THEORETICAL METHODS IN CONDENSED PHASE CHEMISTRY Progress in Theoretical Chemistry and Physics VOLUME Honorary Editors: W.N Lipscomb (Harvard University, Cambridge, MA, U.S.A.) I Prigogine (Université Libre de Bruxelles, Belgium) Editors-in-Chief: J Maruani (Laboratoire de Chimie Physique, Paris, France) S Wilson (Rutherford Appleton Laboratory, Oxfordshire, United Kingdom) Editorial Board: H.Ågren (Royal Institute of Technology, Stockholm, Sweden) D Avnir (Hebrew University of Jerusalem, Israel) J Cioslowski (Florida State University, Tallahassee, FL, U.S.A.) R Daudel (European Academy of Sciences, Paris,France) E.K.U Gross (Universität Würzburg Am Hubland, Germany) W.F van Gunsteren (ETH-Zentrum, Zürich, Switzerland) K Hirao (University of Tokyo,Japan) I Hubac (Komensky University, Bratislava, Slovakia) M.P Levy (Tulane University, New Orleans, LA, U.S.A.) G.L Malli (Simon Fraser University, Burnaby, BC, Canada) R McWeeny (Università di Pisa, Italy) P.G Mezey (University of Saskatchewan, Saskatoon, SK, Canada) M.A.C Nascimento (Instituto de Quimica, Rio de Janeiro, Brazil) J Rychlewski (Polish Academy of Sciences, Poznan, Poland) S.D Schwartz (Albert Einstein College of Medicine, New York City, U.S.A.) Y.G Smeyers (Instituto de Estructura de la Materia, Madrid, Spain) S Suhai (Cancer Research Center, Heidelberg, Germany) O Tapia (Uppsala University, Sweden) P.R Taylor (University of California, La Jolla, CA, U.S.A.) R.G Woolley (Nottingham Trent University, United Kingdom) Theoretical Methods in Condensed Phase Chemistry edited by Steven D Schwartz Albert Einstein College of Medicine, New York City, U.S.A KLUWER ACADEMIC PUBLISHERS NEW YORK / BOSTON / DORDRECHT / LONDON / MOSCOW eBook ISBN: Print ISBN: 0-306-46949-9 0-792-36687-5 ©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at: http://www.kluweronline.com http://www.ebooks.kluweronline.com Progress in Theoretical Chemistry and Physics A series reporting advances in theoretical molecular and material sciences, including theoretical, mathematical and computational chemistry, physical chemistry and chemicalphysics Aim and Scope Science progresses by a symbiotic interaction between theory and experiment: theory is used to interpret experimental results and may suggest new experiments; experiment helps to test theoretical predictions and may lead to improved theories Theoretical Chemistry (including Physical Chemistry and Chemical Physics) provides the conceptual and technical background and apparatus for the rationalisation of phenomena in the chemical sciences It is, therefore, a wide ranging subject, reflecting the diversity of molecular and related species and processes arising in chemical systems The book series Progress in Theoretical Chemistry and Physics aims to report advances in methods and applications in this extended domain It will comprise monographs as well as collections of papers on particular themes, which may arise from proceedings of symposia or invited papers on specific topics as well as initiatives from authors or translations The basic theories of physics – classical mechanics and electromagnetism, relativity theory, quantum mechanics, statistical mechanics, quantum electrodynamics – support the theoretical apparatus which is used in molecular sciences Quantum mechanics plays a particular role in theoretical chemistry, providing the basis for the valence theories which allow to interpret the structure of molecules and for the spectroscopic models employed in the determination of structural information from spectral patterns Indeed, Quantum Chemistry often appears synonymous with Theoretical Chemistry: it will, therefore, constitute a major part of this book series However, the scope of the series will also include other areas of theoretical chemistry, such as mathematical chemistry (which involves the use of algebra and topology in the analysis of molecular structures and reactions); molecular mechanics, molecular dynamics and chemical thermodynamics, which play an important role in rationalizing the geometric and electronic structures of molecular assemblies and polymers, clusters and crystals; surface, interface, solvent and solid-state effects; excited-state dynamics, reactive collisions, and chemical reactions Recent decades have seen