supplied or taken away by the relative motion of molecules and the probability of this is low. The existence of the Ak = 3n rule is more obvious in the case of orrho-NH, for which the value of k is multiple of 3 ; in these levels the Ak = 3/1 rule is equivalent to the ortho-ortho rule. The observed values of AI/I in the NH,-(raregas atom) collisions are plotted in Fig. 25. The existence of the Ak = 311 rule is obvious in the figure. An analysis of the results of CH,OH-He collisions shown in Fig. 23 gives a k “selection rule” in CH,OH which is in sharpcontrast to theoneforNH,(rare-gas atom) collisions. Since the CH,OH molecule has a dipole moment in the plane of ab axes (p, = 0.885 D and & = 1.44 D), the Ak = 1 as well as Ak = 0 transitions are allowed by the dipole selection rules. Indeed, the results of pure gas studies can be explained by assuming both of these transitions to be “ allowed.” However, the experimental results on CH,OH-He collisions indicate that in such collisions the Ak = 0 transitions have higher probabili-

+

+

181

COLLISION-INDUCED ROTATIONAL TRANSITIONS

c

*-T-----

I

1

0

1

,

/

3

,

,

,

6

,

,

,

9

,

,

/

-

,

IAk

1

,

15

12

I

FIG.25. Observed values of A l / I versus I Akl for NH3-rare-gas collisions. Each point represents some hundred nieasurements on a four-level system. The 1XI signs summarize about 50 zero points. The zero values of AI/I in the NH3-Ar and the NH,-Xe collisions are omitted for the sake of simplicity. The existence of the k = 3n rule is obvious. He, A Ar, D Xe (Oka, 1968b).

ties than the Ah- = L- 1 transitions. Thus contrary to the case of NH, where long-range interaction with polar molecules causes only Ak = 0 transitions and short-range interactions with rare-gas atom increase the value of Ak to 5 3 or k 6 , in CH,OH short range interaction tendstodecrease the probability of Ak # 0 transitions with respect to those with Ak = 0. This difference is interpreted as due to the shape of the molecule; in an oblate rotor like NH , , the short-range interaction can efficiently exert a torque around the molecular axis, thus changing the values of k , whereas in a prolate rotor the short-range interaction exerts torque around axes perpendicular to the molecular axis and thus changes the values of J but does not change the values of k easily. N o collision-induced transition between the A and E torsional states of CH,OH has been observed (Lees and Oka, 1969); such transitions are forbidden by the symmetry of the spin wave functions. Several attempts have been made at observing collision-induced transitions betweeen different vibrational levels but none has been successful. The vibrational relaxation is typically a thousand times slower than the rotational relaxation and it is unlikely that a strong signal can be observed by the four-level microwave double resonance method.

T. Oka

182 4. A4

"

Selection Rule"

In order to study the selection rule on M , the quantum number corresponding to the projection of the total angular momentum along a space fixed Z axis, the degeneracy of M levels has to be lifted by an external electric field. This kind of experiment can be performed only for rotational levels with relatively low J values. Since the difference between the pumping and signal frequencies is small, the cavity filter described in Section II,B has to be used. Figure 26 indicates experiments in NH, using (3, 3) rotation-inversion levels

Ir-t N H 3 ( 3 , 3 ) Line

M

E F 3

2

d

0 I 2

3

I' = I

Parity

-t

-+

+79%

*I = t74%

I

+129%

0%

+92%

-06%

FIG.26. Four-level double resonance experiments leading to M-selection rules. A dc electric field was applied to split the (3, 3) levels of NHs into M components. The double resonance experiment was conducted using a cavity filter. The numbers indicate the observed values of AZ/I (Oka, 1969, unpublished).

split by an external electric field. The result of these experiments using pure gas indicates clearly that the M selection rules are A M = 0, 1 and the parity changing transitions are preferred to parity conserving transitions. Foreign gas experiments are rather difficult to perform in this system because they require higher pressures that lead to pressure broadening and to discharge. However, an experiment on a NH,-He, mixture did indicate breakdown of

COLLISION-INDUCED ROTATIONAL TRANSITIONS

183

the M selection rules. Relatively few experiments of this type have been done so far, but the results of such experiments will be of great importance for quantitative understanding of collision processes.

D. MAGNITUDE OF A l j l The great number of observed values of AIjI are related to the rate constants of collision-induced transitions and therefore include quantitative information for these parameters. However, since the value of AIjZ is a function of several rate constants as seen in Eqs. (83) and (84), the values of individual rate constants are not obtained from four-level experiments alone. Since Eqs. (83) and (84) include four rate constants, two more independent experimental values are necessary in addition to the value of A l j l and the pressure broadening parameter. No such complete determination has been done. The closest to it was the interesting experiment by Gordon et al. (1969) in which they determined the ratio of individual rate constants of HCN by the use of modulated double resonance on certain assumptions on k i . This method will be described later. It is hoped that relaxation times T, obtained from various relaxation experiments such as the microwave echo (Jenkins and Wagner, 1968), the decay of emission (Hill, Kaplan, Herrman, and Ichiki, 1967), and the decay of transient nutation (Schwendeman and Brittain, 1970) (see Table 11) are related to the rate constants in a different form than the relaxation time zp obtained from pressure broadening and thus can be used as additional information. So far several experimental results seem to indicate that T is equal to zp within experimental errors. However, I believe that their difference will be revealed with advances of experimental methods. The linewidth is affected by molecular transitions in which the magnitude of the total angular momentum does not change but its direction with respect to a space-fixed axis changes (reorientation). These transitions do not affect the number of molecules in each energy level and therefore do not contribute to relaxation time T I . Thus it will make zP < T . On the other hand the rate equation approach discussed in Section III,E indicates that T~ > z. Gordon, Klemperer, and Steinfeld ( 1968) give an opposite example zp < T. This is an interesting unsettled problem. In any case the experimental determination of individual rate constants has yet to be done. In this section we discuss some of the interesting results of steady state double resonance and interpret semiquantitatively the observed magnitude of A I / I for various systems. 1. Large Signals in H,CO, HCN, and H,CCO

Large values of Ali'l ( - 30 %) have been observed for K-doubling systems of H,CO, HDCO, and H,CCO and for /-doubling systems of HCN and DCN (Oka 1967a). The energy level system and the observed effect for H,CO is

T. Oka

184

h i

ii

I

24.3 cm-’

FIG.27. Rotational energy scheme of H,CO and observed effect of pumping on the signal. The oscilloscope trace “normal” shows the absorption line 927-928under Boltzmannian distribution and the trace “ non-Boltzmann ” shows the same line when the transition 102s-1029is pumped (Oka, 1967a).

given in Fig. 27. The large observed value of A l j l (-30.7 k 2 %) clearly demonstrates that the collisional transfer of molecules from the J = 10, K = 2 levels to the J = 9, K = 2 levels by the dipole-type a transition is very efficient. The other transitions shown in Fig. 14a, that is, the quadrupoletype AJ = 1 transition, y , and the dipole-type AJ = 0 transition, /I, tend to reduce the value ofAI/I. Using the argument given in Section 111, B [after Eqs. (83) and (84)], we can set the upper limit for the rates of these transitions as

k , < 0.48k,

and

k, < 1.4ka

The latter conclusion was rather surprising when the large signal was first observed because it meant that the efficiency of transferring large amounts of rotational energy (24.3 cm-’ for a ) is nearly equal to that of transferring small amounts of energy (0.8 cm-’ for /I). This is explained by noting the fact that the dipole matrix element for the AJ = 1 transitions, a, is proportional to ( J 2 - K 2 ) l i 2 whereas that for the AJ = 0 transitions, /I, is proportional to K . In all the K-doubling (and I-doubling) systems examined, the value of J is

COLLISION-INDUCED ROTATIONAL TRANSITIONS

185

much larger than that of K (and I ) and thus the matrix element favors the CI transition. Since the amount of transferred rotational energy (24.3 cm-') is rather large, the probability for a process in which this much energy is converted to translationalenergy is small. However, rotational resonance between two H 2 C 0 molecules (see Fig. 7) makes the AJ = i-1 energy transfer very efficient. The resonant collision partners are relatively abundant because they may have any value of K ; furthermore, the resonance does not have to be very accurate. Application of the first-order treatment of Andersoq gives the value of k, = 1.2k, (Oka, 1967a). More detailed treatment of this system has been done by Prakash and Boggs (1972), the result of which agrees well with the observed A1;Z. Figure 28 indicates the effect of resonance in H 2 C 0 calculated

Jz

FIG.28. The Boltzmann factor of the collision partner and the transition probabilities as a function of the J value of the collision partner. The peaks for the n transition indicate the resonance shown earlier in Fig. 7. The "width" of the resonance is of the order of 1/2n~, where7-,isthecollision time.--', C L , ; .... ~ " , ~ ' ; - . - . ( , , ( ~ ; - - - / 3 . ( P r a k a s h and Boggs, 1972).

T. Oka

I86

by Prakash and Boggs. It is noted that the resonance enhances the probability very much and that the width of the resonance is of the order of 5 cm-' as discussed in Section II,D. The case of 1 doubling in HCN is special in that this doubling occurs only in the first excited bending vibrational state which is 71 1.7 cm-' above the ground state. Therefore, in contrast to H,CO or NH, where most of the collision partners also have small doubling, most of the collision partners in HCN do not have doubling. As a result of this the resonance condition given in Eq. (14) is not easily satisfied for the transition 8. When molecule 1 in the excited state changes parity by a 8-transition, molecule 2 in the ground state has to change parity by an a-transition. This lack of resonance together with the smallness of the matrix element makes the rate of 8 transition in HCN Pumping J = K + I series Signal J = K series

cm-' 500

7= 7=

400

300

200

too

0

.I

K=I

K.2

K=3

K.4

K.5

K.6

FIG. 29. Rotation-inversion energy levels of NH3 used in a set of experiments, the results of which are shown in Fig. 30 and in Table v. The J = K+ 1 series of inversion doublet levels are pumped, and the J = K series are monitored. Each broken line summarizes four collision-induced transitions occurring between a pair of doublets. The splittings of the inversion doublets are magnified with respect to the rotational separations.

COLLISION-INDUCED ROTATIONAL TRANSITIONS

187

extremely small. Therefore, a large value of AZ/Z is observed. The detailed study of HCN collisions by the Harvard group will be discussed in Section III,E. 2. J , K Depeiideence of AIII in NH,

Unlike K doubling or I doubling, the frequency of the inversion doubling in NH, is dependent on the rotational quantum numbers J and K only weakly through centrifugal distortion. Therefore, any inversion doublings appear in the microwave region and systematic studies of collision-induced transitions over many rotational levels are possible. Figure 29 indicates rotation inversion levels of NH, and one set of four-level double resonance experi-

-I

1.5

I -

0.5

-

1

1

2

3

4

5

8-K 6

FIG.30. Observed values of AZ/Z. The unfilled circles represent the observed values by using a 30 cm cell and the filled circles represent observed values by using a 3 m cell (Oka, 1968a).

ments performed by pumping J = K+ 1 series and monitoring J = K series. The result of observations is shown in Fig. 30. The smallness of the signal ( 1.54 0.35 %) is due to two factors (1) k , is much larger than k, , and (2) Ay is not much smaller than k , . The factor (1) is explained by the fact that the matrix element for the /? transition is larger than that for the a transition for K = J , and the fact that almost all the collision partners (except those

-

T. Oka

188

with K = 0) have inversion doubling and therefore the resonance for the p-transition is almost always satisfied. Anderson (1949) calculated the pressure broadening of the NH, spectrum and concluded that for the pressure broadening of J = K inversion lines (with J > l), the effect of the tl transition is negligible. Yet it is the c( transitions which cause the observableAl/l in the microwave double resonance experiment. Factor ( 2 ) is related to factor (1): in NH,-NH, collisions of J = K series, the probability of the /3 transition is so much higher than that of the c( transition that some c( transitions cannot occur without simultaneously causing a j transition during one collision and thus resulting in a y transition. Many experiments of the kind shown in Figs. 29 and 30 have been done for other systems (Oka 1968a). The observed J, K dependence of the signal such as in Fig. 30 can be explained semi-quantitatively by considering three

2

3

4

5

6

7

0

- J

FIG.31. J dependence of the observed values of All1 for each value of K (Oka, 1968a).

factors (1) the matrix element, ( 2 ) the energy separation of the rotational levels, and (3) the statistical distribution of the collision partners. The observed dependence of AIjZ on the value of J is shown in Fig. 31 for each K value. They show a sharp rise at low J values, reach a maximum at an intermediate J values, and then become smaller for higher J . The sharp increase at low

189

COLLISION-INDUCED ROTATIONAL TRANSITIONS

J values is caused by factor ( I ) mentioned above, enhanced by factor (3); the latter is caused by the fact that NH, molecule can find resonant collision partners most easily at around J 3 which is most densely populated. Then the maximum is reached and the value of AIjI starts to decrease because of factors ( 2 ) and (3). More quantitative analysis of these observed values has yet to be given.

-

3. Variation of AIjI with Collision Partners The magnitude of the effect, AIjZ, varies with collision partners. Table V gives some of typical observed values of AI/Z for the (J = K l)p-(J = K ) , systems of NH, for various collision partners. For these experiments about 1 part of NH, is mixed with about 100 parts of foreign gas so that the NH,NH, collisions are negligible compared with NH,-foreign-gas collisions. For collisions between NH, and polar or nonpolar molecules, the semiquantitative explanation of the kind mentioned above have been given (Daly and Oka, 1970; Fabris and Oka, 1973). However, quantitative explanation has yet to be given.

+

TABLE V

+

OBSERVED VALUES OF AI/I FOR THE FOUR-LEVEL SYSTEMS (J = K l)& OF NH3 FOR SOME TYPICAL COLLISION PARTNERS

= K)s

A41 ( %) Collision partner J= 1

Polar molecules NH3

2

3

4

5

1.46 13.0 4.0 0 7.0

1.19 11.0 0.5 0 3.0

1.06 7.5 0.0 0 1.5

0.80 6.5

0.0 0 0.5

0.63 4.0 0.0 0 0.5

2.3 1.9 1.3 1 .o 1.3

0 0

1.1

CH 4 SF,

5.2 4.7 3.1 1.9 2.8

0 0 0 0 0

0 0 0 0 0

Atoms He Ar Xe

-3.5 5.8 6.3

-4.2 5.2 4.0

-2.2 5.0 2.5

0.0 2.2 1.5

1.1 1.1 0.9

H20

CH3F CHCI3 NO Nonpolar molecules H2 D2 0 2

0 0

T. Oka

190

Here we will discuss some of the more remarkable points about the observed values listed in Table V. The most remarkable change of AIjI is the reversing of the sign of AIjI for He, which clearly indicates the preference of the parity conserving transition over the parity changing transitions. This has been discussed in Section II1,C. The large positive values of AZ/I for Ar and Xe are also surprising. Considering that the rate of the 5 transition (see Fig. 15) is large compared to that of the u transition (because of less strict ‘‘selection rules”), and also that the rate of the 1 transition is large (because of small energy transfer), we have to conclude that the large positive value of AIjI indicates a rather strong preference for the c1 transition over the y transition. As discussed in Section III,C, this contradicts the expectation from longest range dispersion or induction force. Here is a veiy definite experimental result for which we do not even have a qualitative explanation. It is very interesting to note that NH,-H,O collisions give a value of AZjI whichis 10 timeslargerthan NH,-NH, collisions. This is interpreted as due to the reduction of the rate for the 1transition; the H,O molecule does not have strong transitions in the microwave region and the resonance condition for the p transition is not satisfied. The resonance with the u transition is very efficient as is seen from the far infrared rotational spectrum of water shown in Fig. 32. The opposite case, that is, a reduction of AI/I is found when oblate symmetric top molecules are used as collision partners. These molecules have

I

H 2 0 Far I R Spectrum 6“5

5L4

Intensity

3-2

50

100

cm-’

FIG.32. Pure rotational spectrum of H20(Hall and Dowling, 1967). The arrows indicate positions of NH3 pure rotational lines. It is seen that H 2 0 has resonance with AJ = 1 transitions of NH3 but not with A J = 0 transitions of NH3.

COLLISION-INDIJCED ROTATIONAL TRANSITIONS

191

-

many rotational levels with K J which are highly populated. Since these levels are of double parity and have large diagonal matrix elements, the resonance condition for the p transition is satisfied. On the other hand, since they are heavy molecules, the resonance conditions for the a transition are not well satisfied. As a result, the values of AIjI are zero within the experimental uncertainty of & 1 When a prolate symmetric rotor like CH,F is used as a collision partner, the values of AZjl are larger because the rotational levels with high K values are less populated and the rotational constant B is larger.

x.

4. Parallelism hetween NH,-He and NH,-H,

Results

As indicated in the results listed in Table V, He and H, act very differently as collision partners for AK = 0 transitions of NH, . This difference was ex plained as due to the fact that the H, molecule has a quadrupole moment and hence exerts a long-range force, whereas He has no permanent moment. However, if we compare the observed values of AIjl for Ak = 3 transitions of NH, we find a close parallelism between the two cases as shown in Fig. 33. These results indicate that for the strong collisions causing Ak = f3 transitions, He and H, act similarly and the presence of the quadrupole moment in

9t

FIG.33. Parallelism between the observed values of AZ/Z for NHS-HZ collisions and NH3-He collisions. The parallelism exists for Ak = & 3 transitions but not for Ak = 0 transitions (Fabris and Oka, 1972).

192

T. Oka

the latter is immaterial. For such collisions short-range repulsive force must play the dominant role and the “selection rule” is determined more by the mass and velocity of the collision partner than by the symmetry. The values ofAI/Ifor Ar and Xe for the corresponding transitions (see Oka, 1968b) have no parallels with thoseof He and H, .The difference between themagnitudeof AZ/I for the NH,-He collision and for the NH,-H, collisions (the ratios range between 3 and 5), is interpreted as due to the difference of denominator in Eq. (84); the observed pressure broadening parameter is about three times larger for NH,-H, collisions than for NH,-He collisions. It is expected theoretically that the H, molecules in the J = 0 level act differently as collision partners from H, molecules in the other rotational levels. This is because the H, molecule in the J = 0 level does not possess the first order permanent quadrupole moment. This expectation was confirmed by measuring A I / I for AK = + 3 transition in NH, using para-Hi, as collision partners. (Fabris and Oka, 1973). The percentage of J = 0 H, in normal H, is 13% at room temperature, whereas in the para-H, it is 52%. This difference is even larger at dry ice temperature (72.8%). In agreement with expectation, it was further observed that the pressure broadening parameter is larger for normal H, than for para-H, . It is concluded from these experiments that H, molecules in the J = 0 level act like He atoms as collision partners. Therefore the mechanism of collisional pumping or relaxation of interstellar molecules depends very much on whether the interstellar H, molecules are normal or para. It is likely that most on the interstellar H, molecules are in the para form because of their formation on low temperature dust grains (Hollenbach, Werner, and Salpeter, 1971), but this has yet to be confirmed by high resolution ultraviolet observation. 5. Higher Order Solution of the Rate Equarions

When the signal is observed for high AJ values as in H,CO, H,CCO, and HCN, Eq. (83) does not give a good approximation, because the assumption that levels other than the four levels directly related to the double resonance act as a thermal bath is no longer valid. Therefore we need a higher iteration procedure. As we go to higher approximation, the number of independent rate constants k i j increases rapidly, and some assumption to reduce the number is necessary in order to apply it to a practical problem. The approximation used for H,CO, H,CCO, and HCN is shown in Fig. 34. It is assumed that rate constants corresponding to a set of selection rules are equal and independent of J ; therefore all the dipole-type transition ( f H -) corresponding to AJ = 0 and AJ = k 1 are denoted by @ and a, respectively, and all the quadrupoletype transitions ( + -+, - H -) corresponding to AJ = f 1 and AJ = f 2 are denoted by y and 6, respectively. Also for the sake of simplicity of the

COLLISION-INDUCED ROTATIONAL TRANSITIONS Parity

+

J

Dipole

193

“Quadrupole”

-

J-I

5-2

+

t -

5-3

+

5-4 -

Fig. 34. Simplified dipole-quadrupole-type cascading model. The collision-induced transitions a. and /3 are dipole-type transitions, while 6 and y are quadrupole-type transitions (Oka, 1967a).

formula, it is assumed that k,’ = kaL= k , and that the various J levels have the same population. These approximations will be reasonably good for relatively high J values like those used in the experiment ( J = 10 for H2C0, 13 for H,CCO, 12 for HCN, and 1 3 for DCN). Also the errors incurred by the latter two assumptions tend to cancel each other. Then proceeding as in Section III,B, we obtain the following formula (Oka, 1967a), A(J - n) = [2(k,

+ kp + k , + k,)]-’{(k, - k,)[A(J - n + 1 ) + A(J - n - l)] + k , [A(J - n + 2) + A(J - n - 2)]} (86)

where A ( J - 1 7 ) gives the deviation of the population difference between the doublet levels from that at the Boltzmann distribution, A ( J - n ) = Sn(J - n)] - Sn(J - n), . If we assume A ( J + I ) = A(J - I ) from the symmetry of the levels and use A ( J ) = -nhv,/kT given in Eq. (73), we can solve Eq. (86) by an iteration procedure. By applying the analytical solution to the observed values of A I / I , the parameters X

= ( k , - k,)/2(k,

and

Y = k,/2(k,

+ kp + k , + k,)

+ k , + k , + k,)

(87)

were determined as shown in Table VI. The ratio of X to Y gives a measure of the ratio of AJ = I transitions to AJ = 2 transitions. It is seen from Table V I that in H2C0 the ratio is of the order of 10 8, whereas in H2CC0 and in

-

T.Oka

194

TABLE VI VALUESOF X

AND

Y DETERMINED FROM THE OBSERVED AI/I(J - 1) AND AI/I( J

Molecule HzCO HDCO

System

Y"

(102s 10~9)~-(927 + 9zsk@z6 +827) (183, 1 5 + 1 8 3 , 16)p-(173.1 4 + 173,15)-(163,13 + 163,14) (153,1z 1 5 3 , 13)p-(143.11 143, 12)-(133.10+ 133.11) (224, I 8 224, I 9)p-(214. 1 7 21 4. I 8)-(204, I 6 + 204, I 7) 12,-111-101 131-121-1 1 I

0.190 0.020 0.149 0.020 0.126 0.055* 0.098 0.016 0.212 0.077 0.178 0.089 ( ~ ~ 1 , 1 z + ~ ~ ~ , 1 ~ ~ p ~ ( ~ ~ 1 , 1 1 + 1 2 1 , 1 z ) - ( 1 111) 1 , 1 o 0.149 + 1 1 1 ~0.069 +

+

+

HCN DCN HzCCO

X"

- 2)

+

+

The definitions of X and Y are given in Eq. (87). The ratio of X to Y gives the approximate ratio of the probabilities of dipole-type transitions to those of quadrupole-type transitions. This value is anomalous.

