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dissertation entitled

THE STEADY-STATE GREEN'S FUNCTION METHOD AND ITS APPLICATIONS
TO UNIMOLECULAR REACTIONS IN LIQUIDS.
BOND-BREAKING ELECTRON LRANSFER REACTIONS
presen ed y

Olga Borisovna Jenkins

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Chemistry

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THE STEADY-STATE GREEN‘S FUNCTION METHOD AND ITS APPLICATIONS
TO UNIMOLECULAR REACTIONS IN LIQUIDS.
BOND-BREAKING ELECTRON TRANSFER REACTIONS

By

Olga Borisovna Jenkins

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1995

ABSTRACT

THE STEADY-STATE GREEN'S FUNCTION METHOD AND ITS APPLICATIONS
TO UNIMOLECULAR REACTIONS IN LIQUIDS.
BOND-BREAKING ELECTRON TRANSFER REACTIONS

By

Olga Borisovna Jenkins

A new theory for evaluation of the dynamic effects in unimolecular reaction in
liquids is presented. Unlike previous theories which require calculation of time-dependent
population probabilities, our theory is based on steady-state differential or integral
equations for evaluation of the reaction rate constants. New forms of the Green's functions
convenient for evaluation of reaction rate constants, the steady-state Green's functions,
were introduced and their physical meaning was established. Predicated upon these
Green's functions the concept of mean first passage time (MFPT) was generalized from the
"MFPT to a point" to the definition of the "effective MFPT", which was defined as an
average time to reach a region on the reactive surface, rather than a point.

In the frameworks of the suggested steady-state Green's function theory, the
decoupling approximation for solution of the integral equations was designed and the
criterion of its applicability was formulated. We explicitly proved that for reactiOns on
narrow sinks the dynamic effects can be separated from the rest of the reaction, and thus

can be treated independently.

The steady-state Green's function theory as well as the effective MFPT concept
were tested on one- and two-dimensional potential surface reactions. Special attention was
addressed to high activation barrier reactions on narrow as well as wide reaction sinks. The
rate constants for reactions on one- and two-dimensional harmonic oscillator and bistable
potentials were evaluated for all barrier heights.

The bond-breaking electron transfer reactions (BBET) were considered as a special
application of the presented theory. The effect of solvent and intramolecular dynamics on
the rate of concerted as well as consecutive BBET reactions was investigated. The reaction
took place on two-dimensional diabatic potential surfaces with one coordinate the solvent
polarization and the other the breaking bond's displacement. We assumed that the
polarization fluctuations are always overdamped and viewed two classical (overdamped and
energy-diffusion) and one quantum mechanisms of dynamics along the bond-breaking
coordinate. For the classical dynamics along the bond-breaking coordinate we found that
the overall rate for both concerted and consecutive BBET can be considered as a
combination of non-adiabatic rate constants (TST analogs) and various effective MFPT
(purely responsible for the dynamic effects). A special method of analysis was developed
for situations where one dynamics is fast/slow compared with the other.

For the quantum bond-breaking mechanism it was found that the original two-
dimensional surfaces are reduced to one-dimensional diabatic potentials with multiple
channels providing the decay into the products. In this picture the overall rate constant can

never be separated into non-adiabatic and diffusional parts.

ACKNOWLEDGMENTS

It is my pleasure to thank Professor R. I. Cukier, my advisor, for his
encouragement and help during my research work at Michigan State University.

I am grateful to Professor A. B. Doktorov who initially directed my research
interests into the field of stochastic reaction theory. I also would like to express my
gratitude Professors A. I. Burshtein, K. Hunt and A. A. Ovchinnikov for their interest in
my work.

The Department of Chemistry at Michigan State University is also acknowledged

for offering me its resources and support during the preparation of this dissertation.

iv

TABLE OF CONTENTS

Page
LIST OF FIGURES .................................................................. viii
INTRODUCTION ..................................................................... 1
PART I. THE STEADY-STATE GREEN '8 FUNCTION THEORY FOR
UNIMOLECULAR IRREVERSIBLE REACTIONS IN LIQUIDS ............. 6

CHAPTER 1. THEORY, GENERAL OBSERVATION OF IRREVERSIBLE
REACTIONS ........................................................................... 6

CHAPTER II. REVERSIBLE REACTIONS ..................................... 11

CHAPTER III. REACTION RATE CONSTANT S AND THE CONNECTION

BETWEEN THEM ................................................................... 13
I. The reaction rate "constant" ....................................... p ...... l4
2. The integral kernel of the kinetic equation ............................. 15

CHAPTER IV. INTEGRAL FORM OF THE PRINCIPAL EQUATIONS AND

THE STEADY-STATE GREEN'S FUNCTION METHOD .................... l8
1. The Green 's function representation ................................... 18
2. The steady-state Green 's functions ..................................... 19

CHAPTER V. REACTIONS ON THE PINHOLE AND

NARROW SINKS ................................................................... 22
1. Reactions on the pinhole sink ........................................... 22
2. Reactions on the narrow sink ........................................... 25
CHAPTER VI. THE DECOUPLING APPROXIMATION .................... i 30
CHAPTER VII. APPLICATION OF THE STEADY-STATE GREEN'S
FUNCTION METHOD ............................................................. 36
I. The bistable potential .................................................... 36

2. The harmonic oscillator potential ....................................... 41

2.1 One-dimensional harmonic oscillator V(x)=x2 ............... 42
2.2 Two dimensional harmonic oscillator V(x,y)=x2/2.+y2/2 ..... 5l
3. Concluding remarks ...................................................... 56

PART 1] APPLICATION OF THE STEADY-STATE GREEN '8 FUNCTION
METHOD TO BOND-BREAKING ELECTRON TRANSFER

REACTIONS .......................................................................... 59
CHAPTER VIII. INTRODUCTION TO PART II ................................ 59
CHAPTER IX. POTENTIAL ENERGY SURFACES ........................... 65

CHAPTER X. ENERGY-DIFFUSION MECHANISM IN BOND-BREAKING

ELECTRON TRANSFER REACTIONS .......................................... 71
1. The equation of motion and the sink function ......................... 71
2. The reaction rate constant ................................................ 73
3. The energy diffusion operator ........................................... 74
4. Integral formalism ......................................................... 75

5. General form of the non-adiabatic and difi’usional rate constants 77

CHAPTER XI. FAST AND SLOW MODES IN MULTIDINIENSIONAL

KINETIC PROCESSES ............................................................. 80
I. General aspects ........................................................... 80
2. Fast bond-breaking ....................................................... 85
3. Fast polarization .......................................................... 87

CHAPTER XII. GENERAL ANALYSIS OF THE CONCERTED BBET
THEORY ............................................................................... 9O
1. Energy diffusion mechanism and comparison with
other BBET theories . ..................................................... 9O
2. Concluding remarks on the energy diffusion in BBET reactions 97

CHAPTER XIII. QUANTUM MODE IN BOND-BREAKING ELECTRON
TRANSFER REACTIONS ......................................................... 101
I. General remarks and structure of the potential surfaces ............ lOl

vi

2. Interference of the reaction channels in BBET with quantum

bond-breaking ........................................................... 105

3. Concluding remarks .................................................... 124
CHAPTER XIV. CONSECUTIVE BBET ....................................... 126
1. Reaction kinetics in the consecutive BBET .......................... 128

2. Potential surfaces for the intermediate state and non-adiabatic

rate constants ............................................................ 133

3. MFPT on the intermediate state PES .................................. 136

4. Concluding remarks ..................................................... 147
CHAPTER XV. CONCLUSION .................................................. 151

APPENDIX A. PSEUDOSTATIONARY GREEN '8 FUNCTION.
CALCULATION OF Go(x,x') and 1p ............................................. 158

APPENDIX B. CALCULATION OF 1p AND G(x,x' ,3) FOR HIGH
ACTIVATION POTENTIAL BARRIER .......................................... 163

APPENDIX C. THE TIME—DEPENDENT GREEN '8 FUNCTION FOR
HIGH ACTIVATION BARRIER .................................................. 166

APPENDIX D. THE STEADY-STATE GREEN'S FUNCTION FOR
HIGH ACTIVATION BARRIER .................................................. 167

APPENDIX E. EVALUATION OF THE EFFECTIVE MFPT FOR THE
INTERMEDIATES ON TWO-DIMENSIONAL HARMONIC OCSILLATOR
POTENTIAL SURFACE ........................................................... 168

BIBLIOGRAPHY 172

LIST OF FIGURES

Page
Figure 1. Schematic picture of reaction kinetics Pe(t) ...................................... 8
Figure 2. Profile of the bistable potential given by Eq. (1.7.1) ........................... 36

Figure 3. Dependence of the relative MFPT values g(BEA) and gem-ABBA) in Eqs. (1.7.7a)
and (1.7.7b). gdBEA) corresponds to the function g(BEA) calculated for the Boltzmann
initial distribution inside the reactant well, f0. g5(BEA) is calculated for the 5-functional

initial distribution fo' (x) = 5(x + Zi/EA lfia) .............................................. 39

Figure 4. MFPT for the reactants to reach the product well 1p? W and MFPT for the reactant
to reach the top of the barrier, tpTB are shown as the ratios to the Kramers' MFPT 40

Figure 5. Dependence of the reaction rate constants 2e, k and k' on the relative sink
intensity q=K,/do , where do=kleK3T exp(- 1. xp2/(2KBT)) for the reaction on the 6-

function sink located at three different barrier heights on the harmonic oscillation reaction
surface V(x)=(l/2) M2 : a) A xpz/(KBT) = O; b) I. xp2/(KBT) = l and c) hxp2/(KBT) = 2.

 

. . . a: a . .
(— —): generalized probability a: = ' d ,where zerzK, and and =td, lS given as the
a: +3.:

dependence p=a:/do on q; ( ): long-time rate constant k which is the solution of
Eq.(7.24) given as a function p=kldo of q; (- - -): approximation of the real long-time rate

constant k with the form k'=K,Kd /(K,+Kd) , p=k'/do .................................... 44

 

Figure 6. The dependence of the dimensionless pre-exponential factor for the effective
MFPT given in Eq. (1.7.21) on the activation energy for the reaction on Gaussian sink
(1.7.15). The dependence is presented for three different widths of the reaction sink,

0': A“ 4; (6:0 corresponds to the pinhole sink W(x)=Ko 5(x-xp)) .................... 51

Figure 7. Dependence of the reciprocal of the dimensionless effective MFPT for the
reaction on two-dimensional harmonic oscillator potential, T on the ratio of the relaxation
times along the reactive coordinates, tx/ty is given for the two barrier heights: a) EA=1

viii

KBT; b) EA=lO KBT. The reaction region in this case was taken as a line on the reaction

surface ........................................................................................... 54

Fig. 8. Schematic picture of the potentials for each of the coordinates participating in
BBET: a) the orientational polarization (or ET) coordinate; b) the bond-breaking (BB)
coordinate ....................................................................................... 67

Fig. 9. Concerted pathway (a) is sketched in comparison with the consecutive
(b) one ........................................................................................... 70

Fig. 10. Schematic plot of the potential energy surface for concerted BBET
reactions ......................................................................................... 72

Fig. 11. The logarithmic dependence of three dimensionless pre exponential factors (X 12x )
is given for three dynamic rate constants: 1) 2D BBET rate constant Kd, see Eq. (4.5); 2)
BBET rate constant calculated in the "harmonic—like" approximation, see Eq. (6.4); 3) 2D
overdamped rate constant given by Eq. (6.1). These dynamic constants are compared for
three different sets of the reorganization energies for polarization and for bond-breaking, Ax
and A; respectively ............................................................................. 94

Fig. 12. The logarithmic dependence of the dimensionless pre-exponential factors (x TX)
for the 2D BBET rate constant Kd in comparison with the approximation K 3”” of
Eq. (11.12.10) when Ax>>kz. The other dynamic constants, K5” and K3)”, are also

depicted .......................................................................................... 97

Fig. 13. Schematic potential energy surfaces for the quantum mode in BBET reactions.

(a) The high frequency vibration levels in the initial and final state Morse potentials. (b)
Each vibrational state of the reactant (R) and product (P) is characterized by a distinct
polarization potential surface. The physical situation corresponds to an exothermic reaction
(A G 0 <0) ....................................................................................... 103

Fig. 14. Schematic of the potential energy surfaces representing the BBET for quantum
bond-breaking in three different regimes: (a) normal; (b) activationless; (c) inverted... l 12

Fig. 15. The dimensionless non-adiabatic rate constant UK, for the normal regime is
evaluated for the following parameters: A, = leV and AGo = -O.5eV. Three values of the

ix

tunneling parameter S = D, / ha), are used: S=3 (D, = 9kBT), S=5 (D, = 15kBT) and S=8
(D, = 24kBT) .................................................................................. 114

Fig. 16. Dependencies of the dimensionless rate constant '32:" on the number of sinks are
presented for the normal regime at 2., = leV and AG0 =-0.5eV. Two values of the

tunneling parameter S = D, / hwb are used: S=3 (D, = 9kBT) and S=8 (D, = 24kBT). (a)
Comparison of E for different values of 0t (1 — 0t=0.1; 2 — 01:1; 3 - 0t=5; 4 - oc=10) is
made for the two values of the tunneling parameter. (b) The approximate rate constants 5,8
(short dashes) and '50 (long dashes) from Eq.(II. 13.20) are compared with E for S=3 and

0t=5 and 10 .................................................................................... 116

Fig. 17. The dimensionless non-adiabatic rate constant UK, for the activationless regime is
evaluated for A, = leV = 4OkBT and AG0 z —1.eV = 4lkBT. Three values of the tunneling

parameter S = D, / ha), are taken the same as for Fig. 15 ............................... 119

Fig. 18. Dependencies of the dimensionless rate constant 72* on the number of sinks are
presented for the activationless regime for ,1, = leV and AG0 z —leV. Two values of

tunneling parameter S = D, / ha), are used: S=3 (D, =9kBT) and 8:5 (D, =15kBT).
Comparison of E for different values of or (1 - 0t=1; 2 - a=5; 3 - 01:10) is made for the
two values of the tunneling parameter. The solid lines correspond to the Boltzmann initial
distribution, and the dashed lines present 5(x) initial distribution ....................... 120

Fig. 19. The dimensionless non-adiabatic rate constant UK, for the inverted regime is
evaluated for A =leV=4OkBT and AG0 =-62kBT. Three values of the tunneling

I

parameter S = D, / ha), are taken the same as for Fig. 15 ................................ 122

Fig. 20. The dimensionless generalized probability UK, for the inverted regime is
evaluated for 11, = leV=4OkBT and AG, z -62kBT. and two values of the tunneling

parameter S = D, / ha), : (a) S=3; (b)S=8. Dashed lines are used for the 6(x) initial
distribution. The other notations are: l - 0t=1; 2 - oc=5; 3 - 0L=10 ....................... 123

Figure 21 One-dimensional sketch of the potential surfaces used for the
consecutive BBET .......................................................................... :. 127

Figure 22. The projection of the PBS for the consecutive BBET on the (x,z) plane... 138

Figure 23. The dependence of the dimensionless pre-exponential factor 1le; on the sink
activation energy EA23 for three different slopes of the initial distribution line, ql=O, 1 and
10. The continuous line stands for the activation energy for the initial distribution line.
EA21=5kBT and the dashed line represents EA21=1k3T .................................. 142

Figure 24. The dependence of the dimensionless pre-exponential factor 1:le2 on the ratio
of the relaxation times 0t=1:Z ltx for three different slopes of the initial distribution line,
q1=O, l and 10. The activation energy points for the initial distribution and the sink lines
are taken the same, EA23=EA21=1kBT. The continuous line represents the dependencies for
q2=O.2 and the dashed one shows a similar behavior for q2=O.l ....................... 144

Figure 25. The dependence of the dimensionless pre-exponential factor 11x12 on the sink
activation energy EA23 for three different ratios of the relaxation times 0t=1:z/ 'cx=O.1, 1, 10.
The initial distribution line activation energy was taken as EA21=lkBT. The continuous line
stands for the positive slope of the initial distribution line, q2=0.5 and the dashed line
represents q2=-O.5. For comparison, the dotted and dashed line gives TzX12(BEA23) for
q2=0 (when there is no dependence upon or) .............................................. 145

Figure 26. The dimensionless pre-exponential factor 1le for the MFPT on the reactant
PES is presented as a function of the activation energy, EAlz. The continues lines present
the dependencies for q1=l and the lines with the shorter dashes are for q1=0.1 and the lines
with the longer dashes are for ql=0.5. Three ratios of the relaxation times 0t=1rzl tx=O.l, 1,
10 are considered. The curves. for all three values of q] coincide at (1:1 ............... 149

INTRODUCTION

This work has two major goals. First, we present a new theory, the steady-state
Green's function theory, based on steady-state differential and integral equations for
description of irreversible unimolecular reactions. Apart from simplifications in evaluating
reaction rate constants, this new approach also permits generalization of the mean first
passage time (MFPT) concept, which is extensively used in understanding solvent dynamic
effects in liquids. Second, the steady-state Green's function theory has been applied to
some important classes of unimolecular reactions among which are electronic relaxation
processes and electron transfer reactions. Special attention has been paid to bond-breaking
electron transfer reactions which not only are quite common but also present a challenge
from a theoretical point of view.

Unimolecular reactions are a prevalent phenomenon and play a central role in many
chemical processes such as electron transfer reactions, electronic relaxation, and the
breaking and forming of chemical bonds. Unimolecular reaction rate theory has a long and
extended history, and a great number of methods and approaches have been developed in
order to describe diffusion-influenced unimolecular reactions, see Ref .[1 - 25]. While some
attempts have been made in order to progress in the reversible description of reactions, the
irreversible formulation of the theory is widely used as a model for evaluation of reaction
rates. The formulation implies that in order to obtain the chemical rates it is sufficient to
consider the reaction decay from the initially populated reaction state. The reactants which
escape from the initial state potential surface through some reaction region are considered to
be withdrawn from the process.

Following Kramers [1] many studies of bond breaking or forming reactions use the

model of a system moving along a ground-state adiabatic surface over a barrier. The

2

particles are subject to thermal damping, the objective being to escape irreversibly out of the
reactant potential well. Thus, the reaction model may be characterized by a single potential
surface (activated or otherwise) for which one defines an effective reaction region beyond
which the reaction dynamics is no longer interesting as the reaction was assumed to be
irreversible.

Although the general approach to the description of electron transfer (ET) reactions
in solutions is based on the scheme of transitions between two adiabatic or diabatic
surfaces, the irreversible mechanism displayed acceptable results. In this case orientational
polarization of the solvent presents one potential energy surface for each of the reactant and
product. Electronic coupling between the two surfaces provides the reaction region for the
reactants to move onto the product state. The reverse reaction from the product to reactant
surface is neglected for simplicity, which can be justified for activated reactions. Serious
attention was paid to high activation barrier [26, 27] as well as low barrier [6, 7] reactions.
Of course, there are many electron transfer reactions where bonds are broken or formed in
a way which is connected to the electron transfer event [28, 29]. This type of process is
referred to as bond-breaking electron transfer (BBET) [3 O] and is often associated with
realm of organic chemistry. A theory of BBET is intrinsically more complicated then that
for ET as it must involve at least two distinct reaction coordinates: one connected with the
orientational polarization of the solvent and the other a bond-stretch coordinate along the
direction of the chemical transformation. Nevertheless, despite the obvious
multidimensional nature, the irreversible mechanism of reaction decay can also be applied
here by analogy with the straightforward ET.

Another class of unimolecular reactions of interest consists of electronic relaxation
processes involving large amplitude stochastic motion. In this case it is assumed that the
stochastic motion of reactants on the excited-state potential surface leads to irreversible
transitions to the ground state due to the coupling between the ground and excited state

electronic surfaces. The reaction potential may or may not present a barrier to the motion

3

along the reaction coordinates leading to the decay of the initially formed excited state
population. In this case the irreversible mechanism seems to be especially suitable, as the
exotherrnicity of the reaction is extremely large. Theoretical investigations of the electronic
relaxation processes were directed towards both high activated barrier and activationless
reactions [12, l3, 17, 18, 20] dealing with these two types separately. In experimental
observations cis/trans isomerization and photochemical transformations proceed through
high activation barrier [31, 32], whereas photolysis of a large number of organic dye
molecules and isomerization reactions exhibit activationless behavior [31].

As one can see, the variety of reactions falling under the irreversible scheme
(accompanied by a great range of theoretical models) is enormous and it would be pointless
here to describe them in detail. On the other hand, there are a few general features which
are common for all these reactions. All of them are assumed to take place on a single
reaction potential surface from which there exists some mechanism providing the
population decay. This mechanism can often be represented by a reaction region, or sink.
The other important feature for us is that the reactions under study are diffusion-influenced,
so that the dynamical effects on the reaction rates are extremely important.

However, theoretical study of the reaction kinetics and rate constants is difficult,
even for irreversible reactions, not to mention for reversible ones. It is impossible to list
here all of the approaches developed to describe these reactions. Development of
experimental techniques and theory dictate that the reaction models become more and more
complicated. Widely used early one-dimensional potential surface models fail to describe
the gradually increasing number of chemical reactions so that multidimensional potential
surface models become more and more valuable [7, 19, 21, 33, 34] as is the case, for
example, with BBET [30]. Therefore, theoretical studies of the reaction kinetics and the
reaction rates face more and more complications. These complications arise partially due to
the use of a well-accepted technique wherein all reaction rate constants are evaluated

through the evolution of the total population remaining on the reaction surface. Thus, from

4

the theoretical point of view, the highest priority has been to evaluate the time-dependent
population density of the decaying initial state. As a partial solution to the problem,
numerical methods have become more and more popular [8, 16 - 19, 21, 30]. Despite the
usefulness of numerical investigations in interpreting the experimental data, they do not
provide clear parametric dependence of the reaction kinetics and reaction rate constants and
hamper a clear comprehension of the real nature of the physical phenomena. Furthermore,
they may be expensive and time-consuming to perform. This, for example, appeared to be
the case with the modeling of one-dimensional evolrition for the Fokker-Planck
underdamped motion [17, 18]. Therefore, even a simple extension to motion along a
multidimensional surface may be extremely difficult. Note that, in reality, analysis of the
steady-state characteristics alone such as fluorescence yield or reaction rate constants is
sufficient for the explanation of experimental data.

In other words, whether or not it is possible to avoid the solution of the well-
known time-dependent equation for calculations of the steady-state characteristics in
unimolecular reaction has become a significant question. It would dramatically simplify the
evaluation of the reaction rates not only from the analytical, but also from the numerical
perspective. Introduction of steady-state equations instead of the time-dependent ones
would be extremely profitable for investigation of multidimensional problems.

In the first Part of the present work, we deveIOp methods for the direct evaluation
of the reaction rate constants, based on solution of appropriate steady-state (rather than the
traditional time-dependent) equations. A profitable method for calculation of the reaction
kinetics is also suggested. The concept of the mean first passage time is generalized, which
permits evaluation of the mean time to reach a reaction region, rather than a point, on the
reaction surface. Further in Part I we test our generalized system against the well-defined
adiabatic and diabatic models for the reaction potentials. An effective one-dimensional

bistable potential is considered for the case of adiabatic reactions. In the diabatic case the

5

reaction rates were evaluated for one- and two-dimensional harmonic oscillator surfaces
accompanied by a special analysis for high activation barrier reactions.

Part 11 contains a detailed description of charge transfer processes which are
accompanied by forming or breaking of chemical bonds (BBET reactions). BBET is
considered to be a two-dimensional process following either a concerted or a consecutive
mechanism. The main interest is focused on the dynamical influence of the ET as well as
the bond breaking (BB) on the kinetics of such reactions. So, the theory describing this
type of reaction is more sophisticated than the ordinary ET description as the latter one does
not include any knowledge about dynamical influences from the other degrees of freedom
but the electron-transfer driving coordinate. In contrast, as will be shown in Part II, the
dynamics along the second, BB coordinate strongly influences the total kinetics. Two
stochastic (overdamped and energy diffusion) and one quantum mechanism will be
considered for transformations along the BB coordinate.

As will be shown, the nature of the reaction surfaces can provide a distinction
between the concerted and consecutive pathways in BBET. Predicating on study of the
early works in this field [29, 35] we will introduce an intermediate state which can easily
span the two extreme reaction mechanisms. The more stable is the intermediate state, the

more preferable could be the consecutive mechanism in comparison with the concerted one.

PART I. THE STEADY-STATE GREENS FUNCTION THEORY FOR
UNIMOLECULAR IRREVERSIBLE REACTIONS IN LIQUIDS

CHAPTER 1. THEORY, GENERAL OBSERVATION OF IRREVERSIBLE
REACTIONS

The time evolution for the case of irreversible reactions can be modeled as the
stochastic motion of solute particles on a one- or multi-dimensional potential surface with a
position-dependent sink, which provides localized decay of the reaction population. In this
case, the theoretical description of the reactions is based on the solution of the Fokker -
Plank equation for the survival probability distribution function P(5c', \7,t) at time t, with the
reaction surface coordinate SE, and the instantaneous velocity of the Brownian particle ii

[15]:
gm}, 17.1) = £(3c', 13)P(5c', \7,t) — W(5c’)P(5c', it) (1.1.1)

where W0?) = W0S(5c') is a position-dependent sink (max S(5c') = 1) and 1965,17) is a
functional operator describing the stochastic motion. In the case of underdamped motion
the operator 1:65, 9’) is given by the equation: '

k

Lark-(v,€7.>+-‘-(9.V<2),v.)+£(i1fl—fizfa»
m m m

where V,,(i = 55, V) is the gradient, ( , ) is the scalar product, V (55) is the potential energy
surface, m is the effective mass and C is the relevant friction coefficient.

In the case of overdamped motion Eq. (1.1.1) becomes the Smoluchowski
equation, which is defined in the same way as Eq. (1.1.1), but for the survival probability
distribution function P(3c',t) = I P(5c',i5,t)d9 . Thus, the stochastic motion operator along
the reaction coordinate if is given as:

La)=<'7.0(i.+fii.V(z»a DWWZé (1.1.2)
¢(x) .

where ,8 = 1 / [CRT and D = 1/(By) is the space diffusion coefficient, which, in principle,
can depend upon position 55, and tp(ic') = Z0“1 exp(—B V(5E)) is the Boltzmann distribution.

7
In the case of ET in Debye solvents, the diffusion coefficient D is given through

the longitudinal polarization relaxation time TL by the formal identity 7/:er where A is an
elasticity constant for the harmonic potential V (56) = k(ic' — 560)2 / 2 [6].

The boundary conditions for Eq. (1.1.1) require that the kinematic flux at the
boundary 8 of the reaction space under study (which may be, in particular, infinite) is zero,
i. e.:
jP/m = o

where j is the kinematic flux operator defined as LP = —div(jP)

Owing to this condition the following property holds:
[[1123 V)P(f,§,t)d§d9 = U}(ic’)P(f,v',t)clSd\7 = 0 (1.1.3)
S

This is the conservation law for the number of particles in the system (if the reaction does
not take place).

The kinetics of the process, that is the total population remaining on the reaction

potential surface at time t, is defined through the values P(5c', i7,t) or P(Sc', t) as:

P,(t) = ”P(5E,i7,t)didi7 a [Hindi (1.1.4)
In the general case, it is assumed to obey the differential kinetic equation:
iii-Ra) = —k(t)P,(t) (1.1.5)

which is usual for first order reactions, but the rate "constant" depends upon time. Such
equations are called "non-Markovian kinetic equations" [36]. When the rate " constant" k(t)

reaches a constant value over a long time (see Fig. l):
k = lim k(t) = —lim%ln(P,(t)), (1.1 .6)

1-».
Eq. (1.1.5) obtains the form of the mass action law. The constant in Eq. (1.1.6) is called the
long time rate constant and is traditionally identified with the rate constant of the‘quasi-
steady-state exponential decay. However, the existence of the above limit is a necessary,
but not a sufficient condition for existence of exponential asymptotic behavior (e. g. the

steady-state decay) at t-> oo [37, 38] (see also Chap. 11 of this Part). In addition, the

8

absence of the steady-state kinetics as t-> 00 does not mean that it can not occur in a

sufficiently wide intermediate time interval as discussed in Ref. [38].

rpm

1, time t

Figure 1. Schematic picture of reaction kinetics P, (t).

The other rate constant in unimolecular reactions is called the generalized probability

and is defined as:

CD

'=P[ (t)dt= It(—d—1:;——)t(tmv)dt=r (1.1.7)

0
This constant is the reciprocal of the mean survival time rm”. The mean survival time
depends upon the motion along the reaction surface, the form of the reaction region (sink)
and the intensity of the reaction, W0. When the intensity of the reaction sink is infinitely
strong, the mean survival time becomes the mean first passage time (MFPT), rp, i.e.
r m= 1', at Wo->oo [6, 8]. MFPT is the average time that a particle moving randomly
starting from the domain outside the sink takes to leave the domain and appear in the
reaction region for the first time. This time depends only upon the dynamics along the
reaction surface and the form of the sink. Further, we pay special attention to the evaluation

of the MFPT because in many cases it is the key to understanding of the whole decay

9

process. From a practical point of view it is important to distinguish between MFPT to a
point and to a region on the reaction surface.

It is clear that the generalized probability depends strongly on the initial distribution
on the reaction surface. Despite the fact that a: is an integral characteristic, it is very
common to approximate the survival probability using this rate constant instead of the long
time one: Pe(t) = exp(— a: t) = exp(- t / rm) [9, 13]. The legitimacy of this approximation
will be discussed later.

The differential form of kinetic equation (I. 1.5) is convenient for evaluation of the
long time kinetic behavior. In order to calculate generalized probability (1.1.7), the integral-

differential form is more profitable:
2
5-110):- ]; ae(t- r)P,(r)dr (1.1.8)

This form is widely used in reaction theory based on the memory function formalism [3 9]
or the projection operator technique [40] and in the case of non-Markovian encounter
theory to describe reactions in diluted solutions [36].

Applying the Laplace transformation given as:

its) = ]f(t)exp(-st)dt (1.1.9)

to convolution type equation (1.1.8), one can easily connect the Laplace transforms of the

reaction kinetics and the integral kernel :

~ 1
P‘(S) _ s + &(s)

 

(1.1.10)

The integral kernel may also be used to evaluate the generalized probability
as = 13.10)“ = 23(0)

The main goals of Part I are to investigate the conditions of the quasi-steady-state
regime and to develop methods for direct calculation of the long time rate constant k and the
generalized probability a: without solution of time-dependent equation (1.1.1). The other
aim was to generalize the way of calculating the MFPT, t, and obtain the method of

evaluating the MFPT to a reaction region. This was achieved through the introduction of

10

the Green's functions independent of the reaction, which we call the steady-state Green's
functions. The procedure for evaluating the MFPT from one point on the one-dimensional
reaction surface to another is well-known and has been used by many authors (see Ref. [8]
and the references therein for the review). However, we should strongly emphasize that the
previous methods used for evaluating the MFPT can not be extended to the case wherein
one is interested in MFPT from a certain distribution to a spatially-dependent reaction
region (i.e., when the region is not a point on the reaction surface). Such reaction regions
are very common for diabatic reactions [6, 13, 41, 42].

In Chapter II of the Part I we connect the rates of reversible reactions with the rates
for irreversible processes. Thus, under a few assumptions evaluation of irreversible
kinetics could be sufficient to define a more general reversible process. In Chapter III the
link between different rate constants is established and both integral and differential steady-
state methods for their evaluation are derived. The detailed analysis of the principal steady-
state equations is given using the steady-state Green's firnctions in Chapter IV. Chapter V
presents evaluations of the reaction kinetics for both pinhole and narrow reaction sinks.
First, the MFPT to a point is introduced through the steady-state Green's functions. Then
we generalize the concept of MFPT and give a definition for the effective MF PT which is
the mean first passage time to a reaction region, not a point. To complete the steady-state
Green's function formalism, the decoupling approximation as a method of solution of
steady-state integral equations for narrow reaction regions is presented in Chapter VI. The
strict conditions of applicability of the above approximation are also derived. Finally, in
Chapter VII we apply the steady-state Green's function formalism to both adiabatic and
diabatic reaction potentials and use the generalized definition for the MFPT to evaluate the
reaction rates. We consider the bistable potential as the model for adiabatic reactions and the
decay on one- and two-dimensional harmonic oscillators as examples of irreversible

diabatic reactions.

11

CHAPTER II. REVERSIBLE REACTIONS

In the previous Chapter we considered irreversible reactions. Here we show how
the reversible reaction rates are expressed through the irreversible ones at the present state
of the theory of reversible reactions.

The reversible problem is usually formulated by introducing two states - initial and
final - coupled to each other. Then it is considered that the initial state is populated at the
starting moment of the reaction and the coupling to the final state leads to the transition
between these two levels. Thus, the Liouvillian for the reversible problem incorporates the

system of four equations for the density matrix f) which describes the evolution of the

population under the effect of stochastic motion:

gran = [11p11-iV12(p21-p12)

a .

3,102. = 1121022 +IV1.(p21-Pn) . (1.2.1)
503p], = 112,01, — i(U1 — U,)p12 — iV,2(p22 -Pu)

102. =p12 *

where V12 is the coupling between the two states, L),- is the motion operator along the U,-
potential, and L12=L21 describes the motion along the cross-surfaces (U1+ U z)/2.

Currently, the above system has been solved under the assumption that the length
over which the two surfaces are coupled to each other is short compared with the size of the
reaction space. This means that the crossing dynamics are very localized and the off-
diagonal elements of the density matrix vary rapidly [43] and can be eliminated to yield two
closed equations for the diagonal elements of the density matrix [23, 42, 44].

The next assumption traditionally made concerns the form of the reaction kinetics.

The forward km and back k21 rate constants are defined from the population equations:

21d 1V1 __ k12 -k21 [VI
1.1.1.1., 111

12

where M = [tapu are the populations of the initial (i=1) and final (i=2) states, and
N 1+N 3:1.

Form (1.1.12) implies that the reaction kinetics obey the exponential law, and the
reaction rate constants k12 and k2] are, in fact, assumed to be the long-time rate constants
for the decay from the initial and final states respectively. Whereas assumption (1.2.2) is
quite acceptable for high barrier reactions, it is much more difficult to justify for lower
barrier crossings.

Nevertheless, these two assumptions produce the answer which allows for the
forward and backward reactions (initial to final state and vice versa). The rate constant for

the forward reaction can be written in the form:
K12
k = ’
12 12 12 21 21
1+ K, r, + K, r,

 

(1.2.3)

The reverse rate is obtained by interchanging the indexes 1 and 2. Here K j" are the non-

adiabatic rate constants (see Part I, Ch. V) for forward and back reactions and 1?, are the
mean first passage times to the crossing region from the initial state well ( r‘Pz) and from the

21
final state one ( r, ).

The important point made by Eq. (1.2.3) is the fact that it is not necessary to solve
even the system of the two closed equations for p11 and p22 in order to evaluate k12 and

kg 1. Each of the reaction rates K :7 and ri’, can be evaluated from appropriate irreversible

problems. K :2 and r}: (K 3‘ and 1,2,1) can be evaluated from the solution of Eq. (1.1.1) for

the irreversible motion along the initial (final) state potential U} (U 2) with the reaction sink
W(5c')located at the crossing region with the intensity W0 proportional to the strength of the
coupling, V12.

