New trends in Kramers' reaction rate theory

103 Pages·1995·4.321 MB·English

by  Peter Talkner, Peter Hänggi

roy
Understanding Chemical Reactivity
New Trends in Kramers'
Volume 11
Reaction Rate Theory
Series Editor
Paul G. Mezey, University of Saskatchewan, Saskatoon, Canada
Editorial Advisory Board
R. Stephen Berry, University of Chicago, /L, USA
Edited by
John I. Brauman, Stanford University, CA, USA
A. Welford Castleman, Jr., Pennsylvania State University, PA, USA
PETER TALKNER
Enrico Clementi, IBM Corporation, Kingston, NY, USA
Stephen A. Langhoff, NASA Ames Research Center, Moffett Field, CA, USA Paul Scherrer Institute
K. Morokuma, lnst#ute for Molecular Science, Okazaki, Japan Villingen, Switzerland
Peter J. Rossky, University of Texas at Austin, TX, USA
and
Zdenek Slanina, Czech Academy of Sciences, Prague, Czech Republic
Donald G. Truhlar, University of Minnesota, Minneapolis, MN, USA
PETER HANGGI
lvar Ugi, Technische Universitat, MOnchen, Germany
Department of Physics, Universitlit Augsburg,
Augsburg, Germany
.,
1
KLUWER ACADEMIC PUBLISHERS
The titles published In this series are listed at the end of this volume. DORDAECHT I BOSTON I LONDON
-:::w.
r__,
Table of Contents
Library of Congress Cataloglng-ln·Publlcatlon Data
New trends in Kra1ers' reaction rata theor~ I edited by Peter Tal~ner
and Peter Hiinggi.
p. c111. -- (Understanding chuical reactiVIty , v. 11l
Includes Index.
ISBN 0-7923-2940-S (acid-free paperl
1. Che1lcal k1netlcs. 2. Molecular theory. I. Tal~ner, Peter.
II. Hiinggl. Peter. III. Title: Kra.ers· reaction rate theory.
IV. Series.
00502.N49 1995
541 . 3' 94--dc20 94-20434
ISBN 0-7923-2940-6
Preface .,;; I
Published by Kluwer Academic Publishers, List of Contributors
P .0. Box 17, 3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
1. Introduction
the publishing programmes of
Peter Hiinggi and Peter Talkner
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
2. Variational Transition State Theory in Condensed Phases
by Kluwer Academic Publishers,
Susan C. Tucker
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed 3. Anharmonic Barrier Corrections for Kramers'
by Kluwer Academic Publishers Group, Rate-Problem in the Spatial Diffusion Regime
P. 0. Box 322, 3300 AH Dordrecht, The Netherlands.
Peter Talkner
4. The Theory of Multi-Barrier Crossing
P. lung and B. 1. Berne
Printed on acid-free paper
5. Surmounting Fluctuating Barriers
Peter Hiinggi
6. Noise~ Induced Transitions and Chemical Rate Laws
Raymond Kapral 107
7. Invariant Densities and Escape Rates for Maps with
All Rights Reserved Weak Gaussian Noise
© 1995 Kluwer Academic Publishers Peter Reimann and Peter Talkner 143
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
8. First Passage Time Renormalization and Escape from
including photocopying, recording or by any information storage and
retrieval system, without written pennission from the copyright owner. an Imperfect Trap
C. Van den Broeck and J. M. R. Parrondo 177
Printed in the Netherlands
--<:v.
-
vt Table of Contents
Preface
9. Stretched Times and Divergent Time Scales
189
John T. Bendler and Michael Shlesinger
10. A Feynman Path Integral Formulation of Quantum
Mechanical Transition State Theory
197
Gregory A. Voth
11. Recent Developments in the Quantum Mechanical Theory
of Chemical Reaction Rates
225
William H. Miller
247
Index
The theme of 'escape from metastable states', either via noise~assisted hop~
ping· and/or quantum tunneling, is pivotal to many scientific disciplines. It
impacts on such diverse physical, chemical and biological processes as diffu~
sian in solids, chemical reactions per se, nucleation phenomena and transfer
of matter and information in biological complexes, to name only a few.
