Sina Yeganeh and Mark A. Ratner- Effects of anharmonicity on nonadiabatic electron transfer: A model

8/3/2019 Sina Yeganeh and Mark A. Ratner- Effects of anharmonicity on nonadiabatic electron transfer: A model

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Effects of anharmonicity on nonadiabatic electron transfer: A modelSina Yeganeha and Mark A. Ratnerb

Department of Chemistry and Center for Nanofabrication and Molecular Self Assembly, NorthwesternUniversity, Evanston, Illinois 60208-3113

Received 28 September 2005; accepted 6 December 2005; published online 27 January 2006

The effect of anharmonicity in the intramolecular modes of a model system for exothermic

intramolecular nonadiabatic electron transfer is probed by examining the dependence of thetransition probability on the exoergicity. The Franck-Condon factor for the Morse potential is

written in terms of the Gauss hypergeometric function both for a ground initial state and for the

general case, and comparisons are made between the first-order perturbation theory results for

transition probability for harmonic and Morse oscillators. These results are verified with quantum

dynamical simulations using wave-packet propagations on a numerical grid. The

transition-probability expression incorporating a high-frequency quantum mode and low-frequency

medium mode is compared for Morse and harmonic oscillators in different temperature ranges and

with various coarse-graining treatments of the delta function from the Fermi golden rule expression.

We find that significant deviations from the harmonic approximation are expected for even

moderately anharmonic quantum modes at large values of exoergicity. The addition of a second

quantum mode of opposite displacement negates the anharmonic effect at small energy change, but

in the inverted regime a significantly flatter dependence on exoergicity is predicted for anharmonic

modes. 2006 American Institute of Physics. DOI: 10.1063/1.2162172

I. INTRODUCTION

The classical and semiclassical theories of electron

transfer ET developed by Marcus and others13 have beensuccessful in describing and predicting many aspects of the

ET rate constant including its dependence on driving force

the negative free-energy change, G0 Ref. 4 and ontemperature. However, the neglect of quantum-mechanical

tunneling causes the classical theory to disagree with experi-

mental results in significant ways. The Marcus expression

gives a Gaussian dependence on free energy, resulting in too

steep a drop-off in the inverted region. In addition, the theory

predicts that the rate constant approaches zero as temperature

falls to zero while experimental results have shown a nonva-

nishing rate constant at low temperatures.5

These discrepan-

cies can be dealt with by treating the problem quantum me-

chanically, proceeding from the golden rule result of first-

order perturbation theory and in the simplest approach

assuming a single coupled vibrational mode represented by

displaced harmonic potentials with same frequency.6

Several

authors have extended this treatment to include multiple

modes of differing frequency,7,8

and the physically important

case of change in both displacement and frequency has beenaddressed.

9Starting from the Kubo-Lax generating function

for multiple harmonic oscillators with displacement but no

frequency change, Jortner and co-workers1012

formulated

the ET rate constant in terms of a two-mode model with a

high-frequency mode, representing an average of the relevant

quantum modes, and a lower-frequency solvent mode. A very

important early result came in applying the single-mode limit

to the experimental data of DeVault and Chance5

in chroma-

tium, providing an explanation for the temperature depen-

dence of the ET rate.

While the harmonic approximation is generally appropri-

ate for describing the low-frequency solvent modes,13

it is

significantly unrealistic for the quantum modes at large val-

ues of as significantly anharmonic behavior is expected.

Several authors have accounted for anharmonicity using

Morse potentials14

in treatments of nonradiative decay in

crystals,15

proton transfer reactions,16,17

as well as nonadia-

batic ET.11,1821

In Sec. II of this paper we derive a more

compact form of the transition probability for nonadiabatic

ET with anharmonic quantum modes, making use of a finite

sum form of the Gauss hypergeometric functions.22

In Sec.

III the results of computations with this closed form are

compared with the corresponding harmonic results, and the

golden rule results are compared with grid-based wave-

packet propagations using the methods of Kosloff and co-

workers. Finally, the two-mode case of anharmonic quantum

mode and a harmonic low-frequency phonon mode is con-

sidered in Sec. IV, and comparisons are made with the har-

monic results from the treatment by Jortner.12

Finally, the

expected effects of significantly anharmonic modes on theET rate constant are discussed for parameters of interest in

experimental systems such as the exoergicity in the optimal

regime. Regions are identified in which the anharmonic ef-

fect on the rate constant is expected to be largest.

