Direct simulation of proton-coupled electron transfer across...

Direct simulation of proton-coupled electron transfer across multipleregimesJoshua S. Kretchmer and Thomas F. Miller Citation: J. Chem. Phys. 138, 134109 (2013); doi: 10.1063/1.4797462 View online: http://dx.doi.org/10.1063/1.4797462 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i13 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 138, 134109 (2013)

Direct simulation of proton-coupled electron transferacross multiple regimes

Joshua S. Kretchmer and Thomas F. Miller IIIa)Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena,California 91125, USA

(Received 18 January 2013; accepted 10 March 2013; published online 2 April 2013)

The coupled transfer of electrons and protons is a central feature of biological and molecular catal-ysis, yet fundamental aspects of these reactions remain poorly understood. In this study, we extendthe ring polymer molecular dynamics (RPMD) method to enable direct simulation of proton-coupledelectron transfer (PCET) reactions across a wide range of physically relevant regimes. In a system-bath model for symmetric, co-linear PCET in the condensed phase, RPMD trajectories reveal dis-tinct kinetic pathways associated with sequential and concerted PCET reaction mechanisms, and it isdemonstrated that concerted PCET proceeds by a solvent-gating mechanism in which the reorganiza-tion energy is mitigated by charge cancellation among the transferring particles. We further employRPMD to study the kinetics and mechanistic features of concerted PCET reactions across multi-ple coupling regimes, including the fully non-adiabatic (both electronically and vibrationally non-adiabatic), partially adiabatic (electronically adiabatic, but vibrationally non-adiabatic), and fullyadiabatic (both electronically and vibrationally adiabatic) limits. Comparison of RPMD with the re-sults of PCET rate theories demonstrates the applicability of the direct simulation method over abroad range of conditions; it is particularly notable that RPMD accurately predicts the crossover inthe thermal reaction rates between different coupling regimes while avoiding a priori assumptionsabout the PCET reaction mechanism. Finally, by utilizing the connections between RPMD rate the-ory and semiclassical instanton theory, we show that analysis of ring-polymer configurations in theRPMD transition path ensemble enables the a posteriori determination of the coupling regime for thePCET reaction. This analysis reveals an intriguing and distinct “transient-proton-bridge” mechanismfor concerted PCET that emerges in the transition between the proton-mediated electron superex-change mechanism for fully non-adiabatic PCET and the hydrogen atom transfer mechanism forpartially adiabatic PCET. Taken together, these results provide a unifying picture of the mechanismsand physical driving forces that govern PCET across a wide range of physical regimes, and theyraise the possibility for PCET mechanisms that have not been previously reported. © 2013 AmericanInstitute of Physics. [http://dx.doi.org/10.1063/1.4797462]

I. INTRODUCTION

Proton-coupled electron transfer (PCET) reactions, inwhich both an electron and an associated proton undergo reac-tive transfer (Fig. 1(a)), play an important role in many chem-ical and biological processes.1–4 Key examples include thetyrosine oxidation step of photosystem II5, 6 and the proton-pumping mechanism of cytochrome c oxidase.7, 8 Dependingon the chronology of the electron- and proton-transfer eventsand the magnitudes of the electronic and vibrational cou-pling, a variety of reactive processes can fall under the um-brella of PCET;9–13 investigation of the dynamics that governthis full range of behavior provides significant experimentaland theoretical challenges, and the characterization of tran-sitions between different regimes of PCET remains incom-plete. In this study, we extend the ring polymer moleculardynamics (RPMD) method to allow for the direct simulationof PCET reaction dynamics and to characterize condensed-phase PCET reaction mechanisms and thermal rates across awide range of physically relevant regimes.

a)Electronic mail: [email protected]

PCET reactions are typically described (Fig. 1(b)) interms of the following reactant, intermediate, and productspecies:1, 9, 14–16

D − H + A (OU),D− + [H − A]+ (OP),[D − H]+ + A− (RU),

D + H − A (RP).Here, D and A indicate the donor and acceptor molecules,

respectively, and the labels O/R and U/P indicate the oxi-dation state (oxidized or reduced) and the protonation state(unprotonated or protonated) of the acceptor molecule. Thereactions can be categorized among two groups, sequentialand concerted PCET, depending on whether both the electronand proton transfer in a single chemical step (Fig. 1(b)).9, 14–16

Sequential PCET exhibits distinct electron-transfer (ET) andproton-transfer (PT) reaction events separated by a metastableintermediate species; concerted PCET exhibits the transfer ofboth particles in a single reactive step, bypassing the forma-tion of the OP and RU species in Fig. 1(b). Within these two

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134109-2 J. S. Kretchmer and T. F. Miller III J. Chem. Phys. 138, 134109 (2013)

De Dp H Ap Ae

H+

e

p

O

k

D H + A

D + H A

D H[ ] + + A

D + H A[ ] +

p-

O

k

e

U

k

e-

U

k

p

R

k

e

P

k

CPETk

(a)

(b)

FIG. 1. (a) Schematic illustration of a co-linear PCET reaction, where De/Dpand Ae/Ap are the respective donor and acceptor for the electron/proton.(b) Schematic illustration of sequential and concerted PCET reaction mech-anisms, indicating the rate constants for the individual charge transfer pro-cesses. The sequential mechanism proceeds along the horizontal and verticaledges of the schematic, whereas the concerted mechanism proceeds along thediagonal.

broad categories for PCET, there exist a range of couplingregimes that depend on the degree of electronic and vibra-tional non-adiabaticity for the PCET reaction.9–13

Rate theories have been derived and successfully em-ployed to study concerted PCET reactions in a vari-ety of limiting regimes, including (i) the fully non-adiabatic regime1, 17–19 in which the reaction is electronicallyand vibrationally non-adiabatic, (ii) the partially adiabaticregime10, 12, 20, 21 in which the reaction is electronically adia-batic and vibrationally non-adiabatic, and (iii) the fully adi-abatic regime20, 21 in which the reaction is both electroni-cally and vibrationally adiabatic. These rate theories, whichgenerally employ Golden Rule and linear response approx-imations, provide a powerful toolkit for investigating bothconcerted and sequential PCET reactions in many systems.However, the applicability of any given rate theory is limitedto the particular coupling regime for which it was derived, andwithout prior mechanistic information about a given PCETreaction, it can be difficult to know which formulation to ap-ply in practice. Furthermore, with few exceptions,12 existingrate theories do not offer scope for the study of PCET reac-tions with intermediate values for the electronic and vibra-tional coupling, which exist between the limiting regimes forwhich the rate theories have been derived. Methods that en-able the direct simulation of PCET reactions across all elec-tronic and vibrational coupling regimes, including interme-diate regimes, are needed to achieve a unified picture for thedynamics, mechanisms, and driving forces that govern the fullrange of PCET reactions.