Thus, the above mentioned arguments extend the application of the irreversible
approach to evaluation of the chemical rates onto the reversible processes. So, the rest of

the present work will be devoted exclusively to the irreversible reactions.

13

CHAPTER HI. REACTION RATE CONSTANT S AND
THE CONNECTION BEWEEN THEM

It is easy to see that the reaction kinetics given by Eqs. (1.1.5) and (1.1.8) are
equivalent for description of irreversible reactions. However, whereas there is a
complicated integral connection between k(t) in Eq. (1.1.5) and the integral kernel fe(s) of
Eq (I. 1.8), it is easy to connect the long time constant k in Eq. (1.1.6) with 53(5). Actually,
if the exponential asymptotic of the kinetics exists, the following equation may readily be

obtained using Eq. (1.1.10) and the main property of the Laplace transform [45]:
s

 

£11111) P,(t) exp(kt) = “11111313,“ — k) = 112}, S _ k + 53(5 _ k) = const
So, we have:
Imam = const, or 11113 é(s — k) = k
s-> s s->

Thus, the desired long time rate constant is a positive root of the equation :

2"e(—k)=k. (1.3.1)
Analyzing the behavior of Eq. (1.1.10) on a complex plane of s, one can easily deduce that
the point 3 = -k is the extremely right isolated singular point, moreover, it is a simple pole.
This means that this root corresponds to the first order zero of the function 3 + 23(5)

Ifthe properties of the root differ from the above mentioned ones, for example, if it
corresponds to an essentially singular point of Eq. (1.3.1) on the complex plane, the
exponential asymptotic ~exp(—kt) is just an intermediate part of the time scale of the
reaction kinetics. This case, in principle, may occur if Eq. (1.3.1) does not have a positive
root.

Apart from this, it is not difficult to prove that if exponential decay with rate
constant k occurs at all times, then é(3)=é(0)=ae= k. So, the integral-differential form of

Eq. (1.1.8) permits us to reduce the calculation of the steady-state characteristics to the

investigation of the integral kernel 52(s) only.

14

I. e c 'o ”co an ”
To formulate a steady-state differential equation for the long-time rate constant k let
us introduce the modified population probability [45]:
P(SE, 17 t)

n(s,v,t)=-—F—(—t3— , j [ "(2,17,th17 :1 (1.3.2)

which satisfies the equation:

grip-a, V, t) — k(t)n(5c',i7,t) = £(5c',v')n(5c',ii,t) - W(5c')n(fiiit)

Taking into account property (1.1.3), the rate "constant" in Eq. (1.1.5) may be defined
through this new function as
k(t) = [[ W(r)n(ie,i3,t)drdv
This expression is a complete analog of the definition of the reaction rate constants in
bimolecular reaction theory [3 8, 46].
The advantage of using n(3c', 12', t) appears at the limit as t->00. If the long time rate

constant exists, this limit can be carried out directly in Eq. (1.3.2) n(5c',13) = lim n(i,l7,t)

and we can write down the differential equation for the long-time rate constant k
—f.(f,V)n(f,v') + W(5c')n(3c',17) = kn(5c', t») (1.3 .3)
supported by the requirement
[Mandate = 1. (1.3 .4)
The boundary conditions for this differential equation are the same as those for Eq. (1.1.1).
However, it should be mentioned that even if Eq. (1.3.3) looks like an ordinary
eigenvalue equation, its solution for the definition of k and n(ic', TI) does not require
utilization of a property such as completeness of the set of eigenfunctions. This requirement
is substituted by the normalization condition for n(5c', V), which significantly simplifies the
problem. Moreover, as will be discussed below, Eq. (1.3.3) exhibits a simple connection

with the function which determines the integral kernel of Eq. (1.1.8).

15

2. Ike integral heme] of the kinetic equation

In this Section we define the integral kernel of Eq. (I. 1.8) and present the steady-
state differential equation for the generalized probability a. Due to the convolution nature
in equation (I. 1.8), for convenience we generated Laplace transforms (1.1.9) for all values

relevant to the discussion. As a result, we obtained from Eq. (1.1 . l) the following:

£05, v)13(s, 17,3) — W(5c')i>(s,v,s) = seams) — f(f,\7) (1.3 .5)
where f(5c', i?) is the normalized initial-distribution function.
Let us introduce

6(5c',17,s)=—P 0" V") -

13,(s) — (s + 2e(s))P(5E, 17,5) (1.3.6)

This function satisfies the normalization property H €(f,it',s)dfdl7 = l and will be called

here the modified population Laplace transform (note that it is not the Laplace transform of
the modified population probability n(5c', i", t), but a completely drflerentfimction). Now the
Laplace transform of the integral kernel is

5e(s) = [[ W(£)§(5c',v,s)tadv. (1.3.7)

According to definition (1.3.6), Eq. (1.3.5) yields:

L(E,V)§(f, is) — W(f)§(f, 9,3) = 35(56, 15.8) — (s + é(s))f(5c°,v') (1.3 .8)

Even though Eqs. (1.3.5) and (1.3.8) look very similar, a few words should be said
about their differences. First, according to Eqs. (1.1.10) and (1.3.1), the Laplace transform
of the kinetics P,(s) diverges at s->-k, therefore POE, 13,5) has the same feature. So, it is
impossible to take the limit s->-k correctly in Eq. (1.3.5). On the other hand, at s-> -k Eq.
(1.3.8) generates Eq. (1.3.3) for the calculation of the long-time rate constant k. Moreover,
the transition from Eq. (1.3.8) to Eq. (1.3.3) at s-> -k clearly demonstrates the physically

obvious fact that the long-time rate constant, k, does not depend on the initial distribution

16
f(f,i3). The connection between values n(.\?,i?) and 5(E,fi,s‘) is given as
n(f,l7) = 5(ic',i?,—k)
The singularity of P,(s) and [3(2, 13,3) may appear even in the limit s->O, which
would cause difficulties for the calculation of the time averaged rate constant $= 53(0).

Actually, the following steady-state equation, drawn from Eq. (1.3.5) for

FOE, ii): POE, 17, O) is completely legitimate:

M. 2113a: 13) — Wafer) = 413.9). (1.3.9)
but it is not directly applicable to the calculation of P,(O) = l/ae in the strong non-adiabatic
case when Wo -> O and the rate is controlled exclusively by weak non-adiabatic transitions

between the reactants and products, i.e. by perturbations near the transition point. In this

case one has re -> O and, hence, P,(O) diverges [6].
On the contrary, the transition 565,17) = 5(f,17,0) is perfectly legitimate and Eq.

(1.3 .8) results in the following steady-state equation for at

1112315133) — Wane. 9') = -aef(5c',i?), (1.3.10)
which should be completed with normalization condition
[[§(z,v')dzdr=1. (1.3.11)

Now, perturbation series based on 5(55, i") can be easily constructed for a at We -> 0 using
a iterative procedure. In the zero-order perturbation theory for W0 (W0 = O; aa=2e(0)=0)
from Eq. (1.3.10) one can obtain 5“”(2, 17) = (p019). Taking into account the definition of
the generalized probability 2e:

ac: ”megawatts, (1.3.12)

the first order approximation gives:
a“) = a, = K, = ”'W(f)(p(f,15)fidl7, (1.3.13)

where K1- is the non—adiabatic rate constant.

17

Taking into account definition (1.3.13), it is convenient to represent the sink rate in

the form

W(5c') = K, 1110(2) (1.3.14)
where w0(5c')= Wo S(ic’)/Kr is the function normalized with the weight (11(2) 2 [(o(5c',17)d17

and [$05) w0(f)df = 1. Now the function WOO?) defines the true geometric form of the
reaction zone.

It should be mentioned that the legitimacy of the limit 8 -> 0 taken directly in
differential equation (1.3.5) (and even more so, the transition to the field of negative values
of s->-k on the complex plane in Eq. (1.3.8)) is not obvious.

To substantiate that these transitions are correct, we now consider the integral form

of Eqs. (1.3.5) and (1.3.8) using the steady-state Green's function formalism.

18

CHAPTER IV. INTEGRAL FORM OF THE PRINCIPAL EQUATIONS AND
THE STEADY-STATE GREEN‘S FUNCTION METHOD

[. Zhe (Zreen's function representation
The Green's function G(5c', 13, 5c”, ir",s) for the operator 1:665?) satisfies the equation
13(55, 17)G(5c', ", igvgs) - sG(5c',17,5c°',17’;s) = —6(5E — 350607 — v") (1.4.1)
where 5( 55) is the Dirac delta-function. This Green's function describes only the stochastic
motion along the surface, not the reaction, and has the same reflecting boundary conditions

at the edges of the reaction space as those for Eq. (1.3.5). Function G is the Laplace

transform of the conditional probability density function p(ic', 17, 7', it";t) which corresponds
to the conditional probability of realization of values if and 17 at time t if they had values 5:"

and it" at t=0. The Green's function satisfies the steady-state and normalization conditions:

Hope, 9355', v';s)tp(s',v')d2'dv' = -1-(p(55,i3) [ G(e,v,z',v';s)tadv = -1- (1.4.2)
3 S

” .0 .0, ‘

where (p(5c',l7)= limp(x,v,x ,v';t) = lingsGOc’, 15,27,173) is the Boltzmann distribution.
t—>ao s—>

Moreover, it can be easily proved that:

12(55, 1'5)(p(5c', r») = o G(ic’,ii,5c",ii';s) ~ legs, 17) at s -> 0. (1.4.3)
S

Note, that the steady-state condition in Eq. (1.4.2) is only the consequence of the detailed
balance principle G(5c',17, 5c", V’;s)(o(5c", it") = G(5c°’, it", 56, iz';s)(p(1'c', if)

The Green's function G(£,ii,5c",v";s) is the kernel of the integral operator G(s) =
(s - 1:) 4. Thus, considering the second members in both the right and left parts of Eq.

(1.3.5) as the non-homogeneous part of this equation, one can obtain the integral equation

13(55, as) = ”066, 9,217';s)f(3c",15')d5c"di:" - [[ G(3c', 1'5, 55', v';s)W(r')13(5a', $39.1?va
(1.4.4)

By analogy, the integral equation for 5(55, 12°,s) is:

19
5(2, 9,5) = (s + 5:(s))”G(1E,V,5c°’,17’;s)f(5c",17’)df'diz"—
—HG(SE,V,5c",ix";s)W(5c")§(5c",17’,s)dic"d§’. (1.45)
However, according to the second equation in (1.4.3), the Green's function
G(E,v°,i',ii';s) has a singularity at 3 ->0 and this limit is prohibited in Eq. (1.4.1). This
means that the direct transition from the time-dependent Green's function to a steady-state
one is impossible. Thus, we are obliged to question the limit s -> O in Eqs. (1.3.5) and
(1.3.8) which was taken in order to obtain steady-state equations (1.3.9) and (1.3.10), as
far as the latter two equations cannot be represented in the integral form through the

Green's function G at s = O.

I

2. es -tte e

First of all, we are going to prove that the limit transition 3 -> O is legitimate in Eqs.
(1.3 .5) and (1.3.8) and explain the apparent contradiction. For this purpose let us introduce
the modified Green's filnction G(ic',fi,f’,it";s) as in [14]:
6(2, 13, E',17’;s) = 0(3, 17,2", V’;s) — 593—”) (1.4.6)

where the subtracted part is, in fact, the limit of the ordinary Green's function,
06537331733) at s->0. According to Eq. (1.4.1), the above function satisfies the
following equation:

1‘12, 06 (55, t7, 5:", 9";s) — 5505,15, auras) = -5(ie — man: -— a") + tape, V). (1.4.7)
This'equation has the same reflecting boundary conditions as the original differential

equation. The modified Green's function has the following properties:

”magic",v";s)¢(e',v')tte'dt7' = 0 j j G’ (55, are, 17';s)d1'c'd17 = 0. (1.4.8)
Contrary to the initial Green's function, the modified function G—(ic', i5, E',V’;s) allows the
limit transition 5 ->0. Moreover, if the long-time rate constant k exists, the modified

Green's function guarantees the transition to the field of the negative values of s (s -> -k).

20
At s=O, this function turns into a steady-state Green's function C(ic', 17,5521“),
G(55,17,55’,\7') = 5055,55 17,5 = 0). C(55, 13,5539”) has the same properties as the ones
given by Eq. (1.4.8).
Integral equation (1.4.5) can be rewritten in the steady-state form:
are) = (005,15) + ae [ j (7‘ (55, 15,2", 15') f(55', 9")d15’d17’
— j j 6 (55, t7, 2',i7')W(£')g(ie', 13")ta'dv' . (1.4.9)
When the initial distribution f is Boltzmann, the second member on the right hand side is
zero and the integral equation obtains its simplest form.

As one can see, the introduction of the new Green's function, G (55, U,55’,i)",s)
allows us to manipulate the integral equations (Eqs. (1.4.4) and (1.4.5)) in a way which is
the most profitable for us. By doing this we achieved two things. First of all, we proved
that the steady-state differential equations for P(55,ii) and 5 (55, ii) , Eqs. (1.3.5) and
(1.3.8), are completely legitimate. Thus, the fact that the limit transition s->O is prohibited
in integral equations (1.4.4) and (1.4.5) does not affect the differential equations. Second,
we formulated the correct steady-state integral equations. Therefore, now we have two
alternative types of approaches - differential and integral - for evaluation of the desired rate
constants.

For all cases with non-statistical initial distribution, it is convenient to introduce
another modified Green's function which also permits the limit transition 5 ->0

G(55,ii,55',iz";s) = G(55,17, ram) — HG(55,V,5C”,V';s)f(55’,11")d55'd17'
5 565,15, raves) — [j G' (55, 17, 3C", V';s)f(55',it")d5c”d17'. (1.4. lO.a)
It satisfies the difi‘erential equation:
L(55,17)G(55,ii,55',12";s)— s65, 17, 5C", 17’;s) = —5(5e — r956; - v") + foe, t7) (14.10.13)

and has the following properties

”Gay,2',v';s)f(s',v')¢edt7' = o [[G(s,v,r',v';s)d2dt7 = 0. (1.4.11)

21
It is obvious that (7 is only a particular case of the general modified Green's function G
forf(55,17)= (p(55,ii).
The equation for evaluation of the steady-state Green's function
anger) = (3(35, 1333910) is
£05, r)(";(5c',t7, £3?) = —6(55 — 205(9 — a) + f(f,13). (1.4. 12)
The important point here is that the property H L(55,13)G(55,15,f',13')d2d9 = O is only a
consequence of Eq. (1.4.12). So, the previous reflecting boundary conditions can not be
used as the boundary conditions for Eq. (1.4. 12). Instead of them, Eq. (1.4.12) should be
completed with the additional requirement
”Gay, 55',17')d15d13 = o. (1.4. 12a)
The integral equation for the steady-state modified population probability can be obtained

from integral equation (1.4.5) with account of definition (1.4. 10a).

505,17) 2 tp(55,\7) — “C(25): 55',1'2")W(15).§(55’,13')d5c"a'ii'. (1.4.13)
To sum up, evaluation of the generalized probability a can be based either on
differential (1.3.10) or integral (1.4.13) steady-state equations for .5 (55, ii). The differential
method requires use of normalization conditions (1.3.11) whereas the integral one
combines Eq. (1.4.13) with the definition for the survival probability in Eq. (1.3.12).
At the end of this Chapter we would like to mention the existence of one more

problem which is discussed in Appendix A. It concerns the pseudostationary Green's

’ -0'

function Go (55,v, x ,9") which is the direct Green's function for the solution of differential

equation (1.3.10).

22

CHAPTER V. REACTIONS ON THE PINHOLE AND NARROW SINKS

For reactions on one-dimensional potential surfaces a quite narrow reaction sink can
usually be approximated by an infinitely narrow, or pinhole, sink. This presumption
significantly simplifies calculations of the reaction rates in ET and electronic relaxation
processes [6, 12, 13, 14]. The only problem with this simplification lies in drawing up the
appropriate criteria for its application. In Ref. [6] the authors suggested that the pinhole
formalism can be applied when the width of the reaction sink is much smaller then the
width of the Boltzmann distribution on the reaction surface. This case was called the
"narrow reaction window" limit and the sink was taken as a point on the reaction surface.
The criterion of validity of this approximation will be considered in Chapter VI. In this
Chapter we are going simply to assume that it is satisfied with a note that the criterion

formulated in Ref. [6] is not sufficient for the pinhole reaction to take place.

I- R . I . I I . I
For the decay on a pinhole sink we can present the reaction sink in the form

W(x) = K,1//o(x) = 7&3“): — xp) 2 K0601: — xp) where KI=K0(p(Xp) is the nonadiabatic
(P 35,,

rate constant defined in Eq. (1.3.13) and (p(x) is the Boltzmann distribution. From Eq.

(1.4.9) and definitions (1.3.7) and (1.4. 10a) we can easily find the integral kernel:

- K.(1+s] 61x,.x;s)f(x>dx/¢(x.))
w(s = ~ .
1+ K,G(x,,xp;s) / (p(x,)

 

(1.5.1)

According to Eqs. (1.5.1) and (I.4.10.a), the'long-time rate constant k may be obtained

from Eq. (1.3.1) in the form:

1: 1 +G(x2’xr;_k), (1.5.2)
k K Mr.)

r

 

 

23

As one can see, the long time rate constant does not depend on the initial distribution f(x)
and depends only on the Green's function G corresponding to the Boltzmann initial

dist1ibution. The long-time rate constant k can be obtained through the analytical expansion

of the function G(xp, x p,s) to the region of negative real values of s as

G(xp,x,,—k)=-£%-’i-)-s—7;—. (1.5.3)
0

According to Eq. (1.5.1), the reciprocal value of a= 53(0) for the 5-functional sink
is always given as a sum of the reciprocal nonadiabatic rate constant and MFPT, rp. MFPT
in this case is the average time for particles to reach the absorbing point xp having started

from the initial distribution f(x):

=—-+-—E-l—+T. (1.5.4)

As one can clearly see, only the generalized probability is calculated through MFPT, and
not k as sometimes is implied.

The formalism outlined in Eqs. (1.5.1)-(1.5.4) is straightforward to extend to
multidimensional processes if one is interested only in the average time to reach a point on
the reaction surface. Such a problem appears, for example, in multidimensional adiabatic
reactions with high activation barriers [8]. In this case the MFPT is evaluated as the average
time to reach the saddle point having started from some distribution deep inside the reactant
well. However, if the reaction region is considered to be different for a multidimensional
process, then the "narrow reaction window" limit is not so easy to define. We come back to
its definition for multidimensional reactions in Sec. 2 of this Chapter and Part II.

For one-dimensional surface reactions In can be defined through the Green's

functions in Appendix A:

24

_1__ _G___(x,. x,)_

a, 41x1 .I.D(y1_Tu1_I.‘p(z)dzI,f(z)d‘ I“

————I 421214414 1ch

Tp

Dmy 0’),

(1.5.5)
Thus, we have established the physical meaning of the function C(xp, xp) , which

determines the MFPT to the point xp at the given initial distribution f(x).
Taking as an example the initial distribution f(x)=6(x-xo), we can express the time

at which a particle having started at some point x0, first reaches the point xp:

,
3‘;

dy
ate,,x,)-a(x,,x,)_ ,I, 0111 61.9“” Wham“

We.)

 

t,(x,) = (1.5.6)

I :DUcltp (y)I¢(Z)dZ Where 11:0 2 xp

 

The integral form for the case x0 > xp was obtained by Szabo et. a1. [9] by considering
reactions on a black absorbing boundary located at xp. The approach presented in [9] was
developed for Green's functions which described an infinitely fast reaction at the boundary
xp (the Smoluchowski boundary condition for the reaction described by the sink function is
W(x) = KOO(x-xp) where O is a step-function and K0 ->oo). The above Green's functions
shouldn't be confirsed with the ones which are used in the present work. The steady-state
Green's functions do not include any knowledge about the reaction region, they are defined
solely for the reaction potential with the reflecting boundary conditions at the edges of the
reaction space which makes them universal for any reaction region on the given reaction
surface.

Taking into account the above expression for rp(xo) and Eq. (1.5.5), the MFPT for

the pinhole sink located at xp can be represented as:

T? = Irp(xo)f(xo)dro (1.5.7)

where the integration is carried out over all reaction space, which is intuitively sensible.

25
2. Reactions on the narrow sink

Ifwe assume that the reaction takes place in a multidimensional space on a narrow

reaction region which is given by the function wo( 55) in Eq. (1.3.14), then the integral

kernel in Eq. (1.5.1) can be represented in the general form:

53(5) ___ K,(1— sAg(s))
1 + K,g(s)

where
Ag(s) = —H 11.181618»? ;s)f(? 14245:" .
For s=0, Ag gives the time difference between MFPT to reach the zone 1110(55) having
started from an arbitrary initial distribution f( 55) and the one for the Boltzmann initial
distribution (11(55):
At, = Ag(O) -.- —[_[ w,(£)€(e,e )f(55')d1'5d1‘c‘ (1.5.8)

By analogy we can introduce:

g1s1= H 11.121512: 1? 31141:? )f(5c" 14245::
For s=O this time is the average time for particles to reach the reaction zone (not a point) on

multidimensional reaction surface for a first time:

t, = I] maths? 111.12 144? 1424?. (1591
This MFPT is vitally different from all previous average times. We will call it the effective
MFPT. Introduction of the effective MFPT is absolutely necessary for reactions on
multidimensional potential surfaces even if the reaction sink 1110 is infinitely narrow (a line
on the multidimensional surface). The effective MFPT can not be evaluated using the
Smoluchowskii boundary conditions as was done in Refs. [8, 9] As far as the reaction
region has to be narrow, for the one-dimensional case Eq. (1.5.9) gives only a correction to
the MFPT to a point. This correction should include the effective width of the reaction

region and is important only for diabatic potential surfaces, because the traditional adiabatic

26

approach does not require reactive particles to reach a region, but a point. However, the
above definition is extremely important for both adiabatic and diabatic multidimensional
reactions. For irreversible reactions on diabatic surfaces the reaction region is formed by
the area near the line of intersection of the initial and final diabatic surfaces. The reaction
decay occurs through the whole region and the MFPT to a point cannot be of any interest.
For high activation barrier adiabatic reactions on multidimensional surfaces the reaction rate
still can be evaluated through the MFPT to a point, as was mentioned before (in this case it
is the MFPT to the saddle point) [8]. However, for less than extremely high activation
barriers, the particles escape from the metastable state along many trajectories, which may
or may not go through the saddle point. Therefore, again, evaluation of the MFPT to a
region is necessary.

For the narrow zone we can also generalize the expression for the long-time rate

constant k,
1 - - ..
7— = I] Wo(X)G(x.f' rut/.12 141x 142d? .

Now we intend to demonstrate that the condition for the applicability of the
decoupling approximation (suggested in [6] for one-dimensional reactions) is not
sufficient. The condition required small values of the reaction sink width 6 compared with
the width A of the Boltzmann distribution.

Let us consider a decay process on a one—dimensional potential surface V(x) when
the sink located at some point xp has the width 5<<xp and the reaction occurs over a high
activation barrier. That means the requirement fiV (x ,) >> 1 is satisfied, which is equivalent
to the condition A <<xp . However, the relationship between A and 6 is not bounded.

Using Eq. (A14) and asymptotic estimate (A10) from Appendix A, one can obtain the form

of the modified Green's function G(x,x') when x and x' are located inside the reaction
sink (BV‘ (xp) >> 1):

27

 

 

 

' <p(x)
~ D V' ’ 1
G(x,x')=i fl (xpmx) xzx (1.5.10)
1 xSx’
[DfiV'(xp)
For this case, Eq. (1.4.13) may be represented as:
(PUG) W(JC')
= 5 W dx'. 1.5.11
ax) 41x1- D,” )I,( x...) (x'dx1 -—D,,,,'( ,)I (x151x1 ( 1

Despite the required restrictions for Green's function (1.5.10) on the location of x, the
solution of Eq. (1.5.11) is meaningful for the whole reaction space, -oo<x<oo. However it
should be taken into account that the above solution coincides with the correct one only
inside the reaction region. Application of this solution is legitimate because according to
Eq. (1.3.12) the generalized probability as is defined by the part of the solution which is
located only in the reaction region.

Let us accept the condition which was suggested in [6] as the criterion for the
"narrow-reaction window" case:

5<<A. (1.5.12)

Now, in order to be consistent with results (1.5.10) and (1.5.11) we should use the

following representation1 of the Boltzmann distribution:

<p<x1~ ~-;eXP[- wV1xp1+BV'(x.1(x— x.111 (1.5.131

This representation helps us to produce the new differential equation:

5"(x)+BV’(x.)€’(x)-%W(x)€(x)=0. (1.5.141

with the boundary conditions which follow from Eqs. (1.3.12) and (1.5.11):

a: , _ , _
50¢)" ¢(x)x:>fw-Ws 5 (x) (P (x)x_)_>w0-

The equation thus generated obviously. does not correspond to the case of a " narrow
reaction window" limit, as far as its solution depends on the form of W(x). Specifically,

we suppose for simplicity that the sink has the form of a plateau

28
[(0/5 xp—d/ZstxPHS/Z
W06):

0 x<xp—6/2,x>xp+5/2

Then the nonadiabatic rate constant Kr can be defined through Eq. (1.3.13) as:

_ 2K0 , _
, ———_6BV'(xp)Z0 sh(,BV (xp)6/2)exp( BV(xp)). (1.5.15)

Moreover, Eq. (1.5.14) rewritten for the intervals -oo<x<xp -6/2 and xp —5/2 SXSXp -5/2

generates a differential equation with constant coefficients. Its solutions can be obtained

easily and the expression for the generalized probability at 5<< A<< xp is:

2K0 exp(—[3V(xp) + a6 / 2)th(5E / 2)

 

 

 

”5‘20 2 _2 (1.5.16)
a +3:
1+ _ th(6E/2)
2.11

where

 

a sz'(x,), 3:.ja' +4K, /D6.
Note that the nonadiabatic and diffusion constants Kr and ad cannot be separated in this

expression for 21:. At [(0 ->00, as expected, Eq. (1.5.16) may be transformed to the pure

diffusion rate constant

lz-Lsrpz—ZLexprxP-B/Z» (1.5.17)
a a, D01
which corresponds to the reaction with infinite intensity at the point Xp 43/2.

The result of the "narrow reaction window" limit, Eq. (1.5.4) may be obtained from

Eq. (1.5.16) with the additional condition:

6<<—1-= l ~A—A—<<A (1.5.18)

11 fiV'(xp) x,

which is more rigid than the original one in Eq. (1.5.12). Restriction (1.5.18) exists for any

 

form of the reaction sink W(x). When condition (1.5.18) is satisfied, one can neglect the
second member of the exponent in Eq. (1.5.13).

In the case when criterion (1.5. 18) is reversed:

1 l
= — 6 A 1.5.19
[II/'05,) 01 << << ( )

one can obtain the expression for the generalized probability from Eq. (1.5.16):

 

29

4K
at: ' , (1.5.20)
(l+\/1+4K,/aed)' ‘

where at, is the diffusion generalized probability (1.5. 17) and the nonadiabatic rate constant

may be represented in the form:

K0 _ _ ,
K,—Wexp( [II/(x1, 5/2» (15.21)

which follows from Eq. (1.5.15) under the condition (1.5. 19). Therefore, let us emphasize
the fact that although the width of the reaction zone is much smaller than the width of the
Boltzmann distribution, the last result does not correspond to the "narrow reaction
window" limit and strongly depends on the form of the reaction sink. Moreover, the
generalized probability still cannot be represented as the sum of two reciprocals l/Kr and
Has, which is a direct sign that the "narrow reaction window" case is not applicable to the
situation.

One more significant note should be made. For any "narrow", but extensive
reaction sink, a powerful limitation on the diffusion rate is required for the validity of the
"narrow-reaction window" approximation. Roughly speaking, when the reaction sink is
essentially localized in space, but has weak long tails (sharp Gaussian or exponential forms
of W(x) may be given as examples) one can see that when the motion along the reaction
surface becomes somewhat slow, the portion of reactants enduring the reaction decay on
those tails becomes more and more significant. In this case no "narrow-reaction window"

picture can be involved in description of the reaction kinetics, even though the condition

5<< A can still be satisfied.

30

CHAPTER VI. THE DECOUPLING APPROXIMATION

The "decoupling approximation" is a procedure for solving the integral equation for
the probability distribution function (like Eqs. (1.4.4) or (1.4.13)) in order to obtain the
reaction rate constants. The decoupling procedure breaks the connection between the
integral kernel in the integral equation (the Green's function) and the probability
distribution function [6]. This allows us to separate the motion along the reaction surface
from the reaction itself.

One of the first applications of the decoupling approximation to solution of
diffusion-type motion equation for unimolecular reactions was made by Sumi and Marcus
[6]. They called the case when the approximation is applicable the "narrow reaction
window" limit. Some attempts to estimate the range of applicability of this approximation
were also made by Nadler and Marcus [7] and by Sumi himself [47].

Since this method is a very powerful tool for solution of integral equations, the
precise conditions of its applicability are very important. The empirical condition is quite
simple - we can break the connection between the motion and the reaction only if the
reaction decay occurs in a narrow region compared with the whole reaction space. In other
words, if the effective width of the reaction sink W( 55), o is much smaller then the width of
the Boltzmann distribution, A , then the narrow window limit takes place. If this is the
case, then the reaction disturbs the population inside the potential well only in a small,
insignificant area. This statement seems to be very reasonable. However, as was shown in
the previous Chapter, this requirement is not sufficient for the "narrow reaction window"
limit to take place even in a simple one-dimensional case. The problem appears to be more
complicated if one tries to generalize the definition of the "narrow reaction window" for a
multidimensional reaction. Is the same geometrical criterion still applicable in this case?

How do the different ratios between the relaxation times along different coordinates in the

31

system affect the limit? What happens if the relaxation along one of the coordinates is very
fast? These and many other questions must be asked when trying to apply the "narrow
reaction window" limit for multidimensional processes.

Taking all of the above arguments into consideration, our aim here is to introduce
the applicability conditions for the decoupling approximation using the kinematic approach
developed for bimolecular reactions [46, 48] and to discuss some important physical
characteristics which can be evaluated using this method.

In order to evaluate the reaction rate constants we can use the integral equation for
the modified population probability, 5(55,s) (Eq. (14.9)) which in the case of the

Boltzmann distribution is as follows:

was) = 41(55) —- II6(£,r',s)W(£')§(£',s)ta'. (1.6.1)

The above integral equation has to be accompanied by the definition of the integral kernel
&(s) = I W (55’); (i',s)d55'. The limit transition s->O is legitimate in the above equations

(and the following equations as well).

Let us introduce the new firnction 5*(55,s):

5015) = 51(3,S)¢(f). (1.62)
The modified Green's functions satisfies the following equilibrium property:
G'(5c°,5c".s1to(f'1=Masons). (1.6.31

Taking the last two equations into account we can rewrite Eq. (1.6.1) as:

§'(5c',s) = 1 — IG(5c",f,s)W(i')§(55',s)d5' (1.6.4)

with the new definition of the integral kernel:

5:61: I W(f)5*(f,s)<p(i)rfi. (1.6.51
Let us construct a new function which is simply some normalized "zone" encircled

by the reaction sink W(55):
-. _ W(55)5'(55.S)
W") ‘ (W131 518s»

 

(1.6.6)

 

32

   

where ( 5*) denotes the average IdEW(55)§+(55,s)(o(55). Using this function and

definition (1.6.5) we can present Eq. (1.6.4) in the following form:

Ell—:3)+I6E’G(55’,x,s)1y(x',s)=é( —) (1.6.7)
We can introduce the symmetric integral operator.
”(3'36 J3S)=—(;,C,T:j)c-2+G(x’ ,x,S) (1.6.8)

which allows us to rewrite Eq. (1.6.7) in a convenient form of average:

<w(£,s1|H(5c",5czs1l w(r'.s1)=-§s—). (1.6.9)

From the form of Hamiltonian (1.6.8) and the above equation it follows

immediately that the integral kernel 53(s) can be separated into two parts:

1 1 1
.. = - + -
arts) ails) $.(s)

 

(1.6.10)

 

where

.. VIC-5,5) 1 we
53,15) I "4’ “I W121 I Sens) (“HIE-(x ”M” H”)

and G-(55’, 55,s) (or shorter G(s)) denotes an integral operator. Function 111(55, s) itself is

the solution of the following equation:

 

was) = wo(f){1+ K,((wic7(s1w) — 5(s)w(f',s)II (1.6.111
where K, = (W (55)) = Ide(55)q1(55) is a nonadiabatic rate constant and
who?) = (:22) = W?) (1.6.12)

is the original "zone" firnction ul(55, s) in a zero-order approximation, i.e. when

ans): 1.

33
The decoupling approximation uses the above zero-order "zone" function ill/0(5)
instead of the real function w(55,s) in order to evaluate an approximate value for the

integral kernel é(s) in Eq. (1.6.9). In other words, from Eq. (1.6.10) we obtain
1 l l l

83(3) . 52°61 = 7?, I 5:381

 

 

(1.6.13)

where the diffusion part of the integral kernel is

1

2336)

and rp(s) is some "time" which depends only upon the motion on the reaction surface. The

 

 

= e,(s) =(w. (37.9%) = ?1- [I ta,ta"tp(s)W(s)c’1(r',i,s)W(5a') (1.6.14)

exact physical meaning of this value exists for the steady-state value 1,, = lim0( r,(s)). ‘tp is

the MFPT to the reaction region W (55)which was given in Eq. (1.5.8). Limit s->O turns
the integral kernel in Eq. (1.6.13) into the generalized probability a:
§=é+é=é+y (1.6.151
Here ‘Cp is the effective MFPT, i. e. an average time for particles started (in our case from a
Boltzmann distribution) to reach the reaction region W (55) given in Eq. (1.5.9). If the
reaction occurs at one point (a pinhole sink) W(55) = K 06 (f — 550), then the effective MFPT

becomes the MFPT to the point, 2,, and Eq. (1.5.8) produces Eq. (1.5.5).

Using the decoupling approximation formalism we can introduce another important

time which depends on the motion along the potential surface. This is the average time

inside the "zone" 1410(55) if the motion started at some point 55:

4131: 511.1561: I 41511811101. (16161

Then the effective MFPT in Eq. (1.5.9) can be rewritten in the form of an average:

t, = (w,(s)|t,(z)) :2} (1.6.17)

34

Now, coming back to the formalism itself, we can easily construct the first-order

approximation of the "zone" function (1.6.11) predicated on Eqs. (1.6.12) and (1.6.13):

 

wi(f.s)=w.(5c°){l+ K,’ Kim/0(3)}. (1.6.18)

21:3 5) _
This function can be rearranged into a more familiar form using definition (1.6.6) of the

"zone", definition (1.6.2) and property (1.6.3):

was) = not) - é-IdEC—HSE', 55,s)q1(55')0 I&W(2)g(£,s). (1.6.19)

This is the original integral equation, Eq. (1.6.1), after the decoupling approximation has
been used in order to separate the kernel G(55',55,s) from the solution 5(5s). Function
(1.6.18) used in Eqs. (1.6.9) and (1.6.10) easily gives us the first-order correction a‘ (s) to
the integral kernel , then the second-order approximation to the "zone" and so on.

