Semiclassical instanton formulation of Marcus–Levich–Jortner theoryEric R. Hellera) and Jeremy O. Richardsonb)

Laboratory of Physical Chemistry, ETH Zurich, 8093 Zurich, Switzerland

(Dated: 13 May 2020)

Marcus–Levich–Jortner (MLJ) theory is one of the most commonly used methods for including nuclear quan-tum effects into the calculation of electron-transfer rates and for interpreting experimental data. It dividesthe molecular problem into a subsystem treated quantum-mechanically by Fermi’s golden rule and a solventbath treated by classical Marcus theory. As an extension of this idea, we here present a “reduced” semiclas-sical instanton theory, which is a multiscale method for simulating quantum tunnelling of the subsystem inmolecular detail in the presence of a harmonic bath. We demonstrate that instanton theory is typically signif-icantly more accurate than the cumulant expansion or the semiclassical Franck–Condon sum, which can giveorders-of-magnitude errors and in general do not obey detailed balance. As opposed to MLJ theory, whichis based on wavefunctions, instanton theory is based on path integrals and thus does not require solutions ofthe Schrodinger equation, nor even global knowledge of the ground- and excited-state potentials within thesubsystem. It can thus be efficiently applied to complex, anharmonic multidimensional subsystems withoutmaking further approximations. In addition to predicting accurate rates, instanton theory gives a high levelof insight into the reaction mechanism by locating the dominant tunnelling pathway as well as providinginformation on the reactant and product vibrational states involved in the reaction and the activation energyin the bath similarly to what would be found with MLJ theory.

I. INTRODUCTION

The theoretical study of electron-transfer is of essen-tial importance and relevance not only because these re-actions are a key step in many chemical and biologicalprocesses but also because the methods developed to dealwith them can be applied in many other scenarios rang-ing far beyond their original scope. This follows from thefact that electron-transfer reactions are just one exam-ple of the more general set of curve-crossing problems.Hence, contributions to the understanding of electron-transfer reactions have been made with various motiva-tions including electrochemistry, molecular spectroscopy,polaron transport as well as more general atom-transferreactions, which led to different ways of tackling the prob-lem from classical dielectric continuum theory to a fullquantum molecular picture.1,2

Inspired by earlier work,3 Marcus based his theory ofelectron transfer, for which he later won the Nobel prizein 1992,4 first in terms of a dielectric solvent continuum5

and later on a classical statistical mechanical descriptionof the solvent.6 To this day, Marcus theory is probablythe most commonly applied approach for the descrip-tion of electron-transfer reactions and initiated tremen-dous development involving electron and hole transferbetween atoms, molecules or even proteins, in the con-densed phase as well as at interfaces.7,8 Hence, his find-ings had and still have an enormous impact on a mul-titude of scientific disciplines comprising solution chem-istry, solid-state physics as well as biological processes.9

One of the essential insights from Marcus’ classicaltheory was the prediction of the so called “inverted

a)Electronic mail: eric.heller@phys.chem.ethz.chb)Electronic mail: jeremy.richardson@phys.chem.ethz.ch

regime”,6 the existence of which was later confirmed byexperiment,10 where the rate decreases as the thermody-namic driving force grows larger than the reorganizationenergy. Soon, however, it was realized by theory and ex-periment that the neglect of nuclear quantum effects inMarcus theory can lead to dramatic errors of several or-ders of magnitude in the rate, especially in the invertedregime.11–13

Based on the connection to spectroscopy and solid-state nonradiative processes Levich and coworkers putthe theory onto a rigorous quantum-mechanical ba-sis and introduced a quantum statistical mechanicaldescription of outer sphere electron transfer.14 Thiswas done by employing Fermi’s golden rule15 formulafor the quantum transition rate, which is obtained asthe nonadiabatic (weak-coupling) limit from perturba-tion theory.16,17 However, because outer-sphere electron-transfer is typically dominated by the low-frequency sol-vent modes, the resulting quantum effects are rathersmall.

Several years later, crucial advancements were madein particular by Jortner and coworkers by explicitlytaking the reorganization of the inner sphere intoaccount.1,2,18–22 As opposed to the solvent, the innersphere often exhibits intra- and intermolecular rearrange-ments associated with high-frequency vibrational modeswhich are therefore subject to substantial quantum ef-fects. Hence, they treated the inner sphere quantum-mechanically using Fermi’s golden rule while keepingthe classical approximation for the solvent bath. Theresulting Marcus–Levich–Jortner (MLJ) theory consti-tuted a considerable progress for the whole field, as it wasthe first rigorously derived method able to describe nu-clear quantum effects in electron-transfer reactions whichwas valid throughout virtually the whole temperaturerange.23 Thus, the method poses a vital step towards

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the goal of establishing a unified description of electrontransfer in the various fields mentioned above across di-verse time and temperature scales.8,24,25

MLJ theory is broadly applied to the prediction andexplanation of charge-carrier mobilities26 often with theobjective to give a guideline for the synthetic study andreasonable design of high-performance semiconductorsthat can be applied in organic photovoltaics.27–30 Theapplication to large systems can be facilitated by us-ing the theory in conjunction with density-functionaltheory.31 Furthermore it can be applied in the studyof molecular junctions,32 photonics,33 polaritons34 andpolarons35 as well as for the description of spin tran-sitions and phosphorescence.36–40 Besides these techno-logical disciplines, it is also frequently applied for theunderstanding of complex chemistry,41 electron trans-fer in supermolecules42 and biochemistry,43–45 chargetransfer in DNA46–48 and photosynthesis.49,50 As tun-nelling is especially prevalent in the Marcus invertedregime,51 MLJ theory is of particular interest in thestudy of molecular electron-transfer reactions whichare strongly exothermic52–55 or which are initiated byphotoexcitation.56,57

The MLJ description of quantum tunnelling, whichwill be extensively discussed in the later sections, is un-derstood by shifting the Marcus parabolas (free energyalong the bath coordinates) by the quantized energy lev-els of the inner sphere. The rate therefore consists of con-tributions from multiple vibrational channels weightedaccording to a thermal distribution.58 This interpreta-tion appears quite different from the standard pictureof tunnelling in which a particle penetrates a potentialenergy barrier with an energy smaller than the barrierheight. The main disadvantage of the approach is that itrequires wavefunction solutions of the time-independentSchrodinger equation in order to compute the energy lev-els and Franck–Condon overlaps, which severely limits itsusefulness for the description of realistic, anharmonic andmultidimensional systems.

A number of alternative methods which can be used forthe study of electron-transfer reactions and other golden-rule processes59–75 are based on Feynman’s path-integraldescription of quantum mechanics76,77 rather than onwave mechanics. From this set, semiclassical golden-ruleinstanton theory78,79 in particular bears multiple appeal-ing features. Without any prior knowledge about theanalytic shape of the potential, it locates the “instan-ton” in the full-dimensional configuration space of thesystem, which can be thought of as the optimal tun-neling pathway,80–83 and therefore provides direct in-sight into the reaction mechanism. Furthermore themethod was recently extended towards the Marcus in-verted regime,84 which otherwise typically poses a prob-lem for imaginary-time path-integral approaches,85 al-though some extrapolation techniques have been usedsuccessfully to avoid this problem in other methods.67 Byemploying a ring-polymer discretization to the paths,86

the instanton method is able to simulate tunnelling in

multidimensional, anharmonic systems in a computation-ally efficient way and is ideally suited for calculationsin conjunction with high-level electronic structure meth-ods just as in the standard adiabatic formulation of thetheory.81,87–94 Golden-rule instanton theory constitutes asemiclassical path-integral formulation of Fermi’s goldenrule and hence has the potential to be applied in a mul-titude of different fields, just as the golden rule itself.

One of the great strengths of instanton theory is itsfull-dimensional formulation of tunnelling such that itdoes not rely on an a priori choice of the reaction coor-dinate. However, for many relevant reactions, especiallyin the condensed phase, even if one does not know theexact tunnelling path, one already has a good idea ofwhich part of the system under investigation has to beconsidered explicitly and which part can be accountedfor on a coarser level. It is this same separation into aninner and outer sphere which was the cornerstone of MLJtheory. Therefore in this paper, the formalism for a “re-duced” semiclassical golden-rule instanton theory will belaid out, which describes tunnelling within the modes ofthe inner sphere under the implicit influence of either aclassical or quantum harmonic bath. The presence of thebath affects the equations of motion of the inner sphereand renders the resulting reduced instanton non-energy-conserving due to energy exchange between inner andouter sphere. This is analogous to the reduced densitymatrix formalism employed in the study of open quan-tum systems. The resulting instanton picture preservesthe convenient interpretation of quantum tunnelling as aparticle travelling in the classically forbidden region be-low the barrier.

Although the formalisms seem at first glance rather dif-ferent, we will draw a connection between the MLJ andinstanton theories by deriving them both from a commonexpression. In doing so, it will be shown clearly that theinstanton approximation is fundamentally different fromother approximations such as the broadly applied cumu-lant expansion method95 and the semiclassical Franck–Condon sum.11,96 Numerical results demonstrate that in-stanton theory is very accurate over a range of systemsincluding anharmonic modes where these alternative ap-proximations break down.

Some rate theories have the advantage that they arebased on expressions which are simple enough that onecan easily see their dependence on certain parametersand thus gain insight into the behaviour of differentsystems.2,97 We will argue that instanton theory allowsfor a well-balanced combination of easily attainable in-sights, as well providing a realistic molecular simulation.Even when applied to complex anharmonic multidimen-sional potentials, the method uniquely identifies an op-timal tunnelling pathway which provides a simple one-dimensional picture of the reaction, highlighting whichmodes are involved in the tunnelling. In addition to this,the instanton can be analysed to obtain information onthe energies of the initial and final states of the systembefore and after the electron-transfer event, similar to

3

what is computed in MLJ theory as we will show.

II. GOLDEN-RULE RATE

The total Hamiltonian which describes electron trans-fer between a reactant |0〉 and product |1〉 electronic stateis defined by98

H = H0 |0〉〈0|+ (H1 − ε) |1〉〈1|+ ∆(|0〉〈1|+ |1〉〈0|

),

(1)

where the electronic interaction between the states isgiven by the nonadiabatic coupling ∆. Throughout thiswork the electronic coupling is taken to be constant,but the generalization to position-dependent couplings isfairly straight forward.78,99 Furthermore the coupling isassumed to be very weak such that the rates occur in thegolden-rule limit, i.e. ∆ → 0, which is typically the casein electron-transfer reactions.98 A driving force, ε, hasbeen included explicitly in the total Hamiltonian, whichcould describe an internal energy bias or the effect of anexternal field. It is kept separate here for clarity but itcould of course be simply absorbed into the definition ofH1.