the emergence of a novel approach to scientific research, based on the exploitation of fast electronic digital computers Computation provides a method of investigation which transcends the traditional division between theory and experiment Computer-assisted simulation and design may afford a solution to complex problems which would otherwise be intractable to theoretical analysis, and may also provide a viable alternative to difficult or costly laboratory experiments Though stemming from Theoretical Chemistry, Computational Chemistry is a field of research v Progress in Theoretical Chemistry and Physics vi in its own right, which can help to test theoretical predictions and may also suggest improved theories The field of theoretical molecular sciences ranges from fundamental physical questions relevant to the molecular concept, through the statics and dynamics of isolated molecules, aggregates and materials, molecular properties and interactions, and the role of molecules in the biological sciences Therefore, it involves the physical basis for geometric and electronic structure, states of aggregation, physical and chemical transformations, thermodynamic and kinetic properties, as well as unusual properties such as extreme flexibility or strong relativistic or quantum-field effects, extreme conditions such as intense radiation fields or interaction with the continuum, and the specificity of biochemical reactions Theoretical chemistry has an applied branch – a part of molecular engineering, which involves the investigation of structure–property relationships aiming at the design, synthesis and application of molecules and materials endowed with specific functions, now in demand in such areas as molecular electronics, drug design or genetic engineering Relevant properties include conductivity (normal, semi- and supra-), magnetism (ferro- or ferri-), optoelectronic effects (involving nonlinear response), photochromism and photoreactivity, radiation and thermal resistance, molecular recognition and information processing, and biological and pharmaceutical activities, as well as properties favouring self-assembling mechanisms and combination properties needed in multifunctional systems Progress in Theoretical Chemistry and Physics is made at different rates in these various research fields The aim of this book series is to provide timely and in-depth coverage of selected topics and broad-ranging yet detailed analysis of contemporary theories and their applications The series will be of primary interest to those whose research is directly concerned with the development and application of theoretical approaches in the chemical sciences It will provide up-to-date reports on theoretical methods for the chemist, thermodynamician or spectroscopist, the atomic, molecular or cluster physicist, and the biochemist or molecular biologist who wish to employ techniques developed in theoretical, mathematical or computational chemistry in their research programmes It is also intended to provide the graduate student with a readily accessible documentation on various branches of theoretical chemistry, physical chemistry and chemical physics Contents Preface xi Classical and quantum rate theory for condensed phases Eli Pollak I II III IV V VI Introduction The GLE as a paradigm of condensed phase systems Variational rate theory Turnover theory Quantum rate theory Discussion Feynman path centroid dynamics 16 26 34 47 Gregory A Voth I II III IV V VI Introduction The centroid distribution function Exact formulation ofcentroid dynamics The centroid molecular dynamics approximation Some applications of centroid molecular dynamics Concluding remarks Proton transfer in condensed phases: beyond the quantum Kramers paradigm 47 49 52 58 60 63 69 Dimitri Antoniou and Steven D Schwartz I II III IV V VI Introduction Calculation of quantum transfer rates Rate promoting vibrations Position-dependent friction Effect of low-frequency modes of the environment Conclusions vii 70 72 78 82 85 88 viii THEORETICAL METHODS IN CONDENSED PHASE CHEMISTRY Nonstationary stochastic dynamics and applications to chemical physics 91 Rigoberto Hernandez and Frank L Somer, Jr I II III IV V Introduction Nonstationary stochastic models Application to polymer systems Application