-

HCN it is only of the order of 3 2 which means that the convergence of perturbation treatment for collision-induced transitions is rather poor. Indeed in a theoretical study by Rabitz and Gordon (1970) the ratio of k, to k , was shown to be of the order of two which suggests that higher order theory is needed. The ratio of X to Y was also confirmed by the triple resonance method described in Section 111,C,2.

E. SOME OTHERMETHODS In this section we summarize several methods which give information on collision-induced transitions that are independent from those obtained by means of four-level double resonance or pressure broadening experiments.

I. Modulated Microwave Double Resonance As shown in Eqs. (83) and (84), steady state double resonance gives only relative values of rate constants. An obvious extention of themethod to obtain information on absolute values of rate constants is to use the pulse technique. This has been done for three-level double resonance (Hancock and Flygare, 1971). However, a difficulty is experienced in signal averaging and the method has not been applied to four-level systems so far. An ingenious method that can be used to avoid this difficulty has been developed at Harvard University by the use of a modulated microwave double resonance spectrometer (Woods, Ronn, and Wilson, 1966). In the experiment, instead of applying the pumping radiation continuously, it is modulated by a square wave at a frequency v,

COLLISION-INDUCED ROTATIONAL TRANSITIONS

195

that is, the pumping radiation is turned on for a period 1/2v and off for a period 1/2v. The absorption of the signal radiation is then amplified at the modulation frequency and phase detected; only the double resonance modulated part of the population change is detected. A typical time dependence of the population difference in the pumped levels and that in the signal levels is shown in Fig. 35. Thus, by measuring the intensity and the phase shift of the modulated

FIG.35. Typical time dependence of (2)the pumping power, (b) populational variation of the directly pumped levels, and (c) populational variation of the collisionally pumped levels (Gordon, 1967). More detailed time dependence is shown in Fig. 37.

signal, one can measure the collision-induced effect and the delay without sacrificing sensitivity of detection. The intensity measurement at the high pressure region gives information which is similar to that obtained from the steady state measurement, but the pressure dependence of the intensity and the phase shift between the applied modulation and the detected signal gives additional information. The modulation provides a built-in time scale for the dynamic response of the molecular system and by varying the pressure one can cover the region which is appropriate for the determination of rate constants. Assuming a set of linear equations

Gordon (1967) has solved the time dependent behavior of the population n(t) under the condition of the modulated double resonance experiment. The theory was applied for the analysis of the observed modulated microwave double resonance in HCN (Gordon, e t a / . 1969). Using the approximate transi-

T. Oka

196

tion scheme shown in Fig. 34, they were able to determine the ratio of the rate constants for the transitions a, /3, y, 6 to be 1 :0:0.26:0.27. The ratio was in fair agreement with the calculated ratio of 1 ;0.078: 0.226:0.434 (Rabitz and Gordon, 1970). An example of the observed pressure dependence of the intensity and a calculated fit by using Gordon’s theory is shown in Fig. 36. I

0.24

I

I

1

I

I

I

I

I

i

0

PRESSURE lp Hg)

FIG.36. Comparison of the observed pressure dependence of AI/I with the calculated pressure dependence. 0 AJ = 1, AJ = 2, AJ = 3 (Gordon, Larson, Thomas, and Wilson, 1969).

As discussed in Section III,D,.5, the rotational level systems for the discussion of collision-induced transitions in HCN are rather complicated because of strong AJ > 1 transitions. This fact together with extensive use of computer by Gordon et al. (1969) made it difficult to understand qualitatively how the separation of rate constants was possible. Here we apply Gordon’s theory to the simpler case discussed in Section III,B. For the system depicted in Fig. 14b, we find the rate equation,

where

71

= k,

+ k , + k , + k, .

COLLISION-INDUCED ROTATIONAL TRANSITIONS

197

The value of k,f is approximated to be equal to k,&, for the sake of simplicity of the analytical expression. If we define the change of population difference in levels with J as A ( J ) = an, - &,and that in levels with J - 1 as A(J - 1) = an, - an,, we have the equation of motion for the population differences as

A ( J ) = - A(J)(k, + 2k, A(J - I )

=

+ k , + ke) - ( k , - k,) A(J - 1)

- A(J - I)(k, + 2k,

+ k , + k,) - ( k , - k,) A ( J ) K = k , + 2k, + k , + k , to L = k, - k ,

(90)

We know the ratio of already from the steady state measurement. Equations (90) clearly indicate that the additional information we obtain from the modulated double resonance experiment will be the absolute values of K and L. Assuming that the transition probability due to radiation is much larger than the probabilities of collisioninduced transitions and the frequency of modulation, we obtain the time dependence of A ( J ) for on-time - 1/2v < t

10" Hz) are scarcely affected by the nuclear hyperfine interactions (< lo8 Hz). The eigenvalues for the vth vibrational and J t h rotational state are often given (Townes and Schawlow, 1955) for a diatomic molecule as WIJ, J = we(u

+ t)2t + 4)3 + BeJ(J + 1) + +)J(J + 1) + ye(v + +)'J(J + 1)

+ 4) -

- me(0

- DeJ2(J

aexe(v

weye(U

+ 1)2 - Pe(v + J)J2(J + 1)2 + . .

*

.

Alternatively it may be convenient to write the energy as a Dunham series expansion with coefficients, Y p s , in powers of both the rotational and vibrational quantum numbers wu,J

=

1ypq(' + !?)p"J(J + l)I4

P. 4

The Dunham coefficients Ypqare closely related to the vibrational and rotational constants: Y , , % w e , Yo, % B e , etc. (Gordy and Cook, 1970). The hyperfine operator Zhfs accounts for the minute contributions to the molecule's energy levels which arise from the interaction of the nuclear electric and magnetic moments with each other and with the electronic structure of the molecule. The phenomenological constants that characterize these contributions are best measured by the methods of radio-frequency and

MOLECULAR BEAM ELECTRIC RESONANCE SPECTROSCOPY

28 1

microwave spectroscopy. Since nuclei are usually described by their spin and their electric and magnetic moments, it is convenient to express (Svidzinskii, 1964) the hyperfine Hamiltonian as an expansion in multipole moments

XhTs =

c q'

even I

1

Q" +

1m'

odd 1

*

MI

(3)

The Q Aand MAare the nuclear electric and magnetic multipole moments; the q' and m' are the appropriate multipolar characterizations of the electric and magnetic fields with which the nuclei interact. The first few (A < 4) terms (which represent the magnetic dipole, the electric quadrupole, and the magnetic octupole interactions) suffice to describe the atomic and molecular hyperfine structures that have been observed to date. In much of the earlier literature, molecular hfs was conceptualized as a coupling of the nuclear spins and the molecular rotation with each other; the physical model for the individual interaction (" spin-rotation," " spin-spin " and so forth) was usually considered before the operator for the interaction was expressed in the language of angular momentum. This emphasis on the physical picture gave rise to a heuristic operator notation which is still widely used, and in most of the pre-1970 literature the hyperfine operator is written in a form such as (e.g., Kusch and Hughes, 1959)

+ c3

.

.

3(I, J)(I2 * J) + 3(I, * J)(Il * J) - 2(I, IZ)J(J + 1) (25 - 1)(2J + 3)

+ c41, '1, (4)

This Hamiltonian is applicable to a 'C diatomic molecule in which nucleus 2 has spin +,and nucleus 1 has spin > 4.I, and I, are the nuclear spin operators for nuclei I and 2, respectively, and J is the total angular momentum of the molecule excluding nuclear spin. The first term represents the interaction of the nuclear electric quadrupole moment Q , of nucleus 1 with the molecular electric field gradient q, at the nucleus due to the remaining charges in the molecule. [It should be emphasized that (4) is just an approximation to (3), and a more exact form than (4) is needed for the analysis of high resolution experiments (Stephenson et a/., 1970)]. The second and third terms are the spin-rotation interactions for the two nuclei, and c, and c2 are the respective coupling constants. The last two terms describe the tensor and scalar parts of the nuclear spin-spin interaction with coupling constants c j and c4, respectively. The difference between expressions (3) and (4) for %hfs is that the latter expression is valid only for calculating matrix elements of Xhfs which are diagonal in the quantum number J . Although the contributions to the

282

J. C. Zorn and T. C. English

energies from off-diagonal elements in J a r e small, they assume added importance as the resolution of spectrometers becomes greater. The interaction of a diatomic molecule with externally applied electric and magnetic fields E and H is described by the operator

-

Xextfield = -p * E - fE a * E - p 1 ( 5 -a,)* H - p2 - pJ * ( 5 - OJ) H - fH * 5 H

( 5 - a2) H (5)

where p is the permanent electric dipole moment of the molecule, p, and p, are the magnetic moments of the two nuclei, p, is the molecular rotational magnetic moment of the molecule, u1, u 2 ,and uJare the magnetic shielding tensors for the respective magnetic moments, a and 5 are the electric polarizability (susceptibility) and magnetic susceptibility tensors, and' 1 is the unit tensor. The electric dipole moment of the molecule is sensitive to the average expectation value of electron coordinates ( r ) a v ,the rotational magnetic moment to ( r 2 ) , , , and the nuclear electric quadrupole interaction to ( r - 3 ) , , . The rotational constant of the molecule is sensitive to the internuclear separation R through the expectation value ( R - 2 ) , v , the nuclear spin-spin interaction through Thus the measurement of each of these interactions provides complementary information on the structure of the molecule. It is worth noting that the relationship between the Hamiltonians of Eqs. (I), (3), and (4) is not as arbitrary as one might suppose. The correspondence between these forms of the Hamiltonian has been discussed by many authors (Ramsey, 1956; Abragam, 1957; Schlier, 1961). Tabulations and reviews of the phenomenological constants have been given by various authors (Lovas and Lide, 1971; Miller et al, 1973; Lucken, 1969; Davies, 1967; Fraga and Malli, 1968; Rosen and Bourcier, 1970).

B. ENERGY EIGENVALUES Since most molecular beam experiments do not involve changes in the electronic state of the molecule, X'e,ec may ordinarily be neglected in the analysis of the experimental data. The usual procedure is to take the operator X u , to be the unperturbed Hamiltonian, and to consider X" = Xhfs + X e xf i,e l d as a perturbation. The structure of the energy levels due to hyperfine interactions and the presence of external fields may then be found by using time-independent perturbation theory, or by directly diagonalizing the X ' matrix. Eigenvalues of the hyperfine and external field Hamiltonians for 'Z polar molecules have been calculated by many authors; the calculations vary in the number of nuclear quadrupole interactions accounted for, in the choice of representation, and in the approximations made. The combining of results from different papers must be done with caution since the notation and sign

,

MOLECULAR BEAM ELECTRIC RESONANCE SPECTROSCOPY

283

conventions are not uniform; a conspicuous example is the notation for the nuclear electric quadrupole interaction which differs by a factor of four between the earlier papers and the works published after 1950. Moreover, the convention used by workers in molecular beam magnetic resonance spectroscopy to express the spin-rotation interaction still does not agree with the usage in the electric resonance literature (English and Zorn, 1973). The choice of representation which most nearly diagonalizes the #' matrix is governed by the relative sizes of the various interactions (Hughes and Grabner, 1950b). If, for example, the quadrupole interaction of nucleus 1 is largest, then it is advantageous to use the completely coupled representation (English and Zorn, 1967) ~Z,JF,Z2FMF) where F, = I, + J and F = I2 F,. If the external fields are large, then the couplings of the nuclear and molecular moments with the fields are usually the most important, and the completely uncoupled representation I Il M,12M2J M J ) is useful (Graff and Rundfsson, 1963; deLeeuw et al., 1969; Heitbaum and Schonwasser, 1972). Other representations have been chosen to handle intermediate field situations (Graff et al., 1967) and to do calculations when both nuclei in a molecule have comparable quadrupole interactions (Robinson and Cornwell, 1953; Cederberg and Miller, 1969). Eigenvalues of the Hamiltonians for other than ' 2 molecules may be found in many of the papers that deal with the MBER spectroscopy of those molecules (Freund and Klemperer, 1965; Freed, 1966; Cupp et al., 1968; Stern et al., 1970; Wofsy et al., 1970, 1971 ; Gammon et al., 1971a, b; Davies ef al., 1971). An interesting analysis of the Hamiltonian for the H F dimer, a nonrigid molecule that exhibits tunneling, has been given by Dyke, Howard, and Klemperer (1972b).

+

RULESAND TRANSITION PROBABILITIES C. SELECTION

The coupling between the molecule and the radio-frequency field in an MBER experiment is through the molecule's permanent electric dipole moment; the appropriate operator is p Erf.The selection rules and transition probabilities can be derived by considering the effect of the operator on the eigenfunction that describes the molecule as it enters the C field (Hughes and Grabner, 1950b). For diatomic, 'Z molecules with one dominant nuclear spin-molecular rotation interaction the appropriate representation in very weak external fields is I I,JF,f2F M F ) , and the selection rules for microwave transitions between rotational states are (Kusch and Hughes, 1959)

-

AJ= & 1;

AF, = 0 , f 1;

AF=O, 5 1;

AMF=O, f 1

284

J, C. Zorn and T.C. English

Since matrix elements of p a E vanish between states of the same J, hyperfine transitions within a single rotational level are forbidden in first order. Such transitions can occur in second order if states of J & 1 are admixed by the presence of an external field. A dc field as weak as 1 V/cm gives enough admixture to make hyperfine transitions observable; the selection rules for these radio-frequency transitions are : AJ=O; AFl =0, f I , k 2 ; AF=O, f l , f 2 ; AMF=O, + I Alternatively, the radio-frequency field itself can admix the J f 1 states; hyperfine transitions that are made allowable in higher order by the rf field occur at frequencies that are 3, 3, &, ... of the separation of the initial and final molecular energy levels. These are the multiple quantum transitions first found by Hughes and Grabner (1950a) in their MBER study of RbF. The calculation of transition probabilities for the very weak-field MBER spectra requires that the effect of both the rf and the dc fields on the molecular wavefunction be taken into account. Moreover, such a calculation also requires that one know (1) the population of states in the beam as it leaves the A field, (2) the velocity distribution of the beam, (3) the state-selecting power of the B field, and (4) the distribution of rf and do field strengths over the beam while it is in the C field. This problem for 'C molecules has been discussed in part by Schlier (1957) and others; the most complete available calculations of very weak-field transition probabilities, with experimental verifications, are those by Hilborn, Gallagher, and Ramsey (1972). If the external field is so strong that the nuclear and rotational angular momenta within the diatomic 'C polar molecule are decoupled from each other (Fig. 12), the appropriate representation is [Z,M,12M2 J M J ) . The selection rules for either radiofrequency (AJ = 0) or microwave (AJ = & 1) transitions in the strong field case are (Kusch and Hughes, 1959) AMI=O; AM2=O; AMj=O, + 1 In contrast to the situation in very weak fields where the observed spectral lines are invariably superpositions of several M F components, the spectrum available in the strong field is usually composed of a series of well separated lines that may be handled with the more usual methods of molecular beam line shape analysis (Ramsey, 1956; Kusch and Hughes, 1959, English and Zorn, 1973). The component of rf field perpendicular to the dc field induces AM = f 1 transitions while that component parallel to the field induces AM = 0 transitions; M in this instance refers to M , in the very weak field case, to MJ in the strong field case. We note that transitions must both obey the selection rules and meet the observability criterion (Section II1,B) if they are to be observed in the usual MBER spectrometer.

MOLECULAR BEAM ELECTRIC RESONANCE SPECTROSCOPY

285

VII. Molecular Properties of 'E Molecules A. ROTATIONAL AND VIBRATIONAL CONSTANTS An MBER spectrometer can be used to measure the frequency of microwave-induced AJ = 1 transitions between the rotational states of polar molecules. The virtual absence of collisions between molecules in the beam makes it possible to obtain very narrow spectral lines and this facilitates the study of hyperfine structure effects on rotational transitions (Cupp et a/., 1968). The microwave spectra of a number of the more reactive, high-temperature diatomic molecules were first measured with MBER spectrometers (e.g., Carlson et al., 1952; Lee et al., 1953; Lew et al., 1958; Wharton et a/., 1963a; Kaufman et al., 1965; Ritchie and Lew, 1965; Hollowell et a/., 1964). For most molecules which have a useful vapor pressure below 1400"K, however, subsequent experiments with microwave absorption spectrometers have provided most of the values of the rotational (&, D,,a,, etc.) and vibrational ( w e ,wexe,etc.) parameters which characterize the ground electronic state of the molecule. As compared to an MBER spectrometer which works well only for low ( J < 5) rotational states, ;Imicrowave absorption spectrometer has the advantage that it can be used to observe transitions between high rotational states (e.g., J = 50 + J = 51) for which hyperfine structure is small but in which effects such as centrifugal stretching are pronounced. For a summary of the data that are available on high temperature species the reader may consult the reviews by Lovas and Lide (1971) and by Miller et a/.(1973). B. ELECTRIC DIPOLEMOMENTS The electric dipole moment (EDM) of a polar molecule can be determined from observations of the way in which a homogeneous electric field changes the frequencies of the lines in the molecule's radio-frequency or microwave spectrum. Stark effect measurements done with MBER spectrometers yield the most accurate values available for the magnitude of molecular electric dipole moments. (The problem of determining the sign of the electric moment is discussed in Section X,B.) 1. Stark Effect

The Stark energy levels of an idealized, polar, 'C diatomic molecule in an electric field E is expressible in a power series

ws=

r";2]

J ( J + 1) - 3 M J 2 2J(J + 1)(2J + 3)(2J - 1)

J . C.Zorn and T. C. English where p, and B, are the electric dipole moment and the rotational constant for the molecule which is itself in state u, J , M , . The leading term of Eq. (6) is ordinarily the most important contributor to the Stark energy. However, the terms of order E 4 and E 6 are significant in the precision spectroscopy of molecules that have relatively small rotational constants and large EDM’s; the coefficients of these terms, h(J, M,) and g(J, M,), have been given by various authors (e.g., Wharton and Klemperer, 1963; deLeeuw et al., 1969). We note also that the Stark effect of a molecule that has a large centrifugal stretching constant may show a small ( - 10 ppm) deviation from the prediction of Eq. (6); the deviation arises from the nonrigid nature of the rotor (Schlier, 1959; Wharton and Klemperer, 1963). In addition, Eq. (6) does not include the contribution to the Stark energy from the induced moment (Section VII,B,3).

2. Dipole Moment Values Molecular electric dipole moments are measured to an accuracy on the order of 10 ppm with modern MBER spectrometers. The high resolution of these instruments makes it possible to measure the Stark effect on individual positively identified spectral lines, and the low density of the beam assures that there will be no damage to the precision Stark field electrodes by the molecules of the sample. The accuracy of MBER electric dipole moment measurements is far better than is needed to test theoretical calculations for the magnitudes of these moments. The merits of the molecular wavefunction calculations (Matcha, 1970; Green, 1971) or of the semi-empirical models (Rittner, 1951 ; Varshni and Shukla, 1965; Maltz, 1969) can be decided with 1 % measurements. The higher precision is of considerable use, however, to show how the moment dependson thequantum state and on the isotopic composition of the molecule From a measurement of the change of the dipole moment with vibrational state, for example, one can deduce the way in which the moment depends on the internuclear separation (Trischka and Salwen, 1959; Salwen and Trischka, 1959; Herman and Short, 1968; Kaiser, 1970). To obviate the need for careful, absolute determinations of the electric field strength in Stark effect experiments, it is useful to use a standard molecule as a reference against which other dipole moments can be compared. Carbonyl sulfide, 16012C32S, has an intense and uncomplicated spectrum and is a suitable standard provided that the beam spectrometer has an electron bombardment detector. The dipole moment of OCS is known to an absolute accuracy of about 50 ppm from the MBER studies by Muenter (1968) and by deLeeuw and Dymanus (1970). The main drawback of OCS as a calibrant in beam experiments is that a universal detector is required. It may sometimes prove

MOLECULAR BEAM ELECTRIC RESONANCE SPECTROSCOPY

287

more convenient to use other molecules (e.g., BaO, TIF, CsF) with well-known dipole moments as intermediate standards in Stark spectroscopy (Wharton et al., 1962a; Hoeft et al., 1970). Electric dipole moments of the alkali halides have been measured in MBER experiments by many workers (e.g., Lew et a/., 1958; Wharton et al., 1963a; Graff et a]., 1967); a useful summary of the many measurements carried out at Berkeley and elsewhere has been given by Hebert and his collaborators (1968). We also call attention to the studies of dipole moments in group IV/VI diatomic molecules made with the methods of microwave absorption Stark spectroscopy by Hoeft, Lovas, Tiemann, and Torring (1970, 1971). 3. Polarization Effects

The polarization of the electronic charge distribution of the molecule by the external electric field gives rise to the term +E * a * E in the Hamiltonianof Eq. ( 5 ) . The tensor polarizability a relates the deformation of the molecule to the strength and direction of the applied electric field; thus, for example, the field induces an electric dipole moment pind=aE. Since only the anisotropic part of the polarizability will contribute to spectroscopically observable energy differences (Wharton and Klemperer, 1963) a polarizability anisotropy measurement requires the observation of several different molecular transitions in the same Stark field. In the typical MBER experiment the polarization energy is four orders of magnitude smaller than the Stark energy from the permanent electric dipole moment. However, the effects of the anisotropic polarizability were significant in the measurements of the OCS and HCI electric dipole moments (Scharpen et al., 1970; Muenter, 1968; Kaiser, 1970). Muenter (1972) has measured the polarizability anistropy of H F to be 0.220 & 0.020 A3. In homonuclear diatomic molecules there is no Stark energy from a permanent dipole moment to obscure measurements of the polarizability tensor; the polarizability anistropy of D, has been determined by observing the Stark effect on the radio-frequency spectrum as measured in a molecular beam magnetic resonance spectrometer (English and MacAdam, 1970). Other contributions to the Stark energy of a polar molecule come from the anharmonicity of vibration, the centrifugal distortion, and the vibrationrotation interaction (Wharton and Klemperer, 1963; Dijkerman et al., 1972). These contributions, being quite small, are hard to isolate and measure, however, Dijkerman et al. (1972) have pointed out that measurements done in certain selected field strengths facilitate the separation of the different contributions to the Stark energy.