We are interested in studying problems which can besubdivided into an inner sphere, whose molecular struc-tural characteristics will be explicitly taken into account,and an outer sphere, which typically includes the solventdegrees of freedom and will be treated as an effectiveharmonic environment characterized by its spectral den-sity. In the language of open quantum systems, these arecalled subsystem and bath and are here taken to be un-coupled to each other,100 although there is of course stillsome coupling through the nonadiabatic terms in Eq. (1).Hence, the full nuclear Hamiltonian for electronic state|n〉 can be written as

Hn = Hsn + Hb

n, (2)

where

Hsn =

d∑k=1

p2k

2m+ V s

n(q), (3a)

Hbn =

D∑j=1

P 2j

2M+ V b

n (Q). (3b)

The subsystem Hamiltonians, Hsn, only depend on the co-

ordinates q = (q1, . . . , qd) and their conjugate momenta

p = (p1, . . . , pd), while the bath Hamiltonians, Hbn, are

solely a function of the coordinates Q = (Q1, . . . , QD)and momenta P = (P1, . . . , PD). Without loss of gener-ality, these degrees of freedom have been mass-weightedsuch that all subsystem modes are associated with thesame mass, m, and likewise all bath modes with massM .

The harmonic approximation for the bath will beemployed:1

V b0/1(Q) =

D∑j=1

12MΩ2

j (Qj ± ζj)2, (4)

where the plus sign corresponds to the reactant stateand minus sign to the product state. The bath Hamil-tonians thus combine in Eq. (1) to describe a spin-bosonmodel,101 defined by the associated frequencies Ωj anddisplacements ζj, which can be selected such that theyrepresent an appropriate spectral density. This spin-boson model is complemented by the subsystem modes,whose potential-energy surfaces will be kept general forthe derivations in this work such that they can, in prin-ciple, provide a realistic description of an anharmonicmolecule.

The full system is prepared as a thermal equilibriumensemble in the reactant state with inverse temperature

β = 1/kBT and partition function Z0 = Tr[e−βH0

]. The

quantum-mechanical rate expression for a reaction fromthe reactant to the product electronic state in the golden-rule regime can be derived from a perturbation expansionto lowest order in the nonadiabatic coupling ∆ betweenthe two electronic states of an integral over the flux cor-relation function98,102 to give

k(ε)Z0 =∆2

~2

∫ ∞−∞

Tr[e−(β~−τ−it)H0/~ e−(τ+it)(H1−ε)/~

]dt.

(5)

The flux-correlation function is an analytic functionof time, and hence, the rate is independent of theimaginary-time parameter τ ,103 although it has been in-cluded explicitly as it will play a pivotal role in the semi-classical approximations taken later on.

Due to separability of the subsystem and bath, a quan-tum trace can be taken independently over the respectivecontributions. The reactant partition function thus fac-torizes according to Z0 = Zs

0Zb0 into a subsystem part,

Zs0 = Trs

[e−βH

s0

], and a bath part, Zb

0 = Trb

[e−βH

b0

].

The correlation function likewise splits into product ofsubsystem and bath parts.

The rate can thus be rewritten using the convolutiontheorem of Fourier transforms:20

k(ε) =∆2

2π~3

∫Is(v)Ib(ε− v) dv, (6)

where the subsystem and bath lineshape functions are

Is(v) = (Zs0)−1∫ ∞−∞

Trs

[e−(β~−τ−it)Hs

0/~

× e−(τ+it)(Hs1−v)/~]dt, (7a)

Ib(ε− v) =(Zb

0

)−1∫ ∞−∞

Trb

[e−(β~−τ−it)Hb

0 /~

× e−(τ+it)(Hb1−ε+v)/~]dt. (7b)

4

The equivalence to Eq. (5) can easily be checked by sub-stituting Eqs. (7) into Eq. (6) after renaming the integra-tion variable t in the two cases to ts or tb and using

∫ei(ts−tb)v/~ dv = 2π~ δ(ts − tb).The lineshape function of the subsystem expanded si-

multaneously in the position and eigenstate bases can bewritten as

Is(v) = (Zs0)−1∫∫∫ ∞

−∞

∑µ

∑ν

e−(β~−τ−it)Eµ0 /~ e−(τ+it)(Eν1−v)/~ 〈q′|ψµ0 〉 〈ψµ0 |q′′〉 〈q′′|ψν1 〉 〈ψν1 |q′〉 dq′dq′′dt, (8)

where Eµ0 and Eν1 are the internal energy levels and ψµ0and ψν1 are the corresponding wavefunctions of reactantsand products, respectively.

Because of the global harmonic approximation of thebath, the well-known result for the spin-boson model1,101

can be used to cast Eq. (7b) into

Ib(ε− v) =

∫e−Φ(τ+it)/~−(τ+it)(v−ε)/~ dt, (9)

where the effective action of the bath is defined by

Φ(τ) =

D∑j=1

2MΩjζ2j

[1− cosh Ωjτ

tanh 12β~Ωj

+ sinh Ωjτ

]. (10)

Note that this is an analytic function of its argument andcan therefore also be used to describe real-time dynam-ics in Eq. (9). The coordinate dependence of the bathhas been completely integrated out, which is the rea-son why the effective action [Eq. (10)] only depends ontime. Assuming the spectral density of the bath is known,the time-integral of Eq. (9) can be carried out (either byquadrature or by steepest descent) in order to accountfor quantum effects within the solvent.60,104

In cases where the bath represents a polar solvent envi-ronment, which typically comprises long-wavelength po-larization modes, it is often justified to approximate theeffective action [Eq. (10)] by its classical, low-frequencylimit where |Ωjτ | 1 and β~Ωj 1.1 In this case, theclassical bath action is

Φcl(τ) = Λb

(τ − τ2

β~

), (11)

where the bath reorganization energy is given by

Λb =

D∑j=1

2MΩ2jζ

2j . (12)

In these formulas, τ/β~ plays the role of a “symmetryfactor” as described in Ref. 2.

In order to include a quantum harmonic bath withEq. (10), knowledge of the bath spectral density is re-quired to define Ωj and ζj. On the other hand, aclassical harmonic bath [Eq. (11)] can be simpler to em-ploy as it is fully characterized by its reorganization en-ergy Λb and thus requires much less information.

The approach which we will follow in this paper is toevaluate the subsystem and bath lineshape functions us-ing different representations and approximations to de-rive multiple methods for computing electron-transferrates in the golden-rule regime.

For instance, if the relaxation of the inner sphere is as-sumed to play no role in the reaction under consideration,there is no subsystem contribution to the Hamiltoniansin Eq. (2). The subsystem lineshape function Eq. (7a)therefore reduces to Is(v) =

∫e+(τ+it)v/~ dt = 2π~ δ(v).

Employing the classical approximation for the action[Eq. (11)] in the bath lineshape function Eq. (9), pluggingthe lineshape functions into Eq. (6) and performing thefinal time-integral analytically leads the famous Marcusrate equation13

kbMT(ε) =

∆2

~

√πβ

Λbe−β(Λb−ε)2/4Λb

. (13)

It describes electron-transfer reactions which do not in-volve significant rearrangements within the inner sphere(subsystem) and are therefore determined only by theconformational changes in the outer sphere (bath). As iswell known, Marcus theory thus gives the correct classicallimit of the rate in the case of a spin-boson model.1,14,105

In an alternative and more powerful derivation of Mar-cus theory, the trace could have been evaluated over thebath degrees of freedom in Eq. (7b) directly by a classi-cal phase-space integral.85,106 In fact we could treat thesubsystem in the same way to obtain kMT(ε), a theoryequivalent to Eq. (13) but written in terms of the total re-organization energy, Λ = Λs + Λb. This treatment allowsfor anharmonic potential-energy surfaces but reduces togive the same rate formula as long as the free-energy sur-faces themselves are harmonic. In this case, one shouldtreat the driving force ε as a free energy as it can alsoinclude entropic effects.107

In many cases, however, the inner sphere undergoessignificant conformational changes as well and can there-fore not be ignored. Moreover, the molecules in the re-action center commonly exhibit high-frequency modes,which necessitates the explicit consideration of quantumeffects (such as the existence of zero-point energy andpossibility of tunnelling) within an anharmonic environ-ment. We can derive various methods to compute thesubsystem contribution simply by carrying out the sums

5

and integrals of Eq. (8) in different orders. Although allthese approaches give identical results in their exact form,they provide different starting points for taking approxi-mations.

III. MARCUS–LEVICH–JORTNER THEORY

The subdivision of the full nuclear Hamiltonians ofeach electronic state into independent subsystem andbath parts [Eq. (2)] is the foundation on which MLJ the-ory is grounded.18,19 In this approach one then treatsthe subsystem quantum mechanically and the bath clas-sically.

A. Formalism

Here we rederive MLJ theory on the basis of the for-malism laid out in Sec II. Starting from Eq. (8), we firsttake the integrals over positions and time and set τ tobe zero. Consequentially one arrives at Fermi’s golden-rule (FGR) formula108 for the lineshape function of thesubsystem19,20

IsFGR(v) = (Zs

0)−1∑µ

e−βEµ0

∑ν

|θµν |2 δ(Eν1 − Eµ0 − v),

(14)

where θµν =∫ψµ0 (q)∗ψν1 (q) dq are the Franck–Condon

factors and the subsystem contribution to the reactantpartition function in the energy eigenbasis is Zs

0 =∑µ e−βE

µ0 .

Combining this wavefunction representation of thesubsystem part with the bath lineshape function [Eq. (9)]using the classical effective action [Eq. (11)] and perform-ing the final convolution integral in Eq. (6) leads directlyto the Marcus–Levich–Jortner electron-transfer rate the-ory in a system of two crossing potentials of arbitraryshape in a classical harmonic bath20

kMLJ(ε) =∑µ

∑ν

kµν(ε), (15)

with

kµν(ε) =∆2

~

√πβ

Λb

e−βEµ0

Zs0

× |θµν |2 e−β(Λb−ε+Eν1−Eµ0 )2/4Λb

, (16)

which is the most general version of MLJ theory used inthis work. The total rate in Eq. (15) comprises contribu-tions from all reactant and product vibrational channels.