Fundamental theoretical challenges in the description ofPCET reactions arise due to the coupling of intrinsically

quantum mechanical ET and PT dynamics with slower mo-tions of the surrounding environment. New simulation meth-ods are needed to accurately describe this electron-proton-environment dynamics and to efficiently and robustly simu-late long trajectories that bridge the multiple timescales ofthese reactions. In this study, we address these challengesby extending the RPMD method to enable the direct simu-lation of condensed-phase PCET reactions. RPMD22 is an ap-proximate quantum dynamical method that is based on Feyn-man’s imaginary-time path integral formulation of statisticalmechanics.23, 24 It provides a classical molecular dynamicsmodel for the real-time evolution of a quantum mechanicalsystem that rigorously preserves detailed balance and samplesthe quantum Boltzmann distribution.24–26 The RPMD methodhas been previously employed to investigate a wide rangeof quantized reactive and dynamical processes,27–40 rangingfrom gas-phase triatomic reactions27 to enzyme-catalyzed hy-drogen tunneling.28 We have demonstrated that RPMD simu-lations can be extended to accurately and efficiently describecoupled electronic and nuclear dynamics in condensed-phasesystems, including excess electron diffusion,31 injection,32

and reactive transfer.33 Prior validation of RPMD for the de-scription of ET reactions throughout the normal and activa-tionless regimes,33 in combination with prior demonstrationof the method for a range of H-transfer processes,27–30 pro-vides a basis for expecting the method to adequately describethe dynamics of PCET reactions, which will be tested in thecurrent study.

Alternative theoretical methods have previously ad-vanced our ability to simulate and understand coupledelectronic and nuclear dynamics,41–52 and promising newmethods continue to be introduced.53 Established methods in-clude Ehrenfest dynamics,41, 42 mixed quantum-classical tra-jectory surface hopping dynamics,43–48 the ab initio multiplespawning approach,49 and semiclassical methods based onthe Meyer-Miller-Stock-Thoss mapping.50–52 However, de-spite their successes, these methods do not yield a dynam-ics that rigorously preserves detailed balance,54, 55 a featurethat is valuable for the robust calculation of thermal reac-tion rates56, 57 and for the utilization of rare-event samplingmethods.58, 59 Although it is clear that other methods must bepart of the toolkit for understanding PCET reactions, we em-phasize that the formal properties of the RPMD method areideally suited to this goal.

In this paper, we extend the RPMD method to allow fordirect simulation of co-linear, condensed-phase PCET reac-tions across a wide range of physically relevant regimes. Inaddition to providing validation for the simulation method viaextensive comparison with existing PCET rate theories, weanalyze the RPMD reactive trajectories to elucidate a varietyof mechanisms for the concerted charge-transfer process. Thepresented analysis offers a unifying picture for PCET acrossa wide range of physical regimes, and it suggests new PCETregimes that have yet to be characterized.

II. RING POLYMER MOLECULAR DYNAMICS

The RPMD equations of motion for N particles that arequantized using n ring-polymer beads are22, 31



v̇(α)i = ω2n

(q

(α+1)i + q(α−1)i − 2q(α)i

)− 1

mi

∂

∂q(α)i

U(q

(α)1 , q

(α)2 , . . . , q

(α)N

), (1)

where v(α)i and q(α)i are the velocity and position of the αth

bead for the ith particle, respectively, and q(0)i = q(n)i . Thephysical mass for particle i is given by mi, ωn = n/(β¯) isthe intra-bead harmonic frequency, and β = (kBT )−1 is thereciprocal temperature. The potential energy function of thesystem is given by U(q1, . . . , qN).

To allow for the straightforward comparison with PCETrate theories, we quantize only the transferring electron andproton in this study and consider the classical (i.e., 1-bead)limit for the N solvent degrees of freedom. Furthermore,we employ a mixed-bead-number path-integral representationthat reduces the cost of the potential energy surface calcu-lations by utilizing the more rapid convergence of the path-integral distribution for heavier particles.60 We thus obtain themodified RPMD equations of motion:

v̇(α)e = ω2ne(q(α+1)e + q(α−1)e − 2q(α)e

)− 1

me

∂

∂q(α)e

U(q(α)e , q

((α−k) 1nep

+1)p , Q

), (2)

v̇(γ )p = ω2np(q(γ+1)p + q(γ−1)p − 2q(γ )p

)

− 1mp

nep∑l=1

∂

∂q(γ )p

U(q

((γ−1)nep+l)e , q

(γ )p , Q

), (3)

and

V̇j = − 1neMj

np∑γ=1

nep∑l=1

∂

∂QjU(q


(γ )p , Q

), (4)

where ne is the number of imaginary-time ring-polymer beadsfor the transferring electron, me is the physical mass for theelectron, and q(α)e and v

(α)e are the respective position and

velocity for the αth ring-polymer bead of the electron; thecorresponding quantities for the transferring proton are indi-cated using subscript “p.” In Eqs. (2)–(4), it is assumed thatnep = ne/np is an integer number, and

k = α − nep⌊

α − 1nep

⌋, (5)

where �. . . � denotes the floor function. As before, theperiodic constraint of the ring-polymer is satisfied viaq(0)e = q(ne)e and q(0)p = q(np)p , and the intra-bead harmonic fre-quencies are ωne = ne/(β¯) and ωnp = np/(β¯). The posi-tion, velocity, and mass for the jth classical solvent degreeof freedom are given by Qj, Vj , and Mj, respectively, andQ = {Q1, . . . ,QN }.

In the limit of a large number of ring-polymer beads, theRPMD equations of motion yield a time-reversible molecu-lar dynamics that preserves the exact quantum mechanicalBoltzmann distribution.24–26 Equations (2)–(4) introduce noapproximation to Eq. (1) beyond taking the classical limit ofthe solvent degrees of freedom.

Analogous to the classical thermal rate constant,61–63 theRPMD thermal rate constant can be expressed as56, 57

kRPMD = limt→∞ κ(t)kTST, (6)

where kTST is the transition state theory (TST) estimate forthe rate associated with the dividing surface ξ (r) = ξ ‡, ξ (r)is a collective variable that distinguishes between the reactantand product basins of stability, and κ(t) is the time-dependenttransmission coefficient that accounts for recrossing of tra-jectories through the dividing surface. We have introducedr = {q(1)e , . . . , q(ne)e , q(1)p , . . . q(np)p ,Q1, . . . QN } to denote theposition vector for the full system in the ring-polymer rep-resentation. As is the case for both exact classical and exactquantum dynamics, the RPMD method yields reaction ratesand mechanisms that are independent of the choice of divid-ing surface.56, 57, 64

The TST rate in Eq. (6) is calculated using29, 33, 65, 66

kTST = (2πβ)−1/2〈gξ 〉c e−βF (ξ ‡)∫ ξ ‡

−∞ dξe−βF (ξ )

, (7)

where F(ξ ) is the free energy (FE) along ξ ,

e−βF (ξ ) = 〈δ(ξ (r) − ξ )〉〈δ(ξ (r) − ξr )〉 , (8)

ξ r is a reference point in the reactant basin, and29, 67–69

gξ (r) =[

d∑i=1

1

mi

(∂ξ (r)∂ri

)2]1/2. (9)

Here, ri is an element of the position vector r, mi is the corre-sponding physical mass, and d is the length of vector r. Theequilibrium ensemble average is denoted as

〈. . .〉 =∫

dr∫

dv e−βH (r,v)(. . .)∫dr

∫dv e−βH (r,v)

, (10)

and the average over the ensemble constrained to the dividingsurface is denoted as

〈. . .〉c =∫

dr∫

dv e−βH (r,v)(. . .)δ(ξ (r) − ξ ‡)∫dr

∫dv e−βH (r,v)δ(ξ (r) − ξ ‡) , (11)

where

H (r, v) =N∑

j=1

1

2MjV

2j +

ne∑α=1

1

2mb,e

(v(α)e

)2

+np∑

γ=1

1

2mb,p

(v(γ )p

)2 + URP(r). (12)Here, mb,e and mb,p are the fictitious Parrinello-Rahmanmasses for the electron and proton, respectively,25

v = {v(1)e , . . . , v(ne)e , v(1)p , . . . , v(np)p , V1, . . . , VN } is thevelocity vector for the full system in the ring-polymer



representation, and

URP(r) = 1ne

ne∑α=1

1

2meω

2ne

(q(α)e − q(α−1)e

)2

+ 1np

np∑γ=1

1

2mpω

2np

(q(γ )p − q(γ−1)p

)2

+ 1ne

np∑γ=1

nep∑l=1

U(q


(γ )p , Q

). (13)