288

J . C . Zorn and T. C. English

4. Dejection Experiments

Many early measurements of the electric and magnetic moments of molecules were done by measuring the deflection of a molecular beam by a static, inhomogeneous field (e.g., Estermann and Fraser, 1933); these have been described in the reviews by Fraser (l93l), Estermann (1946), Ramsey (1956), and Zorn (1964). Although resonance measurements with Stark and Zeeman effects usually yield the most precise values of these moments, deflection experiments are sometimes quite useful for the understanding of a molecule's structure. Electric deflection experiments make it possible to distinguish between polar and nonpolar species, to obtain rough values for the permanent electric dipole moments of strongly polar species, and to measure the electric susceptibility (polarizability) of nonpolar species. The experiments can be done with either two-wire or multipole fields. Representative papers include those by Wharton et al. (1963b), Buchler et a / . (1964, 1967), Kaiser et al. (1968, 1970), and Dyke et al. ( 1 972a). In addition, Berg and his collaborators (1965) point out that electric deflection experiments can, in certain favorable cases, distinguish 'Z from non-'Z molecules because the electric behavior of a polar molecule changes markedly if there is a net component of the total angular momentum along the internuclear axis. We note that molecules conventionally thought to be nonpolar may exhibit behavior that is characteristic of species with a permanent electric dipole moment. Using the MBMR techniques, Ozier (1971) has observed linear Stark effects and allowed electric dipole transitions in the ground electronic and vibrational state of CH,; the small permanent moment [5.38(10) x Debye] is a centrifugal distortion effect. Using the molecular beam electric deflection technique, A. Muenter and her colleagues have observed that CCI,, CF,, and SiCI, are polar in their v3 excited vibrational states; they ascribe this behavior to a vibrationally induced dipole moment that is possible in certain types of degenerate vibrational states (A. A. Muenter et a / . , 1971). To extract the value of a molecule's dipole moment from the measurement of the deflection of a beam of the molecules is a complicated business. Even if a mechanical velocity selector is used to reduce the spread in molecular velocities, there may be dimers and higher polymers in the beam that will not have the same polar characteristics as the monomer being studied. Moreover, the efective dipole moment of each molecule depends on its quantum state (J, M j ) , so the distribution in rotational states associated with the temperature of the oven will lead to a distribution in effective dipole moment values; the observed pattern of the deflected beam must then be interpreted in terms of the hypothesized J , M j population of the beam. An analysis of this kind is described by Fraser (1931) and by Story (1968), and it is clear that precise values of electric moments are hard to get from electric deflection experiments.

MOLECULAR BEAM ELECTRIC RESONANCE SPECTROSCOPY

289

C . MIXEDALKALIDIMERS Dagdigian, Graff, and Wharton (1971) have studied beams of mixed alkali dimers (e.g., NaLi) using an apparatus that employs a focusing state selector 8.8 m long. The polarizabilities of the M = 0 sublevels of the J = I and J = 2 rotational states of NaLi were measured by observing the voltages required to refocus a supersonic molecular beam with an electric quadrupole field. From these measurements the permanent dipole moment of NaLi was deduced to be 0.46(1) Debye, a factor of two smaller than is predicted by recent calculations (Green, 1971). Beam resonance experiments on the mixed alkali dimers have been done by both magnetic and electric resonance methods. Brooks, Anderson, and Ramsey (1972) used an M B M R apparatus to study the radio-frequency spectrum of NaK and NaLi; they measured the rotational magnetic moment of NaK and also obtained values for nuclear quadrupole interactions in both molecules. An MBER apparatus with a supersonic jet source has been used t o study the electric resonance spectra of NaK, NaLi, NaCs, LiK, and LiRb (Graff, Dagdigian, and Wharton, 1972; Dagdigian and Wharton, 1972). Dipole moments determined from these measurements range from 2.76( 10) Debye for NaK to 4.75(20) Debye for NaCs, where the uncertainties arise mainly from the lack of accurate values for the rotational constants of the molecules. In addition, components of the polarizability tensor and nuclear quadrupole interactions were measured for some of these molecules. Where the comparison is possible one finds substantial disagreement between MBMR and MBER experiments on mixed alkali dimers. The value for eqQ(’Li) in NaLi determined from the MBER experiments, 28 _+ 4 kHz, is far smaller than the 54 k 5 kHz value quoted from M B M R experiments. The origin of the discrepancy is not identified but it may well lie with the known difficulty in analyzing an MBMR spectrum when contributions from many rotational states are present (cf. also Section X,B).

VIII. Quadrupole hfs in

‘Z

Molecules

A. NUCLEAR ELECTRIC QUADRUPOLE INTERACTIONS

The electric quadrupole coupling constant eqQ is the phenomenological quantity that characterizes the first nonvanishing electric multipole term in Eq. (3). This coupling constant is a product of a nuclear property Q (the quadrupole moment) and a molecular property q (the electric field gradient at the nucleus). Thus q can be obtained from a molecular hfs measurement if the value of Q is available, but unfortunately the absolute values of nuclear

290

J . C. Zorn and T. C. English

quadrupole moments areseldom known to more than a few percent (Townes, 1958; Lobner et al., 1970). Conversely, absolute values of Q can be obtained from measurements of eqQ if a reliable calculation (Matcha, 1970) of the field gradient can be made. Values of eqQ, even of moderate precision, are of interest because they can provide a severe test for ab inirio theories that attempt to predict either q or Q . Values of eqQ measured to high precision are of interest because they show dependence of the coupling constant on the quantum state (0,J ) of the molecule and on the isotopic species of the nuclei; from this data it is possible to infer how q varies with the internuclear separation and thus test semi-empirical theories of the molecular structure. In addition, the high precision measurements serve as a test for the adequacy of hyperfine structure theory, for the presence of pseudo-quadrupole and electric hexadecapole interactions, and for the electric polarizability of the nucleus. The interpretations of the quadrupole interactions in LiH and LiF illustrate the use of eqQ values as tests of both molecular and nuclear calculations. The quadrupole hyperfine interaction in 7LiH, known to about 0.1 % from MBER work (Wharton et al., 1962b; Rothstein, 1969) is combined with a carefully calculated value of the field gradient q (Browne and Matsen, 1964; Cade and Huo, 1967; Lucken, 1969), to obtain a value for the quadrupole moment of the 7Li nucleus. The result, Q(7Li) = -4.5 x cm2, has a 10% uncertainty from the q calculation. Then Wharton, Gold, and Klemperer ( I 964) used MBER measurements of the quadrupole hfs in 7Li'9F and 6Li'9F to determine that the ratio of the lithium quadrupole moments is Q(6Li)/ Q('Li) = 0.0176(10), and to deduce therefrom that the quadrupole moment of 6Li is -8.0 x lo-'* cm'. The interpretation of eqQ in a many-electron molecule is illustrated in Kaiser's (1970) report of his MBER study of HCI and DCI. Hartree-Fock calculations of the field gradient at the deuteron in DCI are compared with thevalue for q that is obtained by dividing the measured eqQ(D) by the known value of the deuteron quadrupole moment; the agreement is in the 5-l0% range. Calculations of the field gradient at the chlorine nucleus (Scrocco and Tomasi, 1964; HUO,as quoted in Kaiser, 1970; Lucken, 1969) agree very well with the experimental value, but Kaiser ( I 970) emphasizes that the agreement depends on the particular value assumed for the quadrupole moment of the chlorine nucleus. The dependence of eqQ on the rotational and vibrational state of a polar diatomic molecule can be related to the molecular constants o,,Be, etc. as shown by Zeiger and Bolef (1952). Trischka and Salwen (1959), and Schlier (1959, 1961) have shown that the expectation value of an operator (such as the electric dipole moment or the electric field gradient) can be expressed as a function of internuclear separation R provided that the changes in the

MOLECULAR BEAM ELECTRIC RESONANCE SPECTROSCOPY

291

expectation value with both v and J are measured; the functions p ( R ) and q ( R ) are of particular interest to theorists. Changes of eqQ with v, being on the order of 1 % between neighboring vibrational states, are readily measured, but changes with J, being dependent on Be2/coe2 and therefore small for heavier molecules, have been reliably measured for only a few molecules. In HCI, the chlorine quadrupole interaction of about 67 MHz changes by 1.7 MHz between vibrational states and by 0.02 MHz between rotational states (Kaiser, 1970); the distance derivatives of the chlorine field gradient function, q( R), are in surprisingly good agreement with the unpublished Hartree-Fock calculations of Huo. In InF, the indium quadrupole interaction of about 700 MHz changes by 3 MHz between vibrational states and by 0.01 MHz between rotational states (Hammerle et al., 1972); these changes are in qualitative agreement with the Zieger-Bolef model, but wave functions for InF are not available for a quantitative comparison with theory. Interpretations of quadrupole coupling in heavier molecules are usualy semi-empirical. Many models and controversies (reviewed by Lucken, 1969) are based on the approach of Townes and Dailey (1949) in which the principal contribution to the field gradient is estimated from the parameters that describe the valence electrons (Townes, 1958). In other cases, it is fruitful to use a description in which charges external to the electron shells around the nucleus produce a gradient that is enhanced by the distortion of these electron shells; this is the antishielding effect (Sternheimer, 1967, and papers there cited). The abundant data available from MBER and microwave absorption spectroscopy can be used to test these theoretical descriptions. Fairly good agreement is found for the alkali quadrupole couplings in alkali halide molecules, but the descriptions are not adequate for the halogen quadrupole couplings in alkali halide or thallium halide molecules. De Wijn (1966) has attacked this problem with a combination of Townes-Dailey and antishielding theories that he augments with the assumption that polarizations of the ions in a polar molecule are quenched to some extent by the repulsive forces between the ions; he gets agreement with some of the data. Peters and Todd (1972), using both antishielding and pn polarization effects, fit results from the alkali chlorides and bromides. Still the theoretical problem is not yet well understood (Lucken, 1969; Miller and Zorn, 1969; Van Ausdal, 1972). B. FALSEQUADRUPOLE EFFECTS There are several ways in which a molecular hyperfine energy can depend on nuclear orientation in the same wayas does the energy from the true nuclear quadrupole interaction. The measured value of eqQ = eqQ(True) + eqQ (False) may then have a small contribution from false quadrupole effects (Townes, 1958). One such effect may arise from a deformation of the nucleus

292

J. C. Zorn and T. C. English

by the electric fields within the molecule (nuclear polarization) ;another may arise from a second-order paramagnetic interaction between the nuclear magnetic moment and the electron orbital moment (pseudo-quadrupole effect). One way to test for these effects is to look for a dependence of eqQ(Fa1se) on the environment of the nucleus; this can be done by measuring the way in which the ratio of eqQ for isotopes of the same element, for example eqQ(3SCl)/eqQ(37C1)= R(35/37 Cl), depends on the host molecule of the quadrupole nucleus. In addition, the true and false quadrupole effects are expected to have different dependences on the internuclear separation. Evidence for the polarization of the chlorine nucleus was reported by Gunther-Mohr, Geschwind, and Townes (195 I ) who found in their microwave absorption study that R(35/37 Cl) as measured in GeH3Cl differed by 0.2% from the same ratio measured in CH3Cl. Subsequent quadrupole resonance experiments performed on solid samples show that lattice and thermal effects can produce shifts of comparable magnitude (Lucken, 1969), but measurements of R(35/37 C1) in gas-phase molecules have not shown a significant dependence on the species of the host molecule. For example, the MBER determination of R(35/37 Cl) in TlCl (Hammerle et al., 1969) differs by less than 0.006% from the value measured in atomic chlorine (Holloway, 1956). Although it appeared for a time that R(79/81 Br) was evidence for the polarization of the bromine nucleus by molecular fields (Bonczyk and Hughes, 1967), subsequent MBER measurements on LiBr (Hebert and Street, 1969), TlBr (Dickinson et al., 1970), and again on LiBr (Hilborn et al., 1972) showed that R(79/81 Br) in these molecules agrees with the value in atomic bromine (Brown and King, 1966) to within 0.001 %. Using a Steinwedel-Jensen nuclear model, Bonczyk (1970) showed that the expected polarization is too small to have been observed in the cited experiments. The pseudo-quadrupole contributions to hyperfine energy levels (Foley, 1947; Ramsey, 1953a; Pyykko and Linderberg, 1970) are proportional to the square of a magnetic hyperfine interaction energy divided by the separation between the electronic energy levels. No contributions from the pseudo-quadrupole effect to the hfs of 'Z molecules have yet been conclusively identified, but English (1966), Dickinson (1968, 1970), and Hilborn et al. (1972) have taken pains to show that the expected magnitude of the effect in the alkali and thallium halides is just about the same as the uncertainties quoted for MBER measurements of eqQ in these molecules.

c. NUCLEAROCTUPOLE AND HEXADECAPOLE INTERACTIONS Hamiltonians (Eq. 3) that include hyperfine interactions of no higher order than those from nuclear magnetic dipole and electric quadrupole moments have thus far sufficed to interpret the results obtained with MBER spectro-

293

MOLECULAR BEAM ELECTRIC RESONANCE SPECTROSCOPY

meters. However, nuclear magnetic octupole interactions have been observed in the hfs of atomic iodine (Jaccarino et al., 1954), indium (Kusch and Eck, 1954), chlorine (Holloway, 1956) and bromine (Brown and King, 1966); moreover, tentative identifications of electric hexadecapole interactions had been made for the nuclei of antimony (Wang, 1955) and indium (Mahler et al., 1966). Thus one might expect that the spectra obtained with a high resolution MBER spectrometer would reveal contributions to hyperfine structure that arise from magnetic octupole or electric hexadecapole moments. To explore this possibility, a series of MBER measurements on the halides of thallium and indium were undertaken at the University of Michigan. The results obtained from TIC1 (Hammerle et al., 1969), TlBr (Dickinson et al., 1970), TI1 (Stephenson et al., 1970), and InF (Hammerle et al., 1972) are in excellent agreement with the predictions of the hyperfine Hamiltonian provided that the energy eigenvalue calculations are carried to fourth order in the quadrupole interaction (Fig. 23) and provided that the calculations include

:i:k”A=!,,lh

c

,IKH‘_,

.-

~

t

I

3944 I

39583

39724

I

,

I

I

I

39865

FREQUENCY (KHz)

FIG.23. Transitions ( F , = 4, F = 1) + (F, = 4, F = 1 ) within the J = 3, u = 0, 1,2, 3 states of 205Tl’271and 203T11271. The lines from both species overlap completely in the u = 0 state. For the u > 0 states, the lines separate because the B values (and hence the second-order quadrupole and I J contributions) of the two isotopic species have a different dependence on vibration (D. A. Stephenson, Dissertation, University of Michigan, University Microfilms, 1968).

the off-diagonal contributions of the magnetic moment of the nonquadrupolar nucleus. No evidence for nuclear octupole or hexadecapole interactions was found; in fact, it appears that the nuclear hexadecapole moment for “’In is less than cm4. To better understand the absence of hexadecapole interactions in these spectra, Hammerle and Zorn (1973) have considered the question of hexadecapole magnitudes more carefully. By assuming that the nuclear charge

J. C. Zorn and T. C.English

294

distribution would be described by the parameters of the nuclear potential, they are able to (a) calculate electric quadrupole moments that are in reasonable agreement with experiment, and (b) show that the intrinsic nuclear electric hexadecapole moments are about an order of magnitude smaller than the rough guess cm4) based on the cm4 dimensions of the moment. Moreover, they point out that the spectroscopically observable hexadecapole moment H i s smaller than the infrinsic moment H , by a factor: H

H , = (21 -

+ I)( - 1 y - K (-;

; ;)(A;);

(7)

where Z is the nuclear spin and K = 1 for most ground-state nuclei. The quantities in large brackets are 3-j symbols. The factor may be substantial, for example H / H , = & for a nucleus with I = $. We note that atomic beam magnetic resonance spectra have provided evidence for nuclear hexadecapole interactions in isotopes of dysprosium and holmium. Penselin, Dankwort, and Ferch (1972) report hexadecapole moments that have a magnitude on the order of 0.5 x cm4 (uncertain to about 30%); this is in reasonably good agreement with the predictions for those nuclei as calculated by Hammerle and Zorn (1973).

IX. Magnetic hfs in 'C Molecules A. SPIN-ROTATION INTERACTIONS

-

I J terms in the hyperfine Hamiltonian arise from the interaction of the nuclear magnetic moment with the magnetic field produced by the rotation of charges about this nucleus (Wick, 1948); the expression for the spinrotation interaction constant is given in Eq. (12) below. The interaction is small in 'Z molecules that are in low rotational states since the fields produced by rotation are only a few gauss per rotational quantum. However, if the molecules in the beam have a rotational state distribution that includes many large and different values of J, as is often the case in molecular beam magnetic resonance (MBMR) experiments, the effects of the spin-rotation interaction may complicate the interpretation of the observed spectrum (Nierenberg and Ramsey, 1947; Zeiger and Bolef, 1952; Mehran et al., 1966). The problem arises because the magnetic moment associated with MJ is so small that the magnetic A and B fields in an MBMR spectrometer ordinarily do not give effective J-state selection. However, Ozier and his colleagues (1964, 1965) have used a low temperature source in the MBMR studies of molecular fluorine so that only the lower rotational states were populated; they found a centrifugal effect (Ramsey, 1952) on the spin-rotation interaction constant :

MOLECULAR BEAM ELECTRIC RESONANCE SPECTROSCOPY

295

+

c(F,) = - 156.85(10) kHz - J(J 1)0.0024(10) kHz. Clear evidence for a rotational state dependence of spin-rotation interaction constants has only recently been found in MBER experiments (Muenter, 1972) although it had been anticipated (Trischka, 1948). MBER spectrometers, with their effective rotational state selection, have yielded most of the reliable information on the spin-rotation interactions in polar molecules. The interaction constants, usually designated c1 and c2 in heteronuclear diatomic molecules, range from less than 0.5 kHz to more than 40 kHz; they are typically known to an accuracy of k0.25 kHz. The change of the interaction constant with vibrational state is often clearly seen. Further interpretation of this interaction is discussed in Section X,D. B. SPIN-SPININTERACTIONS The nuclear magnetic moments in a diatomic molecule have a n interaction that is expressed (Ramsey, 1956; Schlier, 1961) by those terms that involve c3 and c4 in the hyperfine Hamiltonian of Eq. (4). The classically expected tensor term of this interaction is characterized by the coupling constant c 3 . The scalar term, c411-1, would not be expected on the basis of a classical dipole-dipole interaction, but it is required to explain the spectra observed in high resolution NMR experiments (Ramsey and Purcell, 1952; Ramsey, 1953b). The tensor coupling has one contribution attributed to the direct interaction of the nuclear spins and another contribution from an indirect electroncoupled interaction between the spins. It is usual to express the tensor coupling constant as a sum c3 = c,(direct)

+ c,(electron

coupled)

The direct part of this constant is defined by the expression c,(direct)

= g1g2pN2 ( u ,

~1

R - 3 I u, J )

where g1and gz are the nuclear g factors, pNis the nuclear magneton, and R is the internuclear separation. By subtracting the calculated c,(direct) from the measured value of c 3 , the size of the electron-coupled contribution to the tensor interaction can be obtained. In the alkali halides, c,(electron coupled) is typically 0.5 kHz or less (English and Zorn, 1967), while values on the order of 1-2 kHz are measured in the thallium halides (Hammerle et a/., 1969; Dickinson er a/., 1970; Stephenson et al., 1970; Ley and Schauer, 1972). The scalar part of the nuclear spin-spin interaction comes wholly from the electron-coupled interaction between the nuclear moments. The scalar coupling constant, c4, is usually less than 0.5 kHz for the alkali halides but it can be as large as 13 kHz in the thallium halides. Ley and Schauer (1972) note

J . C. Zorn and T. C. English

296

that molecules such as TIC1 have excited electronic states that obey Hund’s coupling case c so that generalizations from alkali halide data cannot be made with confidence, but a satisfactory theoretical explanation is yet to be made for this dominance of the electron-coupled interactions to the spin-spin interaction in the thallium halides.