to protein folding Concluding remarks Orbital-free kinetic-energy density functional theory 92 94 104 110 111 117 Yan A Wang and Emily A Carter I II III IV V VI VII Introduction The Thomas-Fermi model and extensions The von Weizsäcker model and extensions Correct response behavior Nonlocal density approximations Numerical implementations Applications and future prospects 119 123 130 133 141 156 166 Semiclassical surface hopping methods for nonadiabatic transitions in condensed phases 185 Michael F Herman I 11 III IV Introduction Semiclassical surface-hopping methods for nonadiabatic problems Numerical calculations of vibrational population relaxation Summary Mechanistic studies of solvation dynamics in liquids 186 187 198 203 207 Branka M Ladanyi I II III IV V Introduction The basics of solvation dynamics Solvation dynamics within the linear response approximation Nonlinear solvation response Summary Theoretical chemistry of heterogeneous reactions of atmospheric importance: the HCl+ ClONO2 reaction on ice 207 209 213 225 229 235 Roberto Bianco and James T Hynes I II III IV Introduction HCl+ ClONO2 → Cl + HNO3 on ice _ C1– + ClONO2 → Cl + NO on ice Concluding remarks 235 236 242 243 Contents Simulation of chamical reactions in solution using an ab initio molecular orbital-valence bond model ix 247 Jiali Gao and Yirong Mo I II III IV V Introduction Methods Free energy simulation method Computational details Results and discussion 10 Methods for finding saddle points and minimum energy paths 248 249 253 256 257 269 Graeme Henkelman, Gísli Jóhannesson and Hannes Jónsson I Introduction II The Drag method III The NEB method IV The CI-NEB method V The CPR method VI The Ridge method VII The DHS method VIII The Dimer method Configurational change in an island on FCC( 111) IX X Results XI Discussion XII Summary Appendix: The two-dimensional test problem Index 269 272 273 279 279 280 281 282 283 284 286 286 287 303 Methods for finding saddle points and minimum energy paths [62] G T Barkema and N Mousseau, Phys Rev Lett 77, 4358 (1996) [63] Polanyi and Wong, J Chem Phys 51, 1439 (1969) 291 292 G Henkelman, G Jóhannesson and H Jónsson Table I Number of force evaluations needed to reach saddle point to 0.01 eV/ tolerance in the force saddle Drag CI-NEB(3) CI-NEB(1) Ridge CPR 10 11 12 13 47 37 146 149 156 153 81 75 285 276 333 654 735 300 351 363 282 294 333 25 25 177 179 151 204 206 163 179 115 126 48 105 189 288 1369 1129 1165 1369 1245 772 781 734 869 884 913 Average Std 115 56 336 184 131 64 901 368 DHS Dimer ART 241 232 240 230 1277 788 1464 785 1443 736 2412 2434 2426 2057 776 526 748 483 1551 736 2612 706 718 521 686 478 80 76 439 94 354 449 430 262 281 510 214 186 304 83 70 246 236 250 380 386 - 1276 810 283 149 236 125 824 662 Methods for finding saddle points and minimum energy paths 293 Table II Number of force evaluations needed to reach saddle point to 0.001 eV/Å tolerance in the force saddle Drag CI-NEB(3) CI-NEB(1) Ridge CPR DHS Dimer ART 10 11 12 13 324 70 323 338 299 293 372 192 597 585 675 999 978 573 855 648 447 687 738 122 45 27 246 314 274 271 309 446 174 237 150 230 3441 288 2382 2047 2112 2187 2144 4090 1995 1610 1859 1861 1901 653 433 1610 1729 1695 2821 2720 1197 1268 1739 2793 1038 969 795 290 1295 1296 1258 4310 4076 1320 1342 1468 1474 1160 1097 328 244 746 546 570 704 588 559 553 816 308 386 562 332 146 336 366 377 742 754 - Average Std 275 102 642 228 242 103 2147 890 1590 788 1629 1182 532 173 436 227 294 G Henkelman, G Jóhannesson and H Jónsson Figure The ‘drag ’ method A drag coordinate is defined by interpolating from R to P with a straight line (dashed line) Startingfrom R, the drag coordinate is increased stepwise and heldfixed while relaxing all other degrees of freedom in the system In a two-dimensional system, the relaxation is along a line perpendicular to the P – R vector The solid lines show thefirst and last relaxation line in the drag calculation The final location of the system after relaxation is shown withfilled circles As the drag coordinate is increased, the system climbs up the potential surface close to the slowest ascent path, reaching a potential larger than the saddle point, and then, eventually, slipping over to the product well In this simple test case, the drag method cannot