With 'New Trends in Kramers' Reaction Rate Theory' this book fills yet
another part of the multifaceted scope which underpins the Understanding
of Chemical Reactivity. Since the publication of the comprehensive review
about reaction rate theory in Rev. Mod. Phys. 52, 251 (1990) the field has
witnessed many major developments and extensions both in experiment and
-theory. In this book the focus will be on the theoretical progress. In doing so,
the editors collected a series of authoritative articles from major practitioners
in the field which as a whole give a representative- although not complete
sample of the novel recent theoretical advances.
As an ine_vitable consequence, the editors recognize that not all readers
will wish to digest the volume in its entirety. We trust, however, that the
reader will be able to choose from the many methods and techniques which
he is interested in, and which he requires to perform his own new research in
this area. There is the consistent underlying theme of noise-assisted barrier
crossing that is running through all of the book. Nevertheless, each chapter
should be considered as self contained. In this spirit the editors share the
confident belief that the future research on the Kramers problem, and related
topics, will be invigorated by the selected contributions herein.
PETER TALKNER and PETER HANGG1,
December 1993
••
List of Contributors
John T. Bendler J. M. R. Parrondn
Department of Chemistry Dep. Fisica Aplicada I
S. Dakota School of Mines & Tech, Uni venidad Cnmpluteru;e
Rapid City, SD 57701 28040-Madrid
U.S.A. Spain
B. I. Berne Pcte.r Reimann
Department of ChemiW)' Lirnhurgs Untversitair Cernrum
Columbia University B-3590 Dlepenbeek
New York, NY 10027 Belgium
U.S.A.
Michael F. Shlesinger
Peter Httnggi Physics Divisinn
Department of Physics Office of Naval Research
Universitlit Augsburg 800 N. Quincy Street
Memminger Str. 6 Arlington, VA m17-5660
D-86135 Augsburg U.S.A.
Germany
Peter Thl.lmer
P. Jung Paul Scherret lnstitut
University of lllinois at Urbana-Champaign CH-5232 Vtlligen
Beckman Imni11.1te Switur1and
Urbana, lL 6l~tH
U.S.A. Susan C. Tucker
Department of Chemistry
Raymond Kapral Univenrity of Cali:fumla
Chemical Physics "''beoly Group Davis, CA 95616
Department of C1temhtry U.S.A,
Univenity nfThronto
Toronto M5S lAl C. Van den Brnt:oCk
c""""
Limlmf&$ Un!vmilau Centrum
B-3590 Diepenbeek:
William R Miller Belgium
Department of Chemistry
University of california Otepy A. VQth
""' Depllrtment nf Chemistty
Chernk:a!. S6ences Division Univ. ofPenru;ylvania
Lawrence Berkeley Labora:tory Philadelphia, PA 19104-6323
Berkeley, CA 94720 U.S.A
U.S.A.
'
Introduction
PETER HANGGI AND PETER TALKNER
The past two decades witnessed a remarkable increase in activity in the
study of noise-driven escape from metastable states. Reactive processes take
place while the system is undergoing interactions with its environment, which
generally consists of a solvent or a solid. The situation is therefore quite
distinct from the case where activated barrier crossing proceeds essentially
as a free flight over a reactive barrier. The latter situation is often modeled
accurately by the so called transition state theory (TST) in one-, or in many
dimensional configuration space. This TST predicts a rate coefficient k which
takes the form [1-3]:
(Z')
kBT
k ~ • Zo exp( -~E,), (I)
where Eb denotes the threshold energy for activation. In (1) Zo denotes
1
the partition function of the metastable state, {3 = (knT)- is the inverse
z#
temperature, and refers to the partition function of the transition state. This
theory a priori cannot account for the influence arising from the surrounding
solvent, unless, of course, the infinitely many bath degrees of freedom are
part of the many-dimensional configuration space. The effects induced by the
coupling to the environment, or more specifically: (i) the role of recrossing
of reactive trajectories, as well as (ii) the influence of any nonequilibrium for
the phase-space area near the threshold energy for reaction, are not accounted
for by the equilibrium theory underlying the TST answer in (1). The physics
of these latter effects modify the TST result in the form of a transmission
coefficient "'• with"' < 1, thus correcting the grande concept in (1) which
itself has survived as the reigning method of calculating rate processes for
many decades.