Thus, the decoupling formalism enables us to build infinite series for the integral
kernel and the survival probability 3:. If the decoupling approximation (see Eqs. (1.6.13)-
(I.6. 15)) works well, these series should converge quickly. In other words, for the zero-
order generalized probability, Eq. (1.6.1), to be a good approximation to the real at, the
first-order correction should be small. It was proved precisely in Refs. [46, 48] that for the
decoupling approximation to hold, the dispersion of the effective MFPT should be much
smaller than its square value:

d0 << Ti. (1.6.20)

The dispersion is evaluated through the time of being inside the reaction region, Eq.
(1.6.16):

d. 53—? =(17.121r3121)—(w.(i1|t.(£1)'. (1.6.211

35
Eq. (1.6.20), accompanied by definition (1.6.21) for dispersion, is the desirable condition

for the decoupling approximation to be applicable to the solution of integral equations of
type (1.6.1).

Thus, in this Chapter we presented the decoupling method for the solution of the
integral form of reaction equations for irreversible unimolecular reactions. It should be
mentioned that the above presentation of the decoupling approximation given through the
kinematic approach is much clearer, simpler and more complete than the one given in [6].
The key stone of this simplicity and completeness is the steady-state Green's function
which allowed us to represent all of the differential and integral equations in the specific
form analogous to the steady-state one in bimolecular reaction theory. It appears that this
form is the most advantageous for evaluation of reaction rates. The decoupled integral
equation for €(55,s) is given in Eq. (1.6.19). A similar expression can be obtained for the
ordinary population probability, but calculations would be somewhat cumbersome. More
than that, the decoupling procedure enabled us to generalize the MFPT concept and evaluate
the effective MFPT, Eq. (1.5.9). Another physically meaningful value obtained by means
of the decoupling formalism is the average time for particles to spend inside the reaction
zone if started from some point on the reaction surface, Eq. (1.6.16). It was this time which
provided us with the criterion of applicability of the decoupling approximation, Eq.
(1.6.20). As one can see, this criterion is much broader than the simple requirement o<<A
and may include not only the geometrical form of the reaction region, but the type of

reaction motion as well.

36

CHAPTER VII APPLICATIONS OF THE STEADY-STATE GREEN'S FUNCTION
METHOD

1. The bistable mtential

In this Section we consider application of our method to adiabatic, but still one-
dimensional potential surface reactions. The standard framework for discussion of chemical

rates in this case is provided by the concept of transitions in the bistable potentials:

flaz 4 a 2
V(x)= x --x, (1.7.1)
1615,, 2

 

where E A is the height of the activation barrier, the frequency in the wells is m02=2 a and

the frequency on the top of the barrier is cob = a, see also Fig. 2.

 

 

 

 

 

Black absorbing point

Figure 2. Profile of the bistable potential given by Eq. (1.7 . 1).

In the case of thermally activated reactions the Kramers' theory gives the reaction
rate constants for all ranges of the friction coefficient 7 [1, 2]. Traditionally this theory is
applied to activated processes which exhibit an effective one-dimensional and Markovian

character. However, these requirements are not always fulfilled, which gave impetus to the

37

development of a more sophisticated theory involving both the non-Markovian nature of the
interaction between the reactants and solvent and the multidimensional nature of the
unimolecular process.

Part of this theory calls for the necessity of considering a position-dependent
friction along the reactive coordinate which, as was shown in [10] arises from the
multidimensional nature of the chemical transformations.

For high friction the Kramers' reaction rate

k. = ”2°”: lexpl—BEA (1.7.21
7! 7

 

was corrected by Compiany [1 l] to take into account the modulated, non-constant friction
“f(x). The interested reader can find the details of these formulations in Refs. [10] and
[11]. Here we just give the final outline and emphasize that the rates have been calculated
only for very high activation barrier reactions. Our goal here is to demonstrate the
application of the steady-state Green's function method to the solution of the problem for
reactions with any barrier height.
The problem starts from the introduction of the second reactive mode, y coupled to
the reaction coordinate x, so that the total reactivezpqtential is
to y y x2

U, =V +——+ —,
(xy) ()0 2 ayz

where V(x) is given by Eq. (1.7.1) and a is the strength of the coupling between x and y.
Each of these coordinates has its own friction (y for x and C for y). Thus, by means of the
adiabatic elimination procedure the y-dynamics is projected onto the x-dynamics [10, 11]
which results in the one-dimensional overdamped equation of type (1.1.1) with the
diffusion coefficient new): (13 y.fl(x))-1, where yen(x)=(y+o x2) and o=ot C/(ony‘fy). So,

the stochastic motion operator of the problem becomes

__c_l_ l i dV(x)
L(x) -— dxfiy,fl,(x)[dr +fi_—d1c ]. (1.7.3)

Initially the reactive particles are distributed exclusively in the reactant (R) potential well, so

the initial distributions is f()(x)= (I (113 / 7r exp(—611,‘,z / 2(x + 24/ E A / flay).

 

38

The rate constant in this process is just the reciprocal of MF PT to the product (P)

well, i.e. to the point 2 i/E‘, / [3a (no nonadiabatic constant, of course). From Eq. (1.5.6)

we get
Z'IEEA/a
_ : -IB1'.5LV)epVMder’ {W(Z )dzIf(z)dZ—- _ 1.0+ r” ([14)

where the total MFPT is the sum of the two separate times

2 85, la 2 1113 la
70 _fi)’ LeflVU’)dyj:e ‘BV(Z)ciz-j:of(z)dz Tcorr_Byo_ LyZefiVU)dy Lei”, (Z)dz;if(2)dz
For the high activation barrier the answer is known [11]: tp0=er’ and tp°°"=o/ a tpKf so

t, = 11304.3), (1.75)
a

where 11,19 =l/kKr and k, is given in Eq. (1.7.2). Unfortunately, analytical integration of
Eq. (1.7.4) is possible only in the high barrier case and produces the above result.
However, this problem can be easily solved numerically and we can analyze the
dependence of rpo and rpm" parameters of the bistable potential EA and a. It is interesting

that both of these times can be re-normalized to extract the dependence on a (or (D 0 and

 

(ob):
t, °=tf'g(pE)= 2’” g(pE)e 55* (1.7.6a)
wow b
can. 47raC pg“
r. =31. g...(BE)- 213....(1315 )e (1.7-6b)
my 0 b

Functions g(BEA) and gc0,,(BEA) tend to 1 when BEA->00. So, when the coupling or is
small, rpm“ is a linear correction to the one-dimensional MFPT. When the coupling is
moderate or strong, the contribution from the other (y) mode is dominant, therefore the
dependence on C, not 7 should be the point of interest. Fig. 3 depicts the form, of the
functions g(BEA) and gco,,(BEA).

In order to demonstrate the influence of the initial distribution on the rate constant,

we took not only the Boltzmann distribution in the R well, but also the distribution
fo'(x)=5(x+24IEA / [3a ). As one can see, the difference between the rate constants for f0(x)

39

and f0'(x) is indistinguishable (never more then 10%). Thus we can conclude that from the
experimental point of view the question of the influence of a non-statistical distribution on
the rate in the case of bistable potential can be left aside. We found the following
interpolation functions for g(e) and gcm(s) which can be easily used for interpretation of
the experimental data (the error of the interpolation is about 0.1%):

g(£)=1/(®(2.75 - 8)F(8) + @(s - 2. 75)F1(8)) (1.7.7a)

gm,(8)=1/(®(4.56 - 8)F°°‘r (8) + @(8 - 4. 56)Ff°” (8)), (1.771))

where @(e) is the step-function.

F(8) = 34.8575 + 0.048184.9684(e—4 2298) _ 9357434 3228(5—4 229s) __ 0.0104 0.0852 _ 0.014 _

£3 82 8

 

 

9.73998 +1.169452 - 0.058253
2.7051 1.5052 0.2894
E(s)=1.+ — —

£3 £2 3

 

 

0.007 0.2246 0.4713
+ +

s3 82 s

Fcon(8) : 01333 _ 0.0043e-l.785(£-l.7116) _ 0 00128-0.595(8-l.7116) _

0.07553 + 0.01171:2 - 0. 001653
E(8)= 1% 17.589 _ 6.9351 _ 0.7658.

 

£3 £2 a

relative MFPT

 

 

 

O
0‘
‘Vfiv rt

 

Figure 3. Dependence of the relative MFPT values g(BEA) and gem-(BEA) in Eqs. (1.7 .7a)
and (1.7.7b). gf(BEA) corresponds to the function g(BEA) calculated for the Boltzmann
initial distribution inside the reactant well, f0. g5(BEA) is calculated for the 6-functional

initial distribution fo' (x) = 6(x + Zw/EA / ,Ba).

40
Thus, we can write down the rate constant for the bistable potential with a medium
- to high activation barrier (BEA>3) with the contribution from the position-dependent
friction:
k = fiab-exM—BEA) (1.7.8)

017'
0.341 1.394 1.777 ZaC 0.297 14.185 27.603

fiEA + (fiEA)2 - (fiEAY ) + (0:60060b (1+ fiEA + (flEAY — (,8ng )

As one can see, the corrections to the original Kramers' constant are significant for the

 

 

 

 

where 747 = 7(1 +

barriers with EA<20-30 KBT.

relative MFPT

 

 

 

Figure 4. MFPT for the reactants to reach the product well rpm” and MFPT for the reactant

to reach the top of the barrier, 11,73 are shown as the ratios to the Kramers' MFPT.

Another interesting question arises from the reversibility of the transitions in the
bistable potentials. As it is well-known [8], for irreversible high activation barrier

reactions, MFPT for the reactants to reach the product well, tpPW is twice as long as the
MFPT for the reactants to reach the top of the activation barrier, rpm. The reason for this
correlation is that the particles which have reached the top have probability 1/2 to fall back

into the R well and probability 1/2 to escape to the products and never come back. Fig. 4

41

presents the situation for all barrier heights. One can see that ‘1?pr is not equal to 2*tpTB for
moderate and low barriers, which is caused by the reversible reaction R -> P. Because
rpm” is the "average" time for particles to reach the product well, it includes the
contribution from the particles which move back into reactant well after they have reached
the P minimum and thus increase the average time erW. So, even though each individual
particle still has the probability (1/2,1/2) to go either to the right or to the left of the barrier,
the reversibly moving particles shift the average probability to the left and on average the
probability for particles on the top to return to the R well is higher then the probability to
escape.

To sum up, the results of this Section extend the traditional Kramers‘ result for
transitions in the bistable potential. In our case the analysis was carried out without
drawing in the harmonic approximations on the top and in the bottom of the potential. The
combination of times (9.6) gives the rate constant for reactions with any barrier height and
includes the contribution from the position-dependent friction. The treatment doesn't
assume that 7311‘ undergoes small variations along the potential. The presence of the
position-dependent friction in the reaction rate is related to the breakdown of the
assumption of the existence of clear-cut reaction coordinates in some unimolecular
reactions. As one can see from Eq. (1.7.8), strong coupling or to another mode (it can be
the solvent or another vibration mode) might lead to significant corrections in the rate, or
might even change the reaction picture completely and make in multidimensional. This
method leads to the change of the power law friction dependence in the rate constant. For
the purely one-dimensional case the dependence of the rate on friction behaves according to
1/y (see Eq. (1.7 .2)). If the coupling to another mode is present, this law changes to the

1/yP (p<1) dependence and the stronger the coupling, the closer p is to O. This effect

becomes more significant for lower barriers.

2. :2k hwmonic oscillator Martial

42

In this Section we consider diabatic two— and one-dimensional harmonic oscillator
potentials as reaction surfaces. We assume that in both cases the irreversible reactions take
place in narrow regions. Here we are going to demonstrate the application of the steady-
state Green's function method to both pinhole and narrow sinks, see Chapters V and V1.

We presumed the reaction motion to be overdamped.

2.1. One-dimensional harmonic oscillator V (x) =x2/2

The reaction model we investigate here is considered for many classes of chemical
reactions. Among the most important are the ETR and electronic relaxation processes. The
demonstration of how this model has been used can be found in Refs. [6, 14, 26, 27] for
the case of ETR and in works [12, 13, 17, 18, 20, 31, 49] for isomerization and
photolysis. The simplest case is when it is assumed that the transitions take place on the
pinhole sink W(x)=K0 5(x-xp) located at the point of intersection of the two diabatic

harmonic oscillator potentials, with x p = 57:73-03 where 11, is the reorganization energy

for the x-coordinate and AGO is the exotherrnicity of the reaction. The generalized
probability, as a: in Eq. (1.5.4) for the pinhole sink, is determined through the MFPT,
which in the case of the Boltzmann initial distribution, ¢(x) = x/fl / 27: exp(—flx2 / 2) has

the form :

{ fiV(x,) \
rp = rxLan + 2f]; Jerf(y)exp(yz)dyJ (1.7.9)

0
where r, is the relaxation time along the potential (r,=1/(BD)), and erf(y) is the error

function (for any other type of initial distribution see Appendix A or Eq. (15.5)).
In order to evaluate the long-time rate constant k we used the following Green's

function (it was also calculated by Sebastian [50]):
Guy-S): _1_1"(-v) ear—002~x'21/4)D.(73x>D.<—Jfix'> mr
’ ’ 27:9 D exp(—/3(x2—x'2)/4)D.(-\/t7x)D.(J/?x') xsx'

(1.7.10)

43
where I‘(z) is the Euler gamma function, Dv(z) is the Parabolic cylinder function and

v = -s/(DB). Equation (1.5.3) for the definition of k through the analytic expansion of

function (1.7.10) has the form :
l
r(-v..)D.,(Jamar—lax.) = 7 (1.7.11)

where vo=k Tx, q = K’ and d0 2 DB exp(-[9’xp2 / 2), the parameter q is the relative sink

 

intensity and do represents the speed of diffusion along the potential. The lowest positive
root of this equation gives the long-time rate constant k for any barrier height V(xp). For

the barrierless reaction Xp =0 and Eq. (1.7.11) can be significantly simplified:

—r(—§) / 111-715 = —3— where p = 5.

fiq do
Fig. 5 presents the dependence of the reaction efficiencies a/do and k /do as functions of the

relative sink intensity q for the barrierless reaction (xp=0), for fix}, =1, and for

\ffi—xp = 2 . For comparison we also calculated the approximate long-time rate constant k'

in the form l/k‘=1/Kr + UK) where the diffusion constant, Kd was the root of Eq. (1.7.1 1)

4 Diffusion Control Limit 4—1-1’

Eflicacy ot the reaction. p

 

40
Relative sink intensity. q

 

Fig. 5a

44

 

 

 

 

 

 

 

 

 

Fig. 5b
‘ meenuuum
;——-—-'_
a 07 /” "’ _ _ #:-
_ 0.6 /
.3 0.5 /
g 0.4 x2
—D—.4
15 0.3 igr
3‘ 0.2
m 0.1
o 10 W 30 4o 55 E ’
WWW 0
Fig. 5c

Figure 5. Dependence of the reaction rate constants a, k and k' on the relative sink
intensity q=K,/do , where do=XZDlKBT exp(- 2. xpz/(ZKBT)) for the reaction on the 5-
function sink located at three different barrier heights on the harmonic oscillation reaction
surface V(x)=(1/2) 2x2 : a) A xp2/(KBT) = 0; b) A. xpz/(KBT) = 1 and c) M92/(KBT) = 2.

(_ _): generalized probability a = —2—B-'f—",where ar=Kr and ad =rd, is given as the
8r 3d

dependence p=ae/do on q; ( ): long-time rate constant k which is the solution of Eq.

(7.24) given as a function p=kldo of q; (- - -): approximation of the real long-time rate

constant k with the form k'=K,Kd /(K,+Kd) , p=k'/do .

 

45

at q->oo (for barrierless reactions Kd=d0). The dependence of the ratio p' = k'/d0 on q is
also given on Fig. 5. k' gives an upper estimate to the real long-time rate constant k and,
obviously, the higher is the activation barrier, the better the approximation.

For large values of K, (q > 1) a: is not a very good approximation to k for any finite
barrier, i.e. Pe(t) = exp(- at) works less effectively as the sink intensity increases. The
biggest deviations occur, of course, at the diffusion control limit (K, >> Kd, or q->oo)
when the generalized probability is exactly the MFPT to xp. Thus, Fig. 5 clearly
demonstrates objections against calculations of k through the MF PT [9]. The ratio aslk at

K, /Kd>1 is always more than unity. For barrierless reactions it has the maximum:

lim3=fl=__q___>_l_.
q->°° k K4 1+qln24->°°1n2

This indicates that the non-exponential reaction decay is always present in such reactions.
However, as one can expect, the gap between ad and K) defrnitely decreases as the barrier
becomes high.

As the barrier goes up, the analysis of the reaction constants becomes easier. The

MFPT becomes independent on the initial distribution as in Eq. (B3) :

 

 

7t ( 1 N
r z 1, 1+—— exp(BV(x )) (1.7.12)
" 01/051 20V<x,.)l "
1 1 1 2n
d — = — = — + 2 / 2 .
an a: k K, r, [3V(xp) exp(fixp )
The relaxation time of the system may be calculated from Eq. (A17) in Appendix A
(see also Appendix B) and is equal to:

I W \ 4
I" = TXLN/E {dwrfcozyexPyZ _ 1112 J WSW t, 1n,/pV(x,)r (1.7.13)

where D=1/ B I", F=expC and C is the Euler constant. The relaxation time 1,, is absolutely

correct for sufficiently large barriers, but it could serve as an estimate for lower barriers.

46

For example, in the case of activationless reaction MFPT to the point xp=0 may be .
evaluated using the second equation in (1.5.6) when x0>>12

e, = t,1n(2 fiV(x0)l'), ,8V(xo) = 3x; /2 >> 1. (1.7.14)
It is easy to see that both I, in Eq. (1.7.13) and IP in Eq. (1.7.14) for barrierless reactions
are of the same order of magnitude as the relaxation time 1,. So, the non exponential stage
of the reaction decay is very deep and the exponential decay corresponds only to a small
part of the kinetic curve.

Above, the chemical rates were evaluated for the reactions on the infinitely narrow,
or pinhole sink. The situation when the sink is narrow but not a point has been considered
in a number of works [6, 13, 30, 42]. The origin of the argument in favor of such a sink
arises from the multidimensional nature of the reaction potential surfaces. While the
coordinate x diffuses under the influence of the potential V(x), there could exists another
reaction coordinate, q which is assumed to be equilibrated under very fast damping. For
the electron transfer reactions the variable x is proportional to a certain component of the
polarization vector which passes through the two minimum energy points associated with
the reactants and products in the polarization coordinate space. The other coordinate, q is
considered as the intramolecular vibration coordinate, which can be immediately
generalized to include more coordinates. The vibration relaxation is usually much faster
then the fluctuations in the polarization, so it is supposed that while the distribution of x on
the reactant's surface may not to be an equilibrium one, that of q is. Thereby one can define
a sink function W(x) which involves at each x a suitable averaging over the population in
the q coordinate. Ifthere is no q coordinate, W(x) is proportional to a delta function which
is peaked at some value of x, xp (the pinhole sink). Otherwise, the Gaussian sink function

is obtained:

 

_ 0:: C738. _ 2) _40+AG.
W(x)—KOJZexpk 22,06 xp) J, rep—W, (1.7.15)

x

47
where M, and M, are the reorganization energies for the x and q coordinates respectively,
X0=7tx + kq. It is obvious that the sink in (1.7.15) can be considered to be narrow when
(FM/Xx <<l.
In order to evaluate the chemical rates for reactions on sink (1.7. 15) the decoupling
approximation described in Ch. VI has to be used. Thus, the generalized probability a: has

the form given be Eq. (1.6.15): Ila: = UK, +rpeff, where the nonadiabatic rate constant is

2
K, = Ide(x)(p(x) = K0 Ef- exp(—,BEA), E, = (A + AG") (1.7.16)
\i 2, 42,
and the effective MFPT, ,peff has to be evaluated using Eq. (1.5.9):
r2,” = j] wo(x><7(x.x' wow >006 14w, 1!. = W(x>/ K,. (1.7.17)

which is the average time for particles to reach the Gaussian region having started from
some distribution. Below, we will calculate the effective MFPT for the Boltzmann initial
distribution, so the steady-state Green's function in Eq. (1.7.17) satisfies Eq. (1.4.2) for
f(x) = (p(x) = W exp(—Bx2 / 2). For the convenience of evaluating integral (1.7.17)
let us introduce a new function:

g(x)=I(-}(x,x’)wo(x' )(p(1d )dx‘. (1.7.18)

Then the evaluation of the effective MFPT is reduced to one integration:

:3” = ] w.(x>g(x>dr.

Function g(x) satisfies the differential equation which can be easily obtained from Eq.

(1.4.2)
L(x)g(x) = (p(x)(1— ty0(x)). (1.7.19)

Instead of the boundary conditions for the above equation, one has to use the condition
Idxg(x)= 0 which comes from the properties of the steady-state Green's function,

(Tux).

For the overdamped stochastic operator for the harmonic oscillator reaction surface,
1 d d
L(X.’) = —— E

[3 dx +fix), Eq. (1.7.19) can be solved using the Fourier transformation
Ix

48

method. The Fourier transforms 7% v) = 3%:- I f (1:)e'mv of the differential operator and

all the functions in (1.7.19) are easy to obtain. As a result, we can express the effective

MFPT through the Fourier transform §( v)

I ,1 °° v3).
efl _ BEA ~ .
Tp — TX 37$}:- 8 L dvg( V) 8Xp[- Zfil: + rvxp]

., 2 ‘ f 2 )
g(v) = exp[-§5]£%Lexp[;fiiz y2 — iyvi—gxPJ- 1). (1.7.20)

The final result for the effective MFPT in the case of the Boltzmann initial distribution is

 

where

 

quite simple:
(3135.45?)

‘ + 3x2
tfthii-y- “7,-1, E,=—-£, 19:54: 1 . (1.7.21)
0y 1‘82)!“ 2 0'0 1+0

It is straightforward to evaluate the above integral for activationless reactions, i.e. for

 

EA=02

e 2 2
1,5(E, = 0) = 1,,an = t,(E, = 0)— t,1n(1+\/1— e ). (1.7.22)

So, a narrow Gaussian sink with the width 6 compared with the pinhole sink located at the
bottom of the potential well, decreases the MFPT for the reaction by the factor ~
ln(1+ m ) .

The integration in Eq. (1.7.21) can also be carried out for the case of high activation
barrier. For the large parameter BEA>>1 the Laplace method can be applied, Ref. [51]. The

efi‘ective MFPT of the first order for a high activation barrier reaction is

1+5 ”5‘ 1 1-8 3 BE (l-e)
‘fiz — F_’ +— 1_2 r-, , =# 1.7.23
T” 1‘ 28 BEA{ (2 ,u) 2s,u( H) (2 14)} ‘1 1+8 ( )

where F(0t,x) is the incomplete Gamma function. The above equation for a pinhole reaction
(e=1) transforms into Eq. (1.7.12). If we assume that the parameter p is small (we will talk
about the validity of this assumption concerning the "narrow reaction window" limit

below), Eq. (1.7.23) can be given in its simplest form:

49

2E0 0'7—
aha-re” ”[2/2+'0+ 2+:) (1.724)

where ‘Cp is given by Eq. (1.7.9).

 

Now we have to address the conditions under which the decoupling approximation
applied to the evaluation of the rate constants in the case of the Gaussian sink function, Eq.
(1.7.15) is valid. In Ch. V the condition of a narrow reaction sink was obtained for a high

barrier reaction through a narrow plateau. This condition reformulated for the Gaussian

sink would call for

 

l
‘1 << <<1, 1.7.25
1. 20E, ( )

which is identical to the requirement u<<1 used in Eq. (1.7.23) in order to obtain Eq.
(1.7.24). This looks like a reasonable extension from the problem presented in Ch. V to the
reaction we are regarding here. Nevertheless, in order to obtain the precise condition we
shell use Eq. (1.6.20) for the dispersion of the effective MF PT .

For the sake of convenience we rewrite Eq (I. 6. 20) in the following form:

Ida: "’-—°———""g’<x)— [Idxw.(x)g(x>]

 

<<1 (1.7.26)
[Idxw.(x)g(x)]
where g(x) is defined in Eq. (1.7.18). The above expression can be presented more
usefully:
(5%,- jdxw,(x)g(x)Q(x) << 1, where Q(x)— — :))—Idxtpo(x)g(x)

As one can see, the desired requirement would be fulfilled if Q(x) was a weak function of x

and Q(x)/rpefl <<1. Having all necessary functions at hand, we can evaluate Q(x) easily:

"ex [_fl(x—eyx,)2] mfg)“
p 2(1— syz) 1+ sy

t/l-ey2 — ill-82
1 5

First of all, it should be mentioned that the general sketchy analysis of the three above

 

Q(x)= 8.1%

 

 

equations let us make the conclusion that whatever the requirement for the one-dimensional

1/_  - .L

 

50

reactions is, it should not contain restrictions on the relaxation time along the reaction
coordinate, Ix. So, only the geometrical and energetic requirements should emerge from the
restriction on the dispersion in Eq. (1.7.26). To proceed further we can use the upper
estimate for the above function, Q(x)~Q(xp) and the assumption that the first derivative of
function Q(xp) in respect of the small parameter c (s=1/(1+0')) would be small compared
with the effective MFPT, Tpefi. After some algebraic transformations one can find that this
presumption is satisfied when
20 EA<1, (1.7.27)

which is exactly the same requirement as the one in Eq. (1.7.25) which was obtained by
comparison of the reactions on harmonic oscillator potentials with the general high
activation barrier reaction case described in Ch. V.

However, Eq. (1.7 .27) does not give us a satisfactory answer when the reaction is
activationless. We have to consider this special case separately and estimate when
Q(x)~Q(xp)=Q(0) is much less then tpeff(BEA=0) given in Eq. (1.7.22). After some

straightforward calculations we obtained

0'<%, (1.7.28)

which is the applicability condition for the decoupling approximation to the activationless
reactions. Fig. 6 depicts how the pre-exponential factor of the effective MFPT varies for
different width of the sink, 6 depending on the height of the absorbing origin, xp.
Summing up, in this Subsection we presented a complete analysis of the reaction
kinetics taking place in the reaction proceeding along a one-dimensional harmonic oscillator
potential surface through a pinhole or narrow Gaussian sink. The effective MFPT for the
reactions on the narrow Gaussian sink was evaluated by means of the decoupling
approximation using the steady-state Green's function formalism. To substantiate the
decoupling approximation calculations we also evaluated the conditions of its applicability

for activated as well as activationless reactions.

51

 

 

 

Figure 6. The dependence of the dimensionless pre-exponential factor for the effective
MFPT given in Eq. (1.7.21) on the activation energy for the reaction on Gaussian sink

(1.7.15). The dependence is presented for three different widths of the reaction sink,
0' = 11% (o=0 corresponds to the pinhole sink W(x)=K0 5(x-xp)).

2.2. Two-dimensional harmonic oscillator V (x, y) =x2/2 + y2/2

Here we consider the overdamped decay on a two-dimensional diabatic surface.
Previously some numerical investigations were performed in order to obtain the rate
constants for this model in reference to reactions involving electron transfer [7 , 30] and in
application to electronic relaxation [19]. The analytical solution seemed to be impossible to
obtain. Here we demonstrate how the steady-state Green's function method enabled us to
resolve this problem.

We assume that the reaction takes place on the line of intersection between the two
diabatic potentials and the reactants decay irreversibly from the reaction surface. For the

harmonic oscillator diabatic potentials the sink line is presented by the Dirac delta function,
5(xox+yoy-c) (see Refs. [30, 52]), where x0 = ([22,, yo = 211’, and c = AG0 - 11x - 2),.

Here A, and 1’1. y are the reorganization energies for the x and y coordinates for the product

 

‘_ .1.

52
diabatic surface and AG0 is the exotherrnicity of the reaction. Thus, the sink function in Eq.
(I. 1.1) for the reaction is W(x,y) = K05(x0x +y0y — c).

As one can see, the sink function for the two-dimensional potential surface is a
narrow sink, but by no means it can be approximated by a point on the surface. Thus, in
order to evaluate the generalized probability we have to calculate the effective MFPT given
in Eq. (1.5.9).

As far as the reaction region in this case is represented by a narrow (in fact,

infinitely narrow) firnction, the generalized probability has the form as in Eq. (1.5.4):

5:70”? where K,= dedyW(x.y)cp(x,y) and

f

t, =7{'r7[[[[dxdydx dy W(x,y)6'(x,x' ,y,y )W(Jd,y)(p(x',)/ ). (1.7.29)

Here 5 (x, x' ; y, y ) is the steady-state Green's function defined in Eq. (1.4.7) when s=0,
and <p(x, y) = fl / 27texp(—fi(r:2 + y’) / 2) is the Boltzmann distribution.

The nonadiabatic rate constant can be evaluated easily:
AG, + A, 2

K, = K0 —fi-—e"flE‘ where 2.0 = A, +2y and E, = 4A
0

47010

By analogy with the previous Section, for evaluation of the mean first passage time

it is convenient to introduce a new function, g(x, y ) which is defined as:
1 - I 1
gm) = E—fldx‘dy G(x,X';y,Y)W(X'.Y)<p(x .y ).

The difi‘erential equation for the above function can be obtained directly from Eq. (1.4.7)

when s=0:
L(x,y)g(x.y> = -,/P-f§%6(xox +yoy — c>e””*"’“’*”“ + <p(x,y). (1.7.30)

1 a —¢(x,y)£ ¢p(x,y)+ 1 _6_ -¢P(X.Y)_€_ @(X.)’)

134.259 are may 4y

the solution of Eq. (1.7.30) can be obtained in a Fourier transform form,

 

For the overdamped operator L(x, y) =

53
g(y,a)) = 2—1-dedyg(x,y)e-W-"m so that the effective MFPT in Eq. (1.7.29) can be
7:

presented in the integral form:

1 so -
rp : i 02 9”" Ida) exp(iimlflwqw). (1.7.31)
”18x0 —oo x0

The final answer can be given as follows:

f F

 

 

 

2 E x+ 21:“
1dr exP(l-€q:(+qu2xzj y r
T,=T,I—t 2 —1> where q=—9- and a=—l (1.7.32)
0 x 1 (x+q2x“) x0 I,
\ 1+q2 J

 

 

where 1,, and ry are the relaxation times along the x and y coordinates respectively. This is
the average time for particles to reach the sink line, 5(x0r+yQy-c ) on the two-dimensional
harmonic oscillator potential surface if they started from the Boltzmann distribution. The
effective MFPT for other types of initial distribution can also be evaluated using equations
similar to Eqs. (1.7.29) - (1.7 .30), but for different steady-state Green's functions which
would include different initial distributions (see Eq. (1.4.12)).

As one can see, the effective MFPT is the function of two parameters - the ratio of
the two relaxation times, or, and the cotangent of the sink line projection on the (x, y )
plane, q. From the above formula one can conclude that when the sink line projection on
the plane (x, y ) is perpendicular to one of the axes, the reaction is effectively one-
dimensional. So, if q=0, the projection is perpendicular to the x axis and ‘11, is a function of
the relaxation time along the x- axis, 1,. When q= 00, then the effective MFPT depends
only upon the relaxation time along the y -axis, 1y. In both these cases Eq. (1.7.32) is an
analog of the one dimensional MFPT to a point given in Eq. (1.7 .9). Unfortunately, one
equation can not be transformed into the other directly and only numerical evaluations

confirm the conclusion.

54

log( 1/ '1‘)

 

 

\

LDQHU'IN

 

 

0:1 0:5 1 5 10. 10g(rx/tz)

(a)

 

 

 

 

 

0‘1 0‘2 0‘s 1 2 5 ”1398“,”)

(b)

Figure 7. Dependence of the reciprocal of the dimensionless effective MFPT for the '
reaction on two-dimensional harmonic oscillator potential, ? on the ratio of the relaxation
times along the reactive coordinates, ‘tx/‘ty is given for the two barrier heights: a) EA=1

KBT; b) EA=10 KBT. The reaction region in this case was taken as a line on the reaction
surface.

55

The dependence of the dimensionless effective MFPT, Y = % e’mi" on the

parameter or is depicted in Fig. 7 for different heights of the activation barrier. Fig. 7a
shows the dependence for a low barrier, E A=2 KBT whereas Fig 7b presents the picture for
a high barrier, EA=10 KBT for a wide range of ratios tx/ty. If both relaxation times are
approximately equal and 0t=1, then the effective MFPT is quasi one-dimensional in the
sense that it depends effectively only upon one relaxation time, but include two-
dimensional geometric parameters. If one of the relaxation times is much faster (shorter)
then the other, the analysis of the two~dimensional situation in the low barrier case is quite
complicated. Both relaxation times play important roles in formation of the dynamic rate.
Thus, we are unable to conclude that there exists a situation wherein one relaxation time
prevails over the other. However this is not the case for high barrier reactions. As one can
see from Fig. 7b, at small values of ct, i.e. when 12,, << I, , the parameter 1/ E tends to a
plateau. Thus, the effective MFPT becomes insensitive to the changes in the slower
relaxation time, ‘ty and the dynamic rate is fully controlled by the faster relaxation time, 1,.

Actually, the situation in the multidimensional reactions is even more complicated
when one relaxation time is faster then the other. The complete analysis of the situation
with the fast/slow modes in two-dimensional reactions is considered in Ref. [52] and will
be discussed here in Part 11, Ch. X1.

When the activation barrier is high, it is possible to obtain a clear analytical answer
for the effective MFPT. Here we carry it out for the two-dimensional reactions, but the
result can be easily generalized to a higher multidimensional case.

From the results in Ch. V (see in particular Eqs. (1.5. 13)-(1.5. 14)), we conclude
that in the case of a high activation barrier reaction, only the first derivatives of the
multidimensional potential along the reactive coordinates near the activation point contribute
to the dynamic rate constant. Thus, the two-dimensional potential can be represented in the

form:

56
WW) = V(X*,y*) + V..(x - x*) + Vy(y - y"),
where Vx = %V(x,y4x* and Vy = gvogyiy“ . (1.7.33)
Therefore, if the line of intersection between the initial and final diabatic potential surfaces
is high for both of the potentials, the line of intersection can be represented in the form
6(xox+y0y-c) where x0=V,, yo=Vy and c=x* Vx+y* Vy . Then, applying the formalism

outlined above in this Subsection to the potentials of type (1.7.33), one can obtain the

 

following effective MFPT:
t=r —— where = —+——°—. 1.7.34
2° 0<v3+vs>e await. 1.) ' ’

This simple result clearly demonstrates that if one relaxation time is much faster (shorter)
then the other, the faster time controls the effective MFPT. So, the slower coordinate does
not contribute to the delivery of the particles to the reaction region when the nonadiabatic
transitions at the region of intersection are fast (in the kinetic regime when K,>>Kd), for

more details, see Part 11, Ch. XI-XII.