This “static” formulation of electron transfer (i.e. timehas been integrated out) results in a rate expression thatrequires knowledge of all internal states of the subsystemHamiltonians, Hs

n. For complex anharmonic molecules,

this is not possible to compute without further approx-imations. Thus, the most commonly employed form ofthe Marcus–Levich–Jortner theory takes the extra ap-proximation that the subsystem potentials for the re-actant and product are displaced one-dimensional har-monic oscillators with identical frequencies, ω. Moti-vated by the fact that in many problems of physical inter-est the subsystem comprises very high-frequency modes,it is often appropriate to assume that the thermal en-ergy is low compared with the energy spacing in thesemodes. Hence, only transitions from the ground vibra-tional reactant state with quantum number µ = 0 haveto be considered. The general expression for the one-dimensional overlap integral of two displaced harmonicoscillator wavefunctions can therefore be further simpli-fied, because only the terms22

|θ0ν |2 =Aν e−A

ν!, (17)

with A = Λs/~ω, have to be taken into account. This re-sults in the well known rate formula for a single quantumharmonic mode in the low-temperature limit21

kMLJ(ε) =

∞∑ν=0

kν(ε), ~ω kBT, (18)

where the rate into the product-state ν is

kν(ε) =∆2

~

√πβ

Λb

Aν e−A

ν!e−β(Λb−ε+ν~ω)2/4Λb

. (19)

This low-temperature rate therefore consists of contribu-tions from multiple, parallel product vibrational channelswith effective driving forces of εν = ε− ν~ω.

As can be seen from the exponential “activation” partof Eq. (19), the major contributions to the rate will typ-ically involve the product vibrational states whose effec-tive driving force εν is approximately equal to the bathreorganization energy. In cases where ε < Λb, this is notpossible, and so then the ν = 0 product state is expectedto dominate. Thus, the dominant product vibrationalstate will depend on the thermodynamic driving forceand transitions to highly excited vibrational states are ofparticular importance for very exothermic reactions andhence especially in the inverted regime.

B. Model example

The simple model which we will use to illustrate MLJtheory is formed of two-dimensional displaced harmonicoscillators with one mode treated quantum mechanicallyand the other classically. The subsystem potentials aredefined by

V s0/1(q) = 1

2mω2(q ± ξ)2, (20)

where the reactant state is associated with the plus signand the product state with the minus sign, and because

6

−20 0 20 40√MQ (

√uA)

−60

−30

0

30

60

Energy

(kcal/mol)

(a)

Inverted regime

−20 0 20 40√MQ (

√uA)

−20

0

20

40

60

Energy

(kcal/mol)

(b)

Normal regime

0 5 10 15 20 25 30ν

0.0

0.5

1.0

1.5

kν(ε)/kMT(ε)

(c)

0 2 4 6 8 10µ

0

5

10

15

20

ν

(d)

0.0

0.06

0.12

0.18

0.24

kµν(ε)/kMT(ε)

FIG. 1: Plots to analyze the MLJ rate for the twoharmonic models defined in Table I. (a,b) Plot of V b

0

and V b1 − ε (black lines) as functions of the bath mode.

The coloured lines are copies of the product potentialshifted by the excitation energies of the quantized

subsystem mode, and in (b) only, also shifted copies ofthe reactant potential. In each case, only every fifth

state is shown. (c,d) State-resolved contributions to theMLJ rate relative to the corresponding Marcus theory

rate for the full two-dimensional model. For theinverted-regime model the reactant is almost alwaysfound in its vibrational ground state and therefore

µ = 0.

d = 1, we drop the mode index. Given the frequency andreorganization energy, the displacements are defined byξ =

√Λs/2mω2. The bath potentials are defined accord-

ing to a D = 1 version of Eq. (4) with ζ =√

Λb/2MΩ2.In particular we will apply the theory to two differentmodels, defined by the parameters in Table I, one ofwhich is in the normal and the other in the invertedregime.

The parameters are chosen so as to illustrate twocommon scenarios. Because the inner sphere typically

TABLE I: Definition of the parameters used in the twoharmonic models studied in this work. The rate is

independent of the masses m and M , which thereforedo not have to be defined. As is common practice, the

value of the frequencies are defined by their relatedwavenumber.

Inverted-regime Normal-regimemodel model

T (K) 300 300Λs (kcal mol−1) 25 50Λb (kcal mol−1) 25 50ε (kcal mol−1) 75 25ω (cm−1) 1000 500Ω (cm−1) 50 50

comprises high-frequency vibrational modes, the low-temperature limit of the MLJ rate [Eq. (18)] can oftenbe applied to a good approximation.1,20 Hence, activatedvibrational reaction channels only have to be consideredfor the product. This case is exemplified by the inverted-regime model. The situation is illustrated in Fig. 1(a)which shows the product potential shifted by the vibra-tional energy gap ν~ω. In Fig. 1(c) the reaction rateis broken down into contributions from the individualproduct channels, which are clearly centered around thedominant vibrational state ν = 15 and rapidly fall off oneither side.

Sometimes, however, a system also requires the con-sideration of activated vibrational states of the reac-tant, which necessitates the use of the general expres-sion Eq. (15). As illustrated in Fig. 1(d), this is the casefor the normal-regime model, where excited vibrationalstates of both the reactant and product make significantcontributions to the rate. Thus, in Fig. 1(b), not onlythe product but also the reactant potential is shifted bythe vibrational energies. The required Franck–Condonoverlap integrals for a subsystem of two displaced har-monic oscillators can be computed with well-known ana-lytic formulas.95 Fig. 1(d) shows that the dominant con-tribution to the rate comes from the reaction channelfrom µ = 3 to ν = 11.

Perhaps even more important than the ability of MLJtheory to predict rates is that it provides this simplepicture of the quantum nuclear effect on an electron-transfer reaction. By viewing the reaction along thebath coordinates and shifting the potential-energy sur-faces by the excitation energies of the subsystem, one ob-tains vibrational-state resolved contributions to the rate,which are centred around a dominant vibrational chan-nel. This is a popular way of understanding reactionsand has for instance been used to explain why rates inthe inverted regime commonly flatten off instead of de-creasing rapidly with driving force as predicted by classi-cal Marcus theory [Eq. (13)].10,12 In the inverted regime,it can easily be seen from Eq. (18) and Figs. 1(a) and(c) that the dominant contribution to the rate originatesfrom the vibrational channel that approximately shiftsthe bath product potential to the activationless regime,where εν ≈ Λb,109 and thus predicts a rate approximatelyindependent of driving force.110

This analysis of the inverted-regime model is based onthe simplification that the rate is fully determined by theexponential “activation” part. In reality, however, ratesin the inverted regime are also affected by the Franck–Condon factors such that they do not actually becomeconstant with driving force. We find that the bath activa-tion energy for the dominant vibrational transition in theinverted-regime model is 0.51 kcal mol−1, which is almostactivationless, but still not negligible relative to the ther-mal energy. In the normal-regime model, where excitedreactant states also play an essential role as illustratedin Figs. 1(b) and (d), the full rate expression Eq. (15) isno longer dominated by an activationless channel at all.

7

We find a significant activation energy for the dominantvibrational channel of 6.64 kcal mol−1, which illustratesthe compromise between minimization of the activationenergy and maximization of the Franck–Condon overlapsthat has to be made. This considerably complicates theinterpretation of the MLJ rate formula even when theharmonic oscillator approximation is employed.

For more realistic systems described by multidimen-sional anharmonic potential-energy surfaces the Franck–Condon factors are practically impossible to obtain andthe subsystem is thus commonly approximated by sim-ple models for which these are known analytically. Thisintroduces unknown errors into the predicted rate, andit is to avoid this problem that we now turn to instantontheory.

IV. REDUCED INSTANTON THEORY

Inspired by the Marcus–Levich–Jortner approach wewill derive a reduced instanton theory, where only the in-ner sphere is treated explicitly in molecular detail whilethe outer solvent shells are accounted for with the har-monic bath approximation.

A. Formalism

In order to derive the semiclassical golden-rule instan-ton rate expression, we start from the time-dependentcorrelation function formulation of the reaction rate inEq. (5). The trace can be split up into a subsystem andbath contribution, where the latter can, due to its har-monic nature, again be replaced with the well known so-lution for the spin-boson model in terms of the effectivebath action Φ(τ). In contrast to the MLJ approach, thetrace in the subsystem coordinates will be expanded inthe position basis, which leads the following expressionfor the rate:

k Zs0 =

∆2

~2

∫∫∫ ∞−∞

K0(q′, q′′, β~− τ − it)

×K1(q′′, q′, τ + it) e−Φ(τ+it)/~+(τ+it)ε/~ dq′ dq′′ dt.(21)

Again in analogy to the dynamics of open quantumsystems, e−Φ(τ+it)/~ plays the role of an influencefunction.77,101 The matrix elements of the quantum prop-agators,

Kn(qi, qf, τn) = 〈qf|e−τnHsn/~|qi〉 , (22)

describe the dynamics of the subsystem variables evolv-ing according to the Hamiltonians Hs

n from the initialpositions qi to the respective final position qf in imagi-nary time τn. The imaginary-time propagators are equiv-alent to quantum Boltzmann distributions and it is thisconnection which allows instanton theory to approximate

the thermal rate in a statistical way using imaginary-timedynamics.

If the imaginary-time propagators and spatial integralswere evaluated by path-integral Monte Carlo calcula-tions and the remaining time integral taken by steep-est descent, one would obtain a version of Wolynes the-ory where the bath is treated implicitly by the influencefunction.59,68 This, however, is not the purpose of thiswork as we wish to derive a semiclassical instanton for-mulation of the rate.

Instead we replace the quantum propagators by thecorresponding van-Vleck propagators111 generalized forimaginary-time arguments87,112

Kn(qi, qf, τn) ∼√

Cn(2π~)d

e−Sn/~, (23)

thus introducing a semiclassical approximation. The re-sulting expression is evaluated by locating the classicaltrajectory, qn(u), travelling in imaginary time u, whichmakes the Euclidean action of the subsystem, Sn, sta-tionary. The action for a path travelling from its initialposition qn(0) = qi to its final position qn(τn) = qf inimaginary time τn is defined as

Sn ≡ Sn(qi, qf, τn) =

∫ τn

0

[12m‖qn(u)‖2 + V s

n(qn(u))]

du,

(24)

where qn(u) = dqndu is the imaginary-time velocity. The

prefactor of the semiclassical propagator is given by thedeterminant

Cn =

∣∣∣∣− ∂2Sn∂qi∂qf

∣∣∣∣ . (25)

By multiplying the two propagators in Eq. (21) to-gether, we obtain the total action

S(q′, q′′, τ) = S0(q′, q′′, β~− τ) + S1(q′′, q′, τ), (26)

as the sum of contributions from two trajectories, oneof which travels on the reactant potential and the otheron the product potential. These trajectories join eachother to form a continuous periodic pathway, called theinstanton. The imaginary times τn associated with thetwo paths are given by τ0 = β~− τ and τ1 = τ .