II. GOLDEN RULE FORMULATION: MORSEPOTENTIALS

The starting point for the nonadiabatic quantum-

mechanical treatments is the Fermi golden rule result,23

aElectronic mail: [email protected]

bElectronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 124, 044108 2006

0021-9606/2006/1244 /044108/9/$23.00 2006 American Institute of Physics124, 044108-1

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WRP =2

PR

2EP , 1

where WRP, the transition probability per unit time from re-

actants to products, is given in terms of the product density

of states DOS, EP, and the matrix element over the re-actant R and product P wave functions. Making theCondon approximation

24that the electronic coupling is inde-

pendent of the nuclear motion, the transition rate can be writ-ten as

25

WRP =2

HRP

2rp2EP, 2

where the electronic coupling matrix element HRP has been

factored out and is assumed to be independent of separation,

leaving the Franck-Condon FC overlap integral over reac-tant and product vibrational states, r and p, respectively.In this single-mode expression, energy conservation requires

that the product vibrational state be given by Enp =+Enr, where En is the energy of the nth vibrational

eigenstate. The energy conservation requirement can be re-laxed in the single-mode case by introducing coarse graining,

assuming a vibrational coupling to environmental modes.12,26

For the multimode case, a more general form is used where

the delta function allows contributions only from energy-

conserving transitions primed and unprimed variables referto the final and initial states, respectively,

WRP =2

HRP

2k

nk=0

n

k=0

Ink,nk

k2

eEnkk

/kBT

Qk

k

En

k

k Enk

k . 3Here the product is over the k different modes, Qk is the

partition function for the kth mode, Ink,nk

kis the FC overlap

integral given by nkk

nk

k , and Enkk and E

nk

kare the vibra-

tional energies. The summations over nk and nk in Eq. 3represent the possible transitions between initial and final

vibrational levels weighted by the fractional population in

each reactant initial vibrational level, as given by

eEnkk

/kBT/Qk. The delta function then picks out combinations

of nk and nk which result in energy-conserving transitions.

As noted by Jortner,12

this delta function should be replaced

by a Lorentzian to account for energy-time uncertainty

broadening resulting from finite lifetimes. Kubo7

and Lax8

were able to reduce Eq. 3 to give a generating function forthe FC factors for multiple harmonic oscillators.

For our treatment of anharmonicity in the quantum

mode, we consider the eigenfunctions of two Morse poten-

tials centered at r0 and r0,

Vr = D1 e rr02, Vr = D1 e rr02, 4

and proceeding in the usual fashion, we take the ranges = and strengths D =D of the potentials to be the same.Unlike the harmonic case, however, in which analytic forms

can be written for the FC factor of oscillators of different

frequencies, the Morse result can only be calculated analyti-

cally when =. The eigenfunctions of this anharmonic

potential14

can be written as27

nN,;r r0 = Mn,NNw + wN/2n

exp N+ 12

wLnN2nNw + w , 5where Ln

k is the generalized Laguerre polynomial and

w = err0, Mn,N = j=0nN 2n + j

j! 1/2

,

6

N= 22D

1 .

The parameter N is physically important because N/ 2 gives

the highest bound level in the Morse potential. The FC over-

lap integral,

In,n = nN,;r r0nN,;r r0dr, 7was first given for Morse eigenfunctions by Fraser and

Jarmain.28,29

The expression simplifies as N=N for our case

of identical strengths and ranges, and we can write in the

notation of Iachello and Ibrahim,27

In,n = Mn,NMn,NN/2n 2

1 + Nnn

m=0

n

m=0

n 1m+mm ! m!

N nn m

N nn m

m

2

1 + m+m

N n n

+ m + m

, 8

with = er0r0.

We now proceed to simplify Eq. 8 in the low-temperature limit where the initial state is entirely in its

ground vibrational level. In keeping with our notation we set

n =0, yielding

I0,n = M0,NMn,NN/2 2

1 + Nn

m=0

n 1mm!