A first theory addressing the problem of the effect of coupling to an envi
ronment for reactions taking place in condensed phases was given by Kramers
in 1940 [4, 5]. Kramers modeled the influence of the environmental coupling
in terms of a macroscopic friction function. His work which served as the
stepping stone for many further developments treated both, the strong bath
coupling limit (i.e. high friction) and the weak bath-coupling limit (i.e. small
friction). Kramers' pioneering piece of work was_ fir.;t met with skepticism, ~
~t
•
'
z
Peter Hiinggi and Peter Talkner Introduction 3
and its general acceptance within both the physics and chemistry of the Kramers problem with inertia. In addition, the interrelationship among
communi~
tics has been slow. This very fact can most likely be traced to the style of such concepts as the the 'stochastic separatrix', the
'Rayleigh~coefficient',
Krarners himself\ namely work containing a whole series of gems, but as 'mean first passage time' and the 'Kramers form-function' is also elucidated
hi.~
a whole being somewhat cOI'Ijusing. see van Kampen in {6], therein.
As mentioned already, Kramcrs provided a rigorous treatment of both the Tile next six chapters rnainly deal with complex metastable systems. In
high-ro~moderate-friction regime and the low-friction regime, which nowa Chapter 4 one addresses the. problem of multiple barrier crossing in periodic
days are referred to as the spatial-diffusive regime and energy-diffusion con potentials which predominantly takes place at weak friction. The multiple
trolled reru;tion regime. respectively. A number of aspects of the barrier hopping rates are calculated in absence and in presence of an external force,
crossing problem were originaUy not addressed by Kramers, A few examples logether with corresponding probability distributions for the lengths of
cor~
are: {i) The problem of bridging the above mentioned two limiting regimes. related jumps.
i.e. the Kramers-tunwver problem. (ii) 'The roie of quantum mechanical tun The ever growing interest in transport in disordered systems spurred many
neling which impacts the escape rate at low temperatures. (iii) The finite theoretical practitioners to investigate rate problems in complex nonequilib
barrier corrections to the leading prefactor of the rare. These corrections to rium systems, which typically lack the condition of detailed balance. The
the leading prefactor depend both on the nonlinear form of the metastable problem of surmounting a ftuctuatlng barrier is the theme of Chapter 5. Here,
potential landscape and the friction mechanism (see in particular Chapter 3). several new phenomena are identified for the average escape time driven
(iv) The problem of escape in stationary metastable nonequUibrium systems sirnultanoously by both thennal white noise and environmental noise of finite
which typically lack: the property of 'detailed balance'. as well as the exten~ correlation (i.e. colored noise), with the latter controlling the temporal fluc
sion of rate theory to multi-dimensional state spaces wlth anisotropic friction tuations of the intervening potential barrier. The role of external colored
andfor discrete time evolution, to name only a few. In recent years, several of (dichotomous) noise-driven escape between attracting states is elucidated in
the&e problems (and others) have been srudled actively. As a result, the field further detail in Chapter 6. The transitions between a locally stable state and
of rate controlled processes witnessed many major new developments, both a stable limit cycle are investigated therein in some detail
in theory and~~ of equal importance- Jn experiment (5, 7]. Escape in complex systems exhibiting chaos can suitably he described In
With Nt.'W Trends in Kromers • Reaction Rare Theory the editors shaH focus terms of a discrete-time stochastic map dynamics. The discretization stems
primariJy on a set of recent theoretical advances which potentiully may guide either from a stroboscopic description of the continuous-time dynamics or a
even future developments, By doing so, we certainly are not able to cover Poincare-se<."tion description. The authors of Chapter 7 present an authorita
all of the exciting contributions which followed the pioneering paper by tive study of escape times and invariant probabilities for bistable maps that
Kramers. The interested reader, however. may consult recent review papers are driven by white noise. The following two Chapters 8 and 9 both deal with
