Computational Chemistry Reviews of Current Trends

Computational Chemistry Reviews of Current Trends Volume 9 Computational Chemistry: Reviews of Current Trends Editor-...

Author: Jerzy Leszczynski

54 downloads 710 Views 13MB Size Report

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

DOWNLOAD PDF

Computational Chemistry Reviews of Current Trends Volume 9

Computational Chemistry: Reviews of Current Trends Editor-in-Charge:

Jerzy Leszczynski, Dept. of Chemistry, Jackson State University, USA

Published Vol. 1: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 2: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 3: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 4: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 5: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 6: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 7: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 8: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski

Computational Chemistry Reviews of Current Trends Volume 9

editor

Jerzy Leszczynski Department of Chemistry Jackson State University USA

\fp World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

COMPUTATIONAL CHEMISTRY: REVIEWS OF CURRENT TRENDS Volume 9 Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-097-1

Printed in Singapore by World Scientific Printers (S) Pte Ltd

PREFACE There is no doubt that many of the readers of this book series wonder how the research/computational facilities will look like in the near future. It seems that the efficiency and compactness of the silicon-based computers are almost exhausted, and for significant progress one has to look for alternative solutions. A very promising future for computing is based on molecular computers. Among the molecular elements required for such computers, the developments rely on the existence of nanostorage devices. A recent Science article (Z. M. Liu, A. A. Yasseri, J. S. Lindsey, and D. F. Bocian; Science, 302, 1543, 2003) reveals silicon-tethered porphyrins that are resistant to heat (up to 400° C) and harsh conditions. Since porphyrins possess well known data storage characteristics, the unique properties of these new derivatives reveal the possibility for their commercial production. It is assumed that porphyrin-based memory chips can be assembled using facilities devoted to the traditional silicon devices, opening the way toward nano components of molecular computers. The research in this area is extensive. To assist the readers, we have decided to collect more chapters for this book that are devoted to the reviews of current advances in nano science. As a result, this volume includes three chapters related to this important research area. The first chapter, written by J. J. Palacios, A. J. Perez-Jimenez, E. Louis, E. San Fabian, J. A. Verges, and Y. Garcia, reviews fundamental issues underlying first-principle quantum theory in atomic and molecular systems. The authors have recently developed a new approach called Gaussian Embedded Cluster Method that is based on the GAUSSIAN98 and GAUSSIAN03 codes. The review provides numerous examples that illustrate the applicability and reliability of this method. This is supported by the study and interpretation of various experiments in the field of molecular electronics. Apparently, an ultimate goal of nano sciences is the development of tools for spatial and chemical control of single molecules. Among the experimental techniques that have emerged in recent years are single molecule fluorescence, optical and magnetic tweezers, and atomic force spectroscopy. All of these techniques allow for mechanical manipulations of single molecules. The proper comprehension of the experimental results requires adequate theory. W. Nowak and P. E. Marszalek present an overview of

V

vi Preface

recent advances and trends in the applications of computer simulations aimed at understanding atomic force microscopic experiments. Those variants of classical molecular dynamics (MD) simulations which are particularly helpful in the interpretation of experimental data obtained for single biopolymer molecules, such as steered MD or biased MD, are reviewed and discussed in detail. In the third chapter, J. Seminario, P. Derosa, L. Cordova, and B. Bozard discuss the application of combined molecular dynamic simulations based on ab initio force fields and signal processing techniques to analyze the dynamic properties of nanocells. The analysis includes several characteristic modes at different temperatures. The authors consider the applications of such techniques for the characterization of new, molecular electronics. The prediction of the excited state properties of molecular systems are among the most challenging tasks of computational chemistry. They are assisted by the development of new methods designed for such tasks. The most accurate approaches are based on the coupled cluster theory. D. Mukherjee and his group (S. Chattopadhyay, D. Pahari, and U. S. Mahapatra) review formulations of linear response theory for excited state potential energy surfaces which are based on multi-reference coupled electronpair approximation categories of methods. This review includes a thorough discussion of the genesis of the developed theory. In addition, applications to the potential energy surfaces of low-lying excited states of the model P4 and Li2 molecules are also provided. Recently progress has been achieved in the theoretical prediction of the magnetic exchange coupling constants in compounds of rare earth transition metals. In the fifth chapter, M. Atanasov, C. Daul, and H. U. Giidel describe a method developed for such tasks. After a comprehensive discussion of the calculation procedure, the authors present examples of its application to various dimers of rare earth metals. My recent web search of the phrase "hydrogen bonding" resulted in over 50,000 entries. This demonstrates the popularity of this topic which was formally recognized more than 80 years ago. Over the years new and intriguing aspects of this phenomenon have been studied. Such studies reveal new classes of hydrogen bonds. One of such new categories, dihydrogen bonds (DHB), are discussed in the last chapter of this volume. S. J. Grabowski and J. Leszczynski present the examples of such interactions existing in crystal structures. These interactions are compared with the conventional hydrogen bonds. A comparison shows that DHBs are often similar to typical H-bonds. The discussion includes the results of ab initio calculations,

Preface vii

decomposition energy, topological parameters, and experimental neutron diffraction results. I would like to thank all authors for the excellent contributions and fine collaborations. The very efficient technical assistance of Dr. Manoj K. Shukla in putting together this volume is greatly appreciated. As always, your feedback is very important to me, please feel free to e-mail your suggestions to [email protected]. Jerzy Leszczynski Jackson, MS December 2003

CONTENTS Preface

v

1. Molecular Electronics with Gaussian98/03 J. J. Palacios, A. J. Perez-Jimenez, E. Louis, E. SanFabidn, J. A. Verges and Y. Garcia

1

2.

Molecular Dynamics Simulations of Single Molecule Atomic Force Microscope Experiments W. Nowak and P. E. Marszalek

47

3.

Molecular Dynamics Simulations of a Molecular Electronics Device: The NanoCell J. Seminario, P. Derosa, L. Cordova and B. Bozard

85

4.

Computation of Excited State Potential Energy Surfaces via Linear Response Theories Based on State Specific Multi-Reference Coupled Electron-Pair Approximation Like Methods S. Chattopadhyay, D. Pahari, U. Mahapatra, D. Mukherjee

121

5.

Modelling of Anisotropic Exchange Coupling in Rare-Earth-Transition-Metal Pairs: Applications to Yb 3 +-Mn 2+ and Yb 3 +-Cr 3 + Halide Clusters and Implications to the Light Up-Conversion M. Atanasov, C. Daul and H. U. Giidel

153

6.

Is a Dihydrogen Bond a Unique Phenomenon? S. J. Grabowski and J. Leszczynski

195

Index

237

Content Index

243

ix

Chapter 1: Molecular Electronics with Gaussian98/03

J. J. Palacios ab , A. J. Perez-Jimenez0, E. Louisa'b, E. SanFabian"'0, J. A. Verges"" and Y. Garcia" *Departamento de Fisica Aplicada, Universidad de Alicante San Vicente del Raspeig, Alicante 03690, Spain b Unidad Asociada del Consejo Superior de Investigations Cientificas, Universidad de Alicante San Vicente del Raspeig, Alicante 03690, Spain

c Departamento

de Quimica-Fisica, Universidad de Alicante San Vicente del Raspeig, Alicante 03690, Spain

d Departamento de Teoria de la Materia Condensada Institute de Ciencia de Materiales de Madrid (CSIC) Cantoblanco, Madrid 28049, Spain

Abstract In this review we discuss the fundamental issues underlying first-principles quantum transport theory in molecular- and atomic-scale systems, making emphasis on the actual numerical implementation of them. For this purpose we focus on the ab initio method named Gaussian Embedded Cluster Method, recently developed by the authors, which is based on the popular quantum chemistry code GAUSSIAN98/03. Various examples that illustrate the applicability and reliability of this method in the study and interpretation of a large range of experiments in the field of molecular electronics are also presented.

1

2 J. J. Palacios et al.

1

The basics

For the past forty years, computers and, more generally, electronic devices have grown more powerful as their basic subunits have shrunk. Investigators in the field of next-generation electronics forsee that during the next 10 to 15 years, as the smallest features of mass-produced transistors shrink below 100 nanometers, the devices will become more difficult and costly to fabricate. In addition, they may no longer function effectively since the laws of quantum mechanics may prevent transistors from working under the same principles as today's. In order to continue the miniaturization of circuit elements down to the nanometer scale new paradigms need to be proposed. For instance, unlike today's field effect transistors, which operate based on the movement of masses of electrons in bulk matter, the new devices should take advantage of quantum mechanical phenomena that emerge on the nanometer scale such as the quantum coherence and the discreteness of the electron charge. There are, essentially, two roads leading towards nanometer-scale ultra-dense circuits: • Solid-state quantum-effect devices (silicon-based devices). • Molecular electronic devices (carbon-based devices). It is not the purpose of this review to analyze which alternative is more likely to succeed in the long run. While the first alternative seems the logical continuation of present technologies, we believe, in our modest and biased view, that molecular electronics may have a better chance. Even if this is not the case, basic research in molecular electronics may contribute to advances in multidisciplinary areas such as biotechnology and medicine. Behind the words "molecular electronics" lies the promising idea that functional units for electronics can be built out of very stable and wellcharacterized carbon-based molecules such as fullerenes, carbon nanotubes, or polyphenylene chains, to name a few[l]. The advantage of using carbonbased molecules resides in that they are fabricated identical by nature, can be obtained in any desired quantity, and can be easily functionalized to meet any required property by means of standard and cheap chemistry techniques. Regardless of the final road taken by electronics, there is little doubt that the functionality of future devices will rely on the electrical and mechanical properties of nanoscopic regions composed of a number of atoms that could range from several thousands down to a single one. Hereby the necessity of quantum chemistry or first-principles computational support to which these notes are devoted.

3

Molecular Electronics with Gaussian98/03

2

Electronic transport at the nanoscale: A quick view of Landauer formalism

The first proposal of using molecules for electronic applications was put forward by Aviram and Ratner in 1974[2]. Since then various ideas to use molecules for electronic applications have been proposed [1]. How to put into practice these ideas, however, remains the main issue. It is important to realize that the coexistence of molecules and metallic electrodes will probably form the basis of future molecular devices since it is very unlikely that electronics will be entirely based on carbon. Keeping this in mind, the simplest molecular electronic unit that can be envisioned is a two-terminal device consisting of two large metallic electrodes, several nanometers apart, joined by a molecule or molecules anchored to them. This proposal was actually realized for the first time in the laboratory by Reed's group using a Mechanically Controlable Break Junction (MCBJ)[3], although early measurements go back to the work of Mann and Kuhn[4]. This system is generically known as a molecular bridge or molecular junction and plays a central role in molecular electronics. Conventional lithography, on the other hand, had already been successful in contacting large molecules like carbon nanutubes[5]. Before Reed's work, various groups had already managed to make nanoscale junctions connecting the two metallic electrodes of the MCBJ by just a chain or even a single atom of the same metallic element [6]. These systems are known as atomic contacts or metallic nanocontacts and, although they are not expected to be of any practical application, constitute an excellent test bed to learn about the world of electrical transport at the atomic scale. It is fair to say that other groups had already been working on molecular and atomic junctions since the early nineties from a different perpective using the scanning tunneling microscope (STM)[6, 7, 8]. The actual fabrication of these electronic devices at the molecular and atomic scale poses new challenges for theoretical physicists and chemists who must develop new theories and, most importantly, implement existing ones to address the present and upcoming questions. The basics to calculate the zero-bias, zero-temperature conductance Q of a molecular bridge or metallic nanocontact were established by Landauer long before the concept of molecular electronics was commonplace. In Landauer's formalism Q is simply related to the quantum mechanical transmission probability T for electrons at the Fermi level to go from one electrode to the other [9] (we will assume spin degeneracy throughout):

g = goT(EF) = —T(EF) = -^YlT^)i

(!)

The fact that the zero-bias conductance is proportional to an intrinsically coherent quantity such as the elastic transmission probability should not

4

J. J. Palacios et al.

Figure 1: Schematic drawing of the scattering of conduction channels in a constriction in a semiclassical picture. Quantum conduction channels are typically associated to specific quantum numbers of the transversal part of the wavefunctions occupied at the Fermi level when the Schrodinger equation is separable in bulk. come to us as a surprise at zero temperature since inelastic scattering events due to vibrations or many-body interactions are strictly prohibited. The fact that the concept of elastic transmission probability or elastic scattering can still be determinant even at finite bias and temperature is what makes Landauer's formula extremely useful. This is esentially due to the fact that inelastic scattering lengths are typically much larger than the extent of the nanoscopic region that determines the conductance of the whole device. The wide success of Landauer's formula lies in its simplicity. However, this simplicity is misleading and still is the source of some debate (for a recent discussion see Ref. [10]). Landauer's formula has been discussed on numerous occasions[9] and we do not intend to do it again in these notes. Instead, we will simply give some plausible arguments in favor of it. The sum in the last expression of Eq. 1 runs over all possible conduction channels deep in the bulk of the metallic electrodes and not only over the conducting channels in the actual nanoscopic region of interest. It is customary to associate quantum conduction channels to specific quantum numbers m, n of the transversal part of the bulk wavefunctions occupied at the Fermi level when the Schrodinger equation is separable in bulk; in other words, \&(f) = ijjm(x) represents the work function of the sample. Unfortunately, the above formulae deduced within Bardeen's formalism are not valid when tip and sample come in closer contact. This occurs, for


Figure 13: Left panel: Normalized conductance derived from CITS measurements by Rogero et al. [48] for the two types of C ^ adsorption sites on Au(lll). Right panel: CITS image by Rogero et aZ.[48] at a bias of +0.6V, where the difference in brightness between on-top and bridge sites is clearly seen. The circle and triangle indicate the points where the curves shown in the left panel were obtained. The Fermi level is set to zero. [Reprinted with permission from Ref. 48 (Figs. 4d, 4c)© 2002 American Institute of Physics]. instance, when the STM tip is used to fabricate nanocontacts (see, e.g., Ref. [6]). Therefore, a computational method incorporating Landauer's formalism is much more useful if one is interested in studying STM experiments beyond the tunneling regime. An efficient implementation of such a theoretical framework has been explained in Sec. 3 of this work, which has also the appealing feature of being an ab-initio method, without the need to introduce ad-hoc parameters to describe the tip-sample interaction[47, 49, 50]. In the following we will compare the results achieved with the GECM with those from a STM study by Rogero et al.[4S] of Ceo adsorbed on Au(lll). To be more specific, what these authors perform is a spectroscopic study named Current Imaging Tunneling Spectroscopy (CITS). This technique measures the tunneling current at each point during a topographic scan for a range of bias voltages applied to the sample-tip system. The numerical differentiation of each curve provides the corresponding conductance profile. From these curves it is possible to construct an image of the conductance at a given bias for each point of the topographic scan. This type of measurements provide a tremendous amount of information about the electronic structure of the sample around the Fermi energy. However, taking full profit of these type of experiments requires a theoretical method able to corroborate the interpretation of the measurements. We give next the details of

Molecular Electronics xuith Gaussian98/03

27

Figure 14: Conductance spectra calculated with our method for the on-top and bridge adsorption sites in the region around the Fermi level (here set to zero). The upper left inset shows the same data for the on-top geometry on a wider energy range, where the fullerene gap can be clearly appreciated. The other insets show the atomic models we have used in the calculations. Rogero et al.'s work, starting with a brief overview of the adsorption of Cgo on metallic surfaces. Later on we will discuss their main achievements as compared with our own findings using the the GECM[51]. Nucleatkm and growth of Ceo monolayers and thin films on Au(lll) surfaces has been widely studied[52, 53, 54, 55, 56, 57]. Although binding between gold and Ceo is weaker than with other metals [55, 58, 57], it is far from negligible; the adsorption energy is estimated around 40-60 kcal/mol[53, 57]. In fact, adsorption of Ceo is able to lift the well-known 23 x y/3 Au(lll) reconstruction, and photoemission studies of Ceo on polycrystalline Au[59, 60] and Au(lll) surfaces{57] revealed energy shifts indicative of LUMO hybridization and charge transfer from Au to the adsorbed fullerene molecule. At high coverages, closed-packed layers grow with the thermodynamically most stable adsorbate phase being a (2\/3 x 2\/3) R30° structure with a nearly perfect lattice matching in which all the molecules are in equivalent surface sites [53]. Apart from this, another superstructure forms with crystallographic directions matching those of the substrate, resulting in fullerene molecules sitting on different adsorption sites[53, 55]. The proposed 11 x 11 Cgo coverage[53, 55] for this superstructure has been recently discovered to be composed of a smaller 2x2 grid by Rogero et al. [48]. The main features that characterize the adsorption of the above-mentioned Ceo adlayers can be drawn from Fig. 13, where we show the CITS data obtained by Rogero et oZ.[48]. The image appearing in the right panel of Fig.


13 was obtained at a bias of +0.6V[48]. The curves plotted in the left panel of Fig. 13 correspond to the conductance profiles measured at the points indicated with the circle and triangle[48]. From the different position and width of the peaks appearing in the conductance profile Rogero et al. deduce the following facts: 1. The relatively weak C6o-Au(lll) binding reflects on the sharp form of the peaks, with the Ceo LUMO resonance shifted towards the Fermi level by charge transfer from the substrate, yielding a HOMO-LUMO gap of 2.3 eV. 2. The two curves appearing in the left panel of Fig. 13 are interpreted as corresponding to a 2x2 superstructure of the adsorbed layer that places the Cgo molecules on two different adsorption sites (bridge and on-top) of the underlying substrate surface: the interaction of the fullerene with the adsorption site being responsible for the two type of spectra. We now proceed to check if the interpretation of the experimental facts by Rogero and coworkers is confirmed by our calculations. With this aim we calculate the conductance profile corresponding to the two geometries that characterize the 2x2 Ceo superstructure (see insets in Fig. 14). The results are plotted in Fig. 14. From this figure it is immediately apparent one of the main facts found by Rogero et al. and previous authors; namely, that the interaction between the molecule and the surface is not very strong. This is reflected in the sharp peaks that reveal the underlying positioning of the Ceo orbitals, from which we estimate the HOMO-LUMO gap to be 2.9 eV, in good accordance with the experimental CITS results [48]. On the other hand, the relative positioning and height of the conductance maxima between the on-top and bridge geometries is also reproduced. The form of the peaks reflects, as suggested by Rogero et al.,[48] the different interaction of the molecules with the adsorption sites as deduced from the Potential Energy Scan (PES) shown in Fig. 15. For each adsorption site we plot DFT calculations when a Ceo molecule approaches the Au(lll) surface towards the on-top and bridge sites, respectively, with either a sixmember ring or a five-member ring facing the surface. The same equilibrium distance (2.75A) is obtained for all the geometries, which coincides with that found in Ceo-gold nanobridges[61] and is consistent with Altman and Colton suggestion of no height differences between adsorbed Ceo molecules on Au(lll)[53, 54]. As seen from Fig. 15 the adsorption energy is smaller when the molecule sits on top of a gold atom (16 kcal/mol) than when the fullerene binds to a bridge site (35 kcal/mol), which explains the narrower and higher form of the conductance maximum corresponding to the on-top geometry with respect to the bridge site. Once the aforementioned values are corrected by considering the lateral interaction of the Ceo monolayer [57],

Molecular Electronics with Gaussian98/03 300

2

r—•—i—•—i—•—i—•—,—•—i—•—i—•—i—•—i

5

0

-

1 o\ \l oil *\\

• t

E/kcalmol ' -JQO

• • o a

• Hexagon facing on-top • Hexagon facing bridge o Pentagon facing on-top o Pentagon facing bridge

\

50 -

-

%^

-50 _100

I—•

1

:

:

'

'

1.5

'

2

•

'

•

'

•

'

'

2.5 3 3.5 Distance C,0-surface / A

'

4

•

'

•—'

4.5 5

>

Figure 15: DFT Potential Energy Scan of a Ceo molecule approaching the two types of Au(lll) surface sites (on-top and bridge). Results with an hexagon and a pentagon of the fullerene facing the surface are included for completeness. The energy of the separated Ceo + surface is set to zero. estimated from the Lennard-Jones potential to be of about 25 kcal/mol[57], they lie within the experimental margins discussed above. The stronger Au-C6o interaction when the molecule sits on the bridge site is consistent with a larger amount of charge transferred from the gold surface to the molecule: 0.8 electron, as compared to the value of 0.5 electron obtained for the on-top site. This gap accounts for the different alignment of the Fermi level, which for the bridge geometry lies closer to the LUMO-derived orbitals, in complete agreement with Rogero et al.'s findings. The amount of transferred charge calculated by us is in accordance with recent results derived from photoemission spectra[57], which give 0.8±0.2 electrons per fullerene molecule adsorbed on Au(lll). Finally, we point out the fact that there exist minor differences in binding energy with respect to the symmetry axis that points towards the surface. Actually, this coincides with the fact that no predominant molecular orientation had been found [48] and is also indicative of the relative weak C6o-Au(lll) interaction, which allows a large degree of rotational freedom on the adsorbed fullerene. As mentioned before, our method is not only applicable to the tunneling regime but also at tip-sample contact distances. This is reflected in Fig. 16 where we plot the conductance of the system for the two types of adsorption sites as the tip moves towards the adsorbed Ceo- The two maxima correspond to conductance channels coming from the first three C6o LUMO-derived orbitals, whose degeneracy has been partially removed due to the interaction with the gold surface[48, 57] in two sets of two (broader

29

30 J. J. Palacios et al. 1.5 |

.

1

.

.

.

1.5 I—.—i—.—i—.—i—.—I

1 B.«

O

- 3 - 2 - 1 0 ft"»'«v

1

D=1E.DA D.15.5A D = is DA

:', : ;ji

Ail -0.5

0

0.5 *

Energy / eV 1.5

|

.

1.5 i—.