N nn m

2

1 +

m

N n + m

. 9

The binomial coefficient can be expanded in its gamma func-

tion representation, and some grouping produces

I0,n = M0,NMn,NN/2 2

1 + NnN n + 1

m=0

n 2/1 + mm!

N n + m

n m + 1N 2n + m + 1 . 10

044108-2 S. Yeganeh and M. A. Ratner J. Chem. Phys. 124, 044108 2006



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The summation can be rewritten using a finite sum form of

the Gauss 2F1 hypergeometric series that holds when n is a

positive integer,30

2F1 n,b;c ;z = m=0

n nmbm

cm

zm

m!, 11

where the Pochhammer symbol is used,

ak=a + k/a.31 After some manipulations of Eq. 10to obtain an expression in the form of Eq. 11, the summa-tion can be written as

m=0

n 2/1 + m

m!

N n + m

n m + 1N 2n + m + 1

=N n

n + 1N 2n + 1

2F1 n,N n;N 2n + 1 ; 21 +

. 12The normalization constants M0,N and Mn,N can be simpli-

fied as well, with the former given by N1/2, and the latterby

32

Mn,N = N n + 1N 2nn + 11/2

. 13

Combining terms, we write the FC overlap integral for

eigenfunctions of two displaced Morses with identical

strengths and ranges as

I0,n

= N/2

2

1 +

Nn N n

N 2n + 1

N 2nN n + 1n + 1N

1/2

2F1 n,N n;N 2n + 1 ; 21 +

. 14Equation 14 was compared for a number of parameterswith the numerical method of Lopez et al.,

33and the values

obtained agreed.

For the multimode case at sufficiently low temperatures

such that the quantum mode is not thermally populated, all

that remains is to substitute I0,n in Eq. 3 for the appropriatemode. For the single-mode case, the simpler form of Eq. 2can be used after finding an expression for n and a suitable

form for the product DOS. From the energy eigenvalues of

the Morse potential, it can be shown that for a transition

from the n = 0 initial state, n is given by energy conservation

as

n = 1

2 2D

D +

1

4, 15

where =2D/ , with the reduced mass for the os-cillator. We use the classical form of the DOS Ref. 34 by

taking n continuous and differentiating with respect to the

energy E,35,36

E =dn

dE= 2

D E1 . 16

Thus, for a single anharmonic mode in the low-temperature

regime, the transition-probability rate is given by

WRP = 2

HRP2I0,n2

2

D 1

. 17

III. NUMERICAL SOLUTION OF THE SCHRDINGEREQUATION

In order to investigate the results of the previous section,

one-dimensional wavefunctions were propagated on a nu-

merical Cartesian grid with equally spaced points. The meth-

ods are only outlined here as numerous reviews have been

written on the subject.3739

The Schrdinger equation given

by

i

r

,tt

= Hr,t 18

has the formal solution

r,t = eiH

t/r,0 , 19

where the Hamiltonian is written in the usual way, H = T+ V.

The potential operator V is local in coordinate space, and its

action is given simply by Vxx. The operation in the

kinetic term T= 2/2m2/x2 can be calculated with theFourier method

40,41by a discrete fast Fourier transform

FFT of the wave function into momentum space followedby multiplication by K

i

2, where Ki is the wave number in

the frequency domain, and finally an inverse FFT back into

coordinate space. With the use of the FFT algorithm, the

calculation of T scales efficiently as ONlog N, where N isthe number of grid points.

39The exponential in Eq. 19 is

then expanded using a global propagator through the New-

tonian interpolation method,38,42

where the sampling points

are taken to be the zeros of the Chebychev polynomials.

The initial wave function was prepared as the ground

eigenstate of the initial potential as obtained through propa-

gation in imaginary time.43

As shown in Fig. 1, two states,

harmonic or Morse, were used and the dependence of the

transition-probability rate on the exoergicity was probed by

lowering the final potential in small increments of . Thepopulation in the initial state was monitored over the course

of the propagation and then fit to an exponential, i2t

=Aect, where c was taken to be the transition-probability

rate. Since the rate expressions arise from the first-order per-

turbation theory result, the population at short times was

used. By applying the single-mode Morse treatment from

Sec. II and the analogous result for harmonic oscillators,44,45

the perturbation theory results are compared with the nu-

merical propagations Fig. 2. The numerical propagationpredicts a smaller transition probability than the golden rule

result at small values of . This numerical calculation is

somewhat artificial since a dephasing process is implicit in

044108-3 Nonadiabatic electron transfer J. Chem. Phys. 124, 044108 2006



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smoothing the delta function from Eq. 3. When becomessubstantially greater than c, the Morse results from these

two treatments agree as the density of states becomes large.