Combining this result with the effective action of thebath according to Eq. (21), the total effective action be-comes

Sr(q′, q′′, τ) = S(q′, q′′, τ) + Φ(τ)− ετ, (27)

where one could employ either the effective quantum bathaction from Eq. (10) or its classical limit Eq. (11). Theonly effect of the bath is to thus alter the total action byadding an extra τ -dependence alongside the driving forceterm. However, as we will show, the simple addition ofthe bath action can lead to significant changes for the

8

instanton path and for our interpretation of the reactionmechanism.

In order to obtain the semiclassical instanton expres-sion for the rate, the integrals over q′ and q′′ as wellas the time-integral will be carried out by steepest de-scent. Therefore it is necessary first to study the pathcorresponding to the stationary point of the effective ac-tion, for which ∂Sr

∂q′ = ∂Sr

∂q′′ = ∂Sr

∂τ = 0. This path is our

definition of the reduced instanton and by analyzing theconsequences of vanishing derivatives, we can understandits properties in the general case.

As in the standard golden-rule instantonformulation,78 the instanton pathway consists oftwo trajectories q0(u) and q1(u), which join smoothlyinto each other at the stationary or hopping point inthe subsystem coordinate space q′ = q′′ = q‡. Becausethe q-derivatives are not altered by the influence ofthe bath, the momentum, given by p′ = −∂S0

∂q′ = ∂S1

∂q′

(equivalent for double primes), is thus still conservedacross the hopping point. In this sense, the instantontherefore remains a periodic orbit of imaginary time β~as in the standard theory.78

However, a major difference occurs due to the bath’sinfluence on the derivative with respect to τ . The sub-system energies of the two trajectories are given by

Esn =

∂Sn∂τn

. (28)

Therefore, in the case without the presence of a bath, thecondition at the stationary point is given by ∂S

∂τ − ε = 0,

where ∂S∂τ = Es

1 − Es0 ≡ ∆Es. Considering ε as a contri-

bution to the product energy as was done in Ref. 78, thisrelationship implies that the reaction conserves energy,i.e. Es

0 = Es1 − ε. The hopping point must therefore be

located on the crossing seam where V0(q) = V1(q)− ε.This no longer holds true once bath modes are added.

Then the condition at the stationary point changes to

∂Sr

∂τ=∂S

∂τ+∂Φ

∂τ− ε = 0. (29)

The presence of the bath will thus affect the stationaryvalue of τ and hence the entire instanton path and thevalue of its action.113 In particular, the energies of thetwo trajectories no longer match Es

0 6= Es1 − ε in gen-

eral. Hence, the presence of the bath renders the reducedinstanton non energy-conserving within the subsystem.However, the energy change in the subsystem is exactlycompensated by an energy change of opposite sign in thebath given by ∆Eb = ∂Φ

∂τ such that ∆Es + ∆Eb− ε = 0.This is in agreement with what one would expect froman open quantum system, in which only the total com-bined energy of subsystem and bath is conserved butnot the individual components. In this theory one doesnot have direct access to the energies of the bath whichwould be needed to fully justify this interpretation for∂Φ∂τ . However, we will show that this definition is correctin Sec. V B.

0 1 2 3q (A)

0

50

100

Energy

(kcal/mol)

Es0

Es1 − ε

∆Es − ε

Λs

0.80 1.00 1.20

-2

0

2

4

E0 E1 − ε

FIG. 2: Potential curves V s0 (q) (blue solid lines) and

V s1 (q)− ε (orange solid lines) of the subsystem as

defined in Eq. (32) with ε = Λ/4 = 31.7 kcal mol−1. Thetwo trajectories of the reduced instanton are shown at

discrete time steps by blue (reactant) and red (product)dots, and their energies, Es

0 and Es1 − ε, are depicted by

the blue and red dashed lines. These energies areseparated by the energy gap ∆Es − ε, which is equal to

the potential-energy gap at the hopping point (q‡,purple dot). The inset shows the instanton for a modelwith the same subsystem but without the presence of abath enlarged from the area framed by the grey dotted

box. Here ∆Es − ε = 0 and therefore energyconservation is satisfied and the hopping point (purple

dot) is located where the potentials cross.

One consequence of the energy jump caused by thepresence of the bath is that the hopping point, q‡, is notlocated on the crossing seam between the two subsys-tem potentials. In fact, because the momenta of the twotrajectories are equal at the hopping point, the energyjump within the subsystem must correspond exactly tothe potential energy difference, ∆Es = ∆V s(q‡), where∆V s(q) = V s

1 (q)− V s0 (q).

In Fig. 2, we illustrate the reduced instanton pathwayfor the anharmonic model discussed in Sec. IV B, as wellas the the energies of the two trajectories as defined byEq. (28).

Note that the reactant or product energy is conservedalong its respective trajectory and is thus identical tothe potential at the turning point, which can be eas-ily seen in the figure as the point with lowest potentialalong the path. The concept of a turning point in thiscontext can be understood by the fact that dynamics inimaginary time are equivalent to real-time dynamics onthe upside-down potential.112 At the turning point, thepaths therefore bounce against the potential which theyare travelling on.

Because the instanton orbit folds back on itself and istherefore not so easy to depict, it is worth describing it ina little more detail. If we first follow the instanton path-

9

way in Fig. 2 along the trajectory q0(u) starting at the(purple) hopping point on V s

0 and with a certain amountof momentum pointing to the left, we find that the pathdescends towards the reactant state minimum, where itbounces against the potential and returns to where itstarted but with momentum now pointing to the right.So far this is equivalent to the instanton pathway in thestandard formulation of the theory.78 Once the hoppingpoint is reached, however, a sudden jump in potentialenergy occurs, which accompanies the transition into theproduct state. This is in stark contrast to the standardformulation, where both trajectories tunnel at the sameenergy, as shown in the inset of Fig. 2 for a case withouta bath. After the state transition, we travel along thetrajectory q1(u), which after reaching the turning pointon V s

1 also returns to the hopping point. A transitionback to the initial hopping point on V s

0 completes theperiodic cycle.

After the instanton pathway has been located, theintegrals in Eq. (21), where the propagators have beenreplaced with Eq. (23), can be carried out by steepest-descent integration around the stationary point. Thus wearrive at the reduced instanton expression for the golden-rule rate

krSCI(ε)Zs0 =√

2π~∆2

~2

√C0C1

C

(−d2Sr

dτ2

)− 12

e−Sr/~,

(30)where all quantities are evaluated at the stationary pointof Sr(q′, q′′, τ) except the reactant partition function Zs

0,which is treated by an equivalent steepest-descent ap-proximation around the minimum of the reactant.87 Theadditional prefactor from the steepest descent integrationin the subsystem positions evaluates to the determinant

C =

∣∣∣∣∣ ∂2S∂q′∂q′

∂2S∂q′∂q′′

∂2S∂q′′∂q′

∂2S∂q′′∂q′′

∣∣∣∣∣ , (31)

where we have used the fact that derivatives of Sr withrespect to the end points are equal to derivatives of S.The bath therefore has no direct effect on C but doesexplicitly appear in d2Sr

dτ2 = d2Sdτ2 + d2Φ

dτ2 as well having animportant effect on the instanton path itself as previouslydiscussed. Apart from these changes, the formula resem-bles the semiclassical golden-rule instanton rate expres-sion derived in previous work78 and gives identical resultswithout needing to treat the harmonic bath explicitly.

Just as for previous golden-rule instantoncalculations,79,84 a ring-polymer discretization schemeof the instanton pathway is employed in order todescribe nonadiabatic reactions for multidimensional,anharmonic systems. By adopting the ring-polymerformalism, the localization of the instanton path, whichis defined as a stationary point of the action in Eq. (27)in the coordinate and τ variables together, reduces toa standard saddle-point search problem which can besolved numerically with well-established optimizationalgorithms. Algorithms for computing the necessary

derivatives of the action as well as detailed informationabout the optimization scheme can be found in Ref. 86.

In our recent extension of the theory,84 we have shownthat ring-polymer instanton theory can equivalently beutilized to compute electron-transfer rates in the Mar-cus inverted regime, where tunnelling effects commonlyplay a particularly important role. The major differencein this regime is, that one of the two paths travels innegative imaginary time, which allows an analogy to thephysics of antiparticles.114 In the computational realiza-tion, this difference manifests itself merely in a slightchange of the optimization algorithm. Hence, whereas inthe normal regime the instanton is a single-index saddlepoint of the ring-polymer action in the combined space ofring-polymer coordinates and imaginary time, in the in-verted regime the instanton path corresponds to a higher-index saddle point of the ring-polymer action. The indexof a saddle point here defines the number of negativeeigenvalues in the second-derivative matrix of the ring-polymer hessian at this point. But since we exactly knowthe index of the desired saddle-point, the instanton canbe optimized with the same routines by using standardeigenvector-following schemes. We thus take uphill stepsin the direction of eigenvectors corresponding to negativeeigenvalues and standard down-hill steps in the directionof eigenvectors associated with positive eigenvalues. Thismethodology can be directly transferred to the reducedinstanton picture without any additional complicationsand hence allows us to apply it to the normal and in-verted regimes alike.

The advantage of the reduced instanton approach isthat the optimization is confined to the inner sphere andτ -coordinates only, whereas the only direct influence ofthe bath on the optimization procedure manifests itselfin an external field in the imaginary-time variable. Thisreduces the computational costs of the simulation andenables it to be applied within a multiscale modellingapproach, where certain parts of a system are treated athigher levels of accuracy than others.