1

1

i

i

Q

/.I n

o

•

. . \[« '

14

D.140A

[AL, . , . JtA-J

m\

D.I«A

-» -! -1 ° '

0.5 -

-0.5

1

i——T—.—i

- 3 J/

o

.

1

i

="-S:S{

| U

0

0.5 Exergi'UV

1 •

Figure 16: Top panel: Conductance profile from tunneling to contact regime of a Ceo molecule adsorbed on top of a gold atom in a Au(lll) surface. Bottom panel: Same as above for the Ceo molecule adsorbed on a bridge site in a Au(lll) surface. Distance (in Amgstron) between gold surface and tip displayed in the legend. The Fermi level has been set to zero in both sets of curves. The insets in both panels plots the same data on a wider energy range at tip-surface distances of 13.5 and 14.5 A.

Molecular Electronics with Gaussian98/03 1.5 I

.

1

X

S O

.

\

A \

_4 [—, 1i

\ \-,

\

O—e On-top 4—A Bridge 11

1

.

1 12

, 1 13 Distance A

31

,

,—Mt— 14 •

\

\ ^

*J\

12 13 Tip-surface distance/ A

14 r

Figure 17: Maximum conductance from the second peak of Fig. 16 for both geometries. The tunneling regime can be distinguished by the exponential decay in the conductance beyond 13A. The inset shows the same data with a logarithmic scale for the conductance. peak) and one (sharper peak) resonances, respectively. These two peaks also appear at tunnel tip-surface distances in the experimental CITS data of Rogero et al. (see Fig. 13). We can check that, as the tip comes in closer contact with the fullerene, their interaction alters the size, width, and positioning of the peaks. The net result is an increase in the conductance to non-negligible values and a larger shift of the LUMO-derived C6o orbitals towards the Fermi level due to the larger amount of charge transferred from the tip atoms to the fullerene as they approach each other. This fact explains the slight closure of the HOMO-LUMO gap as the tip approaches the molecule, which is apparent after inspecting the insets of Fig. 16. On the other hand, the relative shape of the peaks corresponding to the on-top and bridge geometries found at tunneling distances is maintained as the tip approaches the sample. This still reflects the differences in binding strength commented above. Figure 17 shows the value of the conductance maxima of the second peak vs. tip-surface distance. We can see the change of slope due to the different conductance regimes, with the exponential decay typical of tunneling appearing beyond 13A, which agrees well with the experimental value reported by Joachim et al.[50] of 13.2A. The departure from a linear trend in the logarithmic representation of the conductance is related to the abovementioned closure of the Ceo HOMO-LUMO gap. Structural deformation of the fullerene cage has not been considered in our calculations since we


Figure 18: Two atomistic models for the CNT-electrode contact: (a) an open (5,5) carbon nanotube end-contacted to (111) surfaces and (b) a similar nanotube side-contacted. have focused on the change from tunneling to contact regimes, where the deformation of Ceo by the tip is negligible [50]. In summary, we have shown that the so-called GECM method is able to accurately reproduce CITS spectra. The example of Ceo adsorbed on Au(lll) represents just a starting point, but the method looks promising as a valuable tool in the interpretation of STM and CITS spectra up to the contact regime. The main results of this section have been published in Ref. [51]

4.3

Carbon nanotubes with realistic contacts

The controversy on the observed electrical transport properties of carbon nanotubes (CNT's) is due to our lack of control and understanding of their contact to the metallic electrodes. It is clear now that the CNT-electrode contact influences critically the overall performance of the CNT and that it is crucial to lower the inherent contact resistance to achieve the definite understanding of the intrinsic electrical properties of CNT's[62]. In order to determine the relevant factors behind the contact resistance so that this can be pushed down to the quantum limit per channel RQ = I/Go, a big experimental effort has been made both in CNT growth and lithographic techniques[63, 64]. While considerable progress has already been made, it is fair to admit that theoretical support has not been determinant in this regard. A limited number of works have addressed this issue previously [65, 66, 19, 67, 68], but a full analysis of this problem requires the use of the state-of-the-art ab initio techniques to calculate electrical transport like the one we are describing in this review and these are still under developement. As we have already mentioned in previous sections, there are multiple difficulties beyond the usual computational ones in first-principles calculations. The main one is that the actual atomic structure of the electrode (and probably that of the CNT) at the contact are unknown and, most likely, change from sample to sample even when fabricated under the same


Figure 19: (a) Conductance as a function of energy for an N = 10 (5,5) open metallic nanotube end-contacted to Al(lll) surfaces [see Fig. 18(a)]. The Fermi energy is set to zero. Inset: Schematic band structure of the metallic nanotube showing the four states responsible for the resonances, (b) Transmission as a function of energy for the highest conducting eigenchannels.

conditions. It is our believe that atomic-scale modeling, however, can still be of guidance in the interpretation of the experiments and to the future design of operational devices with CNT's. There are two ingredients in this puzzle: The atomic-scale geometry and the chemical nature of the electrode. In order to address this issue we study open single-walled metallic (5,5) CNT's contacted in two representative forms (see Fig. 18) to Al, Ti, and Pd electrodes which are among the most commonly used metals in the experiments. We find that CNT's contacted to Al in end-contact geometry [see Fig. 18(a)] the two CNT bands couple weakly to the electrodes. Moreover, we find that the two bands couple very differently to the electrodes (one of them is almost shut down for transport) and do not mix. For the side-contact geometry [see Fig. 18(b)] the coupling is the same for both bands, but equally weak as for to the end-contact geometry. Figure 19(a) shows Q{E) for a (5,5) metallic CNT composed of N = 10 carbon layers (diameter « 6.8 A and length «s 11.2 A) that has been endcontacted [Fig. 18(a)] to Al(lll) surfaces. The end-carbon-layer-surface

33

34


distance has been optimized to a value of 1.8A. It is important to obtain a correct estimate of this distance since the charge transfer and the contact coupling are fairly sensitive to it and cannot be chosen whimsically. Small variations of a few per cent away from the optimum contact distance increase the energy of the contact by a large amount. In particular, a decrease of the contact distance by a few tenths of an A, which might improve the contact transmission, is energetically prohibited. Four resonances can be seen in the conductance around the Fermi energy (set to zero) in Fig. 19. These resonances can be easily traced back to four extended states of the isolated finite CNT[69]. Two of them (A;i, fo) originate in the bonding (TT) band of the CNT and the other two (fc*,^) in the antibonding (TT*) band (see inset in Fig. 19). The resonances have different widths for different bands indicating that they couple very differently to the electrodes. Moreover, the two bands do not mix with each other. This is more clearly seen in Fig. 19(b) where we show the highest transmission eigenvalues of the transmission matrix. Two independent channels exhibit resonances in the energy window (« 3.5eV) around Ep where only the TT and TT* bands can contribute to transport. This result is consistent with the fact that TT* states, of large angular momentum, do not couple to the low-angular momentum states of the electrode, while 7T states, of low angular momentum, couple more easily[65, 66]. Notice that there is a charge transfer from the metal to the CNT, but this mainly localizes at the end carbon layer (fa 0.2 electrons per carbon atom) and it does not affect the overall band positioning in the center of the CNT. The specific band origin of the resonances is nicely confirmed by their evolution on the length of the CNT presented in Fig. 20. We present the conductance for TV = 8,9,10,11,12, and 13 carbon-layer CNT's. The opposite signs of the group velocity for the TT and n* bands make the quasi-bound states belonging to the TT* band shift down in energies while those belonging to the 7T band shift up as N increases. As expected from a simple particlein-a-box argument applied to finite CNT's[69], for N — 3Z, where I is an integer, we should expect two states with the same wave vector kn but in different bands to coincide at the Fermi energy. Naively one should thus expect Q = 4e2/h[70}. Our results for the contacted N = 9 and N = 12 CNT's show otherwise: Two resonances never coincide at the Fermi level. The reason is that Coloumb blockade prevents two (band and/or spin) degenerate quasibound states to be filled up at the same time and degeneracies are removed. From Figs. 20(b) and (e) we estimate the charging energy to be w 0.3 eV in these CNT's which is smaller than the single-particle level spacing as confirmed by experiments [63]. We have analyzed this Coulomb blockade effect in detail for the N = 9 CNT. For a partially discharged CNT the two resonances labeled fci and kl coincide in energy above the Fermi energy and the conductance reaches there Ae2/h (see Fig. 21). For the neutral system this degeneracy is partially removed and the conductance drops. The spin degeneracy removal due to Coulomb blockade requires technically


Figure 20: Conductance as a function of energy for an N = 8(a), N = 9(b), N = 10(c), N = ll(d), N = 12(e), and AT = 13(f) (5,5) open metallic nanotube end-contacted to Al(lll) surfaces [see Fig. 18(a)]. The Fermi energy 3,

°-2

,

,

-

,

1

1

0

,

Energy (eV)

,

,

1

Figure 21: Conductance as a function of energy for an N = 9 (5,5) open metallic nanotube end-contacted to Al(lll) surfaces [see Fig. 18(a)] for different values of the Fermi energy or charge of the system Q=-3,-2,-1,0,1,2,3. The Fermi energy has been always set to zero so that curves shift downward as Q increases.

35


Figure 22: Same as in Fig. 19, but for an N = 15 (5,5) open metallic nanotube side-contacted to Al(lll) surfaces [see Fig. 18(b)]. challenging open shell calculations and is currently under study. If the interpretation of the different coupling strengths of the CNT bound states with the Al electrodes is correct and angular momentum considerations are relevant, similar couplings should be expected for both bands if no axial symmetry is present. This is the case for the other contact geometry considered in this work [see Fig. 18(b)]. Figure 22 shows results for an AT = 15 CNT side-contacted to Al(lll) surfaces (the CNT-surface distance has been optimized to 2.3A). Conductance resonances come in pairs in the relevant energy window which is what is expected for an N = 15 CNT. More importantly, all of them present similar widths, confirming our expectations. Contrary to the previous geometry, localized end states[69] influence the coupling around leV for this contact geometry where mixing with the CNT extended states takes place. Our results for the coupling strength with Al contacts are consistent with previous studies where jellium models were considered as contacts[66], and with those in Ref. [19], but we do not subscribe previous ab-initio results presented in Ref. [71] based on what it seems to be more realistic contact models similar to ours. We now complete our study for end-contacted CNT's considering Ti, and Pd electrodes (see Figs. 24 and 23). Pd behaves very much like Al, except that the coupling to the CNT is clearly stronger and the overall conduc-


Figure 23: Same as in Fig. 19, but for an N = 11 (5,5) open metallic nanotube end-contacted to Pd(lll) surfaces (see inset).

Figure 24: Same as in Fig. 19, but for an N = 10 (5,5) open metallic nanotube end-contacted to Ti(lll) surfaces (see inset).

37

38 J. J. Palacios et al. tance higher. Apart from this difference in the coupling strength, the bands do not mix and couple, as well, very differently to the electrode [see Fig. 23(b)]. In contrast to Al and Pd electrodes, where Q exhibits resonances, Q presents an oscillatory behavior for Ti around E-p [see Fig. 24(a)], with the two bands exhibiting similar coupling to the electrodes. This is accompanied by band mixing as the anticrossings in the transmission eigenvalues reveal in Fig. 24(b). This is, to our knowledge, the first numerical evidence of what has been hinted at on the basis of indirect first-principles calculations[68] and what has recently been observed in experiments[64]: Early 3-d elements as Ti are probably the best choice for making high-transparency contacts to CNT's compared to more traditional metals such as Al and Pd (apparently due to the presence of d-states at the Fermi energy). Although perfect transparency at the contact is never achieved, our calculations indicate that properly engineered Ti contacts are a good bet for future perfect contacts to CNT's. At this point, however, we can only speculate on the possibility of perfect transparency for other Ti electrode geometries. The optimal contact distance is similar to that of Al but the charge transfer is « 0.4 electrons per C atom at the end layers. There seems to be a correlation between charge transfer and contact transparency if we compare Al and Ti, but this is no longer the case when comparing Al and Pd and we cannot draw a clear correlation between these two quantities at the present time.

5

Open issues

We would not like to finish this review without enumerating some of the issues in nanoscale and molecular tranport that we have not addressed here and that are still open (maybe we should say even more open than those discussed in these notes): • Electron phonon coupling. A number of effects derived from the interaction between the electronic and nuclear degrees of freedom has been observed in a number of cases[61] and it is nowadays the subject of experimental and theoretical scrutinity[72]. This is closely related to how inelastic effects modify elastic transport and how they can be included in Landauer's formalism. • Structural relaxation in a non-equilibrium situation. The way a current can modify the atomic structure of the nanoscopic region that sustains it has been only recently addressed[73, 74] and, although, it seems to have a small effect, it is no clear whether or not it can be completely neglected from the overall picture. • Beyond a single-particle description. Present theoretical work relies heavily on the single-particle description of the electronic structure.


Many-body physics like the Rondo-effect recently observed in single molecules [75] cannot be addressed within these simplified methodologies.

6

Acknowledgements

Financial support from the Spanish MCYT (grants BQU2001-0883, PB960085, and MAT2002-04429-C03), FEDER european funds, and the Universidad de Alicante is gratefully acknowledged.

A

Bethe lattices

In this appendix we discuss how selfenergies for Bethe Lattices (BL) used to describe the leads are calculated. A BL is generated by connecting a site with N nearest-neighbors in directions that could be those of a particular crystalline lattice. The new N sites are each one connected to N — 1 different sites and so on and so forth. The generated lattice has the actual local topology (number of neighbors and crystal directions) but has no rings and, thus, does not describe the long range order characteristic of real crystals. Let n be a generic site connected to one preceding neighbor n — 1 and N — 1 neighbors of the following shell (n + i with i — l,..,N — 1). Dyson's equation for an arbitrary non-diagonal Green function is written as,

(EI-Eo)G n , f c =V B , B _iG n _ u +

Yl

V «,* G a

(20)

i=l,...,JV-l

where A; is an arbitrary site, E the energy, and Vjj is a matrix that incorporates interactions between orbitals at sites i and j (bold capital characters are used to denote matrices). Eo is a diagonal matrix containing the orbital levels and I is the identity matrix. Then, we define a transfer matrix as Ti_i,iGj_ij = Gij

(21)

Multiplying Eq. (1) by the inverse of G n _i i n we obtain,

(El - E 0 )T n _ l i n = V n , n _! + I

Yl Vni J T B _i, n

\»=1,...,JV-1

(22)

/

Due to the absence of rings the above equation is valid for any set of lattice sites, and, thus, solving the Bethe lattice is reduced to a calculation of a few transfer matrices. Note that a transfer matrix such as that of Eq. (2) could also be defined in a crystalline lattice but, in that case it would be useless.

40


Eq. (3) can be solved iteratively, -i

T B _i, n = El-Eo-

Yl

- l

Vn.n-1

^n,iTn,i

(23)

i=l,...,JV-l

If the orbital basis set and the lattice have full symmetry (including inversion symmetry) the different transfer matrices can be obtained from just a single one through appropriate rotations. However this is not always the case (see below). Before proceeding any further we define selfenergies that can be (and commonly are) used in place of transfer matrices, Sij = V y T y

(24)

Eq. (4) is then rewritten as, -i

£«-!,« = VB_i,B EI-Eo

Y,

S ».'

-i

V»-i,»

W

i=l,...,N-l

where we have made use of the general property V n ) n _i = Vn_i „• As discussed hereafter, in a general case of no symmetry this would be a set of N coupled equations (2iV if there is no inversion symmetry). Symmetry can be broken due to either the spatial atomic arrangement, the orbitals on the atoms that occupy each lattice site, or both. When no symmetry exists, the following procedure has to be followed to obtain the selfenergy in an arbitrary direction. The method is valid for any basis set or lattice. Let T\ be the N nearest-neighbor directions of the lattice we are interested in and Vn the interatomic interaction matrix in these directions. To make connection with the notation used above note that the vector that joins site n - 1 to site n, namely, r n - r n _ ! would necessarily be one of the lattice directions of the set 71. The selfenergies associated to each direction have to be obtained from the following set of 2N coupled selfconsistent equations, S Ti = V n [El - E o - (ET - E ^ ) ] - 1 V*, (26) S n = V n [El - E 0 - ( E T - S T i )]" : V t ,

(27)

where i = 1,..., N and fj = — TJ. VTi is the interatomic interaction in the Tj direction, and S x and S ^ are the sums of the selfenergy matrices entering through all the Cayley tree branches attached to an atom and their inverses, respectively, i.e., N

Sr = £ S T i

(28)

Molecular Electronics with Gaussian98/03 41 N

Sf = E s ^ -

(29)

»=1

This set of 2N matricial equations has to be solved iteratively. It is straightforward to check that, in cases of full symmetry, it reduces to the single equation. The local density of states can be obtained from the diagonal Green matrix, j

P

Gn,n= £ I - E 0 - Y, S n

(30)

t=l,..,JV

Finally, as regards the tight-binding parameters, which should include only nearest-neighbors interactions, there is no unique way of choosing them. The ones we generically use were obtained through fittings to the electronic bulk band structures[23], but the reader is free to choose any other criterion that fits better the problem at hand.

B

Green functions and non-orthogonal basis sets

In this appendix we briefly discuss how Eqs. 6 and 7 for the density matrix and the total number of particles in a nonorthogonal basis are derived. Working with a nonorthogonal basis requires the use of the metrics

£ > Q > S^1 < %| =/• a/3

(31)

Changing to an orthogonal basis \<j>a > can be done through the equation |^>=^5^1/2|^> 0

(32)

Then an expression which allows to obtain the matrix elements in an orthogonal basis of a general matrix A from those in the non-orthogonal basis, can be easily derived, Ao = S~1/2ANoS-1/2 (33) Using the above metrics and the definition of the Green's function operator, which includes the operator selfenergies,

[(£ ± iS)I -F- tf} - E^}] (?W = /,

(34)

Eq. (5) for the matrix elements of the Green's function in a nonorthogonal basis inmediately follows.

42


The local charge density is written in terms of the Green's function as [29]

p(r) = - - f

F

K J-oo

Im < r|G (+) (£)|r >

= - I fEFlmJ2Mr)S^G{+\E)S^;(r), J-°°

afrS

(35)

while the total number of particles is in its turn given by [33] N = f drp(r) = Y, Sa0Pap = Tr [P • S]. J

(36)

0a

References [1] A. Nitzan and M. A. Ratner. Electron transport in molecular wire junctions. Science, 300(5624): 1384-1389, 2003. [2] A. Aviram and M. A. Ratner. Chem. Phys. Lett, 29:277, 1974. [3] M. A. Reed, C. Zhou, C. J. Muller, T. P. Burgin, and J. M. Tour. Science, 278:252, 1997. [4] B. Mann and H. Kuhn. J. Appl. Phys., 3:4398, 1971. [5] C. Dekker. Physics Today, 52:22, 1999. [6] N. Agrait, A. Levy Yeyati, and Jan M. van Ruitenbeek. Physics Reports, 377:81-279, 2003. [7] N. Agrait, J. C. Rodrigo, and S. Vieira. Phys. Rev. B, 47:12345, 1993. [8] N. Agrait, G. Rubio, and S. Vieira. Phys. Rev. Lett, 74:3995, 1995. [9] S. Datta. Electronic transport in mesoscopic systems. Cambridge University Press, Cambridge, 1995. [10] Mukunda P. Das and Frederick Green. In J. da Providencia, editor, Proceedings of XXVI International Workshop on Condensed-Matter Theories. Nova Science, New York, 2003. [11] J. A. Torres and Saenz J. J. Phys. Rev. Lett, 77:2245-2248, 1996. [12] A. Garcia-Martin, J. A. Torres, and J. J. Saenz. Phys. Rev. B, 54:1344813451, 1996. [13] Y. Meir and N.S. Wingreen. Phys. Rev. Lett, 68:2512, 1992.


[14] J. C. Cuevas, A. Levy-Yeyati, and A. Martm-Rodero. Phys. Rev. Lett., 80:1066, 1998. [15] A. Hasmy, E. Medina, and P. A. Serena. Phys. Rev. Lett, 86:5574, 2001. [16] N. D. Lang. Phys. Rev. B, 52:5335, 1995. [17] K. Hirose and M. Tsukada. Phys. Rev. B, 51:5278, 1995. [18] M. Brandbyge, J. L. Mozos, P. Ordejon, J. Taylor, and K. Stokbro. Phys. Rev. B, 65:165401, 2002. [19] J. Taylor, H. Guo, and J. Wang. Phys. Rev. B, 63:245407, 2001. [20] M. Di Ventra, S. G. Kim, S. T. Pantelides, and N. D. Lang. Phys. Rev. Lett, 86:288, 2001. [21] M. J. Frisch, G. W. Trucks, H. B. Schlegel, M. A. Robb G. E. Scuseria, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, Jr., R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, A. G. Baboul, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. AlLaham, C. Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, J. L. Andres, C. Gonzalez, M. Head-Gordon, E. S. Replogle, and J. A. Pople. GAUSSIAN98, Revision A.7, Gaussian, Inc., Pittsburgh PA, 1998. [22] J. J. Palacios, A. J. Perez-Jimenez, E. Louis, and J. A. Verges. Phys. Rev. B, 64:115411, 2001. [23] J. J. Palacios, A. J. Perez-Jimenez, E. Louis, E. SanFabian, and J. A. Verges. Phys. Rev. B, 66:035322, 2002. [24] J. J. Palacios, A. J. Perez-Jimenez, E. Louis, and J. A. Verges. Nanotechnology, 12:160, 2001. [25] E. Louis, J. A. Verges, J. J. Palacios, A. J. Perez-Jimenez, and E. SanFabian. Phys. Rev. B, 67:155321, 2003. [26] P. S. Damle, A. W. Ghosh, and S. Datta. Phys. Rev. B, 64:201403, 2001. [27] Y. Xue, S. Datta, and M. A. Ratner. J. Chem. Phys., 115:4292, 1991.