The chosen potentials for this quantum mode correspond

to a stretch with the final state at larger displacement than the

initial state. While the symmetry of the harmonic potential

makes the distinction between a stretching or compression

mode irrelevant in calculating the FC factor and transition

probability, this distinction has an important effect for a

Morse potential.11

The plot for the Morse oscillator in this

case exhibits a flatter dependence on exoergicity and is lower

valued than in the harmonic case, as is expected from the FC

factor with r0r0. For a compression mode with r0r0 the

opposite would be true as the plot would be more sharply

peaked and higher valued than the corresponding harmonic,

as seen in Fig. 3. In addition, the computed transition prob-

ability stops abruptly at D for the Morse potential, as

energy conservation requires a transition from a bound to

unbound state when approaches D. The expression in Eq.

14 holds only for bound states, when n N/ 2. Beyondthis point, the FC factor for a bound/unbound state of the

anharmonic oscillator is required.

IV. TWO-MODE TREATMENT

Although the single-mode treatment was found to be suf-ficient in treating some situations such as the chromatium

data,12

it is often the case that the relevant modes in a system

can be separated into high-frequency quantum modes and a

collection of low-frequency phonon modes. The low-

frequency modes can be treated accurately as an average

FIG. 1. Initial V and final V potentials are shown for a harmonic and b Morse cases with parameters shown. All values are in a.u. unless otherwisestated. The electronic coupling matrix element HRP was taken as 0.02 eV. The reduced mass c was 2000 a .u. The potentials were constructed such that the

frequencies, Morse2D/c and Harmonic =kH/c, would be the same and equal to c 1870 cm1. The Morse potential has 26 bound states. Thereorganization energies are i

H=0.11 eV and iM=0.18 eV. The ground-state eigenfunction for each case is also shown - - -. The exoergicity is

represented as the bottom-bottom vertical separation of the potentials.

FIG. 2. Golden rule results and wave-packet propagations for the transition

probability as a function of for a single-quantum mode from Fig. 1.

Harmonic potentials: golden rule result and wave packet ; Morsepotentials: golden rule result - - - and wave packet . Propagationparameters are given: nr is the number of spatial grid points, dr is the

distance between grid points, nt is the number of time steps taken for each

propagation, dt is the length of a time step, and nint is the number of inter-

polation points taken in the Newton scheme.

FIG. 3. Transition probability for single harmonic mode , Morse stretch-ing mode - - - with r0 r0 =1.5, and Morse compression mode withr0 r0 =1.5 potentials from Fig. 1 are plotted.




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over many modes of similar frequencies, but incorporating

the effect of many quantum modes explicitly changes the

transition probability significantly.46

We focus our attention

on a system with only one relevant quantum mode with fre-

quency c and one collective solvent phonon mode with fre-

quency s. For harmonic oscillators in both modes, Jortner12

has identified several limiting cases in which simpler forms

for the ET transition probability can be written, and we pro-

ceed analogously. Ulstrup and Jortner11

have shown that an-harmonicity and frequency change can be treated by simply

substituting the appropriate FC overlap integrals and energy

eigenvalues into the harmonic form.

a For the zero-temperature limit, when kTsc,

12

WRPHarmonic =

2

HRP2

sm=0

ScmeScm!

SspmeSspm!

, 20where pm is taken as mc/s and Sc and Ss arethe Huang-Rhys factors for the quantum and solvent modes,

respectively, given in terms of the inner- and outer-sphere

reorganization energies,

i and

o, by Sc =

i/c and Ss=o/s with i =cc

2r0 r02 / 2 and o =ss

2z0z0

2 /2. In the summation, the two groups of terms in pa-

rentheses are the squared FC overlap integrals for the quan-

tum and solvent modes, respectively. We can incorporate the

effect of anharmonicity in the quantum mode by

substituting11

I0,m from Eq. 14 to get

WRPMorse =

2

HRP2

sm=0

N/2

I0,m2SspmeSs

pm! , 21

where pm is Emm +Em0/s and Emn is thenth Morse energy eigenvalue. This result can be compared

with that obtained from the zero-temperature limit of Eq. 3with two modes,

WRPMorse =

2

HRP

2m=0

N/2

I0,m2

n=0

Ssn

eSs

n!