The transmission coefficient in Eq. (7) is obtained fromthe flux-side correlation function,56, 57

κ(t) = 〈ξ̇0h(ξ (rt ) − ξ‡)〉c

〈ξ̇0h(ξ̇0)〉c, (14)

by releasing RPMD trajectories from the equilibrium en-semble constrained to the dividing surface. Here, h(ξ ) isthe Heaviside function, ξ̇0 is the time-derivative of the col-lective variable upon initialization of the RPMD trajectoryfrom the dividing surface with the initial velocities sampledfrom the Maxwell-Boltzmann (MB) distribution, and rt isthe time-evolved position of the system along the RPMDtrajectory.

III. PCET RATE THEORIES

A primary focus of this study is to compare the RPMDmethod with rate theories that have been derived for the var-ious limiting regimes of PCET. We thus summarize thesePCET rate theories below.

A. Concerted PCET in the fully adiabatic regime

For the fully adiabatic regime, both the electronic cou-pling and vibrational coupling between the concerted PCETreactant and product states are large in comparison to the ther-mal energy, kBT. The reaction proceeds in the ground vibronicstate, and it is appropriately described using the expression ofHynes and co-workers20, 21

kadCPET =ωs

2πexp

[−G‡ad

kBT

], (15)

where ωs is the solvent frequency, G‡ad is the free-energy

barrier for the reaction calculated from the difference ofthe ground vibronic energy level at its minimum and at itsmaximum with respect to the solvent coordinate, and kB isBoltzmann’s constant.

B. Concerted PCET in the partially adiabatic regime

For the partially adiabatic regime, the electronic cou-pling is large in comparison to kBT, whereas the vibrationalcoupling is small in comparison to kBT. The reaction pro-ceeds in the ground electronic state, and it is appropriatelydescribed using the expression of Cukier10 and Hynes and co-

workers,20, 21

kpadCPET =

2π

¯ V2μν (4πλkBT )

−1/2 exp[−G‡

kBT

], (16)

where λ is the concerted PCET reorganization energy associ-ated with the transfer of both the electron and proton,

G‡ = (λ + G0)2

4λkBT, (17)

G0 is the driving force for the concerted PCET reaction,and Vμν is the vibronic coupling. In this regime, the vibroniccoupling is equal to the vibrational coupling, VPT, such that

Vμν = VPT

= E1 − E02

. (18)

VPT is obtained from the splitting between the vibrationalground state energy, E0, and first excited state energy, E1,calculated on the lowest electronic adiabat. Equation (16) as-sumes that only a single initial and final vibrational states areinvolved in the concerted PCET reaction.10, 20, 21

C. Concerted PCET in the fully non-adiabatic regime

For the fully non-adiabatic regime, both the electroniccoupling and vibrational coupling are small in comparison tokBT. The reaction is appropriately described using the expres-sion of Cukier17, 18 and Hammes-Schiffer and co-workers1, 19

knadCPET =2π

¯∑

μ

Pμ∑

ν

V 2μν (4πλkBT )−1/2 exp

[−G‡μν

kBT

],

(19)where μ and ν index the reactant and product vibrationalstates, respectively, Pμ is the Boltzmann probability of thereactant vibrational state, and

G‡μν =(λ + G0 + �ν − �μ)2

4λkBT, (20)

where �μ and �ν are the respective energies of the reactantand product vibrational states relative to their correspondingground state. In this regime, the vibronic coupling is given by

Vμν = 〈μ|ν〉VET, (21)where 〈μ|ν〉 is the overlap between reactant and product vi-brational wavefunctions, and VET is the electronic coupling.

D. ET rate theories

We also compare RPMD simulations with rate theo-ries that correspond to the electronically adiabatic and non-adiabatic regimes for pure ET. These ET rate theories are sum-marized below.

1. Adiabatic ET

For the electronically adiabatic regime, the electroniccoupling between the reactant and product ET states islarge in comparison to kBT. The reaction proceeds in the



ground electronic state, and it is appropriately described usingEq. (15), except with the free-energy barrier, G‡ad, calculatedfrom the difference of the ground electronic energy level atits minimum and at its maximum with respect to the solventcoordinate.70, 71

2. Non-adiabatic ET

For the electronically non-adiabatic regime, the elec-tronic coupling is small in comparison to kBT. The reactionis appropriately described using the standard Marcus theoryexpression,72–74

knadET =2π

¯ |VET|2(4πλkbT )

−1/2 exp[−G‡

kBT

], (22)

where

G‡μν =(λ + G0)2

4λkBT. (23)

Here, VET, λ, and G0 are, respectively, the electronic cou-pling, reorganization energy, and driving force associatedwith the ET reaction.

IV. PCET MODEL SYSTEMS

Throughout this paper, condensed-phase PCET is de-scribed using a co-linear system-bath model. The model isexpressed in the position representation using the potentialenergy function

U (qe, qp, qs, Q) = Usys(qe, qp, qs) + UB(qs, Q), (24)where UB(qs, Q) is the potential energy term associated withthe bath coordinates, and

Usys(qe, qp, qs) = Ue(qe) + Up(qp) + Us(qs)+Ues(qe, qs) + Ups(qp, qs)+Uep(qe, qp) (25)

is the system potential energy. The scalar coordinates qe, qp,and qs describe the positions of the electron, proton, and sol-vent modes, respectively, and Q is the vector of bath oscillatorpositions.

The first term in the system potential energy functionmodels the interaction of the transferring electron with itsdonor and acceptor sites,

Ue(qe)=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

aDq2e + bDqe + cD, routD ≤ qe ≤ r inD

aAq2e + bAqe + cA, r inA ≤ qe ≤ routA

−μe[

1

|qe − rD| +1

|qe − rA|], otherwise

,

(26)where rD and rA are the positions of the electron donorand acceptor sites. This one-dimensional (1D) potential en-ergy function consists of two symmetric coulombic wells,each of which is capped by quadratic functions to removesingularities.

The second term in the system potential energy functionmodels the interaction between the transferring proton and itsdonor and acceptor sites,

Up(qp) = −mpω

2p

2q2p +

m2pω4p

16V0q4p . (27)

Here, ωp is the proton vibrational frequency and V0 is the in-trinsic PT barrier height.

The next three terms in the system potential energy func-tion model the solvent potential and the electron- and proton-solvent interactions. Specifically,

Us(qs) = 12msω

2s q

2s , (28)

Ues(qe, qs) = −μesqeqs, (29)and

Ups(qp, qs) = −μpsqpqs, (30)where ms is the solvent mass and ωs is the effective frequencyof the solvent coordinate. The solvent coupling parameters,μes and μps, are of opposite sign due to the opposing chargesof the transferring electron and proton.