43

44 J. J. Palacios et al. [28] J. Heurich, J. C. Cuevas, W. Wenzel, and G. Schon. Phys. Rev. Lett., 88:256803, 2002. [29] E. N. Economou. Green's functions in Quantum Physics. Number 7 in Springer Series in Solid State Physics. Springer, 1970. [30] J. Taylor, H. Guo, and J. Wang. Phys. Rev. B, 63:121104, 2001. [31] Y. Fujimoto and K. Hirose. Phys. Rev. B, 67:195315, 2003. [32] N. D. Lang and M. DiVentra. arXiv:cond-mat/0307442. [33] A. Szabo and N. S. Ostlund. Modern Quantum Chemistry. Hill, New York, 1989.

McGraw-

[34] L. F. Pacios and P. A. Christiansen. J. Chem. Phys., 82:2664, 1985. [35] M. M. Hurley, L. F. Pacios, P. A. Christiansen, R. B. Ross, and W. C. Ermler. J. Chem. Phys, 84:6840, 1986. [36] R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, and P. A. Christiansen. J. Chem. Phys., 93:6654, 1990. [37] P. Jelinek, Ruben Perez, Jose Ortega, and Fernando Flores. arXiv:condmat/0211488. [38] R. H. M. Smit, Y. Noat, C. Untiedt, N. D. Lang, M. C. van Hemert, and J. M. van Ruitenbeek. Nature (London), 419:906, 2002. [39] R. H. M. Smit. PhD thesis, Universiteit Leiden, 2003. [40] F. Pauly J. Heurich, J. C. Cuevas, W. Wenzel, and G. Schon. arXiv:cond-mat/0211635. [41] L. G. C. Rego, A. R. Rocha, V. Rodrigues, and D. Ugarte. arXivxondmat/0206412. [42] Y. Noat S. K. Nielsen, M. Brandbyge, R. H. M. Smit, K. Hansen, L. Y. Chen, A. I. Yanson, F. Besenbacher, and J. M. van Ruitenbeek. arXiv:cond-mat/0212062. [43] G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel. Appl. Phys. 40:178, 1982. [44] J. Tersoff and D. R. Hamann. Phys. Rev. Lett, 50:1998, 1983. [45] J. Tersoff and D. R. Hamann. Phys. Rev. B, 31:805-813, 1985. [46] J. Bardeen. Phys. Rev. Lett, 6:2654, 1961. [47] N. D. Lang. Phys. Rev. B, 34:5947-5950, 1986.

Lett,


[48] C. Rogero, J. I. Pascual, J. Gomez-Herrero, and A. M. Baro. J. Chem. Phys., 116:832-836, 2002. [49] V. Meunier and Ph. Lambin. Phys. Rev. Lett, 81:5588-5591, 1998. [50] C. Joachim, J. K. Gimzevski, R. R. Schlittler, and C. Chavy. Phys. Rev. Lett, 74:2102-2105, 1995. [51] A. J. Perez-Jimenez, J. J. Palacios, E. Louis, E. SanFabian, and J. A. Verges. CHEMPHYSCHEM, 4, 2003. [52] R. J. Wilson, G. Meijer, D. S. Bethune, R. D. Johnson, D. D. Chambliss, M. S. de Vries, H. E. Hunziker, and H. R. Wendt. Nature, 348:621-622, 1990. [53] E. I. Altman and R. J. Colton. Surf. ScL, 279:49-67, 1992. [54] E. I. Altman and R. J. Colton. Surf. ScL, 295:13-33, 1993. [55] E. I. Altman and R. J. Colton. Phys. Rev. B, 48:18244-18249, 1993. [56] D. Fujita, T. Yakabe, H. Nejoh, T. Sato, and M. Iwatsuki. Surf. Sci., 366:93-98, 1996. [57] C.-T. Tzeng, W.-S Lo, J.-Y Yuh, R.-Y. Chu, and K.-D. Tsuei. Phys. Rev. B, 61:2263-2272, 2000. [58] T. Hashizume, K. Motai, X. D. Wang, H. Shinohara, Y. Saito, Y. Maruyama, K. Ohno, Y. Kawazoe, Y. Nishina, H. W. Pickering, Y. Kuk, and T. Sakurai. Phys. Rev. Lett., 71:2959-2962, 1993. [59] T. R. Ohno, Y. Chen, S. E. Harvey, G. H. Kroll, J. H. Weaver, R. E. Haufler, and R. E. Smalley. Phys. Rev. B, 44:13747-13755, 1991. [60] S. J. Chase, W. S. Bacsa, M. G. Mitch, L. J. Pilione, and J. S. Lannin. Phys. Rev. B, 46:7873-7877, 1992. [61] H. Park, J. Park, A. K. L. Lim, E. H. Anderson, A. P. Alivisatos, and P. L. McEuen. Nature (London), 407:57-60, 2000. [62] S. Frank, P. Poncharal, Z. L. Wang, and W. A. de Heer. Science, 280:1744, 1998. [63] Hyongsok T. Soh, Calvin F. Quate, Alberto F. Morpurgo, Charles M. Marcus, Jing Kong, and Hongjie Dai. Appl. Phys. Lett, 75:627, 1999. [64] J. Kong, E. Yenilmez, T. W. Tombler, W. Kim, H. Dai, R. B. Laughlin, L. Liu, C. S. Jayanthi, and S. Y. Wu. Phys. Rev. Lett, 87:106801, 2001.

46


[65] H. J. Choi, J. Ihm, Y.-G. Yoon, and S. G. Louie. Phys. Rev. B, 60:R14009, 1999. [66] M. P. Anantram. Appl. Phys. Lett, 78:2055, 2001. [67] A. N. Andriotis, M. Menon, and G. E. Proudakis. Appl. Phys. Lett., 76:3890, 2000. [68] C. K. Yang, J. Zhao, and J. P. Lu. Phys. Rev. B, 66:041403, 2002. [69] A. Rubio, D. Sanchez-Portal, E. Artacho, P. Ordejon, and J. M. Soler. Phys. Rev. Lett, 82:3520, 1999. [70] D. Orlikowski, H. Mehrez, J. Taylor, H. Guo, J. Wang, and C. Roland. Phys. Rev. B, 63:155412, 2001. [71] M. B. Nardelli, J. L. Fattebert, and J. Bemholc. 64:245423, 2001.

Phys. Rev. B,

[72] Yu-Chang Chen, Michael Zwolak, and Missimiliano Di Ventra. arXiv:cond-mat/0302425. [73] Z. Yang and M. Di Ventra. Phys. Rev. B, 67:161311, 2003. [74] M. Brandbyge, K. Stokbro, J. Taylor, J. L. Mozos, and P. Ordejon. Phys. Rev. B, 67:193104, 2003. [75] J. Park, A.N. Pasupathy, J. I. Goldsmith, C. Chang, Y. Yaish, J. R. Petta, M. Rinkoski, J. P. Sethna, H. Abruna, P. L. McEuen, and D. C. Ralph. Nature (London), 417:722-725, 2002.

Chapter 2: Molecular Dynamics Simulations of Single Molecule Atomic Force Microscope Experiments

Wiestew Nowakab and Piotr E. Marszatekb "Institute of Physics, Nicholaus Copernicus University Grudziadzka 5, 87-100 Torun, Poland hDepartment

of Mechanical Engineering and Materials Science CBIMMS, Duke University, Durham, 27708 NC, USA

Abstract The ultimate goal of nanosciences is to develop tools for spatial and chemical control of single molecules. A number of experimental techniques have emerged in recent years which allow for mechanical manipulations of single molecules including single molecule fluorescence, optical and magnetic tweezers, atomic force spectroscopy. Proper understanding of experimental results requires adequate theory. In this paper we present an overview of recent advances and trends in applications of computer simulations aimed at understanding of atomic force microscope (AFM) experiments. Special attention is devoted to those variants of classical molecular dynamics (MD) simulations, such as steered MD or biased MD, which are particularly helpful in interpretation of experimental data obtained for single biopolymer molecules. A short meta-review (review of recent reviews) of advances in the MD technology is also presented.

47

48

W. Nowak and P. E. Marszalek

Glossary of Terms AFM ATP ADP BMD BPTI CHARMM DFT DNA FF FN FPS GTP IMD Ig LCD MD MscL NAMD2 PC PMF RC REMD QM/SMD QSAR RNA SCC-DFTB SMD TMD UV WHAM

Atomic Force Microscope Adenosine TriPhosphate Adenosine DiPhosphate Biased Molecular Dynamics Bovine Pancreatic Trypsin Inhibitor Chemistry at Harvard Macromolecular Modeling Density Functional Theory Deoxyribonucleic Acid Force Field Fibronectin Force Probe Microscopy Guanosine TriPhosphate Interactive Molecular Dynamics Immunoglobulin Liquid Crystal Device Molecular Dynamics Mechanoselective Ion Channel Name of Molecular Dynamics Code Personal Computer Potential of Mean Force Reaction Coordinate Random Expulsion Molecular Dynamics Quantum Mechanical Steered Molecular Dynamics Qualitative Structure Activity Relationship Ribonucleic Acid Self Consistent Charge Density Functional Theory based Tight Binding Method Steered Molecular Dynamics Targeted Molecular Dynamics Ultra Violet Weighted Histogram Analysis Method

1. Introduction Single molecule experiments are somehow at variance with the idea of "standard" chemistry that typically deals rather with myriads of molecules than a single one (Avogadro number is ~1023). However, one should remember, that the ultimate limit of miniaturization is perhaps at a single

Molecular Dynamics Simulations

49

molecule level.1 The quest for better and better computers, digital memories, transducers, sensors, etc. is the driving force of the nanotechnological revolution. Clearly, single molecule manipulations and chemistry performed on a couple of molecules are in focus of many research labs' activities. There are numerous techniques used to observe and manipulate single molecules. Description of those experimental approaches and new results were subject of many reviews and will not be repeated here. Optical methods of fluorescence2"4 based on scanning confocal microscopy, near field scanning optical microscopy and fluorescence correlation spectroscopy5"7 are very popular in biologically oriented studies.8"10 Optical tweezers,11"13 where light is used to manipulate molecules attached to dielectric beads,10 came a long path from the studies of single myosin molecules binding to a single actin filament14 to the observation of DNA packing inside lambda phage.15'16 Sometimes magnetic tweezers, known for longer time,17 are better suited for performing advanced research on single molecules.18'19 The first single-molecule AFM experiments, initiating the field of force probe spectroscopy (FPS), were performed in 1994 by E.-F. Florin, V. Moy and H. Gaub.5 In their experiments the unbinding force of the individual ligand - receptor pair in the important biotin-streptavidin complex was measured using Atomic Force Microscope (AFM). The principle of FPS is rather simple: a sharp tip of the AFM cantilever is attached to a single molecule while the other end of the molecule is firmly attached to a moving support. The motion of this support is controlled by the application of voltage to the piezo device. The elongation of the molecule requires a force and the cantilever bends under this force. Its minute deformation is usually detected by measuring changes in the intensity of the laser beam reflected from the tip part of the cantilever and projected onto a split photodiode. Since a proper calibration procedure allows for a precise determination of the cantilever force constant, and the position of the piezo stage is strictly controlled, a dependence of the molecule size with respect to the applied force may be determined with high accuracy. The basic principles of AFM experiments are presented in Fig. 1

50

W. Nowak and P. E. Marszatek

Principles of tinglt^molerule fartv m - ]10

58


Further studies (2001) performed by the Schulten group were devoted to the mechanism of refolding of the single 127 domain. During 2 ns SMD simulations two refolding patterns have been observed: seperated A'-G strands reformed the majority of their native hydrogen bonds, while for A and B strand 2 ns time was too short to recover their hydrogen bonding.''' Since not all Ig units of titin are identical, SMD was used to study the differences in the unfolding mechanism between some domains, namely II and I27.112 SMD suggests that, in contrast to 127, II does not have unfolding intermediates. Details of folding-refolding dynamics of the II domain were described by Gao et al. in a separate paper.113 Interestingly, in this work the important stabilizing effect of the disulfide bridge was easily discovered by modeling oxidized and reduced forms of the protein. The S-S bridge limits the maximum extension of the II titin unit. However, very recent AFM results published by Li and Fernandez114 put in doubt the special role of S-S bridges in II modules of titin. More analytical models of protein elasticity probed by AFM and related to Ig domains were presented by Zhang and Evans.115 Best et al.116 used engineered titin modules to test their method of searching for the unfolding mechanism. Some summary information on titin SMD studies may be found in the paper by Isralevitz et al.' 17 Very recently Best et al. have claimed,118 based on MD simulations and analysis of protein mutagenesis data, that a stable intermediate is indeed involved in 127 unfolding,119 but it is related rather to side chain interactions than to hydrogen bond ruptures, as was previously assumed. The unfolding intermediate has also been found in AFM experiments by Schweiger et al.120 done for the Ig rod domain of filamin from Dictyostelium discoideum. The report on AFM experiments performed on the isolated, huge titin molecule was published by Kellermayer et al.121 3.1.2 Fibronectin and other elastic proteins Fibronectin (FN) is another interesting protein extensively studied both by the AFM and MD methods. It is found in the extracellular matrix in all vertebrates. Elastic FN fibrils connect cells, they may stretch four times as long as their relaxed length, and play a role in signalling. Two identical subunits form a dimeric molecule consisting of more than 20 modules on each part. There are three types of monomers (I-III), a module FN-III10 was most often studied computationally.122 The first AMF measurements done by Rief et al.123 have shown that FNIII modules of titin exhibit 20% lower unfolding forces than Ig domains. At the same time Oberhauser et al., who studied FNIII modules from tenascin124, have found that these forces are 50% smaller, and that tenascin FN modules refold twice as fast as those from fibronectin. Since


59

the overall fold of both types of FNIII modules is very similar, observed differences in rupture forces (75-200 pN) are hard to explain without atomistic simulations of dynamics. This hierarchy in elasticity of individual FN modules may have a physiological role.23 Early simulations of FN suffered from limited simulation times and a rather crude treatment of the solvent. Kramer et al.125 used the SMD method to study the behavor of the FNIIIio module under an external force. A small water sphere was used in the simulations. Since timescales were much shorter than in the experiments, a few hundred pN forces had to be applied to unfold this P-structure and to observe that unfolding starts from the rupture of the G strand.125 Simulations have shown that residues in the loop 78-80 (RGD) are "strategically" located close to a membrane protein integrin, and act as a mechanoswitch during unfolding. Further studies by the same group gave a better picture of the mechanism of FNIII unfolding for different modules.126 Paci and Karplus90 prefered to use an implicit solvent model together with the biased MD method (yi). A few dozen of the unfolding computer experiments (0.2-1 ns) were carefully analyzed. Differences between FNIII9 and FNIII 10 folding pathways were highlighted. For the first time an unfolding path for the FIII9 - FNIII I0 dimer was presented. Perhaps the most advanced simulations of FNIII modules were performed by Gao et al.122 They used a huge (370 A long) water box with 130 K atoms to avoid artifacts caused by a spherical solvent shell which undergoes a large deformation during the stretching of the protein. This study suggests that there are at least three distinct unfolding pathways, but their discrimination, based solely on fast SMD simulations, is not possible. A good discussion of FN dynamics may be found in the recent review by Tajkhorshid et al.95 In 2003 Li and Makarov suggested127 that the current time scale gap between AFM experiments and MD simulations (simulations are typically 106 times too fast) may be eliminated by describing the unfolding dynamics in terms of diffusive motion along a predefined unfolding coordinate R in the presence of external driving potential and the potential of mean force G(R). The other degrees of freedom are hidden in a viscous force. The PMF G(R) is computed from a series of equilibrium MD runs with the fixed values of R, and friction coefficients are extrated from a series of SMD simulations. It is notable, that the authors were able to calculate the unfolding free energy barrier of titin at zero force, which was almost identical to that obtained experimentally. Two distinct unfoding pathways of titin are predicted by this model. Mechanical properties of elastin, yet another protein found in the extracellular matrix, especially in relation to the temperature factor, were also computationally studied,128"132 but more traditional techniques of MD

60

W. Nowak and P. E. Marszatek

simulations were used in this case. Spectrin repeats and its two mutants were recently studied by AFM and SMD techniques as well.133 3.1.3 SMD simulations indirectly related to AFM experiments The SMD method was used in several important studies of ligand-protein interactions134 for systems which have not yet been subject to any AFM type experiments. Since the number of such applications will grow, we present a brief account of the recent papers. In 1999 Kosztin et al. presented unbinding pathways of the retinoic acid hormone from its nuclear receptor h-RAR. Three hypothetical paths were selected. The SMD calculations indicated that one of those is not probable and that alternative pathways may be used by the hormone for binding and unbinding. A detailed analysis of possibile close contacts, salt bridges and hydrogen bonds along the path has improved our understanding of this medically important receptor and may help to develop new drugs. In 2001 Heymann and Grubmiiller136 performed a detailed study of forced unbinding of spin-labeled dinitrophenyl hapten from the monoclonal antibody AN02 using a variant of the SMD method called force probe molecular dynamics (FPMD). Large flexibility of the binding pocket region has been found. A large (6 kcal/mol) entropic contribution was calculated to the ligandprotein bond. Interestingly, no differences in unbinding forces were found in two Y33F and I96K mutants of the antibody. Mechanosensitive channel with large conductance (MscL) can be gated by tension in the cellular membrane. Gullingsrud et al, following their initial, classical 3 ns MD study of the gating mechanism,137 used SMD in a recent 10 ns study138 to add radial external forces (70 pN) in the critical part of the channel to induce its opening. In contrast to the previous calculations by Ma et. al. in which TMD was used139, the authors have found that the iris-like motion is probably involved in the process of the channel opening. Both TMD and SMD confirm tandem-like mechanism of MscL sensitivity to the membrane tension. GTP hydrolizing G proteins are the part of the signal transduction system that control proliferation, differentiation and metabolism. Among them Ras p21 attracted particular attention, since it works like a molecular switch, changing states between the GDP-bound inactive form and the GTP-bound active form. The conformational changes related to this process were studied by Ma and Karplus140, Diaz et al.141 using TMD. Recently Kosztin et al.142 added an "artificial" force attached to two carefully selected atoms, in order to estimate the work related to the transition between tense (T) and relaxed (R) states of the so called switch II region in h-RAS protein. It has been found that the force generation proceedes in two steps via a short lived (< Ins)


61

metastable T state formed just after the GTP hydrolysis, and the amount of work related to this process is around 1 kcal/mol. Cell-cell adhesion is mediated by proteins. The interaction between the Tlymphocyte adhesion receptor CD2 and its ligand CD58 was studied by Bayas et al.143 using the SMD method. Two regimes - constant force and constant velocity - were used to enforce detachment of these two proteins. The mechanism of the forced detachment depends strongly on the loading rate. At faster loading rates (70 pN/ps) the proteins unfold before detachment, at slower rates (7 pN/ps) they just separate without significant initial distortions. These calculations clearly show that multiple conditions in the SMD protocol should be tried during studies of protein complexes, before conclusions on interactions mechanisms are reached. Elastic properties of a P-selectin construct consisting of a lectin domain and epithelial growth factor-like domain was studied using SMD by Lu and Long144. Groups of hydrogen bonds were identify to rupture during forced dissociation of this complex, which is important in inflammatory proceses. Best et al.145 posed an interesting question whether proteins, which unlike titin, are not designed for mechanical function can withstand large forces before unfolding. An artificial complex of barnase with the 127 domain of titin was studied by AFM and computationally (3-6 ns SMD trajectories). The results indicate that barnase unfolds at much lower forces than the titin fragment. Since unfolding rates of both proteins in solution are comparable, this indicates that the mechanism of mechanically forced unfolding may be different than the chemically induced one. 3.2 Biased JMD The first application of an interesting variant of enforcing conformational transitions was proposed by Paci and Karplus.90 As the authors say, their method is based on the unpublished work by P. Ballone and S. Rubini, who used biased forces to study the crystallization process. The idea of applying very small perturbations for inducing desirable conformational changes, also related to the earlier work by Harvey and Gabb,146 is the following. The reaction coordinate (RC) has to be selected first. Sometimes it is very complex and difficult to determine or even guess. Usually we do not know how the studied process proceedes. Let us assume that the unfolding of a protein induced by a force attached to C-terminus of the protein backbone is to be studied, and RNc is the distance between the N-terminus and the C-terminus. For example, Paci and Karplus, in their study of the relation of the unfolding process to the topology of the protein,147 used as the reaction coordinate p(t) = R2NC • It is worth to mention,

62


that these authors used the implicit Gaussian model for the solvent. They claim that the adiabatic relaxation of the solvent is necessary for fast unfolding studies, and that such a simplified approach is closer to the real AFM experiments. Time dependent bias force is: Fb = 7(P -Pmax),

(4)

but only when p(t) < pn,ax(t), otherwise the force is set to zero. The parameter pmax (t) is the maximum value of p reached by the reaction coordinate at times less or equal to t. Thus, the force is nonzero only when thermal fluctuations tend to decrease the selected distance (RC). One can guess that if an energy barrier appears on the selected structural transformation, then the bias force is large, in the other regions its value is not high. The values of y parametres are adjusted by a trial and error procedure, depending on the process studied and assumed timescale, suitable to the forced structural transition under study. In this paper147 values of y around 0.0012 kcalmol'1 A"4 were used in biasing forces applied in order to unfold a range of P-sandwich and a-helical proteins. It is worth to note that 1 kcalmol 1 A"1 = 69.4786 pN. BMD has been used in several iluminating studies. Recently, Paci et al.148 have applied BMD (together with the constant force MD simulation) to reveal atomistic details of unbinding of fluorescein from hepten-antibody complex. The same system was simulated by Heynman and Grubmixller,'49 using the SMD method, so the two approaches may be compared. The biased MD technique is useful in generating molecular states, which are not easy amenable to experiment. For example, Paci et al.150 used this method to obtain nearly 400 clusters of partially unfolded states of human <xlactalbumin. A clever selection of the radius of gyration as a reaction coordinate, based on the idea of Marchi and Ballone,151 and using an implicit solvent made the effective generation and analysis of so many semi-stable protein conformations possible.