LEmm Em0 + n s , 22

where the delta function has been replaced by the Lorentzian,

xLx =//x2 +2.12,47 Note that for the Morse thesummation over m is only over the bound levels formally,

the full spectrum of possible transitions should be repre-

sented by the sum over bound levels and an integral over the

dissociative states, but the latter will typically be small in

comparison and is ignored throughout. The results of bothexpressions are shown in Fig. 4. The substitution of Morse

FC factors and eigenvalues into Eq. 20 Jortners12 Eq. 3gives the same result as the more general form of Eq. 22;the similarity of the two results can be understood by rewrit-

ing the delta function,

Emm Em0 + n s

=1

s Emm + Em0

s n , 23

where the delta function is nonzero only when n =pm, andso the summation over m and n is subsumed into a single

sum over coupled values of m and pm. The rounding ofpm or pm to the nearest integer gives width to the deltafunction,

48as does the substitution of the Lorentzian, and so

the results are nearly identical.

The oscillatory structure in Fig. 4 is an interesting fea-

ture that will always be present at zero temperature when

there are two modes of significantly different frequencies and

io.11

For the harmonic oscillator, the peaks occur at in-

teger multiples of c shifted by o. The Morse oscillator

plot shows similar spacing at small values of , but for

larger values of exoergicity the peaks become more closely

spaced together, corresponding to the DOS approaching a

continuum at the dissociative limit. In addition, as expected

from the stretching nature of the quantum mode, the anhar-

monic system has a smaller transition rate near the optimal

regime and broader dependence on the exoergicity. At higher

temperatures, broadening of the peaks is expected as excited

states in the solvent contribute more allowed combinations

of m and n which are energy conserving.

b For the low-temperature regime with skTc, the solvent mode is treated classically, resulting in

the following expression for harmonic oscillators in both

modes:12

WRPHarmonic =

2

HRP

2 m=0

exp o m c24okT

eScScm

m!4okT1/2. 24

The corresponding expression for a Morse oscillator can be

written once again by a simple substitution of the appropriate

FC overlap integrals and energies,

FIG. 4. The zero-temperature transition probability for two modes with

values from Fig. 1 for harmonic and Morse oscillators in the quantum mode

and with the solvent mode parameters shown. The solvent frequency s is

approximately 30 cm1 and o =0.006 eV. The results of Eq. 20 andEq. 21 - - - as well as with a Lorentzian, Eq. 22, for a harmonic and Morse mode are plotted. For the Lorentzian, =1105 hartree.




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WRPMorse =

2

HRP

2

m=0

N/2

exp o Emm + Em024okT

I0,m

24okT1/2. 25

For the Morse result with a Lorentzian and the exact form for

the solvent modes, we make use of Eq. 3:

WRPMorse =

2

HRP

2

m=0

N/2

I0,m2

n=0

n=0

In,nH 2

1 es/kT

esn/kT

LEmm Em0 + n n s , 26

where In,n

His the FC overlap integral for the harmonic oscil-

lators in the solvent mode given by45,49

In,n

H 2 = n ! n ! SsnneSs

j=0

n Ss

j

j ! n j ! n n + j!2 . 27

Figure 5 shows the result for the transition probability at

150 K. The quantum and solvent modes have characteristic

temperatures, Tc = /k, of 2700 and 43 K, respectively, so

in this regime the low-temperature approximation for quan-

tum modes and classical approximation for the solvent

modes are appropriate. Although broader than in Fig. 4, os-

cillations are still visible; Ulstrup and Jortner placed the con-

dition on the presence of these peaks as 2okTc,which is satisfied for the parameters in Fig. 5 with the small

value of o chosen. The low-temperature regime described

here is valid for many temperatures of interest in biological

systems.

c For higher temperatures such that skT c,the full form of the Morse FC overlap integral In,n is re-

quired as the molecular modes become thermally excited.