such as [5, 7-10], which in several a'tpects complement our account, and transport properties of random waikers on a set of discrete lattice points. The
provide additional insight into topics covered herein, issue of' escape from a region embedded on a self-similar structure which
The following ten chapters mainly address classical a~pects of the theme contains a trap is tack1ed in Chapter 8 by use of most appealing methods such
·escape from metastable states •. Chapter 2 provides an appealing introduction as the decimation and the renormalization procedures, To point to one main
inro the Kramers rate theory. addressing both Markovian friction a"' well as resuJt, therein, the authors succeed to identify the conditions for the escape to
non-Markovian, i.e. meroory~friction controUed escape. By use of Variational obey an exponential law. In clear contrast, the authors of Chapter 9 describe
Transitloo State Theory (VTST). in which the bath degrees of freedom are the ubiquitous nonexponential behavior as it occurs in disordered systems
placed on equal footing with the reactive degrees of fret"..dom, one is able to such as in glasses. Even though individual hops behave Arrhenius-like, the
extend the original Kramers theory to memory-frk.1ion and to nonlinearlties averaging over all configurations can give rise to a nonexponentiai relaxation
of the reaction coordinate, and to nonharmonidties of the solute (bath) and in the form of a stretched-exponential. being characterized by a time-scale
system-bath interactions, Tile application of VTST, however~ restricts tbe which obeys as a function of temperatUre a VogeJ-Fulcber-type Jaw.
regime of validity for the rate to the regime of spatial-diffusion controlled The last two chapters deal with the influence of quantum mechanics on
escape, Le. the modifications lmplied by the very weak friction (i.e. energy reaction rates. Both chapters dio;c:u..<ts the problem of a quantum generalization
diffusion controlled escape) are not covered within VTST. In many practical of transition state theory. By use of the concept of the centroid--conditioned
applications the reaction time scales are of exponential, although observable free energy the author of Chapter 10 puts forward a path-integral formula
finite duration. This in turn implies that the noise strength is not asymptoticany tion of quantum-transition-state-theory for reactions occuning in condensed
small. so that finite banier corrections must be taken into accuunt for the value phases [for an alternative. unified approach to the qu.antum-Kramers-rate.
tl>ple·l'!-
oflhc tmnsmissiOII fll:ltor ~' Thls in Cbap<u 3 for lhc case wbicb unifonnly covers tbe whole temperature regime from absolute zero up
..
........
'
4 Peter Hiinggi and Peter Talkner
to room temperatures, note the periodic orbit approach in presence of dissi Variational Transition State Theory in Condensed Phases
pation described in Section IX.D in {5]; or the more detailed representation
given in [11]]. A rigorous quantum rate theory in terms of dynamically exact
S-matrix reaction probabilities is formulated in the final Chapter 11. Newest
developments for the evaluation of the fundamentally important quantity, SUSAN C. TUCKER
termed the 'cumulative reaction probability' are reported therein, and illus
trated with applications to elementary hydrogen transfer reactions. Up to a
factor given by the density of states for the reactant states, this cumulative
reaction probability equals the 'microcanonical' escape rate at given total
energy.
In conclusion, the selected contributions, provide the reader with a tour
through the complexity of the state-of-the-art reaction dynamics, and they
hopefully will put him at the forefront of rate related present and future
Abstract. Variational transition state theory is a powerful tool for tackling the problem of acti
research activities. vated barrier crossing in condensed phases, enabling the exact reduction of infmite-dimensional
problems to much more tractable few-dimensional problems. Variational transition state theory
may be used to go beyond Kramers' theory and consider the effects of memory friction and
solute nonlinearities on the rate of barrier crossing. In addition, since variational transition
References
state theory is a Hamiltonian based fonnalism, it is not tied to the limitations of generalized
Langevin equation dynamics; thus it can be used to treat nonlinear solute-solvent interactions.