Interactions between the transferring electron and protonare modeled via the capped coulombic potential:

Ue(qe) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

− μep|qe − qp| , |qe − qp| > Rcut

− μepRcut

, otherwise

. (31)

The potential energy term UB(qs, Q) models the har-monic bath that is coupled to the PCET reaction. The bathexhibits an ohmic spectral density J(ω) with cutoff frequencyωc,75, 76 such that

J (ω) = ηωe−ω/ωc , (32)where η denotes the friction coefficient. The continu-ous spectral density is discretized into f oscillators withfrequencies:56, 77

ωj = −ωc ln(

j − 0.5f

)(33)

and coupling constants:

cj = ωj(

2ηMωcf π

)1/2, (34)

such that

UB(qs, Q) =f∑

j=1

⎡⎣1

2Mω2j

(Qj − cjqs

Mω2j

)2⎤⎦ . (35)Here, M is the mass of each bath oscillator, and ωj and Qj arethe respective frequency and position for the jth oscillator.

We have developed system parameters to modelcondensed-phase PCET reactions throughout a range of dif-ferent physical regimes. Specifically, System 1 models thefully non-adiabatic regime, Systems 2a-2f model the transi-tion between the fully non-adiabatic and partially adiabatic



regimes, and Systems 3a-3e model the transition between thepartially adiabatic and fully adiabatic regimes. Full details ofthe parameterization are provided in Appendices A and B.

We also employ a system-bath model to investigate pureET in this study, with a potential energy function:

UET(qe, qs, Q) = Ue(qe) + Us(qs) + Ues(qe, qs)+UB(qsQ), (36)

that is obtained by simply removing the proton-dependentterms in Eqs. (24) and (25). Systems 4a-4g model the tran-sition between non-adiabatic and adiabatic ET. Full detailsof the parameterization for the ET reactions are provided inAppendices A and B.

V. CALCULATION DETAILS

Calculations on System 1, Systems 2a-2f, and Sys-tems 4a-4g are performed at T = 300 K; calculations onSystems 3a-3e are performed at the lower temperature ofT = 100 K to clearly exhibit the transition between the par-tially adiabatic and fully adiabatic regimes for PCET. For allsystems, the harmonic bath is discretized using f = 12 degreesof freedom.

A. RPMD simulations

In all simulations, the RPMD equations of motion areevolved using the velocity Verlet algorithm.78 As in previousRPMD simulations, each timestep for the electron and pro-ton involves separate coordinate updates due to forces arisingfrom the physical potential and due to exact evolution of thepurely harmonic portion of the ring-polymer potentials.79 Theelectron is quantized with ne = 1024 ring-polymer beads inall systems, while the proton is quantized with np = 32 ring-polymer beads for Systems 1 and 2a-2f and with np = 128 forSystems 3a-3e. The larger number of beads for Systems 3a-3eis necessary due to the lower temperature.

Two collective variables are used to monitor the PCETreaction mechanism in the RPMD simulations. The progressof the electron is characterized by a “bead-count” coordinate,fb, that reports on the fraction of ring-polymer beads that arelocated on the electron donor,

fb(q(1)e , . . . , q

(ne)e

) = 1ne

ne∑α=1

tanh(φq(α)e

), (37)

where φ = −3.0/rD. The progress of the proton is character-ized using the ring-polymer centroid in the proton positioncoordinate,

q̄p(q(1)p , . . . , q

(np)p

) = 1np

np∑γ=1

q(γ )p . (38)

1. RPMD rate calculations for concerted PCET

The RPMD reaction rate is calculated from the prod-uct of the TST rate and the transmission coefficient(Eq. (6)). The FE profiles that appear in the TST rate expres-

sion (Eq. (7)) are obtained using umbrella sampling and theweighted histogram analysis method (WHAM), as describedbelow.80, 81

For System 2f, the 1D FE profile used in the rate calcula-tion is obtained in the proton centroid coordinate, F (q̄p), us-ing the following umbrella sampling protocol. Nine indepen-dent sampling trajectories are harmonically restrained to uni-formly spaced values of q̄p in the region [−0.20 a0, 0.20 a0]using a force constant of 1.3 a.u. Additionally, 18 independentsampling trajectories are harmonically restrained to uniformlyspaced values of q̄p in both the region [−1.10 a0, − 0.25 a0]and in [0.25 a0, 1.10 a0] using a lower force constant of1.0 a.u. to ensure extensive overlap among the sampled dis-tributions. The equilibrium sampling trajectories are per-formed using path-integral molecular dynamics (PIMD) withmb,e = 2000 a.u. and mb,p = 1836.1 a.u., which allows for atimestep of 0.1 fs. Each sampling trajectory is run for 10 ns,and thermostatting is performed by re-sampling the veloc-ities from the MB distribution every 500 fs. We note thatthis choice of the Parrinello-Rahman masses, mb,e and mb,p,allows for a large timestep in the sampling trajectories buthas no affect on F (q̄p) or any other equilibrium ensembleaverage.25, 26

For all PCET systems other than System 2f, the 1D FEprofile used in the rate calculation is obtained in the electronbead-count coordinate, F(fb), using the following umbrellasampling protocol. Ninety-three independent sampling trajec-tories are harmonically restrained to uniformly spaced val-ues of fb in the region [−0.92, 0.92] using a force constant of20 a.u.; seven independent sampling trajectories are harmon-ically restrained to uniformly spaced values of fb in both theregion [−0.991, − 0.985] and in [0.985, 0.991] using a higherforce constant of 5000 a.u.; nine independent sampling trajec-tories are harmonically restrained to uniformly spaced valuesof fb in both the region [−1.0, − 0.992] and in [0.992, 1.0]using a higher force constant of 10 000 a.u.; 32 independentsampling trajectories are harmonically restrained to the val-ues of fb ∈ { ±0.93, ±0.935, ±0.94, ±0.945, ±0.95, ±0.955,±0.96, ±0.962, ±0.965, ±0.967, ±0.97, ±0.974, ±0.976,±0.978, ±0.98, ±0.982} using a force constant of 500 a.u.For Systems 1 and 2a-2e, an auxiliary restraining potential isintroduced for the PIMD sampling trajectories to restrict thesystem to the concerted channel, as described in Appendix C.Each sampling trajectory is run for 10 ns using a timestep of0.1 fs, with mb,e = 2000 a.u. and mb,p = 1836.1 a.u. Ther-mostatting is performed by re-sampling the velocities fromthe MB distribution every 500 fs.

For System 2f, the transmission coefficient (Eq. (14)) iscalculated using RPMD trajectories that are released fromthe dividing surface associated with q̄p = 0. A total of6000 RPMD trajectories are released. Each RPMD trajec-tory is evolved for 400 fs using a timestep of 1 × 10−4 fsand with the initial velocities sampled from the MB distri-bution. Initial configurations for the RPMD trajectories areselected every 10 ps from long PIMD sampling trajecto-ries that are constrained to the dividing surface. The sam-pling trajectories employ mb,e = 2000 a.u., mb,p = 1836.1a.u., and a timestep of 0.1 fs. Thermostatting is performedby re-sampling the velocities from the MB distribution every



500 fs. The sampling trajectories are constrained to the divid-ing surface using the RATTLE algorithm.82

For all PCET systems other than System 2f, the trans-mission coefficient is calculated using RPMD trajectories thatare released from the dividing surface associated with fb = 0.A total of 4500 RPMD trajectories are released for Systems1 and 2a-2e, and at least 10 000 trajectories are released forSystems 3a-3e. Each RPMD trajectory is evolved for 300 fsusing a timestep of 1 × 10−4 fs and with the initial ve-locities sampled from the MB distribution. Initial configu-rations for the RPMD trajectories are selected every 10 psfrom long PIMD sampling trajectories that are constrainedto the dividing surface. The sampling trajectories employmb,e = 2000 a.u., mb,p = 1836.1 a.u., and a timestep of 0.1 fs.Thermostatting is performed by re-sampling the velocitiesfrom the MB distribution every 500 fs. The sampling trajec-tories are constrained to the dividing surface using the RAT-TLE algorithm. For Systems 1 and 2a-2e, the same auxiliaryrestraining potential used in the calculation of F(fb) is intro-duced for the PIMD sampling trajectories to restrict the sys-tem to the concerted channel, as described in Appendix C;throughout this paper, the RPMD trajectories used to calcu-late the transmission coefficients are not subjected to auxiliaryrestraining potentials.