3. Concluding remarks

The major portion of the present Part was to devoted a new steady-state theory for
description of unimolecular irreversible reactions. The main idea behind this theory was to
create a simpler method for calculation of reaction rates influenced by solvation dynamics.
This method avoids use of the traditional time-dependent equations (see Eq. (1.1.1)) for
evaluation of such rate constants as long-time rate constant (1.1.6) and survival probability
(1.1.7). Instead, we constructed a scheme (Ch. 111) based on steady-state differential and
integral equations which, of course, does not include time as a variable (and thus decreases
the dimension of the differential and integral equations to be solved) and provides us with a

consistent technique for investigating the kinetic parameters. The theory can be applied to

57

both overdamped and underdamped mechanisms of stochastic motion, and diabatic as well
as adiabatic irreversible reactions.

In order to prove the legitimacy of the suggested equations, the steady-state Green's
functions were introduced in Ch. IV. These functions describe only the motion along the
reaction potential and do not include any dependence upon the reaction decay (in the form
of boundary conditions, for example, [9]). More importantly, they provide the correct
transformation from the time-dependent to steady-sate equations but do not retain the
property of conservation of particles inside the reaction space, Eqs. (1.4.7)-(1.4.12).

From a practical point of view, the steady-state Green's functions allowed us to
simplify evaluations of the chemical rates, especially in the case of narrow reaction regions.
We were able to build a complete theoretical description of such reactions, starting from the
generalization of the MFPT concept and concluding a consistent introduction of the
decoupling approximation in Ch. V and VI.

The MFPT defined in previous works [8] was formulated in order to evaluate the
average time for particles started their stochastic motion from some point in space to reach a
point on the surface. Its definition through the steady-state Green's functions is given by
Eq. (1.5.5). The effective MPFT which we introduced in Eq. (1.5.9) is the time to reach a
region if started from some distribution. Even though the regular MFPT is a very valuable
constant in one-dimensional reactions, its use is quite restricted, especially in the case of
multidimensional processes. The decoupling procedure, which is a method of solving
integral equations, allows to separate the reaction motion along the surface from the
mechanism of transitions. Thus, if the procedure is applicable, the rate constant consists of
two parts - effective MFPT and equilibrium rate constant. The first has now been defined
through the steady-state Green's functions, and calculations of the second are
straightforward and can be accomplished by means of the Golden Rule ore transition state
theory. One of the greatest advantages of utilization of the steady-state functions in the

decoupling approximation is that it makes the procedure simple and as a consequence, its

58

applicability conditions become clear and could be formulated precisely, see Eqs. (1.6.20),
(1.6.21).

As a conclusion, in the present Chapter we have shown some applications of the
steady-state Green's function method to both adiabatic and diabatic overdamped reaction
models. These models are widely used to describe a broad range of chemical reactions. For
example, both the classical outer-sphere electron transfer and photochemical quenching
reactions employ diabatic potentials. For this purpose we considered reactions on one- and
two- dimensional harmonic oscillator surfaces. Isomerization reactions and other processes
involving reorganization of chemical bonds, such as dissociation of molecules are usually
described by adiabatic models. The bistable potential can be used in this case, even though
the diabatic approach may be of some interest as well. In all of these cases, application of
the steady-state Green's function method opens the door to simple, clear calculations of
dynamical effects on the reaction rates.

So, after we have developed the steady-state theory for unimolecular reactions in
Part I and have demonstrated its basic applications, our next step is to widen the level of its
application and consider a more sophisticated reaction model than the one employed above.
In other words, in the following Part we intend to present a theory for multidimensional
processes and analyze features of the multidimensional dynamics in detail. Thus, Part H is
devoted exclusively to the application of the steady-state Green's function method to
description of bond-breaking electron transfer reactions in liquids which are intrinsically a

two-dimensional process.

PART II APPLICATION OF THE STEADY-STATE GREEN'S FUNCTION METHOD
TO BOND-BREAKIN G ELECTRON TRANSFER REACTIONS

59

CHAPTER VIH. INTRODUCTION TO PART H

The dynamical effects on the reaction rates for one-dimensional potential energy
surfaces (PES) have been discussed in the previous Chapter for the bistable and harmonic
oscillator potential models. As was discussed in Part 1, reactions on one-dimensional
potential surfaces have been the subject of the great interest for over fifty years, see Ref.
[2, 6, 8, 13, 53]. However, the multidimensional nature of many reaction PES forced
development of the kinetic theory towards investigation of the multidimensional surface
processes. Indeed, for reactions where the intrinsic potential surfaces are multidimensional,
there is no necessity to follow the lowest energy path and hence it is impossible to extract
this single, reaction coordinate which could justify the existence of some effective one-
dimensional reaction potential. So, we can anticipate that the investigation of the dynamics
of a reacting system on its multi-dimensional surface can demonstrate a richer behavior.

In the present Part of the dissertation we demonstrate the use of the steady—state
Green's function theory on reactions along two-dimensional potential surfaces, namely
bond-breaking electron transfer reactions. The majority of theoretical studies of ET
consider that no bonds are formed or broken during or consequent to the electron transfer
event. In this case ET is a simple chemical reaction. It has been studied extensively and the
one-dimensional approach in this case has shown quite satisfactory results [6, 23, 42, 54,
55]. However, there are many electron transfer reactions where bonds are broken or
formed in a way that is connected with the electron transfer event [28, 29, 56, 57, 58, 59,
60, 61]. We will refer to this circumstance as bond-breaking electron transfer (BBET). For
such processes, the level of the theory has lagged behind, in our view, that developed for
the "standar " ET, where no bonds are broken or formed. A theory of BBET should be
intrinsically more complicated than that of ET, as it must involve at least two distinct
coordinates: one connected with reorganization of the solvent (outer sphere) 0r bond

lengths (inner sphere), and the other a bond-stretch coordinate along the direction of the

60

chemical transformation. Only a few attempts have been made to approach the theoretical
description of BBET [30, 35, 52, 62]. The purpose of this Part is to present an extended
theory for BBET reactions which can include dynamical effects along both the solvent and
bond-breaking coordinates.

There is great diversity in the kinds of BBET reactions that have been studied.
Several examples that exemplify reactions of our interest are as follows [29, 54, 56, 63,
64]:

I) Dissociative electron transfer: e. g. reactions of aryl and alkyl halides (RX) with

various nucleophiles (N').
BBET
RX+N'—>R-+X'+N-

In this mechanism, the transfer of the electron and the bond breaking are considered
to be a concerted process. The RX bond must be stretched at the transition state,

which is also a non-equilibrium state in e. g. the solvent polarization coordinate.

II) SRNI (consecutive) reactions : e.g. reactions of aryl and alkyl halides with

various nucleophiles.
ET BB
RX+N'—>R"X+N'—>R'+X'+N'

This is nucleophilic substitution that proceeds by first an electron transfer step, and
then, decomposition of the radical anion. In this mechanism, the transfer of the

electron and the bond breaking are considered to be a consecutive process.

111) SN2 (concerted) reactions.
RX + N' —> (X--R--N)' —> RN + X'
Here, the electronic charge redistribution and bond rearrangement occur in a

concerted fashion. Nevertheless, these reactions may (though this is controversial

61

[64, 65]) be viewed as a single electron “transfer” with associated bond

rearrangement.

IV) Interval ence charge transfer in the presence of a counteri on.
AA+ + N“ —> MA + N‘

In this case, a self-exchange reaction (AA+—> A+A ) is rendered asymmetric due

to the presence of the counterion. The counterion forms a contact ion-pair with the

electron transfer reagent and the bond breaking is in this ion-pair coordinate.

Clearly, these “bonds” are much weaker than those of reactions I-HI.

Reactions 1 and II have been extensively studied by Saveant and his coworkers[29,
56, 63, 64] where they discussed the experimental verification of both concerted and
consecutive mechanisms and the conditions under which to expect one or the other.
Reaction IV has been discussed by Hupp and co-workers [59]. Again, the point was made
that this reaction can be viewed as either a concerted or a consecutive process. A study of a
reaction with a scheme similar to IV involving counterion effects on rates of charge transfer
reactions in radical anions has recently been carried out [61]. The SN2 reaction (III) has
been treated as a bond making/breaking event on an adiabatic surface under the influence of
equilibrium and non-equilibrium solvation [66, 67, 68, 69]. It was shown that a two
coordinate description is necessary (one coordinate corresponds to the reactive path of the
chemical transformation such as isomerization or dissociation, the other is responsible for
description of the solvent effects). In polar solvents with a charge-transfer reaction, this
solvent coordinate is conventionally taken as a solvent polarization. In the limit of a very
sluggish bath, the rate will be determined by slow polarization fluctuations, a case familiar
from the solvent-controlled adiabatic limit of electron transfer theory.

Thus, the picture of BBET shows great diversity and, in order to incorporate all of
the different features of the reaction, the theory has to be fairly broad. First of all, we have

to distinguish between two different mechanisms which are so common in the field - the

62

concerted and consecutive pathways of BBET. So, the first issue to discuss is the nature of
the surfaces that characterize BBET. One formulation must be found that can span the

extremes of the consecutive and concerted reaction paths. By analogy to what was used to

distinguish between SN] and 8N2 mechanisms [65], we suggest that this can be done by

introducing a third, intermediate state Ro'X in addition to the initial, RX and the final, RX-'

states. Therefore, the general geometric picture for BBET should include three two-

dimensional potential surfaces - one for each of the states. From the common physical

sense it is clear, that if the intermediate state is sufficiently stabilized, it will lead to an

intermediate of sufficiently low energy to favor the consecutive mechanism. Conversely, if
it is of high energy, the concerted pathway should be favored.

After the question of the geometry of the two-dimensional potential surfaces is settle
we have to consider what possible stochastic processes can be responsible for the motion
of reactive particles along the chosen potentials. It is conventional to treat the electron
transfer as driven by a combination of inner-sphere vibration degrees of freedom and outer-
sphere solvent polarization modes. The influence of both these processes on the electron
transfer have been thoroughly investigated [70, 71, 72]. The inner-sphere vibrations
usually are treated as a much faster processes compared to the polarization changes [6]. By
this assumption, the vibration modes only contribute to the equilibrium, not the dynamical
reaction rate constant. On the other hand, polarization changes can be sufficiently slow that
dynamical effects on the rate of electron transfer are associated with these fluctuations.
Moreover, it is traditional to treat the polarization fluctuations as an overdamped diffusion
process, though this may not always be the case [73, 74]. Thus, as far as our major
concern is the dynamical influence on BBET, we will consider only the slow polarization
coordinate. The general problem with the included vibrational part is analyzed in Ref. [30]

For the bond-breaking part of the whole event, in principle, both overdamped and
underdamped mechanisms could be applicable. If the frequency of oscillations of the bond

is much slower then the frequency of collisions with the solvent molecules, the

63

overdamped mechanism is appropriate. This might be the case with weak, non-covalent
bonds such as the ones in intervalence charge transfer IV.

The dissociation dynamics of proper intramolecular bonds have been studied
starting with Kramers’ work [1]. In the absence of solvent perturbation, the bond would
perform pure oscillatory motion. However, the solvent forces do affect the bond
oscillations, by producing occasional transitions between the energy levels in the bond
potential. This is a low friction regime in the sense that the latter forces cause only a small
variation of the energy during the time of an oscillation. The effect of the solvent consists in
the gradual change of the distribution over the different energy values inside the potential
well for the bond-breaking coordinate, and can be modeled as a random walk on the energy
levels. This is Kramers’ energy diffusion regime [1]. It was proven rigorously [2] that, for
the friction y to be low, it should satisfy the following inequality y <<cozk377De where to, is
the frequency of oscillation inside the bond-breaking potential, and D, is the bond
dissociation energy. In other words, the force along the bond-breaking coordinate should
be much larger than the thermal one. The energy-diffusion mechanism has proven to give
satisfactory results for the bond-breaking process in dissociation-recombination reactions
[8].

However, for the methyl halides discussed by Saveant, the bond frequency is very
high, around 600 cm '1 [75]. This calls into question a classical treatment of the bond-
breaking coordinate. It seems more reasonable to consider this coordinate as a quantum
system, so the bond transformation takes place through a tunneling process. Two
coordinate treatments with one quantum and the other classical are well established [76].

The plan of our work on BBET theory consists of the following sections. First of
all, in Chapter IX we establish the forms of PBS which are mostly suitable for'BBET.
Then, the choice of the surfaces naturally divides the theory into the concerted and
consecutive parts. As was mentioned above, in the concerted case, the bond-breaking and

electron transfer events take place simultaneously. Thus, the reactants, moving under the

64

influence of damping forces overcome some potential barrier and escape into the product
potential well diverting from the intermediate state. For the solvent coordinate (the electron
transfer mode) we shall always assume overdamped motion, so a Smoluchowskii high-
friction diffusion equation will be used to describe the damping forces. For the bond-
breaking mode we will consider three alternatives: overdamped, extremely underdamped
(Kramers' energy—diffusion) and quantum regimes. In Chapter X we develop the theory for
physically important and challenging energy diffusion mechanism along the bond-breaking
coordinate. As far as we are dealing here with a two-dimensional problem, it is quite
probable to observe a situation when the motion along one coordinate is much faster then
the one along the other. Hence, the dynamics may change from being two-dimensional to
effectively one-dimensional. Chapter X1 is devoted to solution of this problem. The case
where both coordinates are overdamped was accounted in Subsection 2.2 of Chapter VII.
Thus in this Part, in Chapter XII we will discuss the properties of the chemical rates
evaluated there and compare them with the ones evaluated for the energy-diffirsion
mechanism. The kinetics of BBET in the case when transformations along the bond-
breaking coordinate have to obey quantum rules are evaluated in Chapter XIII. This
Chapter closes the concerted part of the BBET theory.

The consecutive mechanism of BBET is discussed in Chapter XIV. In this case
particles moving out of the reactant potential surface have to pass some intermediate state
before they can reach the product well. Thus, at least three surfaces are involved in the
reaction. Though, in principle, the total kinetic picture of the BBET is completely different
then in concerted reaction, the dynamics along each of these potential surfaces are
analogous to the ones that are discussed in the framework of the concerted scheme. Chapter
XV contains concluding remarks concerning the steady-state Green's function theory in
unimolecular reactions as well as advantages and consequences of its application to the

irreversible processes.

65
CHAPTER IX. POTENTIAL ENERGY SURFACES

As has been discussed in the previous Chapter and exemplified in reactions I-IV,
BBET involves both ET and bond-breaking. PES for the BBET reaction may be regarded
simply as the addition of the surfaces for the two individual processes. This does neglect
the possibility of a coupling between them, which is certainly a possibility which should be
considered. But, clearly, the qualitative features we wish to explore should be evident in
this simplified model. According to the scheme outlined in Ch. VH1, we intend to consider
three reaction states for BBET reactions - the initial, RX, the intermediate, R-‘X , and the
final, RX-' state. Each of states has to be represented by a two-dimensional diabatic
potential surface. The coupling between them provides transitions from one state to another
and we say that the reaction has taken place when the particles from the initial state
(reactants) appear into the final potential well.

For ET reactions it is well known [71, 72] that PES can be considered to be

parabolic in the polarization contributions

VP(x) = x2 /2 (II.9.1.a)
V,(x)=(x—xo,)2 /2+AGO, (II.9.1.b)
VP(x) = (x — x0,,)2 / 2 + AGO, (11.9.1 .c)

where AGOI and AGop are the exothermicities, and x is the coordinate which characterizes
the outer-sphere solvent polarization motion. The subscripts R, 1 and P denote the reactant
(initial), intermediate and product (final) states respectively. The solvent polarization
coordinate x is defined by
x2 / 2 = 35 “P(r) — Pf(r)|2dr
c

where P(r) and P? (r) are the non-equilibrium and equilibrium solvent orientational
polarization and c is the Pekar factor c=eeo'1 -eo’1 with see and so the high frequency and

static dielectric constants. The outer-sphere reorganization energies are defined as:
r, = x3, / 2 = 3min) - P3(r)|2dr, k = I,P
c

66

2,, zm—JZ. (11.9.2)
The schematic picture of the diabatic surfaces for the electron transfer coordinate is given in
Fig. 8 a.

Now, the diabatic surfaces for the bond-breaking process have to be constructed in
order to incorporate them into Eqs. (11.9. 1)-(II.9.3). For the gas phase, Morse potentials
would be appropriate for a description of the bond dissociation motion of the reactants, RX
and intermediates, R-‘X. For the dissociative state, RX-', the repulsive branch of the
Morse potential would be used. However, we are dealing with a liquid state problem and
the requisite potentials are the potentials of mean force. The solvent environment, of
course, readjust all of these potentials, especially the dissociative one. Nevertheless, with a
fairly good accuracy we can state that the product and intermediate state potentials keep
their shape [29]. We believe that this is not the case with the dissociative potential V p(z).
The caging effect of the solvent molecules creates a shallow minimum on the repulsive
surface, so that the products do not immediately separate and disappear into the bulk, but
form a stable state. Thus, we choose the following form of the bond-breaking part of the

reaction surfaces:

VR(z) .—. D,(exp(—bz) — 1)2 (11.9.3 .a)
V,(z) .—. D,(exp(—b(z — d,)) - 0,)2 (11.93.15)
V,(z) = D,(exp(—b(z — d,)) — 0P)2 (11.9.3 .c)

where De is the dissociation energy of the bond, and 01 and 6p are the displacements of the
intermediate and final state potentials. Fig. 8 b depicts the potentials given in Eqs. (11.9.3)
These Morse potentials should mimic bond-breaking reaction in the liquid.
For future calculations it is convenient to introduce the reorganization energies for
the transitions from one state to another:
22R, = D,(l + bd, - 0,)2 (11.9.4.a)
2,3,, = D,(1+ 0d,, - 0,)2 (11.9.41))
2”,, = D,(b(d,. — d,) + a, — 0,)2. (11.9.4.0)

67

 

 

 

 

 

ad
ES
0)
5
0 1‘01 xOP polarization
(3)
p51] 1
9' P
g R
301
: 5 c5?
0 d1 dp bond-breaking
(b)

Fig. 8. Schematic picture of the potentials for each of the coordinates participating in

BBET: a) the orientational polarization (or ET) coordinate; b) the bond-breaking (BB)

coordinate.

68

If the points of intersection lie sufficiently below the dissociation energy of the
Morse potentials, a harmonic approximation of them would be reasonable [30]. Expanding

the Morse potentials around their equilibrium bond distances, we get:

VR(z) = Debzz:2 (11.9.5.a)
V,(z) = Debz(z—zo,)2 (11.9.5.b)
VP(z) = Deb2(z — 20,.)2 (11.9.5.c)

with z0K=(l-oK)/b. For the sake of simplicity these potentials have equal spring constants.
The majority of ET reaction analyses make the same assumption. For our case the form of
the product surface is not very important, because we consider irreversible reactions. The
only effect it produces consists in the equation for the line of intersection between the
surfaces and the harmonic approximation does not affect it dramatically. The form of the
intermediate state is more important for the kinetics of consecutive BBET. For this reason
in Chapter XIV we consider not only the case of approximately equal spring constants, as
in Eqs. (11.9.5), but also a situation when the intermediate state is much shallower then the
initial one (i.e. the bond of an intermediate is much weaker then the bond of a reactant).

Now, adding the potentials for the bond-breaking to the electron transfer term, we
obtain the two-dimensional reaction surfaces

VK(x,z)=VK(x) + VK(z), K=IL I, P. (11.9.6)

The lines of intersection between these diabatic surfaces can be evaluated form the
set of equations V,(x,z)=Vj-(x,z), where i,j =R, 1, P. Each of these lines has a minimum
point which corresponds to the activation energy of the transitions. These minimum points
can be obtained by Lagrange's method, the undetermined multiplier accounting for the
constraint V ,-=V}-. Thus, we have three sets of coordinates for the points on the surfaces
which determine the activation energies of transitions: (x1311; 2111*). (xRp*, sz *), '(xp1*.
2191*):

3cm": ZAflA—G-glfii, zR,*=1/22,R,A—G;—'E-i’-, l,=2fl+lw (II.9.7.a)
I I

69

AG — A0,... + 2 AG — A6,. + A,
le* : 23am 01 - 1P , ZIP* : ‘V 24:1,, 01 ' ‘P ,

21m 21m
’11? : item + 1:1? (II-97b)
A +2 A
36“,“ = 211,, —G£——P, zRP* = 21,“, ME, 1,, = 11,, + 1,”. (II.9.7.C)
21F 21?

The relative position of these points in the reaction space determines which of the two
mechanisms - concerted or consecutive - is most suitable for the reaction. Fig. 9
schematically presents the two different pictures. The concerted pathway takes place when
V g(xRp“, sz *) < V R(xR1*, zRI“) and the consecutive mechanism is preferable when this
inequality is reversed.

For the sake of simplicity, in the next four chapters devoted to the concerted

mechanism of BBET we will use the notation: A, a 11,}, and A, a 2m.

70

 

 

 

 

a:
9
g l
0
reaction coordinate
(a)
0:
56
O
8
reaction coordinate
(b)

Fig. 9. Concerted pathway (a) is sketched in comparison with the consecutive (b) one.

71
CHAPTER X. ENERGY-DIFFUSION MECHANISM 1N BOND-BREAKING

ELECTRON TRANSFER REACTIONS

In this Chapter, our aim is to combine both solvent polarization dynamics
(coordinate x) as described by high friction, overdamped diffusion, and bond-breaking
dynamics (coordinate 2) as described by low friction, energy diffusion, in one concerted
process and investigate their joint influence on the BBET reaction dynamics. Hence, we
assume here that the reactants undergo direct transitions into the products and essentially
ignore the existence of high energy intermediate state. When the electron transfer and
breaking of a bond occur in a concerted manner, we would like to understand how the
relaxation times for the coordinates r, and r, , and the reorganization energies 2x and A,
contribute to the total kinetic rate. As in Part I, here we will consider only irreversible
mechanism of the reaction, as far as the rate constants for reversible reactions can be
evaluated through those ones for irreversible, see Ch. 11 and Eq. (1.2.2) in particular. The
schematic picture of the reaction potential VR(x,z) is given on Fig. 10. This Chapter
presents an integral approach which provides the overall rate constant in terms of explicitly
defined equilibrium and diffusion rate constants [52] by means of the steady-state Green's

function method.

1. e e tion 0 otion the sink nction

Since we will deal with energy-diffusion along the z coordinate, it is convenient to
use its energy “image” ,E, to describe the dynamics. Then the time-evolution equation for

the joint probability P(x,E,t).is given as (see Eq. (1.3.1))
aP(x, E, t)/ at = [L(x) + L(E)]P(x, E, t) — W(x, E)P(x, E, t) (11.10. 1)

where the diffusion operator for the polarization coordinate is
_é: 1 .
6x 906) ’

 

L(x)=-’§-l€§x-¢(x) Tx 5 TL (1110.2)

72
with Q the longitudinal dielectric relaxation time, B = 127(37‘, and (p( x) = 4.3/27! [Bl/(x)
the Boltzmann distribution along the polarization coordinate. The dynamical operator for

the bond-breaking coordinate comes from Kramers’ energy diffusion problem for the

harmonic oscillator potential [1]:

(.4
6E

+16), with D(E)=—£-,r.=-1— (1110.3)

6
L = __
(E) 6ED( fir, y

where y is the friction. The energy range is 0 to cc, and the Boltzmann distribution for the
bond breaking is (p(E) =13e‘”. We will assume that reactants are initially in thermal
equilibrium,

P(x, E, 0) = tp(x) . (0(E). (II. 10.4)
The boundary conditions for Eq. (11. 10.1) require that the fluxes of the particles at the

boundary x,z = {:00} are zero.

 

 

 

 

 

 

Fig. 10. Schematic plot of the potential energy surface for concerted BBET reactions.

To define the sink function W(x, E), we start from the definition of the thermal

equilibrium rate constant for a nonadiabatic, irreversible reaction [30]:

__ (22:51 [2117,2117z e-fiH18(AV(x)+ AV(z))

K
e rt ’ Idl‘xdfze‘fifll

 

(1110.5)

73
where r_,(r,) denote the phase space coordinates dpx dx (dpz dz). H1 is the Hamiltonian

corresponding to the initial state potential VR(x,z), and AV is the difference in potentials for

the initial and final state surfaces. Using the energy coordinate E = p? / 2 a + D, 3:2 and
introducing the identity 1 = jdE6(E - 1110‘,» in Eq. (1110.4) yields:

K, = [dEjdx W(x,E)(p(E)tp(x) (11. 10.6)
where

27th ®(E- E(x))

W(X’EFW JE—Etx) '

(1110.7)

Here, 6) is the step function, V,fis the strength of coupling between the initial and final

states and the activation energy is defined as

 

2
.. A 0 2 — 271
E(x)- 2,,[ G + 0 ”J (1110.8)

' 271, 422,,
In Eq. (11. 10.8), 2,, is the reorganization energy for the polarization coordinate, A, = De is

the bond-breaking reorganization energy, which coincides with the dissociation energy,
and 10 = Ax + 12.

For a high activation barrier reaction, in the sense that 0E,“ (x) >> 1 for all x, Eq.

(11. 10.6) can be approximated by

W(x,E)=% [3% 5(E—E‘(x)), (11109)

that is a sink line on the potential energy surface in the x—-E coordinates.

W
Before introducing the form of the reaction rate constant, let us emphasize that, in

order to derive Eq. (11. 10.9), we utilized the assumption that the activation barrier of the

reactions under consideration is high, BEG >> 1, where Ea is the activation barrier energy.

This is the second major assumption that we apply in our considerations - the first one is

74

the assumption of separability of the motions along x and z coordinates. The high activation
barrier assumption permits several significant simplifications in the calculation of the
reaction rate constant.

It was explicitly proved [45] (see also Appendix B or Eq. (1.7.12) and the
following) that the high activation barrier kinetics are essentially mono-exponential. Thus,

the rate constant can be defined simply by using the probability distribution function P:

K" = [:20 [5° [:0 dxdE P(x,E,t). (11.10.10)

This is the rate constant of mono-exponential unimolecular decay, traditionally represented

as Pe(t) = e—K’ where Pe(t) is the survival probability.

For extremely low friction y --) 0, Kramers noted that the motion of a particle inside
a one-dimensional potential well is mainly oscillatory with occasional transitions between
the energy levels. Thus action (or energy) is almost a constant of the motion, and is the
appropriate variable with which to obtain a stochastic description. The lower the particle
sits inside the well, the more difficult it is for the external fluctuations to move it on to a
higher level. The energy-dependent diffusion coefficient D(E) in Eq. (11. 10.3) reflects this
picture. As a particle reaches energy levels close to the barrier, it gains energy more readily
and finally, when it has enough energy to surmount the barrier, it irreversibly escapes from
the well. The reaction rate constant of this process can be expressed through the mean first

passage time for a particle from the bottom of the well E = 0 to reach the barrier E = E, [8]

1 E, dEe‘fi-E E -pE'
K=—; T = —— e dE'. (11.10.11)
t'p p Jo D(E) i0 .

With the assumption of a high activation barrier reaction, the asymptotic value of the above

integral yields the result [1]

K = pyE e‘l’Ea. (11.10.12)
0

75
Note that an implication of Eq. (11.10.12) is that the energy diffusion coefficient D(E)

appears as the constant D(Ea) = Ea/li'rz. This means that only the dynamics around the
barrier contribute to the rate constant, and the diffusion coefficient may as well be taken as
a constant for high activation barrier reactions. This conclusion permits significant

simplification of the operator, Eq. (1110.3) to

E,_a_ (E3 1
finals") 6E (pus)

 

L(E) = (11.10.13)

We will use the above form of the energy-diffusion operator, as it is consistent with our

assumption of high activation barrier.

4. t ' l'
The differential equation, Eq. (11101) with initial condition Eq. (1110.4) can be
written in the integral form:
P(x, E, t) = (p(x)(p(E) — Ede]: [:dx'dE' G(x, x'; E, E';t — T)W(x', E’)P(x', E', 1)
(11.10.14)
with the Green’s function defined by the equation:
L(x, E)G(x, x’;E, E',t) = %G(x’ x';E, E',t). (11.10.15)
Here, because of the global separability of the reaction potential V(x,E) = V(x) + V(E) and,
hence, L(x,E) = L(x) + L(E), the Green’s function G(x, x’;E, E',t) is a product of Green’s
functions for the independent operators along the x and E coordinates:
G(x, x';E, E',t) = G(x, x’,t) - G(E, E',t) (11.10.16)
where
L(x)G(x, x’,t) = 50(x, x',t)/5t C(x, x',0) = 6(x — x') (11.10.17a)
and

L(E)G(E, E',t) = aG(E,E',t)/at G(E, E', 0) = 6(E - E'). (11.10.1715)

76

It is convenient to represent Eq. (11.10.14) in a different form, introducing a new
variable, 5(x,E,s)= E(x,E,s)/ [39(3), where 13(x,E,s) and Pe(s) are the Laplace
transforms of the population probability distribution function and survival probability,
respectively. Then for the high barrier case, Eq. (11.10.14) transforms to the following

expression, (see Eq.I.4.13):
g(x, E, s) = tp(x)(p(E) — EOJ-wdx’dE' G(x, x';E, E',s)W(x', E') g(x', E',s) (11.10.18)
with the very convenient redefinition of the reaction rate constant, as given in Eq. (1.3.12):

K =1irn I?dxdEW(x,E)§(x,E,s). (11.10.19)

In principle, the Green’s function used in Eq. (11.10.18) should differ from the
Green’s function used in Eq. (11.10.14). However, because we are dealing with the high
activation barrier case, this difference is of the order of the small parameter (BEA—1, which
should be neglected for the sake of consistency.

As far as the sink function W(x,E) is a line on the potential surface (formed from
the intersection of the initial and final state surfaces), with an infinitely small width, as in
Eq. (1110.8), the decoupling approximation, described in Ch. VI can be applied to the
integral equation, Eq. (11.10.18). Taking into account definition of the rate constant in Eq.

(11.10.19), the result of the decoupling can be represented in the form
K = K,/(1+ K, Mg) (11.10.20)

with the equilibrium (non-adiabatic) rate constant

K, = TidxdEW(x,E)tp(x)tp(E) (11.10.21)

-®O

and the diffusion rate constant, Kd = l/rp, where rp is the effective MFPT, originally

presented in Eq. (1.5.9):

e, = 7:7 [“20”] j M' dEdE' K(x, E)G(x, x'; E, E',t)K(x’, E')(p(x', E') (11.10.22)

with the Green’s function is defined in Eq. (11.10.16).

77
The Eqs. (11.10.20)-(II.10.22) give the expressions for the BBET rate constant in a

general form. The important issue here is that, with the given delta function form of the
reaction sink (Eq. (11108)), the rate constant can be divided into separately calculable
parts — a diffusive part K d and an equilibrium rate constant part K ,.

The analytical solution for the Green’s functions, with the assumption of a high
activation barrier, is presented in the Appendix C. Now, we will discuss the results of their

utilization.

5. eneral rm 0 the non-adiabatic and di sional rate constants

When relaxation along both coordinates is fast, the reaction is controlled by the
equilibrium rate constant, Eq. (11.10.21). This regime, ofien called the kinetic control limit,
corresponds to the physical situation where the rate constant is given by the Golden Rule
expression for a nonadiabatic transition or by transition state theory for an adiabatic
reaction. It assumes thermal equilibrium prevails in the initial state. Thus, the motion inside
the well is so fast that it re—establishes the Boltzmann distribution inside it afier each escape
event. In terms of the reaction constants given by Eqs. (11.10.21) and (11.10.22), this
means that K, << Kay and K = K,. Calculation of the equilibrium rate constant is

straightforward, taking into account the form of Eq. (1110.8):
2
[3 AGO+20
( 21114;) ( )

l3 42
K = j e o . 11.10.23
’ L h j 47:2, ( )

The other extreme regime possible in this situation corresponds to the inequality K,

 

 

>> Kd and is called the diffusion-control limit. In this case, the intensity of the sink is so
large that the motion inside the potential well determines the reaction rate. In principle, this
regime coincides with Kramers’ one-dimensional problem, which can be formulated as the
mean first passage time from the bottom of a one-dimensional potential well to the point E,
[8, 53]. For multidimensional processes we have to consider the mean first passage time to

a line on the multidimensional surface. For a correctly chosen (adiabatic) potential surface,

78

this time gives the rate constant that corresponds to the multidimensional Kramers rate
constant, and is given by Eq. (11.10.22). Eq. (11.10.22) is a general recipe for calculation
of the mean first passage time to a line on the multidimensional surface. We note that the
Green’s function in Eq. (H.10.22), in the case of high activation barrier, can be considered
as the one given by Eq. (11.10.15) for any two-dimensional potential V(x, y), i.e. not
necessarily separated into two independent ones, as assumed in this article.

Returning to the calculation of the mean first passage time for the BBET reaction
given by Eq. (11.10.22) with Green’s firnction (11.10.16), we require the Green’s functions

in Eq. (11.10.17), for the case of high activation barrier. These are obtained in Appendix D:
p(E-E'+E;t/t,)’

GE(E, E' ,=1) 2% e 45“,“ (11.10.24)

 

 

Gx(x,x',t)= "3— e (11.10.25)

where x* and E; define the minimum of the sink function given by Eq. (1110.8). E, is the

activation energy along the z (or E) coordinate, and the activation energy along the x

coordinate is E; = x *2/2. E; and x* are [30]:

 

 

15;: #320210? (11.10.26a)
x* = 91 (AGO + 10). (11.10.26b)
1] 2 A,

By utilizing the form of the Green’s functions, Eq. (11.10.24) and Eq. (11.10.25),
in Eq. (11.10.16) and then in Eq. (11.10.22),'we can represent the effective MFPT as:

(40 2 )2
T _ 31:3 231: a _LLFO_ {7:11 dzInt(z I)“ _ [3(2 + x * tr, Kathi
p - 42 822E“ p 4ftz «er0)

(11.10.27a)

 

79

where

’12
1nt(z,t) : 2 Z 2
2,: + 22,221 + 22,1

 

 

 

exp[_ p(2,.-:2 + 22,1) (22,0100 + 20)2 + 2.g(2,,22 + 22,1) - 2222,,/2j(2100 + 2,, ))
82, 2,2 + 22,2,1 + 22,1
(11.10.27b)
Unfortunately, we were not able to calculate the mean first passage time for BBET
analytically from this expression for arbitrary parameter values. Thus, the following
discussion will be devoted to an analysis of the possible limiting cases, and accompanied

by a numerical analysis of Eq. (11.10.27).

80

CHAPTER X1. FAST AND SLOW MODES IN MULTIDIIVIENSIONAL
KINETIC PROCESSES

1. General Asmcts

When one of the modes in a 2D problem is much slower than the other, it is
possible to reduce the problem to an effective 1D one. As was noticed in Ref. [21], this
reduction is an “adiabatic elimination procedure,” which has the Born-Oppenheimer
approximation as its analog in quantum mechanics. Let us take, for example, 1: as the slow
mode. Thus, diffusion along the x coordinate is much slower than along E (or 2), which, in
terms of the relaxation times, can be expressed as 12 <<rx. In this case, the contribution of
the operator L(x) in Eq. (1110.1) is small compared to that of L(E), so that we can neglect

it, resulting in the following expression:

6P(E, x, t)/ at = L(E)P(E, x, t)- W (E, x)P(E, x, t). (11.1 1.1)
In this differential equation, x is a parameter, entering the solution P(E,x, t) only from the
sink firnction. The separation performed in Eq. (H.1 1.1) indicates that the dynamics along
the x coordinate sees only the equilibrated picture from the coordinate E created by its fast
motion.