locate the saddle point Methods for finding saddle points and minimum energy paths 295 Figure The Climbing Image Nudged Elastic Band method, CI-NEB An elastic band is formed with three movable images of the system connected by springs and placed between the fixed endpoints, R and P The calculation is started by placing the three images along a straight line interpolation The images are then relaxed keeping only the the component of the spring force parallel to the path and the component of the true force perpendicular to the path The image with the highest energy is also forced to move uphill along the parallel component of the true force to the saddle point 296 G Henkelman, G Jóhannesson and H Jónsson Figure The conjugate peak refinement (CPR) method Points along a path connecting R and P are generated, one point at a time through a cycle of maximization and then minimization First, the maximum along the vector P – R isfound, y1 Then, a minimization is carried out along a conjugate vector (small dashed line) to give location x1 on the path In the second cycle (shown in inset) the maximum along an estimated tangent to the R – x1 – P path (solid line in inset) is found, y2, and then energy is minimized along a conjugate vector (small dashed line in inset) to give a fourth point along the path, etc Methods for finding saddle points and minimum energy paths 297 Figure The Ridge method A pair of images on each side of the potential energy ridge is moved towards the saddle point First, the maximum along the vector P – R is found, point a in the inset Then the two images an formed on each side of the maximum, points x´0 and x´0 , and are displaced downhill along the gradient to points x″0 and x″1 This cycle of maximization between the two images, and the downhill move of the two images along the gradient is repeated, with smaller and smaller displacements until the saddle point is reached 298 G Henkelman, G Jóhannesson and H Jónsson Figure The method of Dewar, Healy and Stewart (DHS) Initially, a pair of images is created at R and P In each cycle, the lower energy image is pulled towards the higher energy one and then allowed to relax keeping the distance between the two fixed Eventually, the two images straddle the energy ridge near the saddle point Methods for finding saddle points and minimum energy paths 299 Figure The calculation of the effective force in the Dimer method A pair of images, spaced apart by a small distance, on the order of 0.1Å is rotated to minimize the energy This gives the direction of the lowest frequency normal mode The component of the force in the direction of the dimer is then inverted and the minimization of this effective force leads to convergence to a saddle point No reference is made to the final state 300 G Henkelman, G Jóhannesson and H Jónsson Figure Application of the dimer method to a two-dimensional test problem Three different starting points are generated in the reactant region by taking extrema along a high temperature dynamical trajectory From each one of these, the dimer isfirst translated only in the direction of the lowest mode, but once the dimer is out of the convex region a full optimization of the effective force is carried out at each step (thus the kink in two of the paths) Each one of the three starting points leads to a different saddle point in this case Methods for finding saddle points and minimum energy paths 301 Figure On-top view of the surface and the seven atom island used to test the various saddle point search methods The shading indicates the height of the atonis The initial state is shown on top The saddle point configutration and thefinal state of the 13 transitions are also shown, with the energy of the saddle point (in eV) indicated to the left The first two transitions correspond to a uniform translation of the intact island Transitions 3-5 correspond to a pair of atoms sliding to adjacent FCC sites In transitions and the pair of atoms slides to the adjacent HCP sites and the remaining atom slide in the opposite direction to HCP sites In transitions and 9, a row of three edge atoms slides into adjacent FCC sites In transitions 10 and 11 a pair of edge atom moves in such a way that one of the atoms is displaced away from the island while the other atom