4. Free energy profiles in pulling experiments Relating single molecule measurements to thermodynamics is not an easy task, since AFM experiments are not performed on ensambles of molecules and the processes studied are not in equilibrium. Free energy profiles along selected reaction coordinate or potential of mean force (PMF) have a special importance at a single molecule level.152'153 In 1997 Balsera et al.154 presented a one-dimensional stochastic model for the reconstruction of PMF along the path of force induced conformational

Molecular Dynamics Simulations 63

transitions in a molecular system. The model, based on suggestions by Evans and Ritchie,155 is useful in the analysis of ligand unbinding. Dissipation is related to a friction coefficient. The accuracy of the potential energy function evaluated is determined from fluctuations which are proportional to the amount of irreversible work deposited into the system by external forces. In this paper uncertainties 8F in the calculated forces are estimated to be about 50 pN for a 10 ns simulation. Let us assume that the initial state / is in equilibrium with the heat bath at temperature T (unextended molecule). The parameter X, related to a controllable degree of freedom, describes a transition of the system from state / to state / (extended molecule). Changes in X result in expenditure of some amount of work W, changes of free energy of the system AG and the exchange of heat with the bath. In 1997 C. Jarzynski156 proved an equality that relates the irreversible work W, performed during /->/transition of the system, to the equilibrium free energy difference between states / a n d / , AG: <e-W/kT> =

e-AG/kT

( 5 )

This equation, further analysed by Jarzynski,157'158 is very important and became the basis of estimates of thermodynamic quantities in single molecule experiments. In 2001 Hummer and Szabo159 presented a simple recipe on how to reconstruct free energy surfaces from multiple nonequillibrium pulling experiments. Perhaps one of the first applications of this formalism to SMD computer experiments was the work done by Jensen et al.160, who studied conduction of glycerol through the membrane aquaglycerolporin channel. PMF calculated from Jarzynski's equality gives a reasonable understanding of molecular events in this highly selective system. The calculated highest energy barrier of 7.3 kcal/mol compares well with the experimental Arrhenius activation energy 9.6 ±1.5 kcal/mol.160 A similar methodology has been used by Amaro et al.161 in their research of ammonia transfer in the HisF enzymatic protein, which is important in histidyne synthesis. A total of 40 ns of the simulation time (many trajectories of 1.2 ns each) was used in order to collect reasonable statistics. Park et al.' 62 very recently applied Jarzynski's equality to free energy calculations from SMD of helix-coil transitions of deca-alanine. The limited number of 10 trajectories was generated and the second order cumulant expansion of rigorous formulas yielded the most accuarate estimates of free energy changes. The authors notice, that the average of the exponential work (Eq. 5) is dominated by trajectories corresponding to small W values, that only arise rarely. Practical applications are limited to slow processes, for which the fluctuations of W are comparable to temperature. A related paper by

64


Zuckerman and Woolf163 gives a new theory of errors inherent to nonlinear averages, such as those used in free energy calculations. It is worth to note that Jarzynski's equality has not been proven experimentally until recently.164 The work by Liphardt et al.164 that verifed Jarzynski's equality was named one of the top 10 scientific breakthroughts in 2003 by "Science" magazine. The authors tested Jarzynski's equality by performing irreversibly and reversibly mechanical stretching of a single RNA molecule between two conformations. Application of this equality to the irreversible work trajectories recovers the AG profile of the stretching process to within kT/2 of its best independent estimate, i.e. the mean work of reversible stretching. Other approaches relating thermodynamics to SMD were suggested in the literature. As we mentioned before, in 1999 Gullingsrud et al.100 proposed three time series analysis methods, aimed at the reconstruction of PMF from SMD data on forces and positions. One technique, based on WHAM, views stretching as a quasi-equilibrium process. It has been found not to be very promising in the interpretation of AFM experiments. The other two, one based on van Kampen's Q-expansion and the second on the least squares minimization of the Onsager—Machlup action with respect to the choice of PMF, assume a Langevin description of the dynamics in order to account for the non-equilibrium character of SMD data. A MD trajectory is equivalent to a particle moving in one-dimensional potential PEM W(z) which has to be found in the course of analysis from the equation: 7z

= F(z,t)-^^-

+ cr{(t).

(6)

oz Here y is a friction coefficient, calculated from the velocity autocorrelation function, F(z,t) is the applied external force, and o£(t) is the Gaussian white noise; the fluctuation-dissipation theorem gives a relation o2 =2ykT, where T is temperature and k is Boltzmann's constant. The methods were tested on artificial potentials and a phospholipid membrane monolayer system.100 In AFM experiments a cantilever tip picks up a protein molecule attached to the surface in a random fashion. Therefore some force spectrograms are difficult to interpret, since the mode of coupling is not known "a priori". The same difficulty is related to the problem of selecting RC in SMD or BMD simulations. In principle one may try all possible (and reasonable) models of performing the desired conformational transition, but this is not feasible. In certain studies, such as ligand dissociation from the site buried in an enzyme, the random expulsion molecular dynamics (REMD) scheme proposed by Wade et al. may help.165 The random force is acting on the center of mass of


65

the ligand. The direction of the force is kept constant for a chosen time interval, unless the ligand encounters rigid parts of the cavity. Then a new direction of the force is chosen. The substrate probes different regions of the protein until it finds the exit pathway. Ludemann et al.166 used the REMD method to determine the most probably exit path of camphor from the cytochrom P450cam active site. The points determined from REMD, located outside the protein, were then used to attach a force, which extracts the ligand from the protein. The magnitude of the SMD unbinding forces is reasonable (500-1000 pN) and compares well with the forces obtained in earlier studies of similar systems.83'135

5. The most recent results The progress in MD simulations of single molecule experiments and forced conformational transitions is very fast. In 2003 Brockwell et al.167 have shown that pulling geometry to a large extent determines forces observed during unfolding. In all-P protein, E21ip3, unolding forces differ by an order of magnitude, depending on the force direction chosen. Such an anisotropy may have physiological role. Lorenzo et al.168 calculated molecule-membrane interaction forces. To illustrate the principle of the method a small aminoacid alanine was modeled on a us timescale in an explicit solvent environment. Non-specific interactions at water-membrane interface were determined. Hummer and Szabo169 developed a simple procedure for extracting kinetic data from pulling experiments. This approach was tested on computer simulation data for titin. Special attention should be devoted to simplified mechanical models of proteins. For example, Cieplak et al.170171 used Go-like models to perform comparative forced unfolding of various secondary structures. Thermal and mechanical unfolding may be studied in that way. The sequence of events during the unfolding of titin is similar to those observed in fully atomistic simulations.171 However, availability of large thousand-processors clusters, make record breaking, huge SMD simulations possible. Brockman et al.172'173 have performed ns simulations (280 000 h CPU time) of dynamics of an extremely interesting system - a dual-domain molecular motor Fô-ATPase. This enzyme may work in two modes. The off-membrane Fi domain utilizes the transmembrane proton gradient to synthetize ATP from ADP and P;. Alternatively, it may hydrolize ATP and release energy. In both modes torsional motion of the central part of the enzyme is involved.172 It is known

66


that the isolated ATPase may work as a molecular nanomachine. Brockman et al. studied a sequence of conformational transitions in Y\ subunit induced by an external torque applied to the cental "y-stalk". It has been found that some parts of the enzyme ((3E subunit) behave as a prestretched "springs" during the enzymatic cycyle.173 Schulten at al.95 reported recently SMD simulations of both model Fo (112 K atoms) membrane and Fj (327 K atoms) off-membrane subunits of this ATPase. The principal goal of the work was to understand events that couple proton translocation across the membrane with the mechanical rotation of the Cio subunit oligomer, which is in the contact with the central y-stalk. This simulation was not able to predict all steps in the activity cycle of Fo, but helped to check numerous hypotheses on the mechanism of the ATPase activity. The TMD method was used by Ma et al.174'175 to generate trajectories for the bovine mitochondrial Fl-ATPase, related to elementary 120 degrees rotations of the central y subunit. This part has a special "ionic track" of Arg and Lys residues that plays a role in the guiding motions of |3 subunits. Excited state dynamics opens new challenges and opportunities for computer simulations.176 The first application of ab inito MD to isomerisation phenomena in bacteriorodopsin was done by Hayasi et al.177 According to the simulations, the protein plays a crutial role in the kinetics and selectivity of the photoisomerisation of the retinal. It discriminates from many photoisomerization paths the one which initiaties the process of light signal transduction. We stress the future role of the modeling of excited states, since optomechanical experiments on single molecules have been already performed in H.Gaub's laboratory.178'179 A peptide with multiple photoactive azobenzene groups incorporated in the backbone was synthetized. Cis to trans-azo switching was induced by UV irradiation and the changes in single molecule length were registered by the AFM technique. The molecule was shortened by light and at the same time it was able to perform some mechanical work. Thus, optomechanical single molecule energy conversion was demonstrated.179 It seems that performing SMD simulations of this very interesting system is only a matter of time.

6. MD results for stretching of polysaccharides Polysaccharides are not so popular in biophysical or chemical studies as proteins or nucleic acid, albeit their physiological role is enormous.180


67

Mechanical properties of simple polysaccharides were studied by AFM single molecule spectroscopy. In 1997 Rief et al,181 studied dextran linked to a gold surface and tried to explain the force-extension curves using MD simulations for 5 ring model systems and CHARMM force field. Li et al.182 measured by AFM the difference in a-(l,4) and P-(l,4) linked polysaccharides (amylose vs. cellulose). Heymann and Grubmuller183 performed MD simulations of stretching 16-mer models of these two polysaccharides, using EGO code and constant velocity pulling regime both in vacuo and in TIP3P water. In order to correctly reproduce the experimental plateau on the forceextention curves of amylose, these authors had to introduce artificial conformational constraints to set up the anti-parallel orientation of individual a-(l,4)-glucose units. In 1998 Marszaiek et al.24 demonstrated that polysaccharide elasticity is governed by chair-boat transitions of gluocopyranose rings. AFM experiments have shown that oxidation of the ring nearly eliminates the enthalpic component of the elasticity of dextran and amylose. The same group put forward a hypothesis that stretching of the pyranose ring may serve as a part of molecular signaling: it is pleausible that the fragment of pectin, namely a-(l-4)-D-galactouronan, acts as a two state elastic switch.25 Quantum chemical optimization of geometries of standard 4 Ci, boat and 'C 4 conformations of galatouronic acid, performed on the B3LYP/6-31* level, supported this hypothesis. The AFM force spectroscopy technique may be used for analytical purposes. In 2001 Marszaiek et al.34 have shown that individual molecules of floridean starch may be effectively identified as contaminats in the solutions of agarose or ^.-carrageenan due to distinct molecular fingerprints of these polysaccharide species. Chair-boat transitions in a single polysaccharide molecule may be observed in force-ramp AFM experiments.184 In this regime a pulling force is linearly increasing and sudden geometrical changes of the pulled molecule may be easily identified. An analytical two-state model developed by Marszaiek et al.184 allows for the quantitative characterization of fundamental ring transitions. Extensive quantum mechanical studies of barriers to forced transitions in sugar rings were performed by O'Donoghue and Luthey Schulten.185 The HF calculations show that the twist-boat conformation of aD-glucose is the most probable stable minimum in a stretched polysaccharide and the barrier for the chair to boat transition in a-D-glucose ranges from 14 to 15 kcal/mol. Recently, results of single molecule AFM measurements of elasticity of heparin have been published.186 Elastic properties of polysaccharides with 1-6 links are more complex than those with 1-4 links. Recenly we have found by stretching single pusrulan molecules in the AFM instrument that P-l—>6 linkages produce an unusual hookean-like elasticity.187 Pusrulan [a P~(l—>6)-linked glucan, (see Fig. 4)] is

68


similar to dextran [an oc-(l—>6)-linked glucan] but equatorial position of glycosidic bonds (CpOi) in pustulan, and the axial one in dextran, lead to clear differences in mechanical properties of a and P-(l-6)-glicosidic linkages. In contrast to cellulose whose 1-4 links allow for fairly free rotations of the sugar rings in the polymer chain pustulan experiences rotational restrictions during the stretch, and therefore its force-extension relationships strongly deviate from the simple freely rotating chain-like relationship for cellulose. Additional work is used to rotate the aglycone bond (O6-C6) about the C6-C5 bond, because this rotation contributes to the extension of pustulan but is absent in cellulose. The elasticity of dextran is different from both cellulose and pustulan and suggests that its oc-(l—^-linkages induce not only the rotation about the C6-C5 bond but also trigger various ring instabilities. We applied SMD simulations to investigate the deformations of the pyranose ring induced by forces attached to the ring at carbon atom d and through a rotatable C6-C5 bond. Using NAMD2 code188 running on a linux cluster in parallel mode we stretched pustulan and dextran decamers immersed in water boxes of such a size that at least a 6 A layer of TIP3P water molecules was present at each step of the simulations. After a 1-5 ns equilibration at 300K (dielectric constant e = 1) the structures were stretched by applying force (spring constant of 68 pN/nm) to the Oi atom of the last sugar and moving this spring at a constant velocity of 10"4 A /fs. To check the effect of the pulling speed on the results a control 200 ns simulation in vacuum with e = 80 were also performed Selected results187 are presented in Fig. 4. A comparison of AFM data and the detailed atomistic picture emerging from computer modeling, indicates that one can control the orientation of an atomic "crank": the O6-C6 bond and reorient it mechanically from the gauche-trans (gt) or gauche-gauche (gg) position to the less favourable energetically but more extended trans-gauche (tg) position. These rotations produce a hookean-like elasticity of pustulan but do not affect the chair conformation of the pyranose ring. However, deforming the pyranose ring by oc-1—>6 linkages, using the O6-C6 cranks and axial C r O | levers, such as those in dextran, not only reorients the O6-C6 bond to the tg position but also produces ring instabilities, which flip the pyranose structure to a boat-like conformation. Studies of polysaccharide elasticity and mechanical properites of organic

rings expand single molecule mechanochemistry and are important for future nanotechnological applications.

Molecular Dynamics Simulations 69

4600 .

3500^ R > ^ ^ \ - ^ ^ ^ ^

2500 •

Ug-75*

R7

£ 0,-(yA 5.42 O 1600 •

< ^

i

1 tg-!74» 6.12

jmrnsf

^

j l l TT

1000 •

^ ^ ^

500 •

,^"^1^^

-500 f . . . . . . . . . i . . . . T . . . . i . . . . . . . . . . .

3.5

4.0

B :::r^

4.5

i,.,..,,,.!.........

5.0

5.5

6.0

6.5

._^^*"111" ;

o H -12O -; 5 -iao J

A- ^ m iTirtfli- •

2

t3 ISO -I Ptms r .•S 1 2 O :

j

'MHitPIIH^

° oo: o -

-18O 3.S

-^BSi*Wfifli* **.O

>*.S

5.O

S.3

©.O

6.S

Normalized Extension per Ring/A

Fig. 4 (A) Results of AFM measurements and SMD simulations for pustulan, a p(I—^-linked glucan, dark gray trace - 5 ns in explicit water, gray trace - 200 ns in vacuum e = 80, black curve - experimental AFM data. (B) The torsional angle (black dots) changes indicate a gt —> tg flip of pustulan rings in the region exhibiting "hookean" elasticity. Constant values of tj and t2 torsional angles (horizontal lines in the middle) show that the chair to boat transition do not occur during the stretching of pustulan.

70


1. Quantum molecular dynamics of amylose stretching Quality data may be obtained from forced MD simulations of single molecules only when the force field (FF) is not flawed. The results of modeling are force field dependent. The development of FFs was centered around equilibrium benchmark structures so the quality of the potentials applied to molecules stretched far from energy minima is not necessarely the best. Thus, there is high demand for a fully quantum-mechanical, forcefield free, calculation for streteched biopolymers. In a recent paper189, which is a result of a collaborative effort among Yang, Marszalek, Nowak and their students, Lu et al189 have presented a practical solution to this long lasting problem. We used for the first time a quantum mechanics based methodology to investigate how a polysaccharide immersed in water, responds to a stretching force. The amylose fragment composed of ten sugar rings was placed in a TIP3P water box (30 A x 30 A x 80 A; 8004 atoms) and was treated by the self consistent charge density functional theory-based tight-binding method (SCC-DFTB).190 Explicit long range van der Waals forces were used. The total energy of the system including the interactions between amylose and water was calculated with the divide-and-conquer linear-scaling QM/MM model developed by the Yang group191 Our tests performed for amylose dimers indicate a very high similarity of potential energy surfaces calculated by the SCC-DFTB method and the DFT B3LYP method (6-31 lg++** basis set was used). The results of the amylose stretching are presented in Fig. 5.

Molecular Dynamics Simulations OH

1 0,70

,

1

0.75

,

1 0.80

,

! 0.85

71

OH

,

1 0.90

,

, 0.95

,

1 100

1

1 -60 1.05

Normalized extension (x/IJ Fig. 5 Results of SMD simulations of the amylose stretching. Black trace: the experimental force-extension relationship obtained by AFM. Dark gray trace: the force-extension relationship obtained by QM/SMD simulations. Light gray trace: the O1-C1-C2-O2 dihedral angle averaged over the glucose units. The extension, x, was normalized by the molecule length, lc, determined at the force of 1620 pN.

The QM/SMD simulations represent the mechanical properties of amylose in water remarkably well. The force-induced chair to boat transitions (represented by the dihedral angle shown in Fig. 5) coincide with the plateau region of the experimental force-extension curve, as detailed analysis of trajectories shows. The increasing tension forces the neighboring glucose rings to rotate around the O1-C4' bonds. The rings orient themselves in a parallel fashion, even though they were initially in an anti-parallel orientation. The elasticity of amylose to a large extent depends on a complex pattern of the breakage/creation of hydrogen bonds. Without a proper solvatation it is not possible to reproduce the value of the plateau force. These calculations,189 though quite time-consuming, bring a promise for doing realistic simulations of nanomechanical interactions in any biologically important glycoproteins on a routine basis, without the need of tedious optimization of force field parameters.

72


Concluding remarks The new methods and protocols of performing MD simulations - TMD, SMD, BMD - greatly help to interpret single molecule experiments. The progress is very fast and trends are clear: forced transitions may bridge mechanics of molecules with chemistry. Better simulation protocols and better hardware will reduce the gap in timescales between SMD or BMD simulations (ns) and AFM experiments (ms). The findings of computational efforts, sometimes unexpected, lead to new ideas, sugest new mutagenesis work and new experimental tests. Piconewton forces observed in AFM or optical tweezer setups are ultimate mechanical signals sent up from the nanoworld to keen observers who are now well prepared to receive and interpret them.

Acknowledgements This work was supported by the NSF (WN and PEM), KBN and GRUMK (WN).

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Robertson, N. & McGowan, C.A. A comparison of potential molecular wires as components for molecular electronics. Chem. Soc. Rev. 32, 96-103 (2003). Plakhotnik, T., Donley, E.A. & Wild, U.P. Single-molecule spectroscopy. Ann. Rev. Phys. Chem. 48, 181-212 (1997). Kuhnemuth, R. & Seidel, C.A.M. Principles of single molecule multiparameter fluorescence spectroscopy. Single Molecules 2, 251-254 (2001). Orrit, M. Single-molecule spectroscopy: The road ahead. J. Chem. Phys. 117, 10938-10946(2002). Florin, E.L., Moy, V.T. & Gaub, H.E. Adhesion Forces between Individual Ligand-Receptor Pairs. Science 264, 415-417 (1994). Medina, M.A. & Schwille, P. Fluorescence correlation spectroscopy for the detection and study of single molecules in Biology. Bioessays 24, 758-764 (2002). Bohmer, M. & Enderlein, J. Fluorescence spectroscopy of single molecules under ambient conditions: Methodology and technology. Chemphyschem 4, 793-808 (2003). Haran, G. Single-molecule fluorescence spectroscopy of biomolecular folding. J. Phys.-Cond. Mat. 15, R1291-R1317 (2003). Forkey, J.N., Quinlan, M.E., Shaw, M.A., Come, J.E.T. & Goldman, Y.E. Threedimensional structural dynamics of myosin V by single-molecule fluorescence polarization. Nature 422, 399-404 (2003).