2. RPMD rate calculations for ET prior to PT

For System 1, we calculate the rate for both the sequen-tial and concerted PCET mechanisms. For the ET step in thesequential mechanism, we calculate the forward and reverseET reaction rates between the OU and RU species (kUe andkUe− , Fig. 1(b)). The symmetry of the system requires thatkPe = kUe− . The 1D FE profile used in the rate calculation forthe ET reactions is obtained in the electron bead-count coor-dinate, FSET(fb), using the same umbrella sampling protocoldescribed for the calculation of F(fb); however, in the calcu-lation of FSET(fb), an auxiliary restraining potential is intro-duced for the PIMD sampling trajectories to restrict the sys-tem to the ET channel, as described in Appendix C. The in-dependent sampling trajectories used to calculate FSET(fb) areeach run for 15 ns.

The transmission coefficients (Eq. (14)) for the forwardand reverse ET reactions are calculated using RPMD trajec-tories that are released from the dividing surface associatedwith fb = 0.18. A total of 12 000 RPMD trajectories are re-leased. Each RPMD trajectory is evolved for 300 fs using atimestep of 1 × 10−4 fs and with the initial velocities sam-pled from the MB distribution. Initial configurations for theRPMD trajectories are selected every 10 ps from long PIMDsampling trajectories that are constrained to the dividing sur-face. The sampling trajectories employ mb,e = 2000 a.u.,mb,p = 1836.1 a.u., and a timestep of 0.1 fs. Thermostat-ting is performed by re-sampling the velocities from the MBdistribution every 500 fs. The sampling trajectories are con-strained to the dividing surface using the RATTLE algorithm.The same auxiliary restraining potential used in the calcula-tion of FSET(fb) is introduced for the PIMD sampling trajec-tories to restrict the system to the ET channel, as described inAppendix C.

3. RPMD rate calculations for PT prior to ET

For the PT step in the sequential mechanism in System 1,we calculate the forward and reverse PT reactions betweenthe OU and OP species (kOp and k

Op− , Fig. 1(b)). The symmetry

of the system requires that kRp = kOp− . The 1D FE profile usedin the rate calculation for the forward and reverse PT reac-tions is obtained in the proton centroid coordinate, FSPT(q̄p),using the same umbrella sampling protocol described for thecalculation of F (q̄p).

The transmission coefficients (Eq. (14)) for the forwardand reverse PT reactions are calculated using RPMD trajec-tories that are released from the dividing surface associatedwith q̄p = 0.21 a0. A total of 10 500 RPMD trajectories arereleased. Each RPMD trajectory is evolved for 300 fs with atimestep of 1 × 10−4 fs and with the initial velocities sam-pled from the MB distribution. Initial configurations for theRPMD trajectories are selected every 10 ps from long PIMDsampling trajectories that are constrained to the dividing sur-face. The sampling trajectories employ mb,e = 2000 a.u.,mb,p = 1836.1 a.u., and a timestep of 0.1 fs. Thermostatting isperformed by re-sampling the velocities from the MB distri-bution every 500 fs. The sampling trajectories are constrainedto the dividing surface using the RATTLE algorithm.

4. Two-dimensional FE profiles

For the purpose of analysis, we calculate the two-dimensional (2D) FE profile for System 1 in the electronbead-count and proton centroid coordinates, F (fb, q̄p). The2D FE profile is constructed using PIMD sampling trajec-tories that are harmonically restrained in both the fb and q̄pcoordinates. A total of 4553 sampling trajectories are per-formed, in which the coordinates fb and q̄p are sampledusing a square grid. The coordinate fb is sampled using 93windows that are harmonically restrained to uniformly spacedvalues of fb in the region [−0.92, 0.92] using a force con-stant of 20 a.u.; seven windows are harmonically restrainedto uniformly spaced values of fb in both the region [−0.991,−0.985] and in [0.985, 0.991] using a higher force constant of5000 a.u.; nine windows are harmonically restrained to uni-formly spaced values of fb in both the region [−1.0, − 0.992]and in [0.992, 1.0] using a higher force constant of 10 000a.u.; 32 windows are harmonically restrained to the values offb ∈ {±0.93, ±0.935, ±0.94, ±0.945, ±0.95, ±0.955, ±0.96,±0.962, ±0.965, ±0.967, ±0.97, ±0.974, ±0.976, ±0.978,±0.98, ±0.982} using a force constant of 500 a.u. For eachvalue of fb, the coordinate q̄p is sampled using nine windowsthat are harmonically restrained to uniformly spaced values ofq̄p in the region [−0.20 a0, 0.20 a0] using a force constant of1.3 a.u., and 10 windows that are harmonically restrained touniformly spaced values of q̄p in both the region [−0.70 a0,−0.25 a0] and in [0.25 a0, 1.10 a0] using a lower forceconstant of 1.0 a.u. No auxiliary restraining potentials are em-ployed for the calculation of F (fb, q̄p). Each sampling tra-jectory is run for 2.5 ns using a timestep of 0.1 fs, withmb,e = 2000 a.u. and mb,p = 1836.1 a.u. Thermostatting isperformed by re-sampling the velocities from the MB distri-bution every 500 fs.



We additionally calculate the 2D FE profile for System1 in the electron bead-count and solvent position coordinates,F(fb, qs), for sampling trajectories corresponding to the con-certed PCET reaction. To generate F(fb, qs), the harmoni-cally restrained sampling trajectories used to calculate F(fb)for System 1 are utilized.

5. RPMD transition path ensemble

As we have done previously,28 we analyze the transi-tion path ensemble58 for the RPMD trajectories in the currentstudy. Reactive trajectories are generated through forward-and backward-integration of initial configurations drawn fromthe dividing surface ensemble with initial velocities drawnfrom the MB distribution. Reactive trajectories correspondto those for which forward- and backward-integrated half-trajectories terminate on opposite sides of the dividing sur-face. The reactive trajectories that are initialized from theequilibrium Boltzmann distribution on the dividing surfacemust be reweighted to obtain the unbiased transition pathensemble.58, 83, 84 A weighting term, wα , is applied to each tra-jectory, correctly accounting for recrossing and for the factthat individual trajectories are performed in the microcanoni-cal ensemble. This term is given by83

wα =(∑

i

∣∣ξ̇ (r)i∣∣−1)−1

, (39)

where the sum includes all instances in which trajectory αcrosses the dividing surface, and ξ̇ (r)i is the velocity in thedividing surface collective variable at the ith crossing event.The reweighting has a minor effect on the non-equilibrium av-erages if the reactive trajectories initialized from the dividingsurface exhibit relatively little recrossing, as is the case forthe systems studied in this paper. Non-equilibrium averagesover the RPMD transition path ensemble are calculated byaligning reactive trajectories at time 0, defined as the momentin time when the trajectories are released from the dividingsurface.