1. H. Eyring, J. Chern. Phys. 3, 107 {1935). Other recent advances include the development of new formalisms for improved optimization
2. P. Pechukas, Ann. Rev. Phys. Chem. 32, 159 (1981). of the transition state dividing surface, the incorporation of intramolecular solute modes, and
3. K. J. Laidler and M. C. King, J. Plrys. Chem. 87, 2657 (1983). the development of centroid-based quantum transition state theory analogs.
4. H. A. Kramers, Physica (Utrecht) 7, 284 (1940).
5. For an extensive survey of Kramers-like approaches see: P. H!inggi, P. Talkner, and M.
Borkovec, Rev. Mod. Plrys. 62, 251 (1990).
6. N. G. van Kampen, Ber. BWJSenges. Plrys. Chem. 95, 225 (1991); in particular seep. 227.
7. G. R. Fleming and P. Hfulggi, Activated Barrier Crossing, World Scientific, Singapore
1. Introduction
(1993).
8. B. J. Berne, M. Borkovec, and J. E. Straub, J. Phys. Chem. 29, 371 (1988)
9. A. Nitzan,Adv. Chem. Plrys. 70 (part 2), 489 (1988).
10. V.I. Melnikov, Phys. Rep. 209, I (1991). A great majority of chemistry and biology occurs in solution, and it is there
11. P. H!inggi and W. Hont.scha, Ber. BWJSenges. Plrys. Chern. 95, 379 (1991). fore critical to understand the detailed nature of solvent effects on chemical
reactivity. Our basic understanding of the effect that solvents have on rates
of chemical reactions involving an activated complex goes back to Kramers'
elegant treatment of 1940 [1, 2]. However, Kramers treated only an extreme·
ly simplified physical model, and much modem research has been directed
towards the treatment of more realistic chemical and physical systems [2-7].
Recent research has shown that variational transition state theory (VTST) pro
vides a powerful tool for extending Kramers' theory to include the effects of
anharmonic potentials (nonlinear forces), slowly relaxing solvents (memory
friction), and quantum mechanics.
We begin with a short review of Kramers' theory. After a discussion
of memory friction, we review Grote-Hynes theory (GH), which extends
Kramers' theory to this more general case [8]. We then introduce transition
state theory (TST), emphasizing the connection of various formulations of
TST to the Kramecs' and Grote-Hynes theories. The bulk of the chapter then
"· .,
focuses upon recent formulations of TST which illustrate the power of this
. .. . formalism for going beyond the Kr&.mers' and Grote--Hyn~s results.
•
<
~,:, ,:
6 Susan C. Tucker Variational Transition State Theory in Condensed Phases 7
1.1. Kramers' theory Only in the last two decades, during which the advent of faster, more
tunable lasers has enabled experimentalists to probe the details of chemical
In 1940 Kramers published an expression for the rate of noise-assisted escape
reactions in condensed phases, has Kramers' theory come under experimental
of a particle from a metastable well [1]. He used a Langevin equation to
scrutiny. It is now clear that while Kramers' theory provides the foundation
describe the classical stochastic dynamics of a particle of mass _m moving on
of our understanding of nonequilibrium solvent effects on chemical reaction
a one-dimensional potential, W(q), and coupled to a heat bath [9-11]. The
rates, the simple physical model on which it is based frequently breaks down.
Langevin equation,
First, the Langevin equation is based on the assumption that the particle of
.. 8W(q) . R(t) .
interest moves more slowly than do the bath modes, and that, consequently,
Bq , (I)
mq=~ ~m11q+
there is a large separation in the time scales of these motions. It is now widely
accounts fof both the frictional (dissipative) and random forces exerted on accepted that this separation is invalid for many problems of physical interest.