73

10. Mehta, A.D., Rief, M., Spudich, J.A., Smith, D.A. & Simmons, R.M. Singlemolecule biomechanics with optical methods. Science 283, 1689-1695 (1999). 11. Molloy, J.E. & Padgett, M.J. Lights, action: optical tweezers. Contemporary Physics 43, 241-258 (2002). 12. Coirault, C , Pourny, J.C., Lambert, F. & Lecarpentier, Y. Optical tweezers in biology and medicine. M S-Medecine Sciences 19, 364-367 (2003). 13. Schwille, P. & Kettling, U. Analyzing single protein molecules using optical methods. Curr. Opin. Biotech. 12, 382-386 (2001). 14. Finer, J.T., Simmons, R.M. & Spudich, J.A. Single Myosin Molecule Mechanics Piconewton Forces and Nanometer Steps. Nature 368, 113-119 (1994). 15. Purohit, P.K., Kondev, J. & Phillips, R. Mechanics of DNA packaging in viruses. Proc. Natl. Acad. Sci. 100, 3173-3178 (2003). 16. Bustamante, C , Smith, S.B., Liphardt, J. & Smith, D. Single-molecule studies of DNA mechanics. Curr. Opin. Struct. Biol. 10, 279-285 (2000). 17. Smith, S.B., Finzi, L. & Bustamante, C. Direct Mechanical Measurements of the Elasticity of Single DNA-Molecules by Using Magnetic Beads. Science 258, 1122-1126(1992). 18. Zlatanova, J. & Leuba, S.H. Magnetic tweezers: a sensitive tool to study DNA and chromatin at the single-molecule level. Biochemistry and Cell Biology-Biochimie Et Biologie Cellulaire 81,151-159 (2003). 19. Romano, G., Sacconi, L., Capitanio, M. & Pavone, F.S. Force and torque measurements using magnetic micro beads for single molecule biophysics. Optics Comm. 215, 323-331 (2003). 20. Basche, T., Nie, S. & Fernandez, J.M. Single molecules. Proc. Natl. Acad. Sci. USA 98, 10527-10528 (2001). 21. Marszalek, P.E., Lu, H., Li, H.B., Carrion-Vazquez, M., Oberhauser, A.F., Schulten, K. & Fernandez, J.M. Mechanical unfolding intermediates in titin modules. Nature 402, 100-103 (1999). 22. Li, H.B., Linke, W.A., Oberhauser, A.F., Carrion-Vazquez, M., Kerkviliet, J.G., Lu, H., Marszalek, P.E. & Fernandez, J.M. Reverse engineering of the giant muscle protein titin. Nature 418, 998-1002 (2002). 23. Oberhauser, A.F., Badilla-Fernandez, C , Carrion-Vazquez, M. & Fernandez, J.M. The mechanical hierarchies of fibronectin observed with single-molecule AFM. J. Mol. Biol. 319, 433-447 (2002). 24. Marszalek, P.E., Oberhauser, A.F., Pang, Y.P. & Fernandez, J.M. Polysaccharide elasticity governed by chair-boat transitions of the glucopyranose ring. Nature 396,661-664(1998). 25. Marszalek, P.E., Pang, Y.P., Li, H.B., El Yazal, J., Oberhauser, A.F. & Fernandez, J.M. Atomic levers control pyranose ring conformations. Proc. Natl. Acad. Sci. (USA) 96, 7894-7898 (1999). 26. Fisher, T.E., Marszalek, P.E. & Fernandez, J.M. Stretching single molecules into novel conformations using the atomic force microscope. Nat. Struct. Biol. 7, 719724 (2000). 27. Zhang, L , Wang, C , Cui, S.X., Wang, Z.Q. & Zhang, X. Single-molecule force spectroscopy on curdlan: Unwinding helical structures and random coils. Nano Letters 3, 1119-1124(2003).

74


28. Onoa, B., Dumont, S., Liphardt, J., Smith, S.B., Tinoco, I. & Bustamante, C. Identifying kinetic barriers to mechanical unfolding of the T-thermophila ribozyme. Science 299, 1892-1895 (2003). 29. Hansma, H.G., Pietrasanta, L.I., Auerbach, I.D., Sorenson, C , Golan, R. & Holden, P.A. Probing biopolymers with the atomic force microscope: A review. J. Biomaterials Sci.-Polymer Ed. 11, 675-683(2000). 30. Jandt, K.D. Atomic force microscopy of biomaterials surfaces and interfaces. Surf. Sci. 491, 303-332 (2001). 31. Demers, L.M., D. S. Ginger, S.-J. Park, Z. Li, S.-W. Chung & Mirkin, C.A. Direct Patterning of Modified Oligonucleotides on Metals and Insulators by Dip-Pen Nanolithography. Science 296, 1836-1838 (2002). 32. Durig, U., Cross, G., Despont, M., Drechsler, U., Haberle, W., Lutwyche, M.I., Rothuizen, H., Stutz, R., Widmer, R., Vettiger, P. "Millipede" - an AFM data storage system at the frontier of nanotribology. Tribology Letters 9, 25-32 (2000). 33. Ludwig, M., Rief, M., Schmidt, L., Li, H., Oesterhelt, F., Gautel, M. & Gaub, H.E. AFM, a tool for single-molecule experiments. Appl. Phys. - Mat. Sci. & Proc. 68, 173-176 (1999). 34. Marszalek, P.E., Li, H.B. & Fernandez, J.M. Fingerprinting polysaccharides with single-molecule atomic force microscopy. Nat. Biotech. 19, 258-262 (2001). 35. Bustamante, C , Bryant, Z. & Smith, S.B. Ten years of tension: single-molecule DNA mechanics. Nature 421, 423-427 (2003). 36. Strick, T.R. et al. Stretching of macromolecules and proteins. Rep. Prog. Phys. 66, 1-45 (2003). 37. Clausen-Schaumann, H., Seitz, M., Krautbauer, R. & Gaub, H.E. Force spectroscopy with single bio-molecules. Curr. Opin. Chem. Biol. 4, 524-530 (2000). 38. Engel, A., Gaub, H.E. & Muller, D.J. Atomic force microscopy: A forceful way with single molecules. Curr. Biol. 9, R133-R136 (1999). 39. Zhuang, X.W. & Rief, M. Single-molecule folding. Curr. Opin. Struct. Biol. 13, 88-97 (2003). 40. Rief, M. & Grubmuller, H. Force spectroscopy of single biomolecules. Chemphyschem 3, 255-261 (2002). 41. Leckband, D. Measuring the forces that control protein interactions. Ann. Rev. Biophys. Biomol. Struct. 29, 1-26(2000). 42. Riener, C.K., Stroh, CM., Ebner, A., Klampfl, C , Gall, A.A., Romanin, C , Lyubchenko, Y.L., Hinterdorfer, P. & Gruber, H.J. Simple test system for single molecule recognition force microscopy. Analytica Chimica Ada 479, 59-75 (2003). 43. Best, R.B., Brockwell, D.J., Toca-Herrera, J.L., Blake, A.W., Smith, D.A., Radford, S.E. & Clarke, J. Force mode atomic force microscopy as a tool for protein folding studies. Anal. Chim. Ada 479, 87-105 (2003). 44. Ikai, A., Afrin, R., Sekiguchi, H., Okajima, T., Alam, M.T. & Nishida, S. Nanomechanical methods in biochemistry using atomic force microscopy. Curr. Prot. & Pept. Sci. 4, 181-193 (2003). 45. Williams, P.M. Analytical descriptions of dynamic force spectroscopy: behaviour of multiple connections. Analyt. Chim. Ada 479, 107-115(2003).


75

46. Haile, M. Molecular Dynamics Simulation: Elementary Methods, (John Wiley, 1992). 47. Frankel, D. & Smit, B. Understanding Molecular Simulation, (Academic Press, 2001). 48. Field, M.J. A Practical Introduction to the Simulation of Molecular Systems, (Cambridge University Press, 1999). 49. Schlick, T. Molecular Modeling and Simulation - An Interdisciplinary Guide., (Springer, New York, 2002). 50. Kholmurodov, K.T., Altaisky, M.V., Puzynin, I.V., Darden, T. & Filatov, F.P. Methods of molecular dynamics for simulation of physical and biological processes. Physics of Particles and Nuclei 34, 244-263 (2003). 51. Werner Ebeling, L.S.-G., Yuri M. Romanovsky. Stochastic dynamics of reacting biomolecules, (World Scientific, Singapore, 2003). 52. Karplus, M. Molecular dynamics of biological macromolecules: A brief history and perspective. Biopolymers 68, 350-358 (2003). 53. Hansson T, O.C., van Gunsteren W. Molecular dynamics simulations. Curr. Opin. Struct. Biol. 12, 190-196(2002). 54. Marx, D. & Hutter, J. Ab initio molecular dynamics: Theory and Implementation, in Modern Methods and Algorithms of Quantum Chemistry, Vol. 1 (ed. Grotendorst, J.) 301-449 (John von Neumann Institute for Computing, Julich, 2000). 55. Carloni, P., Rothlisberger, U. & Parrinello, M. The role and perspective of a initio molecular dynamics in the study of biological systems. Ace. Chem. Res. 35, 455464 (2002). 56. Karplus, M. & McCammon, J.A. Molecular dynamics simulations of biomolecules. Nat. Struct. Biol. 9, 646-652 (2002). 57. Moraitakis, G., Purkiss A. G, Goodfellow J. M. Simulated dynamics and biological molecules. Rep. Prog. Phys. 66, 483-406 (2003). 58. Zhou, Y.Q. & Karplus, M. Interpreting the folding kinetics of helical proteins. Nature 401, 400-403 (1999). 59. Lazaridis, T. & Karplus, M. Thermodynamics of protein folding: a microscopic view. Biophys. Chem. 100, 367-395 (2003). 60. Vendruscolo, M. & Paci, E. Protein folding: bringing theory and experiment closer together. Curr. Opin. Struct. Biol. 13, 82-87 (2003). 61. Mayor U., Guydosh, N.R., Johnson, CM., Grossmann, J.G, Sato S., Jas G.S., Freund S.M., Alonso D.O., Daggett V., Fersht A.R. The complete folding pathway of a protein from nanoseconds to microseconds. Nature 421, 863-867 (2003). 62. Fersht, A.R. & Daggett, V. Protein folding and unfolding at atomic resolution. Cell 108, 573-582(2002). 63. Daggett, V. & Fersht, A.R. Is there a unifying mechanism for protein folding? Trends inBiochem. Sci. 28, 18-25 (2003). 64. Daggett, V. Molecular dynamics simulations of the protein unfolding/folding reaction. Ace. Chem. Res. 35, 422-429 (2002). 65. Warshel, A. & Parson, W.W. Dynamics of biochemical and biophysical reactions: insight from computer simulations. Quart. Rev. Biophys. 34, 563-679 (2001). 66. Warshel, A. Molecular dynamics simulations of biological reactions. Ace. Chem. Res. 35, 385-395 (2002).

76


67. Warshel, A. Computer simulations of enzyme catalysis: Methods, progress, and insights. Ann. Rev. Biophys. Biomol. Struct. 32, 425-443 (2003). 68. Karplus, M. Aspects of protein reaction dynamics: Deviations from simple behavior. J. Phys. Chem. B 104, 11-27 (2000). 69. van Speybroeck, V. & Meier, R.J. A recent development in computational chemistry: chemical reactions from first principles molecular dynamics simulations. Chem. Soc. Rev. 32,151-7 (2003). 70. Bala, P., Grochowski, P., Nowinski, K., Lesyng, B. & McCammon, J.A. Quantum-dynamical picture of a multistep enzymatic process: Reaction catalyzed by phospholipase A(2). Biophys. J. 79, 1253-1262 (2000). 71. Simonson, T., Archontis, G. & Karplus, M. Free energy simulations come of age: Protein-ligand recognition. Ace. Chem. Res. 35,430-437 (2002). 72. McCammon, A.J. & Wolynes, P.G. Theory and simulation - Enlarging the landscape - Editorial overview. Curr. Opin. Struct. Biol. 12, 143-145 (2002). 73. Pande V.S., Baker, I., Chapman, J., Elmer, S.P., Khaliq, S., Larson, S.M., Rhee, Y.M., Shirts, M.R., Snow, C D . , Sorin, E,J., Zagrovic, B. Atomistic protein folding simulations on the submillisecond time scale using worldwide distributed computing. Biopolymers 68, 91-109 (2003). 74. Daggett, V. Long timescale simulations. Curr. Opin. Struct. Biol. 10, 160-164 (2000). 75. Straub, J.E., Guevara, J., Huo, S.H. & Lee, J.P. Long time dynamic simulations: Exploring the folding pathways of an Alzheimer's amyloid A beta-peptide. Ace. Chem. Res. 35, 473-481 (2002). 76. Elber, R. Novel methods for molecular dynamics simulations. Curr. Opin. Struct. Biol. 6,232-235(1996). 77. Paci, E. High pressure simulations of biomolecules. BBA-Protein Struct, and Mol. Enz. 1595, 185-200(2002). 78. Wong, C.F. & McCammon, J.A. Protein flexibility and computer-aided drug design. Ann. Rev. Pharm. Toxicol. 43, 31-45 (2003). 79. Sulpizi, M., Folkers, G., Rothlisberger, U., Carloni, P. & Scapozza, L. Applications of density functional theory-based methods in medicinal chemistry. Quant. Struct.-Act. Relat. 21, 173-181 (2002). 80. Banci, L. Molecular dynamics simulations of metalloproteins. Curr. Opin. Chem. Biol. 7, 524 (2003). 81. Giorgetti, A. & Carloni, P. Molecular modeling of ion channels: structural predictions. Curr. Opin. Chem. Biol. 7, 150-156(2003). 82. Dwyer, D.S. Molecular modeling and molecular dynamics simulations of membrane transporter proteins. Methods Mol. Biol. 227, 335-50 (2003). 83. Grubmuller, H., Heymann, B. & Tavan, P. Ligand binding: Molecular mechanics calculation of the streptavidin biotin rupture force. Science 271, 997-999 (1996). 84. Elber, R., Karplus, M. A method for determining reaction paths in large molecules: Application to myoglobin. Chem. Phys. Letters 139, 375-381 (1987). 85. Elber, R. Reaction Path Studies of Biological Molecules, in Recent developments in theoretical studies of proteins., Vol. 7 (ed. Elber, R.) (World Scientific, Singapore, 1996). 86. Elber, R., Ghosh, A. & Cardenas, A. Long time dynamics of complex systems. Ace. Chem. Res. 35, 396-403 (2002).


77

87. Schlitter, J., M. Engels, P. Kruger, E. Jacoby & A. Wollmer. Targeted molecular dynamics simulation of conformational change - application to the T to R transition in insulin. Mol. Sim. 10, 291-308 (1993). 88. Leech, J., Prins, J.F. & Hermans, J. SMD: Visual Steering of Molecular Dynamics for Protein Design. IEEE Comp. Sci. Eng. 3, 38-45 (1996). 89. Izrailev, S., Stepaniants, S., Balsera, M., Oono, Y. & Schulten, K. Molecular dynamics study of unbinding of the avidin-biotin complex. Biophys. J. 72, 15681581 (1997). 90. Paci, E. & Karplus, M. Forced unfolding of fibronectin type 3 modules: An analysis by biased molecular dynamics simulations. J. Mol. Biol. 288, 441-459 (1999). 91. Galligan, E., Roberts, C.J., Davies, M.C., Tendler, S.J.B. & Williams, P.M. Simulating the dynamic strength of molecular interactions. J. Chem. Phys. 114, 3208-3214 (2001). 92. Torrie, G.M. & Valleau, J.P. Nonphysical sampling distribution in Monte Carlo free energy estimation: umbrella sampling. J. Comput. Phys. 23, 187-199 (1977). 93. Izrailev, S., Stepaniants, S., Isralewitz, B., Kosztin, D., Lu, H., Molnar, F., Wriggers, W. & Schulten, K... Steered molecular dynamics, in Algorithms for Macromolecular Modelling (eds. Deuflhard, P. et al.) (Springer Verlag, New York, 1998). 94. Isralewitz, B., Baudry, J., Gullingsrud, J., Kosztin, D. & Schulten, K. Steered molecular dynamics investigations of protein function. J. Molec. Graph. Model. 19, 13-25(2001). 95. Tajkhorshid, E., A. Aksimentiev, I. Balabin, M. Gao, B. Isralewitz, J. C. Phillips, F. Zhu & Schulten, K. Advances in Protein Chemistry, in Protein Simulations, Vol. 66 (eds. Eisenberg, D. & Kim, P.) (Elsevier, 2003). 96. Izrailev, S., Crofts, A.R., Berry, E.A. & Schulten, K. Steered molecular dynamics simulation of the rieske subunit motion in the cytochrome bc(l) complex. Biophys. J. 77, 1753-1768 (1999). 97. Prins, J.F., Hermans, J., Mann, G., Nyland, L.S. & Simons, M. A virtual environment for steered molecular dynamics. Future Generation Computer Systems 15, 485-495 (1999). 98. Stone, J., Gullingsrud, J., Grayson, P. & Schulten, K. The 2001 ACM Symposium on Interactive 3D Graphics: A system for interactive molecular dynamics simulation, pp 191-194 (ACM SIGGRAPH, New York, 2001). 99. Grayson, P., Tajkhorshid, E. & Schulten, K. Mechanisms of selectivity in channels and enzymes studied with interactive molecular dynamics. Biophys. J. 85, 36-48 (2003). 100.Gullingsrud, J.R., Braun, R. & Schulten, K. Reconstructing potentials of mean force through time series analysis of steered molecular dynamics simulations. J. Comp. Phys. 151, 190-211 (1999). 101.Stepaniants, S., Izrailev, S. & Schulten, K. Extraction of lipids from phospholipid membranes by steered molecular dynamics. J. Mol. Model. 3, 473-475 (1997). 102.Rief, M., Gautel, M., Oesterhelt, F., Fernandez, J.M. & Gaub, H.E. Reversible unfolding of individual titin immunoglobulin domains by AFM. Science 276, 1109-1112(1997).

78


103.Card on-Vazquez, M., Oberhauser, A.F., Fowler, S.B., Marszalek, P.E., Broedel, S.E., Clarke, J. & Fernandez, J.M. Mechanical and chemical unfolding of a single protein: A comparison. Proc. Natl. Acad. Sci. (USA) 96, 3694-3699 (1999). 104. Carrion-Vazquez, M., Oberhauser, A.F., Fisher, T.E., Marszalek, P.E., Li, H.B. & Fernandez, J.M. Mechanical design of proteins-studied by single-molecule force spectroscopy and protein engineering. Prog. Biophys. & Mol. Biol. 74, 63-91 (2000). 105.Li, H.B., Carrion-Vazquez, M., Oberhauser, A.F., Marszalek, P.E. & Fernandez, J.M. Point mutations alter the mechanical stability of immunoglobulin modules. Nat. Struct. Biol. 7, 1117-1120 (2000). 106.Lu, H., Isralewitz, B., Krammer, A., Vogel, V. & Schulten, K. Unfolding of titin immunoglobulin domains by steered molecular dynamics simulation. Biophys. J. 75,662-671(1998). 107.Lu, H. & Schulten, K.. Steered molecular dynamics simulation of conformational changes of immunoglobulin domain 127 interpret atomic force microscopy observations. Chem. Phys. 247, 141-153 (1999). 108.Lu, H. & Schulten, K. The key event in force-induced unfolding of titin's immunoglobulin domains. Biophys. J. 79, 51-65 (2000). 109.Lu, H., Krammer, A., Isralewitz, B., Vogel, V. & Schulten, K. Computer modeling of force-induced titin domain unfolding, in Elastic Filaments of the Cell, Vol.481 143-162(2000). 110.Isralewitz, B., Gao, M., Lu, H. & Schulten, K. Forced unfolding or mutant titin immunoglobulin domains with steered molecular dynamics. Biophys. J. 78, 28A28A (2000). 111.Gao, M., Lu, H. & Schulten, K. Simulated refolding of stretched titin immunoglobulin domains. Biophys. J. 81, 2268-2277 (2001). 112.Gao, M., Lu, H. & Schulten, K. Unfolding of titin domains studied by molecular dynamics simulations. J. Muscle Res. Cell. Motil. 23, 513-21 (2002). 113.Gao, M., Wilmanns, M. & Schulten, K. Steered molecular dynamics studies of titin II domain unfolding. Biophys. J. 83, 3435-3445 (2002). 114.Li, H. & Fernandez, J.M. Mechanical Design of the First Proximal Ig Domain of Human Cardiac Titin Revealed by Single Molecule Force Spectroscopy. J. Mol. Biol. 334, 75-86 (2003). 115.Zhang, B. & Evans, J.S. Modeling AFM-Induced PEVK Extension and the Reversible Unfolding of Ig/FNIII Domains in Single and Multiple Titin Molecules. Biophys. J. 80, 597-605 (2001). 116.Best, R.B., Fowler, S.B., Toca-Herrera, J.L. & Clarke, J. A simple method for probing the mechanical unfolding pathway of proteins in detail. Proc. Natl. Acad. Sci. (USA) 99, 12143-12148 (2002). 117.Isralewitz, B., Gao, M. & Schulten, K. Steered molecular dynamics and mechanical functions of proteins. Curr. Opin. Struct. Biol. 11, 224-230 (2001). 118.Best, R.B., Fowler, S.B., Herrera, J.L., Steward, A., Paci, E. & Clarke, J. Mechanical unfolding of a titin Ig domain: structure of transition state revealed by combining atomic force microscopy, protein engineering and molecular dynamics simulations. J. Mol. Biol. 330, 867-77 (2003). 119.Fowler SB, Best RB, Toca Herrera JL, Rutherford TJ, Steward A, Paci E, Karplus M, Clarke J. Mechanical unfolding of a titin Ig domain: Structure of unfolding


79

intermediate revealed by combining AFM, molecular dynamics simulations, NMR and protein engineering. J. Mol. Biol. 322, 841-849 (2002). 12O.Schwaiger, I., Kardinal, A., Schleicher, M., Noegel, A.A. & Rief, M. A mechanical unfolding intermediate in an actin-crosslinking protein. Nat. Struct. Mol. Biol. 11, 81-85 (2004). 121.Kellermayer, M.S.Z., Bustamante, C. & Granzier, H.L. Mechanics and structure of titin oligomers explored with atomic force microscopy. BBA-Bioenergetics 1604,105-114(2003). 122.Gao, M., Craig, D., Vogel, V. & Schulten, K. Identifying unfolding intermediates of FN-III10 by steered molecular dynamics. J. Mol. Biol. 323, 939-950 (2002). 123.Rief, M., Gautel, M , Schemmel, A. & Gaub, H.E. The mechanical stability of immunoglobulin and fibronectin III domains in the muscle protein titin measured by atomic force microscopy. Biophys. J. 75, 3008-3014 (1998). 124.0berhauser, A.F., Marszalek, P.E., Erickson, H.P. & Fernandez, J.M. The molecular elasticity of the extracellular matrix protein tenascin. Nature 393, 181185(1998). 125.Krammer, A., Lu, H., Isralewitz, B., Schulten, K. & Vogel, V. Forced unfolding of the fibronectin type III module reveals a tensile molecular recognition switch. Proc. Natl. Acad. Sci. (USA) 96, 1351-1356 (1999). 126.Craig, D., Krammer, A., Schulten, K. & Vogel, V. Comparison of the early stages of forced unfolding for fibronectin type III modules. Proc. Natl. Acad Sci. (USA) 98, 5590-5595 (2001). 127.Li, P.C. & Makarov, D.E. Theoretical studies of the mechanical unfolding of the muscle protein titin: Bridging the time-scale gap between simulation and experiment. J. Chem. Phys. 119, 9260-9268 (2003). 128.Li, B., Alonso, D.O.V., Bennion, B.J. & Daggett, V. Hydrophobic hydration is an important source of elasticity in elastin-based biopolymers. J. Am. Chem. Soc. 123, 11991-11998(2001). 129.Li, B., Alonso, D.O.V. & Daggett, V. The molecular basis for the inverse temperature transition of elastin. J. Mol. Biol. 305, 581-592 (2001). 130.Li, B., Alonso, D.O.V. & Daggett, V. Stabilization of globular proteins via introduction of temperature-activated elastin-based switches. Structure 10, 989998 (2002). 131.Li, B. & Daggett, V. Molecular basis for the extensibility of elastin. J. Muscle Res. CellMotil. 23, 561-573 (2002). 132.Li, B. & Daggett, V. The molecular basis of the temperature- and pH-induced conformational transitions in elastin-based peptides. Biopolymers 68, 121-129 (2003). 133.Altmann, S.M., Grunberg, R.G., Lenne, P.F., Ylanne, J., Raae, A., Herbert, K., Saraste, M., Nilges, M. & Horber, J.K.H Pathways and intermediates in forced unfolding of spectrin repeats. Structure 10, 1085-1096 (2002). 134.Verkhivker, G.M., Bouzida, D., Gehlhaar, D.K., Rejto, P.A., Freer, S.T. & Rose, P.W. Complexity and simplicity of ligand-macromolecule interactions: the energy landscape perspective. Curr. Opin. Struct. Biol. 12, 197-203 (2002). 135.Kosztin, D., Izrailev, S. & Schulten, K. Unbinding of retinoic acid from its receptor studied by steered molecular dynamics. Biophys. J. 76, 188-197 (1999).