B. Model example

We will employ the newly formulated reduced instan-ton method along with MLJ theory to compute reactionrates of an anharmonic subsystem of two bound Morseoscillators in a multidimensional harmonic bath. Thesubsystem is defined by the potentials (depicted in Fig. 2)

V sn(q) = De

n

(1− e−αn(q−ξn)

)2

, (32)

where α0 = 1.5 A−1

and α1 = 1.4 A−1

determine thelength scales, ξ0 = 1.0 A and ξ1 = 1.5 A are the equi-librium positions, and De

0 = 115 kcal mol−1 and De1 =

80 kcal mol−1 are the dissociation energies of reactantsand products. The (product) reorganization energy of

this subsystem is therefore Λs = V1(q(0)min) − V1(q

(1)min) =

10

82.2 kcal mol−1, where q(n)min is the minimum of Vn(q). The

reduced mass is chosen to be m = 1.10 u. The well fre-quencies of the two Morse oscillators obtained by har-monic analysis are ωn = αn

√2De

n/m, which results infrequencies of ω0 = 2358 cm−1 and ω1 = 1835 cm−1

for the reactant and product well respectively. TheSchrodinger equation for the Morse oscillator can besolved analytically to give the bound-state energies

Eµ0 = ~ω0(µ+ 12 )− ~ω0χ0(µ+ 1

2 )2 (33)

and likewise for Eν1 , where the dimensionless anharmonic-ity parameters of the Morse oscillators are defined byχn = α2

n~/2mωn and in this case have values of 0.015and 0.016 for the reactant and product potential respec-tively.

The bath is defined by the discretized spectral density

J(Ω) =π

2

D∑j=1

c2jMΩj

δ(Ω− Ωj), (34)

and the D = 100 bath modes were chosen according toRef. 115 as

Ωj =j2

D2Ωmax, j ∈ [1, D]. (35)

The effective mass of the bath modes M does not haveto be specified, as the rate is independent of this choice.The frequency spectrum is bounded from above by themaximum frequency Ωmax = 3000 cm−1 and thus has thedensity116

ρ(Ω) =D

2√

ΩΩmax

. (36)

The couplings were chosen to emulate a Debye spectraldensity defined by

JDe(Ω) =ηΩcΩ

Ω2 + Ω2c

, (37)

with characteristic frequency Ωc = 500 cm−1 and η =25 kcal mol−1. Hence the coupling constants cj are de-termined by the formula

c2j = MΩj2

π

JDe(Ωj)

ρ(Ωj), j ∈ [1, D], (38)

which are related to the shifts ζj in Eq. (4) by cj =MΩ2

jζj . The reorganization energy of the bath is then

obtained by Eq. (12), which in our case results in Λb =44.8 kcal mol−1. The total reorganization energy of thesubsystem and bath combined is therefore given by Λ =Λs+Λb = 127.0 kcal mol−1. The temperature for the ratecalculations was chosen to be 300 K.

Similar models were studied with MLJ theory inRef. 117. In order to make use of analytical formulasfor the Franck–Condon factors, however, in that workthe reactant’s subsystem mode was assumed to be in the

0.0 0.5 1.0 1.5 2.0ε/Λ

0

4

8

12

16

log 1

0[k(ε)/kTST(0)]

TST

Cumulant

SFC

MLJ

Exact

rSCI (cl. bath)

rSCI (QM bath)

FIG. 3: Rates calculated by various methods for ananharmonic mode in conjunction with a harmonic bath

are shown for different values of the driving force ε,including: the reduced semiclassical instanton [rSCI,

Eq. (30)] with either a quantum or classical bath;classical golden-rule transition-state theory [TST,Eq. (A2)]; the second-order cumulant expansion

[Eq. (B6)]; the semiclassical Franck–Condon sum [SFC,Eq. (C3)]; Marcus–Levich–Jortner theory [MLJ,

Eq. (15)] and exact quantum mechanics [Eq. (A1)]. Ineach case, the results are given relative to the classical

golden-rule TST rate at ε = 0.

low-temperature limit. Although it only makes a minordifference, here, we include as many reactant states as isnecessary to converge the rate, and perform the Franck–Condon overlap integrals numerically. Where necessary,we take the continuum states of the Morse oscillator intoaccount, by extending the MLJ formula given in Eq. (15)in the same way as explained for the exact quantum rate[Eq. (A1)] in Appendix A.

The reaction rates for this model system computedwith various methods as a function of the driving forceε are presented in Fig. 3. The exact (Fermi’s goldenrule) and MLJ rate calculations are based on the knowl-edge of the analytic expressions for the energy levels andwavefunctions of the Morse oscillator (see Appendix A).Thus, the only approximation made by MLJ is to treatthe bath classically. It can be seen from Fig. 3 that in-stanton theory, which does not require knowledge of theeigenstates nor even global knowledge of the potentialalong the subsystem mode, is virtually identical to theexact result when employing a quantum bath with theeffective action from Eq. (10). This excellent agreementwas expected from the results and analysis seen in pre-vious instanton studies of electron transfer.70,79,84 Whenusing a classical bath with the action given by Eq. (11),it is slightly less accurate, although then very similar toMLJ theory as they both suffer from the assumption ofa classical bath.

11

The classical golden-rule transition-state theory (TST)rate, outlined in Appendix A, constitutes the classicallimit of the quantum rate and would reduce to Marcustheory in the case of a subsystem consisting of displacedharmonic oscillators. The deviation of the TST rate fromthe exact, MLJ and instanton rates underlines the impor-tance of nuclear quantum effects, which causes the clas-sical rate to differ from the exact results by more thanseven orders of magnitude in some cases. The differencesare most extreme for the largest driving forces in the in-verted regime.

For ε > Λ, the instanton analysis predicts a negativevalue of τ , which is a clear indication that the invertedregime has been reached, and it requires a subtly differ-ent ring-polymer optimization scheme.84 Despite this, itis noteworthy that the observed turnover in the rate (i.e.the point at which the rates start to decrease with grow-ing driving force) actually occurs at a slightly smallerdriving force. This is predicted correctly by all methodstested apart from classical golden-rule TST. In instantontheory, this effect is caused by the prefactor in Eq. (30)as the effective reduced action in the exponential has itsminimum at ε = Λ.

The fact that in the inverted regime the MLJ, instan-ton and exact quantum rates almost coincide, revealsthat practically all the quantum effects in this regimeoriginate from the subsystem and not from the bath. Al-though the quantum subsystem still plays a dominantrole in the normal regime, it is clear that there is also asmall quantum effect from the bath, which explains thesource of the error of MLJ and likewise of rSCI theorywhen employing a classical bath.

In Sec. V, we will further elaborate on the relationshipbetween rSCI and MLJ theory that is apparent from theresults in this section.

C. Comparison with alternative approximations

In this subsection, we will compare the instanton ap-proach with two other approximate methods for includ-ing quantum effects into electron-transfer rates, namelythe cumulant expansion and the semiclassical Franck–Condon sum. As well as discussing the accuracy of thesevarious methods, we will also focus on the computationaleffort required for their calculation.

In Fig. 3, we present the rates obtained with thesecond-order cumulant expansion118–120 described in Ap-pendix B. To enable a direct comparison with MLJtheory, we employed a classical bath in its calcula-tion, although like with rSCI it would also be possibleto use the effective action of a quantum bath. Thismethod is not only commonly used for the study ofelectron-transfer reactions in anharmonic systems,121–128

but also for the simulation of optical spectroscopy,129

vibrational lineshapes95 and the description of energy-transfer processes.130 In practice often further approxi-mations are invoked to obtain analytical expressions for

the rate131 before the method can be applied to complexproblems.

The advantage of the cumulant expansion over an ex-act (FGR) or MLJ calculation is that it does not requireknowledge about the excited state’s vibrational eigen-states, but only about its potential-energy surface. How-ever, although the method is exact for displaced har-monic potentials,95 the results in Fig. 3 clearly demon-strate, that, as opposed to instanton theory, the rates ob-tained by the second-order cumulant expansion can differsignificantly from the exact rates for the Morse oscillatormodel, with the worst case being at zero driving force inthe normal regime.132

Moreover the rate expression of the cumulant expan-sion does not satisfy the detailed balance relation forthermal rates,107,133

k0→1 Zs0 = e+βε k1→0 Z

s1, (39)

in anharmonic subsystems or even in a subsystem oftwo displaced harmonic oscillators of different frequency.Here, k0→1 and k1→0 are the rate constants of the for-ward and backward reactions. Note that this relationwould normally be written with total partition functions,but here we have already used the fact that in our caseZb

0 = Zb1 . Detailed balance is however obeyed by Fermi’s

golden rule, MLJ theory, all forms of instanton theoryand even classical golden-rule TST.

Another method that does not obey detailed balancefor anharmonic subsystems is the “semiclassical Franck–Condon sum” (SFC). In fact, the rates computed withinthis approximation do not even fulfil detailed balance fora subsystem of two displaced harmonic oscillators of thesame frequency if the driving force is different from zero.Originally the method was developed to describe spec-tral line shapes of solids134,135 and later used to describeelectron-transfer in biological systems.43,136 The deriva-tion of the method for models with a harmonic bath asconsidered in this paper is outlined in Appendix C. In thiscase we treat the bath itself within the SFC approxima-tion as this can be done with a closed-form expression.Like the cumulant expansion, it requires knowledge ofthe vibrational eigenstates of the reactant but not of theproduct. In accordance with the findings of Siders andMarcus,11,96 it is accurate near the activationless regime,works fairly well in the inverted regime and gives signifi-cant errors of more than four orders of magnitude in thenormal regime.

Ultimately, the major intrinsic problem of the MLJmethod is that it relies on knowledge of the wavefunc-tions of the reactant and product states and is there-fore practically impossible to apply to complex multidi-mensional problems. The cumulant expansion and SFCmethods only go part of the way to improving this situ-ation as they require wavefunctions only for the reactantstate. However, numerical integration over the coordi-nates would still require both potentials to be evaluatedover a large grid. Typically therefore at least the reactantpotential is approximated by a low-dimensional harmonic

12

oscillator, which introduces an unknown additional errorinto the predicted rate.

In contrast, instanton optimizations only require in-formation along the tunnelling pathway, which is locatedclose to the hopping point and thus minimizes the compu-tational effort. This advantage of instanton theory overthe wavefunction-based methods increases in significancewith growing dimensionality of the subsystem. The rea-son for this is that the instanton pathway always remainsone-dimensional, whereas the number of points needed toevaluate the potential-energy surfaces on a grid grows ex-ponentially with subsystem size. It can thus be appliedin principle to complex systems without making extraapproximations.

It is therefore worth noting that although our instan-ton approach as well as the SFC method and a num-ber of other theories are labeled “semiclassical”, theyclearly employ quite different approximations. Not onlyis semiclassical instanton theory superior in accuracy, itis also applicable to more complex multidimensional an-harmonic problems.