6. RPMD rate calculations for pure ET

The RPMD rates for pure ET are calculated for Systems4a-4g. For Systems 4a-4e, the 1D FE profile used in the ratecalculation is obtained in the electron bead-count coordinate,FET(fb), using the same umbrella sampling protocol describedfor the calculation of F(fb); however, no auxiliary restrainingpotentials are introduced for the PIMD sampling trajectories.

For Systems 4f and 4g, the 1D FE profile used in the ratecalculation is obtained in the solvent coordinate, FET(qs), byreducing the 2D FE profile in the electron bead-count and sol-vent coordinates, FET(fb, qs). The 2D FE profile, FET(fb, qs),is constructed using PIMD sampling trajectories that are har-monically restrained in both the fb and qs coordinates. A to-tal of 5809 sampling trajectories are performed, in which thecoordinates fb and qs are sampled using a square grid. Thecoordinate fb is sampled using 93 windows that are harmoni-cally restrained to uniformly spaced values of fb in the region[−0.92, 0.92] using a force constant of 20 a.u.; seven windows

are harmonically restrained to uniformly spaced values of fbin both the region [−0.991, − 0.985] and in [0.985, 0.991] us-ing a higher force constant of 5000 a.u.; nine windows areharmonically restrained to uniformly spaced values of fb inboth the region [−1.0, − 0.992] and in [0.992, 1.0] using ahigher force constant of 10 000 a.u.; 32 windows are harmon-ically restrained to the values of fb ∈ {±0.93, ±0.935, ±0.94,±0.945, ±0.95, ±0.955, ±0.96, ±0.962, ±0.965, ±0.967,±0.97, ±0.974, ±0.976, ±0.978, ±0.98, ±0.982} using aforce constant of 500 a.u. For each value of the fb coordi-nate, the qs coordinate is sampled using 37 windows that areharmonically restrained to uniformly spaced values of qs inthe region [−9.0 a0, 9.0 a0] using a force constant of 0.03 a.u.Each sampling trajectory is run for 2.5 ns using a timestepof 0.1 fs, with mb,e = 2000 a.u. Thermostatting is performedby re-sampling the velocities from the MB distribution every500 fs.

For Systems 4a-4e, the transmission coefficient (Eq. (14))is calculated using RPMD trajectories that are released fromthe dividing surface associated with fb = 0. A total of 3000RPMD trajectories are released for Systems 4a-4c, 6000 tra-jectories for System 4d and 4500 trajectories for System 4e.Each RPMD trajectory is evolved for 300 fs using a timestepof 1 × 10−4 fs and with the initial velocities sampled fromthe MB distribution. Initial configurations for the RPMD tra-jectories are selected every 10 ps from long PIMD samplingtrajectories that are constrained to the dividing surface. Thesampling trajectories employ mb,e = 2000 a.u. and a timestepof 0.1 fs. Thermostatting is performed by re-sampling the ve-locities from the MB distribution every 500 fs. The samplingtrajectories are constrained to the dividing surface using theRATTLE algorithm.

For Systems 4f and 4g, the transmission coefficient is cal-culated using RPMD trajectories that are released from thedividing surface associated with qs = 0. A total of 1500 tra-jectories are released for Systems 4f and 4g. Each trajectoryis evolved for 700 fs using a timestep of 1 × 10−4 fs andwith the initial velocities sampled from the MB distribution.Initial configurations for the RPMD trajectories are selectedevery 10 ps from long PIMD sampling trajectories that areconstrained to the dividing surface. The sampling trajecto-ries employ mb,e = 2000 a.u. and a timestep of 0.1 fs. Ther-mostatting is performed by re-sampling the velocities fromthe MB distribution every 500 fs. The sampling trajectoriesare constrained to the dividing surface using the RATTLEalgorithm.

B. PCET rate theory calculations

Expressions for the thermal reaction rates for concertedPCET are provided in Eqs. (15)–(21). Since the current paperconsiders only symmetric PCET reactions, the driving force,

G0, is zero in all cases.

The concerted PCET reorganization energy, λ, iscalculated using the following result for symmetricsystems:85–87

λ = 〈U 〉reac, (40)



TABLE I. Values of the electronic coupling, VET, vibrational coupling, VPT,and reorganization energy, λ, for the system-bath model systems for PCET.a

System VET VPT λ

1 5.0 × 10−6 1.8 × 10−6 1.84 × 10−22a 5.0 × 10−6 4.6 × 10−7 9.71 × 10−32b 5.0 × 10−5 3.6 × 10−7 9.47 × 10−32c 5.0 × 10−4 2.1 × 10−7 9.78 × 10−32d 5.0 × 10−3 9.6 × 10−8 9.22 × 10−32e 2.5 × 10−2 4.5 × 10−7 9.32 × 10−32f 1.0 × 10−1 4.8 × 10−6 8.47 × 10−33a 3.3 × 10−2 1.5 × 10−8 3.27 × 10−23b 2.7 × 10−2 2.3 × 10−5 3.29 × 10−23c 1.8 × 10−2 8.5 × 10−4 3.34 × 10−23d 1.5 × 10−2 2.2 × 10−3 3.35 × 10−23f 1.5 × 10−2 2.8 × 10−3 3.33 × 10−2aAll quantities reported in atomic units. For Systems 1 and 2a-2f, VET is calculated asdescribed in Sec. V B; for Systems 3a-3f, VET is calculated from the splitting betweenthe ground and first-excited adiabatic electronic state energies with qs = qp = 0.

where U is the concerted PCET energy gap coordinate,

U = URP( − q(1)e , . . . ,−q(ne)e ,−q(1)p , . . . ,−q(np)p , qs, Q)

−URP(q(1)e , . . . , q

(ne)e , q

(1)p , . . . , q

(np)p , qs, Q

), (41)

and 〈. . . 〉reac denotes the equilibrium ensemble average inthe reactant basin. The ensemble average is calculatedfrom a 50 ns equilibrium PIMD trajectory, where the elec-tron and proton are initialized and remain in the reactantbasin. The sampling trajectories employ mb,e = 2000 a.u.,mb,p = 1836.1 a.u., and a timestep of 0.1 fs. Thermostattingis performed by re-sampling the velocities from the MB dis-tribution every 500 fs. Values for the reorganization energy inthe various systems are presented in Table I.

The free-energy barrier for PCET in the fully adiabaticregime, G‡ad in Eq. (15), is calculated from the differenceof the ground vibronic energy level at its minimum and at itsmaximum with respect to the solvent coordinate. The adia-batic vibronic states are obtained as a function of the solventcoordinate in the range −4 a0 ≤ qs ≤ 4 a0. For each valueof qs, the system Hamiltonian associated with Usys(qe, qp, qs)(Eq. (25)) is diagonalized using a 2D discrete variable repre-sentation (DVR) grid calculation in the electron and protonposition coordinates, qe and qp, respectively.88 The grid spansthe range −30 a0 ≤ qe ≤ 30 a0 and −1.5 a0 ≤ qp ≤ 1.5 a0,with 1024 and 20 evenly spaced grid points for the electronand proton position, respectively.