Now, let us introduce a time re, as the time by which exponential (equilibrium)
kinetics with some long-time rate constant has been established along the fast E coordinate
(this time was introduced in Appendix A, Eq. (A17)). It was proved that this time should
be of the same order as the relaxation time 1,. Therefore, by the time 1,, an equilibrium
profile (not necessarily Boltzmann) has been established along the E coordinate. To utilize
this conclusion, let us make a statement that, as we will see later, is correct: for all t > re
(and r, < < rx), namely, that the population probability distribution function can be given

as follows:

81
P(E,x,t) = R(E,x) - Q(x,t). (1111.2)
The equilibrium profile is represented by the steady-state function R(E, x) and all the
time dependence is due to the dynamics along the slow coordinate x .
We can obtain R(Ex) and the long-time rate constant corresponding to this process

from the steady-state differential equation

L(E) R(E, x) - W(x,E)R(E, x) = -K(x)R(E, x) (11113)
with the boundary conditions for the E coordinate as in Eq. (1110.1) and an additional

condition for R(E,x), which lets us define the long-time rate constant K (x):
[ dE R(E,x) = 1. (11.1 1.4)
This method was discussed in detail in Ref. [45], see also Chapter 111. It should be
emphasized that Eq. (1111.3) is not an eigenvalue problem, but an ordinary differential
equation with some boundary conditions supported by Eq. (1111.4) as an additional
condition necessary for the definition of the long-time rate constant.
Assuming that we know the forms of R(E,x) and K (x) by solving Eqs. (1111.3)

and (1111.4), we can use the representation of Eq. (1111.2) directly in Eq. (11.101), to

obtain
R(E, x)-§;Q(x, t) = Q(x, t)[L(E)R(E, x) - W (x, E)R(E, x)] + L(x)R(E, x)Q(x, t).(II. 1 l .5)

Then, taking into account that the expression in square brackets is the left part of Eq.
(1111.3), integrating Eq. (1111.4) over all E space, and using Eq. (1111.4), we obtain a
1D differential equation for the slow coordinate with the effective sink function from Eq.
(1111.3):

gate, 1) = L(x)Q(x, t) — K(x)Q(x, t). (11.1 1.6)

To sum up, we note that the above is a general scheme to reduce any 2D problem
with fast and slow modes to a 1D one that contains dynamics only along the slow mode.

The most difficult part of this scheme is the solution of differential equation, Eq. (1111.3),

82
in order to get the effective sink function K (x). However, the fact that we are dealing with

the high-activation barrier case permits some significant simplifications of this problem.
The key point is that, for hi gh-activation barrier reactions, the long-time rate constant K is
approximately equal to the generalized probability a = fdt P¢ (t) which is much easier to
calculate. The reason for this is that the time interval for the non-exponential stage in the
high-activation barrier reactions is so small, compared to the time scale of the reaction, that
the exponential kinetics can be considered to take place at all times. Then, the long-time rate
constant and generalized probability are identical. The proof can be given easily. If the
kinetics is exponential, i. e., Pe(t) = e‘K’ where K is the long-time rate constant, then the
generalized probability, 21:, is equal to K.

The equation for the definition of the generalized probability is usually much easier

to solve than the one for the long-time rate constant (Eq. (11.1 1.3)):
L(E)R(E, x) — W(x,E)R(E, x) = —5:(x)f(E) (11.1 1.7)
or, in integral form:
R(E,x)= f(E)- [dE G(E,E)W(E‘,x)R(E,x) (1111.8)
with the rate constant (see Eqs. 3.5) and (1110.20)):
52(x) = IdE W(E, x)R(E, x).
0

Here, f(E) = (p(E) is the initial distribution and G—(E, E') is the steady-state Green’s
function that satisfies the equation (for high barrier)

L(E)C-}(E,E)= —6(E-E). (1111.9)
This Green’s function can be defined easily (see Appendix A). Ifthe sink function W(x,E)
has the form

W(x,E) = K06(E - E“ (x)), (11.11.10)

that is the case in Eq. (1110.8), then solution of Eq. (1111.8) is very straightforward:

83

 

 

5e(x): 1+ K252121426) (11.11.11)

where
K,E(x) = IdEW(x, E)(p(E) = K0(p(E * (x)) (11.1 1.12a)
K,,.(x) = 5(E¢5:;g1)(x)). (11.11.1211)

The probability-distribution function also can be found easily from Eq. (1111.8):

 

R(E, x): (p(E)|:1—K ”(55; 1 +i((§71§*1)5*)ia (11.11.13)

In the case of the regular diffusion-type operator L(E): DEE— Etp——(E) 6E -—(—-¢E),th

Green’s function G—(E, E) for high activation barrier has the form, see Eq. (1.5.10):

 

' 1 «P(E) E>E,
(7(E,E')=t MEI/(E ”(15) (11.11.14)
1 ,
—— E<E
,BDEVTE“)

 

where V '(E *) = dV(E)/6E| E“ and E *is the energy at the activation barrier. Then Eq.
(H.11.12b) can be rewritten as
KdE = pDEV'(E * (x))(p(E * (x)). (11.11.15)
Thus, for high activation barrier reactions, the sink function in Eq. (H. 1 1.6) for the

slow mode can be calculated as

K (x) = 52(x) 2 1+ K1155"; a (x) (11.1 1.16)

 

with K,E(x) and KdE(x) defined in Eqs. (H.11.12a) and (H.11.15).

84
As we can see from Eq. (11.11.16), the general form of the sink function K (it) as a

function of the slow coordinate is very complicated, so that an analytical solution of Eq.
(H.116) is hard to achieve. However, if the dynamics along the fast coordinate E is so fast

that we can talk about the kinetic control limit for this coordinate (K ,5 << K 2113), then the

sink function K(x) expression simplifies dramatically:

K(x) = K,,(x) = Ko(p(E * (x)). (11.1 1.17)
Moreover, the probability distribution function R(E,x) from Eq. (11.11.13) simplifies to
R(E,x)ztp(E), and therefore it coincides with the Boltzmann distribution along the fast
coordinate.

This analysis demonstrates that the separation of a multidimensional reaction into
fast and slow processes can reduce the problem to consideration of a one dimensional
process only. However, the case can still be very complicated if the dynamics along the fast
coordinate is not fast enough to compete with the equilibrium rate constant, in other words,
K,E z KdE or K,E > Kdg. In this case the sink function given by Eq. (1111.6) has the form
in Eq. (H.11.16) or K (x)z Kdg respectively. We will come back to this situation in the
next Chapter when we will discuss the total picture of 2D reactions. However, when K, z
Kdg, Eq. (H.116) is very unlikely to be solved analytically.

When K ,5 << KdE and K (x)z K0 (p(E*(x)), then the situation becomes similar to
the one originally described in Ref. [6] for the fast and slow modes, for the case of
harmonic oscillator potentials. If we assume, in accordance with Ref. [6], that V(x) = x2/2
and V (E) = E2/2, then the sink function of Eq. (11.11.10) transforms into (see Ref. [30] for
details):

2
E*(x)=[AG°+A°;xVu'] (11.11.18)
E

Kc _ t
W(x,E)—Vi:6(E E (XI)), 4

where 25 and 2, are the reorganization energies for the E and x coordinates, and 20 =

2E+2, .The effective sink function K(x) given by Eq. (11.11.17) has the Gaussian form:

85

~ _ f 2
K(x)=K0 ’figexp[_fi(400+lo x 21") (11.11.19)

425

 

that coincides with the form of the sink function in Ref. [6] obtained by using transition
state theory. Furthermore, if ,825 —9 0, then K (x) in Eq. (11.11.16) transforms into a 5-
functional sink function K (x)~6(x-x*) with x*=(AGo+2,J2/ 1/22, and we encounter the
problem of 1D harmonic oscillator which was considered in detail in Section 2.1 of Chapter
VH. However, one should remember that application of this simple sink function K (x) for

the slow mode is legitimate only if the condition K,E << KdEis satisfied.

2. Fast bond breaking.

This situation corresponds to the case when the relaxation time for the bond-
breaking coordinate is much faster then the polarization relaxation time, i.e. rz<<rx .
Hence, the scheme, described in the previous Section can be applied to analyze the reaction
kinetics.

Here, the fast bond-breaking dynamics creates the effective sink K (x) for the
polarization to transfer the electron. Then, for the effective sink function given by Eq.

(H. 1 1.16), we can calculate K,E and Kdg from Eqs. (11.11.12a) and (11.11.15):

 

 

 

 

— 2
,1 _
K,E(x)= K0 [3 exp -fl x(x x0) ; 350 = AGO +10 (11.11.20)
4712, 222 22,,
_— 2
KdE(x)= fif" exp|:- [31x21 x0) ]. (11.11.21)
Z Z

As one can see, both of these “constants” have a Gaussian form. However, the limit [321—9
0 in order to get a 5-function sink is not allowed in this situation, because originally, in
order to get a constant diffusion coefficient D(E)=D(Ea)=Ea/[irz , we required that the

reorganization energy for the bond-breaking mode be much higher than the thermal energy,

86
22>>1/B .Thus, the above rate “constants” have to be Gaussian, and the effective sink

function K (x) has the form

K01/4’82
K(x - 71’ z

) ' 1+ K02, «Em/472,732,)

 

 

— 2
exp|:— “4:; x0) ] (11.11.22)

It can not be reduced to a delta function but can be treated as a narrow function that permits
the decoupling approximation solution, when 22/71,<<1. Finally, for fast bond-breaking,

the reaction rate constant K can be represented as
K=Kf/(1+Kf/Kj) (11.11.23)

where the equilibrium and diffusion rate constants are calculated for the “narrow” sink

function K(x) in Eq. (H.11.22):

K01} 4 [32
K? = j 2:801:70): " z

1 + K01, «Em/42132,)

 

 

2
CXP[_p(AG40 1:10) ] (11.11.24a)

1_r,_ 1
K3 , (K32

 

 

i: ‘1’” “1990435,” 01306 )<p(x' ). (11.11.2411)

As one can see from Eq. (H. 1 1.23), the equilibrium rate constant now depends on the fast
relaxation time, so it has gained some dynamic features and cannot be obtained from the
classic transition state theory.

The diffusion rate constant is given through the effective MF PT to the narrow zone
K (3:) when the dynamics along the x coordinate start from the Boltzmann distribution.
Here, we give the frnal result for the diffusion rate constant using the Green's functions for
high activation barrier given in Eq. (H. 10.25):
1 , 2, 47:2,, /1_p2§AGO+20)2\*
K31 x 4or \lfiMGo + 40?) 240 4412) J

,,p[0(400 + 4012 (1+ 2, 4% +140; 4,12)
420 22,, 420

 

 

 

(11.11.25)

87
To sum up, for the case of a narrow reaction sink function, i.e. when 22/2,<<1 ,
and for the reaction controlled by the bond-breaking process (t,<<r,), the total rate

constant has the form K=KfK§/(K,x +Kj) where K, and Kd are given in Eqs.

(11.11.23) and (11.11.25).

3. Fast mlarization.

Above we considered the situation when the energy-diffusion coordinate was
characterized by a faster relaxation time than the polarization one. In this Section we
consider the reverse situation — the polarization fluctuations are faster then the bond-
breaking dynamics. In this case the scheme developed in Section 1 has to be applied when
x refers to the energy-diffusion coordinate E, and E refers to the polarization coordinate x.
Then, the separation in Eq. (H.112) has the form P(x,E,t)=R(x,E) Q(E,t), and the

effective sink firnction defined by Eq. (H.11.11) becomes
K203)
1+ Km(E)/ de(E)‘

 

12(15): 53(E) = (11.11.26)

For further analysis we assume that the t, is so fast, that kinetic control, K ,x<<de, along

the fast polarization coordinate takes place. Then, according to Eqs. (11.11.12a) and
(H.11.17)

8(5) = 82(5) = fig—Ae

The effective sink function is the sum of two Gaussians with different shifts. The situation

 

-3142 (Ea/2, / 22,170)2 + e—%(JE—Jlx / 21230 )2 ]
X x

simplifies considerably when the polarization reorganization energy is small, 2,12,<<1.:

 

~ K
K = 6 E-E 11.11.27
(5) W ( a) ( , )
with
E =fi=(AGO+2,)2
" 2 42,

The analog of Eq. (1111.6) for the energy-diffusion coordinate has the form

88

 

5Q(E.t)/é’l = L(E)Q(E,t)- K(£)Q(EJ). (II-1128)
For the sink function of Eq. (11.1 1.27) the reaction rate constant can be obtained
immediately
K = KiKi/(KlE + K5 1, K5 = K0 5 exp[-fiEa]
47:2

2

(11.11.29)

 

K5 = ”f“ exp[—73E.]

2

where E, = (AGO + 2,)2 / 42,. The above diffusion rate constant precisely corresponds
to the Kramers result for energy diffusion.

At this point it is interesting to compare two possible mechanisms for the bond-
breaking, energy diffusion and overdamped diffusion. All the difference in the rate
constants describing these two different mechanisms is due to the pre-exponential factor in
the dynamic rate constant. For overdamped motion, the analog of Kd presented in Eq.

(11.11.29) is (see Eq. (1.5.17)):

V' at:
K,= :02”) fiexp[-[3V(z*)]

Z

 

for a general potential V(z), where 2* is the point of transition and 1,00 is the overdamped
relaxation time. For the harmonic oscillator potential with the transition taking place at the

point corresponding to energy Ea, the above rate constant is given by Eq. (1.7.12)

K, 1(1),, 4%:- exp[—,BE,] (11.11.30)

2
Of course, the relaxation time tZOD now has a completely different physical meaning, in

 

that it is now proportional to the friction 7, but its value still could be comparable with
tz=lly for the energy-diffusion mechanism. As one can see from Eq. (H.11.30), the
dependence of the pre-exponential factor upon the activation energy changes from the linear

for the energy-diffusion mechanism to a slower square-root one for the overdamped

motion.

89

To summarize, the scheme elaborated here presumes that the relaxation times for the
x and E coordinates are on very different scales. According to this scheme, after the
reaction has started, the faster motion along one of the coordinates (with shorter relaxation
time) shapes the "conditions" of the ensuing reaction. So, by some short time, the fast
relaxation equilibrates the distribution along its own coordinate in such a way that it forms a
sink through which the population of the slow coordinate decays and yields the product. i

The rate of this reaction is controlled exclusively by the slower relaxation time.

90
CHAPTER XH. GENERAL ANALYSIS OF THE CONCERTED BBET THEORY

I. Ener di sion mechanism and com rison with other BBET theories

In this Chapter we analyze the rate constant for the BBET concerted reaction taking
place on a 2D potential surface, with polarization fluctuations as one coordinate and the
bond-breaking process as the other. We have assumed that the reaction is irreversible and
occurs through a high activation barrier. As we discussed in Chapter VH1 two mechanisms
are possible for motion of the reactive particles along the bond-breaking coordinate. The
first one is the overdamped diffusion motion (OD mechanism) that was discussed
previously, see Ch. VII, Sec.2.2 and Ref. [30]. The other, energy-diffusion mechanism
(ED), was the subject of preliminary discussion in the same paper and now is considered in
detail here. We now discuss the general results for the ED mechanism in the 2D BBET
problem, and compare them to the results previously obtained for the ED and OD
mechanisms [30].

The general result for the rate constant for the OD 2D irreversible reaction in the

case of high activation barrier can be represented as K = KrKgD/(K, + K g D ) where K, is

the same as in Eq. (H. 10.23) and

K50 = 1 (AG0 + 20)2 exp[—B(AG° + 20 )2
871/10 420

 

 

] (1112.1)

701)

with the 2D relaxation time given as (see also Eq. (17.34))

1 ”'4 + ’12 (1112.2)

= on '
700 2'01} A0 T:

 

Here, 1'90 is the relaxation time for overdamped diffusion motion along the bond-breaking

coordinate, IZOD=1/(fi Dz) ~ y.
As one can see, the form of the total rate constant K for the OD case is exactly the
same as for the ED 2D one given in Eq. (H. 10.20). Naturally, because we are dealing with

two different dynamic mechanisms, the only difference we can observe lies in the dynamic

91

(diffusion) rate constants. Therefore, in our following discussion, we will not take into
account the equilibrium rate constant K, because it is well defined, and is not affected by
changes in the BBET dynamics. All our attention in the this Chapter will be devoted to the
dynamic rate constant that exhibits a very complex behavior.

The results obtained before [30] for the ED mechanism for a 2D ED BBET reaction
were based on the transformation of energy-diffusion operator in Eq. (1110.3) using the

substitution E=e2/2 along with some further simplifications:

 

 

 

1 d d 1 l d d l 6’ d
L = —E—+ L = —— — z —— 1112.3
(E) Br, 6E (67E B)—) (e) Brzedee(8e+fi) fltf de(de +fi)( )
2 AG + 2
where If = r,/ 27rflEa, t2 =1/y and Ea =-f[—(;.TA9-) .

This “harmonic-like” transformation gave a result for K ED very similar to the one given by

Eq. (1112.1):

ED_ 1 (400%? _fi(AGo+/lo)2
Kd -TED\[ 87:20 exp[ 410 :1 (1112.4)

with the only difference, that TED now is defined through a different bond-breaking
relaxation time If = r, /4/27r/3E, ~1/y.

_1_:_2_,_+ A, (1112.5)
TED 1071: A01':

 

 

There are of course dangers in changing the differential operator L(e) into the
harmonic form as in Eq. (11. 12.3). However, the final operator
L(e) = (1/ Brf)6/6e(6/6e + 13)) reproduces Kramers’ result (11.1 1.21) for the one
dimensional energy diffusion process. Nevertheless, the legitimacy of using it in the 2D
problem is not clear. Therefore, we will also compare the result given in Eq. (H. 12.4) with

the one obtained in the theory of concerted BBET and with that given by Eq. (H. 10.27).

We will also compare the results obtained by numerical solution of Eq. (H. 10.27)

with those of the previous Chapter obtained for widely different values of the two

92

relaxation times, and where we also required that the faster part of the reaction was under
kinetic control. As the result, the final rate constant Kd was determined through the slow
coordinate, so that the long relaxation time controlled the dynamic rate (see Eq. (11.11.25)
for Tz<<1fx and Eq. (11.11.29) for t,<<t,). The discussion presented in this Chapter acts as
a compliment to that of Ch. X1 , and completes the description of the concerted 2D ED
BBET reactions. The total, combined picture of the BBET reaction as a two dimensional
process will be given later in Section 2 of this Chapter.

As we were not able to calculate dynamic rate constant of Eq. (11.10.27)
analytically, we used a numerical integration procedure to compare it with K ,9 D and K513.
First of all, let us consider a situation when no heat is emitted or absorbed by the system
during the reaction, i.e. AG0=O. Then, in terms of different ratios for the reorganization
energies 2x and 2,, we can investigate how the change in relaxation times affects the

dynamic rate constants (see Fig. 11a,b,c). Fig.11 presents the logarithmic dependence of

the pre exponential factors xtx (in units of the polarization relaxation time tx) for the
dynamic rate constants Kd = xexp[—13(AGO + 20)2 /(420)] (or K30, or K50), on the
ratio of the relaxation times, log( t, /rx) (or log(tz/IQD), or log(rz/ 150) respectively).

It is easy to see from the analytical expressions for K50 and K30 (see Eqs.
(1112.1) and (11124)), that the 2D relaxation time is controlled by the shorter of the
times, 1,. and 12. So, in Figs. 11a,b,c we can call the area where log(tx/tz)<0 the
polarization control region and the area where log(rxltz) >0 the energy-diffusion control
region.

It is important to note that this behavior of the 2D rate constant Kd for separation of
the fast and slow modes is just opposite to the one presented in Chapter XI. Here, the
shorter (faster) relaxation time controls the dynamic 2D rate (as we will see further in this
Section), whereas in Chapter X1 the slower relaxation time determined the rate. This is not

a contradiction but the revealment of different features of the complicated 2D process at

93

different circumstances. The complete picture given in the next Section will clarify the
situation.

Here we analyze the behavior of the rate constant K d for three sets of the
reorganization energies: i) 2,,<2z ( 2x=l kBT, 2z=20 kBT, see Fig.11a); ii) 2,~2,(
2x=2z=20 kBT, see Fig.11b) and iii) 222, ( 2x=50 kBT, 2z=10 kBT, see Fig. 1 1c). In the
last case we satisfy the assumption 2z>>kBT, made in Chapter X.

As one can see from Fig. 11a, in the case of small polarization reorganization
energies, 71,, the region of the energy-diffusion control is very wide. In fact, the energy-

diffusion rate

K, =-Tl—,0E,e‘”"’° (1112.6)

describes the BBET process not only when rz< Tx, but even when they are comparable, r,
a: 1,. That can easily be explained by the small contribution of the first member in Eq.
(11. 12.5) to the 2D relaxation time. In this region Kd and K550 coincide, so that the rate

K ED is a good approximation to expression (H. 10.27).

 

log(x 1ix)
)szlkBT
12:20 kBT 2 i
1 _ IQ)
Ky
-M 3. E log(tz/Ix)
K’ci‘D —1

 

Fig. 11a

94

log(x Tx)
kx=20 kBT 2.5»

Kz=20 kBT 2 .

 
    

 

 

 

 

 

 

1.5
Kd .
j 13)
“i7
. . j k .
-3___=.z¢_41—-/ 1 210g(tz/Ix)
Fig.11b
log(x Ix)
)ux=10 kBT 2i
kz=50 kBT 1 . 5’
Kd
_______,.1—
KSD 0 . 5’
_‘2 _‘1 f Eiugfiz/Ix)
Fig. 11c

Fig. 11. The logarithmic dependence of three dimensionless pre exponential factors (X 12x)
is given for three dynamic rate constants: 1) 2D BBET rate constant Kd, see Eq. (4.5); 2)
BBET rate constant calculated in the "harmonic-like" approximation, see Eq. (6.4); 3) 2D
overdamped rate constant given by Eq. (6.1). These dynamic constants are compared for
three different sets of the reorganization energies for polarization and for bond-breaking, Xx

and 2.2 respectively.

95
It should be noted that Eq. (11.10.27) can be significantly simplified when A, 01;, <

1 for an arbitrary ratio of r; and rx :

 

 

fl(AG0+/7.0)2
Kd=xe 4,10 (II.12.7a)
with
AG +
,C=iz.%exp[3’1?£[1+23]Ii 0 30). (II.12.7b)
A0 72 0 72 4’10

When Ax=0 Kay and K ED coincide completely on the whole scale of the ratio rz /rx and the
reaction is controlled by the bond-breaking process with the Kramers rate constant as given
in Eq. (11.11.29).

The diffusion rate constant for overdamped motion in this case differs significantly

from the one for energy-diffusion:

l flEK - E“
K00 = ,l—a— " 4. 11.12.23
d Tim 27: e ( )

So, as one can see, the asymptotic behavior of the pre-exponential factors for K; and K C? D

 

are ~ 21 and \lkz respectively.
Now, for Ax < A, the polarization fluctuations should control the BBET rate for

smaller polarization relaxation times ( r,/ rz <1). As we can see from Fig.11a, in this case

K g D and K (I150 are close to each other and give the same pure polarization control limit

when 1x << rz :

 

 

 

AG +2 )2 1
K3D=KfD=X(Tx/Tz—>O)e m 4‘}. 0 where yawn—>0): ’1‘ ,ijGOW)
0 07x 87th
(11.129)

However, as we can see from the figure, the real behavior in the BBET reaction at 13,/rz

<<1 described by K; in Eq. (11.10.27) doesn't correspond to the pure polarization control
situation, as is the case for K90 and K531) . The BBET rate constant Kd still exhibits some

influence of the bond-breaking process on the total reaction rate. Unfortunately, the

absence of the analytical solution for [(4 doesn't let us analyze the reason for this behavior.

96

Nevertheless, we can assume, that one of possibilities is that the 2D relaxation time for the

BBET dynamic constant of Eq. (11.10.27) has a different structure from the one for [(5)0
or K50 (see Eqs. (H.122) or (II.12.5)).

The situation when lx~ A, is presented in Fig 11b. Eq. (11.10.27) doesn't permit
an analytic analysis of this case. However, one can see that the region of the energy-
diffusion control (TX/1'2 >1, Kd= K ED as in Eq. (11. 12.7)) is smaller, and valid only when

rz is about an order of magnitude smaller then ‘tx. In this case K50 is a good

approximation for K; only on a small interval. As the ratio rx /rz decreases, the difference

D

between Kay and K5: decreases dramatically and tends to be more then one order, Kd

/ K dED ~ 10 for smaller polarization relaxation times, 7,, <9. So, the estimation of the total

D

rate constant for BBET reaction by K5: fails as we go into region of dominating

polarization fluctuations.
The case corresponding to the situation when Ax> A, is depicted on Fig. 11c. As

one can see, the energy-diffusion control of the reaction is never present for reasonable

ratios 1“,! r, and K ED

doesn't approximate the 2D ED BBET rate Kd. When the polarization
reorganization energy is much larger then the bond breaking one, lz<<lx, and the
relaxation times are comparable, in other words, when in addition to the requirement v<<1

(Fl, MI), the inequality v a<<l («Fr/Ix) is valid, we obtain an asymptotic behavior of

the rate constant in Eq. (11.10.27):

 

 

 

 

 

2
Kd = ZBXp[_ fi(AG:A: ’10) :i
_ 1.. le(AGo+/lo)2(2+va) _1_ _g 2 ,
where x- Tx\/ 2,01%) F(2’2’1’2+va)
ml fivao + lo? M] (11.12.10)
4110 4

 

_1_ nmAGo +1.? 11133
rx 1x va

97

From Fig. 12 it is clear that approximation in Eq. (11.12.10) to the constant K; works for
the ratios rz/rx<1. Also, because the picture shows the dependence of (z TX) on the ratio of
the relaxation times, we can point out that both K ,9 D and K50 tend to a constant value
when rx is small and control the reaction (see Eq. (II. 12.9)). The 2D ED BBET rate Kd for

these conditions decreases logarithmically, as it is given by the asymptotic law in Eq.

(11.12.10).

Xx =50 kBT
A. z=10 kBT

 

 

 

 

-‘2 -1‘.5 -‘1 -o‘.5 0.5 iiogfizh")

Fig. 12. The logarithmic dependence of the dimensionless pre-exponential factors (x TX)
for the 2D BBET rate constant Kd in comparison with the approximation K 3”" of Eq.

(II. 12.10) when kx>>}uz. The other dynamic constants, K 5” and K 5 D , are also depicted.

2. (Zoncluding rem/cs on the energy diffigion in BBET reactions

Thus, being intrinsically multidimensional, the BBET reactions exhibit quite
complex behavior. Two relaxation times, rJr and rz, provide different contributions to the
dynamic part of the reaction rate constant. When both of these times are comparable, then
the total rate constant has the form K=Kr Kd/(K,+Kd) where K r and Kd are the chemical
transformation and diffusion rate constants given by Eqs. (11.10.23) and (11.10.27),

respectively. The behavior of the diffusion rate constant [(4 is shown in Figs 11 and 12.

~ 98
When one time is faster then the other, then the picture of the BBET process

becomes more involved. An apparent contradiction was found between the results of
Chapter XI, where the slow relaxation time controls Kay, and Section I of the present
Chapter, where the fast relaxation time controls [(4. The following considerations will

resolve the apparent contradiction. Let us assume that we have two coordinates, x and y.

To be specific, consider the relaxation along x as fast, characterized by If“, and along y as

slow, characterized by If”. Now, we should emphasize that in this article we have

,0 -1
calculated the generalized probability a: 2 [IO dtP,(t)] . Pe(t) is the survival probability in

the reaction that, for high activation barrier reactions, coincides with the asymptotic rate

constant K = -d In Pe(t)/dt. Thus, we shall focus on a

If rf“‘<< 1:10" so these dynamics take place on very different time scales, we can
roughly divide the whole 2D process into two stages. In the first stage the If” dynamics is

important and the y-dynamics is so slow that it can be treated as static. As discussed in

Chapter XI, the fast dynamics at this stage equilibrates the distribution along its coordinate

by some time I, , which is still short compared with rj’“. The second stage takes place

after the fast coordinate has been equilibrated and the slow relaxation time If” will control

the reaction rate.

Making use of this assumption suggests that we may split the total generalized
probability as a = ae,+ a, in order to describe these two stages separately. 2e, should be
calculated when the slow mode is taken as static, i.e. the Green's fianction for it is just a 5-
function G(y,y ,t)= 6(y— y') and the dynamics is described only by G(x,x',t). In the
second stage the fast coordinate has equilibrated, so that G(x,x’,t)=<p(x) and G(y, y’, t)
defines the rate. It is straightforward to apply the scheme described in Chapter XI to both
of these stages. We again assume that the transitions take place along the line on the
surface: K(x,y)=Ko6(x0(x-x')-yo(y-y*)) where (x‘, y‘) are the coordinates of the
transition state point (see Eq. (II.9.7a) or Ref. [3 0]). Then,

99

 

 

 

 

 

 

'- 4: :1: =1: :1: I“
K —{x — — x — ( - \
K (”WP +1.??ka yiiy y), ”Loy y)J
331:4 ldy .. ... : (II.12.lla)
x0 (x —yO(y-y )\
(ka x0 l
f K
aez—K, i1 KS’ZWl’ (II.12.11b)

 

where (px and (py are the Boltzmann distributions for the x and y coordinates, K, is given in

Eq. (11.10.23) and 6, and (‘3, are defined in Eq. (c1).

The analysis of the above expression for an and a2 reconciles the two views
presented in this Chapter of the whole BBET process, when one relaxation time is much
shorter then the other. In particular, if the sink strength K0 is small compared with the
diffusive rate in the first stage, as controlled by 1/ 1?“, then a] is negligibly small
compared to 322 and the process becomes a one-dimensional reaction diffusion process

along the y coordinate. This case leads to the result given in Eq. (11.11.23):
1/a1e=1/ee2 =1/K, +1/K; where 1/K3; ~ 1"“ (11.12.12)

y

The dynamic part of the rate expression is controlled by the slow relaxation time If”. This

situation corresponds to the picture presented in Chapter XI.

However, if kinetic control does not apply during the first stage, as is favored by a

strong reactive sink K0 relative to 1/ rfa‘", then a1~ 1/ If“ and 331 >> an. So the total rate

x

constant is defined mostly during the first stage (3:2 is just a small correction) and is
controlled by If“ . The behavior of the rate constant for this case is described above in this
Chapter through the 2D Kd expression.

We conclude with a few remarks regarding the assumptions made in this work.
First, we considered an irreversible reaction such that after the reactive particles have left
the potential well they escape‘out of the reaction region. This assumption is quite
reasonable, taking into account that the product surface is dissociative, and that the

escaping particles undergo three-dimensional motion obeying bimolecular kinetics. The

100

next two assumptions are more severe. In our theory we assumed that the bond-breaking
and solvent polarization coordinates are independent. This idealization may be questioned.
It would be natural to think that the solvation energy depends on the bond excursion, as the
strength of the coupling between a solvent dipole and the charge distribution of the reacting
particle are affected by the bond excursion. Lastly, we only consider high activation barrier
reactions. This restriction permitted us to make important simplifications in the reaction
model. The first one concerned the energy-diffusion operator that originally had an energy-
dependent diffusion constant D(E) (see Eq. (II.10.3)). For the high activation barrier
reactions it can be taken as a constant. Secondly, high activation barrier Green's functions
are very straightforward to work with. Finally, the definition of the rate constant for high
activation barrier reactions coincides with the definition of the generalized probability, as in

Eq. (11.10.10).

101

CHAPTER XIII. QUANTUM MODE IN BOND—BREAKING ELECTRON TRANSFER
REACTIONS

1. General remgrks and structure at the mtential surfaces

In the previous two Chapters we considered the situation when the transformations

 

along the bond-breaking coordinate were described classically, i.e. classical dynamics and
interactions with solvent molecules produced the dissociation. This picture is legitimate
when the bond frequency is much smaller or comparable with the thermal frequency of
collisions of solvent molecules, which for room temperatures is about 200 cm'l. If the
bond frequency is much higher, then a classical treatment of the bond-breaking can not be
applied to the process. As an example, the methyl halides discussed in Refs. [29, 63, 64]
have bond frequencies around 600 cm'l, which is sufficiently high that the classical
dynamics for the bond-breaking coordinate may be questioned. Nevertheless, the ET part
of the reaction can still be described by the overdamped changes in solvent polarization.
Thus, we have to incorporate the quantum nature of one coordinate into the classical
treatment of the other. This type of approach is well known and has been developed to
introduce high frequency inner-sphere vibrations into the outer-sphere ET reactions. First,
Jortner and Bixon [77] introduced a simple model in order to involve the quantum mode
into ET dynamics. Following their work Barbara and coworkers [60, 78] used this model
to interpret experimental data, in particular, to address the issue why rates could be found
that are larger then possible for standard solvent- controlled ET.

In principle, the above situation corresponds to the limit of zero friction in the
Kramers' energy diffusion treatment of the BB coordinate. If the quantum dissociation of
the bond takes place in the concerted BBET, there is no transition from one vibrational level
to another inside the initial and final Morse potential wells. In the classical regime, these

transitions are initiated by collisions with solvent molecules. So, it is stated that the

102

collisions provide the friction which causes the system to overcome the reactive barrier and
yield the products. Now the frequency of the bond vibrations, cab is so high, that these
collisions are ineffective and can not supply enough energy to promote the population of
the bond-breaking coordinate from the lower vibration levels onto the activation barrier.
Thus, the dissociation of the bond takes place by means of tunneling. However, we should
remember, that the low-frequency solvent polarization mode has to respond to all changes
in the bond structure.

Hence, it is not difficult to find the place of BBET with a quantum BB mode in the
total 2D picture of the BBET reactions which is our designation in the present Part of the
dissertation. From the previous Chapter we learned that the BBET reaction can be
described as a 2D process with two dynamic coordinates: the energy diffusion BB and the
polarization fluctuation ET coordinates with the relaxation times 113 and 13x respectively. We
also found out that when one relaxation time is much faster (shorter) then the other, the
reaction dynamics may be reduced to a 1D process which is controlled by either the fast or
the slow relaxation time. Which time controls the dynamics depends strongly on the
relation between the fast time and the speed of the non-adiabatic transitions in the reaction
region given by the non-adiabatic rate constant K,.

To emerge from the energy-diffusion to the quantum mechanism of the bond-
breaking we can put 13 ~ 7 ->0 where y is the friction. This immediately provides TE <<
1x. Then, if the non-adiabatic transitions are slow compared with t5, then the population
along the BB coordinate is always at equilibrium and all dynamical effects are provided by
the slow solvent polarization relaxation, Ix . This all have been discussed in Ch. XI, Sec.
1,2. However, there we assumed that the BB coordinate supplies only one potential surface
for polarization changes for each reactants and products. This might not be the case when
the friction is truly zero and instead of the classical treatment we have to use the quantum
approach. In this case, the correct view on the BBET with quantum mode demands a

slightly different picture, which we are going to consider here.