the particle by the heat bath. Here q is the particle, or system, coordinate, Also, the Langevin equation is based on linear response and thus can not
with mass m. For chemical reactions, q is the one-dimensional, possibly describe nonlinear solute-solvent interactions. Second, Kramers considered
curvilinear, solute reaction coordinate. The bath, or solvent, is accounted the escape rate only in the limit of an infinitely high barrier for which the
for by a fluctuating random force R(t) and the damping parameter, 'rJ, which solute anhannonicity may be neglected and the stationary points of the solute
determines the degree of dissipation. The damping parameter is a microscopic potential modelled as hannonic (parabolic). Third, Kramers' result is based
analog of the viscosity and provides a measure of the coupling between the upon classical dynamics, and, as such, it is not rigorously applicable to many
particle (solute coordinate q) and the bath (solvent). chemical reactions of interest.
By recasting the Langevin equation in terms of a Fokker-Planck equation
[1 0, 11] Kramers was able to find the escape rate in both the weak and strong
1.2. Memory friction
damping limits. We will be concerned with the strong damping limit, as
TST applies primarily to this case. Also, the majority of chemical reactions In order to treat systems which do not satisfy the time scale separation condi
in liquids appear to fall in this regime [12]. Important recent developments tion of the Langevin equation, Kubo introduced on phenomenological grounds
applicable to the weak and intermediate damping regimes are thus beyond a generalized Langevin equation (GLE) [21, 22],
our scope, and the reader is referred to [13--17]. J .
t
Kramers' expression in the strong damping, or spatial diffusion, limit is .. 8W(q)
oq
mq= -m dq(t-r)q(r)+R(t). (4)
[I, 2]
[(ry' lz)'/' ry] 0
I -+w (2)
kK=w-l ~- ktdTST The GLE accounts for frictional forces which act on the same time scale
4 2
as the system motion by using a time-dependent friction kernel 1(t). This
where ktdTST is [18-20]
integral equation approach to including a time-dependent friction is often
Wo -i3W:j: called 'memory friction'. In the limit of an infinitely fast bath or, equivalently,
ktdTST =- e (3)
211" an infinitely slow particle, the friction acts instantaneously and t(t) = 21}6(t),
w+
where b(t) is the Dirac delta function. Under these conditions the original
as will be discussed explicitly below. Here is the barrier height of the
w+ Langevin equation is recovered. We adopt the terminology 'ohmic friction'
one-dimensional system potential W(q), w = (m-1W11(qo))112 and =
0 to denote this friction kernel. However, the reader should be aware that this
(m -tw''(q+)) 112 are the frequencies of the system potential of mean force
usage is not universal. Some authors use 'ohmic friction' to denote any friction
1
[20] at the well minimum and at the saddle point, respectively; f3 is (kbT)- , kernel for which t(t = 0) i- 0. The time-dependent rafidom force R(t) is
where kb is Boltzmann's constant and T is temperature. The prefactor in
assumed to satisfy a Gaussian probability distribution and to have an average
Kramers' result is always less than unity, indicating that the damping effect
value of zero. It determines the friction kernel via the Fluctuation-Dissipation
of the bath always decreases the rate relative to the 1dTST result. In the strong theorem [21, 22],
damping regime, this slowing of the rate is attributed to 'collisions' which
(R(t)R(O)) = mk,T -y(t). (5)
hinder the progress of the system in its escape over the barrier - hence the
designation 'spatial diffusion regime'. In the limit of very large damping,
While the GLE provides greater flexibility than the Langevin equation, it
w+,
» 1
71 K.ramers' result gives kK ex: ,- ; the rate is inversely proportional is not completely general. Specifically, the GLE cannot be derived fro~ a
to the_d:amping constant'" - · tza.