V. INSTANTON FORMULATION OF MLJ THEORY

Although MLJ and reduced instanton theory can bothbe derived from Eq. (5), the resulting methods and rateformulas [Eq. (15) and Eq. (30)] look rather distinct fromeach other and thus lead to quite different interpretationsof the reaction. Marcus–Levich–Jortner theory relies onthe wavefunction picture of quantum mechanics and com-putes the rate as a sum over reactant and product stateswhich will be dominated by one particular reaction chan-nel as shown in Fig. 1(d). Instanton theory, on the otherhand, is based on the path-integral formalism of quantummechanics and is dominated by a path which describesthe mechanism during the electron-transfer event.

Another fundamental difference between MLJ theoryand the rSCI approach presented in Sec. IV is that, inrSCI, the focus is shifted from the bath modes to the sub-system. The standard MLJ picture as shown in Fig. 1interprets the reaction in terms of the activation energy inthe bath and includes the effect of the subsystem throughthe shift that they give to the bath potentials. The com-putation of the reduced instanton approach, however, iscarried out directly in the subsystem modes under the in-fluence of the bath. This reflects more appropriately thecomputational effort put into the calculation of subsys-tem and bath, as typically the subsystem will be treatedin much more detail or on a higher level of theory.

Both interpretations can be useful, but it is not imme-diately obvious that they can be reconciled, although thecommon foundation in Eq. (5) suggests that both meth-ods must be related. This idea is reinforced by the factthat the rates obtained for the double Morse oscillatormodel, shown in Fig. 3, are practically identical whenboth methods treat the bath classically. In the followingwe will show that a different derivation of the semiclas-

sical instanton approximation leads to an equivalent for-mulation but which can be used to give the same insightsas MLJ theory.

A. Formalism

The objective of this section is to derive an instantonformulation of MLJ theory. The bath is thus assumedto be classical and for simplicity both subsystem andbath are kept one-dimensional here. The formulas do,however, generalize straightforwardly to the multidimen-sional case.

In order to show the relation with MLJ theory moreclosely, the convolution formula [Eq. (6)] will again serveas the starting point. The expression for the lineshapefunction of the bath in Eq. (7b), will be evaluated by aclassical phase-space integral, which is one dimensionalin both the position and momentum coordinate. Aftercarrying out the integrals in momentum and time, this re-sults in the one-dimensional classical configuration-spaceintegral

Ibcl(ε− v) = 2π~

(Zb

0

)−1

√M

2πβ~2

×∫

e−βVb0 δ(∆V b − ε+ v) dQ, (40)

where the Hamiltonians of the bath [Eq. (3b)] have beenreplaced by their classical analogues and the indepen-dence with respect to τ appears naturally. In addition,we define the potential energy difference in the bath∆V b(Q) = V b

1 (Q)−V b0 (Q), although we suppress the Q-

dependence to avoid clutter. For the harmonic bath po-tential, ∆V b = −2MΩ2ζQ = −ΛbQ/ζ. The Q-integralcould of course easily be carried out immediately to givethe Marcus theory lineshape. However, In order to ob-tain a picture of the reaction from the point of view ofthe bath, we leave it for later.

Using this classical result for the bath lineshape func-tion in Eq. (6) and performing the convolution integralleads the approximate rate formula

k(ε) ≈(Zb

0

)−1 ∆2

~2

√M

2πβ~2

∫I(ε−∆V b) dQ, (41)

where we define the subsystem lineshape functionweighted by the bath thermal distribution

I(ε−∆V b) = Is(ε−∆V b) e−βVb0 . (42)

Note that the effect of the convolution manifests itselfin a change of the argument of the subsystem lineshapefunction Is, which now implicitly depends on the bathcoordinate Q via ∆V b.

Viewing the expression for the reaction rate with animplicit dependence on the bath coordinates is also the

13

idea that enables the illustration of the MLJ rate byshifted potentials along the bath modes, as shown inFig. 1. In fact, if Eq. (14) is used for the subsystemlineshape function and the remaining Q-integral over thedelta-function in Eq. (40) is taken, the standard MLJ rateformula [Eq. (15)] is recovered.

Here, we seek to treat the subsystem part with semi-classical instanton theory. Note that both the MLJ andinstanton version of the subsystem lineshape functionemerge from Eq. (8). The difference is induced by the or-der in which the sums and integrals in Eq. (8) are taken.Whereas in MLJ theory the configuration-space integralsare taken before the sums over states are carried out, ininstanton theory these steps are taken in reversed orderleading to a path-integral instead of a wavefunction for-mulation of the reaction rate. Only in the path-integralformulation is it possible to take the steepest-descent in-tegration which leads to semiclassical instanton theory.The instanton subsystem lineshape function is thus givenby137

IsSCI(ε−∆V b) =

√2π~Zs

0

√C0C1

C

(−d2S

dτ2

)− 12

× e−S(τ)/~−(∆V b−ε)τ/~, (43)

where again all quantities are evaluated at the stationarypoint of the exponent

[S(τ)/~ + (∆V b − ε)τ/~

]in the

subsystem coordinates q′, q′′ and imaginary time τ si-multaneously. Using this approximation in Eqs. (42) and(41) defines the instanton formulation of MLJ theory.

Here we show that this approach gives the same resultas the reduced instanton theory derived in Sec. IV A. Byemploying Eq. (42) for the subsystem lineshape functionin Eq. (41), the effective action in the exponent becomes

S(Q, τ) = S(τ) + (∆V b − ε)τ + β~V b0 . (44)

Due to the harmonic nature of the bath, the stationarypoint in the bath coordinates can be solved for analyti-cally. This defines the hopping point at which the elec-tron transfer dominantly takes place. Within the classi-cal limit, it is given by

Q‡ = ζ

(2τ

β~− 1

). (45)

Evaluating Eq. (44) at this point therefore leadsS(Q‡, τ) = Sr(τ). So the exponent becomes identical tothat of reduced instanton theory (with a classical bath)and hence the value of τ at the stationary point is thesame too.

The rate expression for this instanton version of MLJtheory is obtained by performing the remaining Q-integral by steepest-descent and using the classical par-tition function Zb

0 = (β~Ω)−1 to give

kSCI(ε) =∆2

~2

√β~MΩ2 ISCI(ε−∆V b)

(d2SdQ2

)− 12

,

(46)

where all quantities are evaluated at the stationary pointQ = Q‡ including the system lineshape function, whichimplicitly depends on Q through ∆V b.

In order to verify the equivalence of this rate expressionwith Eq. (30), we make use of the rules of consecutivesteepest-descent integrations78,138

d2SdQ2

=∂2S∂Q2

− ∂2S∂Q∂τ

(∂2S∂τ2

)−1∂2S∂τ∂Q

. (47)

Because the spatial subsystem and bath coordinates areindependent, the partial derivatives involving Q can beeasily evaluated. After rearranging, this results in

∂2S

∂τ2

d2SdQ2

= β~MΩ2 ∂2Sr

∂τ2, (48)

where ∂2Sr

∂τ2 = ∂2S∂τ2 − 2Λ/β~.

Using these expressions in Eq. (46) shows that this ap-proach is therefore identical to rSCI [Eq. (30)], which isnot surprising as all we have done is carry out the samesteepest-descent integrations but in a different order.

Following this procedure for the displaced harmonic-oscillator models defined in Table I, we obtain the in-stantons depicted in Fig. 4. Panels (a) and (b) show

I(ε − ∆V b) computed with the semiclassical instantonapproximation as a function of Q for the inverted andnormal-regime model. As expected, the function is cen-tered around Q‡ and is well approximated by a Gaussian.From instanton theory, we have therefore obtained a re-duced picture of the reaction, but this time the focusis along the solvent coordinate and hence can provide asimilar interpretation to that from MLJ theory.

B. Analysis and Mechanistic Insights

In addition to the formal connection between SCI andMLJ discussed in Sec. V A, we will show that, as wellas the insight into the tunnelling pathway, it is possibleto use instanton theory to extract very similar informa-tion about the reaction as is offered by MLJ theory, suchas the bath activation energy and the dominant reac-tant and product vibrational states. We thus suggestthat instanton theory may be used instead of MLJ the-ory for understanding and interpreting electron-transferreactions in complex anharmonic systems.

In Table II, we present numerical values of the reac-tion rates for the two models in the normal and invertedregimes models defined in Table I computed with differ-ent methods as well as a number of values obtained fromthe instanton calculation which we will describe later. Acomparison of the accuracy of the approaches has alreadybeen carried out in Sec. IV B and thus here we simplynote a couple of points which are special to this case.The fact that for the inverted-regime model the rSCIrate is even slightly closer to the quantum rate than theMLJ rate can be attributed to a fortuitous error cancel-lation, as MLJ theory is, in principle, the more accurate

14

0

4

8

12I(ε

−∆V

b) (a)

Inverted regime

0

4

8

I(ε

−∆V

b)/10

−8 (b)

Normal regime

−30 −20 −10 0√MQ (

√uA)

−40

−20

0

20

Energy

(kcal/mol)

Q = Q‡(c)

∆V b(Q‡)− ε

−20 −10 0 10√MQ (

√uA)

−20

−10

0

10

Energy

(kcal/mol)

Q = Q‡

(d)

∆V b(Q‡)− ε

−1.0 −0.5 0.0 0.5 1.0√mq (

√uA)

0

20

40

Energy

(kcal/mol)

Es0

Es1

∆Es

(e)

−2 −1 0 1 2√mq (

√uA)

0

10

20Energy

(kcal/mol)

∆V s(q‡)

Es0

Es1

∆Es

∆V s(q‡)

(f)

FIG. 4: Insights from instanton theory into the reaction mechanism for the inverted and normal-regime model. (a,b)Eq. (42) as a function of Q. (c,d) Plot of the potential energy curves (including the driving force, ε) along the bathmode. The location of the hopping point along the classical mode Q‡ and the potential energy differences at this

point ∆V b(Q‡)− ε are indicated. (e,f) Plot of the potential energy curves along the subsystem mode together withthe optimized ring-polymer instanton corresponding to Q = Q‡, which was used to compute the subsystem

contribution to the rate. The instanton energies in the subsystem [Eq. (28)] (dashed lines) and the correspondingenergy difference ∆Es are indicated. The energy difference can be measured equivalently as ∆V s(q‡) at the hopping

point (q‡, purple dot).

method in this case. Because both models consist of dis-placed symmetric harmonic oscillators, the second-ordercumulant expansion is exact in these cases and thereforenot shown. In contrast, the rate obtained with the SFCmethod, while showing decent agreement with the ex-act rate for the inverted-regime model, exhibits an errorof almost one order of magnitude for the normal-regimemodel. This is in agreement with the findings in Refs. 11and 96.