The vibronic coupling in the partially adiabatic regime(Eq. (18)) is obtained from the splitting between the groundand first vibrational states calculated for the potential definedby the ground adiabatic electronic state; the ground adiabaticelectronic state is calculated for a frozen solvent configurationfor which the reactant and product concerted PCET states aredegenerate.20, 21 The calculation of the vibronic coupling inthe partially adiabatic regime thus requires two tasks that in-clude (i) the calculation of the adiabatic electronic states as afunction of the proton coordinate for a frozen solvent config-uration and (ii) the calculation of the proton vibrational statesfor the potential defined by the lowest adiabatic electronic

state. To complete task (i), the adiabatic electronic states areobtained as a function of the proton coordinate in the range−1.5 a0 ≤ qp ≤ 1.5 a0, with qs = 0. For each value of qp,the system Hamiltonian is diagonalized using a 1D DVR gridcalculation in the electron position coordinate. The grid spansthe range −30 a0 ≤ qe ≤ 30 a0 with 2048 evenly spaced gridpoints. To complete task (ii), a polynomial of the form:

Uad(qp) =6∑

i=0c

(i)ad |qp|i (42)

is fit to the lowest adiabatic electronic state in the range−1.5 a0 ≤ qp ≤ 1.5 a0. The vibrational energies, E0 and E1,are calculated for the fitted potential in Eq. (42) by diagonal-izing the 1D DVR Hamiltonian in the proton position coor-dinate. The grid spans the range −1.5 a0 ≤ qp ≤ 1.5 a0 with2048 evenly spaced grid points. The values of the vibrationalcoupling, and hence the partially adiabatic vibronic coupling,are presented in Table I. The coefficients for the polynomialfit to the lowest adiabatic electronic state (Eq. (42)) are pre-sented in Appendix D (Table X).

The vibronic coupling in the fully non-adiabatic regime(Eq. (21)) is obtained from the product of the electroniccoupling and the overlap of reactant and product vibrationalwavefunctions. The vibrational wavefunctions are calculatedfor the potential defined by the reactant and product diabaticelectronic states; the diabatic electronic states are calculatedfor a frozen solvent configuration for which the reactant andproduct concerted PCET states are degenerate.1, 17–19 The cal-culation of the vibronic coupling in the fully non-adiabaticregime thus requires three tasks that include (i) the calculationof the electronic coupling, (ii) the calculation of the diabaticelectronic states as a function of the proton coordinate for afrozen solvent configuration, and (iii) the calculation of thevibrational energies and wavefunctions for the potential de-fined by the reactant and product diabatic electronic states. Tocomplete tasks (i) and (ii) for Systems 1 and 2a-2f, the elec-tronic coupling and diabatic electronic states are obtained asa function of the proton coordinate for qs = 0 using the lo-calization procedure described in Appendix E. The electroniccoupling (Eq. (E7)) is found to be nearly constant over thephysical range of qp, so we employ a constant value of VETthat corresponds to the qp = 0 value. For Systems 2e and 2f,the localization procedure does not yield fully localized dia-batic states, which contributes to the breakdown of the fullynon-adiabatic rate calculation. The values of the electroniccoupling are presented in Table I. To complete task (iii), thereactant and product diabatic electronic states (Eqs. (E5) and(E6)) are computed for a uniform grid of 2048 points in therange −1.5 a0 ≤ qp ≤ 1.5 a0, and the reactant and productvibrational energies and wavefunctions are then obtained bydiagonalizing the 1D DVR Hamiltonian in the proton positionon this grid.

C. ET rate theory calculations

Expressions for the thermal reaction rates for ET are pro-vided in Eqs. (15) and (22). The free-energy barrier for ETin the electronically adiabatic regime, G‡ad in Eq. (15), is



TABLE II. Values of the electronic coupling, VET and reorganization en-ergy, λ, for ET systems that vary between the adiabatic and non-adiabaticregimes.a

System VET λ

4a 1 × 10−6 7.184b 1 × 10−5 7.454c 1 × 10−4 7.444d 1 × 10−3 7.374e 4 × 10−3 7.264f 1 × 10−2 7.184g 2 × 10−2 7.30aλ is given in units of a.u. × 10−2; all other parameters are given in atomic units.

calculated from the difference of the ground electronic en-ergy level at its minimum and at its maximum with respectto the solvent coordinate. The adiabatic electronic states areobtained as a function of the solvent coordinate in the range−8.0 a0 ≤ qs ≤ 8.0 a0. For each value of qs, the system Hamil-tonian associated with Eq. (36) is diagonalized using a 1DDVR grid calculation in the electron position coordinate, qe.The grid spans the range −30.0 a0 ≤ qe ≤ 30.0 a0 with 2048evenly spaced grid points.

The electronic coupling, VET in Eq. (22), is obtained fromthe splitting between the ground, ε0(qs), and first excited,ε0(qs), adiabatic electronic state energies,

VET = 12

[ε1(qs = 0) − ε0(qs = 0)]. (43)

The ET reorganization energy, λ, is calculated usingEq. (40),85–87 where U is now the ET energy gap coordi-nate,

U = UETRP( − q(1)e , . . . ,−q(ne)e , qs, Q)

−UETRP(q(1)e , . . . , q

(ne)e , qs, Q

). (44)

The ensemble average is calculated from a 50 ns equi-librium PIMD trajectory, where the electron is initialized andremains in the reactant basin. The sampling trajectories em-ploy mb,e = 2000 a.u. and a timestep of 0.1 fs. Thermostattingis performed by re-sampling the velocities from the MB dis-tribution every 500 fs. The values of the reorganization energyare presented in Table II.

VI. RESULTS

The results are presented in two sections. In the first, weanalyze the competition between the concerted and sequen-tial reaction mechanisms for PCET. In the second, we studythe kinetics and mechanistic features of concerted PCETreactions across multiple coupling regimes, including thefully non-adiabatic (both electronically and vibrationally non-adiabatic), partially adiabatic (electronically adiabatic, but vi-brationally non-adiabatic), and fully adiabatic (both electron-ically and vibrationally adiabatic) limits.

-0.8 -0.4 0 0.4 0.8fb

-0.4

0

0.4

− q p /

a 0

0

5

10

15

20

25

OU

RPOP

RU

FIG. 2. Reactive RPMD trajectories reveal distinct concerted (red), sequen-tial PT-ET (purple), and sequential ET-PT (orange) reaction mechanisms forPCET in System 1. The trajectories are projected onto the FE surface in theelectron bead-count coordinate, fb, and the proton centroid coordinate, q̄p,with contour lines indicating FE increments of 2 kcal/mol.

A. Sequential versus concerted PCET

We begin by investigating the competing PCET reactionmechanisms in System 1. Figure 2 presents the 2D FE pro-file for this system along the electron bead-count, fb, and theproton centroid, q̄p coordinates. The FE profile exhibits fourbasins of stability corresponding to the various PCET reactant(OU), intermediate (OP and RU), and product (RP) species(Fig. 1(b)). Distinct channels on the FE surface connect thevarious basins of stability. Due to the symmetry of the reac-tion, the two channels associated with the PT step of the se-quential pathway (connecting OU to OP and RU to RP) areidentical, as are the the two channels associated with the ETstep of the sequential pathway (connecting OU to RU and OPto RP). A single channel on the FE surface connects OU toRP, bypassing the intermediate species.

Also plotted in Fig. 2 are representative samples from theensemble of reactive RPMD trajectories for PCET in Sys-tem 1. The trajectories cluster within the channels on theFE surface, providing a direct illustration of the concerted(red) and sequential (purple and orange) reaction mechanismsfor PCET. Such distinct clustering of the reactive trajectoriesneed not be observed in general systems that undergo PCET;we note that the RPMD method makes no a priori assump-tions about the preferred reaction mechanism or the existenceof distinct sequential and concerted reaction mechanisms forPCET.