103

 

 

 

 

 

 

a».
25'
E
O
'
(a) BB coordinate, z
at.

$3

0

5

AGO

 

 

polarization, x
Fig. 13. Schematic potential energy surfaces for the quantum mode in BBET reactions. (a)
The high frequency vibration levels in the initial and final state Morse potentials. (b) Each

vibrational state of the reactant (R) and product (P) is characterized by a distinct polarization

potential surface. The physical situation corresponds to an exothermic reaction (AG0<0).

104

To proceed farther, we invoke the central assumption made in Ch. IX: the coupling
between the bond-breaking and electron transfer coordinates is negligible, so the
coordinates are independent and the total potentials are regarded simply as an addition of
the surfaces for two individual processes. Figure 13 provides a schematic view of the
solvent polarization potential surfaces corresponding to the distinct vibrational states of the
bond breaking coordinate. For high-frequency vibrational modes of the bond one can
assume that kBT<< 72(1),, , so that intramolecular vibrational excitations of the reactants RX
are negligible. The ET process, which now involves only overdamped polarization
changes, is described now in terms of a superposition of parallel channels RX(ni=0) ->

RX-"(nf) onto the vibrational levels [If of the final state. Here we consider transitions only

from the ground vibration level in the initial state (ni=0) since it is the most populated level
and the population decay from it is supposed to give the greatest contribution. So, instead
of having one initial and one final state for polarization fluctuations, we have multiple
states, each corresponds to a particular vibrational level of the quantum mode, as depicted
on Fig. 13.

Modification of the energetic parameters for different final states RX°'(nf) is
straightforward. For the ni=0 -> n ET process (for simplicity we will use n=nf), the
energy gap is

AG, =AGo-nhwb. (II.13.1)
Thus, we have the following set of potentials which will be responsible for the reaction

changes in the system:
V R = x2 / 2
(II. 13.2)
VP, 2 (x- x0)2 /2 + AG,
where lx=x02/2 is the reorganization energy of the solvent.
As we consider irreversible reaction, the reaction kinetics will be formed by the

decay of the ground vibrational state population of the reactant, RX through the multiple

sinks onto the set of vibrational states of the product, RX". These transitions are provided

105
by the coupling of the solvent polarization surfaces in Eqs. (11.132), 27th}, / h as well as
by the overlap between the bond vibrational quantum states which is described by the
Franck - Condon factors [77]
|(0|n)|2 = (S" / nl)exp(—S), where S: D, Mia), (II. 13.3)

where DC is the reorganization energy (the dissociation energy) of the quantum, bond-
breaking mode.

Thus, the position-dependent sink function, W(x) can be expressed as a sum over
all channels involving the transitions from the ground state of RX to the all exited levels of

final state vibrational manifold of the quantized vibrational degree of freedom:

W(x) =

 

2”,?” 2|(0ln)|25(V,,,,(x) — VR(x)). (1113.4)

Or, we can define a partial sink W0"(x) located at some point xn“ which, according to Eqs.

(II.13.1)-(II.13.4), is

ZnW
0,.= N— —-&”-|(0| "ll
W°"(x) = K0,,6(x - x, *), where (11. 13.5)
* AGO, + A
" 22,
and the total sink function is given as
W(x): 2W”(x). (II.13.6)

Thus, we have defined all geometric parameters of the model for the quantum bond-
breaking coordinate in BBET reactions. Now we will consider the dynamical influence of

the solvent on the generalized probability of this process.

2. ter rence o t e reaction channels in with the an bond-brea 'n
Our aim here is to find out the influence of the overdamped dynamics of the ET on

the BBET reaction when the bond-breaking coordinate obeys the laws of quantum

106

mechanics. In the early works investigating the dynamic effects on reactions with multiple
channels provided by a quantum mode [77, 79] it was assumed that the reaction channels
are independent. This implied that each channel forms its own rate and the total survival
probability is just the superposition of all independent reaction rates. Later [14], it was
explicitly proved for the case of two 5-functional channels that this is not the case for
reactions which are influenced by solvation dynamics. So, the picture for the reaction on a
one-dimensional surface with multiple sinks is much more intricate then it was assumed at
first. In order to develop this model further some numerical simulations [60, 78] were
performed to fit experimental data. Here we want to proceed beyond simple numerical
simulations and present application of the steady-state Green's function theory to the above
mentioned model.

According to Eq. (1.3.10), the survival probability for the reaction on a single PES

can be obtained by solution of the steady-state differential equation:

£0050) — W(x):(x) = - 2210:)

with the operator of motion f(x) = )i-j—x(§x— + B %) along the potential V R(x),where

TX is the average relaxation time, and the sink function is defined in Eq. (11. 13.6). For the
polarization fluctuations in Debye solvents 12x should be taken as the longitudinal relaxation
time, 12L. However, there are some other arguments in favor of different definitions of the
relaxation time which are supposed to improve the description of the electronic relaxation in
solvents [78].

Taking into account the 6 functional form of the sink function, Eq. (11135), the
integral formulation (see Chapter IV) of the problem should be more profitable:

g(x) = 020:) — [G(m )W(rd )5(:) (1113.7)
where 010:) = N exp(-fi VR (x)) is the Boltzmann distribution inside the potential V R(x) and
G(x, it )is the steady-state Green's function for an arbitrary initial distribution f(x) defined
by the equation

f.(x)G(x,1t) = -5(x - e )+ f(x). (11. 13.3)

107
The generalized probability in the reaction is given as

ae=[W(x)§(x)dx. (1113.9)
Equations (II.13.7)-(II.13.9) are the one-dimensional analogs of Eqs. (1.4.13), (1.4.12)
and (1.3. 12) respectively.
For the sink function as in Eq. (1113.6), from Eqs. (1113.7) and (1113.9) we
derive:

€(x1)= <p(x,)— ZKO.G<x,,x.>é<x.) (11.13.10)

and

ae=ZK,,§(x,). (11.13.11)
It is more convenient to represent the system of Eqs. (11.13.10) and (1113.1 1) in a matrix
form and obtain the final answer for the generalized probability of BBET with a quantum
mode:

2=Koli where E=(I+F’)“ifi (11.13.12)

where I is the identity matrix and

 

 

 

 

 

 

—K01-l "(P(xmy [(01611 K02 ~ K0116»:

.. K x . ~ ~ ~

K0: °2 , (,5: M M) , and F: K" G” K02 22 K°"G2" (11.13.13)
K0"_ _¢(xo" )_ LKOIG~n1 K02 Gn2 KOnénn‘

where Ga“ =G(x,., xj)is the steady- state Green's function defined in Eq. (H. 13.8). Note,

the initial distribution is included in the steady-state firnction, so different Green's functions
are required for different distributions. For the case of one-dimensional PES these
functions are defined in Appendix A, see Eq. (A13).

As the simplest example we consider the case of two channels. Solution of Eq.
(11.13.12) with the account of Eq. (11.13.13) gives us the generalized probability of the

form:

108

Krl + KrZ + KrlKr2(Gll/(p(xl ) + Cfizz/(p052). G12/‘P(x1)‘ ~21/‘p(3‘72 ))

 

 

E : ~ ~ ~ ~ ~ ~
1+ KrlGll/¢(xl ) 'l' Krszz/(P(x2) ‘l' Kr1Kr2(Gr1/§0(x1)Gzz/(P(x2) " G1: (P(x1)G21/(P(x2 ))
where
V2 7: _ S" AG + A 2
Km = K0,,(p(x,,) = -% lie Smexp(—EA,,), where E," = m 21x ‘) (1113.14a)

is the non-adiabatic rate constant for the n-th channel. As one can see from this example,
for reactions taking place on multiple channels, it is impossible to separate the non-adiabatic
part of the generalized probability from the dynamical part unless the reaction takes place at
high barrier (at all sink). If we take into account the form of the steady-state Green's

function for high barrier reactions (Eq. (1.5. 10)), we can easily obtain

_ Kr] + KrZ Kr]

$ — z
1+ r1,,(K,.l + K,2(p(x1)/ (p01:2 )) 1+ er,,

where ID is the MFPT to the sink point of the lowest energy, 1,, = 1', $81); So, the final
x1

expression shows that only the first sink located at the lowest point contributes into the

rate.
In order to observe different limiting cases possible in the reactions under study, let
. . . . . . - V 2 7r . . .
us introduce some "tune" of non-adiabatic transrtrons, v ‘ = 7";- %— which 1s, 1n fact,

the average time of perturbation (non-adiabatic) transitions which take place in the absence
of the diffusion influence (the transitions that are normally described by the Golden Rule
constant). The stronger is the electronic coupling VRp between the reactant and product
surfaces, the shorter is the time of non-adiabatic transitions

It is easy to prove, that if for any i'th sink on the surface the inequality

_1’7» 1, _ (11.13.14)

K0101
holds, i.e. the diffusion relaxation time is fast compares with the time of the non-adiabatic

transitions, the electronic coupling VRp is weak and the kinetic control limit takes place. In
other words, a: = K, = Kor'p' = 2 K", where Km is the non-adiabatic rate constant on the n-

109

the channel, and the generalized probability, indeed, is a sum of rates from the individual
channels as was assumed in by Jortner and Bixon [77]. So the transitions do not disturb
the population on the reactive surface significantly. This is the result of perturbation theory
over the small parameter of the interstate interaction, 1).

The other limit occurs when inequality (H. 13.14) is reversed, hence the electronic
coupling V RP is comparatively strong and the non-adiabatic transitions are fast. This is the

diffusion control limit and the generalized probability is given as follows:
6;“ G

In

aede=§*G“ffi, where G: , (11.13.15)

G (3,,

n1

e' is the unit vector. For only two channels, the above equation transforms into
K,(n = 2) = G11 / $09) + (322 / ”’09) ’ 92 / $901921 I “’02). (11. 13.16)
011/(P(x1)' Gzz /(P(x2)-G12 l‘p(x1)'Gzr /‘P(x21)

It is easy to deduce that this rate constant, in general, can not be represented as a sum of
diffusion rate constants on the independent channels. This would be possible, however, if
all non-diagonal elements of the G-matrix were zeros, i.e. Gr; / (p(x,.)=0. As we will 'see
later, usually this is an unreasonable assumption. It works only in the case of a high
activation barrier in the normal regime. In all the other situations the non-diagonal elements
differ significantly from zero.

As was obtained in Ch. V, Eq. (1.5.5), the ratio G”. / (p(x,.) is the MFPT to the

point x,- if start occurred from some initial distribution f(x). This MFPT here will be
denoted as rp(x,.). From Eq. (15.6) we can conclude that G, / q)(x,.)— GU. / (p(x,.) is the

MFPT from the point it} to the point x,- (or rp(xj— > x,) in our notation). The other

average time which will appear in the diffusion constant is the difference between two
MFPTs - r,(x,., x1.) = rp(x,.) — rp(xj- > x,) - which measures how much longer it takes to
reach the point x,- if started from a Boltzmann distribution (rp(x,.)) and if started from

another point, xj- , (r, (xj— > x,)). Thus, the structure of the diffusion rate constant for two

channels given in Eq. (H. 13. 16) obtains a clear physical interpretation:

110

rp(x2- > x,)+ rp(x,— > x2)

K401 : 2) : rp(x1)-rp(x2)— re(x,,x2)re(x2,x,)'

 

So, it easy to see, that the channels would be independent if there was no difference

between two MFPTs participating in the problem - rp(x,) and rp(xj.— > x,), which means
that the steady-state Green's function should be insensitive to the choice of initial
conditions and r,(x,.,xj) = 0. As has been shown many times in this dissertation, this
happens, indeed, when all decay takes place over a high activation barrier, or, in other
words, if all sinks are positioned high. This is the only situation were the assumption about
independence of the channels is legitimate.

Thus, the rate is controlled not only by the diffusion towards the sink points, but
also by the motion between them. One can imagine that the decay on multiple channels
produces some "holes" in the population at the points of the sinks. However, these holes
are not highly localized, so that the decay at one point disturbs the population near the
others and therefore the population at each sink gradually responds to the changes at all the
others. Since the joint efficiency of the multiple channels is not a sum of the partial ones,
they can not be considered separately. This is why we have to consider the interference
between the multiple channels.

For evaluation of the rate constant in Eq. (H. 13.12) for BBET reactions with the
quantum bond-breaking we will need the steady-state Green's functions for the harmonic
oscillator potential VR(x). We are interested in two different initial distributions - the
Boltzmann, War) and the 5-functional one located at the bottom of the well, 6(x). The last
distribution was taken since there are a few experimental studies which refer to it as to a
probable distribution in photochemical reactions on excited potential surfaces[12, 13, 28].
From Eq. (A13) one can obtain:

(1) for Wx) :

111

~ e-Bxflz (41323—1): VET/:2 . \
G(xl.,xj): “V73? 1n2+\/7EL [ e (1+erf(y))dy— [e (1—erf(y))dyJ

0 0
(II.13.17a)
(lag/22
-2«/;(~)(x,-xj) fey dy
”Ari/2
(2) for 6(x):
.. e-nxf I: fix? / 2,2 112x} / 2
G(x,,x.)=rx-T— 2®(x,.) e dy—2®(x,.)@(x,—x.) eyzdy
J Zfl l! J fllez
(II.13.17b)
#6sz /2 fix} / 2
-®(x. — x.) e (1 -erf(y))dy- je” (1—erf011dy
fixj/z 0

We will be interested in three distinctive regimes of the reaction, namely, the
normal, activationless and inverted, see Fig. 14 a-c. It is convenient to introduce the ratio
of the relaxation time ‘Cx to the characteristic time of the non-adiabatic transitions 0, a=tx/u.
This parameter is supposed to reflect the strength of the electronic coupling V Rp compared
to the solvent relaxation. Even though we can not separate the non-adiabatic part of the

generalized probability from the diffusional part, this parameter will help as to estimate

energy
7::

n=3
=2
n=1

n=0

 

 

(Fig. 14a) 0 polarization, x

112

 

 

 

 

 

g; R
2 P
n:
n:
=1
2 n=0
(Fig. 14b) 0 polarization, x
a R P
2
0
‘ =2
=1
n:
(Fig. 14c) 0 polarization, x

Fig. 14. Schematic of the potential energy surfaces representing the BBET for quantum

bond-breaking in three different regimes: (a) normal; (b) activationless; (c) inverted.

113

which part prevails at the different values of 01. Moreover, the matrix F in Eqs. (11.13.12),
(H. 13.13) becomes function of the parameter or.

For our further investigations we will evaluate the dimensionless generalized

probability E = v - as which can be evaluated from the following expression:
'eezu-ir',(i+fi)¢ (11.13.18)

If or<<1 (or inequality (H.13.14) takes place), the generalized probability a: is
supposed to tend to its kinetic control limit, Kr. Otherwise the dynamics along the potential
play an important role. For a>>1 we should expect the diffusion control limit when the
generalized probability is given by Eq. (11.13.15). In this case Kr>>Kd. So, these two
simple limits will aid us in interpreting the results of calculations of the chemical rates in
BBET reactions with the quantum bond-breaking mode.

The exotherrnicity of the BBET reaction, AGO and the frequency of the quantum
bond-breaking mode, (0}, determine the exotherrnicities for the set of final polarization
states, AGO" in Eq. (1113.1). Combined with the reorganization energy for the x
coordinate, 2.x, they define the activation energies E A" for the sink points (Eq.
(H. 13.14a)). Thus, these three parameters assign the energetic geometry of the reaction and
hence specify the dynamical behavior. The ratio of the dissociation energy of the bond, D6
to its vibration energy ha), determine the efficiency of tunneling and so specify the non-
adiabatic rate constant, see Eqs. (H.133) and (H.135) (in all our further calculations we
will take arb=600 cm'lz3kBT at room temperature, which is the approximate bond
frequency in methyl and aryl halides [29]). Therefore, with the parameter or which
measures relative relaxation time of the system we have five variables that influence the
kinetics in BBET reactions with quantum bond-breaking.

The reaction surfaces for the normal regime are sketched on Fig.14a. We have
discussed the case of high activation barrier for this regime earlier, so only low barrier
reactions will concern us here. Taking Ax=1 eVz40kBT and AGO =-0.5 eVz-ZOkBT ,

which are typical values for the reorganization energy for polarization and exotherrnicity,

114

we would get the desired low energy activated reaction (the activation energy of the lowest
point is about 2.5kBT). First we investigate the behavior of the generalized probability at
the two limiting cases. For kinetic control (or<<1) depicted on Fig. 15, the rate constant is a
sum of the partial non-adiabatic rate constants given by Eq. (II.13. 14a). Thus, the more
sinks we take into account, the larger is the rate. However as channels go up higher, the
partial rates decrease proportionally to the activation factor, exp(—BEA,,), which steadily
decreases contribution from the higher channels. The other contribution to the non-adiabatic
rate constant Kr comes from the Franck-Condon factor. As can be easily seen from Eq.
(H. 13.3), depending on the value of S the factor has a maximum at some n which means
that the n-th vibrational level of the final state has the maximum overlap with the ground
vibrational level of the initial state. The smaller is S, the closer this level with the maximum

overlap is to the ground vibrational level of the final state.

 

 

 

 

 

UK: S=3
0. 0051
0.004 »
0.003 -
0.0021 S=
0.001» / S=8
2 ti 3 a 10
number of channels

Fig. 15. The dimensionless non-adiabatic rate constant UK, for the normal regime is
evaluated for the following parameters: 11, =1eV and AGo = —0. SeV. Three values of the
tunneling parameter S = D¢ / 7100,, are used: S=3 (D, = 9kBT), S=5 (D, =15kBT) and S=8
(D, = 24kBT).

115

However, in the normal regime these two contributions partially counteract one
another and their combination produces the simple picture presented on Fig. 15. for three
different values of S. Namely, the smaller is the gap between the final state vibration levels,
S, the larger is the rate constant K,..

In contrast, the diffusion limit (a>>l) produces a rate constant that is not additive
(see Eq. (1113.15)). Calculations show that in the normal regime this rate does not depend
upon the number of sinks and is defined, essentially, by the MFPT to the lowest sink
point. For the parameters chosen above we evaluated the following: (a) for the Boltzmann
initial distribution utilization of the steady-state Green's function in Eq. (11.13. 17a) gives
Kd=0.066 Ix ; (b) for the 6-functional initial distribution at the bottom of the potential Eq.
(H.13.17b) yields Kd=0.062 Ix. The second constant is fractionally smaller then the first
one. This is very easy to understand, as the Boltzmann distribution provides some
population at all point on the surface before the reaction starts, whereas the particles from
the bottom for the 5(x) distribution have to spend some time spreading inside the potential,
which makes the MP PT for this case longer and the rate constant smaller.

For the intermediate values of or we have to evaluate the dimensionless generalized
probability given by Eq. (11.13.18). Fig. 16a shows two sets of dependencies of the
constant 5 for two values of S for the Boltzmann initial distribution. For the normal
regime the difference between the rate constants for the two different initial distributions
(p(x) and 5(x) is negligible, so they are indistinguishable on the picture (which is not
surprising because the difference between the rate constants in the diffirsion control limit
for the two distributions is minute). First of all it should be mentioned that in the normal
regime the saturation of the rate occurs at a very small number of channels (4-5), so that
accounting for a larger number of sinks does not change the chemical rate (influence of the
activation factor). Another characteristic feature of the rate for multiple channels is that it

increases more (compared with the rate for a single channel, n=1) for the smaller values of

116

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'55
0.006; 1
0.005} 2
0.0041 S=3
E 3
0.003» 4
0.0021
C / 1 2
C 3
0.001; S=8 4
0 2 A 4 6 A 8 10
(a) number of channels
E _
23113
' E
0.004» .. (L -... ._ .......
28113
0.0035» """"""""""" L "”
0.003 1 --r ----------
3‘30
0.0025; (1:10
6 s 10
0'0015,’ number of channels

 

(b)

Fig. 16. Dependencies of the dimensionless rate constant 5 on the number of sinks are

presented for the normal regime at A, =1eV and AGo = —0.SeV. Two values of the
tunneling parameter S = D, / her, are used: S=3 ( D, = 9k3T) and S=8 (D, = 24kBT). (a)
Comparison of as for different values of or (1 - 01:01; 2 — 01:1; 3 - 01:5; 4 - 01:10) is
made for the two values of the tunneling parameter. (b) The approximate rate constants few
(short dashes) and 750 (long dashes) from Eq. (H. 13.20) are compared with 55 for S=3
and a=5 and 10.

117

S and or. The reason is quite simple - we don't observe any boost in the rate for a bare
diffusional case, therefore all changes are caused by the influence of the non-adiabatic part
as it is the only part which is effected by aggregation of additional channels. Hence, the
stronger is the dominance of the non-adiabatic part in the transition rate, the faster is the
relative growth of E with the increasing number of channels (up to the saturation limit, of
course). As was discussed previously (see also Fig. 15), smaller values of S make the non-
adiabatic rate constant larger which causes sharper variations in the dimensionless
generalized probability E. Finally, a similar effect can be achieved by decreasing or. The
smaller is or, the relatively faster is the diffusion relaxation time ‘tx compared with the time
of non-adiabatic transitions, 1), the closer is the situation to the kinetic control limit.

The reaction model considered here was applied to ET reaction with quantum mode
in Ref. [77], where the overall rate constant was considered to be a sum of the rate
constants from the independent channels. In other words, the correct generalized

probability a: in Eq. (H. 13. 12) was approximated by the following expression:

 

 

1_ _1_ 1
23 —Z[Km+Kd.l (11.13.19)

_1_ =

3.18 n n n
where an is the generalized probability for decay on the n-th sink. This approximation is
equivalent to the assumption that the matrix F” in Eq. (11.13.13) is diagonal, i.e.
G(x,,xj) = 0. The form of the steady-state Green's functions given by Eqs. (H.13.17)
suggests that this assumption is highly debatable.

An alternative approximation of the correct generalized probability may be given by
another simple formula which separates the non-adiabatic and diffusional rate constants:
E0 = K,Kd / (K, + K d), where Kd is the correct rate at the diffusion control limit evaluated
from Eq. (11.13.15). From Eqs. (11. 13.12)-(II. 13.13) and the case of two channels

considered above it is clear that neither of these approximations are really applicable to the

118

multi-channel model. However, in order to give a qualitative picture we evaluated the

dimensionless constants
E = 03 and a "UE " r d (1113 20)
J8 JB 0 0 K, '1' Kd ' .

 

which were compared with the correct one for the normal regime (the result is shown on
Fig. 16b). It is easy to see, that the difference between the correct and approximated values

'is quite significant (apart form the case of one sink, n=l, when Eqs. (11.13.20) are both,

obviously, exact). The approximate formula for E0 overestimates the correct rate by at least
30%, and approximation with E ,8 is even worse. Naturally, approximation EO should

work better then E J3, since the first one partially takes into account the interaction between

individual channels in the form of the diffusion Kd constant from Eq. (H. 13.15), whereas
the last one neglects it completely. This neglect may resolve either in overestimation (as in
Fig. 16b) or in underestimation of the correct rate depending on the contribution of the off-
diagonal elements in the matrix 1:" from Eq. (H.13.13) into the rate. One can expect that if
this discrepancy is significant in the normal regime, it should only increase for the
activationless and inverted reactions when the dynamic influence on the rate is stronger.

The energy scheme for the activationless regime for BBET with quantum bond-
breaking is depicted on Fig. 14b. Starting with analysis of the two limited regimes, on Fig.
17 we present the behavior of the dimensionless non-adiabatic rate constant which
describes the reaction when 01<<1. As one can see, the general features of its behavior are
very similar to the ones for the normal regime except that now the rates are much higher as
it should be for activationless processes. Again, one does not need to take very many sink
into account, as long as the activation factor exp(-,BEA,,) rapidly decreases contributions
from the activated channels at energies 3 kBT and higher.

Whereas the behavior of the non-adiabatic rate constants for the normal and
activationless regimes are similar, the rate constants for the diffusion control limit (or>>1)
are very different. Unlike previously, Kd for the activationless reaction strongly depends

upon the choice of the initial distribution. One should observe crucial difference between

119
diffusional rate constants for (p(x) and 5(x) . Because the Boltzmann distribution spreads
reactive particles along the whole surface, the diffusional rate constant for Kd can not vary
strongly when the sinks approach the bottom of the well (but, of course, it should not be a
constant as for normal regime). This is not the case for 5(x) which provides exceptionally
localized concentration of particles at the start of the reaction. So, the first two or three
sinks located beside it should deliver an extremely intensive decay whose speed should be

significantly higher then the one for the Boltzmann initial distribution. This difference

should be somehow reflected in the generalized probability for intermediate values of or.

UKr S:
0.25'

 

 

 

 

 

number of channels

Fig. 17. The dimensionless non-adiabatic rate constant UK, for the activationless regime is
evaluated for A, = leV = 40kBT and AG0 z —1.eV = 41kBT. Three values of the tunneling

parameter S = D, lharb are taken the same as for Fig. 15.

When the relaxation time 1x and the average time of non-adiabatic transitions 0 are
comparable, evaluation of generalized probability (H.13.12) is necessary. In Fig. 18 we
present the result of calculations of the dimensionless constant E for an activationless
reaction. The rate constants were evaluated for both Boltzmann and 6(x) initial distributions

for Ax=1eV=40kBT, AG()=-41kBT, S=5 and three different values of or. For the saturation

120

to take place in the this regime we still only need a few sinks (6-7). So, we can conclude

that as for the normal regime, as the barrier for the n-th channel reaches 3-4 kBT, all higher

sinks can be excluded from consideration.

 

 

 

 

 

 

 

E

L ,---"""’-" 1

1 /
0.2» /

1 /

: ///"' ----------- 2
0 15» ..

1 S=3 / ------------- 1

l // ’———-_ 3
0.11 /

i _-_—'"m‘2

. A ____3
0.051 4..

1 S=5

‘ 2 4 E a 10
number of sinks

Fig. 18. Dependencies of the dimensionless rate constant E on the number of sinks are

presented for the activationless regime for A, = leV and AGo 2: —leV. Two values of
tunneling parameter S = D, / her, are used: S=3 (D, = 9kBT) and S=5 (D, = ISkBT).
Comparison of E for different values of a (l — 01:1; 2 - 01:5; 3 — 01:10) is made for the
two values of the tunneling parameter. The solid lines correspond to the Boltzmann initial

distribution, and the dashed lines present 6(x) initial distribution.

As one can see, there is a difference between the two initial distributions. However,
for S=5 this difference is insignificant. Only smaller gaps between the vibration levels
(S=3) can provide a noticeable difference. The smaller is the spacing between the bond
vibrational levels in the final state (the smaller is S), the bigger is the density of the sinks
near the bottom of the potential where 6(x) is taken at the start, hence the more significant is

this difference. This difference rises even more as the value of or increases and, therefore,

121

the diffusion influence on the rate becomes stronger. Note, that the localized distribution
increases the value of the rate constant compared with that of the equilibrium one, which is
in accordance with our argument concerning the diffirsion control limit for this regime.

The most interesting case of the model under consideration is the inverted regime
when the excited vibrational states of the products become significant and the transitions
into some of them are always activationless (see Fig. 14c). So, these activationless
channels should provide the maximum contribution to the rate as the activation factor for
them is minimal. Let us first consider the non-adiabatic rate constant K,. Fig. 19 shows it
for kx=40kBT, AGo=-62kBT and S=3, 5 and 8. As one can see, in contrast with the two
previous cases, the biggest value of S provides the largest rate. As was mentioned in the
discussion for the normal regime, the Franck-Condon, tunneling factor in the sink
functions (see Eq. (H. 13.4)) has a maximum at some values of n which depend upon S. So
the ground vibration state of the reactants has the maximum overlap with a few exited
vibrational levels of the products. Hence, in the inverted regime both the activation and
tunneling factors may increase the efficiency of the transitions into the higher vibrational
states, whereas previously they worked in the opposite directions and the higher tunneling
factor used to be suppressed by the activation one.

Similar consequences of mutually supportive factors can be observed for the
generalized probability of reaction in the inverted regime. A typical picture for the
dependence of chemical rates on the number of sinks for this scheme is presented in Fig.
20. First of all, the overall growth of the rate for the reaction on multiple channels
compared with that of on one sink is about two orders of magnitude. This is an enormous
adjustment, which completely changes the total picture of the reaction. Even though the rate
in the inverted regime can be slightly smaller then the one for the activationless'reaction
with all other parameters apart from the exotherrnicity AGO being equal (compare Fig. 18
and F1. 20 for S=3), this decrease is almost insignificant compared with the changes in the

exothermicity.

122

 

    

 

2 4 15 a 1‘0 12 14
number of channels

Fig. 19. The dimensionless non-adiabatic rate constant UK, for the inverted regime is

evaluated for A, =1eV==40kBT and AG0 z—62kBT. Three values of the tunneling

parameter S = D, / 710),, are taken the same as for Fig. 15.

In the traditional Marcus' picture the rate constants have maximum values for
activationless regime and then they should diminish with the growth of activation barrier
for inverted reactions as fast as the activation factor exp(—BEA). This does not happen if
transitions into excited vibration levels of the products are possible. The rate for multiple
channel reactions should tend to almost a plateau as the exothermicity increases. Second,
the largest S provides the biggest growth of the rate which is just the opposite to what we
observed for the normal and activationless regimes. Thus, there should exists some
Franck-Condon factors (as happened with S=8 for our choice of parameters) for which the
reaction rates in the inverted regime would even increase instead of the familiar Marcus'
turnover.

The reason why the larger S better stimulates growth of the generalized probability
lies in the behavior of the non-adiabatic rate constants discussed above. Because both
activation and tunneling factors can enhance each other for the sinks at the bottom, the

relative value of K, increases faster for the set of factors which are most mutually

123

supportive. As the diffusion process itself is insensitive to the intensity of the non-adiabatic

transitions, the larger values of Kr always boost the total rate higher.

0.11
0.081
0.061
0.04
0.021

 

 

(a)

 

931

0.3»
0.25»
0.2» S=3
0.15»
0.11

 

 

A

2 4 5 8 1‘0 1‘2 194
(b) number of sinks

Fig. 20. The dimensionless generalized probability UK, for the inverted regime is
evaluated for A, =1eV= 401137" and AGo z -62kBT. and two values of the tunneling

parameter S = D, / ha), : (a) S=3; (b)S=8. Dashed lines are used for the 6(x) initial

distribution. The other notations are: 1 - or=1; 2 - a=5; 3 - or=10.

124

Another feature accented on Fig. 20 is the discrepancy in the rate constants for the
two different distributions we used. The saturated rates for 6(x) are somehow higher then
the ones for the Boltzmann initial distribution . This can be easily explained by the
argument used in the discussion of the normal regime. The reaction decay through the sinks
located at the bottom of the potential well becomes significantly faster when at the start of
the reaction all particles were concentrated at one point at the potential minimum. If the
distribution is spread beyond the narrow area near the minimum, the particles need more

time to reach the most effective sinks at the bottom which decreases the chemical rate.

3. (Zoncluding remgks
In this chapter we explored the BBET reactions with quantum bond-breaking. We

assumed that the reactive transitions occurred form the ground vibration level of the bond in
the reactant state onto all vibrational levels in the product state. This mechanism is the last
part of our investigation of the concerted BBET reactions. So far in Part II we have
presented the theory for a broad range of mechanisms possible in the concerted BBET
reactions. Predicating on the assumption of small coupling between the ET and BB modes
we have considered overdamped, extremely underdamped (energy diffusion) and quantum
mechanisms of the BB transformation. Which mechanism would be applicable to a real
process depends upon the frequency of the breaking bond. The higher is the frequency, the
weaker are collisions with the solvent, and the smaller becomes the friction until it reaches
its limit y->0 and the bond dissociation turns into a quantum process. The ET coordinate
has been treated here as an overdamped process in accordance with the well-established
technique [6].

Thus, when the friction is low, but not zero, the energy diffusion dynamics
describes the BB coordinate. As the friction decreases further, the high friction ET mode
takes over the dynamical changes and the BB coordinate just provides the static population

for the overdamped ET. If the friction is not zero, population from the ground vibration

125

level of the bond decays by means of ET onto the ground vibration level of the product
state. The reaction rates for this picture of BBET were evaluated in Ch. XII, Sec.2.
However, if there is no friction and BB is quantum, the decay is considered to occur onto
the complete set of the final vibration levels. As far as each final vibration level cuts its own
sink on the reactant potential surface, we have to deal with a reaction on multiple channels.
Even if the total picture of the multiple channel reactions is very complicated, a few
general conclusions could be made. As was demonstrated in the present Chapter, the rates
for reactions through multiple sinks are significantly higher then their would be if we took
into account only the sink leading onto the ground vibration level of the bond in the
products. The deviations become high (5-10 times) for activationless reactions and increase
enormously (up to a few orders of magnitude) for the inverted regime. This happens
primary due to the extensive growth of the non-adiabatic part of the rate constant for
multiple channels. In principle, this rise can lead to a complete change in the Marcus'
turnover dependence for the rate constants on the exotherrnicity. So that instead of
decreasing, the rate would grow as the exotherrnicity increases for the inverted regime.
This can happen because the activationless channels in the inverted reaction can correspond
to the states with the maximum overlap between the ground vibration level in reactant and
some exited vibration levels of the product. At last we should note, that it is not
straightforward to trace how the change of characteristic relaxation time for polarization
effects the total rate, as there is no direct proportion between the changes in time and the

reaction rate as it is veiled by the non-adiabatic transitions through the multiple channels.

126

CHAPTER XIV. CONSECUTIVE BBET

In the present Chapter we conclude the theory of BBET with a discussion of the
consecutive reaction mechanism. As was outlined in Chapters VIH and IX, the structure of
the PES is the characteristic that spans the differences between the consecutive and
concerted pathways in the reaction. For the concerted pathway, we assume that the
intermediate state is of high energy, and thus unstable. So, the reactants undergo a chemical
transformation directly to the products and therefore, only two PBS (for reactants and
products) participate in the chemical act. We used a diabatic surface viewpoint as is
standard in weakly-coupled electron transfer reactions. For the consecutive mechanism,
one has to take into account the existence of a stable intermediate state and hence consider at
least three potential surfaces, see Fig. 21. Thus, the reactants have to spend some time
moving along the stable intermediate surface before they can reach the product well. Thus
the overall reaction scheme is as follows:

RX+N' —) RX-‘+No -) R0+X‘+N0 (1114.1)
5.3;...) me}? step
In other words, in the consecutive reaction the anion radical is formed upon the initial step
and then the final transforms to the corresponding radical and the anion of the leaving
group. The latter step itself can be viewed as a particular case of dissociative electron
transfer where the unpaired electron located in one part of the molecule dissociatevely
reduces a bond belonging to the same molecule.

This picture is analogous to what has been suggested by experimental studies of
different organic halides as a method of distinguishing between the two mechanisms. It
was observed that with aromatic halides the mechanism is consecutive in all investigated
cases [80], whereas the bond-breaking and electron transfer are concerted with simple

aliphatic halides and with perfluoroalkyl halides, at least in polar solvents [56]. It was

127
concluded that this pattern reveals the role of the 11* orbital where the incoming electron
may be stabilized in transition. Thus, the 11* orbital can be responsible for formation of the
intermediate surface. The lower the energy of this orbital (as in aromatic halides), the

higher the probability of the consecutive mechanism.

energy
7:

 

E12

 

 

 

 

Figure 21 One-dimensional sketch of the potential surfaces used for the consecutive BBET.