80


136.Heymann, B. & Grubmuller, H. Molecular dynamics force probe simulations of antibody/antigen unbinding: entropic control and nonadditivity of unbinding forces. BiophysJ. 81, 1295-1313 (2001). 137.Gullingsrud, J., Kosztin, D. & Schulten, K. Structural determinants of MscL gating studied by molecular dynamics simulations. Biophys. J. 80, 2074-2081 (2001). 138.Gullingsrud, J. & Schulten, K. Gating of MscL studied by steered molecular dynamics. Biophys. J. 85, 2087-2099 (2003). 139.Kong, Y., Y. Shen, T. E. Warth & Ma, J. Conformational pathways in the gating of Escherichia coli mechanosensitive channel. Proc. Natl. Acad. Sci. (USA) 99, 5999-6004 (2002). 14O.Ma, J. & Kaplus, M. Molecular switch in signal transduction: Reaction paths of the conformational changes in ras p21. Proc. Natl. Acad. Sci. (USA) 94, 1190511910(1997). 141.Diaz, J.F., Wroblowski, B., Schlitter, J. & Engelborghs, Y. Calculation of pathways for the conformational transition between the GDP and GTP bound states of the Ha-ras-p21 protein. Calculations with explicit solvent simulations and comparison with calculations in vacuum. Proteins Structure Function and Genetics 28, 434-51 (1997). 142.Kosztin, I., Bruinsma, R., O'Lague, P. & Schulten, K. Mechanical force generation by G proteins. Proc. Natl. Acad. Sci. (USA) 99, 3575-3580 (2002). 143. Bayas, M.V., Schulten, K. & Leckband, D. Forced detachment of the CD2CD58 complex. Biophys. J. 84, 2223-2233 (2003). 144.Lu, S. & Long, M. Forced extension of P-selectin construct using steered molecular dynamics. Chinese Sci. Bull. 49, 10-17 (2004). 145.Best, R.B., Li, B., Steward, A., Daggett, V. & Clarke, J. Can non-mechanical proteins withstand force? Stretching barnase by atomic force microscopy and molecular dynamics simulation. Biophys. J. 81, 2344-2356 (2001). 146.Harvey, S.C. & Gabb, H.A. Conformational transition using molecular dynamics with minimum biasing. Biopolymers 33, 1167-1172 (1993). 147.Paci, E. & Karplus, M. Unfolding proteins by external forces and temperature: The importance of topology and energetics. Proc. Nat. Acad. Sci. (USA) 97, 65216526 (2000). 148.Paci, E., Caflisch, A., Pluckthun, A. & Karplus, M. Forces and energetics of hapten-antibody dissociation: A biased molecular dynamics simulation study. J. Mol. Biol. 314, 589-605 (2001). 149.Heymann, B. & Grubmueller, H. AN02/DNP-hapten unbinding forces studied by molecular dynamics atomic force microscopy simulations. Chem. Phys. Letters 303,1-9(1999). 150.Paci, E., Smith, L.J., Dobson, C M . & Karplus, M. Exploration of partially unfolded states of human alpha-lactalbumin by molecular dynamics simulation. J. Mol. Biol. 306, 329-347 (2001). 151.Marchi, M. & Ballone, P. Adiabatic bias molecular dynamics: a method to navigate the conformational space of complex molecular systems. J. Chem. Phys. 110,3697-3702(1999). 152.Keller, D., Swigon, D. & Bustamante, C. Relating single-molecule measurements to thermodynamics. Biophys. J. 84, 733-738 (2003).


81

153.Fox, R.F. Using nonequilibrium measurements to determine macromolecule freeenergy differences. Proc. Natl. Acad. Sci. (USA) 100, 12537-12538 (2003). 154.Balsera, M., Stepaniants, S., Izrailev, S., Oono, Y. & Schulten, K. Reconstructing Potential Energy Functions from Simulated Force-Induced Unbinding Processes. Biophysical! 73, 1281-1287(1997). 155.Evans, E. & Ritchie, K. Dynamic strength of molecular adhesion bonds. Biophys. J. 72, 1541-55 (1997). 156.Jarzynski, C. Equilibrium free energy differences from nonequilibrium measurements: a master equation approach. Phys. Rev. E 56(1997). 157.Jarzynski, C. Equilibrium free energies from nonequilibrium processes. Ada Phys. Pol. B 29, 1609(1998). 158.Jarzynski, C. How does a system respond when driven away from thermal equilibrium? Proc. Nat. Acad. Sci. (USA) 98, 3636 (2001). 159.Hummer, G. & Szabo, A. Free energy reconstruction from nonequilibrium singlemolecule pulling experiments. Proc. Natl. Acad. Sci. (USA) 98, 3658-3661 (2001). 16O.Jensen, M.O., Park, S., Tajkhorshid, E. & Schulten, K. Energetics of glycerol conduction through aquaglyceroporin GlpF. Proc. Natl. Acad. Sci. (USA) 99, 6731-6736(2002). 161.Amaro, R., Tajkhorshid, E. & Luthey-Schulten, Z. Developing an energy landscape for the novel function of a (beta/alpha)8 barrel: ammonia conduction through HisF. Proc. Natl. Acad. Sci. (USA) 100, 7599-604 (2003). 162.Park, S., Khalili-Araghi, F., Tajkhorshid, E. & Schulten, K. Free energy calculation from steered molecular dynamics simulations using Jarzynski's equality. J. Chem. Phys. 119, 3559-3566 (2003). 163.Zuckerman, D.M. & Woolf, T.B. Theory of a systematic computational error in free energy differences. Phys. Rev. Lett. 89, 180602 (2002). 164.Liphardt, J., Dumont, S., Smith, S.B., Tinoco, I. & Bustamante, C. Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski's equality. Science 296, 1832-1835 (2002). 165.Ludemann, S.K., Lounnas, V. & Wade, R.C. How do substrates enter and products exit the buried active site of cytochrome P450cam? 1. Random expulsion molecular dynamics investigation of ligand access channels and mechanisms. J. Mol. Biol. 303, 797-811 (2000). 166.Ludemann, S.K., Lounnas, V. & Wade, R.C. How do substrates enter and products exit the buried active site of cytochrome P450cam? 2. Steered molecular dynamics and adiabatic mapping of substrate pathways. J. Mol. Biol. 303, 813-830 (2000). 167.Brockwell, D.J., Paci, E., Zinober, R.C, Beddard, G.S., Olmsted, P.D., Smith, D.A., Perham, R.N. & Radford, S.E. Pulling geometry defines the mechanical resistance of a beta-sheet protein. Nat. Struct. Biol. 10, 731-7 (2003). 168.Lorenzo, A.C., Pascutti, P.G. & Bisch, P.M. Nonspecific interaction forces at water-membrane interface by forced molecular dynamics simulations. J. Comp. Chem. 1A, 328-339 (2003). 169.Hummer, G. & Szabo, A. Kinetics from nonequilibrium single-molecule pulling experiments. Biophys J 85, 5-15 (2003).

82


17O.Cieplak, M., Hoang, T.X. & Robbins, M.O. Thermal folding and mechanical unfolding pathways of protein secondary structures. Proteins-Structure Function and Genetics 49, 104-113 (2002). 171.Cieplak, M., Hoang, T.X. & Robbins, M.O. Folding and stretching in a go-like model of titin. Proteins-Structure Function and Genetics 49, 114-124 (2002). 172.Bockmann, R.A. & Grubmuller, H. Nanoseconds molecular dynamics simulation of primary mechanical energy transfer steps in Fl-ATP synthase. Nat. Struct. Biol. 9, 198-202 (2002). 173.Bockmann, R.A. & Grubmuller, H. Conformational Dynamics of the F(l)-ATPase beta-Subunit: A Molecular Dynamics Study. Biophys. J. 85, 1482-91 (2003). 174.Ma, J.P., Flynn, T.C., Cui, Q., Leslie, A.G.W, Walker, J.E. & Karplus, M. A dynamic analysis of the rotation mechanism for conformational change in F-lATPase. Structure 10, 921-931 (2002). 175.Ma, J.P, Flynn, T.C., Cui, C , Leslie, A.G.W., Walker, J.E. & Karplus, M. A dynamic analysis of the rotation mechanism for conformational change in F-lATPase (vol 10, pg 921, 2002). Structure 10, 1285-1285 (2002). 176.Doltsinis, N.L. & Marx, D. First principles molecular dynamics involving excited states and nonadiabatic transtitions. J. Theoret. Comput. Chem. 1, 319-349 (2002). 177.Hayashi, S., Tajkhorshid, E. & Schulten, K. Molecular dynamics simulation of bacteriorhodopsin's photoisomerization using ab initio forces for the excited chromophore. Biophys. J. 85, 1440-1449 (2003). 178.Hugel, T., Holland, N.B., Cattani, A., Moroder, L., Seitz, M. & Gaub, H.E.Singlemolecule optomechanical cycle. Science 296, 1103-1106 (2002). 179.Holland, N.B., Hugel, T., Neuert, G., Cattani-Scholz, A., Renner, C , Oesterhelt, D., Moroder, L., Seitz, M. & Gaub, H.E. Single molecule force spectroscopy of azobenzene polymers: Switching elasticity of single photochromic macromolecules. Macromolec. 36, 2015-2023 (2003). 180.Ramesh, H.P. & Tharanathan, R.N. Carbohydrates - the renewable raw materials of high biotechnological value. Crit. Rev. Biotechnol. 23, 149-73 (2003). 181.Rief, M., Oesterhelt, F., Heymann, B. & Gaub, H.E. Single molecule force spectroscopy on polysaccharides by atomic force microscopy. Science 275, 12951297(1997). 182.Li, H., Rief, M., Oesterhelt, F., Gaub, H.E., Zhang, X. & Shen, J. Single-molecule force spectroscopy on polysaccharides by AFM - nanomechanical fingerprint of alpha-(l,4)-linked polysaccharides. Chem. Phys. Letters 305, 197-201 (1999). 183.Heymann, B. & Grubmueller, H. Chair-boat transitions and side groups affect the stiffness of polysaccharides. Chem. Phys. Letters 305, 202-208 (1999). 184.Marszalek, P.E., Li, H.B., Oberhauser, A.F. & Fernandez, J.M. Chair-boat transitions in single polysaccharide molecules observed with force-ramp AFM. Proc. Natl. Acad. Sci. (USA) 99, 4278-4283 (2002). 185.O'Donoghue, P. & Luthey-Schulten, Z.A. Barriers to forced transitions in polysaccharides. J. Phys. Chem. B 104, 10398-10405 (2000). 186.Marszalek, P.E., Oberhauser, A.F., Li, H.B. & Fernandez, J.M. The force-driven conformations of heparin studied with single molecule force microscopy. Biophys. J. 85, 2696-2704 (2003). 187.Lee, G., Nowak, W., Jaroniec, J., Zhang, Q. & Marszalek, P.E. Nanomechanical control of glucopyranose rotamers. J. Am. Chem. Soc, 126, 6218-6219 (2004).


83

188.Kale, L., Skeel, R., Bhandarkar, M., Brunner, R., Gursoy, A., Krawetz, N., Phillips, J., Shinozaki, A., Varadarajan, K., Schulten, K. NAMD2: Greater scalability for parallel molecular dynamics. J. Comp. Phys. 151, 283-312 (1999). 189.Lu, Z., Nowak, W., Lee, G., Marszalek, P.E. & Yang, W. The elastic properties of single amylose chains in water: a quantum mechanical and AFM study. J. Am. Chem. Soc, (2003) in press. 19O.Cui, Q., Elstner, M., Kaxiras, E., Frauenheim, T. & Karplus, M. A QM/MM implementation of the self-consistent charge density functional tight binding (SCC-DFTB) method. J. Phys. Chem. B 105, 569-585 (2001). 191.Liu, H., M. Elstner, E. Kaxiras, T. Frauenheim, J. Hermans & Yang, W. Quantum mechanics simulation of protein dynamics on long time scale made possible. Proteins: Structure, Function and Genetics 44, 484-489 (2001).

Chapter 3: Molecular Dynamics Simulations of a Molecular Electronics Device: The NanoCell

Jorge Seminario, Pedro Derosa\ Luis Cordova6 and Brian Bozardc Chemical Engineering Department, Texas A&M University, College Station, Texas 77843, USA "Institute for Micromanufacturing/Department of Physics, Louisiana Tech. University, Ruston, LA71272, USA ^Department of Electrical Engineering, University of South Carolina, Columbia, SC 29208, USA (Former affiliation for all authors) cSpace

and Naval Warfare (SPAWAR) Systems Center, Charleston, SC 29419-9022, USA

Abstract Molecular dynamics simulations using ab initio force fields and signal processing techniques are combined to analyze the dynamic properties of a minimum unit of a field programmable random molecular array also called nanocell. We analyze stretching, angular, and torsional internal modes at several temperatures of operation and their correlations as well as the possibility of signal modulation and processing at terahertz frequencies. The proposed scenario is another alternative for molecular electronics, which is being developed at very low frequencies of operation using the electronic states of molecules, clusters and other nanoscopic systems.

85

86 J. Seminario et al.

1. Introduction The use of small molecules to complement silicon-based electronic devices is under intensive development [1-7]; each molecule occupies an area between 105 to 106 [2, 5] times smaller than conventional electronic devices and therefore provides hopes to go beyond the limits set by the use of bulk silicon devices. On the other hand, molecules would allow us to develop several other applications directly related or reminiscent of biological systems. All this impetus has created the new field of molecular electronics (moletronics). Moletronics focuses on the development of electronic devices using small molecules with feature sizes on the order of one or a few nanometers [2, 5]. These very small feature sizes are well below the minimum feature size of present electronic devices (-45 nm in the year 2004). Notice that a high doping of 1018 cm"3 corresponds to an average separation between nearest impurities of 10 nm thus a radical change of technology would be needed when approaching such a size. The ultimate goal of miniaturization is to use the minimum amount of atoms per electronic function. However, this goal poses a strong challenge since molecules with dimensions smaller than 10 nm cannot be assembled as standard semiconductor devices using modern lithographic techniques; the alternative is the use of other techniques such as molecular self-assembly and vapor deposition, which bring a strong random component into the field. Deposition techniques yield patterns resulting in minimum units that can be thoroughly analyzed but it will be required that they can be reconfigured (programmed) in order to compensate for their lack of addressability [8, 9]. Thus, programmability of devices at the molecular level compensates for their lack of addressing. Addressing is one of the most important properties of standard microelectronics. Fortunately, programmability of non addressable molecular devices can be achieved using molecules featuring highly nonlinear current-voltage characteristics like negative differential resistance (NDR) [1]. Experiments showing NDR and their use as memories have been already reported [10, 11], and a strong effort is presently underway in order to demonstrate programmable molecular devices [1,6, 12]. Addressability during the construction and operation of molecular devices is a challenge. Both self-assembling [5, 13] and direct assembling [14] have been proposed for the construction and they depend on the chemical affinity of terminal groups attached to the "electrically active" part of the molecule. Due to the non-deterministic characteristics of these processes, molecules are expected to be arranged in quasi-random configurations, which together with their small size, make them no directly addressable. The conversion, reconfiguration, or reprogramming during operation of a randomly constructed

Molecular Dynamics Simulations of a Molecular Electronics Device 87

circuit relies on the use of molecules with highly nonlinear features that allow the system to have alternative states able to be set externally after construction to perform a specific function [1, 8,9, 12]. Molecular circuits will be created by chemical means inside externally addressable units; however the inside of these units cannot be addressed except by the external programming; these small units are called nanoCells. The nanoCell is one of the most important and viable concepts for the development of electronics beyond the deterministic or lithographic approach of present C-MOS integrated circuits. Very few scenarios have been proposed to the use of molecular-based devices in order to bypass problems undermining miniaturization or scaling-down processes for integrated circuits through the introduction of a revolutionary approach in which molecules or nanoclusters of atoms are used to perform logical operations complementing or extending the capabilities of deterministic ones. The nanoCell concept takes advantage of the great ability of chemistry to synthesize molecules with specific characteristics with tolerances of fractions of one nanometer; such a tolerance is far from the reach of any IC fabrication technique, which is limited to practically two dimensions. However molecules are created in a single flask in quantities in the order of Avogadro's number ~6xlO23 and they are not individually addressable or interconnectable thus it is impossible to read or write any information in a deterministic manner. A way out of this problem is to address a molecular circuit constructed by chemical means, of a size such that is addressable by deterministic techniques (like present lithography), and indirectly program the individual molecules for performing specific and more complex functions than their microelectronics counterparts. This almost unthinkable approach is possible if the molecules (assembled almost randomly) can be chosen with programmable features. It has been already shown in the literature that even two single molecules with strong non linear characteristics can perform programmable functions thus compensating for the lack of addressability of single molecules [1, 8, 9, 12]. The applications of these nanoCells can be easily extended interconnectiong long biomolecules like DNA or one dimensional wires. This macroscopic systems in one dimension (or backbone) and nanoscopic in the other two allow the transmission of vibrations along the molecule backbone due to the presence of actual bands where vibrational modes become part of a one-dimensional continuum and thus signals entered in one place of a long molecule can be read and processed in other places far away in the molecule as it was already suggested ealry in 1941 [15]. In this work, we study one of the possible minimum programmable structures, a square consisting of four active molecules interconnected by four Au clusters. This arrangement is one of the two most likely allowed to take

88 J. Seminario et al.

place when self-assembling on a substrate (the other one being a triangle) [12]. We focus this study on the thermal stability of the system and the correlation among vibrations in the molecules. Ab initio calculations, molecular dynamics simulations, and signal processing techniques are combined to achieve this aim. MD simulations provide time dependant information of the system, and signal processing techniques are used for the analysis of the time-domain signal in order to show the strong correlations among the dynamics of internal degrees of freedom and to suggest a new scenario for the use of molecular arrangements as signal processing devices. A detailed analysis of the frequency spectra yields correlations between several local vibrations that can be used to include vibrational signals externally introduced through the contacts of the molecular device. Although these correlations between internal modes are not surprising, the question is if using the natural amplitude oscillations of the vibrational modes, which can be experimentally detected, will allow the detection of mixed signals in localized sites. If this were possible, several single operations could be implemented, enabling molecular processors dealing with signals in the THz range of frequencies. We propose that signals can be encoded as the movement of atoms that can be detected observing the changes in the electrical characteristics of the molecule. For instance, when the displacement, ? f of atoms associated to a vibrational mode changes the dipole moment of the molecule ^, a signal proportional to the so called infra red (IR) intensity is obtained from IR Intensity u)cv = 0 V/,/i

(4)

As we emphasized above, we would rather use the less general QM in our present article in the numerical implementation. To see the explicit form of the CEPA-like approximations, we write the above equation in long hand as follows: \{XiW\^) I

+

to|[#,m^> II

+ T,AXi\exp(-T>l)exp(Tl/)\(j>tt)H^Cv IV

+

!<Xj|[[ff,TV]|^>]c M III +

other t e r m s

=0

(,s K

'

There are several features of the equations which should be emphasized here: (a) the combining coefficients cM are not frozen at some preassigned values, but are iteratively updated to their relaxed values; (b) the first three terms have explicitly connected algebraic structure in case T^ is connected; (c) among the quadratic terms appearing in III, the terms with one-body excitations are still doubles while the product of doubles or of singles and doubles lead to virtual functions \i which are more than doubly excited with re-

130 S. Chattopadhyay et al.

spect to (j)^; (d) the fourth term contains two pieces, one containing T" and the other containing TM, which may be individually disconnected but the entire term containing the matrix element {xi\ exp(—TM)exp(T")|0M)H'A1^ is a connected entity [27; 28]. The aspect (d) is what we emphasized earlier as generating a connected entity from two individually disconnected pieces, and which could not have been derived from the MR-CISD equations. For a proof of this assertion, we refer to our earlier publications [27; 28]. This feature of the SS-MRCC equation, retained in generating the SS-MRCEPA equations is very important for our purpose, since in any approximation or truncation scheme, each term of the two pieces must be treated on the same footing to ensure connectivity. Thus the term IV has to be computed in a manner which uses the same approximation for the matrix elements H^. Using the same kind of considerations as were used to generate the various CEPA-like approximations from the SRCC-SD equations, we confine the rank of cluster operators TM to at most two-body excitations. Thus for each <j>n, the x;' s used in the projection in Eq.(5) above must be at most doubly excited with respect to k) satisfies the Schrodinger equation (12)

H\il>k) = Ek\il>k)

In close analogy with the SR-CCLRT, we include in each excitation operator S% all the virtual excitations from |0M), and retain the cluster components TM of the multi-reference ground state (eq(l)) in our Ansatz for \tpk}- In addition, we also want to change the relative weights of the various |^M)'s in the function 1^*)- With this choice, we can represent S% as

SZ = *fl + SZ

with S» = Y,*?kY?' m

(13)

where Y™^ creates a virtual function \Xm) from \4>n)- It is evident that there are several S£'s which excite to the same \xm) from several | ^ ) ' s . Thus, we have to invoke a set of sufficiency conditions to determine them. We have

£[exp(T^)|^> + exp(r")5f |^>(^|ff M |^)

+exP(r'X°|<M + E e x P( r A I )^^^l^> + exp(r")S^QH M |^) + exp(T")a^gH M |^>]c M = Efcêxp(T^^|^)^

(14)

where Q is the full virtual space projector. Following the same manipulations leading to the SS-MRCC equations [27; 28], we interchange the labels H and v in the above equation in the first three terms and equate terms with same fi. For each virtual space projection onto (x;| exp(-T^) we have

(Xl\%S%\^ + (x^QH^K + {xilHMsfcp + ^{Xi\ exp(-T") e x p C n S j ^ X ^ l i ^ l ^ V

+ (xi\ exp(-T") eMTv)\^)(^\HvSvk\4>v)cv +{Xi\ exp(-T*) exP(Tn\^)(^\Hu\x\HAv)sfcv = Ekcxsf

(19)

which on simplification generates,

£<e* + ^ ( ^ I ^ I ^ ) ^ = EkdXk vm

v

(20)

136 5. Chattopadhyay et al.