takes its place In transitions 12 and 13 a single atom gets displaced 302 G Henkelman, G Jóhannesson and H Jónsson Figure The frequency at which the various saddle points for the surface island transitions (illustrated in figure 8) are found with the Dimer method The lowest saddle points are found with the highest frequency Also shown are the number of iterations required to go from the intial state to the saddle point to within a force tolerance of 0.001 eV/Å For the more practical 0.01 eV/Å tolerance, the average number of force evaluations was a little under 300 The error bars show the standard deviation Index Centroid methods, 47–62 density, 48 distribution function, 51 molecular dynamics, 58–62 potential of mean force, 53 quasi-density operator, 52 time correlation functions, 57 variables, equilibrium, 53 dynamical, 54 irreversible GLE, 96-104 numerical solution parabolic barrier, 5-6 space-dependent friction, 4, 14, 82– 85, 96 spatial diffusion regime, 22 turnover theories, 16-20 Grote-Hynes theory, 5–6, 12, 71 Hohenberg-Kohn theory, 119 energy density functionals, 119 HK equations, 121 variational optimization, 157 Density matrix methods, 26 diffusion, coefficient of, 24 on surfaces, 22, 34, 271, 282–283, 294 Kohn-Sham potentials, 119, 122 Laplace inversion methods, 27-28 Energy density functionals, linear response, 133–138 non-local approximation, 141–156 Marcus’ theory, 72-73 Miller-Schwartz-Tromp rate formula, 30, 74 exponential resummation of evolution operator, 74–75 mixed quantum-classical methods, 32, 248 Fluctuational barrier preparation, 78 molecular orbital valence bond method, 252-265 friction kernel, 5, 71, 83 colored, 92 stationary, 95 Polymerization, 104–105 dense, 109–110 iGLE model, 106–108 Generalized Langevin equation, 3,70, 94 energy diffusion regime of GLE, 18, 20–21 Gaussian noise, 95 potential of mean force, 3, 13, 17, 95 centroid, 53, 59 303 304 for polymers, 106, 109 for surface diffusion, 23 umbrella sampling calculation, 255 protein folding, 110 Rate promoting vibrations, 78–80 Rayleigh quotient method, 9–10 reactive-flux method, 8–9 Saddle points, methods for finding, 272–285 solvation dynamics, 209–221 dielectric relaxation, 225 linear response, 213–224 nonlinear response, 225–226 solvation influence spectrum, 214– 216 solvation velocity, 213 role of static correlations, 220– 222 time-domain methods, 217–219 Spectral density, 6, 24, 75, 94 low-frequency branch, 85–87 surface-hopping methods, 187–198 initial value representation, 192 Herman-Kluk propagator, 192 non-adiabatic coupling vector, 188, 190 prefactor of propagator, 77, 186, 196 Thomas-Fermi model, 123 gradient expansion, 125 Variational TST, 11-15 variational quantum TST, 30–32 vibrational population relaxation, 198 Water quantum, 62 proton transfer in, 62, 257 Wolynes’ formula, 22, 73 Weizsäker model, 124, 130–133 Zwanzig Hamiltonian, 4, 71 Progress in Theoretical Chemistry and Physics S Durand-Yidal, J.-P Simonin and P Turq: Electrolytes at Interfaces 2000 ISBN 0-7923-5922-4 A Hernandez-Laguna, J Maruani, R McWeeny and S Wilson (eds.): Quantum Systems in Chemistry and Physics Volume 1: Basic Problems and Model Systems, Granada, Spain, 1997 2000 ISBN 0-7923-5969-0; Set 0-7923-5971-2 A Hernandez-Laguna, J Maruani, R McWeeny and S Wilson (eds.): Quantum Systems in Chemistry and Physics Volume 2: Advanced Problems and Complex Systems, Granada, Spain, 1998 2000 ISBN 0-7923-5970-4; Set 0-7923-5971-2 J.S Avery: Hyperspherical Harmonics and Generalized Sturmians 1999 ISBN 0-7923-6087-7 S.D Schwartz (ed.): Theoretical Methods in Condensed Phase Chemistry 2000 ISBN 0-7923-6687-5 KLUWER ACADEMIC PUBLISHERS – NEW YORK/ BOSTON / DORDRECHT / LONDON / MOSCOW ... and related species and processes arising in chemical systems The book series Progress in Theoretical Chemistry and Physics aims to report advances in methods and applications in this extended... theories are discussed in Section V and the review ends with a Discussion of some open questions and problems II II.1 THE GLE AS A PARADIGM OF CONDENSED PHASE SYSTEMS THE GLE In Kramers’39 classical... possible to present a theory which provides the necessary concepts and insight needed for understanding rate processes in condensed phases? Although classical molecular dynamics computations