Just as in the reduced instanton formalism derived inSec. IV, the instantons computed in the subsystem co-

ordinate space, shown in the bottom panels of Fig. 4,consist of paths qn(u) whose energies are not equal butdiffer by the amount ∆Es. We will show that this energyjump is a good approximation to the difference in energiesbetween the dominant reactant and product vibrationalstates in the MLJ sum, i.e. Eν1 − Eµ0 . As explained inRef. 84, in the normal regime, the trajectories travel inopposite directions away from the hopping point q‡, butin the same direction when in the inverted regime. Thisoccurs because τ < 0 in the inverted regime such that theproduct trajectory travels in negative imaginary time and

15

TABLE II: Computed quantities for the harmonicmodels defined in Table I. The reduced instanton rateswith classical bath and the corresponding values τ at

the stationary point of the reduced action Sr wereobtained from ring-polymer instanton optimizations

with 256 beads equally distributed between bothelectronic states. The contributions to the total

effective action from subsystem Sn and bath Φcl arealso given. The exact rate is obtained from integrationof the flux correlation function of the full system.78 As

described in Appendix C, the SFC approximation isused for both subsystem and bath in order to compute

the corresponding rates. For both models, the MLJ rateincludes contributions from excited reactant states. All

rates, including the Marcus rate for the full systemkMT(ε), are given relative to the Marcus rate for the

bath of the respective model only kbMT(ε).

Inverted-regime Normal-regimemodel model

Es0/~ω 0.02 3.08

Es1/~ω 15.46 10.91

τ/β~ −0.12 0.36S0/~ 2.538 6.787S1/~ −9.171 10.700Φcl/~ −5.501 19.367V b0 (Q‡) (kcal mol−1) 0.34 6.55

kex(ε)/kbMT(ε) 5.159 · 1016 5.484 · 10−8

kMLJ(ε)/kbMT(ε) 5.144 · 1016 5.366 · 10−8

krSCI(ε)/kbMT(ε) 5.152 · 1016 5.360 · 10−8

kSFC(ε)/kbMT(ε) 4.221 · 1016 49.856 · 10−8

kMT(ε)/kbMT(ε) 0.615 · 1016 0.762 · 10−8

thus in the opposite direction from its momentum.The energy jump in the bath is indicated in Figs. 4(c)

and (d) and can be defined from the potential energydifference at the hopping point [Eq. (45)]

∆Eb = ∆V b(Q‡) = Λb

(1− 2

τ

β~

), (49)

which is seen to be equal to ∂Φcl

∂τ and just justifies iden-tifying this term as the energy jump in the bath inSec. IV A. At the stationary point we have ∆Es +∆Eb−ε = 0, which confirms that the total energy is conserved.

As well as predicting the energy jump, we can also pre-dict the reactant and product vibrational states whichdominate the MLJ sum. In instanton theory the ener-gies of the two trajectories qn(u) making up the reducedinstanton, defined by Eq. (28), indicate the energies withthe largest contributions to the thermal rate. In this har-monic system we can relate the energies directly to thevibrational quantum numbers as the energy levels areknown. This would of course not be possible in a com-plex system, although knowledge of the energy in thesubsystem before and after the reaction, which providessimilar insight, would still be available.

As one can read from the table, for the inverted-regime

model, the instanton energies correspond to a transi-tion from the reactant ground vibrational state µ ≈ 0to the product state ν ≈ 15. The dominant vibrationalchannel in the normal-regime model is predicted to in-volve an excited reactant vibrational state µ ≈ 3 andthe product state ν ≈ 11. A comparison of these valueswith Figs. 1(c) and 1(d) reveals that, for both models,the instanton energy picks out the same dominant vi-brational channel as MLJ theory. Note that instantontheory does not actually quantize the reactant and prod-uct wells as it relies solely on imaginary-time trajectorieswhich exist only in the classically forbidden regions. Itdoes not therefore give integer values for the dominantstates. This is however not a serious concern as there isno particular relevance of the individual state with thelargest contribution because typically MLJ theory pre-dicts that a cluster of states are involved and thus anyprediction within the cluster is practically as good.139

Furthermore, for a subsystem in conjunction with aclassical harmonic bath, the activation energy of the bathmodes can be easily recovered using Eq. (45) to give

V b0 (Q‡) = Λb

(τ

β~

)2

, (50)

which should be evaluated at the stationary value of τ .The values for the bath activation energy obtained fromrSCI theory are also given in Table II and are in goodagreement (i.e. with an error less than the thermal en-ergy) with the results obtained from MLJ in theory givenin Sec. III B.

In the inverted-regime model, as can be seen inFig. 4(c), the bath activation energy is thus substantiallylower than it would be if there were no subsystem, forwhich it would correspond to the point where the poten-tials cross. The presence of the subsystem therefore leadsto a significant speed-up of the reaction, which explainswhy the rate for the full system in Table II is many or-ders of magnitude larger than the corresponding reactiontaking place in the bath only. However, the rate is notonly dependent on the bath activation energy but alsodepends on the action of the subsystem instanton, as canbe seen from Eq. (44). As previously discussed, the sta-tionary value of the bath configuration, Q‡, is associatedwith an energy jump ∆V b(Q‡)−ε, that must be compen-sated by ∆Es = ∆V s(q‡) with an equal magnitude butopposite sign in the subsystem in order to satisfy energyconservation. Panels (c) and (e) of Fig. 4 illustrate thatminimizing the bath activation energy causes the tun-nelling pathway in the subsystem to lengthen which in-creases the system action. The elongation of the path canbe understood from Fig. 4(e) which shows that, in orderto reach a point where the potential-energy difference be-tween the subsystem potentials exactly compensates theenergy jump in the bath, the system has to travel “up-hill” to the left. Hence, in general a compromise has tobe made between minimizing the bath activation energyand the subsystem action. Only in one particular case isthe magnitude of the energy jump at the reactant mini-

16

mum in the bath (including the driving force) identical tothe potential-energy difference at the reactant minimumin the subsystem such that an activationless reaction be-comes possible. In this special case, the energy jump in

the bath is Λb−ε and in the system is ∆V s(q(0)min), which

for our example where V s0 (q

(0)min) = V s

1 (q(1)min), leads to the

requirement ε = Λ. The rates in the inverted regime aremuch faster than in the fully classical treatment, becausethe reaction within the subsystem can proceed via quan-tum tunnelling, as depicted in Figs. 4(e) and (f), insteadof relying on thermal activation. Altogether, this impliesthat, although the turnover curve in the inverted regimeis not as steep as it would be according to the classicaltheory, it does not become independent of ε.

On the other hand, in the normal regime, the bathactivation energy in Fig. 4(d) is seen to be higher thanit would be without the presence of the subsystem. Thiscauses the rate for the full system to decrease relativeto the electron-transfer reaction in the bath only. Thetunnelling effect increases the rate relative to a classicalcalculation, although typically not as dramatically as inthe inverted regime. This can also be understood froman analysis of the instanton tunnelling trajectories as wasexplained in Ref. 84.

Thus, we were able to show how practically all in-sights from MLJ theory including, first and foremost, thedominantly contributing reactant and product energiescan equally be obtained from reduced instanton theory.Semiclassical instanton theory further allows one to at-tain this understanding of the reaction even in complex,anharmonic systems. This information is complementedby the localization of the optimal tunnelling pathway inthe subsystem, which can be interpreted as the reactionmechanism in configuration space.

VI. CONCLUSIONS

We have developed an instanton formulation of MLJtheory, which focuses on the subsystem while including aclassical or quantum harmonic bath implicitly. This pro-vides a practical method to complement the simulationof electron-transfer reactions of multidimensional anhar-monic subsystems by the effect of a solvent environment.Thus, the method is ideally suited to study problems thatnecessitate multiscale modelling.

Electron-transfer rates have been calculated and com-pared to results from several other methods for an asym-metric anharmonic model and the results demonstratethat reduced semiclassical instanton theory is in excel-lent agreement with either the exact rate or the MLJrate depending on whether the bath is assumed to beclassical or not. Thus we argue that semiclassical instan-ton theory can be reliably employed in situations whichhave previously been simulated by MLJ theory.

In addition to MLJ theory, we have also compared ourapproach to the second-order cumulant expansion, a pop-ular method commonly used to describe electron-transfer

and optical transition rates, and to the semiclassicalFranck–Condon sum. The results obtained with boththese approximations exhibit severe errors, especially inthe normal regime and in fact, unlike instanton theory,neither the cumulant expansion nor the SFC approxima-tion satisfy the detailed balance relation [Eq. (39)]. Thisunderlines the fact that, although both the SCI and SFCmethods have been termed “semiclassical”, the approxi-mations are quite unrelated.

We also compared and contrasted the insight that MLJand instanton theories can offer into the mechanism ofelectron-transfer reactions. The traditional MLJ pictureis shown along the bath coordinates, in which the sub-system has an effect by shifting the reactant and prod-uct potential by their respective internal energy levels.Although undoubtedly simple and intuitive in one di-mension, this picture quickly becomes convoluted when amultidimensional anharmonic subsystem has to be con-sidered. There is also little insight given into the tun-nelling dynamics of the subsystem itself.

Instanton theory, on the other hand, automatically lo-cates a unique reaction coordinate which describes theoptimal tunnelling pathway of the subsystem modes. Inanalogy to the dynamics of open quantum systems, theaddition of a bath changes the instanton pathway in thesubsystem such that, due to energy exchange betweensubsystem and bath, the reduced instanton exhibits anadditional jump in energy at the hopping point. Al-though the energy in the subsystem is therefore not con-served by the electron-transfer reaction, the excess energyis absorbed by the bath such that the total energy is con-served as it of course should be. This picture of tunnellingunder the barrier along a reaction coordinate reflects thetypical situation of practical simulations, where the focusis on the subsystem under the influence of a surroundingsolvent bath.

Nonetheless, we have also discussed how instanton the-ory is connected to MLJ theory by deriving them bothfrom a common expression. This shows that in principlesimilar insights can be extracted from either method. Inparticular, we show that instanton theory can success-fully predict the same dominant initial and final vibra-tional state of the system before and after the electron-transfer event as MLJ theory.

Instanton theory overcomes the main disadvantage ofMLJ theory, which is that it requires knowledge of the en-ergy levels and wavefunctions of the subsystem. Becauseof this, applications of MLJ theory are often limited to aharmonic-oscillator approximation, which introduces anuncontrolled error when simulating an anharmonic sys-tem. Hence, although rSCI (with a classical bath) istechnically an approximation to MLJ theory, in manyanharmonic cases it will lead to more accurate resultsdue to its ability to account for anharmonicity along thetunnelling pathway. In conjunction with a ring-polymerdiscretization, instanton theory can be applied directly tomultidimensional anharmonic problems. The applicationof this theory to electron-transfer reactions, spin transi-

17

tions and energy-transfer processes of molecular systemsin combination with high-level ab-initio electronic struc-ture methods, as has been used in previous instantonstudies,92–94 will be integral part of future work.