The above MR-CCLRT equations can be written compactly as a matrix eigenproblem. Once the cluster amplitudes of TM for all /i's have been found out by solving the SS-MRCC equations, we can get the associated excited state energies relative to the state \ip) by solving the MR-CCLRT equations. An attractive feature of the MR-CCLRT is that the computed excitation energies are size-intensive [29] in the sense that they become asymptotically equal to the sum of the fragment excitation energies in the limit of noninteracting fragments. We will now develop the corresponding MRCEPA-LRT using exactly the same approximations on MR-CCLRT as has been used in the SS-MRCC theory to arrive at the SS-MRCEPA theories. To derive the working equations for the SS-MRCEPA based response theories, we start with the parent MR-CCLRT equations (18) and (20) for the excited states and make approximations akin to what have been used to arrive at the SS-MRCEPA(O) and SS-MRCEPA(D) theories from the SS-MRCC theory for the ground state. Both the MRCEPA-LRT schemes can be subsumed in the same type of working equations, with appropriate definitions of certain dressed operators. In our SS-MRCEPA-like methods, we ignore more than double excitations out of any given ^ leading to a xjj a n d quasi-linearize the terms in [exp(—TM)exp(T")] accordingly. The working equations of the virtual space projections of the CEPA-based MR-CCLRT Eq. (18) for both the CEPA-based schemes can be simplified as follows

m m vm

+ Y.(xlMTl/-T»)\>1)H) o n the same footing, and we have two distinct options to choose our orbitals: (a) we may choose either of the two determinants to generate the HF orbitals, or (b) we may use the orbitals obtained from a CAS-SCF calculation. It turns out that for the molecules studied by us in this paper, there is the following rough general trend: the PES of the ground state obtained either by the SS-MRCEPA or as the lowest root from the MRCEPA-LRT is improved generally, by going over the CAS-SCF orbitals. For P4 model, the results of the ground state obtained by CEPA(O) method show improvement by around 0.3-0.9 mH and for CEPA(D) this varies from around 0.03-0.45 mH. At the square geometry (highly degenerate case), this improvement is maximum. The performance of the MRCEPA-LRT for the excited PES with the CAS orbitals is not uniformly better however, over the entire PES. For the P4 model, the results obtained by either of the two variants of MRCEPA-LRT are altered by 1-3 mH relative to the actual choice of the orbitals, as described above. Similarly, for L12 molecule, the ground state energies obtained using CAS orbitals are improved by around 0.4-1.5 mH


and 0.42-0.8 raH in case of CEPA(O) and CEPA(D) methods respectively. In the dissociation region, the improvement in either method using CAS orbitals over the HF one is negligibly small (around 0.006 mH). However, for Li2 molecule, the performance of excited PES exhibited by either of the two methods of MRCEPA-LRT are changed between 0.87-3.4 mH relative to the actual choice of the orbitals. Thus, there is no significant loss in accuracy even if we use the HF orbitals, but it saves the time for optimization. Also, the CAS-SCF orbitals for the ground state tend to over-emphasize the importance of the nondynamical correlation for the ground state, which can get altered substantially for the excited states. This is why, we will present here the results obtained with HF orbitals using one of the two model functions. The actual model function used by us is described below for each system. To assess the performance of our methods for studying the low-lying PES, we compare our results obtained with those from full CI (FCI) calculations in the same basis. The required molecular integrals and the different CI results were generated by the electronic structure package, GAMESS. The two sets of unknowns, the cluster amplitudes and the combining coefficients, are coupled in the SS-MRCEPA equations. They are solved in a nested iterative loop. In the micro-iteration the cluster amplitudes are solved for fixed coefficients till a local convergence is reached. The coefficients are then updated by diagonalizing the matrix H^ in the macroiteration. Once these unknowns in the MRCEPA are determined, we construct the matrix of Ji, and get the required eigenvalues and vectors from the MRCEPA-LRT eigenproblem. It should be noted that, unlike in the case of SR-CCLRT, our MRCEPALRT equations are defined in a space of overcomplete basis: each virtual function \xm) appears several times-generated from various | ^ ) ' s , with attendant coefficients d™k. If the number of linearly independent functions in the Hilbert space is Nh, then only Nh roots are meaningful. The rest of the roots are extraneous. These are easily diagnosed from the very small value of the norm of these functions. A. P4 Model System Our first test case is the simultaneous stretching of both H — H bonds in P4 (rectangular i/ 4 model). The P4 model introduced by Jankowski and

Computation of Excited State Potential Energy Surfaces 141 -2.000 -j

1

-2.025 -2.050 •

-2.075 ^

-2.100-

3 ^

• -2.125-

> g

' -2150"

, \

1

s^ A

i

\

/ \

/ \ \

/ /

\ \

g -2,75:

/ /

\

-2.200 -

.

-2.225 -

^

/

MRCEPA(0)-LRT MRCEPA(D)-LRT

/ —

F

C

|

\

/

\

,

-2.250 -

-2.275 -I

/

^ ^ ^ ^ - ^ ^ ^ ^ ^ ^ ^ ^

, 2

,

1 3

,

, 4

,

R (a.u.)

, 5

,

, 6

Fig. 1 : PES of some low-lying states of the P4 model systerr (Energy of the 11A2 state shifted down by 0.25 a.u.)

Paldus [45; 46] and later also studied extensively by several workers [28; 42; 47; 48; 49] is a rather simple test system and thus well suited to explore the validity and efficacy of our formalism. The geometries considered here are the same as in Ref.[45]. The geometry of P4 model is specified by the distance of approach of the two H2 molecules, R, which is varied from the pure square geometry and then gradually elongated to the rectangular geometry. The configurational quasi-degeneracy involving '/^

•

//

"185"

if

MRCEPA(0)-LRT

/

190

MRCEPA(D)-LRT

2'A,/ -1.95-

FCI

/

-2.00 2

3

4

5

6

R (a.u.) Fig. 2 : PES of some low-lying states of the P4 model systerr

The results for SS-MRCEPA(O) and SS-MRCEPA(D) based linear re-

Computation of Excited State Potential Energy Surfaces 143

sponse methods for the PES of states of various symmetries are plotted in Figs. 1 and 2. The corresponding FCI results are also shown. In order to show PES of several states in the same graph, we have sometimes shifted down the energies of some higher-lying states by a fixed amount, as explained in the captions for these graphs. From this we can see that: (i) The FCI values and the results obtained by the MRCEPA-LRTs for the ground state, l 1 ^ ! as well as excited state, 11A2 are very close and uniform for all points on the PES. For the I1.02 state, the scenario is slightly different. For this state, the performance of our methods is also quite good. However, for very small values of R, our results are found to be slightly away from the corresponding FCI values. (ii)At small values of R, the performance of our MRCEPA-LRT for I1.©! state is quite uniform and the deviation is fairly small with respect to the FCI results. But the deviation of MRCEPA-LRT results from the FCI values is not that small for this state at larger values of R. (iii) Although the performance of MRCEPA-LRT for 21Ai state is generally good at small and intermediate values of R, it becomes somewhat worse with increase in the values of R. Both Figs. 1 and 2 show that at larger values of R the deviation of MRCEPA(O)-LRT from the FCI for the 2xAx and l 1 ^ states is smaller than the MRCEPA(D)-LRT, the reason for which is not clear to us at this point. B. Li2 Molecule The Li2 molecule is a well studied system [28; 42; 50; 51] to test the efficacy of the various SR and MR methods. It is now well known that the ground state of the Li2 molecule requires a two-determinantal description in the zeroth order. In this calculation we have used a two-dimensional CAS defined by the two determinants (pi = lojlo^2o| and fa — \a'1g\a2u2u'2u. This is a good example to test the performance of our methods within a small reference space, since there is a quasi-degeneracy of the reference determinants at large internuclear distances in the PES of the ground state and there are intruders to <j>2 at the intermediate distances. We have used a small basis (DZ) [52] for the study of the Li2 PES of some low-lying electronic states, using D2h point group at each step of calculation. The configuration <j>\ has been chosen as the HF function in our calculations. Using these, we can compare the potentiality of our MRCEPA-LRT with the FCI values for all the states studied.

144 5. Chattopadhyay et al. -14.845 -i

1

-14.850 -

^ * ^ *

-14.855-

/

-14.860^

/«!/r:rrrTr^

-14.865-

/

>, -14.870-

\ " \ \ 1

35

5 •j - 1 4 - 8 7 5 "

I / /

1 -14.880-

I /


/ /

\

-14.885-

/ /

/

I

\

•v'v

FCI

/

\l/

-14.890-

'

i

4

'

i

6

•

i

8

•

i

10

•

R (a.u.)

i

12

'

I

14

'

I

16

'

Fig. 3 : PES of some low-lying £ states of Li2 molecule (Energy of the £u* state shifted down by 0.06 a.u.)

We plot the results in Figures 3-5 to check the uniformity and accuracy of the PESs of various low-lying excited states obtained using our newly developed MRCEPA-LRT alongwith the corresponding FCI values. Again, as in the case of P4, we have sometimes shifted down the energies of the PES for some higher-lying various states by a fixed amount, as explained in the captions of the corresponding figures. We have tagged the symmetries of the different energy states according to the I?ooft point group, though we have performed the calculation within the £>2/i framework. It is evident

Computation of Excited State Potential Energy Surfaces 145

that the MRCEPA-LRT results are in close proximity to the FCI values and quite uniform throughout the PES, including the points of intruders and degeneracies. The performance of our SS-MRCEPA based linear response methods for Li2 molecule is quite good for low-lying £ states as is evident from the Figs. 3 and 5. From Fig. 4, it is clear that the performance of our newly developed MRCEPA-LRT to compute the PESs of Ug and n u states is not so satisfactory as that of the performance for the previously mentioned low-lying £ states. To underline the performance of the MRCEPA-LRT vis-a-vis the parent MR-CCLRT in predicting the excitation energies, we present in Table-I the vertical excitation energies of the Li2 molecule at the equilibrium ground state geometry obtained from both MRCEPA(O)-LRT and MRCEPA(D)-LRT as well as the full-blown MR-CCLRT methods alongwith with the FCI results.

Table-I FCI, MRCEPA(O)-LRT, MRCEPA(D)-LRT and MRCCLRT vertical excitation energies (in mH) for a few low-lying singlet excited states of Li2 at the equilibrium distance (R=5.0512 a.u.) of the ground state. Values within the parentheses indicate the deviation from FCI. ~

Methods

£+

II M

ng

S~

S~

FCI MR-CCLRT

68.203 68.23 (-0.027)

109.723 109.56 (0.163)

112.709 112.524 (0.185)

173.4 173.598 (-0.198)

206.726 206.896 (-0.17)


69.397 110.633 113.451 174.455 207.755 (-1.194) (-0.91) (-0.742) (-1.055) (-1.029) 68.221 109.543 112.502 173.713 207.144 (-0.018) (0.183) (0.207) (-0.313) (-0.418)

From the table, it is evident that the numerical accuracy achieved by the MRCEPA-LRT is quite close to that of the parent MR-CCLRT approach. This indicates that SS-MRCEPA based LRT (MRCEPA-LRT) is a very useful and good approximant of the full-blown SS-MRCC based LRT (MRCCLRT), although SS-MRCEPA neglects most of the complicated nonlinear terms of the parent SS-MRCC method using a series of physically motivated graded approximation schemes. The method works very well to compute the PES of various low-lying exited states for ground states with

146 S. Chattopadhyay et al. -14.760 -i

1

.14765.

I i I

-14.770-

3.

> ^

u

-14.775-

MRCEPA(O)-LRT MRCEPA(D)-LRT FCI

!l

\l

' 1

-14.780-

\

-14.785-

\

-14.790-

-14.795 -I

"'^:^?'

\.

1 4

,

, 6

,

s^^"***^

1 8

,

, 10

,

, 12

,

, 14

,

, 16

R (a.u) Fig. 4 : PES of some low-lying n states of Li2 Molecule

varying degrees of MR character and displaying real or avoided crossings. The general trend of the results for both P4 and Li2 systems is very encouraging indeed. We do not want to draw definite conclusions regarding the relative performances of MRCEPA(0)-LRT and MRCEPA(D)-LRT from these pilot calculations. Both perform quite well. As Table-I indicates, the excitation energies computed by MRCEPA(D)-LRT are of better quality at the ground state equilibrium geometry of Li2, though MRCEPA(D)LRT sometimes show greater variations at some other geometries.

Computation of Excited State Potential Energy Surfaces 147 -14.66 -i

.1467.

1

-14.683

MRCEPA(O)-LRT MRCEPA(D)-LRT FCI

-\ " \ \

'

\

>£- -14.69-

\ \ \

5K

O>

v

,

, 6

,

, 8

,

, 10

,

, 12

,

, 14

,

, 16

(a.u.) 2 ' states of Li2 Molecule Fig. 5 : PES of someRlow-lying

4

Summary

In this paper, we proposed two variants of linear response methods developed for excited states from the state specific multi-reference CEPA theories for the associated ground state. These theories are designed to generate excited state energies of a system whose ground state is multi-reference in character and also show intruders. Around the quasi-degenerate configurations, the effective hamiltonian based methods are expected to work well,


but they will fail in the regions of the PES where intruders are present. But our SS-MRCEPA based LRT performs with equal efficacy in the entire range of the PES for the various excited states. Numerical test calculations on P4 model and Li2 molecular systems support the above facts. A comparison of our results with those obtained from the FCI in the same basis indicates that the computed excited state energies are very accurate and uniform in nature throughout the entire range of the PES, and this is generally valid for states with various spatial symmetries. Thus our MRCEPA-LRT serves as a good model to compute accurate and uniform PES of the excited states. This fortifies our belief that the MRCEPA-LRT methods are useful and good approximants of the parent MR-CCLRT, despite the neglect of a host of terms present in the parent theory. The results are encouraging for the systems we have studied using our MRCEPA-LRT, which leads us to express cautious optimism about its general potentiality. 5

Acknowledgments

DM wishes to thank the CSIR, INDIA for financial support [Proj. No. 01 (1624)/EMR-II)]. DP thanks the CSIR, INDIA for the research fellowship. This article is dedicated to P Ca rsky, I Hubac and M Urban on the happy occasion of their reaching sixty.

Bibliography 1. I. Shavitt (1975) in Methods of electronic structure theory, Vol. 3, H. F. Schaefer III (ed.) Plenum Press, NY. 2. R. J. Bartlett and W. D. Silver (1975) Int. J. Quantum Chem. S9, 183; J. A. Pople, J. S. Binkley and R. Seeger (1976) Int. J. Quantum Chem. S1O, 1. 3. W. Meyer (1973) J. Chem. Phys. 58, 1017; W. Meyer (1989) Theor.Chim. Ada 35, 277; A. C. Hurley (1976) in Electron Correlation in Small Molecules, Academic Press, London; R. Ahlrichs, H. Lischka, V. Staemmler and W. Kutzelnigg (1975) J. Chem. Phys. 62, 1225; W. Kutzelnigg (1977) in Methods of Electronic Structure Theory, H. F. Schaefer III (ed.) Plenum Press, NY. 4. R. J. Gadanitz and R. Ahlrichs (1988) Chem. Phys. Lett. 143, 413. 5. P. G. Szalay and R. J. Bartlett (1995) Chem. Phys. Lett. 214, 481; L. FustiMolnar and P. G. Szalay (1996) J. Phys. Chem. 100, 6288.

Computation of Excited State Potential Energy Surfaces 149 6. P. G. Szalay (1997) in Recent Advances in Coupled-Cluster Methods, Ed. R. J. Bartlett, World Scientific, Singapore and references therein. 7. J. Cizek (1966) J. Chem. Phys. 45, 4256; J. Cizek (1969) Adv. Chem. Phys. 14, 35; J. Paldus, J. Cizek and I. Shavitt (1972) Phys. Rev. A 5, 50. 8. R. J. Bartlett (1989) J. Phys. Chem. 93, 1697; R. J. Bartlett (1995) in Modern Electronic Structure Theory, D. R. Yarkony (ed.) World Scientific, Singapore. 9. R. J. Bartlett (1981) Ann. Rev. Phys. Chem. 32, 359. 10. J. Noga and R. J. Bartlett (1987) J. Chem. Phys. 86, 7041; E. E. Scuseria and H. F. Schaefer III (1988) J. Chem. Phys. 152, 382; K. Raghavachari, J. A. Pople, E. S. Replogle and M. S. Head-Gordan (1990) J. Chem. Phys. 94, 5579; P. Piecuch, S. A. Kucharski and R. J. Bartlett (1999) J. Chem. Phys. 110, 6103. 11. R. J. Buenker and S. D. Peyerimhoff (1974) Theor. Chim. Acta 35, 33; P. E. M. Siegbahn (1980) J. Chem. Phys. 72, 1674; R. J. Buenker and S. Krebs (1999) in Recent Advances in Multireference Methods K. Hirao (ed.) World Scientific, Singapore and references therein. 12. S. R. Langhoff and E. R. Davidson (1974) Int. J. Quantum Chem. 8, 61; E. R. Davidson (1974) in The World of Quantum Chemistry, R. Dauel and B. Pullman (eds.) Reidel, Dodrecht; J. P. Francois and R. Gybels (1990) Chem. Phys. Lett. 172, 346. 13. J. P. Malrieu (1982) Theor. Chim. Acta 61, 163. 14. M. R. A. Blomberg and P. E. M. Seigbahn (1983) J. Chem. Phys. 78, 5682; W. D. Laidig and R. J. Bartlett (1984) Chem. Phys. Lett. 104, 424. 15. J. P. Daudey, J. L. Heully and J. P. Malrieu (1993) J. Chem. Phys. 99 1240; I. Nebot-Gil, J. Sanchez-Marin, J. P. Malrieu, J-L. Heully and D. J. Maynau (1995) J. Chem. Phys. 103, 2576. J. P. Malrieu, I. Nebot-Gil and J. Sanchez-Marin (1994) J. Chem. Phys. 100, 1440. 16. S. Chattopadhyay, U. S. Mahapatra, B. Datta and D. Mukherjee (2002) Chem. Phys. Lett. 357, 426. 17. S. Chattopadhyay, U. S. Mahapatra, P. Ghosh and D. Mukherjee (2002) in Low-Lying Potential Energy Surfaces, Eds. M. R. Hoffmann and K. G. Dyall ACS Symposium Series No. 828, Washington, DC. 18. B. Brandow (1967) Rev. Mod. Phys. 39, 771; I. Lindgren (1974) J. Phys. B, 7, 2441. 19. D. Mukherjee, R. K. Moitra, and A. Mukhopadhyay (1977) Mol. Phys. 33, 955; I. Lindgren (1978) Int. J. Quantum Chem. S12 33,; I. Lindgren, and D. Mukherjee (1987) Phys. Rep. 151, 93; D. Mukherjee, and S. Pal (1989) Adv. Quantum Chem. 20, 561. 20. B. Jeziorski, and H. J. Monkhorst (1981) Phys. Rev. A 24, 1668; B. Jeziorski, and J. Paldus (1988) J. Chem. Phys. 88, 5673. 21. T. H. Schucan and H. A. Weidenmiiller (1972) Ann. Phys. 73, 108; G. Hose and U. Kaldor (1979) J. Phys. B 12, 3827. 22. J. P. Malrieu, Ph. Durand and J. P. Daudey (1985) J. Phys. A: Math. Gen.