pn•••Hwniltnnian, <><!'"P!)n tho JiWt af linear """"""" ~1• It tliuo
·' "•
~
8 Susan C. Tucker Variational Transition State Theory in Condensed Phases 9
remains valid only for infinitesimal displacements in the system coordinate. term were not included in the Hamiltonian. However, as a consequence of
The GLE does rigorously describe finite motions in the system coordinate this simpleminded replacement of V by W, the Hamiltonian is no longer a
for very simple model Hamiltonians involving a harmonic bath bilinearly true Hamiltonian. Unlike a simple molecular potential function, the potential
coupled to the system coordinate, i.e. of mean force depends both upon temperature and upon an average of the
equilibrium solvent configurations. Hence, for real systems W(q) is likely
p2 np'l, In n
+ ; + + z w}x}- L: + to vary with solvent properties such as viscosity, and such variations must
H = ; ~ V(q) ~ CjXjQ l/;:hm(q), (6)
be explicitly included in Equation (6) when analyzing experiments. Towards
J J J
this end, Huston, Zichi and Rossky have derived a general expression relating
in the limit n oo [25-28]. To simplify, we have used (and will continue to
-t
the friction kernel and the potential of mean force, see [32].
1 2
use) mass weighted coordinates. The system coordinate q is really m- 1 q,
and similarly for the bath coordinate Xj. Here Pq(Pi) is the momentum conju
1.3. Grote-Hynes theory
gate to q(xj ), and V ( q) is the bare system potential in the absence of the bath.
The coupling constant determines the interaction between the jth bath
Cj Kramers' theory for the escape rate in the strong damping regime was first
mode, having frequency Wj, and the system coordinate q. Straightforward extended to the case of memory friction by Grote and Hynes [8]. Subsequent
evaluation of Hamilton's equations of motion for this Hamiltonian yields the
derivations have been given by Hanggi and Mojtabai [33], Pollak [34], and
GLE with the identification [25]
Dakhnovskii and Ovchinnikov [35] (see Section 2.3).
d
n To incorporate memory friction, Grote and Hynes used a GLE to describe
m
~ltl-~E~~·~t the dynamics of the motion along the system coordinate [8]. Like Kramers,
n.....,.oo _ W-
J J they modelled the system potential (of mean force) as a parabolic barrier. To
for the friction kernel. Note that the expression in Equation (7) is a periodic proceed, Grote and Hynes used the stable states picture of Northrup and Hynes
function with period T, and that to rigorously recover the GLE one must take to find the particle escape rate directly from the GLEdynamics [36]. The basic
1'( t) in the limit that the period T goes to infinity. idea of the stable states picture is as follows. Two sutfaces, SR and Sp, are
The GLE equivalent Hamiltonian commonly includes a shift term (as it placed on the reactants and products side of the barrier, respectively (Figure 1) .
does in Equation (6) [25], These sutfaces are placed such that they represent absorbing boundaries for
,....,cj
the particle flux, i.e. they are placed far enough from the barrier region that
2
(8) particles crossing from the barrier region to reactants (products) through SR
Vsruft(q) = ~ w q ·
2 2 (Sp) do not return to the barrier region. The rate of escape of the particle
J J
over the barrier is then given by the correlation function of the forward flux
When this shift term is included in the Hamiltonian, the coupling that gives rise
passing through surface SR at time zero, j(SR), with the forward flux passing
to the frictional forces does not also induce a change in the potential of mean
through the sutface Sp, j(Sp, t), at all later times, i.e.
force along the solute coordinate [25, 29-31}. Thus, for this Hamiltonian, the
I
00
potential of mean force along q, W(q) (see Equations (1) and (4)), is simply
k .. - dt (j(SR)J(Sp,t))R, (9)
V(q), the one-dimensional system potential in the absence of the bath. This
implicit absence of potential of mean force effects, also known as equilibrium 0
solvation or free energy effects, is appropriate for some problems, such as where ( ) R indicates an average over an equilibrium distribution of reactants.