Summing up, in order to discuss the consecutive mechanism we have to consider
three two-dimensional PES, one for each of the reactive states - reactants, intermediates and
products. The form of these surfaces was discussed in Chapter IX, see Eq. (H.9.6) and
Fig. 8. Note that in our representation of the consecutive pathway we do not assume that
the electron transfer (or the bond-breaking event) takes place first, as sometimes is implied
[81]. In this case the BBET reaction is effectively split into two individual reactions - the
ET and the dissociation of the bond. Each of these can be treated independently as a simple
reaction where the rate constants can be evaluated separately from the traditional ET and
bond dissociation theories (which usually involve only a single, one-dimensional PES.)
Here we assume that the intermediate state involves transformation of both the electron
transfer and bond-breaking coordinates, so we always have to deal with changes in both

dimensions. Thus, according to scheme (1114.1), the first step in our approach

128

characterizes the transitions from the state of reactants VR(x,z) to the intermediate state
V1(x,z). At this point the electron is transferred from one species to the other and

simultaneously the bond in the RX molecule stretches. Secondly, the R-X bond breaks and

the electronic rearrangement occurs to form the leaving anion and the R- radical, i.e. the

intermediates undergo the final transformation into the product state V p(x,z), see Fig. 21.
We are interested in the overall rate of reaction (1114.1) and the conditions for which the

first or the second step controls the rate.

As in all of the studies presented in the dissertation, in this Chapter we assume that
the BBET reaction is irreversible. For the consecutive mechanism this means that there is
no significant flux of particles coming from the product state onto the intermediate surface,
whereas the reverse reaction from the intermediate state into the reactants is still possible.
For this reaction scheme we have to consider at least two population probability
distributions which are exposed to reaction changes - p1 1(x,z, t) and p 2 2(x,z, t) that
correspond to populations of the reactant and intermediate potential surfaces respectively.
The following system of equations should describe the reaction process if we assume the

relaxation of the off-diagonal density matrix elements to be fast, as in Ref. [23, 43]:

a .
é”. : [11p11-1W12(X,Z)(P11"p22)

a _ .
#63:; : [’ZZPZZ +111V12(x,2)(p11 'P22)-IW23(x:z)p22

(1114.2)

where L11(x,z) and L 22(x,z) are the operators of motion along the reactant and
intermediate state potentials respectively. W12 is the reaction region for the transitions from
V R to V1 surface and back and W 2 3 provides only irreversible transitions from the

intermediate state into the products. In the simplest case the reaction regions can be

129

considered as lines of the intersection between each pair of potential surfaces (so the sinks
are infinitely narrow). Precisely this form of reaction regions was used when we evaluated
the reaction rates for the concerted BBET, see for example Eq. (H. 10.9).

The overall rate constant in the consecutive BBET reaction can be defined through

the changes in the product state population:

E =[Idtfldxdzp11(x,z,t)]- = [jjdxdzpn(x,z,s = 0)]-1. (1114.3)

To proceed further we have to employ some knowledge about the form of the
reaction region. If we accept that the reaction sinks are narrow (for example, they are
simply lines on the potential surfaces), the integral form of differential system (1114.2)

similar to the one given by Eq. (1.4.4) becomes more profitable:

p”(x,z,s) = (0R(x,z)/s— ”dx' dz' Gn(x,x' ;z,i ,s)W12(x’ ,z’ )(pn(x' ,z’,s)—p22(1rJ ,z’ ,s))

p,,(x,z,s) = [[212 (12' G,,(x,2 ;z,z',s)W12(x' ,2 )(p,,(2 ,2 ,s) —p,,(2 ,2 ,s))
{[22 dz' 0,,(x,x1 ;2,2 ,s)W,,(2 ,2 )p,,(2 ,2 ,s)

where G 1 1 and G22 are the ordinary Green's functions defined in Eq. (1.4.1). We assumed
that the reactants were initially distributed statistically, p,,(x,z,t = 0) = (pR(x,z). The
decoupling procedure analogous to the one described in Chapter VI, can be applied to the
above integral equations if all sinks are narrow. As a result we can write down a very

simple expression for the overall chemical rate:

1 1 l l K
—=——+r+— ——+r +1 , K“’=-—1-'=-. 1114.4
a K], 1 K“! [K2, 21 12] K.“ ( )

Here K1, and K 2,. are the non-adiabatic rate constants for the forward reactions (R)->(I) and

(I)—>(P) and K. 1,. is the non-adiabatic rate constant for the reverse reaction (I)->(R):

130

K], = [jdxde,,(x,z)(p,(x,z) (11.4.5a)
K_,, = ”dxde12(x,z)(p,(x,z) (11.4.5b)
K2, = [[dxde,,(x,z)¢,(x,z). (11.4.5c)

The first non-adiabatic rate constant determined by the shape of the reactant surface was
evaluated earlier when we considered the concerted mechanism and is given by Eq.
(1.7.16) or (11.10.23). Later in this Chapter we will evaluate the other two non-adiabatic
rate constants which depend on the choice of the intermediate state PES (through the
Boltzmann distribution (p , (x, 2)) .

11, I12 and 121 in Eq. (1114.4) are different mean first passage times which

describe the stochastic motion from one potential to another (for analogy see Eq. (1.7.29)):

 

Tl:

K1, “dedzdx’dz'W12(x,z)G-H(x,x';z,z‘)Wn(x’,z’)(pR(Jc,z‘) (1114.6)

is the MFPT from the Boltzmann initial distribution inside the reactant potential, (pR(x, z) to
the sink line onto the (1) surface, W,2(x,z). The steady-state Green's function used in the
above expression is the same as the one defined in Eq. (1.4.7) (at s=0) for the Boltzmann
initial distribution. Thus, the above time deals purely with motion along the reactant state
PBS and does not depend upon stochastic mechanisms for any other surfaces. In fact, this
is the MFPT which appeared in all rate constants for the concerted BBET; in other words,
we have been evaluating this time in all previous Chapters of this Part. MFPT (1114.6) has
been evaluated for three different types of motion along the bond-breaking coordinate (we
always assumed that the polarization fluctuations for the electron transfer coordinate are
overdamped). For the overdamped bond-breaking dynamics MFPT (H. 14.6) is evaluated
in Eq. (1.7.32), for the extremely underdamped regime this time is given by Eqs.
(H.10.27). Furthermore, the discussion about the reaction scheme for the case when one of
the coordinate relaxation times is fast/slow (see Chapter X1) in the concerted mechanism is

applicable to the MFPT in Eq. (1114.6). In Chapter XIII we also considered the case

131

wherein the bond-breaking event was a quantum process. Even though the total picture of
the reaction for this case would be more complicated than the one presented by the kinetic
scheme in Eq. (11. 14.4), MFPT 1:] can still be approximated by the Eq. (11.13.15).

The other two MFPTs in rate constant (11. 14.4) describe motion along the
intermediate state potential. Again, both of these constants can be evaluated separately and
independently from the type of motion inside the (R) potential. 1:12 and 121 depend only on
the motion inside the intermediate state potential and the geometry of the two lines of

intersection - one with the reactant state and the other one with the product state potentials.

1 ~
1,, = Fflfidxdzdx’ dz' W23(x,z)G2‘§(x,r ;z,f )W23(1f ,z’ )(p,(x' ,2 ). (1114.7)
2r

This MFPT is the average time to reach the reaction region W 2 3 having started from the
distribution along the line W12. Thus, this is the time for particles escaping from the
reactant state through the cut W12 spent on the (I) surface moving along the potential before
they approach the region W 2 3 and decay irreversibly into the product well. The steady-
state Green's function G3 can be defined using Eq. (1.4.12) for (0 being a statistical
distribution along the line W12. The other MFPT is the average time of the reverse

movement:

1 -,
2,, zFflfldxdzdx'dz‘ W,,(x,z)G,~,2(x,x1 ;z,z‘ )W,,(2 ,2 )¢,(2 ,2 ). (1114.8)
-lr

This time describes the motion of particles reflected from the region W23 and moving
towards the line W12. The steady-state function G22; should be evaluated using Eq. (1.4.12)
for the initial distribution spread along the line W2 3.

Now, coming back to the form of rate constant (H. 14.4) we would like to say a few
words about its structure and the physical picture it represents. The constants in expression
(H. 14.4) are deliberately rearranged to emphasize a simple combination of reaction rates for

the two steps in consecutive BBET reaction. The first step, which describes the escape of

132

the reactants from the (R) surface, is a combination of the first two constants in the right
part of Eq. (1114.4), 1/ E 12.. 1 = 1 / K” + 1“,. By analogy, the superposition of the first two
constants in the round brackets forms the escape rate from the (1) surface into the products,
1/ a: 14? = l / K2, + 12,. This is the rate for the second step. Finally, the reverse reaction
from the intermediates onto the (R) potential is governed by the time In. Thus, in total we
have three fluxes which equilibrate the reaction - the flux of the reactants moving onto the
intermediate surface and creating some population on it. The new-born population of
intermediates then can either disappear into the (P) well or create a flux back to the (R)
surface. Particles from the (R) well are delivered to the sink onto the (1) surface with the
speed E 11.. ,. The equilibrium between the (R)->(I) and (I)->(R) fluxes maintains some
population in the W12 transition region which is proportional to K94. Thus, the UK“)
serves as a population weighting factor for the rest of the rate constants which are formed
by the transformations on the (1) surface. Thus, the overall rate for the consecutive BBET

reaction can be expressed as follows:

i= 1 + l,[ 1 +112] (1114.9)
8 a112—+1 Kg air—41>

 

where all rate constants can be evaluated independently. Now, from the above expression it
is easy to reconstruct the traditional picture for consecutive reactions. As we have two
steps, it is often asked which of them controls the reaction. First, let us assume that there is
no reverse reaction (I)->(R), i.e. the MFPT 112 is not regarded in the above constant.
Then, we have two competing processes - formation of the intermediates from reactants
and decomposition of the intermediates. From basic kinetic theory we know that the

controlling step should be the one which occurs slower, i.e. with the smaller rate. This is

exactly what follows from the above expression. The smaller of the constants E 11.. , and
a316,, makes the largest contribution into the total rate. The only restriction is that the

constant for (I)->(P) transitions must be weighted with the Boltzmann factor K99. Addition

of the dynamical part for the reverse process (I)->(R), 112, makes the overall picture

133

slightly more complex in the sense that now one has to compare not two rate constants but
three. If the time of return from the intermediates into the reactant state is slower than the
rate of escape into the products, then the third rate constant, In may have a chance to
dominate the overall reaction rate.

Later in this Chapter we will investigate the chances for each of the three rates to be
in control. Note, that from all of the constants determining kinetics (H. 14.4) we have to
concentrate on evaluation of only two parameters - 112 and 121 as the non-equilibrium

constants K 5,. are easy to determine and 11 has been evaluated previously.

2. Eatential surfices tor the intermediate state and non-adiabatic rate constants

In this Section we establish the from of the intermediate PES and evaluate the non-
adiabatic rate constants K. 1,, and K 2,, in Eq. (1114.4). The two MFPT which describe the
dynamics along the (I) surface, 112 and 1321 will be calculated separately in the next Section
of this Chapter.

Assuming that harmonic approximation (11.9.5) of the Morse potentials is
acceptable, we can construct the following two-dimensional potential surfaces centered at

the bottom of the intermediate PES (see Eq. (11.9.1) and (H.9.6)):

VP(x,z) = (x+ x01)2 /2 +(z + 20,)2 /2 — AGl (II.14.lOa)
V,(x,z)=x2 /2+22 /2 (H.14.10b)
V,(x,z) = (x— x,,.)2 /2 +(z- 20,.)2 /2 + A0,. (1114.106)

For the above choice of the PES in the consecutive BBET we have in total three lines of
intersection. The minimum of each line is identified with an activation energy point, so we
have three barriers to compare E .42 1, E .42 3 and E A) 3. The first two points correspond to the
minimum points for the intersection lines of the (I) state with the (R) and (P) potentials, and

the last one is the minimum point on the line between the (R) and (P) states:

134

@G, — 2,)2 (AGl — AG, — 1,.)2
E421 = 4). , E423 : 4/1
I IP

AG. 1 3 -
and E,,,=( ‘+ P) —AG,.
' 42.

P

 

 

For the consecutive mechanism to take place the line of intersection between the (R) and (P)
surfaces has to lie higher then the lines ((R),(I)) and ((I),(P)). In other words we have to
require EA13> EA21, EA; 3 . If this condition fails and E413 lies lower then either 15,421 or
EA23, we shell say that the concerted BBET takes place and the reaction scheme behaves in
accordance with the theory presented in Chapters IX - XIH of this Part.

The lines of intersection for the intermediate PES can be defined from the equalities
V R: V; and VP: V1. These lines contour the reaction sinks from the (1) surface. The first cut
is formed by the (R) and (1) surfaces and provides the flux of particles from the reactant

state onto the intermediate surface and back:
W12(x,z)=Ko,6(x,x+z,z—c,), (11.14.11)

where x1 =(/2,1,,, 21:112421 and c1: AG] —1, (A, = 1,, 111,1). Here MK are the
reorganization energies for the polarization coordinate, Eq. (H.9.2). The reorganization
energies for the bond-breaking coordinate, 111 and 11p correspond to )leI and A111) which

are given by Eqs. (H.9.4). The other line, obviously, is the intersection between the (I) and

(P) surfaces:
W23(x,z)= K025(x2x +zzz-c2) (11.14.12)

with x2 =4/2A—d—f2l—xr, z2 =J2—E—4/2A—d, and c2 = AG1 -AG2 -}.,P where
,1 11> = 11,, + 2,, + 2,? + 1,? — 2 2,2,1, — 2m . Through this cut the particles escape
from the reaction space.

Thus, after we have determined the three PBS and the reaction sinks, it is
straightforward to evaluate the non-adiabatic rate constants from Eqs. (H. 14.5b) and

(11.14.5c):

135

 

 

 

AG —2 3
K—lr = K01 2701] exp(_flE.421)’ E421 : ( ill, I) (II-14133)
[3 (AG1 — AG, — 2”.)2
K,= , e —E,, E = “ . 1.14.13
-r 0.- 27M”, Xp( fi .413) A23 4).”, (I b)

Both of these constants have the same form as Marcus' ET rate constant, but the
reorganization energies 1 contain the contribution of the bond displacements in the (I) and
(P) states. This is the only difference which distinguishes the non-adiabatic rate constants
in BBET from those in ET theory. The frequency factors K01 and K 0 2 are proportional to
the coupling between the intermediate potential and the reactant or product PES:
K01=27rV,2,,/h and K02 =27rVIQP/h.

Now, after we have evaluated all of the non-adiabatic rate constants (K1, is given
by Eq. (1.7 .16) or (H. 10.23)) it is interesting to rewrite rate constant (H. 14.4) in the form

which separates the non-adiabatic and dynamical parts:

_1_ 1

+ .
E K, K‘qKz,

 

+ exP(fiEArz)[Z1 + X21 + 112 exp(p(EA23 — EA21))] (II-1414)

where we took 1', = x, exp(flEm), r2, = 12, exp(fiE .121) and In :11, exp(fiEM). In Eq.
(H. 14. 14) we separated the perturbation part from the rest of the factors. This separation
should make analysis of the consecutive reactions easier in case the non-adiabatic
transitions are weak or strong. In the first case only the first two constants would
contribute to the rate as the stochastic motion is comparatively fast. In the case of strong
non-adiabatic transitions the dynamic part (the combination of mean first passage times in
parentheses) is responsible for the chemical rate. As one can see from Eq. (H. 14. 14), in the
dynamic part it is essential to compare not the mean first passage times themselves, but
their pre-exponential factors, Xi]: This makes the dynamical analysis much more involved
as the Boltzmann factors do not influence on the competition between the MFPTs. Thus,

knowledge of the activation barrier heights would not be sufficient to predict the controlling

136

step (as a large barrier can make the appropriate MFPT very slow and hence dominant, this

would easily make the high barrier reaction step (R)->(1) always in control as ‘1'] would be

dramatically longer than all other average times).

3. WflTon the intermediate state PES

Now, the subject of our primary interest here is the evaluation of the effective
MFPTs 112 and 121. Note, that we have assumed that there is no distribution of particles on
the intermediate state surface at the start of the reaction. All particles appear on the (1)
surface as the result of transitions from the reactant state through the region W12. Moving
along the (1) surface these particles can either escape into the product well through the other
cut, W23 or come back to the pumping region, Wu and return to the (R) state. Thus, the
population of the (I) state is balanced by three processes: pumping of particles onto the
surface from the (R) state and escape form both of the reaction regions formed by the lines
of intersection of the (1) surface with the (R) and (P) PES, W12 and W 23. Both times 112
and 1:21 are the average times for particles to reach one line on the (1) surface if they have
started from the other one. Therefore, even though the two times are quite different the
routine for their evaluation is the same. Let us now evaluate the effective time 112 using Eq.

(11. 14.7). The initial distribution for it is statistically spread along the cut W12 :

fn(x,z) = %2L6(qlx + z — 91)exp([3EA21 --fi(x2 + 22) / 2) (11.14.15)

where q 1 = x1/ 21 and 01= c1/ 21. This distribution should determine the form of the
steady-state Green's function G1}: which then shall determine the desired effective MFPT.
However, we do not intend to obtain the Green's function itself, instead, we will use a
method similar to the one outlined in Sec. 2.2 of Ch. VH (see Eqs. (1.7 .18)-(I.7.20)). Here

we assume that fluctuations along both dimensions are overdamped. For details we refer

137
the reader to Appendix E where we evaluate the effective MFPT 112 in greater detail. The

final result for this MFPT is presented in Eq. (E9) and it can be rewritten as follows:

 

 

( mp 139,2 x+q§xafl ] exp(_39§F(x)) 1
fl .. ——
x

 

“2:2,; (1+q§)1+q§+x+q5x _ 2(1+q§) (111416)
0 J1_(x+q§x“)2 J1_ (”611(12):“)2
(1+4?)2 (1+43X1+q§)

where

2(1+q§)(x+qrqzx“ )0 /9- -(x+qrqex“)2(l+03/9§ (1+Q2)/(1+‘I1))
F(x)=

(1+qf)(1+q§)- (“41612:“)

As one can see, the final answer for the MFPT has five parameters. First, or is the ratio of
the relaxation times along the reactive coordinates, a= rz/ 17x, i.e. it reflects the relative
speed of the overdamped relaxation on the (1) surface. The other four parameters are purely
geometric and are determined by the form of the sink lines W12 and W23. given by Eqs.

(11.14.10) and (11.14.11):

q,=x,/z,=mq=x,/z,=(JT_,/T)/(JT_J1—,) (II.14.l7a)

0, = c, /z, = (AGl — 2,)/J22,, 0, = c, /z, = (AG, — AG2 - 2,,)/(er,, — J22,,,).

(H.14.17b)
0,2 022 . . .
Note, that EA21 =——2— and E423 :—_T are the activation energies for the
2(1+<11 ) 2(1 +612)

transition points for the sinks W12 and W23 respectively, which are defined by the phase
factors 61 and 92 (the activation energy points are the lowest points on the intersection
curves). The parameters q 1 and q; are the tangents of the projections of the sinks W12 and

W23 on the (x,z) plane, see Fig.22. The Figure depicts the contour plot for all three PES

138
and the lines of intersection between (R),(I) and (I),(P) surfaces. The dashed circles

represent energy contours for the activation energy points E21 and E 2 3 .

 

>4

 

 

Figure 22. The projection of the PES for the consecutive BBET on the (x, 2) plane.

If q1=q2=0, i.e. the reorganization energies for polarization fluctuations are zero (or
q1=q2=oo and there is no bond-breaking), then Eq. (11.14.16) degenerates to the one-
dimensional MFPT which is the average time to start from point x1 and arrive at point x;
moving along the x coordinate (or start at 21 and reach 22 moving along 2). The expression
for this time is given by Eq. (1.5.6).

Before we analyze the MFPT in Eq. (II. 14.16) further, we would like to discuss
the range of physical parameters participating in the process. Although we have four
geometric parameters in the MFPT under study, we actually have six energies which define
q1, q;, 01 and 02. From the form of the PBS in Eq. (H.14.10) we can see that one has to

think about a set of four reorganization energies - two for each of the coordinates, 1,1 and

139
Ayp, y=x,z as well as about two exothermicities, A01 and AG;. This allows us a
substantial flexibility in the choice of parameters for 1:12 and 1:21.

However, first of all, in the selection of the appropriate energies we have to
remember that our choice is restricted by the requirement to deal with the concerted
reaction, that is the activation energy for the direct transitions from (R) to (P), E A I 3 should
be higher then the other two activation points. Of course, in every particular case this
problem is somehow resolved, but here we would like to make a few general comments
that should help to predict in advance what mechanism prevails in the reaction. To begin
with, very high values of AG 1 should make the intermediate state relatively unstable and
hence act in favor of a concerted reaction. Similarly, a very exothermic reaction (large
negative AG 2) should provide a low barrier for transitions (R)->(P) and therefore may
switch the reaction onto the concerted pathway.

The situation is more complicated with the selection of the reorganization energies.
Let us consider the probable laws for the polarization coordinate first. From Eqs. (H.9.2) it
is clear that the reference point for measuring the reorganization energies for orientational
polarization is the state of reactants. The more significant is the charge redistribution in the
system after it leaves the reactant state, the larger is the expected reorganization energy.
Now, looking at the reaction scheme in Eq. (H.14.l) that we adopted for consecutive
reactions, one may notice that it is reasonable to expect approximately similar
reorganization of the charges in the intermediate and product states compared with the initial
charge distribution in RX+N'. Indeed, the nucleophyl radical N- has lost its electron after

the first stage and simply keeps moving inside the reaction cage in the course of the further
transformations. Thus, its contribution to the charge redistribution is completed at the first

step and therefore is the same for both 3.x] and le energies. A similar argument can be

used in description of the anion radical RX". Even though the second step in the

consecutive reaction may involve electronic rearrangement, the relative contribution to the

two reorganization energies may be expected to be the same. Thus, we are allowed to

140
assume that the reorganization energies 2x1 and Axp have very similar values. If they are
slightly different, we can not generally tell which one should be larger or smaller, as this
matter should be subject to the molecular structure of the RX molecule.

As the polarization energies have been admitted to be similar, they do not provide a
substantial contribution into the competition between the consecutive and concerted
mechanisms. Hence, our last concern is the ratio between the bond-breaking reorganization
energies 2.21 and 22p. If we think for a while about the BBET as a simple lD bond-breaking
process, we can use Fig. 21 ( mentally substituting z coordinate for x) to establish some
features which would be required for the consecutive mechanism to occur. From a basic
geometric point of view we can state that if the transition point 21‘, lies to the left from the
transition point from the (I) to (P) state, 2;, we should encounter a consecutive reaction.
This may be achieved simply by requiring 24 < 112p, which is quite reasonable as we take
into account Eqs. (H.9.4a) and (H.9.4b) for 1,1 (1,31) and 112])(12109). As we see from the
form of the bond-breaking PES chosen in Eqs. (11.9.3), the parameters O'K reflect how
strong the structural rearrangements in the reacting species are at the bond dissociation
(oK>1). It is normal to expect that the system in the (I) state has overcome intermediate (as
the state itself is called) changes in the original bond length. At this point the bond which is
about to be broken should be just stretched sufficiently to overcome the future dissociation
which is awaiting the molecule at the final step. Thus, the state with the broken bond, the
(P) state should present a bigger displacement for the potential surface, i.e. one may take
Gp>01>l which provides the desired behavior of the bond—breaking reorganization energies
and gives the consecutive mechanism a good opportunity. Note, even though all of the
above characteristics may be satisfied, we still have to check every time for the right order
in the activation energy set '

Predicated upon the above argument, we can make a few very useful conclusions

which will narrow the range of the parameter values we have to take into account. First, as

all reorganization energies are positive, q1 must be positive. Second, due to 11,, z 2,? we

141

should have (12 z 0 or at least quite small (however, it may be positive as well as negative).
It is much more difficult to restrict the choice of the phase factors 01 and 02 as the first one
is determined by three energy parameters and the other one by all six of them. The only
assumption that would be reasonable and would ensure a consecutive reaction is that the
phases are of a different sign (then the activation energy points for (R)->(I) and (I)->(P)
transitions lie in different quarters of the (x,z) plane that in 1D projection corresponds to
x,’2 <0 < x53 ,as shown on Fig. 21). Finally, the phase factors define the activation
energies E212 1 and E222 3 which should be small enough to assure low activation barriers as
the intermediate state is not supposed to be very stable.

Now we can turn to the numerical analysis of expression (11.14.16) for the MFPT
on the intermediate potential surface. As we are now investigating the average time 1:12 to
travel from W12 to W23, the particles decay on the line 8(q2 x+z-02). Then, if 11,, z [1,?
and hence q2=0, the decay line is perpendicular to the z axis and the MFPT in Eq.
(11.14.16) loses all dependence upon the relaxation time ‘Cx. In other words, the dynamics
on the (I) surface become insensitive to the relaxation along the polarization coordinate, and
the diffusion rate is determined only by the bond relaxation as can be seen from the

following:

{ [3(p2x [ fix { 2 (P121 1 ]\
“116725) _ CXP W1‘p2‘p‘ “W2 /2+ 2(1 +£1.01]

l
1 =1
12 “Ex «ll-x2 \[l- xz/(l+q,)

Even though this average time is one-dimensional in terms of the relaxation parameters, it
clearly depends upon the two-dimensional geometry layout. As now we have'only three
parameters to follow, the analysis of the MFPT becomes somewhat easier.

As established in Eq. (H. 14. 14), for comparative analysis of the diffusion rates it is

adequate to investigate the behavior only of the pre-exponential factors. Fig. 23 depicts the

142
dependence of the dimensionless pre-exponential factor (12 1121:17. 11; exp(fififlfl on the
activation energies EA23=023 for the absorbing line. W23. We compare two sets of curves
for two different values of the activation energies for the line of initial distribution.
EA21=013/(1+q13). In each set we considered the MFPT for q1=0, 1 and 10 Fig. 23 also

shows geometry of the projection of this example on the (x,z) plane.

 

 

 

 

 

Figure 23. The dependence of the dimensionless pre-exponential factor 1le 2 on the sink
activation energy 8,123 for three different slopes of the initial distribution line, q1=0, 1 and
10. The continuous line stands for the activation energy for the initial distribution line,

Ejm=5kBT and the dashed line represents E A21=1kBT.

As one can see from Fig.23, the pre-exponential factor for the MFPT, am has a maximum
at some point when the activation energy Bag) is very small, essentially IkBT. This is,
actually, typical behavior for all pre-exponential factors for the average times describing
overdamped motion which transports particles over a barrier, independently from what
initial distribution was involved in the dynamic process. For comparison the reader can turn

to Fig. 4 for the adiabatic overdamped reactions or to Fig. 6 for reactions on the diabatic 1D

143

harmonic oscillator surface (in Part I). All differences in the pre-exponential factor arising
from the parameters are maximized at low barriers. As should be expected, the lower
energy initial distribution decreases the MFPT, since the lower the start of the particles
occurs, the less time it takes for them to reach the bottom of the well, spread statistically
inside the potential and then move towards the sink line. A similar argument may by used
to explain why the higher slope of the q) projection decreases the MPFT (and hence
increases the rate constant). Since in the case under consideration (q2=0) only the motion
along the z axis counts, then the shorter is the relative distance between the initial
distribution and the sink in z direction, the shorter is the MFPT. From Fig. 23 it is clear
that, as the parameter q] increases, the activation energy point moves along the circle EA21
which brings the initial distribution line somewhat closer to the sink line q2. This shortens
the average distance |q1q2| (clearly, the area near the minimum of the potential is the most
influenceable) and decreases the average time 112.

However, as the activation energy of the absorbing sink increases, the dependence
upon all of the parameters in the initial distribution disappears (as both the activation energy
E A21 and the slope of the initial distribution line, q1 are parameters for the initial
distribution). As the barrier for this harmonic oscillator potential increases, the pre-
exponential factor for all of these parameters decreases as 1/ J73; , see Eq. (1.7 . 12).

If q1¢0 and the sink line W23 is not parallel to the x axis, then we have to consider
the overdamped motion along the polarization coordinate. As we already know, the largest
effect on the rate operates through a set of parameters which provide the lowest activation
energy points EA21 and EA23 possible. Taking this into account, Fig. 24 presents the
dependence of the pre-exponential factor 112 7112 on the ratio of the relaxation times 01:12 Itx
for EA21=EA23=lkBT We study here behavior of the diffusion process only for moderately
small values of q2, q2=0.1 and 0.2. As one can see, the pre-exponential factor in this case
is very sensitive to the acceleration of the diffusion along the z-coordinate (when 01 < 1, the

relaxation time 1:2 is shorter then IX and the relative diffusion along the bond-breaking

144

coordinate is faster). There are, obviously, some influences of the reaction parameters on
the rates when the polarization relaxation is faster (01> l ). but the differences for various
slopes q] are small and the deviations for q2=0.1 and q2=0.2 are barely noticeable. This
occurs because the MFPT for the motion from one line to another is much more sensit1ve
towards changes in the sink parameters than to variations in the initial conditions. Since in
this example we considered the sink lines to be almost parallel to the x axis. then the
relaxation time along 2 coordinate is still the most dominant of the two times 12 and rx. This
is why even if the motion along x (01> l) is fast, it does not affect the pre-exponential factor

as much as the fast z-motion (a<1).

   
 

q,=10

12
\j . .
.... (12:0 l

p...’

wt
1
E
1.
lq

 

5.421 ' EAz3=lkBT

 

 

l A - .1

0.1 1 10. 100. mm

 

Figure 24. The dependence of the dimensionless pre-exponential factor mm on the ratio
of the relaxation times 01:12 Irx for three different slopes of the initial distribution line,
q1=0, 1 and 10. The activation energy points for the initial distribution and the sink lines
are taken the same, EM3=EA21=1kBT The continuous line represents the dependencies for

q2=0.2 and the dashed one shows a similar behavior for q2=0. 1.

145

Finally. Fig. 25 presents the example where both lines on the surface have
significant slopes and thus are not parallel to any of the two axes. Similarly to Fig. 23. we
considered the dependence of the pre-exponential factor on the activation energy of the
sink. We took q1=1 and two different values of q2: 1) q2=0.5, which makes the sink line
mn along the initial distribution line and so they intersect somewhere at a high energy point;
2) at q2=-0.5 the two lines are pointed towards each other (see Fig.25) and intersect near
the potential minimum. Hence, as the both relaxation times contribute to the MFPT we

considered three different values of the ratio r2 lrx: a=0.1, 1, 10.

 

 

 

 

    
 
 

/ d ------ 3
--

 

 

1;: ’7’, -35-ID--‘-'-“~~——~-. a-l
,, -—’ (rt-0.1 A A A A
o 1 2 3 4 s 65423

Figure 25. The dependence of the dimensionless pre-eprnential factor 1:2sz on the sink
activation energy 5.423 for three different ratios of the relaxation times 01:12/ rx=0.1, 1, 10.
The initial distribution line activation energy was taken as EMI=1kBT. The continuous line
stands for the positive slope of the initial distribution line, q2=0.5 and the dashed line
represents q2=-0.5. For comparison, the dotted and dashed line gives 1:211 2(BEA23) for

q2=0 (when there is no dependence upon a).

146
First of all, for this choice of parameters the MFPT depends on both relaxation

times, i.e. it is a two-dimensional diffusion relaxation process. Even though typically the
pre-exponential factors still exhibit a maximum at the barrier energies close to lkBT, this is
not necessarily the case.

To begin with, let us consider the sink line with a positive slope, q2=0.5. Here we
have a situation that has been expected: for all ratios of the relaxation times the pre-
exponential factors have their maxima at very low barriers, EA23z0.5kBT. Since we count
the pre-exponential factor in the units of the relaxation time t2, then the smaller or
correspond to the faster diffusion and vise versa. The faster diffusion ‘Cz decreases the
MFPT from one line to another, which is the case for a=0.1 on the picture. The situation
for large or is more involved. As one can see, an increase in the relaxation time ratio by a
factor of 10 has far greater influence on the MFPT for low barriers than a corresponding
reduction. Indeed, in the range EA23=0 - 2kBT the MFPT for 01=10 slows down compared
to the MFPT for or=1 more dramatically than the same MFPT speeds up at or=0.1. This
happens because we are considering the mean first passage times between two lines.
Looking at the diagram in Fig. 25 for q2=0.5 one can notice that at EA23~0.5kBT the
average traveling distance from W12 to W23 in the x direction is shorter then the distance
along the z axis (the domain near the potential minimum is the one that really counts). So,
one can expect that the shorter distance provides the most intensive paths for the particles
traveling from one line to the other. Therefore, the average time should be more sensitive to
the changes in the relaxation time along this direction (in this case the x coordinate). Thus,
the MFPT responds more to slowing the particles down along x then to the speeding them
up along 2.

Finally, at the negative value of the sink slope, q2=-0.5, we observed the'situation
when the pre-exponential factor has a minimum at a low barrier instead of the traditional
maximum. For the accepted choice of parameters this happens when 'cz<'cx (or=0.1), see

Fig. 25. The explanation can be given by employing similar arguments to the ones used in

147

the previous paragraph when we discussed geometry of the problem with q2=0.5. The
difference is that now the average distances are shorter for the z direction. Thus, the MF PT
for ot=0.1 has a richer behavior then the one for 01:10. The reason why the first MF PT has
a minimum instead of a maximum also lies in the geometry of the problem. As we drag the
sink line from EA23=O higher, it intersects the initial distribution line at the minimum point.
This configuration is optimal for the passage of particles from one line to the other and

therefore the pre-exponential factor for this MFPR is minimal.

4. (loncluding remarks

To sum up, in the present Chapter we have concluded the theory of BBET reaction
with the discussion of the consecutive mechanism that may be an alternative to the
concerted BBET considered in Chapters IX-XIII. The major distinction between the two
mechanisms is the structure of the potential energy surfaces. Sirnplisticly speaking, in order
for the consecutive mechanism to outweigh the concerted scheme, the intermediate state
potential surface has to be of low energy. In this case, the particles escaping from the
reactant state would have to travel primarily through the intermediate potential in order to
reach the product well. In the concerted BBET the intermediate state is accepted to be quite
high in energy and thus unstable, so that the direct route from reactants to products is more
economical.

In our study we have established that there are three competing processes which
may determine the overall reaction rate in consecutive reactions, see Eq. (H. 14.9). These
processes are: i) the transport of the reactants form the (R) well onto the (1) surface over the
potential barrier EA12 (with the dynamical part defined by 11); ii) irreversible escape of the
intermediates into the product well over the barrier 13.423 inside the (1) potential (with 121);
iii) reverse movement of the particles form the (I) state back into the products over the other
potential barrier formed inside the intermediate state, EA21 (with 112). As was discussed

above, in principle, any of these three transformations can control the whole reaction,

148
depending on which of them is the slowest (see Eq. (11.14.14) and the following

discussion).