150 5. Chattopadhyay et al.

18, 809; J. L. Heully and J. P. Daudey (1988) J. Chem. Phys. 88, 1046. 23. D. Mukhopadhyay, B. Datta and D. Mukherjee (1992) Chem. Phys. Lett. 197, 236; B. Datta, R. Chaudhuri and D. Mukherjee (1996) J. Mol. Struct: THEOCHEM 361, 21. 24. A. Landau, E. Eliav, Y. Ishikawa and U. Kaldor (2000) J. Chem. Phys. 113, 9905. 25. J. P. Malrieu, J. P. Daudey and R. Caballol (1994) J. Chem. Phys. 101, 8908; J. Meller, J. P. Malrieu and J. L. Heully (1995) Chem. Phys. Lett. 244, 440; J. Meller, J. P. Malrieu and R. Caballol (1996) J. Chem. Phys. 104, 4068. 26. J. Masik, and I. Hubac (1997) in Quantum Systems in Chemistry and Physics: Trends in Methods and Applications, R. McWeeny et al. (eds.) KA, Dordrecht; I. Hubac, J. Prittner and P.Carsky (2000) J. Chem. Phys. 112, 8779. 27. U. S. Mahapatra, B. Datta, B. Bandyopadhyay and D. Mukherjee (1998)Adv. Quantum Chem. Vol. 30 D. Hanstrop and H. Persson (eds.) Academic Press Inc. San Diego. 28. U. S. Mahapatra, B. Datta and D. Mukherjee (1998) Mol. Phys. 94, 157; U. S. Mahapatra, B. Datta and D. Mukherjee (1999) J. Chem. Phys. 110, 6171. 29. S. Chattopadhyay, U. S. Mahapatra and D. Mukherjee (1999) in Recent Advances in Multireference Methods K. Hirao (ed.) World Scientific, Singapore; S. Chattopadhyay, U. S. Mahapatra and D. Mukherjee (2000) J. Chem. Phys. 112, 7939. 30. H. J. Monkhorst (1977) Int. J. Quantum Chem. S l l , 421; E. Dalgaard and H. J. Monkhorst (1983) Phys. Rev. A28, 1217. 31. H. Nakatsuji and K. Hirao (1978) Chem. Phys. Lett. 59, 362; H. Nakatsuji and K. Hirao (1988) J. Chem. Phys. 68, 2053; H. Nakatsuji and K. Hirao (1979) Chem. Phys. Lett. 67, 329, 334; K. Hirao (1993) J. Chem. Phys. 79, 5000. 32. D. Mukherjee and P. K. Mukherjee (1979) Chem. Phys. 39, 325; B. Datta , P. Sen and D. Mukherjee (1995) J. Phys. Chem. 99, 6441. 33. H. Sekino and R. J. Bartlett (1984) Int. J. Quantum Chem. S18, 255; P. Piecuch and R. J. Bartlett (1999) Adv. Quantum Chem. Vol. 34, 295 and references therein. 34. H. Koch and P. J0rgensen (1990) J. Chem. Phys. 93, 3333; H. Koch, R. Kobayashi, A. Sanchex de Mears and P. J0rgensen (1994) ibit 100, 4993. 35. S. Pal (1989) Phys. Rev. A 39, 39; S. Pal (1992) Int. J. Quantum Chem. 41, 443. 36. D. Ajitha and S. Pal (1997) Phys. Rev. A 56, 2658. D. Ajitha, N. Vaval and S. Pal (1999) J. Chem. Phys. 110, 2316. 37. S. Tenno, S. Iwata, S. Pal and D. Mukherjee (1999) Theor. Chem. Ace. 102, 252. 38. P. G. Szalay (1995) Int. J. Quantum Chem. 55 , 15. 39. P. G. Szalay, T. Muller and H. Lischka (2000) Phys. Chem. Chem. Phys. 2,

Computation of Excited State Potential Energy Surfaces 151 2067. 40. I. Martin, C. Lavin, Y. Perez-Delgado, J. Pitarch-Ruiz and J. Sanchez-Marin (2001) J. Phys. Chem. A 105, 9637. 41. D. Mukhopadhyay, S. Mukhopadhyay, R. Chaudhuri andD. Mukherjee (1991) Theor. Chim. Ada 80, 441. 42. U. S. Mahapatra, B. Datta and D. Mukherjee (1999) Chem. Phys. Lett. 299, 42; U. S. Mahapatra, B. Datta and D. Mukherjee (1999) J. Phys. Chem. A 103, 1822. 43. P. Ghosh, S. Chattopadhyay, D. Jana and D. Mukherjee (2002) Int. J. Mol. Set. 3, 733. 44. A. Deb, S. Chattopadhyay, D. Pahari and D. Mukherjee (2003) (to be communicated). 45. K. Jankowski and J. Paldus (1983) Int. J. Quantum Chem. 18, 1243. 46. J. Paldus, P. E. S. Wormer and M. Benard (1988) Coll. Czech. Commum. 53 1919. 47. S. Zarrabian and J. Paldus (1990) Int. J. Quantum Chem. 38, 761; K. Jankowski, J. Paldus and J. Wasilewski (1991) J. Chem. Phys. 95 , 3549. 48. J. P. Finley, R. Chaudhuri and K. F. Freed (1995) J. Chem. Phys. 103, 4990. 49. K. Jankowski and I. Grabowski (1995) Int. J. Quantum Chem. 55, 205. 50. U. Kaldor (1991) Theo. Chim. Ada 80, 427; U. Kaldor (1990) Chem. Phys. Lett. 140, 1. 51. I. Garcia-Cuesta, J. Sanchez-Marin, A. Sanchez de Mears and N. Ben Amor (1997) J. Chem. Phys. 107, 6306. 52. T. H. Dunning Jr. (1970) J. Chem. Phys. 53, 2823.

Chapter 5: Modelling of Anisotropic Exchange Coupling in Rare-Earth -Transition-Metal Pairs: Applications to Yb3+-Mn2+ and Yb^-Cr34 Halide Clusters and Implications to the Light Up-Conversion

M. Atanasova, C. DauP, H.U. GiideP a Institute of

General and Inorganic Chemistry Bulgarian Academy of Sciences Acad. G. Bontchev Str. Bill, 1113 Sofia, Bulgaria b Departement

de Chimie, Universite de Fribourg, Ckdu Musee 9, CH-1700 Fribourg, Switzerland

c Departement fur

Chemie und Biochemie, Universitdt Bern, Freiestrasse 3, CH-3000 Bern 9, Switzerland

Abstract A procedure for calculating magnetic exchange coupling constants in rare earth (RE)-transition metal (TM) dimers is presented. In a first step RE-TM transfer (hopping) integrals between orbitals carrying unpaired (magnetic) electrons on RE and TM are determined from a MO-calculation utilising the concept of orbital exchange pathway and effective Hamiltonian theory. In a second step, many-electron wave-functions for ground or excited electronic states on RE and TM are constructed in which spin-orbit coupling (which dominates for the RE) is fully (variationally) taken into account. Finally, the data from steps 1 and 2 are combined to obtain numerical values for kinetic exchange integrals (within Anderson's superexchange theory) utilising a nonHeisenberg (orbital dependent) exchange operator. The MO's corresponding

153

154 M. Atanasov, C. Daul and H. U. Giidel

to the d(Mn) and f(Yb) magnetic orbitals have been used to also calculate ferromagnetic contributions to the exchange coupling. The model is applied to Yb3+-Mn2+ dimers with corner, edge and face shared octahedra and Cl" and Br" ligands. For that purpose we utilise a spectroscopically adjusted (to spectra of single nucleous octahedral Mn2+, Cr3+ and Yb3+ complexes) parameterisation of the Extended Huckel theory. The expediency of the approach is discussed based on a comparison between calculated and experimental (available from inelastic neutron scattering experiments) anisotropic exchange parameters in Br3Cr3+-Ji(Br3) Yb3+Br3 dimers. Ferromagnetic exchange parameters for the ground state and the lowest emitting excited state increase from corner to edge to face Mn-Cl-Yb sharing thus varying in the same way as the efficiency of the up-conversion found for such species. This lends support for an exchange mechanism for this phenomenon postulated previously.

1. Introduction Interactions between transition metals (TM) and rare earths (RE) in solids have been of great interest because of their ability to combine photons of low energy to photons of higher energy (up-conversion, UC) '"2. It was discovered that pairs of Mn2+ and Yb3+ bridged by three or one Cl" ions are able to convert IR radiation at about 10300 cm"1 ( corresponding to the 2F7/2->2F5/2 excitation at Yb3+ ) into red (14700 cm"1, CsMnBr3:Yb - face sharing 3) or orange (16200 cm"1, Rb2MnCl4:Yb - corner sharing 4) luminescense, both corresponding to the "^Tig—>6Aig transition within the Mn2+ ion. Based on spectra in high resolution an exchange UC mechanism has been postulated. Surprisingly, no theoretical work in that direction have been done yet, in spite of the fact, that theories of excitations in pairs of exchange coupled TM are well documented5"10 and tested by experiment n"18. In this study we describe a procedure allowing to treat exchange interaction within a cluster of one TM and one RE, bridged by common ligands. Utilising MO wavefunctions and energies we base our model on the concept of localized magnetic orbitals allowing to treat both cluster ground and excited states and to account for both antiferromagnetic (kinetic) and ferromagnetic (potential) exchange. After an introduction into the modelling of exchange interactions using Anderson's exchange theory including ions in non-degenerate (Section 2.1) and degenerate (Section 2.2) electronic states, we focus on one example known from literature 19, that of CoCl2(H2O)2:Mn2+ with Mn2+ and Co2+ ions in the pair possessing ground states without and

Modelling of Anisotropic Exchange Coupling 155

with orbital degeneracy, respectively. For didactical purposes and including mathematical details as well, we derive expressions for the anisotropic exchange coupling tensor of a Mn2+-Co2+ pair for a general (without assuming any symmetry) bridging geometry. In Section 3 we further introduce a model for treating exchange coupling in TM(Mn2+)-RE(Yb3+) pairs. To approximate the model parameters we make use of a spectroscopically adjusted extended Hueckel theory which we introduce in Section 3.3. As a first application an elaborated example - that of MnC^4" and YbCl63" octahedra sharing one Cl" ligand will be presented in detail (Section 3.6.). In Section 4 we present additional applications to edge and face shared octrahedra of the same ions both in their ground state and in the lowest emitting excited state. In Section 4.2 the relevant aspects of our results regarding the mechanism of up-conversion will be discussed.

2. Modelling Exchange Interactions: Anderson's Exchange Theory 2.1. Exchange interactions for orbitally non-degenerate states Electrons of d (TM) and f(RE) orbitals are quite localized. Therefore, a good starting point in describing exchange coupling between electrons on different centers is to consider unperturbed d and f orbitals and to correct their wavefunctions affected by the surrounding ligands. Taking two TM centers with unpaired electrons occupying d^ orbitals on each (Fig.l) we consider magnetic orbitals dli and dl2 composed of d^1 and d^2 orbitals on centers 1 and 2 and (smaller) tails of the ligand (pz) (pL1 and = -(l/2)a(-l",-l/2)+(l/2)b (O',l/2) -(l/2)c (1 ',-1/2) Osz' | l/2>=-(l/2)aJ(l",l")(-l",-l/2)+(l/2)b J(0',0') (O',l/2) -(l/2)c J(l',l')(l',-l/2) Ja= 2 =(l/2)a(-l",l/2)+(l/2)b(O',-l/2) +(l/2)c (l',l/2) Os x '|l/2>=(l/2)aJ(l",l")(-l",l/2)+(l/2)b J(l',l')(l',l/2) inT 2 S=0 (spin-flip) transition of a transition metal with two unpaired electrons(such as Ni2+ in octahedral coordination) and S=l/2 ion (such as Cu2+ or Yb3+(pseudo-spin >/,)).

(58)

182 M. Atanasov, C. Daul and H. U. Gildel

The two states interact with S=l/2 charge transfer states in which one electron has jumped from TM to RE, L/ 2+ / + /~|,L/, + / + /i,or from RE to TM, d*d-d*\, d*dXd' • This mixing is the cause both for the exchange 1

1

2|

1 2

2

stabilization of »p£ over the »?£, = d*d^f*

(described by the kinetic

exchange integral jkn ) and the intensity of the *F^, - » ¥ ^ , pair transition. As shown in Appendix A.3, using both perturbation theory and applying explicitly the operator H'(eq.52), the ^F^, - ^ ^ transition probability, is (neglecting higher order two-electron terms) proportional to the square of the matrix element (yfD \H\^) •

W*->-^^W^ + W^>]~

(59)

Let us compare this result with Eq.(60) for the exchange integral Jn:

Jk =-h

2[

!

,

*

]

h 2f

1

|

1

We see that in addition to the sum over squares of the hd2fand common to \ fofi, H^SD) '2 (hd2f.hd}f)

an( ^

*î2 > t ' i e r e ^s

an

1 (60) hdlf,

interference term

which appears in the intensity expression. This can add or

subtract from the first one, to show that, in general, | (w°D //j*F^) 12 and J\2 will not be interrelated. However, as it frequently appears, one of the two terms, that for hd2f or hdxfy is small or zero and both | Nf^ M Y ^ , ) I2 and Jj2 are dominated just by one leading term. It then follows that for such cases the intensity of pair transitions will be proportional to J\2. Our results show, that this is the case for corner (see Table 5) and face sharing; in both cases, the exchange Mn2+-Yb3+ is governed by one single dominant contribution - the one between the d^ (Mn2+) - ti(O') f^ (Yb3+) orbitals: in the case of edge sharing several competing terms occur. In Fig.7 we have plot the Jkzz values for the complexes studied; we see that Mn2+-Yb3+ antiferromagnetic coupling is largest for corner sharing; it is tempting to relate


this result with the experimental finding, that up-conversion is most efficient in this case; the underlying electronic excitation involves a change of the electronic state created after the first absorption 6Ai(Mn2+)-2F7/2(Yb3+) - » 6Ai(Mn2+)-2F5/2(Yb3+) to a state where simultaneously Yb 3+ turns back into the ground state 2F5/2, while Mn 2+ (6Ai) becomes excited into the ''Tig state.

• Jzz A

y . y ^

\

MnCl7Yb

/\

-4 -

/ MhBrj^Ns/

-8 '

-12 -

MnCl.Yb

exc. state

\ \

MnClYb \

\

gr. state

\*

Fig.7 Antiferromagnetic J^ (kinetic) exchange coupling constants for Mn +-Yb3+ dimers with octahedrally coordinate Mn2+ and Yb3+ and three, two and one sharing halide atoms (face-, edge- and corner shared octahedra). Turning to the bilinear dependence of the intensity of pair transitions on terms of the form hlf,h\fwn& hlf.h2f we notice the following; integrals of


TM-RE hopping, hxf and hlf, are expressed in terms of hdl and hfl metalligand hopping integrals as follows:

(61)

hdf~_htL==_±£==

J I.

\Q

t gr. state

- ^ZZ

12 -

/ /

g .

/ /MnClYb

4" -

7 exc.state

Jl x* o vu MnBr3Yb

y S

MnCl2Yb

MnCl3Yb Fig.8 Ferromagnetic J^ (potential) exchange coupling constants for Mn2+-Yb3+ dimers with octahedrally coordinate Mn2+ and Yb3+ and three, two and one sharing halide atoms (face-, edge- and corner shared octahedra).

Since coefficents cM and cfl (Eq.46) depend linearly on hdi and hji as well, one can expect that integrals of potentials exchange Jf2 and the transition probability | (w^ H^¥^ \2 will show-up again in a symbathic behaviour,


they both increase with h. Here cross terms, hlf .h2f with contributions from different ligand orbitals are automatically included (which is the missing terms in the comparison of | (*F£ I t f 1 ^ ) ^

with

û)-

R is

striking, that

values of J?z plotted in Fig. 8 follow the same trend as the one found experimentally for the efficiency of light conversion; they increase from face to edge and corner sharing (cf with the ./,* plot, Fig 7) . We attribute this difference to mixed terms (h d l f . hdlf) which do not appear in J\2, but which are important in the case of edge sharing. The correlation does not change qualitatively when comparing exchange integrals for the ground and for the lowest emitting excited state of Yb3+ (see potential exchange terms in Table 6).

5. Conclusions 1) Exchange coupling between TM(Mn2+) and RE(Yb3+) has been studied experimentally with the aid of understanding the mechanism of up-conversion of light [3,4]. The analysis of the experimental data up to date has lead to the proposition that the phenomenon is governed by exchange coupling between the TM and RE. The modelling of exchange coupling between Mn2+ and Yb3+ lends support to this hypothesis; we find that both kinetic and potential contribution to the exchange coupling constant are largest for a corner sharing Mn2+-Yb3+ dimer. We further get the interesting result, that ferromagnetric exchange integrals exactly mirror trends born out by experiment: efficiency of up-conversion increases from face to edge to corner shared Mn-Yb ions. 2) We propose a procedure of calculating exchange coupling constants based on the following steps: (i) In a first step RE-TM transfer (hopping) integrals between orbitals carrying unpaired (magnetic) electrons on RE and TM are determined from a MO-calculation utilising the concept of orbital exchange pathway and effective Hamiltonian theory, (ii) In a second step, manyelectron wave-functions for ground or excited electronic states on RE and TM are constructed in which spin-orbit coupling (which dominates for the RE) is fully (variationally) taken into account, (iii) Finally, the data from steps 1 and 2 are combined to obtain numerical values for kinetic exchange integrals (within Anderson's super exchange theory) utilising a non-Heisenberg (orbital


dependent) exchange operator. The MO's corresponding to the d(Mn) and f(Yb) magnetic orbitals have been used to also calculate ferromagnetic exchange integrals as well. Our method has been implemented at the level of an extended Huckel model (EHT), where a restricted number of parameters has been adjusted to fit spectra in high resolution of separate MnCl64" and YbC^3" chromophores. However, the procedure is quite general and can be used in combination with any method (DFT or ab-initio) which is able to yield reasonable account for the magnetic orbitals involved. In this respect we have to state, that available basis functions and/or functionals within the DFT methodology are not sufficient yet to do this. We motivate further work for improving DFT for rare earths.

Appendices Appendix 1 A

Derivation of the orbital operator (O) for exchange coupling in the case of orbital degeneracy. The orbital operator O, representing the exchange coupling tensor J within the 1=1 basis "J(U) J(l,0) J(l-l)" J = J(0,l) J(0,0) J(O,-l) (A.1.1) j ( - u ) j(-i,o) j ( - i - i ) can be expanded into a series of orbital operators l»ly,lz acting on angular momentum eigen functions for 1=1, mi=0, ±1 and the identity operator 1 as follows: 6 = alz2 +b(UE+Ux)+c(IyIz+lzly) + d(lx2-ly2) +e (Uy+M*) + f 1

(A. 1.2)

The coefficients a to f can be determined in the following way; we substitute the matrices of the operators 1*, ly and lz and 1 2 1 :


l

o -4= o V2

x

= —

0

4i

o ^ L o 4i

—

1= —

J i

y

o -^ o

V 2

0

r

—

l

n r

[i o ol

z

4 i

o 4= o

= 0 0 0

-, p oo

1 = 0 1 0

o o - i o

-^(b-ic) V2 f

i (A.1.3)

into eq (A. 1.2) to obtain, after collecting terms: a+f

o

d-ie

6 = 4=0> + ic) ^ ( - b + ic) < A - L4 ) V2 V2 d + ie -pr(-b-ic) a+f V2 The comparison with eq (A. 1.1) yields the following connection between the parameters a to f and the components of the exchange tensor: J(l,l)=J(-l,-l) = a+f; J(l,O)=J'(l,O)+iJ"(l,O)= 1 ( b _ i c ) =J( 0 ,l)*; J(l,-l)=J'(l,-l)+iJ"(l,-l)= (d-ie)=J(-l,l)*; J(O,-I)=J'(O,-l)+iJ"(O,-l)= - L (_b+ic)=J(-l,0)'; V2 We thus obtain

(A. 1.5) J(O,O)=f

a=J(l,l)-J(O,O); f= J(0,0); b= b)

£(2)(^) = 0 and, since (S=3/2): S.JI2=E(S-1)-E(S);

Jl2

~ 3L

( TD)

(

ro)J"

_A2 [

albLAE(a,^b)

*

a2\hE(a2-*b)

2A£(i->a 2 ) (A.3.6)

AE(bâ,)_

1

1

(A.3.7)

AE(b^>a2)

First order corrections for *F^, and W^, are given by:

tf" =£ TD

fl

^

i2AE(ai-+b)

*(«, ->b)-M Kl

^

^(a2 ->6) +

' \2AE(a2^b)

*& ipp_>«,)_ I

^

K2

y(b_>a2)

(A.3.8)

192

M. Atanasov, C. Daul and H. U. Giidel

¥£(1) =4= +

—

¥(a, -»*>)+4=

—

V(a2 ->*>) +

1 K* »F0->fl.)+ - L ^ *P(6-»fl2) V2AE(6-+a,) ' V2A£(6â 2 )

(A-3-9>

To calculate the transition dipole *Fj^—>*F^,, the following matrix element of the dipole operator p has to be evaluated:

n ( ¥£->¥£)= («?£•> |jP^|vpW)}+(^> 1^1^")

(A.3.10)

Direct substitution from eq.A.3.2(for the zero order wave functions g-7J

Computational Chemistry Reviews of Current Trends

COMPUTATIONAL CHEMISTRY Reviews of Current Trends

Computational chemistry: reviews of current trends