ACKNOWLEDGEMENTS

This work was financially supported by the Swiss Na-tional Science Foundation through SNSF Project 175696.

Appendix A: Quantum and classical rate formulas for ananharmonic subsystem mode in conjunction with aharmonic bath

In order to put the instanton and MLJ results shownin Fig. 3 into context, we also present the exact quan-tum rates for this system and their classical limits. Thisenables us not only to directly check the quality of theresults obtained with the approximate methods, but bycomparison with the classical rates also allows an esti-mation of the relevance of nuclear quantum effects. Inthis section, we harness the formal framework laid outin Sec. II to derive the required rate formulas makinguse of the fact that we can analytically integrate out thecoordinate-dependence of the harmonic bath.

If the wavefunctions of the subsystem are known, asis the case for the two crossing Morse oscillators usedin Sec. IV B, the trace over the subsystem degrees offreedom in Eq. (7a) can be evaluated exactly in thewavefunction representation. Thus, the exact quantum-mechanical rate can be computed by the formula

k(ε)Zs0 =

∆2

~2

∫ ∞−∞

dt∑∫µ

e−βEµ0

∑∫ν

|θµν |2

× e−(τ+it)(Eν1−Eµ0−ε)/~−Φ(τ+it)/~, (A1)

where sums are taken over the bound states of the Morsepotential and integrals are carried out over the ener-gies of the energy-normalized continuum states. Expres-sions for the wavefunctions can be found in Refs. 140and 141. Numerical integration was used to obtain theFranck–Condon overlaps and to perform the integral overtime. In order to make the latter converge easily, theimaginary-time variable τ was chosen appropriately (i.e.using the value obtained from the instanton optimiza-tion).

The classical limit of this rate can be obtained in asimilar way except that the trace in Eq. (7a) is evalu-ated by a classical phase-space integral106 and the clas-sical limit of the effective bath action is used [Eq. (11)].For a one-dimensional subsystem, this gives the classical

golden-rule transition-state theory rate

kTST(ε)Zs0 =

∆2

~3

√m

2Λb

∫e−βV

s0 (q)

× e−β(Λb−ε+∆V s(q))2/4Λb

dq, (A2)

where the reactant partition function is computed by aclassical phase-space integral. The remaining integral inthe subsystem mode can either be taken numerically, aswas done to generate the results in Fig. 3, or by steepestdescent, which would be an excellent approximation inthis case.

The quantum-mechanical rate [Eq. (A1)], as well as theMLJ rate [Eq. (15)] correctly reduce to the TST expres-sion in Eq. (A2) in the high-temperature or low-frequencylimit, while instanton theory [Eq. (30)] reduces to thesteepest-descent version of it.

In the special case that the subsystem consists of dis-placed harmonic oscillators, Eq. (A2) reduces to the Mar-cus theory expression [Eq. (13)] except that in this casethe reorganization energy should be the sum of the sub-system and bath reorganization energies.

Appendix B: Cumulant expansion

Another approximate way of computing correlationfunctions and therefore also to calculate electron-transfer reaction rates is the so called “cumulantexpansion”,118–120 which in our formulation will be ap-plied to the lineshape function of the subsystem Eq. (7a).

In its conventional formulation τ is set to zero and werewrite Eq. (7a) as

Is(v) =

∫ ∞−∞

eivt/~R(t) dt, (B1)

where the correlation function is

R(t) = (Zs0)−1

Trs

[e−(β~−it)Hs

0/~ e−itHs1/~]. (B2)

The time-dependent terms inside the trace can equallybe rewritten as a time-ordered exponential according to

e+itHs0/~ e−itHs

1/~ = T e−i∫ t0

∆V sI (t′)dt′/~ where T is the

time-ordering operator and we make use of the inter-

action picture to give ∆V sI (t) = e+iHs

0t/~ ∆V s e−iHs0t/~,

where ∆V s = Hs1− Hs

0 = V s1 (q)−V s

0 (q).95 This exact ex-pression can then be expanded in a time-ordered powerseries with respect to ∆V s

I .Motivated by the analytic solution for the correlation

function of a system of displaced harmonic oscillators[Eq. (9)], one makes the ansatz R(t) = exp [−Γ(t)] wherethe exponent is defined as a sum of cumulants

Γ(t) =

∞∑j=1

Γj(t), (B3)

18

where Γj(t) is of jth order in ∆V sI . Comparing the two

expansions, the first two terms in Eq. (B3) are given by95

Γ1(t) =i

~(Zs

0)−1∫ t

0

dt1 Trs

[e−βH

s0∆V s

I (t1)], (B4a)

Γ2(t) = 12Γ2

1(t) +1

~2(Zs

0)−1

×∫ t

0

dt1

∫ t1

0

dt2 Trs

[e−βH

s0∆V s

I (t1)∆V sI (t2)

],

(B4b)

whereas higher cumulants are neglected in the expan-sion. The expressions in Eqs. (B4) can be evaluated by

expanding the traces in the energy-eigenstate basis of Hs0.

Performing the time-integrals analytically results in theequations

Γ1(t) =it

~(Zs

0)−1∑µ

e−βEµ0 ∆V s

µµ, (B5a)

Γ2(t) = 12Γ2

1(t) + (Zs0)−1∑µ

∑µ′

e−βEµ0 |∆V s

µµ′ |2

× 1 + i(Eµ0 − Eµ′

0 )t/~− ei(Eµ0−Eµ′0 )t/~

(Eµ0 − Eµ′

0 )2, (B5b)

where ∆V sµµ′ =

∫∞−∞ ψµ0 (q)∗∆V s(q)ψµ

′

0 (q) dq and theseintegrals over the one-dimensional subsystem coordinateare evaluated numerically. The terms in the sum ofEq. (B5b) with µ = µ′ can be evaluated by L’Hpital’srule. For the case of displaced harmonic oscillators, thisexpansion of the correlation function up to second ordergives the exact result, as all higher order terms vanish.95

In the general, anharmonic case, however, the quality ofthe approximation is unclear. One could of course extendthe method to higher orders, but the series is unlikely toconverge quickly to correct result.

The computational advantage of the cumulant expan-sion over the golden-rule formula is that only the eigen-states of the reactant electronic state need be known.It is thus perhaps most useful when computing absorp-tion spectra from a ground electronic state to an excitedstate, for which the ground state is well approximated bya harmonic oscillator, but not the excited state. However,a significant knowledge of the product potential-energysurface is still required in the region where the wave-function overlaps in Eqs. (B5) are sizeable, which can beexpensive to compute.

This result for the subsystem’s lineshape function canbe easily combined with the lineshape function of theharmonic bath from Eq. (9) by performing the convolu-tion integral in v and integrating over the resulting deltafunction. The rate expression based on the second-ordercumulant expansion for the subsystem part is thereforegiven by

kCE(ε) =∆2

~2

∫ ∞−∞

eiεt/~ e−Γ1(t)−Γ2(t)−Φ(it)/~ dt, (B6)

where either a quantum or classical bath can be em-ployed by using the respective expressions for the actionsin Eqs. (10) and (11) and the final time-integral is carriedout numerically.

Note that this cumulant expansion leads to a com-pletely different approximation from that of Wolynestheory59 even though the latter can also be thought ofas a type of cumulant expansion. In contrast to the ap-proach described here, Wolynes theory carries out thetime integral by the method of steepest-descent and com-putes the short-time limit of the correlation function bypath-integral sampling. For the systems studied in thiswork, Wolynes theory would give similar results to thoseof instanton theory (identical in the case of a harmonicsystem), although for certain more complex systems ithas been shown to break down.70,85 Unlike the cumulantexpansion and instanton theory,84 it is also not directlyapplicable to the inverted regime, although an extrap-olation method which extends it in this way has beensuggested.67

Appendix C: Semiclassical Franck–Condon Sum

The “semiclassical Franck–Condon sum” is an alterna-tive way of approximating the electron-transfer rate andcan be obtained from Eq. (5) by neglecting the commu-

tator between H0 and H1 in both subsystem and bath,setting τ to zero and evaluating the trace in the reactant’seigenfunction basis.96,106,107 For the lineshape function ofthe one-dimensional subsystem, this results in

IsSFC(v) =

2π~Zs

0

∑µ

e−βEµ0

∫ψµ0 (q)∗ δ(∆V s(q)−v)ψµ0 (q) dq,

(C1)where by virtue of neglecting the commutators,we were able to make the classical approximation∫

eiHs0t/~ e−i(Hs

1−v)t/~ dt ≈ 2π~ δ(Hs1 − Hs

0 − v). Becausethe kinetic part vanishes in the difference of the Hamil-tonians the final expression can be written in terms of∆V s(q) ≡ V s

1 (q)− V s0 (q).

The same strategy is used to deal with the bath. How-ever, as described in the appendix of Ref. 96, becausethe bath is harmonic, the sums and integrals can be per-formed analytically to give

IbSFC(ε− v) =

√πβ~2

χΛbe−β(Λb−ε+v)2/4χΛb

, (C2)

which has the same form as that of Marcus theory ex-

cept for the correction factor, χ =∑Dj=1 Λb

j γj coth γj/Λb,

which is defined in terms of the reorganization energyassociated with a single bath mode Λb

j = 2MΩ2jζ

2j and

γj = β~Ωj/2.

Following the formalism laid out in Sec. II and per-forming the convolution integral in v first, this leads the

19

rate equation

kSFC(ε)Zs0 =

∆2

~

√πβ

χΛb

∑µ

e−βEµ0

×∫ψµ0 (q)∗ e−β(Λb−ε+∆V s(q))2/4χΛb

ψµ0 (q) dq, (C3)

where the integrals over the anharmonic subsystem modehave to be carried out numerically.

In the special case in which all modes are displacedharmonic oscillators, all degrees of freedom can be as-signed to the bath. Then, the rate formula is directly

given by kSFC(ε) = ∆2

~2 IbSFC(ε). It is easy to see that this

is in error because it predicts results symmetric aroundε = Λ, whereas the true result is known to be significantlyskewed unless in the classical limit.11,20,51 One way to un-derstand the causes of this error has been explained interms of WKB theory in Ref. 142.

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