Since the most interesting behavior of the reaction rate given by Eq. (11.14.14) is
based on the dynamics, for this analysis we will assume that the non-adiabatic transitions
are fairly strong so that they do not control the reaction rate (so we are in the diffusion
control limit). Therefore, out aim here is to estimate which of the three MFPT involved in
the reaction kinetics have better chances in controlling total rate (1114.4). As has been
discussed previously, for comparison of the dynamical factors it is adequate to contrast the
pre-exponential factors for the average times rl, 1:21 and 112, see Eq. (11.14.14). In Section
3 of this Chapter we presented a few graphic dependencies for the pre-exponential factors
for the MF PT, 121 and 112 on the intermediate potential surface. Since these MFPT are the
average times to travel between two lines on the surface, these graphic dependencies may
be refered to the behavior of In as well as 121, as they both describe similar processes.
The only detail which should be remembered in order to contrast the factors, is that the pre-
exponential factor for 112 has to be multiplied by the Boltzmann factor of the difference
between the two activation barriers presented for particles by the (1) surface, exp(B (EA23 -
EA21)). Thus, if for the intermediates the barrier to escape into the products is higher than
the barrier to return back into the reactants (so that EA23 - EA21>0), the time of the
transitions (I)->(P), 112, may have a significant advantage over the time 121 in controlling
the overall rate, see Eq. (11.14.14). Naturally, particles have better chances to escape from
a potential well over a low barrier which makes the appropriate MFPT shorter (and the rate
of the reaction smaller). In the consecutive reactions the slower rate controls the process,
therefore the MFPT over a higher barrier has better chances to dominate the overall
reaction. Hence, if EA23 - EA21>0, 112 is slower than 1:21 and the transitions (I)'->(P) are
dominant. If EA23 - EA21<0, then the reverse reaction (I)->(P) may be the one which

prevails for transformation of the intermediates.

149

To complete the discussion we present Fig.26 where we demonstrate how the
MFPT inside the (R) well, I], normally behaves depending on the height ofthe activation
barrier for the (R)->(I) transformation. The expression for this MFPT, which is the average
time to reach the sink line W12 on a 2D harmonic oscillator surface having started form the
Boltzmann distribution, is given by Eq. (1.7.32). The average time t] has only two
geometric parameters - the barrier height, EA” and the slope of the sink line, q] (in the
notations for this Chapter). It would be reasonable to expect that the reaction (R)->(I)
would take place over a moderately high barrier, say EA12>10 kBT, which has to be

somehow significantly higher than the barriers for the reactions (I)->(P) and (I)->(R), E A2 3
and EA21. If the activation energy EA12 is high, then 112 can be estimated as 1/JE_,,2 .

 

 

 

 

Figure 26. The dimensionless pre-exponential factor 1le for the MFPT on the reactant
PES is presented as a function of the activation energy, EA12. The continues lines present
the dependencies for q1=l and the lines with the shorter dashes are for q1=0.1 and the lines
with the longer dashes are for q1=0.5. Three ratios of the relaxation times 01:1:Z/ rx=0. 1, l,

10 are considered. The curves for all three values of q) coincide at or=1.

150

So, we have to compare the dimensionless pre-exponential factor 1le to the
parameters 12le and 12x12 depicted on Figs. 23-25. From Fig. 26 (see also Sec. 3 of Ch.
VH) it is straightforward to see that the factor thl at E A12>10 kBT can quite easily become
smaller than the other two factors, as its value at a given energy is comparable with those of
for 12le and “mm, but we have to compare the factors at different activation energies. In
other words, the pre-exponential factor x] has to be taken at much higher energy then 112
or x21 and hence can be much smaller than the other two (again, for the consecutive
dynamics the largest pre-exponential factor should control the whole reaction). Therefore, it
should not be too hard for the dynamics on the intermediate surface to control the overall
reaction rate. If this is the case, then the question of which barrier on the (1) surface is
higher should give a good estimate for further selection of the dominant MFPT.

Of course, the detailed picture of the consecutive reaction on a 2D potential surface
is much more complicated. One has to consider not only the geometric layout of the
potential surfaces including the height of the activation barriers are and so on, but also the
two-dimensional dynamics. For the 2D case, in general, the answer to the question what
dynamics control the overall reaction, is hard to prognosticate The difficulty is that the
dynamics act in an unpredictable way, especially for the intermediate surface where the
barriers are low and we can not foretell which relaxation time has the greatest effect. For
the precise answer, one would have to evaluate all MFPT relevant to the discussion using
Eq. (H. 14.16) for the motion on the (1) surface and Eq. (1.7.32) for the dynamics inside
the (R) well.

151

CHAPTER XV. CONCLUSION

Unimolecular reactions represent a wide range of chemical processes, their variety
is enormous - everything from simple isomerization to ET in complicated biological
systems. However, despite their diversity, they all take place in a closed reaction space, so
that the reactants and the products co-exist in the same cage during the reaction. Exploration
of all these processes is united under unimolecular reaction theory which roughly splits into
two major parts - adiabatic and diabatic approaches. The influence of stochastic motion on
the reaction rates for both approaches is the most intense area of interest at the present
moment.

One of the traditional methods for investigation of reaction dynamics involves
solution of a time-dependent Fokker-Planck equation for Brownian motion (either
underdamped or overdamped). The solution generates a population probability function
which decays in time in some area on the potential surface. In this theoretical scheme the
reaction rate constants are determined through the W population probability
function. A few one-dimensional adiabatic as well as diabatic models have been extensively
developed from this method and in some cases the rate constants have been evaluated
analytically. However, we require investigation of more complicated reactions, such as
multidimensional surfaces processes. A few attempts have been made to obtain the rate
constants analytically, but the principal difficulty is that the traditional scheme of equations
is too complicated to be resolved even for two-dimensional PES reactions. Partially this is
the cause of the time dependence in the intermediate calculations in the traditional evaluation
scheme for the rate constants. As a consequence, numerical methods have been more and
more extensively used in the theory.

In this dissertation we present a new, steady-state theory for evaluation of the rate

constants in unimolecular reactions which at no point involves evaluation of time-dependent

152
functions. It is a theory based on the steady-state analogs of the traditional Fokker-Planck

differential as well as integral equations and therefore eliminates time as a variable from all
steps in the process of evaluating reaction rate constants, see Chapters III and IV.
Transition from the time-dependent to steady-state formulation of the theory for
unimolecular reactions was not trivial. So, in order to prove the legitimacy of the steady-
state approach, in Chapter IV we introduced the steady-state Green ’3 fimctions defined
only by the motion on PES. These functions are dramatically different from all other
Green's functions used in unimolecular theory previously (all the details are described in
Chapter IV and Appendix A). They not only provide the correct transition from the time-
dependent to steady-state equations, more importantly, they have a straightforward physical
meaning. In the simplest interpretation a steady-state Green's function defines MF PT from
some initial distribution (either a function or a point) on PBS to a point. Furthermore, the
specific properties of these functions allowed us to generalized the concept of MFPT and
introduce a method for evaluation of the efi'ective MFPT, which is the time for particles to
reach a region on PBS (in contrast to a single point) having started from a distribution. The
effective MFPT is a major dynamical characteristic in multidimensional reactions and may
be the only average time of interest in one-dimensional processes (such as adiabatic reaction
over not extremely high barriers and diabatic reactions on sinks which are wider then a
pinhole).

Among the most common reactions are the reactions taking place on narrow
reaction sinks. In the frameworks of the steady-state method we introduced the decoupling
approximation for solution of integral equations for this case, as outlined in Chapter VI. It
is consistent and very simple to use, yet became possible only due to the steady-state
Green's function technique. Moreover, the steady-state approach enabled us not only to
design the decoupling scheme, but also formulate the criterion of its applicability. It was
proved explicitly that for narrow sink reactions the general evaluation of the rate constant

can be significantly simplified as it is possible to separate the dynamical effects from the

153

reaction itself. As a result, the rate constant for this case consists of two parts which can be
evaluated separately: equilibrium (non-adiabatic) rate constant and effective MF PT (the
average time for particles to reach the narrow sink). The procedure for evaluation of the
effective MFPT requires utilization of the steady-state Green's functions.

Furthermore, as a logical extension for Part I, the steady-state Green's function
theory was applied to a few reaction models. First, we considered one-dimensional
systems in order to test our theory on well developed examples. In Chapter V we resolved
the problem of high activation barrier reactions for narrow and moderately wide reaction
regions for a broad range of potentials. In Chapter VH overdamped reactions on one- and
two-dimensional harmonic oscillator potential surfaces as well as the transitions in a
bistable potential were analyzed carefully using our method

For the bistable potential we evaluated the reaction rate of escape from the reactant
well under the influence of the position-dependent fiiction, ‘Yefi(X) over all range of baniers.
The barriers did not have to be high and the evaluations did not involve the harmonic
approximation on the top and in the bottom of the potential. If the barrier is not very high,
the reverse movement of the particles from the product well into the reactant well
significantly decreases the reaction rate.

As an example of irreversible decay on diabatic surfaces we considered reactions on
one- and two-dimensional harmonic oscillator surfaces. We assumed that the one-
dimensional reaction takes place on a pinhole sink, which is located at the point of
intersection with the final state diabatic surface, so that the reaction rate constants were
evaluated by using the MFPT to the point. For the more sophisticated case of a two-
dirnensional harmonic oscillator we were also able to obtain an analytical expression for the
rate constant. For the two-dimensional surface the reaction sink was taken to be the line of
intersection between the two diabatic surfaces. In this case the rate constant was determined
through the effective MFPT, which was found to be a complicated function of the

relaxation times along each of the reaction coordinates. For low barriers both relaxation

154

times seem to contribute to the reaction, even if one is considerably faster then the other.
However, for high barrier reactions the dynamic influence is more straightforward, so that
the faster relaxation time controls the reaction rate.

In order to extend application of the steady-state Green's function theory to more
complicated systems, in Part H we developed a theory for bond-breaking electron transfer
reactions, or BBET. The reaction scheme for these processes in outlined in Chapter VHI.
These reactions are intrinsically two—dimensional as they require transformation of the
reaction system along both electron transfer and bond- breaking coordinates. To our
knowledge, this theory is the first consistent theory aimed at description of such reactions.

As was established, it is necessary to introduce some intermediate state in order to
distinguish between two possible mechanisms observed in experiments - the consecutive
and concerted. The difference between the mechanisms is provided by the structure of the
two-dimensional diabatic potential surfaces for reactants, intermediates and products, as
discussed in Chapter IX. Treating these two mechanisms separately, our BBET theory is
based on the diabatic formalism and is aimed at understanding the dynamical effects on the
overall reaction rates.

The concerted mechanism takes place if the intermediate state is of high energy and
thus unstable. In this case, one has only two diabatic PES to take into account as the
reactants overcome direct transformation into the products. The theory presented in
Chapters X-XIH concentrates on the complex multidimensional dynamics in the concerted
BBET. Assuming that the polarization fluctuations are always overdamped with the
relaxation time 1x, we viewed two classical and one quantum characters of the dynamics
along the bond-breaking coordinate. The classical overdamped and energy diffusion
dynamics are appropriate for description of extremely low or moderate frequencies of the
breaking bond, whereas the quantum nature of the coordinate should be considered for the

bond frequencies of 600 cm'1 and higher.

155

The energy difiitsion regime appeared for us to be of a special importance, as the
bond period is usually short with respect to the solvent collision time. We explored this
energy diffusion mechanism (characterized by the relaxation time tz) in Chapters X and XI,
by means of the steady-state Green's function method, and succeeded in obtaining the
techniques required to consider overdamped diffusion along one coordinate (ET) and
energy diffusion along the other (BB). We found that for our choice of the two-
dimensional PBS, the overall rate constant in the concerted BBET can still be separated in
two parts, 1/K=er + tp - where one is responsible for the chemical transformation (non-
adiabatic) and the other describes the dynanrics along the two coordinates (the effective
MFPT). The effective MF PT, tp was found to have a very complex dependence upon the
two relaxation times 1,, and 12. As the dynamic scales become different a rich variety of
behavior becomes possible as to which time controls the overall reaction rate, see Chapter
XH. A special method of analysis was developed for the situation where one dynamics is
fast/slow compared with the other one. As a result we obtained that even though the overall
rate consists of the two independent parts, the purely dynamical question which time
controls 11, can be answered only through comparison of the fastest relaxation time with K,.
Despite the fact that the overdamped regime along the BB coordinate has many features
which are similar to those of the energy diffusion dynamics, the comparative analysis of the
two mechanisms was also performed in Chapters XI and XII.

The steady-state Green's function method transpired to be extremely useful for
developing the theory for the quantum bond-breaking coordinate in the concerted BBET as
discussed in Chapter XIH. For frequencies that are high enough to call into question the
classical treatment of the BB coordinate we incorporated the quantum effects in the
dynamical picture on the BBET reactions. In this case the original two-dimensional
structure was reduced to the set of one-dimensional diabatic potentials for the polarization

fluctuations (ET coordinate) which provided population decay from the reactant diabatic

156

surface through multiple channels into the products. In this picture the overall rate constant
can never be separated into non-adiabatic and diffusion parts.

We found that, generally, one can expect an increase in the reaction rate when going
from the classical (energy diffusion) to the quantum treatment for high bond frequencies.
This occurs because the multiple channels created by quantum vibration levels of the
products usually depopulate the state of reactants much faster than the single sink between
the two diabatic surfaces in the classical treatment. For a broad set of examples we
demonstrated that there is a strong interference between the channels in quantum BBET so
that they can not be treated independently. In other words, the overall reaction rate in
quantum BBET is not additive with respect to the partial rates on each cannel. We found
that one can increase the reaction rate dramatically by taking into account presence of the
multiple channels, especially if the ET happens to take place in the inverted Marcus'
regime. This increase (in some cases up to a few order of magnitude) happens primarily
due to extensive growth of the non-adiabatic part of the rate constant for multiple channels.
The interference between the non-adiabatic and diffusion parts in this case is so strong that
it can cause changes in the Marcus' turnover dependence of the rate constant on the
exothermicity. So, the reaction rate, in principle, does not decrease for high exothermicities
in the inverted regime, instead, it can even grow with increase of AGO.

Consecutive BBET occurs if the intermediate state is sufficiently stabilized, which
leads to participation of an additional (intermediate) PES, which for this mechanism has to
depend upon both the ET and BB coordinates. First, as in the concerted BBET the overall
reaction rate in the consecutive mechanism can be under complete dynamical control.
Moreover, all of the features of the dynamical effects obtained for the concerted reactions
are exhibited in the consecutive BBET. Only now two-dimensional dynamics alOng each of
the two surfaces (for reactants and intermediates) have to be considered as these dynamics
compete with each other for control over the total rate. Thus the overall picture of

consecutive BBET is extremely complicated, especially if some of the relaxation times are

157

faster/slower than others. The steady-state analysis of this situation is presented in Chapter
XIV. Three effective MFPT were found to be in competition. One effective MF PT is
responsible for transitions from reactants to the intermediates and two others are formed by
the motion on the intermediate surface (one towards the products and the other one back to
the reactants). Using the decoupling approximation from Chapter VI we evaluated all three
effective MFPT and predicted the conditions at which each average time may control the
overall reaction rate. In particular, we found that if the overall rate in the consecutive BBET
is strongly influenced by dynamics, the motion on the intermediate surface generally has
better chances to control the dynamical rate of the consecutive reaction.

This dissertation is devoted to the creation and development of a new, steady-state
theory for the investigation of dynamical effects in unimolecular reactions. The greatest
advantage of this theory is that it suggests an alternative technique for evaluation of the rate
constants in closed reaction space processes. The technique is based on steady state
diflerential and integral equations. Removal of time as a variable decreases the dimension
of the base equations and opens up new possibilities in analytical solution of multiple
problems, especially for multi-dimensional reactions. As a demonstration we developed a
theory for description of bond-breaking electron transfer reactions that are intrinsically two-
dimensional. The theory may be applied to underdamped as well as overdamped dynamics,
and both adiabatic and diabatic potential surfaces. We believe that our introduction of the
steady-state equations is the first theoretical study which allows calculation of the
unimolecular rate constants directly through the time-independent equations. It is expected
that our approach, because of its specific technique and relative mathematical simplicity will

prove fruitful for many further applications.

APPENDIX A. PSEUDOSTATIONARY GREEN‘S FUNCTION. CALCULATION OF
G0(x,x') and ‘tp

158

APPENDIX A: PSEUDOSTATIONARY GREEN‘S FUNCTION. CALCULATION
OF Go(x,x') AND tp.

As we mentioned in Ch. IV, there is a problem concerned with solution of steady-
state equations (1.3.9) and (1.3. 10) using the formal Green's function defined through the
equation:

Zoe, 17)G,,(5e, 17, it", v") = —5(2 — man; — 12"). (A1)
It is obvious, that this function can be applied to the solution of this steady-state problem.
As one can see, the above equation can not be considered as a limiting case of Eq. (1.4.1) at

s->0, as far as the formal Green's function Go (we will call it pseudostationary function)

”.9”,

has no links with the Green's function G(x,v,x ,17';s). Moreover, the latter function

diverges at s->0 and satisfies the boundary conditions which differ from the ones for the

formal function 6005,15, x", it") . Indeed, G0 does not retain the property of concervation of

particles, but according to (A1) satisfies the following condition:

J] i(2,V)G,(r, v, 21, rpm = 1. (A2)

It means that the number of particles in the reaction system is not conserved anymore,
because Eq. (A2) implies the appearance of a constant flux of particles into the reaction
space. So, it is possible to set the kinematic flux to zero at only one boundary. The

additional condition for the solution of Eq. (A1) should be taken as:

”000113, amend; = const. (A3)
Thus, it is obvious, that Go(x',17,5c",17’) differs dramatically from the steady-state Green's
functions G or G defined in Ch. IV. From the other hand, G0 is obviously appropriate
for solution of the steady-state differential equations (1.3.9) and (1.3.10) as well. The

integral form of Eq. (1.4.6) may be represented as:

159

501:, v): (p(x, v) + EHG0 (x,v, 17 it" ,17')fx ”’ ,ii')df'dv' —
{10 (“,v, " r" 17'1th )4(x .v'ww (A4)
which is exactly the same as the one in Eq. (1.4.9).

At the first glance, it seems strange that such completely different functions as Go
and G end up with the same differential equation for 566,13). However, if we analyze the
boundary conditions and the properties of both functions, e can conclude that equation (A4)
is built in such a way that all differences cancel and both solutions for 130?, 17) are identical.
Preference to one or another method can be dictated only by convenience in calculation of
the different Green's functions.

It is easy to find out the connection between Green's functions G and Go, which

may be represented as:

G(x,v, “ it" ,"')= G ,,(55 r ,2' ,r')- j j G 0,05 17 5c" 2) f v ,ii’)d1'c"dv" (A5)

Thus, the desired Green's function G can be obtained through the pseudostationary
Green's function Go , as far as Go can be defined more easily than G. However, as we
will show further in Appendix A, calculations involving Go produce some restrictions to
the form of the Boltzmann distribution 0302', 17).

Let us calculate the Green's function Go(x,x') from Eq. (A1) for one-dimensional
motion using overdamped operator (1.1.2) . Direct integration under condition (A3) (for the

case when const=0) and assuming that flux at the boundary x=-oo is zero yields

1 ¢(Z)dz- ®(x- 16)] (A6)

Gotx x): 40011 ,1,”

,.D(y)¢ 0'),

Notice, that at the integration by parts we used the condition:

lim :13 ¢(x)de—dz= 0. (A7)

y—>aoy (x)

160

For the further transformations we used the Boltzmann distribution (0(x) in the form:

we): —Z'—exp<-BV(x» (V(x)->00 at x->:oc). (48)
0

Carrying out the transformation:

14004 = —‘-1 exp(—13V(x»dx = iexp1—4Vtx111expmtm) — V(x))idx.
y 20 y 20 y

substitution of variable z=x-y and using the Taylor's series expansion over the small

parameter 2 in the integrand, we obtain:

-Z‘—Oexp(-/3 V0111 explfiWty) — V0 + zmdz ~ inn—131(0)) 1 expo-11 V'(y>z)dz.
O 0

Or, finally, we have the estimate:

 

141x14: ~ exp(- 4V0» (A9)

KT
ooZ 0V'(y)

By analogy with the above procedure, we can analyze another integral:

y—x' ’th) V00)

 

_1_}: 20 exP(-BV(y)) J'e dz ~ 20 exp(_flV(y))J'e-[3V’(y)zdz
y—mo
or, shorter:
y
Idx __efiV (y)_ 1 (A10)
,KPO‘M—mo HV’(y) (p' 0;)

Therefore, condition (A7) is fulfilled when:

V'(y)’2—->0asy—)oo, V(y) ~ yl” (e>0). (All)
y—No

The most rigid condition tums out to be the requirement of convergence of the first integral

in Eq. (A6), because we have:

I61

IV’O’W dy —) cons! asy —> 00, V(y) ~ yzie (8 > 0) (A12)
y—voo

which means, that it diverges even for the harmonic potential. Meanwhile, abstracting from

that problem i. e. assuming that the function V(y) increases sufficiently at y->-_l~ac (V (y)~y“,

n>2), we obtain the steady-state Green's function using Eq. (A5) and integrating by parts:

G(x’x'lw“) lam—fi— ULIfUdZ lbw—_ycT (”MM 6"“ “bow (y)
(A13)
where:
:io—dy— Do) U), I mad: I f(2)dz

Clearly, the above formulas require the above mentioned restriction for the form of the
reaction potential V(x). However, carrying out the transformation of identity, both of them

may be rewritten in the form:

G(x, x') = <p<x){ I

d
«BUT/’75.! <p(z)dz I f(2)dz + I

<p(z>dz f(2)dz+
,cho) do)I i

x

._dy__.. —@ — ' :
+ I D(mchpewz (x x)»

_“’(){l1)ty><,v)lf()d‘£13<y>)<:)(y)l“’(")dz 0“ ”lbw (y) B}

(A14)

where:

I ¢(2)dz I f(2)dz+ I—— I <p<z>dz I f(2)dz (A15)

3100):) (y) Dow (y)_

Now expressions (A14) and (A15) require sufficiently weak behavior of reaction

potential V(y) as y->ioo For example, at f(z)=(p(z) it is sufficient, that:

162

V(y)y_;nKBT1n|y13” (5 > 0). (A16)

Thus, for all potentials which grow up not very sharply at y->ioo the calculation of the
function G(x,x’) should be carried out by direct solution of Eq. (1.4.12) without usage of

function Go(x, x'), because it does not require such a rigid restriction as the one presented

by Eq. (A12). MFPT to the point xp is given by Eq. (1.5.5). For the averaged relaxation

time Te we have:

_G-(x ,0) (7(0,x) 1 "' dy °°
‘ <p(x,,) (13(0) Diirpmiwz) } p

where Bo is the value of B at f(x)=(p(x).
The most simple expression for G(x, x’) and tp can be obtained for the case
f(x)=5(x-xo) immediate from Eqs. (A13) and (1.5.5):
dy

G(x, x’) = -(p%-)-{®(x - x0);—

xo-iy—w dz—o - 'x—dX- A18
W)+£¢(y)["’(z) 0‘ “imi' ( )

APPENDIX B. CALCULATION or 1,, AND G(x, P ,5) FOR HIGH
ACTIVATION POTENTIAL BARRIER.

163

APPENDIX B CALCULATION OF ‘Cp AND G(x, x';s) FOR HIGH
ACTIVATION BARRIER

Let us assume, that f(x)=6(x), then from the first equation in Eq. (1.5.6) at x0=0 we get:

x

_’ dy x” dy °°
T — —— —— dz.
P {Dorm {D(y)<p<y)£‘p(z) (Bl)

According to asymptotic estimate (A10) we should keep only the first member which
grows exponentially when BV(x)>>1. The second integral changes like I V ’(xp)"dx , and
P

can not compete with the first one. Then, ‘te in (A17) is:

~ mIDU—T—fp (”be M (132)

SO, at high activation potential barrier we have Tp>>te and using the next order at the

asymptotic expansion in Eq. (A10) we can write:

1', z %epV<xv) I (1 + ,8V”(xp)x2 /2)e‘/3V'("P)xdx
0

z—ZO 1+ ——-LV( V0?” 3
DBV'(xp)( I3V'(x Mel} (B)

It should be mentioned that the correction term in the pre-exponential part
I! I 2
V (xp) / (fiV (xp) ) is of the same order Of magnitude as (I3V(xp))'1 <<1.
Let us now assume that f(x)=(p(x). Taking into account Eq. (1.5.5) for 1,, has the

form:
f"
I ~1 dy _1_2

1
P D .zro—(yfbli MM“ 3") (B4)

This expression makes clear that the estimation of MFPT given in Eq. (B3) is still
valid for (p(x). Moreover, we can conclude that the result Obtained for Tp is legitimate for

any initial distribution f(x) if its width is not significantly larger than the one for (g(x).

164

Now we calculate the non-steady-state functions (709,09) and C(xp,xp,s)

assuming that the second member on the left side of Eq. (I.4.10b) is the first order

correction on the value s. Representing the steady-state Green's function as:

G(x,x’;s) .—. G(x,x')+(}m(x,x';s) (B5)
we can rewrite the appropriate differential equation for it as:

6 (x)i-1——Gm(x, x’) = sé(x,x').

DECO 3x fix) (B6)

Moreover, the function G(x, x';s) satisfies the same boundary conditions and

condition (1.4.1 1) that the steady-state function G(x, x’) . Afier integration of Eq. (B6) we

 

arrive at
6(1)“: ,x';s) s an J(pg-(396)
p :_ ——dzE
we.) 0.1:”ny «2(2)

El xp§(y2x')d _ngyx')d °° d2 0 g(yax') y dz 7

Di! (My) y I My) ’M’) iii... (My) if“) (B )
where

g(y,x') = f G(z,x')dz = —T G(z,x')dz. (BS)

Moreover, g(y, x) —) 0 at y —) :00. At the condition BV(xp)>>l, the biggest contribution to

Eq. (B7) is given by the first integral in the right hand side:

 

 

 

G—“Kx ,0;s) s x'§(y,0) G“’(x ,x ;s) s x’§(y,x )
p z— (1 ; p P z— —L-d 9
W.) D! <p<y> y woe.) D! (M) (B )

where g is function g at f (x) = tp(x).
Using expression (A14) at 1:20 in the second definition in Eq. (B8), and integrating

by parts Eq. (B9), we obtain:

I65

 

 

;(1).(xp,0 S) zi if]?

dy ’3 , " ,
W(x) Dz Zia—(Biw‘fl‘ l +1420): <p(:)U‘p(w)dwl [RI—fl" "1

2
z—sr

(B 10)
The comparison of expressions (B 10) and (A17) shows that the function
(7 (x p,o; s) achieves its steady-state value when St, << 1.

To consider the function G~(xx s) it is sufficient to take into consideration the

P’ xP’
case f(x)=6(x). So, using expression (A18) at x0=0, x20, x'=Xp in the second definition in
(B8), we can obtain the first order correction for the Green's function under consideration

from the second expression in Eq. (B9) integrating by parts:

2

Gm(xp,xp;s)~__i-2 \
mp) “' D lily— mI‘PW‘y) WT) (lfL (”WWI - (811)

 

Changing the upper integration limit y for xp20 in the first term and regarding the

definition of 13p and ‘te for the case of a high activation potential barrier we have:

IG‘M ml
| rp(x,,)

From Eq. (BIO) we can conclude, that function G(xp,x xp,s) as well as G(xp ,0,s) reach

 

<Zsrcrp. (B12)

their steady-state values at STe<<1.

APPENDIX C. THE TIME-DEPENDENT GREEN‘S FUNCTION FOR
HIGH ACTIVATION BARRIER

166

APPENDIX C. THE TIIVIE-DEPENDENT GREEN'S F UNCION FOR
HIGH ACTIVATION BARRIER

The case of high activation barrier reactions for 1D motion was investigated in
Chaper V. There, it was shown that for any monotonically increasing potential V (x) (it

should grow monotonically starting from some point), the dynamical operator

5 a
L :D — '3V(x)— 3V0) Cl
0‘) x a e a e ( )

for high barrier transforms into
LHBOC) = 1),,[52/(29 + flV'(xp)] (C2)
where V’ denotes the derivative and x1) is the point of location of the reaction sink on the
surface, V (xp)> V(xo) and the only requirements are: i) fiV(xp)>>1 and ii) [3 V’ (xp)>> 1.
Therefore, the Green's functions given in Eqs. (11.10.17) can be defined through
the operator in Eq. (C2)

LHB(x)G(x, x’,t) = %G(x, x’,t), G(x, x', O) = 5(x - x'). (C3)

A straightforward evaluation of the above firnction yields the expression

[3( — '+/3V'(x )Dz)2
G(x,x’,t)=7;=exp - x x p x

47ert 4 prt

 

(C4)

The above Green's function was used to evaluate Eqs. (11.10.27).

APPENDIX D. THE STEADY-STATE GREEN'S FUNCTION FOR
HIGH ACTIVATION BARRIER

167

APPENDD( D. THE STEADY-STATE GREEN'S FUNCTION FOR
HIGH ACTIVATION BARRIER

The steady-state differential equation given in Eq. (II.11.7) was formulated in Ref.
30. Its integral analog is given by Eq. (H.11.8). If the reaction does not occur at a high

barrier then the Green’s function in Eq. (II.11.8) should be defined through the following

equation
L(x)G(x, x') = -5(x — x’) + f(x) (D1)
where fix) is the initial distribution. For the dynamical operator
1
z. — a _j 2
(x)—- D(x)<p(x)— dx —(-—(px ) (D )

the above equation is easy to solve and yields

- ,_ ‘ °° y
G(x’x)_(p(x)II,d D<y)<p (y__)_I, W I D<y)<y(y)I “2"” 9"“ x’IDoypty)
(D3)
with
A—IDU); (”Maw If(2)dz

Here a and b are the lower and upper boundaries respectively (ice are included).

For a high barrier where fiV(E)>>1, Eq. (D1) transforms into Eq. (11.1 1.9) and the
general Green's function in Eq. (D3) transforms to Eq. (11.11.14). (Note that the
dynamical operator changes to the form of Eq. (C2).)

APPENDIX E. EVALUATTION OF THE EFFECTIVE MFPT FOR THE
INTERMEDIATES ON TWO-DIMENSIONAL HARMONIC OSCILLATOR
POTENTIAL SURFACE

168

APPENDIX E. EVALUATION OF THE EFFECTIVE MFPT FOR THE
INTERMEDIATES ON TWO-DIMENSIONAL HARMONIC OSCILLATOR
POTENTIAL SURFACE

In order to evaluate the effective MFPT 1:12, first we represent Eq. (II. 14.7) in the

form:

n. =7<Lfldxde23(x,z>g(x,z) (121)

where

g(x,z) = El—de' dz (3;;(xx ;z,z' )W2,(r ,z' )(p,(r ,2 ).
-lr

Taking into account the form of the initial distribution on the (1) surface for I12, Eq.

(11.14.15), the original equation for the steady-state Green's functions, Eq. (1.4.12) can be

transformed into the following one in order to evaluate the desired function g(x,z):

 

(L. + L.)g(x,z) = 13.0w) - f23(x,2), L. = [,1 fig”; Wu), # = x,z (E2)

where IX and 152 are the coordinate relaxation times and fl; is given by Eq. (11.14.15) and

f3 is as follows:

f23(x>z) = %€P’6(q2x + z ‘ 92)exp(,6EAz3 “M362 + 22) / 2) (E3)

2
where q2= x1/21 and 92: c2 /zz.
Because the operators of motion in Eq. (E2) are independent, the Fourier

transformation is a convenient method for solution of this equation. As the result of

169

applying the Fourier transformation f( 77, v) = 31—dedzf(x,z)exp(—ixv —izn) to Eq.
7:

(E2) one can obtain a first order partial differential equation for the fiinction §( n, v):

 

5 v c? 2 i
[l—+— a +[n—+—Hg(n,V)= f (n,v)- f1(n,V) (E4)
1 on 1 av [31 rx
where
9(n+qV)_ (qn- W] -
’ z r =1,2
f.(n V): WI 1+q. 240w.) 1
where q,-= x,- /z,- and 9,: c,- /z,- are defined through the parameters of the sink lines Wij given
2
by Eqs. (11.14.11) and (11.14.12) (212 + x12 =k1, 222 + x22 =11}: and E421 =——9‘—-2—,
2(1 +611)
922
E423: ——2)-
2(1 + qzz)

The boundary conditions for the above differential equation come from the requirement for
the steady-state Green's function which originated function g, Eq. (1.4. 12a). Thus we have
to call for the restriction §(0,0) = O.
In order to solve Eq. (E2) one has to evaluate the principle integral of the
n 8

homogeneous part [—a—-+—-§-}r(ry,v)= O, which in this case is quite simple:
1) r

r = 17”" v’“ ". Substitution of variables 17 = r"v°‘ (where a = r2 / I) in differential
equation (E4) yields a first order differential equation

§;§(r,v)+[1+r ’V —]g(r V)=r(f2(r v)— fr(r v»/v (ES)

fl [30

This ordinary differential equation is easy to solve. Substituting back the variables

r = 17”“ v'“ " into the solution of Eq. (E5) we obtain:

n+2

Zfi

 

g(n, V) = r.exp(-n+ 1%ij —(goz(n, v x)- gor(77: v )0) (E6)

170

where

Q(nx“ +qz,vx) +“(nx +q,vx)\
1+ +q1 23(1‘l'q1)I,

 

30107, V x)— ‘ ex[DI-'19; =1,2- (E7)

It is easy to show that function (E6) can be utilized for calculation of the effective

MFPT in Eq. (El). Thus, the final expression for evaluation of 1712 obtains the form:

 

r12: 1I “[3 exp(fiEn) I dng(n,q2n)eXP(19 n) (E8)

Taking into account the form of fianction g from Eq. (E6) we arrive at the final expression
for the MFPT 112:

_\
CD

1

x I_ l30§b§(x) I ex I_ fieibfim I)
1'12 = Ize‘ma’II-iiE

2(1+ 6122 )a2(x) 2(1+ q§)a.(x)

Jams) ' Jame) (£9)

 

where
b1(x)=1-%f::—‘:§C: and b2(x):1_ 2:613:25“
(x+q1q2xa)2 x+q3xa
WH- (1+q3)(1+q§) and “g(x): 1’ (14,22):

Thus, Eq. (E9) gives the mean time for particles to reach a line on a two-dimensional
harmonic oscillator potential surface if they started moving from some other line. In the
specific case of 112 we had the starting line taken as 5(q2x+ z -92) and the sink line at
which the particles arrive was 6(q1x+ z -91), see Eqs. (11.14.11) and (11.14.12). In order

to get the other effective MFPT, 1:21 we have to switch the direction of motion, i.e. the

171
particles would have to start at 6(q2x+ 2 -9y line and finish on 5(q1x+ 2 ~91). Eq. (E9) can
still be used for evaluation of 123, but the indexes l and 2 have to be substituted for 2 and 1
respectively and the activation energy for the reaction (I)->(P), EA23 should be replaced

with the activation energy for the sink into the (R) well, EA21-

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