X. Stark-Zeeman Spectroscopy of

‘C

Molecules

The interaction of a molecule with an applied magnetic field is described by the terms pJ*(1 - uJ)* H - p1* (1 - a,)* H - JLZ * (1 - U) * H - t H *

5* H

(8)

in the Hamiltonian of Eq. (2). The rotational magnetic moment of the molecule is denoted by pJ;p, and p2 are the magnetic moments of the two nuclei. Both the magnetic shielding of the nuclei (ui) and the magnetic susceptibility of the molecule (5)arise from changes in the electron distribution caused by the external magnetic field. Since the molecule has axial rather than spherical symmetry, the effects of the shielding and susceptibility tensors depend on the orientation of the molecule relative to the applied field. As is also true for the electric polarizability discussed in Section VII,B,3, only the anisotropic parts of the shielding and the susceptibility are observable with the spectroscopic techniques discussed here. For the discussion of the magnetic characteristics of the molecule, we follow the paper of Drechsler and Graff (1961) and the valuable review by Schlier (1961). A. SPECTROSCOPY IN COMBINED E AND H FIELDS

-

If a strong homogeneous magnetic field is superimposed on the strong electric C field ( E 500 V/cm and H 2 kG, typically) in an MBER spectrometer (Fig. 24), the Zeeman interactions of the nuclei can be separated cleanly from the interaction of the rotational magnetic moment and from the effect of the molecular susceptibility. Then, from a knowledge of pJ and the anisotropy of it is possible to evaluate the quadrupole moment of the molecule’s distribution with respect to an origin at either nucleus or at the center of mass of the molecule. It is the essence of Stark-Zeeman spectroscopy that the strong electric field keeps MJ as a good quantum number so that the representation I Z,MIZzM,JM,) is appropriate (Section V1,B). The interaction of the molecule’s rotational magnetic moment (typically lo-’ nuclear magnetons per

-




t Y)

z

t

2 vI i-3v,

R branch

I l l l l l l l l l l l l l , , l I I I I I I lo360

10370

WAVELENGTH

6)

FIG.1. Two CO, bands in the spectrum of Mars (Belton et al., 1968). Solar lines are flagged with circle-dot symbols.

curve of growth [see Goody (1964) for a detailed discussion of this concept] is illustrated in Fig. 2. The best spectroscopic determination of total atmospheric pressure is obtained from analysis of CO absorptions detected by Kaplan et al. (1969) and displayed in detail in the remarkable Connes atlas of the planets (Connes

MARS ATMOSPHERE

+ +

FIG.2. Curve of growth for the lines of Fig. I , v, 2y2 band. The inset includes additional data with a larger air mass,

327

+ 3y3 band, 0 2yI + 3ys 1967 data, 0 1965 data.

et al., 1969). The atlas shows both strong and weak bands of CO. Individual lines are resolved and the data are ideally suited to quantitative spectroscopic analysis. The CO abundance, derived from the weak 3-0 band, is 5.6 cm atm, and the mixing ratio of CO relative to CO, is 8 x (by number orvolume). The corresponding surface pressure is 5.3 mb, close to the observed partial pressure of CO, (Kaplan et al., 1969). Hunten (1971), considering all of the possible errors, concludes that the spectroscopic data imply a fractional abundance for CO, in the range 80-100 %. Molecular oxygen has also recently been detected as a minor constituent of the Martian atmosphere. Two groups (Carleton and Traub, 1972; Barker, 1972), working in the vicinity of the Fraunhofer A band at 7619 A, found definite evidence for Martian 0,. The abundance was reported as 10.4 I cm atm, corresponding to a mixing ratio of 0, relative to CO, of 1.3 x (Carleton and Traub, 1972). Spectra, obtained with a three-element FabryPerot system (PEPSIOS), are reproduced in Fig. 3. This figure gives the observed spectrum of Mars, a synthetic spectrum indicating contributions due to absorption by terrestrial 0, , and, finally, a ratio spectrum illustrating the contributions of Mars to the observed spectrum. Two lines, "Q(9) and pP(9), are clearly evident in the Martian spectrum. The relative strengths of these

M . B. McElroy

328

fl

SYNTHETIC SPECTRUM

6

RATIO SPECTRUM

1.O(

01

0.9 MARS Reflection spectrum near 7635 8,

FIG.3. Reflection spectrum of Mars showing the absorption lines of 0,.Reproduced courtesy of Drs. N. P. Carleton and W. A. Traub.

lines are in excellent agreement with theoretical prediction, confirming beyond reasonable doubt their identification as intrinsic Martian features associated with 0,. An earlier paper by Belton and Hunten (1968) claimed a tentative indentification of Martian 0, , with an estimated abundance of 20 cm atm. Two lines were detected in the ratio spectrum. One of the lines was displaced from its expected position, however, and the validity of the measurement is now suspect. On the other hand, there are reasons to suspect that the abundance of Martian 0 2 and , CO, may be time variable. Additional data are required in order to clarify this interesting possibility. Discoveries of H,O in the Martian atmosphere have been reported persistently by large numbers of astronomers over the past 100 years. An illuminating historical summary is given by Schorn (1971). He concludes that Kaplan

MARS ATMOSPHERE

329

et al. (1964) made the first effective discovery of Martian H 2 0 , in the same sense that Columbus made the first effective discovery of America. The presence of trace quantities of H,O in the Martian atmsophere is now beyond dispute, due largely to the efforts of Schorn and colleagues at the University of Texas (Schorn, et a/., 1967; Schorn, 1971; Tull, 1971; Little, 1971). The band most commonly used to study Martian H,O corresponds to the transition 21 1 + 000, and many studies have focused attention on the strong isolated line at 8189 A. The observations are normally taken during periods of maximum planetary Doppler shift to avoid contamination of spectra by features associated with terrestrial H,O. There is evidence that the abundance of Martian H 2 0 is variable, sometimes falling below the spectroscopic detection threshold of 10 precipitable microns, or about 5 x l O I 9 molecules cm-2. There are also indications that the observed H,O exhibits a strong latitudinal dependence and that it is sometimes confined to a single hemisphere. Density tracings by Little (1971) from a typical photographic plate are reproduced in Fig. 4. The Martian lines are clearly indicated and their expected positions are marked by the conventional astronomical symbol d. There is evidence in these spectra for a relatively large amount of H,O in the northern hemisphere. From the data shown here, Little (1971) estimates a H 2 0 abundance of 21 precipitable microns in the north polar region, 24 precipitable microns in the equatorial region, and less than 10 precipitable microns in the south polar region. Schorn (1971) argues that the abundance of atmospheric H,O may be directly related to the extent of the polar cap and raises the possibility that Martian H,O may cycle seasonally through the polar caps, now believed to be composed primarily of frozen CO, (Leighton and Murray, 1966; Leovy, 1966; Neugebauer et a/., 1971). Thedetailsof theMartian hydrologic cycle represent a fascinating topic. As yet quantitative models are lacking, but they may soon emerge as the Mariner 9 data are processed and eventually assimilated by planetary physicists. Ozone is unexpectedly rare in the Martian atmosphere. The gas is readily detected through its strong ultraviolet continuum centred at 2550 A. Broadfoot and Wallace (1970) observed the reflection spectrum in the wavelength range 2550-3500 A using a rocket-borne spectrometer. Their spectrum refers to the entire planet and suggests an upper limit of 2 x cm atm to the planetary average abundance of 0,. Positive detections of Martian 0, have since been reported by Barth and Hord (1971) and Lane et a/. (1972). The ultraviolet spectrometer on Mariner 7 found evidence for the 2550 A absorption feature in spectra obtained over the south polar region of Mars in 1969 (Barth and Hord, 1971), and these observations could be explained by the presence of about lo-, cm atm of atmospheric 0,. On the other hand, there were no indications of 0, at other locations on the planet, although the ultraviolet experiment should be sensitive to an abundance of approximately

M . B. McEIroy

330 Southf

FIG.4. The H,O absorption line at A8189 in the spectrum of Mars (Little, 1971).

cm atm. Broida raised the interesting possibility that the observed polar 0, might be trapped in the solid CO, of the polar ice cap and this idea was subsequently supported by detailed laboratory experiments (Broida et al., 1970). Observations by Mariner 9 (Lane et al., 1973) indicate the 0, is apparently absent at all times in the equatorial region of the planet, again subject to the detection theshold of about cm atm, and that it is deficient also in the south polar region during local summer. Ozone was observed in the north polar region and later appeared in the southern hemisphere at latitudes above 50" S with the approach of the Martian autumnal equinox. The implications of the Mariner data remain unclear, however. In particular, it is difficult to assess the extent to which the ultraviolet observations can be explained by atmospheric 0, , or by 0, trapped in frozen CO, on the planetary surface or in CO, ice crystals in the atmosphere.

MARS ATMOSPHERE

331

Noxon (personal communication, 1972) has proposed an ingenious observation which might resolve this paradox. He suggests an observation of the infrared atmospheric bands of 0, (the transition a'A, -+ X3C,-) in the Martian dayglow. It is well known that O,('A,) is a primary product of gas phase photolysis of 0,. Thus, observation of the corresponding airglow should provide an alternate means for specific detection of atmospheric 0,. Preliminary calculations suggest that the observation may be feasible with current ground based telescopic capabilities. In any event, present limits on the abundance of atmospheric 0, imposed by ultraviolet spectroscopy provide powerful constraints on attempts to model the photochemistry of Martian CO,. These constraints will be discussed in more detail in the following section. Other gases expected to be present in significant trace amounts in the Martian atmosphere include nitrogen, argon, and neon. These gases cannot be studied by absorption spectroscopy, except possibly by very difficult measurements in the far ultraviolet. Spectroscopy of the Martian dayglow provides the most promising remote technique for their detection but positive results have not yet been attained. The Mariner spectra show no indication NO, N, or CN of spectral features which might be associated with N,, Nz+, (Barth et al., 1971), and the most prominent emissions expected from A and Ne are outside the spectral range covered by the Mariner instruments. A discussion by Dalgarno and McElroy (1970) of likely excitation processes for N, and N,' emissions suggested an upper limit of about 5 % to the possible mixing ratio of N,, and a similar result was reported recently by McConnell (1972). The absence of NO, N, and CN emissions should provide additional limits to the possible N, abundance. A detailed photochemical model for Martian nitrogen will be required, however, in order to interpret these data. Such a model is currently under study (McElroy and Strobel, 1973) but final results are not yet available. The only available limits for A and Ne are those afforded by ground based infrared spectroscopy. In particular, the current uncertainties in data for the partial pressure of CO, and for the total atmospheric pressure would allow for undetected inert gases provided that their combined mixing ratio should not exceed about 20%. Further refinements in our knowledge of the bulk composition of the Martian atmosphere, and specifically information on the inert gas component, will probably require direct sampling of the atmosphere. The Viking payload, scheduled for launch in 1975, includes a neutral mass spectrometer (Nier et al., 1972) and may provide the first opportunity to carry out the necessary measurements. Our present knowledge of bulk atmospheric composition is summarized in Table I. The discussion so far has emphasized constituents which are present primarily in the lower atmosphere of Mars. The ultraviolet spectrometers on

M . B. McElroy

332

TABLE I SUMMARY OF PRESENT INFORMAnON ON COMPOSITION OF

THE MARTIAN

Constituent

ATMOSPHERE" Abundance (cm atm)

I800

N

5.6

10.4 3, variable 0.4 10-4

0. Vibrational relaxation is likely to occur inefficiently at Martian pressures and the ionosphere should then contain important concentrations of Oz+ with u > 1. The observed densities could be explained if the recombination coefficient for vibrationally excited 0,' were less than that for 0, (21 = 0). This idea merits further attention in view of its potential importance for the Martian ionosphere. It would be useful to determine the vibrational distribution of 0,' formed by reactions (32) and (33) and to study the dependence of k,, on the vibrational development of 0,+ . In the interim, there remain serious difficulties in attempts to develop an inclusive model for Martian ionization. Vibrationally excited 0, + offers attractive possiblities. On the other hand, there are indications from the Mariner dayglow observations, discussed next, that calculated values of 9 may indeed be low. It is already difficult, however, to account for the low values observed for the exospneric temperature of Mars (cf. McElroy, 1970; Stewart, 1972). These difficulties would be compounded by the additional heating associated with larger values of 9. +

M . B. McElroy

348

V. The Martian Dayglow A typical averaged spectrum of the Martian dayglow observed from Mariners 6 and 7 is shown in Fig. l l (Barth et al., 1971). The spectral resolution of the Mariner instruments is 10 A for the wavelength interval 1100-1800 A, and 20 A for the wavelength interval 1900-4000 A. The spectrum is dominated by molecular emission features, the strongest of which are associated with the Cameron (a3n-X'C) and fourth positive (A'n-X'C) bands of CO, the Fox-Duffendack-Barker ( ~ 2 n , - ~ zand l l , )doublet (B2C,'-~zI-I,) bands of C 0 2 + ,and the first negative (B2C+-XZCf)bands of CO'. Also evident are the strong Lyman u line of H, the ultraviolet lines of 0 at 1304, 1356, and

B 2 ~ - R2n

o- co: 0

n

1

> In

r-

C O O 0

~

~

z

1"

W

'z- 200

2500

'D HI N I

Y

N

1"'

0 0n

P

r

11''

-

N O

I-

I'V

6

I'I"

d O

-

C1

2000

~

1,'

3000 WAVELENGTH

(A)

3500

4000

co A ' n - x ' L +

;;08

400

5 300

-

O

S

?

>-

r-

g 200 w I-

100

0 1200

1400

WAVELENGTH

-1600

(A)

I800

FIG.1 1 . Ultraviolet spectra of the upper atmosphere of Mars, 1100-1800 8, at 10 k, resolution and 1900-4000 8, at 20 k, resolution. The spectra shown are the result of the summation of four individual observations (Barth ef al., 1971). From McConnell ( I 972).

MARS ATMOSPHERE

349

2972 A, and the C lines at 1561 and 1657 A. The Mariner spectra obviously offer an abundant source of information on the physical conditions which pertain in the upper regions of the Martian atmosphere. All of the specific emission features identified in Fig. 1 1 originate from levels in the atmosphere above 100 km. The excitation mechanisms for Martian dayglow have been discussed extensively in the recent literature (Barth et af.,1969, 1971, 1972; McConnell and McElroy, 1970; Dalgarno et al., 1970; Anderson and Hord, 1971 ; Thomas, 1971 ; McElroy and McConnell, 1971a,b, Stewart, 1972; McConnell, 1972). According to current ideas, most of the emissions in Fig. 1 1 are excited by processes involving CO, and C 0 2 + .Important exceptions are the resonance lines of H and 0 which are believed to be produced mainly by resonance scattering of sunlight. As indicated earlier, analysis of the 1216 and 1304 A data gives direct information on the abundance and height distributions of H and 0. The Fox-Duffendack-Barker bands give some information on CO,', but much of this emission apparently results from ionization of CO, . The fourth positive bands may be interpreted to place limits on the abundance of CO, and the first negative bands place limits on CO'. The remaining airglow features are excited primarily from CO, and give information on the thermal structure and energy budget of the upper atmosphere (cf. Stewart, 1972). We confine our detailed discussion here to four representative emissions; the Cameron bands of CO, the fourth positive bands of CO, the 11356 line of 0 and the Fox-Duffendack-Barker bands of C 0 2 + .A summary of the various excitation mechanisms for all of the major features in the Mariner spectra, with some indications of their relative importance, is given in Table VI. The absorption spectrum of CO, and the solar flux are shown in Fig. 12 which also indicates energy thresholds for a number of important absorptive processes. The rate (cm-3 sec-') for electron excitation of a given airglow emission is given by Rij

=

s

f(E)Qi jni ~(E)dE

where the integration covers the entire range of energies from threshold to infinity. Heref(E) defines the number of electrons (cmP3)with energy between E and E + A E , Qij is the cross section for the particular process of interest, n i is the number density of target molecules, and v(E) is the electron velocity at energy E. The energy distribution function is most readily computed from the simple formula (Dalgarno et al., 1969; Wallace and McElroy, 1966)

350

M . B. McElroy TABLE VI EXCITATION MECHANISMS FOR PROMINENT FEATURES OF ULTRAVIOLET DAYGLOW OF MARS Emission"

THE

Mechanism"

H(2P-2S)

om-3~) o(5s-3~)

CO(ALn - X 1 X)

C( 3D-3P)

C(3P-3P)

Resonance scattering Resonance scattering Electron impact on 0 Electron impact dissociation of CO, Electron impact on 0 Photon dissociation of CO, Electron impact dissociation of COZ Recombination of CO,+ Resonance scattering Electron impact dissociation of CO, Photon dissociation of CO, Resonance scattering Electron impact dissociation of COZ Photon dissociation of C 0 2 Resonance scattering Electron impact dissociation of C O , Photon dissociation of C O , Recombination of C 0 2 Photon ionization of C 0 2 Electron impact ionization of CO, Resonance scattering by CO, + Photon dissociation of C 0 2 Electron impact dissociation of CO, Recombination of CO,+ Resonance scattering by C 0 2 + Photon ionization of CO, Electron impact on CO, +

CO, +(B2C-X'

n)

O( 'S-l P)

COz+(A2H-X2

n)

Emissions are treated in order of ascending wavelength. Mechanisms for individual emissions are given in approximate order of importance.

wherep(E) defines the total number of electrons produced ( ~ m sec-I) - ~ with energy greater than E, and dE/dx is the instantaneous energy loss function defined by dE/dx = niQij(E)eij (4 I a) ij

Here the summation extends over all constituents i and processesJ, threshold e i j r involved in energy loss at energy E. Equation (41) assumes that energy

MARS ATMOSPHERE

35 1

PHOTON ENERGY ( e V )

10' WAVELENGTH

(d)

FIG. 12. Variation of the CO, absorption cross section and solar flux with wavelength. The solar flux has been averaged over 50 8, intervals. Also shown are the thresholds for various dissociative processes. The dotted line b the photoionization cross section for CO, . From McConnell ( 1 972).

loss occurs continuously. A more complicated expression (cf. Shea et al., 1968) applies if the discrete nature of the loss processes is considered. Equation (41) also assumes that energy loss occurs locally, i.e., that electrons are produced and thermalized in essentially the same atmospheric regions. Modifications for nonlocal effects are discussed by Banks and Nagy (1970), Nagy and Banks ( 1 970), and Stolarski ( I 972). To date, all applications to Mars have employed the approximate expression (41) for f ( E ) . The energy loss functions dE/dx were estimated by Henry and McElroy (1968) and Stewart (1972) but should be revised in light of recent laboratory data (cf. McConkey et al., 1968; Ajello, 1971 ; Freund, 1971 ; Mumma et al., 1971 ; Wells et al., 1972). The contribution of the various excitation processes for the Cameron bands is illustrated in Fig. 13. The figure also includes selected data from Mariner 9

352

M . B. McElroy

00

1

I

I 1 1 1 1 1 1

lo3

I

I

I

I I1111

104

I

lo5

I 1 1 1 1 1 1

106

SLANT INTENSITY ( RAYLEIGHS)

FIG.13. A comparison of calculated CO Cameron band slant intensities with the Mariner 9 results (cf. Barth et at., 1972). The solar zenith angle used in the calculations was 27". From McConnell (1972).

(Barth et al., 1972; Barth, personal communication, 1972).The Mariner instrument viewed the bright limb of Mars tangentially and the theoretical intensities in this and subsequent figures are adjusted to allow for the appropriate geometry. The computations used the solar fluxes of Hall and Hinteregger (1970) and recent laboratory data for photon (Lawrence, 1972) and electron (Freund, 1971 ; Wells et al., 1972) induced dissociation of CO, in the channels hv(e)

+Co2

-

co(a311)

+ 0 + (e)

The quantum yield of CO(a311) from recombination of CO,' was taken as unity. In view of the various uncertainties, both in thetheoryand observations, the calculations show satisfactory agreement with the observational data. The peak value of the observed intensity exceeds the computed value by about a factor of 2. Results for the fourth positive system, and for the 1356 8,feature are shown in Fig. 14. The contributions to fourth positive emission due to photon and electron dissociation of CO, were computed with cross sections measured recently be Gentieu and Mentall (1972) and Mumma et al. (1971). Cross sections for production of O(%) are less well known. We used the shape function obtained by Ajello (1971) for the energy dependence of 1356 8, emission excited by electron impact on CO, . The cross section was taken as 2 x lo-''

353

MARS ATMOSPHERE

801 10'

I

1

I 1 1 1 1 1 1

I

I

I

I

IIIII

I

I 1 1 1 1 1 1 1

lo3

102

lo4

I

1

1

I 1 1 1 1

105

SLANT INTENSITY (RAYLEIGHS)

FIG.14. CO fourth positive slant intensities calculated as described in the text. Also shown are measurements of Mariner 9. The I356 01 emission calculated for 2 % 0 is compared with the Mariner 6 and 7 (A)results. From McConnell(l972).

cm2 at 100 eV in order to fit the Mariner data. A somewhat larger value 6.8 x l o - ' * cm2 was reported by Wells et al. (1972). The evidence for the larger value is indirect, however, and the apparent discrepancy is not considered serious. Electron impact on 0 is a minor source of O(5S),except at higher altitudes. The energy dependence of the cross section for this process was estimated with procedures described by Dalgarno et al. (1969). The cross section was then normalized by fitting to observed intensities of 1356 A emission in the terrestrial dayglow. The predicted intensities for 1356 A emission agree well with the Mariner observations, as might be expected in view of the procedures used to estimate the relevant cross sections. The computed intensities for fourth positive emission are less than observed values, however, by about a factor of 3. The discrepancy here might indicate possible omission of an additional important excitation mechanism, for example

-

CO:

+e

-

CO(A'n)

+ O(3P)

(42)

This reaction could be important if the associated rate constant should exceed about 3 x cm3 sec-'; it was introduced earlier (McElroy, 1968) in connection with a weak ultraviolet feature observed by Barth et af. (1967) in

M . B. McEIroy

354

the night airglow of Venus. Reaction (42), however, cannot account for the observed vibrational population of CO(A'll) in the Martian atmosphere. According to Barth et al. (1971) more than 40% of the radiating CO(A'll) molecules have vibrational quantum numbers u 2 2. Reaction (42), with ground state CO,', may lead to excitation of u = 0 and u = 1 , but levels with u 2 2 are excluded on energetic grounds. Thomas (1971) analyzed the fourth positive bands to set upper and lower bands to the abundance of upper atmospheric CO. Emissions to the u = 0 level of the CO ground state, X'C' are significantly suppressed in the Mariner spectra, indicating that self-absorption by atmospheric CO is important. Analysis of this effect suggests a lower bound to the CO mixing ratio of about 0.3 % at 135 km. An upper bound is imposed if one recognizes that resonance scattering is a minor source of emission in the fourth positive bands. This consideration leads to an upper limit of about 1 % to the mixing ratio of CO at 135 km. These limits are consistent with CO concentrations calculated by McElroy and McConnell (1971a) in their discussion of CO, photochemistry. In particular, Thomas' limits are consistent with the idea that the upper atmosphere of Mars is extremely turbulent, characterized by eddy diffusivities in the range 108-109 cmz sec-I. Predicted intensities for the Fox-Duffendack-Barker bands are compared with observation in Fig. 15. The electron impact contribution was obtained I

I

I

I

1 1 1 1

I

I

I

I

I

I

1

1

I I l l

I

1

I

,

Tx = 360' K

2% 0 AT 1CFX)SPHERIC PEAK

I

I

I

I

1 1 1 1

,

,

,

,

104

106

SLANT INTENSITY (RAYLEIGHS)

FIG.15. Calculated contributions to the C 0 2+ (A-X) slant intensity are compared with the Mariner 6 (0, A) and 7 (0, 0) measurements. From McConnell(1972).

MARS ATMOSPHERE

355

with cross sections measured by Rapp and Englander-Golden (1965) and McConkey P t al. (1968). According to these data, the quantum yield for population of C02'(A2n) is approximately 0.2 over an extensive range of impact energies. Somewhat larger quantum yields, 0.3, are indicated by the experiments of Wauchop and Broida (1971) who studied photoionization of CO, at 584 A. There are no experimental data at other wavelengths. In particular, data are lacking for the wavelength band 150-450 8, whichislikelyto dominate for Mars (cf. Fig. 12). For present purposes we assume that the quantum yield for production of C02+(A2n)associated with photoionization of CO, is comparable to that observed for electron impact ionization, and adopt a wavelength independent quantum yield of 0.2. There is no obvious indication in the observational data shown in Fig. 15 of a change in emission scale at higher altitudes which might be attributed to COz+.The resonance scattering contribution in Fig. 15 was computed with an ionospheric model in which the mixing ratio of 0 was taken as 2 % at 135 km. The densities of C 0 2 + are therefore somewhat smaller than values indicated in Fig. 9 and the 0 densities are somewhat larger than values derived earlier from analysis of the A1 304 observations. The differences are not serious, however, and can probably be included in the combined uncertainties of the observational and theoretical data. The Fox-Duffendack-Barker bands strongly support the conclusion derived earlier, that CO,' is not the dominant component of the Martian ionosphere.