This general equation can be obtained by similar manipula-

tions of Eq. 8 to give

In,n =N/2n

N 2n + 1 2

1 + Nnn N 2nN 2nn + 1N n + 1N n + 1

n + 11/2

m=0

n 21 +

m N n n + mm + 1n m + 1N 2n + m + 1 2

F1 n,N n n + m;N 2n + 1 ; 21 +

. 28

Equipped with this result, transition rates at any temperature

can be calculated within the classical approximation for thesolvent modes. First, for comparison, the harmonic oscillator

expression can be reduced to12,49

WRPHarmonic =

2

HRP

2exp Sc2v+ 1

4okT

m=

ecm/2kTIm2Scvv+ 1

exp o m c24okT

. 29

where v= expc/kT 11 and Im is the modified Bessel

function. We write the Morse ET transition probability,

WRPMorse =

2

HRP

24okT1/2

m=0

N/2

m=0

N/2

exp o Emm + Emm24okT

Im,m

2eEmm/kT

Qm, 30

where Qm is the Morse partition function, which we take as

FIG. 5. The transition probability in the low-temperature regime. The same

potentials as in Fig. 4, at a temperature of 150 K. The classical approxima-

tion for the solvent modes with Eq. 24 for the harmonic quantummode or Eq. 25 - - - for a Morse quantum mode as well as the full sumover both modes using Eq. 3 for the harmonic and Morse mode are plotted. =2.5105 hartree.




7/9

j=0N/2expEmj/kT. For the Lorentzian result, we use the

full two-mode form of Eq. 3,

WRPMorse =

2

HRP

2m=0

N/2

m=0

N/2

Im,m2

eEmm/kT

Qm

n=0

n=0

In,n

H 21 es/kT

esn/kT

LEmm Emm + n n s . 31

The results of these expressions at 300 K are given in Fig. 6.

There is significantly more broadening at this temperature as

is expected from the 1/T temperature dependence in the

Gaussian.

Examining Figs. 5 and 6, we note the following things.

In Fig. 5, there is good agreement between the classical ap-

proximation for the solvent modes and the fully quantum-

mechanical treatment of the solvent modes. In Fig. 6, there is

also good agreement between the two forms; at this higher

temperature, the numerical calculation of Eq. 27 for thesolvent modes is avoided for large values of n and n by

using the recursion relations for the harmonic oscillator FC

factors.45,50

V. CONCLUSIONS

Applying the theory of nonradiative multiphonon transi-

tions to nonadiabatic ET has yielded forms for the transition

probability that incorporate anharmonicity in the molecular

mode. The anharmonicity is treated by substituting the

Morse FC overlap integral and energies into the analogous

relations for harmonic oscillators. At low temperatures and

small values of o, interesting oscillatory behavior in the

transition probability as a function of exoergicity is exhib-ited. Regardless of whether the mode corresponds to a stretch

or compression, significant deviations are seen from the har-

monic result for a relatively anharmonic mode. For increas-

ing values of, as the density of states for the Morse oscil-

lator rapidly increases, the peaks will no longer be equally

separated by approximately c but will move closer to each

other. As other authors have pointed out,11,18

in a system with

two anharmonic quantum modes with opposite displace-

ments, a stretching and compression mode, the changes in

the FC factors from the harmonic values will roughly cancel

and anharmonic effects will disappear. However, at low tem-

peratures and small o such that the oscillatory structure dis-cussed in Sec. IV is present, anharmonic effects will be vis-

ible through the nonuniform spacing of peaks. We are not

aware, however, of a condensed-phase experimental system

which possesses these characteristics.

Turning our attention to a more realistic system with

o =0.16 eV at 150 K, the transition probability for a har-

monic quantum mode is contrasted with stretching and com-

pression anharmonic modes in Fig. 7. One significant feature

is in the inverted regime: the transition probability for the

Morse compression mode falls off steeply as a Gaussian aswould be predicted from a classical harmonic oscillator treat-

ment, while the harmonic oscillator and Morse stretchingmode are described by exponential decay. The sharp drop

leads to an ET rate that is several orders of magnitude

smaller than for a harmonic or stretching mode, even at

=0.5 eV where the inverted regime is just beginning. In ad-

dition, the stretching mode has a significantly broader opti-

mal regime than the harmonic case, and in the inverted re-

gime this results in a rate increase of an order of magnitude

relative to the harmonic case. At this larger value of o, all

oscillatory structure is smoothed outfor distinct peaks a

temperature of 10 K is required. For this single anharmonic

mode, we see that the optimal exoergicity opt can be or-

dered as: optcompression

optstretch.