when the heat bath is an ordered crystal (the case for which the Hamiltonian By following the flux of trajectories for the full time that they are in the
Equation (6) was devised) [25]. For many other problems, such as a reacting interaction region (the region between SR and Sp ), and counting as reactive
molecular solute in a liquid, the potential of mean force will not equal the only those trajectories which exit to products, this method will account for
bare solute potential V(q). The most straightforward mapping of such a the slowing of the rate with increased damping. For example, a trajectory
realistic system onto the model Hamiltonian, Equation (6), is accomplished by which enters the interaction region and subsequently undergoes a series of
replacing the bare solute potential V( q) with the actual potential of mean force collisions with the bath ('dissipative motion'), may be reversed in direction
W ( q) [29, 31]. The shift term, Vshift. then leaves this precalculated potential of and recross SR towards reactants (see Figure 1). Such a trajectory will not
mean force unaltered in the presence of the bilinear solute-solvent coupling, contribute to j(Sp, t) or to the rate. In the absence of dissipation (collisions),
and the mathematics follow unchanged from the case where W(q) = V(q). the trajectory pictured in Figure 1 would have continued on to products and
This replacement allows the inclusion of more realistic equilibrium solvation contributed to the escape rate. It is thus clear that dissipative motion will lead
effects than .would·. arise· from the biliaear. coupliogj i.e. - Vsbift, if the shift to a reduced rate of escape. \ ~ ,
"'1
10 Susan C. Tucker Variational Transition State Theory in Condensed Phases 11
Sp
SR responding to the unstable, divergent motion. Thus, an expansion of Cv(t) as
a sum of exponentials, i.e.
wl-z(-
L
c.(t) ~ CnAne-A.t + C,A,e+A,t),
(II)
n
is dominated by this component in the long time limit. We will see below that
this divergent component may be associated with motion along an unbound
collective coordinate comprised of both system and bath components. The
integrated value of Cv(t) will also be dominated by this component, and the
I
rate is found to be [8]
I
I
I
k C< _.!.._ lim Cv(t) = ~.
(12)
I
88 Jci
w+ Cv(t1 dtl w+
t--+oo )
I
I I The constant of proportionality is just the 1dTST rate constant. Thus, denoting
the stable states rate as the GH result, we have [8]
A,
koH = w+ ktdTST· (13)
I
The decrease in the rate due to solvent damping is accounted for by the
prefactor >.rfw+, which is the ratio of the divergent, or reactive, frequency
r >.r to the system barrier frequency, w+. The reactive frequency is found from
I the GLE (with W(q) parabolic) by recognizing that the Laplace transform of
I the velocity autocorrelation function is [8, 38]
• A
c.( .
(14)
A)~
1
A'+i(A)A-w 2
This expression for (\ ( >.) has a divergence when
Fig. I. The stable states picture. A trajectory (solid line) which crosses SR into the interaction A2 + i(A),\- wl2 ~ 0 (15)
region undergoes dissipative motion and returns to reactants. In the absence of a bath, this
trajectory would have continued on through Sp to products (dotted line). or
w1
2
(16)
,\, ~ -l,+i(A,)
Grote and Hynes [8] followed the trajectories in the interaction region by
using a GLE to define the probability, e(q, Q, t \ qo, tj0), that a particle will
Equations (13) and (16) comprise the Grote-Hynes (GH) result, and Equa
q
have position q and velocity at time t, given that it had position qo = S R
tion (16) is often called the Grote-Hynes equation [8}. The GH result reduces
and velocity Q- at time t = 0. From this distribution one can find the flux
0 to the Kramers' result in the limit of ohmic, or instantaneous, friction for
as a function of time and solve for the escape rate. The escape rate is found
which the GLE reduces to the Langevin equation. The GH result was sub
to depend upon the long time limit of the system velocity autocorrelation
sequently rederived by Hanggi and Mojtabai using the flux over population
function, Co(t) ~ (q(t)q(O)) [37], i.e.
method [33}. This derivation does not require Cv(t) to be expandable as a
I . Co(t) sum of exponentials [2, 33].
k 1 (10)
wt The GH theory provides a significant advance to Kramers' theory by allow
ss "'- t->=ill f~ C.., (t ') dt' ·
ing for memory friction. Memory friction has been shown to be considerable
When the dynamics of q are determined by the OLE with W(q) parabolic, less effective at slowing the rate than is instantaneous, or ohmic, friction; see,
the frequency .spectrum of c.(t) bas one positive frequency eompoaeul COl:· for example. [8. 31, 39].In fact. for many cases of interest the conection to the