-

VI. Evolution of the Martian Atmosphere As mentioned previously, the Mariner observations indicate that hydrogen atoms are escaping from Mars at a significant rate, 1.8 x lo8 cm-' sec-'. The atoms are supplied by a complex chemical sequence initiated by photodissociation of H,O (cf. Section I l l ) . For every 2 hydrogen atoms which escape, 1 oxygen atom is liberated. The potential supply of oxygen by this mechanism is large. All of the oxygen now present in the atmosphere as CO, could be produced in only lo8 yr and the observed abundance of 0, would be formed in as little as 2 x lo5 yr. A detailed evolutionary model for Mars must account for the missing oxygen. We shall argue here that oxygen also escapes, and that the escape rate for 0 is essentially that required to balance the observed escape rate of H. The escape of H is in fact controlled by escape of 0 in such a manner that photolysis of H,O leads to no net increase in the abundance of atmospheric 0. The analysis described here follows McElroy (1972). The bulk composition of the Martian atmosphere can be understood in general terms by analogy with the Earth. The Earth's atmosphere and ocean,

356

M . B. McElroy

according to current ideas, were formed as a result of outgassing from the planetary interior. Outgassing occurs primarily in hot springs and fumaroles and the major products are H,O, CO, and N,. Water is more abundant than CO, by a factor of 45, N, is less abundant by a factor of 33.3Terrestrial H,O is present mainly in the ocean, CO, on the ocean floor as CaCO,, and N, is predominantly in the atmosphere. Precipitation of CO, requires the presence of liquid H,O. Thus we can readily account for the dominance of CO, in the atmospheres of Venus and Mars if we postulate ab initio the absence of liquid H,O. Venus is too hot for liquid H,O, Mars is too cold. These arguments suggest that N, should be a minor constituent of both the Venus and Mars atmospheres, in agreement with observation. If relative outgassing rates for CO, and N, on these planets are comparable to values inferred for Earth (cf. Rubey, 1951) we would expect mixing ratios for N, relative to CO, of approximately 3% in the atmospheres of both Venus and Mars. We shall argue that the mixing ratio for Mars is probably less than this value. The reduction for Mars is expected as a consequence for atmospheric escape. The escape mechanisms for 0 and N are essentially different from the mechanism which regulates escape of H. At Martian temperatures, significant numbers of H atoms have velocities greater than 4.9 km sec-' and are therefore free to escape the planet's gravitational field. The escape rate of H is readily predicted by the classic Jeans theory (Jeans, 1925; Chamberlain, 1963). It depends simply on the density of H atoms at the base of the exosphere, and on the value for exospheric temperature. Escape of 0 and N occurs as a consequence of a number of specific reactions which take place in the exosphere. These reactions lead to production of fast atoms. If the velocity vector is oriented properly, the atoms can escape without undergoing further collisions. We shall refer to the escape mechanisms for 0 and N as nonthermal, in contrast to the thermal process envisaged by Jeans. The escape rates for nonthermal mechanisms depend on the rates of specific atomic and molecular reactions. Fast oxygen atoms are produced in the Martian exosphere by dissociative recombination of 0,'. The dominant reaction is

-

02++e O('P)+O(*D) (43) according to Biondi (1969). Each of the 0 atoms in (43) is released with an initial energy of 2.5 eV and can readily escape if the velocity vector is oriented in the upward hemisphere. Likewise, 0 atoms produced by

-

COz+ + e CO(X'C) + O(3P, 'D or 'S) (44) can also escape. The critical escape energy for 0 is 1.99 eV and the comparable values for CO, C, and N are 3.47, 1.47, and 1.74 eV, respectively.

These factors were computed on the basis of relative numbers of molecules.

MARS ATMOSPHERE

357

The computed escape rate for 0, due primarily to (43), is 6 x lo7 atoms cm-, sec-’, averaged over the entire planet. This value is approximately equal to half the ionization rate of CO, in the sunlit exosphere. We assume that atoms are emitted with equal probability in the upward and downward hemispheres. The computed escape rate for 0 can balance an escape rate for H equal to 1.2 x lo8 cm-2 sec-’, in excellent agreement with the observed H escape. The apparent balance between 0 and H escape is not accidental. We may argue that the thermal escape of H is in fact regulated by the nonthermal mechanism for 0. Any imbalance in 0 and H escape rates would lead to a rapid change in the oxidation state of the atmosphere. Consider, for example, the case in which the H escape is larger than the present values. Then oxygen must accumulate in the atmosphere and there will be a resulting increase in the abundance of atmospheric 0, . The chemistry of the C02-H,O system will adjust accordingly. The rate for production of H, will decrease with a consequent reduction in the rate of supply of H atoms to the upper atmosphere. This process will continue until the postulated imbalance is removed. The system is similarly stable with respect to perturbations in the other direction. A decrease in H escape leads to modification of lower atmospheric CO and 0, with consequent increase in production of H, . The production of upper atmospheric H will increase with appropriate adjustment in the H escape rate until, again, the imposed imbalanceisremoved. Molecular hydrogen acts as a powerful buffer which supplies escaping H atoms at precisely the rate required to balance escape of 0. The buffer is relatively stable. The time constant associated with H, is approximately lo3 yr, in contrast to the much shorter times, 3 yr, which apply for 0, and CO. Dissociative recombination of N 2 + ,either by N 2 ++ e

or N,+ f e

-

-

N(4S)+N(4S)

(45)

N(4S)+N(2D)

(46)

can provide important escape fluxes of N. The atoms in (45) are produced with an initial energy of 2.91 eV, and the corresponding value for (46) is 1.73 eV. If we consider a mixing ratio of 3 x for N , and a turbopause at 90 km, we calculate that the atmosphere of Mars would lose essentially all of its nitrogen in less than 5 x lo8 yr. It follows that Mars is probably deficient in N, at the current epoch. If the atmosphere evolved by rapid outgassing at an early phase in the planet’s history, then the time evolution of the N, mixing ratio, p , would be given by p ( t ) = po exp{ -(4.5 x I O * ~ / N ) ~ }

(47)

358

M . B. McElroy

where p o is the initial value of p, N is the total column density (cm-’) of C O , (2 x and t is the elapsed time in years. In this case, the N, mixing ratio should be reduced by a factor of lo5 in 5 x lo9 yr and we would expect N, to be at most a trace constituent of the present atmosphere. On the other hand, if outgassing occurred at a steady rate over geologic time, the N, mixing ratio should be constant with time and approximately equal to 10% of the ratio of outgassing rates, N, to CO, . With Rubey’s (1951) estimate for the corresponding ratio for Earth, assuming that this value is applicable also to Mars, we calculate a mixing ratio of N, relative to CO, of about 3 x in the present atmosphere of Mars, under the assumption that outgassing has occurred at a uniform rate. Thus, measurements of N, could provide important clues to the mode of origin for the atmosphere. A large mixing ratio would suggest slow, steady outgassing. A small mixing ratio might indicate an early origin for the atmosphere with subsequent evolution of N, associated with nonthermal escape. Quantitative conclusions, however, require specific information on the relative rates for outgassing of N, and CO, on Mars. It seems possible that these additional data could be derived from a measurement of isotopic composition, in particular an accurate determination of the N14-N15 ratio. Additional potential sources of energetic exospheric N include predissociation of N, (Brinkmann, 1971) and electron dissociation of N, . Predissociation is less important than dissociative recombination. Electron dissociation could be more important, however, and measurements, or calculations, of the energy distribution for product atoms as a function of impact energy would be desirable and should permit a more quantitative treatment of N escape. Additional nonthermal loss mechanisms have been discussed also for C . These include CO++e hv+CO e+CO hvfCO2 e+C02

and COz++e

-

C+O

(48)

c+o

(49)

C+O+e c+o2

C+O,+e

C+02

(53)

The possible magnitudes of the associated escape rates are discussed by McElroy (1972). It is of interest to compare relative outgassing rates for H,O and CO, on Mars with corresponding values for Earth. If we assume that most of the

MARS ATMOSPHERE

359

carbon atoms evolved by Mars are still present as CO, in the atmosphere, and further assume that the present rates for escape of H and 0 provide a measure of the rate at which H,O has entered the Martian atmosphere over geologic time, then the ratio of H,O to CO, evolved would be about 45, essentially the same as the ratio inferred for Earth by Rubey (1951). The total amount of H,O evolved per unit surface area of Mars is less, however, than that for Earth by a factor of lo3.

VII. Concluding Remarks We have attempted here to provide a brief summary of present research as it relates to the atmosphere of Mars. The subject matter was selected to emphasize the considerable contributions of atomic and molecular physics. We sought to identify major uncertainties which might be resolved by appropriately directed laboratory research. We require additional information on a number of key reactions in the CO, chemical scheme. In particular, further work on the reactions OHfHOt H+H02

and CO+HO,

-

HZOSO, Hz+O,

COZfOH

would be desirable. The possible role of vibrationally excited 0,' in the Martian, and perhaps terrestrial, ionospheres needs further attention. We need data on the rates for production of individual vibrational states of 0,' by the reactions O++CO,

and CO,+ $0

-

O,++CO

o,+ +co

and we need information on the dependence of 0,' recombination on vibrational quantum numbers. In connection with interpretation of Martian dayglow, we need data on several of the processes which may contribute significantly to the observed Cameron and fourth positive bands of CO, . In particular we require clarification of the relative importance of various energetically possible channels for recombination of CO, +. In connection with the evolutionary models discussed in Section VI, we need information on the energy distribution of nitrogen atoms formed by electron induced dissociation of N,, and detailed data on the specific paths associated with recombination of 0 2 +N, ,', and CO'.

360

M . B. McElroy

Planetary science is a relatively new interdisciplinary activity. Progress depends in large measure on the availability of relatively expensive deepspace probes, as the reader may readily infer from the frequent references in this article to various Mariner experiments. It is hoped that the present review should convey at least the flavor of contemporary planetary research and that it should point to the important role which atomic physics has played and should continue to play in the future development of the subject.

ACKNOWLEDGMENT This work was supported by the National Science Foundation, Atmospheric Sciences Division, under Grant GA 33990X to Harvard University.

REFERENCES Adams, W. S. (1941). Astrophys. J . 93, 16. Adams, W. S., and Dunham, T. (1937). Publ. Astron. SOC.Pac. 49, 209. Adams, W. S., and St. John, C. (1926). Astrophys. J. 63, 133. Ajello, J. M. (1971). J . Chem. Phys. 55, 3169. Anderson, D. E., and Hord, C. W. (1971). J. Ceophys. Res. 76, 6666. Banks, P. M., and Nagy, A. F. (1970). J . Ceophys. Res. 75, 1902. Barker, E. S. (1972). Div. Planetary Sci. Meeting, Amer. Astron. SOC.,Hawaii. Barth, C. A., and Hord, C. W. (1971). Science 173, 197. Barth, C. A., Pearce, J. B., Kelly, K. K., Wallace, L., and Fastie, W. G. (1967). Science 158, 1675. Barth, C. A., Fastie, W. G., Hord, C. W., Pearce, J. B., Kelly, K. K., Stewart, A. I., Thomas, G. E., Anderson, G. P., and Raper, 0. E. (1969). Science 165, 1004. Barth, C. A., Hord, C. W., Pearce, J. B., Kelly, K. K., Anderson, G. P., and Stewart, A. I. (1971). J . Ceophys. Res. 76, 2213. Barth, C. A., Hord, C. W., Stewart, A. I., and Lane, A. L., (1972). Science 175,309. Belton, M. J. S., and Hunten, D. M. (1968). Astrophys. J . 153, 963. Belton, M. J. S., and Hunten, D. M. (1969). Science 166, 225. Belton, M. J. S., a d Hunten, D. M. (1971). Icarus 15, 204. Belton, M. J. S., Broadfoot, A. L., and Hunten, D. M. (1968). J . Ceophys. Res. 73,4795. Biondi, M. A. (1969). Can. J. Chem. 47, 1711. Brinkmann, R. T. (1971). Science 174, 944. Broadfoot, A. L., and Wallace, L. (1970). Astrophys. J. 161, 303. Broida, H. P., Lundell, 0.R., Schiff, H. I., and Ketcheson, R. D. (1970). Science 170,1402. Carleton, N. P., and Traub, W. A. (1972). Science 177, 988. Carleton, N. P., Sharma, A., Goody, R. M., Liller, W. L., and Roesler, F. L. (1969). Astrophys. J. 155, 323. Chamberlain, J. W. (1962). Astrophys. J . 136, 582. Chamberlain, J. W. (1963). Planet. Space Sci. 11, 901.

MARS ATMOSPHERE

36 1

Clark, I. D. (1971). J . Atmos. Sci. 28, 847. Clark, 1. D., and Noxon, J. F. (1970). J . Geophys. Res. 75,7307. Clerc, M., and Reiffsteck, A. (1968). J . Chem. Phys. 46, 107. Cloutier, P. A., McElroy, M. B., and Michel, F. C. (1969). J . Geophys. Res. 74, 6215. Connes, P., Connes, J., and Maillard, J. P. (1969). Atlas des spectres infrarouges de Venus, Mars, Jupiter, et Saturn. ” CNRS, Paris. Dalgarno, A., and McElroy M. B. (1970). Science 170, 167. Dalgarno, A., McElroy, M. B., and Moffett, R. J. (1963). Planet Space. Sci. 11, 463. Dalgarno, A., McElroy, M. B., and Walker, J. G. G. (1967). Planet. Space Sci. 15, 331. Dalgarno, A., McElroy, M. B., and Stewart, A. 1. (1969). J . Atmos. Sci. 26, 753. Dalgarno, A., Degges, T. C., and Stewart, A. I. (1970). Science 167, 1490. Detwiler, C. R., Garnett, P. L., Purcell, J. D., and Tousey, R. (1961). Ann. Geophys. 17, 263. devaucouleurs, G . (1954). “Physics of the Planet Mars.” Faber and Faber, London. Donahue, T. M. (1965). Planet. Space Sci. 14, 33. Dunharn, T. (1952). I n “The Atmospheres of the Earth and Planets” ( G . P. Kuiper, ed.), p. 288. Univ. of Chicago Press, Chicago, Illinois. Evans, J. V. ( I 962). J . Geophys. Res. 67, 4914. Evans, J. V. ( I 965). Planet. Space Sci. 13, 1031. Fehsenfeld, F. C., Schmeltekopf, A. L., and Ferguson, E. E. (1967). J . Chem. Phys. 46, 2802. Fehsenfeld, F. C., Dunkin, D. B., and Ferguson, E. E. (1970). Planet. Space Sci. 18, 1267. Felder, W., Morrow, W., and Young, R. A. (1970). J . Geophys. Res. 75,7311. Ferguson, E. E., Fehsenfeld, F. C., Goldan, P. D., and Schemltekopf, A. L. (1965). J . Geophys. Res. 70, 4323. Fjeldbo, G., Kliore, A. J., and Seidel, B. L. (1970). Radio Sci. 5, 381. Freund, R. S. ( I 97 I). J . Chern. Phys. 55, 3569. Gentieu, E. C., and Mentall, J. E., (1972). Trans. Amer. Geophys. Un. 53,459. Gierasch, P. J. (1972). Earth Extraferrestriul Sci. 1, 1971. Giver, L. P., Inn, E. C. Y., Miller, J. H . , and Boese, R. W. (1968). Astrophys. J . 153, 285. Goody, R. M. (1 964). “ Atmospheric Radiation.” Oxford Univ. Press, London and New York. Goody, R. M. (1969). Annu. Rev. Astron. Astrophys. 7, 303. Hall, L. A., and Hinteregger, H . E. (1970). J . Geophys. Res. 75, 6959. Hanson, W. B. ( I 963). Spuce Res. 3, 282. Hanson, W. B., and Johnson, F. S. (1961). Mem. Soc. Roy. Sci. Liege [5] 4, 390. Henry, R. J. W., and McElroy, M. B. (1968). In “The Atmospheres of Venus and Mars” (J. C. Brandt and M. B. McElroy, eds.), p. 251. Gordon & Breach, New York. Herr, K. C . , Horn, D., McAfee, J. M., and Pimentel, G. C. (1970). Astrophys. J. 75,883. Hochanadel, C. J., Ghorrnley, J. A., and Orgren, P. J. (1972). J . Chem. Phys. 56,4426. Hord, C . W. (1972). Icarus 16, 253. Hunten, D. M. (1968). In “The Atmospheres of Venus and Mars” (J. C. Brandt and M. B. McElroy, eds.), p. 147. Gordon & Breach, New York. Hunten, D. M. (1971). Space Sci. Rev. 12, 539. Hunten, D. M., and McElroy, M. B. (1970). J . Geophys. Res. 75, 5989. Ingersoll, A. P., and Leovy, C. B. (1971). Annu. Rev. Asrron. Astrophys. 9, 147. Jeans, J. H. (1925). “The Dynarnical Theory of Gases.” Cambridge Univ. Press, London and New York. Kaplan, L. D., Munch, G., and Spinrad, H. (1964). Astrophys. J . 139, 1. Kaplan, L. D., Connes, J., and Connes, P. (1969). Astrophys. J . 157, 187.

362

M . B. McElroy

Katakis, D., and Taube, H. (1962). J . Chem. Phys. 36,416. Kaufman, F. (1969). Annu. Rev. Phys. Chem. 20,4590. Kliore, A.J., Cain, D. L., Levy, G. S., Eshleman, R., and Drake, F. D. (1965). Science 149, 1243. Kliore, A. J., Cain, D. L., Fjeldbo, G., Seidel, B. L., and Rasool, S . I. (1972). Science 175, 313. Kuiper, G . P. (1 947). Astrophys. J. 106, 25 1 . Kuiper, G. P. (1952). In “Atmospheres of the Earth and Planets” (G. P. Kuiper, ed.), p. 306. Univ. of Chicago Press, Chicago, Illinois. Lane, A. L., Barth, C. A,, Hord, C. W., and Stewart, A. I. (1973). Icarus 18, 102. Langley, K. F., and McGrath, W. D. (1971). Planet. Space Sci. 19, 413. Lawrence, G. M. (1972). J. Chem. Phys. 56, 3435. Leighton, R. B., and Murray, B. C. (1966). Science 153, 136. Leovy, C. B. (1966). Science 154, 1178. Little, S. J. (1971). In “Planetary Atmospheres” (C. Sagan, T. C. Owen, and H. I. Smith, eds.), p. 241. Reidel, Dordrecht, The Netherlands. McConkey, J. W., Burns, D. J., and Woolsey, J. M. (1968), Proc. Phys. Soc., London (A?. Mol. Phys.) 1, 71. McConnell, J. C. (1972). In “Physics and Chemistry of Upper Atmospheres” (B. M. McCormal and G. Weill, eds.), In press. McConnell, J. C., and McElroy, M. B. (1970). J. Geophys. Res. 75, 7290. McElroy, M. B. (1968). J. Geophys. Res. 73, 1513. McElroy, M. B. (1969). J . Geophys. Res. 74, 29. McElroy, M. B. (1970). Ann. Geophys. 26, 643. McElroy, M. B. (1972). Science 175,443. McElroy, M. B., and Donahue, T. M. (1972). Science 177,986. McElroy, M. B., and Hunten, D. M. (1969). J . Geophys. Res. 74, 5807. McElroy, M. B., and Hunten, D. M. (1970). J . Geophys. Res. 75, 1188. McElroy, M. B., and McConnell, J. C. (1971a). J . Atmos. Sci. 28, 879. McElroy, M. B., and McConnell, J. C. (1971 b). J. Geophys. Res. 76, 6674. McElroy, M. B., and Strobel, D. F. (1973). To be published. McElroy, M. B., L’Ecuyer, J., and Chamberlain, J. W. (1965). Astrophys. J. 141, 1523. McElroy, M. B., Kong, D., and Leverett, B. (1973). To be published. Marshall, J. V. (1964). Comm. Lunar Planet. Lab. 2 , 167. Mumma, M. J., Stone, E. J., and Zipf, E. C. (1971). J. Chem. Phys. 54,2627. Nagy, A. F., and Banks, P. M. (1970). J. Geophys. Res. 75, 6260. Neugebauer, G., Munch, G., Kieffer, H., Chase, S. C., and Miner, E. (1971). Astrophys. J . 76, 719. Nier, A. D. C., Hanson, W. B., McElroy, M. B., Seiff, A., and Spencer, N. W., (1972). Icarus 16, 74. Pang, K., and Hord, C. W. (1971). Zcarus 15,443. Parkinson, T. M., and Hunten, D. M. (1972). J . Atmos. Sci. 29, 1380. Pettingill, G. H., Counselman, C. C., Rainville, L. P., and Shapiro, I. I. (1969). Astron. J. 74,461. Rapp, D., and Englander-Golden, P. (1965). J . Chem. Phys. 43, 1464. Rasool, S. I. (1963). AIAA. J. 1, 6. Rasool, S. I., and Stewart, R. W. (1971). J . Atmos. Sci.28, 869. Reeves, R. R., Harteck, P., Thompson, B. A., and Waldron, R. W. (1966). J. Phys. Chem. 70, 1637. Rishbeth, H. (1968). Rev. Geophys. 6, 33.