51We cannot in general predict

the position of

opt

harmonic

relative to the Morse optimal exoer-gicities because this relation depends on the magnitude of the

horizontal displacement r0 r0. For harmonic oscillators athigh temperatures for which Jortners two-mode form con-

verges to the Marcus expression, the optimal regime is given

FIG. 6. The transition probability with excited initial states in the quantum

modes at 300 K same values as Fig. 4. The full form of Eq. 3 using theappropriate partition functions is plotted for harmonic and Morse potentials =1104 hartree as well as Eq. 29 for the harmonic and Eq. 30 for the Morse - - - mode.

FIG. 7. Logarithmic plot of transition probability in the low-temperature

regime at 150 K. Same as Fig. 5 with o =0.16 eV. Using Eqs. 24 and25, a harmonic mode , Morse stretching mode - - -, and Morsecompression mode are plotted.




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by optharmonic =o +i

H with iH= kHr0 r0

2 /2. This relation-

ship can be used to approximate the optimal regime at lower

temperatures. The Morse reorganization energy is given by

iM=D1 er0r02, but opt

Morse o +iM is a poor approxi-

mation to the actual optimal region. This is due to the fact

that the governing factor for the optimal region is the exoer-

gicity at which energy conservation requires the maximum

FC factor within o, and this will not necessarily peak nearthe activationless exoergicity given by i. While the har-

monic oscillator FC factor does peak close to iH, the maxi-

mum in the Morse FC overlap integral is not well approxi-

mated by

i

M

. An alternative prediction of the location of theoptimal region is not possible without specific parameters.

We can predict that the ET transition probability at the opti-

mal regime for the Morse compression mode will decrease

much more gradually than the harmonic and the stretching

mode as a function of increasing displacement r0 r0, sug-gesting a possible role for anharmonic compression modes in

nonadiabatic ET reactions with large inner-sphere reorgani-

zation.

In Fig. 8 the transition probability for a system with two

quantum modes and one solvent mode is plotted; the two

Morse quantum modes are chosen to be stretching and com-

pression modes, while the two harmonic modes are identical.

The combination of compression and stretch causes theMorse and harmonic results to be nearly identical at low

values of, but at higher exoergicities the Morse result is

dominated by the stretching mode, and a more gradual de-

crease in transition rate is seen. Even at the higher o, anhar-

monic effects are noticeable. This effect is even more pro-

nounced at larger values of i.

It is interesting to consider the case of displaced-

distorted harmonic oscillators in the quantum mode where

both displacement and frequency change are considered. Us-

ing the appropriate FC factors,52

the effect of frequency

change can be compared to the anharmonic effect, but while

frequency change does slightly shift the W vs plot to

higher or lower exoergicities depending on the sign of RP, the shape of the relation is generally the same. Thus,displaced-distorted harmonic oscillators cannot account for

the anharmonic deviations of broadening or narrowing for

Morse stretching or compression modes discussed earlier.

In this paper, we have written the FC factors for Morse

oscillators in condensed form using the hypergeometric 2F1function, and the expected deviations from harmonic behav-

ior for anharmonic stretching and compression modes havebeen discussed. We predict that significant differences in the

electron transfer rate constant are expected, especially in the

inverted regime. As seen in the logarithmic plot in Fig. 7, the

Morse stretching and compression modes will differ by many

orders of magnitude from the harmonic result. In addition, at

large values ofi the Morse compression mode is expected

to have a much larger transition rate than the harmonic and

Morse stretching modes. It is expected that these anharmonic

effects will be significant and experimentally observable in a

properly constructed system.

ACKNOWLEDGMENT

The authors would like to thank Abraham Nitzan for

helpful discussions. This work was supported by the Chem-

istry Division of the NSF and the MURI/DURINT program

of the DoD.

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9/9

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compressionopt

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Sina Yeganeh and Mark A. Ratner- Effects of anharmonicity on nonadiabatic electron transfer: A model

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