MARS ATMOSPHERE

363

Rogers, A. E. E., Ash, M. E., Counselman, C. C., Shapiro, I. I., and Pettingill, G. H. (1970). Radio Sci. 5, 465. Rubey, W. W. (1951). Geol. Soc. Amer., Bull. 62, 1 1 1 . Schorn, R. A. (1 971). In “Planetary Atmospheres” (C. Sagan, T. C. Owen, and H. J. Smith, eds.), p. 223. Reidel, Dordrecht, The Netherlands. Schorn, R. A., Spinrad, H., Moore, R. C., Smith, H. J., and Giver, L. P. (1967). Astrophys. J. 147, 743. Shea, M. F., Sharp, R. D., and McElroy, M. B. (1968). J. Geophys. Res. 73,4199. Simonaitis, R., and Heicklen, J. (1972). J. Chem. Phys. 56, 2004. Slanger, T. G., and Black, G. (1971).J. Chem. Phys. 54, 1889. Slanger, T. G . , Wood, B. J., and Black, G. (1972). J. Chem. Phys. 57, 233. Stewart, A. I. (1972). J. Geophys. Res. 77, 54. Stewart, R. W. (1971). J. Atmos. Sci. 28, 1069. Stolarski, R. S . ( I 972). J . Geophys. Res. 77, 2862. Strickland, D. J., and Donahue, T. M. (1970). Planet. Space Sci. 18,661. Strickland, D. J., Thomas, G. E., and Sparks, P. R. (1972). J . Geophys. Rex 77,4052. Stuhl, F., and Niki, M. (1971). J. Chem. Phys. 55, 3943. Thomas, G . E. (1971). J. Atmos. Sci. 28, 859. Tull, R. G. (1971). In “Planetary Atmospheres” (C. Sagan, T. C. Owen, and H. J. Smith, eds.), p. 237. Reidel, Dordrecht, The Netherlands. Ung, A. Y., and Schiff, H. I. (1966). Can.J. Chem. 44, 1981. Urey, H. C. (1959). In “Handbuch der Physik” (S. Flugge, ed.), p. 363. Springer-Verlag. Berlin and New York. Wallace, L., and McElroy, M. B. (1966). Planet. Space Sci. 14,677. Warneck, P. (1964). J . Chem. Phys. 41, 3435. Wauchop, T. S . , and Broida, H. P. (1971). J . Geophys. Res. 76, 21. Wells, R. A . (1969). Science 166, 862. Wells, W. C., Borst, W. L., and Zipf, E. C. (1972). J . Geophys. Res. 77,69. Widing, K . G., Purcell, J. D., and Sandlin, G. D. (1970). Solar Phys. 12, 52. Zipf, E. C., Stone, E. J., Mumma, M. J., Wells, W. C., and Borst, W. L. (1971). Trans. Amer. Ceophys. Un. 52, 308.

This Page Intentionally Left Blank

Author Index Numbers in italics refer to the pages on which the complete references are listed. A

Aberth, W., 238, 241, 271, 314 Abjodun, R . A., 69, 70, 72, 122 Abragam, A,, 282, 301, 314 Abrams, R. L., 134, 204 Accad, Y., 2, 44 Adanis, W. S., 325, 360 Ajello, J. M., 351, 352, 360 Akins, D. L., 133, 175, 202, 204 Anderson, C. H., 264, 289, 299, 306, 314, 315, 318

Anderson, D. E., 335, 349, 360 Anderson, G . P., 324, 331, 333, 338, 348, 349, 354, 360 Anderson, J. B., 266, 314 Anderson, P. W., 138, 139, 141, 144, 145, 146, 149, 161, 188, 204 Andres, R. P., 266, 314 Andrick, D., 221, 222, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 238, 240, 241

Ash, M. E., 324, 362 Aubrey, B. B., 268, 315 Audoin, C., 262, 314 Auerbach, D., 263, 314

B Bacher, R . F., 16, 44 Bagus, P. S., 7, 12, 25, 26, 27, 44, 45 Baird, J . K.,313, 314 Baker, J. M., 271, 319 Balling, L. C., 238, 241 Bandel, H. W., 233, 238, 241 Banks, P. M., 351, 360, 362 Bardsley, J. N., 228, 242 Barker, E. S., 327, 360 Barriol, J., 133, 162, 168, 206 Barth, C. A., 324, 329, 330, 331, 333, 338, 348, 349, 352, 353, 354, 360, 362 Barton, G. W., 270, 316 Bassel, R. H., 71, 88, 89, 122, 126 Bates, D. R., I I , 44,48, 71, 115, 122 Bauche, J., 18, 44

Bauer, R. K., 270, 314 Baz, A. I . , 112, 122 Beaty, E. C., 80, 81, 82, 125 Becker, E. D., 301, 314 Becker, G., 263, 314 Bedding, D., 309, 314 Bederson, B., 53, 122, 217, 218, 238, 239, 240, 241 Beers, Y., 312, 314 Bell, K. L., 64, 68, 70, 71, 122 Bellet, J., 200, 205 Belton, M. J. S., 324, 326, 328, 336, 360 Bely, O., 75, 79, 114, 122 Bennewitz, H. G., 249,260, 262, 264, 268, 308, 310, 311, 314 Benson, R. C., 245, 298,316 Berg, R. A., 261, 288, 314, 321 Bergrnann, K.,133, 202, 204 Bergstrorn, I., 24, 44 Bernhard, F., 314 Bernstein, H, J., 301, 319 Bernstein, R. B., 255, 262, 263, 265, 314, 317, 318, 321

Bethe, H. A., 63, 122 Bevensee, R. M., 90, 122 Bhagavantam, S., 155,204 Bhatia, A. K., 228, 242 Bickel, W. S., 24, 25, 45 Bickes, R. W., Jr., 314 Biedenharn, L. C., 93, 95, 122 Biondi, M. A,, 356, 360 Birman, A,, 83, 85, 87, 122 Birnbaum, G., 129, 149, 204 Bitsch, A., 222, 227, 229, 230, 231, 232, 233, 234, 235, 236, 238, 241 Black, G., 336, 337, 363 Blankenbecler, R., 105, 126 Blatt, J. M., 93, 95, 122 Bloom, M., 135, 204 Boeckh, R. v., 297, 314 Boese, R. W., 326, 361 Boggs, J. E., 133, 149, 185, 205 Boldt, G., 36, 41, 44 Bolef, D. I., 290, 294, 321 365

Author Index

366

Bonczyk, P. A., 271, 292, 314 Bonham, R. A., 72, 73, 83, 122 Born, M., 62, 122 Borst, W. L., 332, 351, 352, 353, 363 Bransden, B. H., 83, 85, 88, 89, 122, 223, 232, 233, 234, 235, 241 Bratsev, V. I., 75, 76, 125 Braunstein, R., 254, 314 Breivogel, F. W., Jr., 255, 314, 317 Brennen, W., 134,204 Brewer, R. G., 134, 198, 204, 206 Bridges, T. J., 201, 204 Brillouin, L., 18, 44 Brink, D. M., 152, 153, 204 Brink, G. O., 271, 314 Brinkmann, R. T., 358, 360 Brittain, A. H., 134, 183, 198, 199, 204, 206

Broadfoot, A. L., 326, 329, 360 Broida, H. P., 133, 134, 202, 204, 205, 206, 330, 355, 360,363 Bromander, J., 24, 44 Brombach, E. E. A., 263,314 Brooks, P. R., 263,315, 317 Brooks, R. A., 264, 289, 294, 298, 315, 318

Brown, H., 292,293, 315 Brown, R., 40,44 Browne, J. C., 290, 315 Buchta, R., 24, 44 Buckingham, A. D., 153, 155, 204 Buchler, A., 264, 288, 315 Bullard, E. C., 232, 237, 241 Bunge A., 12, 44 Bunge, C. F., 12, 25, 44 Burak, I., 201, 206 Burhop, E. H. S., 208, 226, 242 Burke, P. G., 9, 44, 48, 66, 70, 76, 99, 104, 105, 107, 109, 110, 114, 117, 118, 119, 120, 121, 123, 124, 126, 208, 211, 226, 234, 238, 241 Burkhardt, E. G., 201, 204 Burns, D. J., 351, 355, 362 Burns, R. P., 270, 316 Byron, F. W., 87, 123 C

Cade, P., 290, 315 Cain, D. L., 324, 343, 362

Callaway, J., 121, 126, 230, 231, 234, 235, 241, 242

Callear, A. B., 134, 204 Carleton, N. P., 326, 327, 336, 360 Carlson, R. O., 285, 315, 318 Carrico, J. P., 313, 320 Carrington, A,, 245, 301, 315 Carrington, T., 131, 132, 133, 134, 204 Cartwright, D. C., 65, 66, 69, 70, 75, 76, 79, 80, 81, 82, 126 Casimir, H. B. G., 135, 204 Castillejo, L., 58, 109, 123 Cederberg, J. W., 283, 294, 315, 319 Celotta, R. J., 268, 315 Chamberlain, G. E., 216, 241 Chamberlain, J. W., 324, 335, 356, 360, 362

Chan, S. I., 298, 315 Chandrasekhar, S., 8, 44 Chang, E. S., 234, 242 Chang, T. Y., 201, 204 Chantry, G. W., 245, 315 Chase, S. C., 329, 362 Chen, J. C. Y., 87, 123 Cheo, P, K., 134, 204 Cheung, A. C., 166,206 Chisholm, C. D. H., 15, 17, 19, 44 Clark, I. D., 337, 338, 361 Clarke, E. M., 226, 242 Clementi, E., 27, 44 Clerc, M., 337, 361 Cloutier, P. A., 346, 361 Cohen, J. B., 133, 171, 204 Cohen, M., 15, 44 Cohen, V. W., 245, 316 Coleman, J. P., 83, 85, 88, 89, 122 Collins, R. E., 218, 239, 240, 241, 268, 315 Comer, J., 224, 241 Condon, E. U., 3, 6, 44 Connes, J., 326, 327, 328, 361 Connes, P., 326, 327, 328, 361 Cook, R. L., 245, 280, 301, 313, 316 Cooper, J., 149, 204 Cooper, J. W., 21 I , 234, 241 Cornwell, C. D., 283, 320 Cottrell, T. L., 129, 204 Counselman, C. C., 324, 362,363 Cowan, R. D., 2, 10, 11, 42, 44 Cox, A. P., 160, 204 Cox, D. M., 66, 78, 89, 124

367

AUTHOR INDEX

Crapo, L. M., 294, 306, 319 Crompton, R. W., 233, 241 Crosby, D. A , , 264, 318 Crossley, R . J. S., 2, 15, 44 Cupp, R. E., 277, 278, 279, 283, 285, 304, 305, 315, 320 Curley, E. K., 68, 78, 89, 124, 226, 242 Curnutte, B., 147, 206

D Dagdigian, P. J., 266, 267, 289, 315, 316 Dailey, B. P., 291, 320 Dalgarno, A,, 3, 15, 17, 19, 44, 46, 331, 345, 349, 353, 361 Daly, P. W., 133, 162, 170, 171, 189, 204 Damburg, R., 66, 110, 113, 118, 123 Damgaard, A,, I I , 44 Dankwort, W., 294, 319 Das, C., 12, 44 Das, T. P., 298, 315 Datz, S., 270, 315, 319 Davenport, J., 134, 206 Davies, D. W., 282, 315 Davies, P. B., 267, 283, 306, 307, 315, 319 Day, R. B., 58, 69, 123 Degges, T. C., 349, 361 De Heer, F. J., 64, 65, 125 Deichsel, H., 218 242 deleeuw, F. H., 274, 283, 286, 306, 315 Demaison, J., 133, 162, 168, 206 Demkov, Yu, N., 59, 123, 256, 316 Demtroder, W., 133, 202, 204 Derr, V. E., 265, 277, 278, 279, 285, 287, 288, 298, 304, 312, 316, 320, 321 Desaintfuscien, M., 262, 314 Detwiler, C. R., 333, 336, 361 devaucouleurs, G., 323, 361 De Wijn, H. W., 291, 315 Dickinson, H. 0.. 104, 123 Dickinson, J. T., 245, 254, 255, 265, 276, 279, 281, 292, 293, 295, 315, 317, 320, 321

Donnelly, D. P., 268, 272, 315, 319 Doughty, N. A., 115, 123 Dousmanis, G. C., 299,321 Dowling, J. M., 190, 205 Drake, F. D., 324, 343, 362 Drechsler, W., 250, 275, 296, 298, 300, 315 Dress, W. B., 313, 314 Dunham, T., 325,360, 361 Dunkin, D. B., 343, 361 Dunning, F. B., 272, 315 Duxler, W. M., 211, 230, 231, 234, 235, 241

Dyke, T. R., 267, 275, 283, 288, 306, 307, 315,319

Dymanus, A., 264,270,283, 286, 301, 302, 305, 306, 315, 317, 318, 321

E Eck, T. G., 293, 318 Edlkn, B., 3, 32, 44 Edminston, C., 10, 44 Edmonds, A. R., 151, 153,204,209, 241 Eisenbud, L., 117, 126 Eisinger, J. T., 285, 287, 318 Eisner, P. N., 268, 315 Eissa, H., 68, 71, 122 Eissner, W., I I , 44 Elford, M. T., 233, 241 Enemark, E. A,, 92, 93, 116, 118, 123 Englander-Golden, P., 355, 362 English, T. C., 245, 254, 255, 261, 262, 264,266, 271,279,283, 284,287,292, 295, 297,315,316,321 Erhardt, H., 228, 241 Eshleman, R., 324, 343,362 Estermann, I., 288, 316 Evans, J. V., 345,361 Evenson, K. M., 133, 162, 170, 171,202, 204, 206

Eyb, M., 221, 227, 228 ,230, 231, 234, 235, 236, 238, 240, 241

Dijkerman, H., 266, 270, 287, 297, 298, 315

Dillon, T. A,, 149, 204 Dolder, K. T., 2 18, 224, 225, 227 ,228, 229,230. 231. 232. 233, 234, 235, 241 Donahue, T. M., 332, 336, 338, 340, 341, 342, 343, 361, 362, 363

F Fabricand, B. P., 285, 315,318 Fabris, A. R., 162, 171, 172, 180, 189, 191, 192, 204 Fain, V. M., 256, 316

368

Author Index

Falconer, W. E., 288, 315, 317 Fano, U., 212, 213, 226, 241, 242, 247, 316 Farago, P. S., 53, 123 Fastie, W. G., 324, 349, 353, 360 Feautrier,N., 110, 111, 120, 123, 126 Feenberg, E., 69, 70, 71, 123 Fehsenfeld, F. C., 340, 343, 361 Felder, W., 337, 361 Fenn, J. B., 266,314 Ferch, J., 294, 319 Ferguson, E. E., 340, 343, 361 Feshbach, H., 101, 123 Filseth, S. V., 134, 206 Fink, E. H., 133, 175, 202, 204 Finney, A. A., 285, 318 Fisk, G. A., 264, 268, 269, 309, 310, 316 Fite, W. L., 66, 68, 77, 78, 79, 88, 89, 119, 124, 125 Fjeldbo, G., 343, 361. 362 Flegel, W., 266, 270, 287, 297, 298, 315 Fleming, R. J., 71, 123 Flygare, W. H., 134, 194, 198, 201, 205, 206, 245,298,316 Flynn, G. W., 160, 204 Foland, W. D., 58, 82, 83, 124 Foley, H. M., 148, 204, 292, 301, 316 Fox R. E., 71, 125 Fraga, S., 282, 316 Franck, J., 130, 204 Franzen, W., 222, 225, 229, 230, 231, 237, 238,241, 242 Fraser, J. W., 270, 316 Fraser, R. G. J., 288, 316 Freed, K. F., 283, 303, 316 Frenkel, L., 133, 201, 204 Freund, R. S., 252,268, 272, 274, 283, 302, 303, 316, 320, 351, 352,361 Freund, S. M., 133, 162, 170, 171, 201, 202, 204,205, 264,268,269, 302, 306, 309, 310, 316 Friedburg, H., 249, 260, 316 Friedrich, H., 12, 44 FrohIich, H., 136, 204 Froese Fischer, C., 5 , 10, 12, 22, 23, 30, 31, 44 Frosch, R. A., 301, 316 Frost, L. S., 233,241 Fuller, G. H., 245, 316 Fundaminsky, A., 48, 115, 122

G Gailitis, M., 105, 106, 107. 112, 114, 115, 121, 123 Gallagher, A., 92, 93, 1 16 ,I 18, 123 Gallagher, D. F., 110, 118, 120, 123, 226, 241 Gallagher, J. J., 265, 277, 278, 279, 283, 285, 287, 288, 298, 304, 305, 312, 315, 316,320,321 Gallagher, T. F., Jr., 262, 265, 284, 292, 315,317 Gammon, R. H., 268, 272, 283, 302, 303, 304,316,320 Gaponov, A. V., 256, 316 Garnett, P. L., 333, 336. 361 Garret, W. R., 238, 241 Garstang, R. H., 2, 31, 44, 45 Gehenn, W., 239, 240, 241 Geiger, F. E., Jr., 285, 287, 318 Geltman, S., 66, 81, 82, 103, 110, 118, 119, 120, 123, 124, 125,212, 226,241, 242 Gentieu, E. C., 352, 361 Gerjuoy, E., 71, 87, 88, 89, 122, 123, 126 Geschwind, S., 292, 317 Ghormley, J. A., 341,361 Ghosh, A. S., 89, 123 Gibson, J. R., 218, 224, 225, 227, 228, 229, 230, 231,232, 233, 234, 235, 241 Gierasch, P. J., 323, 361 Gilbert, J., 198, 204 Ginsberg, A. P., 288, 315 Giordmaine, J. A., 264, 316 Giver, L. P., 326, 329, 361, 363 Glauber, R. I., 83, 85, 123 Glaze, D. J., 312, 316 Godfredson, E., 15, 44 Godfrey, J. T., 149, 204 Godfrey, P. D., 306, 316 Gold, L. P., 265, 277, 278, 285, 287, 288, 290, 298, 299, 321 Goldan, P. D., 343, 361 Golden, D. E., 228, 233, 238, 241 Goldschmidt, Z. B., 3, 45 Goldstein, M., 218, 239, 240, 241 Comb&, P., 11,45 Goodrich, G. W., 272, 316 Goody, R. M., 323, 326,360,361 Gordon, J. P., 260, 316

3 69

AUTHOR INDEX

Gordon, R. G., 133, 147, 149, 162, 170, 183, 194, 195, 196, 204, 205 Gordon, R. J., 266, 316 Gordon, W., 49, 124 Gordy, W., 245, 280, 301, 313, 316 Goudsrnit, S., 16, 44 Grabner, L., 253, 254, 275, 283, 284, 316, 317

Griff, G., 250, 254,266, 270,275, 283, 287, 296, 297, 298, 300, 314,315, 316 Graff, J., 266, 289, 315, 316 Gray, C. G., 148, 151, 204 Green, E. F., 270, 316 Green, F. T., 264, 318 Green, S., 286, 289, 317 Green, T. A., 74, 124 Giinther, F., 263. 317 Gunther-Mohr, G. R., 292, 305, 317 Gupta, R., 222, 225, 229, 230, 231, 237, 238, 241, 242 Gwinn, W. D., 134, 198, 204

H Haerten, R., 264, 268, 308, 310, 311, 314 Hagstrum, H. D., 271, 317 Hahn, Y., 105, 124 Hall, L. A., 346, 352, 361 Hall, R. T., 190, 205 Harneed, S., 2, 45 Hamrnerle, R. H., 265, 266, 291, 292, 293, 294, 295, 317 Hancock, J. K., 134, 194, 198, 201, 205 Hansen, J. E., 43, 45 Hanson, W. B., 331, 345, 361, 362 Happer, W., 245, 317 Hara, K., 312, 317 Harrington, H . W., 134, 198, 204 Harris, F. E., 18, 19, 25, 45, 46, 121, 124 Harrison, G., 255, 313, 317 Harteck, P., 337, 362 Hartree, D. R., 12, 21, 45 Hartree, W., 5 , 12, 21, 45 Hebert, A., 292, 317 Hebert, A. J., 252, 255, 274, 285, 287, 314, 31 7

Heddle, D. W. O., 69, 124 Hedges, R. E. M., 134, 204 Heicklen, J., 336, 363 Heindorff T., 229, 242

Heitbaurn, J., 283, 298, 317 Hellwig, H., 312, 317 Helrner, J. C., 263, 317 Henry, R. J. W., 345, 351, 361 Heppner, R. A., 272, 315 Herbst, E., 302, 316 Herman, F., 10, 45 Herman, R. M., 286, 317 Herrnann, G. F., 134, 183, 198, 205 Heroux, L., 24, 45 Herr, K. C., 324, 361 Herschbach, D. R., 264, 266, 268, 269, 309, 310, 312, 316, 318 Herzberg, G., 172, 205 Herzenberg, A., 2, 44 Herzfeld, K. F., 129, 205 Hey, P., 31, 45 Hibbert, A., 9, 44, 117, 123 Hidalgo, M. B., 81, 82, 123, 124 Higginson, G. C., 71, 123 Hilborn, R. C., 265,284, 292, 317 Hill, R. M., 134, 183, 198, 205 Hils, D., 240, 242 Hinnov, E., 29, 36, 45 Hinteregger, H. E., 346, 352, 361 Hinze, J., 13, 45 Hochanadel, C. J., 341, 361 Hofft, J., 229, 242 Hoeft, J., 287, 317 Hoeper, P. S., 222, 225, 230, 231, 237, 238, 242 Hofrnann, H., 221, 240,241 Hollenbach, D. J., 192, 205 Holloway, J. H., 292, 293, 317 Hollowell, C. D., 285, 287, 317 Holt, A. R., 83, 85, 87, 88, 89, 124 Honig, V., 290, 318 Horak, Z., 15, 16, 20, 25, 45 Hord, C. W., 324, 329, 330, 331, 333, 335, 338, 348, 349, 352, 354, 360, 361, 362 Horn, D., 324, 361 Hostettler, H. U.,265, 317 Hougen, J. T., 141, 142, 205 Houriez, J., 200, 205 Howard, B. J., 267, 275, 283, 307, 315, 319

Hughes, H. K., 246,317 Hughes, V. W., 244, 253,254,260,264,271, 275, 281, 283, 284, 292, 312, 314, 316, 317, 318

Author Index

3 70

Huiszoon, C., 305, 317 Hunt, J., 88, 124 Hunten, D . M., 323, 324, 326, 327, 328, 336, 338, 339, 340, 341, 360, 361, 362 Huo, W. M., 290, 315 Hylleraas, E., 13, 45 I Ichiki, S. K., 134, 183, 198, 205 lkenberry, D., 298, 315 Ingersoll, A . P., 323, 361 lnrnan, F. 285, 318 Inn, E. C. Y., 326, 361 Innes, F. R., 17, 44 Inokuti, M., 64, 124 lonov, N., 270, 321 lwai, T., 134, 205

J Jaccarino, V., 293, 317 Jacobus, F. B., 263, 317 Jahn, H. A., 155,205 James, L. W., 293, 318 James, M. G., 2, 44 Jeans, J. H . , 356, 361 Jen, C. K., 298, 317 Jenkins, J. L., 134, 183, 198, 205 Joachain, C. J., 87, 123, 124 Jobe, J. D., 64, 65, 68, 71, 74, 124 John, T. L., 120, 124 Johns, J. W . C., 201, 202, 204 Johnson, F. S., 345, 361 Johnson, N. B., 270 Johnson, S. E., 133, 206 Jones, E. M., 263, 315, 317 Jones, R. H., 264,317, 319,320 Jucys, A. P., 6, 12, 45, 46

K Kaiser, E. W., 286, 287, 288, 290, 291, 308, 317 Kakati, D., 263, 317 Kaminskii, D. L., 261,317 Kaminsky, M . , 270, 271, 317 Kaneko, M., 134, 205 Kang, 1. J., 58, 82, 83, 124

Kaplan, D. E., 134, 183, 198, 205 Kaplan, L. D., 326, 327, 328, 361 Karplus, R., 205 Karule, E., 110, 118, 123,238, 239, 242 Katakis, D., 337, 362 Kaufman, F., 338, 362 Kaufrnan, M., 254, 255, 265, 270, 274, 285, 287, 308, 313, 318, 321 Kauppila, W . E., 66, 68, 77, 78, 79, 88, 89, 119, 124, I25 Kavetskis, V., 12, 46 Kelly, H. P., 3, 43, 45 Kelly, K. K., 324, 331, 333, 338, 348, 349, 353, 354, 360 Kernpf, R. A., 279, 283, 285, 304, 305, 315 Kennedy, D. J., 64,122, 124 Kenney, C. N . , 256, 320 Kern, C. W., 301, 318 Ketcheson, R. D., 330, 360 Khare, S. P., 67, 124 Khosla, A,, 266, 318 Kieffer, H., 329, 362 Kim, Y. K., 64,124 King, J. G., 264, 270, 292, 293, 315, 317, 318 Kingston, A. E., 64, 84, 122, 124 Klais, O., 264, 308, 314 Klapisch, M., 18, 44 Klar, H., 202, 204 Kleinpoppen, H., 53, 124, 226, 240, 242 Klernrn, R. A., 25, 46 Klemperer, W., 133, 183, 204, 206, 252, 254, 255, 261, 264, 265, 267, 268, 269, 270, 272, 274, 277, 278, 283, 285, 286, 287, 288,290, 298,299, 302, 303, 304, 305, 306, 307, 308, 309, 310, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321 Kley, D., 134, 205 Kliore, A. J., 324, 343, 361, 362 Knauer, F., 245, 318 Kobayashi, T., 312, 317 Kohn, W., 54,90, 124 Kolenkow, R. J., 299, 318 Kollath, R., 217, 225, 229, 230, 231, 232, 233, 236, 237,242 Kong, D., 340, 362 Kramer, K. H., 262, 263, 314, 318, 321 Krauss, M . , 10, 44, 79, 124 Krishnaji, 300, 318 Kronig, R. de L., 301, 318

371

AUTHOR INDEX

Krueger, V. R., 264, 317 Kuiper, G. P., 323, 325, 332, 362 Kukolich, S. G . . 134, 198, 201, 205 Kupperniann, A , , 65, 66, 69, 70, 75, 76, 79, 80, 81, 82, 126 Kusch, P., 244, 245, 260, 264, 265, 281, 283, 284, 293, 318, 319 Kuyatt, C. E.,215, 216,220, 221, 226, 241, 242

L LaBahn, R. W., 211, 230, 231, 234, 235, 241, 242 Labuhn, F., 36, 45 Laine, D. C . , 245, 261, 263, 317, 318 Lamkin, J. C., 110, 11 I , 126 Landau, L. D., 139, 140, 150, 158,205 Lane, A. L., 329, 330, 349, 352, 360, 362 Langley, K. F., 338, 362 Larson, P. E., 133, 162, 183, 195, 196, 204 LaTourrette, J. T., 271, 319 Laurie, V. W., 287, 320 Lawrence, G . M., 29, 36, 41, 45, 46, 352, 362 Lawrence, T. R., 299, 318 Layzer. D., 2, 13, 15, 16, 17, 20, 25, 45, 90, 124 Leckenby, R. E., 307,318 L'Ecuyer, J., 324, 362 Le Dorneuf, M., 120, 126 Lee, C. A., 285, 315, 318 Lee, S . S . , 306, 319 Lee, Y. T., 266, 312, 316, 318 Leech, J. W., 48, 115, 122 Lees, R. M., 133, 162, 169, 170, 171, 175, 178, 179, 181, 205 Leigh, R. S., 90, 108, 124 Leighton. R. B., 329, 362 Lemaire, J., 200, 205 Leovy, C. B., 323, 329, 361, 362 Lesk, M. E., 268, 272, 283, 302, 303, 304, 316, 320 Leverett, B., 340, 362 Levy, D. H., 245, 301, 315 Levy, G. S . , 324, 343, 362 Levy, J. M., 134, 198, 201, 205 Lew, H., 264, 265, 270, 271, 285, 287, 314, 318, 320 Lewis, H. R., 271, 319

Lewis, M. N., 15, 16, 20, 25, 45 Ley, R., 253, 295, 297, 298, 299, 301, 314, 318 Lide, D. R., 199,206, 245, 282,285, 318 Lifshitz, E. M., 139, 140. 150, 158, 205 Liller, W. L., 326, 360 Linderberg, J., 292, 319 Lipscomb, W. N., 299, 301, 318, 320 Lipsky, L., 103, 124 Lipsky, S., 69, 79, 126 Lipworth, E., 264, 313, 318, 320 Little, S. J., 329, 330, 362 Litovitz, T. A,, 129, 205 Lobner, K. E. G., 290, 318 Lowdin, P. O., 4, 10, 45 London, F., 135, 136,205 Long, R. L., 66, 78, 89, 124 Longuet-Higgins, H. C., 141, 205 Loomis, F., 131, 206 Lovas, F. J., 245, 282, 285, 287, 317, 318 Lucken, E. A. C., 282, 290, 291, 292, 318 Lundell, 0. R., 330, 360 Lundin, L., 24, 44

M MacAdam, K. B., 287,297,315 McAfee, J. M., 324, 361 McCavert, P., 90, 91, 92, 93, 104, 116, 124 McColm, D., 260,318 McConkey, J. W., 351, 355, 362 McConnell, J. C., 331, 332, 333, 336, 337, 338, 343, 348, 349, 351, 352, 353, 354, 362 McCoubrey, J. C., 129, 204 McCusker, M. V., 240, 242 McDowell, M. R. C., 223, 232, 233, 234, 235, 237, 238,241,242 Macek, J. H., 114, 124 McElroy, M, B., 324, 331, 332, 333, 336, 331, 338, 340, 341, 342, 343, 344, 345, 346, 347, 349, 351, 353, 354, 355, 358, 361, 362, 363 McGowan, J. W., 68, 78, 89, 124, 226, 242 McGrath, W. D., 338,362 McKellar, A. R. W., 201, 202, 204 McKoy, V., 25, 45 Madison, D. U., 80, 81, 83, 124 Mahler, R. J., 293, 318

3 72

Author Index

Maillard, J. P., 326, 361 Majorana, E., 254, 318 Malli, G., 282, 316 Maltz, C., 286, 318 Manor, P. J . , 134, 204 Manz, J., 137, 141, 205 Marantz, H . , 133, 201, 204 Marcus, P. M., 90, 125 Mariella, R. P., 312, 318 Marple, D. T . F., 309, 318 Marschall, H., 213, 242 Marshall, J . V., 332, 362 Martins, P. de A. P., 115, 124 Martinson, I., 24, 25, 44, 45 Massey, H . S. W., 48, 55, 83, 84, 109, 115, 117, 122, 124, 125, 208, 209, 226, 232, 237, 241, 242 Matcha, R. L., 286, 290, 318 Mathews, C. W . , 245, 320 Matsen, F . A., 290, 315 Matsui, T., 312, 317 Meerts, W . L., 301, 302, 318 Mehr, J., 232, 237, 242 Mehran, F., 294, 298, 318 Melendres, C. A,, 287, 317 Menotti, F., 309, 314 Mentall, J. E., 352, 361 Michel, F . C., 346, 361 Michels, H . H., 121, 124 Mielczarek, S . R., 64, 65, 79, 126, 216, 221, 226, 241, 242 Miller, C. E., 264, 282, 283, 285, 315, 318 Miller, J . H . , 326, 361 Miller, K . J . , 79, 124 Miller, P. D., 313, 314 Miller, R. C., 264, 265, 291, 318 Miller, T. A,, 245, 301, 315 Miller, W . F., 64, 125 Millman, S., 245, 275, 318, 319 Milne, T. A., 264, 318 Miner, E., 329, 362 Minnhagen, L., 26,45 Minturn, R., 270, 315, 319 Mittleman, M. H . , 70, 87, 124, 125 Mizushima, M., 149, 204 Monter, B., 266, 270, 297, 298, 315 Moffett, R. J . , 345, 361 M o h r , C . B. O., 83, 84, 109, 117, 124 Moiseiwitch, B. L., 48, 68, 70, 71, 83, 84, 85, 87, 88, 89, 116, 122, 124, 125

Moore, C. B., 133, 175, 202,204 Moore, C. E., 27, 38, 39, 42, 45 Moore, R. C., 329, 363 Moores, D. L., 114, 115, 118, 122, 125, 238, 239, 240, 242 Moran, T . I . , 264, 308, 309, 314, 319 Mori, Y . , 134, 205 Morrison, D. J . T., 73, 74, 75, 77, 78, 79, 89, 125 Morris, D., 285, 287, 318 Morrow, W . , 337, 361 Moser, C . M., 3, 12, 25, 44, 45 Mott, N . F., 5 5 , 125, 208, 209, 242 Moustafa Moussa, H . R., 64, 65, 125 Miiller, G., 264, 268, 308, 310, 31 I , 314 Muenter, A. A,, 288, 295, 319 Muenter, J. S., 267, 274, 283, 286, 287, 288, 305, 306, 315, 317, 319, 320, 321 Munima, M. J., 332, 351, 352, 362, 363 Munch, G., 329, 361, 362 Murphy, J . S., 149, 205 Murray, B. C., 329, 362 N Nagy, A. F., 351, 360, 362 Nakano, H . H . , 237, 242 Nakase, T., 312, 317 Nelissen, L., 263, 320 Nesbet, R. K., 3, 43, 45, 121, 125, 228, 242 Nethercot, A. H., 298, 320 Neugebauer, G., 329, 362 Neumann, R. M., 283, 306, 315 Nicolaides, C., 41, 45 Nier, A. D. C., 331, 362 Nierenberg, W . A,, 294, 319 Niki, M., 336, 363 Norcross, D. W., 1 1 8, 125, 238, 239, 240, 242 Normand, C. E., 233, 242 Novick, S., 267, 307, 319 Nowak, A. V., 201, 206 Noxon, J. F., 337, 361 Nussbaumer, H . , 1 1 , 44, 45

0 Ochkur, V. I., 73, 75, 76, I25 Odabasi, H . , 10, 20, 21, 45

3 73

AUTHOR INDEX

Okiisz, I., 9, 17, 19, 25, 37, 38, 45 Opik, U., 90, 125 Oka, T., 133, 159, 161, 162, 164, 165, 168, 169, 170, 171, 172, 174, 175, 176, 177, 178, 179 180, 181, 182, 183, 184, 185, 187, 188, 189, 191, 192, 193, 197, 200, 201, 202, 203, 204, 205, 206 Olander, D. R . , 264, 317, 319, 320 O’Malley, T. F., 103, 105, 124, 125, 212, 242 Opal, C. B., 80, 81, 82, 125 Oppenheim, I . , 135, 204 Oppenheimer, J. R., 67, 125 Opykhtin, V . , 92, 116, 126 Orgren, P. J., 341, 361 Ormonde, S., 66, 117, 118, 119, 123, 127, 21 I , 234, 241 Ott, W. R., 66, 68, 77, 78, 79, 88, 89, 119, 124, 125 Ottinger, C., 133, 160, 202, 205 Ozier, I . , 288, 294, 306, 319, 321

Pickett, M. M., 198, 206 Pirnentel, G . C., 324, 361 Platzrnan, R. L., 64, 125 Player, M. A., 313, 319 Poe, R. T., 211, 230, 231, 234, 235, 241, 242 Polanyi, J. C., 134, 202, 205 Polder, D., 135, 204 Pople, J. A., 301, 319 Poppe, D., 133, 202,205 Prakash, V., 133, 149, 185, 205, 300, 318 Presnyakov, L., 92, 116, 126 Propin, R., 66, 123 Protopopova, N. G . , 256, 316 Purcell, E. M., 295, 320 Purcell, J. D., 333, 336, 361, 363 Pyykko, P., 292, 319

Q Quinn, W. E., 271, 319

P Pang, K . , 324, 362 Parker, J. H . , 274, 319 Parkinson, E. M., 19, 44 Parkinson, T. M., 339, 362 Paul, W., 249, 260, 262, 314, 316, 319 Pauly, H. 245, 262, 264, 265, 266, 319 Pearce, J . B . , 324, 331, 333, 338, 348, 349, 353, 354, 360 Pearl, J. C., 268, 272, 315, 319 Pekeris, C . L., 2, 44, 45, 46 Pendlebury, J. M., 271, 319 Penner, S., 261, 319 Penselin, S., 294, 319 Percival, I. C., 50, 58, 93, 95, 105, 109, 123, 125 Perrin, R., 84, 125 Persky, A . , 319 Pery, A,, 271, 319 Peter, M., 256, 263, 319 Peterkop, R., 48. 66, 123, 125 Peters, A . W., 291, 319 Petley, B. W., 312, 319 Pettingill, G . H., 324, 362, 363 Phelps, A. V . , 233, 241 Phillips, M., 3, 45

R Rabi, I . I., 245, 254, 285, 318, 319 Rabitz, H. A,, 133, 147, 149, 170, 194, 196, 205 Radford, H. E., 245, 319 Raible, V., 226, 242 Rainville, L. P., 324, 362 Ramsauer, C., 217, 225, 229, 230, 231, 232, 233, 234, 236, 237, 242 Ramsey, N. F., 244, 254, 260, 264, 265, 271, 277, 279, 282, 284, 288, 289, 292, 294, 295, 298, 299, 301, 306, 313, 314, 315, 317, 318, 319, 320, 321 Rank, D. M., 203. 206 Rao, K . N., 245, 320 Raper, 0. E., 324, 349, 360 Rapp, D., 355, 362 Rasool, S. I., 323, 343, 362 Read, F. H., 224, 241 Reeves, R. R., 337, 362 Reichert, E., 218, 229, 239, 240, 241, 242 Reiffsteck, A., 337, 361 Reinhard, H. P., 319 Reitz, J. R., 1 11, 126 Reuss. J . . 263, 320

3 74

Author Index

Reuss, R., 252, 320 Rice, J. K., 65, 66, 79, 80, 81, 82, 126 Rishbeth, H., 343, 362 Ritchie, R. K . , 270, 285, 320 Rittner, E. S., 286, 320 Robb, W. D., 9, 44, 117, 123 Robbins, E. J . , 307, 318 Robertson, A. G., 233, 237, 241, 242 Robertson, H . H., 117, 125 Robinson, G. W., 283, 320 Rodberg, L. S., 58, 69, 123 Roesler, F. L., 326, 360 Roger, J. M., 133, 162, 170, 171, 206 Rogers, A . F. E., 324, 363 Ronn, A. M., 133, 161, 169, 194, 199, 206 Roothan, C . C. J., 7, 13, 46 Rose, M. E., 52, 125 Rosenblum, B., 298, 320 Rosendorff, S., 83, 85, 87, 122 Ross, M., 96, 114, 125 Rostoker, N., 90, 124 Rothstein, E., 290, 299, 320 Roussy, G . , 133, 162, 168, 206 Rubey, W. W., 356, 358, 359, 363 Rubin, K., 239, 240, 241, 242 Rudge, M. R. H . , 54, 5 5 , 58, 67, 73, 74, 75, 77, 78, 89, 90, 91, 92, 93, 97, 104, 108, 116, 120, 123, 124, 125 Runblfsson, 6., 250, 266, 283, 298, 316 Russek, A., 103, 124 Russell, A. M., 275, 297, 298, 320 Russell, T. W., 314

S Sakurai, K., 133, 206 Salmona, A., 116, 125 Salop, A., 237, 242 Salpeter, E. E., 192, 205 Salwen, H., 286, 290, 320, 321 Sanche, L., 220, 221, 228, 242 Sandars, P. G. H . , 255, 313, 317, 319, 320, 321 Sandlin, G. D., 336, 363 Saraph, H. E., 115, 121, 125 Satchler, G . R., 152, 153, 204 Satten, R. A,, 293, 317 Savadatti, M. I . , 134, 205

Savage, B. D., 36,41, 45,46 Schaefer, H. F., 25, 46 Scharpen, L. H., 287, 320 Schauer, W., 253, 295, 297, 298, 299, 301, 318 Schawlow, A . L., 161, 206, 280, 301, 313, 32 I Schermann, J. P., 262, 314 Schey, H. M., 66, 70, 76, 107, 109, 120, 123 Schiff, B., 2, 44, 46 Schiff, H. I . , 330, 337, 360, 363 Schiff, L., 313, 320 Schlecht, W., 202, 204 Schlier, C.. 249, 250, 256, 260, 262, 282, 284, 286, 290, 295, 296, 298, 307, 314, 320 Schlosser, H., 90, 125 Schmeltekopf, A. L., 340, 343, 361 Schneider, B., 67, 125 Schneider, W. G . , 301, 3 / 9 Schonwasser, R., 283, 287, 298, 316, 317 Schorn, R. A., 328, 329,363 Schugerl, K., 263, 317 Schultz, A., 267, 320 Schulz, G. J., 71, 125,220,221, 227, 228, 242 Schulz-Culde, E., 31, 46 Schutten, J . , 64, 65, 125 Schwartz, C . , 107, 120, 125 Schwartz, H. L., 252, 320 Schwarz, R. F., 299, 321 Schwarz, W. H . E., 237, 242 Schwendeman, R. H., 134, 183, 198, 199, 204, 206 Schwinger, J., 205 Scrocco, E., 290, 320 Seaton, M . J . , 3, 46, 48, 50, 58, 69, 70, 72, 93,95, 104, 105, 109, 113, 114, 115, 116, 117, 121, 122, 123, 124, 125, 126 Seibt, P. J., 133, 169, 171, 206 Seidel, B. L., 343, 361, 362 Seiff, A , , 331, 362 Seiler, G . J . , 121, 126 Shamey, L. J., 31, 44 Shapiro, I . I . , 324, 362, 363 Sharma, A,, 326, 360 Sharp, R. D., 351, 363 Shaw, G . , 96, 114, 125 Shea, M. F., 351,363

375

AUTHOR INDEX

Shelton, W. N., 80, 81, 83, 124 Sheniming, J., 121, 125 Sheorey, V. B., 87, 115, 123 Shimizu, F., 200, 206 Shimizu, T., 133, 200, 201, 205 Shimoda, K., 133, 201, 206 Shirley, J. H., 256, 257, 258, 259, 320 Shironas, V., 12, 46 Shoemaker, R. L., 134, 198, 204,206 Short, S., 286, 317 Shortley, G . H., 3, 6, 44 Shukla, R . C., 286, 321 Sibincic, Z., 13, 37, 38, 46 Siekhaus, W. J . , 264, 319, 320 Sil, N. C . , 89, 123 Silverstone, H. J., 18, 46 Simonaitis, R., 336, 363 Simpson, J. A,, 64,65,79, 126,219,226,242 Sinanoglu, O . , 16, 17, 18, 19, 25, 26, 37, 38, 39, 40, 41, 45, 46 Sinfailam, A. L., 228, 242 Sinha, M. P., 267. 320 Sinha, P., 89, 123 Skillman, s., 10, 45 Skinner, B. G . , 84, 124 Slanger, T. G . , 336, 337, 363 Slater, J. C., 10, 46 Slevin, J. A,, 240, 242 Smith, A. C . H . , 272, 315 Smith, D. H., 270, 320 Smith, E. W., 149, 204 Smith, H. J., 329, 363 Smith, K . , 48, 66, 99, 117, 123, 125, 126, 263, 271, 315, 319 Smith, M. W., 15, 46 Smith, S. J., 48, 66, 78, 89, 116, 124, 125. 240, 242 Snow, G . A , , 58, 69, 123 Sobel’man, I., 116, 126 Southworth, G . C., 163, 206 Sparks, P. R.. 333, 363 Spencer, N. W., 331, 362 Spinrad, H., 329, 361, 363 Spruch, L., 105, 124 Stauffer, J. L., 264, 288, 315 Steele, R., 46 Steelhammer, J. C.. 69, 79, 126 Stein, T. S . , 313, 320 Steinfeld, J. I., 133, 134, 183, 198, 201, 204, 205, 205

Stephenson, D. A , , 245, 254, 255, 279, 281, 292, 293, 295, 315, 317, 320, 321 Stern, O . , 244, 245, 318, 320 Stern, R. C., 268, 272, 283, 302, 303, 304, 316, 320 Sternheimer, R. M., 291, 320 Stevens, R. M., 299, 320 Stewart, A. I., 324, 329, 330, 331, 333, 338, 343, 347, 348, 349, 351, 352, 353, 354, 360, 361, 362, 363 Stewart, R. W., 343, 362, 363 St. John, C., 325, 360 St. John, R. M., 64, 65, 68, 71, 74, 124 Stolarski, R. S., 351, 363 Stoke, S . , 252, 320 Stone, E. J., 332, 363 Stone, P. M., 1 1 1, 126 Story, T. L., 287, 288, 317, 320 Strandberg, M. W. P., 256, 263, 319 Strauch, R. G . , 265, 277, 278, 279, 285, 287, 288, 298, 304, 312,316, 320, 321 Street, K., Jr., 255, 285, 287, 314, 317 Strickland, D. J., 332, 333, 363 Strine, G . L., 312, 314 Strobel, D. F., 331, 362 Stroke, H. H., 293, 317 Stuhl, E., 336, 363 Sturrock, P. A,, 263, 317 Sucher, J., 58, 69, 123, 124 Sugar, R . , 105, 126 Sugden, T. M., 256, 320 Sullivan, J., 83, 85, 88, 122 Sullivan, T., 133, 201, 204 Sunshine, G . , 238, 241 Suryanarayana, D., 155, 204 Sutton, D. G . , 201, 206 Svidzinskii, K. V., 281, 320 Swartz, J. C., 271, 320 Swirles, B., 12, 45

T Tai, H . , 88, 89, 126 Takami, M., 133, 201, 206 Takayanagi, K., 139,206 Tanaka, I., 134, 205 Tantilla, W., 293, 318 Taube, H., 337, 362

376

Author Index

Taylor,A. J.,66, 117, 118, 119, 121, 123, 126,238, 241 Taylor, E., 270, 315, 319 Taylor, J. B., 268, 320 Temkin, A., 90, 110, 111, 126,228, 242 Thaddeus, P., 139, 206 Thibault, J., 200, 205 Thomas, B. K., 87, 123 Thomas,C. H., 133, 162, 183, 195, 196, 204 Thomas, G. E., 324, 333, 349, 354, 360,363 Thompson, B. A., 337,362 Thompson, D. G., 117, 126, 21 1, 236, 237, 238,242 Thompson, D. P., 15, 16, 20, 25, 45 Thomson, G. P., 168,206 Tiemann, E., 287,317 Todd, J. E., 291,319 Toennies, J. P., 130, 132, 202, 206, 245, 262,264,265, 266,314, 319 Torring, T., 287, 317 Tomasevich, G., 315 Tomasi, J., 290, 320 Tonutti, M., 283, 287, 298, 316 Tousey, R., 333, 336, 361 Townes, C. H., 161, 166, 203, 206, 260, 280, 290, 291, 292, 298, 299, 301, 305, 313, 316, 317, 320, 321 Trajmar, S., 65, 66, 79, 80, 81, 82, 126 Traub, W. A., 327, 336, 360 Trefftz, E., 9, 12, 44, 46 Trischka, J. W., 254, 264, 271, 274, 286, 290, 295, 308, 309, 314, 318, 319, 320, 321 Truhlar, D. G., 65, 66, 69, 70, 75, 76, 79, 80, 81, 82, 126 Tsao, C. J., 147, 206 Tscherner, M., 266, 300, 316 Tull, R. G., 329,363

U Uberoi, R. S., 121, 126 Ung, A. Y., 337,363 Unland, M. L., 134, 198, 206 Urey, H. C., 323,363

V Vaillancourt, R. M., 198, 204 Vainshtein, L., 92, 116, 126

Van Ausdal, R. G., 291,292, 293,295,317, 321 Van Blerkom, J., 79, 126 Van Kranendonk, J., 148,204, 206 Van Regemorter, H., 110, 111, 120, 123, 126 Van Vleck, J. H., 138, 206, 301, 305, 317, 32 1 van Wachem, R., 264, 270, 283, 286, 315, 321 Varshni, Y. P., 286, 321 Veillard, A., 44 Velasco, R., 133, 160, 202, 205 Veldre, V., 48, 125 Venkates. H. G. R.., 263.. 319 Vetter, M., 290, 318 Visconti, P. J., 240, 242 Vizbaraite, I., 12, 46 Vo Ky Lan, 110, 1 1 1 , 120, 123, 126 von Meyenn, K., 256, 321 von Zahn, U . , 319 Vriens, L., 64, 65, 79, 126

W Waech, T. G., 255,263, 321 Wagner, P. E., 134, 183, 198, 205 Wahl, A. C., 12, 44 Waldron, R. W., 337, 362 Walker, J. G. G., 345, 361 Wallace, L., 329, 349, 353, 360, 363 Wang, J. H. S., 134, 198, 201,205 Wang, T. C., 264, 293, 316, 321 Warneck, P., 337, 363 Warner, B., 11, 46 Warren, R. W., 274, 319 Watson, K. M., 87, 123 Wauchop, T. S., 355, 363 Webb, T. G., 119, 123 Weingartshofer, A., 229,238, 242 Weisheit, J., 3, 46 Weiss, A. W., 10, 15, 24, 25, 32, 36, 37, 38, 42, 46 Weiss, R., 271, 275, 277, 313, 321 Weisskopf, M. C., 313, 320 Weisskopf, V. F., 138, 206 Welch, W. J., 203, 206 Welge, K. H., 134, 205, 206 Wells, R. A., 324, 363 Wells, W. C., 332, 351, 352, 353, 363

311

AUTHOR INDEX

Werner, M. W., 192, 205 Westhaus, P., 19, 40, 46 Westin, S., 223, 242 Wharton, L., 254, 255, 261, 263, 265, 266, 267, 270, 274, 277, 278, 285, 286, 287, 288, 289, 290, 298, 299, 308, 31 3, 314, 315, 316, 318, 321 Whitaker, W., 118, 119, 123 White, R. L., 299, 321 Wick, G. C., 294, 321 Wicke, B. G., 272,283, 302, 304, 316 Widing, K . G., 336, 363 Wiese, W. L., 15, 46 Wigner, E., 143, 206 Wigner, E. P., I 12, 11 7, 126 Wiley, W. C., 272, 316 Williams, A,, 84, 87, 125 Williams, J . F., 68, 78, 89, 124 Willmann, K., 228,229, 236,238, 242 Wilmers, M., 239, 240, 241 Wilson, E. B., Jr., 133, 142, 160, 161, 162, 169, 171, 183, 194, 195, 196, 204, 206 Winnewisser, G., 200, 206 Wiskott, D., 213, 242 Wofsy, S. C., 274,283,305,306,315,321 Wood, B. J . , 336, 363 Wood, R. W., 130,204, 206

Woodall, K. B., 134, 202, 205 Woodgate, G. K., 255, 321 Woods, R. C . , 194, 206 Woolsey, J. M., 351, 355, 362 Wright, S. J., 255, 313, 317

Y Yi, P. N., 306, 321 Yonezaki, G., 312, 317 Young, R. A., 337, 361

Z Zacharias, J. R., 245, 264, 270, 318, 319 Zandberg, E., 270,321 Zapesochnyi. I. P., 71, 126 Zare, R . N., 8, 10, 13, 23, 35, 46, 133, 160, 202, 205, 267, 320 Zecca, A,, 228, 241 Zeiger, H. J., 260, 290, 294, 316, 321 Zipf, E. C., 332, 351, 352, 353,362, 363 Zorn, J. C., 245, 254, 255, 261, 264, 266, 268, 270, 271, 272, 279, 281, 283, 284, 287, 288, 291, 292, 293, 294, 295, 309, 31 I , 315, 316, 317, 318, 319, 320, 321

Subject Index A Adiabatic state inversion, 254 Aluminum(I), wave functions for, 34 Aluminum(II), correlation energies for, 33 Argon sequences, ionization potentials for, 27 Asymptotic degeneracy, persistence of, 20-23 Atoms correlation in excited states of, 1 4 3 Hartree-Fock model of, 4-7, 9, 18, 24, 28-29, 36-37 theoretical and experimental energies for, 38, 39 3d shell collapse in, 40-43 Aufbau principle, 6

B Boltzmann population, in collision-induced rotational transitions, 163 BOMC (Born-Oppenheim minus core) approximation, 69, 79 Bonham approximation, 72-79 Bonham-Ochkur-Rudge approximations, 72-79 Born approximation Coulomb projected, 81 in electron-atom elastic scattering, 21 1 in electron-atom excitation cross sections, 63-67, 79 second-order approximations and, 83-89 Born-Ochkur approximation, 72-79 Born-Oppenheimer approximation (in collision theory), 67-69 orthogonalized, 69-72 C Carbon dioxide reactions, on Mars, 335-342

C fields, in molecular beam electric resonance spectroscopy, 245, 273-279, 283 Charge expansion method, in correlation calculations, 13-15 Collision-induced rotational transitions see also Collision process Boltzmann population in, 163 AI/I variation in, 189-191 double resonance experiments in, 199-202 experimental results in, 160-202 Hamiltonian operator and, 134-137 J selection rule in, 175-179 J, K dependence and, 187-189 k selection rule in, 180-181 large signals in, 183-187 magnitude of AI/I in, 183-194 modulated microwave double resonance in, 194-198 M selection rule in, 182-1 83 operating principle in, 162-169 parallelism between NH, - He and NH3 - H, results in, 191-192 parity rule in, 172-175 rate equations and, 192-194 relaxation time in, 198-199 “ selection rules ” in, 160, 169-183 study methods for, 132-134 theory of, 134-160 time-dependent equation in, 144-150 Collision process see also Electron-atom collisions; Low energy electron-atom collisions; Molecule(s) experimental methods in, 129-134 first-order perturbation in, 146-147, 159 Hamiltonian operator in, 134-137, 139-140 intermolecular potential in, 150-1 54 local symmetry and “selection rules” in, 154-160 among molecules, 127-128 378

379

SUBJECT INDEX

overall symmetry in, 139-144 resonance factor in, 148-149 second-order interaction in, 148-149, 155-158 strong vs. weak collisions in, 137-139 symmetry in, 139-144, 154-160 time-dependent equation in, 144-150 transition probability in, 149 Conservation theorem in collision theory, 53-54 Correlation, core-valence, 9 Correlation effects, 4-8 asymptotic degeneracy in, 20-23 calculations for, 9-19 charge expansion method in, 13-15 collapse of 3d shell in, 40-43 correlation orbitals and, 30-32 excitation energies in, 43 in excited states of atoms, 1-43 large and novel aspects of, 3 large vs. small, 2-3 multiconfiguration self-consistent field in, 12-13 orbital polarization in, 25-29 oscillator strength in, 35 pair correlations and, 3 6 4 0 pair theories in, 16-19 “plunging” term in, 20-23 series perturbations in, 32-36 SOC (superposition of configurations) calculation in, 6, 9-12 specific, 19-43 Z-expansion framework in, 19 Correlation energy see also Correlation effects for AI(1I) states, 33 defined, 4 Correlation orbitals, 30-32 Coulomb approximation, 1 1 Coulomb free wave functions, 49-50 Coulomb projected Born approximation, 81

D Dayglow, of Mars, 333, 348-355 Detectors, in molecular beam electric resonance spectroscopy, 268-270 Diabiatic transition, 254

Direct scattering, in low-energy electronatom collisions, 215-219, 229-240 Discontinuous trial functions, in scattering theory, 90 Dunham series expansion for molecular energies, 280

E Effective electric dipole moment, 255-260 e-H collision, model of, 66 e-H scattering problem, 101-102 boundary conditions for, 50-53 Electric dipole moments, 255-260, 313 of polar molecule, 285-288 Electron-alkali scattering, 239 Electron-atom elastic collisions see also Collision process differential cross section in, 207-240 direct scattering in, 209-21 1 double resonance in, 228 evaluation in, 221-225 low-energy, 207-240 resonances in, 211-214 theory of, 208-214 Electron-atom excitation cross sections see also Electron-atom collisions BOMC approximation in, 69 Born approximation in, 62-67 Born-Oppenheimer approximation in, 67-68 calculation of, 47-122 close-coupling method in, 104-105, 117 correlation method in, 117-1 18 discontinuous functions in, 90-91 exchange scattering amplitudes in, 76 Green functions in, 84-85 for helium, 59-60 high-energy approximations in, 62-93 hydrogen, 77 integral identities and reciprocity relations in, 56-59 low-energy approximations of, 115-121 low-energy theory in, 93-1 15 minimum principle in, 105-107 orthogonalized Born-Oppenheimer approximations in, 69-72 partial cross section in, 100-101 partial wave boundaryconditions in, 93-97

380

SUBJECT INDEX

polarization effects in, 109-1 11 pseudostates in, 110-111, 118-121 regional trial functions in, 89-90, 108 reliability criteria in, 60-62 resonance theory in, 99-104 second-order approximations in, 83-89 threshold behavior and, 61, 111-1 15 total, exchange, and differential cross sections in, 61-62 variational principles and, 54-56, 97-99, 120-1 21 Electron bombardment detectors, 271 Electron helium scattering, 231-235 Electron neon scattering, 236-237 Electronic quadrupole moment, 299-301 Elementary particles, electric dipole moments of, 313 Energy eigenvalues, in molecular beam spectroscopy, 282-283 Ethylene oxide, rotational energy levels of, 162 Excited states, correlation in, 1-43

Hartree-Fock orbitals, 17 Hartree-Fock sea, 18 Hartree-Fock Slater approximation, 26, 35 Helium differential cross section for electron excitation, 81 elastic scattering of electrons by, 227228,230-233 excitation of by electron impact, 59-60 High-energy approximations in electronatom excitation, 62-93 High-energy theory in electron-atom excitation, 49-62 reliability criteria in, 60-62 Hydrogen, on Mars, 335, 355 Hydrogen atom excitation by electron impact, 77, 87, 113 Hydrogen molecules, collisions of, 138 Hydrogen sulfide, rotational and hyperfine structure of, 304-305

I

F Feshbach method, scattering amplitude and, 101-103 Fields, in molecular beam electric resonance spectroscopy, 272-279 Fine structures, in low-energy electronatom elastic collisions, 218-221 First-order exchange (FOE) method, 7172, 76, 79 Free radicals, molecular beam spectroscopy Of, 301-302

Infrared-infrared double resonance, 201 Infrared-microwave double resonance, 199-200 Intermolecular potential, 150-1 54 Ionosphere, of Mars, 343-347

K Kang-Foland method, 82-83 Kohn variational principles, 89 see also Variational principles

G Glauber approximation, 88-89 Green functions, electron-atom excitation cross sections and, 84-85, 105-106 H Hamiltonian operator, for interacting molecules, 134-135, 139-140, 280-282 Hartree-Fock energies, total, 42 Hartree-Fock model, 4-7, 9, 18, 24, 2829,3637

L Lasers, infrared, 199-200 Low-energy approximations, in electronatom excitation cross sections, 115121 Low-energy electron-atom elastic collisions, 207-240 see also Collision process; Electronatom collisions direct scattering in, 215-218 evaluation techniques in, 221-225

SUBJECT INDEX

experimental results in, 225-240 experimental techniques in, 215-221 fine structures and, 218-221 phase shift in, 222 recoil technique in, 217-218 resonances in, 225-229 Low-energy theory, in electron-atom excitation cross sections, 93-1 15

M Magnetic shielding anisotropy, 300-301 Magnetic susceptibility anisotropy, 299300 Many-body perturbation theory, 43 Many-electron theory (MET), 9, 18 Mariner 6, 332, 335 Mariner 7, 332, 335 Mariner 9, 325, 332, 335 Mars argon on, 331 atmosphere composition of, 325-335 atomic and molecular processes in atmosphere of, 323-360 atomic carbon on, 333 Cameron bands on, 351-352 carbon dioxide on, 330-331, 335-342 dayglow of,333, 348-355 Earth's atmosphere and, 355-356 evolution of atmosphere on, 355-359 excitation processes on, 350-351 Fox-Duffendack-Barker bands on, 349 hydrogen on, 335, 355 ionosphere of, 324, 343-347 neon on, 331 nitrogen on, 331, 356-358 oxygen on, 321-328, 356-357 ozone on, 329 space probes of, 323-324 spectroscopic studies of, 325-326 water on, 328-329 MBER, see Molecular beam electric resonance MBER time standards and voltage, 312313 MBMR (molecular beam magnetic resonance) spectrometer, 245 MCSCF proced tire, 12-1 3 MET, see Many-electron theory MET calculation, 37

38 1

Metastable molecules, 302-304 beams of, 268 detectors for, 271-272 Methyl deuteride, hfs and Stark effect for, 305-306 Microwave C fields, 276279 Minimum principle, in electron-atom excitation cross section, 105-107 Modulated microwave double resonance, 194-1 98 Molecular beam electric resonance diabatic and adiabatic transitions in, 254 multiple resonance methods in, 254-255 observability criterion in, 253 principles of, 247-248 Stark energy and, 255-260 two-wire fields in, 260 Molecular beam electric resonance spectromer, 246 see also Molecular beam electric resonance spectroscopy configurations of, 247-252 flop-in and flop-out operation of, 250252 molecular spectra observations with, 252-255 Molecular beam electric resonance spectroscopy, 243-314 beam design in, 262-263 of combined E and H fields, 296-297 deflecting and focusing fields in, 263 deflection experiments with, 288 detectors in, 268-273 dipole moment values in, 286-287 energy eigenvalues in, 282-283 energy levels and transitions in, 280-284 experimental methods in, 264-279 false quadrupole effects in, 291-292 fields in, 272-279 magnetic hfs in 'C molecules and, 294296 of metastable molecules, 302-304 miscellaneous experiments and applications of, 312-314 mixed alkali dimers in, 289 multipole fields in, 260-262 of non- 'c diatomic molecules, 301-304 nuclear electric quadrupole interactions in, 289-291

382

SUBJECT INDEX

Neon sequences, ionization potentials for, nuclear octupole and hexadecapole 27 interactions in, 292-294 Nitrogen, on Mars, 354-358 polarization effects in, 287 Non-Boltzmann population, in rotational of polyatomic molecules, 304-307 levels, 131, 201 quadrupole hfs in 'C molecules and, Non-'C diatomic molecules, MBER studies 289-294 Of, 301-304 of reaction products, 310-312 Nuclear electric-quadrupole interactions, rotational magnetic moments in, 298289-291 299 Nuclear octupole and hexadecapole interselection rules and transition probaations, 292-294 bilities in, 283-284 'C molecules in, 294-304 sources in, 264-268 Stark effect in, 285-287 0 state selection with electric fields in, 255-263 Ochkur approximation, 73, 75 time and voltage standards for, 312-313 OCS molecule, 159-160 Open shell systems, 3 vacuum systems in, 279 of vibrational state populations, 307-312 Orbital polarization, correlation effects in, 25-29 Molecular beam spectrometer, principle of, Orbitals, expansion of, 7 248-252 see also Molecular beam electric resoOscillator strength, transfer of, 35 Oxygen, on Mars, 356-357 nance spectroscopy Molecular Hamiltonian operator, 134-135, 139-140, 280-282 P Molecular spectra, observations of with MBER, 252-255 Pair theories, in correlation calculations, Molecule(s) see also Collision process 16-19 Paramagnetic molecules, beam spectrocollisions of, 127-128 scopy of, 301 Hamiltonian operator for, 134-135, Parity rule, in collision-induced rotational 139-140, 280-282 transitions, 172-175 metastable, 268, 271-272, 302-304 Partial cross section, Breit-Wigner form polar, 247 of, 100 quadrupole moment of, 299-300 Partial wave boundary conditions, crossrotational magnetic moment of, 298-299 section expressions and, 93-97 lC, 285-289 Persistence effect, correlation energy and, sorting of by quantum state, 247 20-23 van der Waals, 307 Perturbation technique, vibrational Morrison-Rudge model, 78 energies and, 136 Multiconfigurational self-consistent field Polarization effects, in MBER spectro(MCSCF) method, in correlation scopy, 287 calculations, 12-13 Multiple resonance methods, 254-255 Polar molecule, electric dipole moments Of, 285-288 Polyatomic molecules, MBER studies of, N 304-307 Pseudonatural orbital technique (PSNO), 10,26 Neon, elastic scattering of electrons by, Pseudostates, 110 236-237

SUBJECT INDEX

Q Quadrupole moment, electronic, 300-301 Quadrupole transitions, 252

R Radio-frequency C fields, 275-276 Reaction products, MBER studies of, 310-312 Reciprocity relations, in electron-atom excitation cross sections, 56-59 Recoil technique, scattering and, 217-218 Regional trial functions, method of, 8993, 108 Resonance, in electron-atom elastic collisions, 21 1-214, 224-229 Resonance experiments, energy levels in, 132 Resonance factor, defined, 148 Resonance theory, in electron-atom excitation cross sections, 99-104 Ritz formula, 32 Rotational angular momentum average, 140 Rotational energy, of symmetrically split levels, 166-167 see also Collision-induced rotational transitions Rotational levels collision-induced transitions in, 127-203 hyperfine levels and, 252-253 non-Boltzmann distribution in, 131 Rotational magnetic moments, 298-299 Rotational transitions resistance in, 141 study method for, 132-134 theory of, 134-160 Rudge approximation, 73-74 Rydberg orbitals, 7, 32-36, 41-42

S Scattering amplitude Feshbach method in, 101-103 variational principles for, 54 Scattering angle, 232 Scattering chamber, 216-217

383

Scattering, electrons by atoms, elastic Born approximation in, 211, 236 crossed beam technique in, 217 differential cross section in, 229-230 direct, 209-211, 215-219, 229-240 d wave in, 236-238 partial waves in, 208 recoil technique in, 217-218 resonance, 229 Scattering, electrons by ions, elastic, 114 SCF, see Self-consistent field Schrodinger equation, 4, 98 conservation theorem and, 53 of Hamiltonian operator, 138 Second Born approximation, 85 " Selection rules," for collision-induced rotational transitions, 160, 169-183 Self-consistent field (SCF) equations, 5 , 7, 9, 12-13 Series perturbations, correlation effects and, 32-36 'C molecules magnetic hfs in, 294-296 molecular properties of, 285-289 quadrupole hfs in, 289-294 Stark-Zeeman spectroscopy of, 296-304 Slater type orbitals (STO), 7, 9 SOC, see Superposition of configuration method Spectrometer beam resonance, 245-246 molecular beam magnetic resonance, 245-246 see also Molecular beam electric resonance spectrometer Spectroscopic 3d orbital, 30 Spectroscopy, 243-314 see also Molecular beam electric resonance spectroscopy Spin change cross section, 53 Spin-rotation interactions, 294, 300-301 Spin-spin interactions, 295-296 Split levels, rotational energy of, 166-167 Stark effect, 285-286, 305 Stark energy, 255260,285-287 Stark transitions, 252 Stark-Zeeman spectroscopy, of 'C molecules, 296-304 Stark-Zeeman transitions, 252 Static C fields, 273-275

384

SUBJECT INDEX

Superposition of configurations (SOC) method, 6, 9-12,28, 32, 34-35 Surface ionization detectors, 269-271 T 3d shell, collapse of, 40-43 Transitions, rotational, see Rotational transitions

U

Van der Waals molecules, 307 Variational principles, in electron-atom excitation cross sections, 54-56, 61, 89-90,97-99, 120-121 Vibrational energies, perturbation technique for, 136 Vibrational state populations, MBER studies of, 307-312 Vibrational states, radiative lifetimes of, 309-310 Viking probe, 331

Unitarized approximations, 115-116 V

Vacuum systems, for molecular beam spectrometers, 279

Z Z-expansion framework, in correlation energy, 19