We now demonstrate that the concerted PCET mecha-nisms is dominant in System 1 by computing the RPMD re-action rates for both the concerted and sequential processes.Figures 3(a) and 3(b) illustrate the FE profile and transmis-sion coefficient that together determine the RPMD reactionrate for concerted PCET (Eq. (6)). As was previously foundfor ET reactions,33 the FE profile exhibits a sharp rise as afunction of fb due to the formation of a ring-polymer con-figuration in which the electron spans the two redox sites(Fig. 3(a), inset), and it exhibits more gradual changes in therange of |fb| < 0.97 due to solvent polarization. For the di-viding surface fb = 0, the transmission coefficient plateausat a value of approximately 0.1, indicating that fb is a rea-sonably good reaction coordinate for the process. These



0

10

20

30

-1 -0.5 0 0.5 1

FS

ET( f

b)

fb

(a) (b)

(c) (d)

(e) (f)

0

10

20

30

-1 -0.5 0 0.5 1

FS

ET( f

b)

fb

(a) (b)

(c) (d)

(e) (f)

0

0.2

0.4

0 100 200 300

κ(t)

t / fs

(a) (b)

(c) (d)

(e) (f)

0

10

-1 -0.5 0 0.5 1

FS

PT(− q

p)

−qp / a0

(a) (b)

(c) (d)

(e) (f)

0

10

-1 -0.5 0 0.5 1

FS

PT(− q

p)

−qp / a0

(a) (b)

(c) (d)

(e) (f)

0

0.5

1

0 100 200 300

κ(t)

t / fs

(a) (b)

(c) (d)

(e) (f)

0

10

20

30

-1 -0.5 0 0.5 1

F( f

b)

fb

(a) (b)

(c) (d)

(e) (f)

0

10

20

30

-1 -0.5 0 0.5 1

F( f

b)

fb

(a) (b)

(c) (d)

(e) (f)

0

0.2

0.4

0 100 200 300

κ(t)

t / fs

(a) (b)

(c) (d)

(e) (f)

5 10 15

0.98 1

(a) (b)

(d)

(e) (f)

5 10 15

0.98 1

(a) (b)

(d)

(e) (f)

0 5

10

0.98 1

(a) (b)

(d)

(e) (f)

0 5

10

0.98 1

(a) (b)

(d)

(e) (f)

FIG. 3. (a) The 1D FE profile in the electron bead-count coordinate, F(fb),utilized in the RPMD rate calculation for the concerted PCET reaction.(b) The corresponding transmission coefficient for the concerted PCET re-action. (c) The 1D FE profile in the electron bead-count coordinate, FSET(fb),utilized in the RPMD rate calculation for the ET reactions prior to PT inthe sequential PCET mechanism. (d) The corresponding forward (red) andreverse (blue) transmission coefficients for the ET reactions prior to PT.(e) The 1D FE profile in the proton centroid coordinate, FSPT(q̄p), utilizedin the RPMD rate calculation for the PT reactions prior to ET in the sequen-tial PCET mechanism. (f) The corresponding forward (red) and reverse (blue)transmission coefficients for the PT reactions prior to ET. All FE profiles areplotted in kcal/mol.

results combine to yield a RPMD rate of kCPET = (2.1 ± 0.7)× 10−20 a.u. for the concerted reaction mechanism inSystem 1.

Figures 3(c)–3(f) present the components of the RPMDrate calculation for the sequential PCET reaction mechanismin System 1. For the ET step of the sequential mechanism,Figs. 3(c) and 3(d) report the FE profile in the electron bead-count coordinate and the forward (red) and reverse (blue)transmission coefficients associated with fb = 0.18. For thePT step of the sequential mechanism, Figs. 3(e) and 3(f) re-port the FE profile in the proton centroid coordinate and theforward (red) and reverse (blue) transmission coefficients as-sociated with q̄p = 0.21 a0. The oscillations observed in κ(t)for the PT step correspond to the vibrational motion of thetransferring proton. These results combine to yield the RPMDrates for the various individual steps in the sequential PCETreaction (Table III).

For the reaction mechanism that involves sequential ETfollowed by PT, the reaction rate is given by1

kep = kUekRp

kRp + kUe−, (45)

TABLE III. RPMD rates for the forward and reverse ET and PT reactionsin the sequential mechanism.

Rate constant

kUe (3.6 ± 2.6) × 10−21kUe− (1.1 ± 0.4) × 10−15kPe (1.1 ± 0.4) × 10−15kOp (2.9 ± 0.3) × 10−13kOp− (9.6 ± 1.7) × 10−8kRp (9.6 ± 1.7) × 10−8

aAll rates are given in atomic units. The notation for the rate constants is defined inFig. 1(b).

which numerically yields kep = (3.6 ± 2.6) × 10−21 a.u. Simi-larly, for the reaction mechanism involving sequential PT fol-lowed by ET, the reaction rate is given by1

kpe = kOpkPe

kPe + kOp−, (46)

which numerically yields kpe = (3.4 ± 1.4) × 10−21 a.u. Thecomputed values for kep and kpe are equal to within statisticalerror, as is consistent with microscopic reversibility in thissymmetric system.

Comparison of the reaction rate for the concerted and se-quential PCET mechanisms (Table IV) reveals that the reac-tion rate for the concerted mechanism is approximately sixtimes larger than that of the sequential mechanism; the RPMDmethod thus predicts that the PCET reaction in System 1proceeds predominantly via the concerted reaction mecha-nism. We note that although the reaction dividing surfaceswere selected to minimize trajectory recrossing, the rates re-ported here for the various sequential and concerted stepsare rigorously independent of this choice of dividing sur-face; the mechanistic analysis provided here thus avoids anyTST approximations or prior assumptions about the reactionmechanism.

Having established that the concerted mechanism is fa-vored for System 1, we now analyze the RPMD trajectorieswith respect to the solvent fluctuations and interactions thatgovern the concerted PCET reaction mechanism.

Figure 4(a) presents the 2D FE profile in the electronbead-count and solvent coordinates, F(fb, qs), computed forthe concerted pathway as is described in Sec. V A 4. TheFE profile exhibits two basins of stability corresponding tothe PCET reactant and product species (OU and RP, respec-tively), separated by a barrier that corresponds to the divid-ing surface in the fb coordinate. Also plotted in Fig. 4(a) arerepresentative samples from the ensemble of reactive RPMDtrajectories (red) and the non-equilibrium average over the

TABLE IV. Reaction rates for the full ET-PT, PT-ET, and concerted PCETmechanisms calculated using RPMD and Eqs. (45) and (46).a

Rate constant

kep (3.6 ± 2.6) × 10−21kpe (3.4 ± 1.4) × 10−21kCPET (2.1 ± 0.7) × 10−20

aAll rates are given in atomic units.



(a)

(b)

(c)

OU

DS

RP

OU

DS

RP

de(t)

dp(t)

dep(t)

-0.8 -0.4 0 0.4 0.8fb

-2

0

2q s

/ a 0

0

5

10

15

20

25

30(a)

(b)

(c)

OU

DS

RP

OU

DS

RP

de(t)

dp(t)

dep(t)

(a)

(b)

(c)

OU

DS

RP

OU

DS

RP

de(t)

dp(t)

dep(t)

(a)

(b)

(c)

OU

DS

RP

OU

DS

RP

de(t)

dp(t)

dep(t)

Ene

rgy

(a)

(b)

(c)

OU

DS

RP

OU

DS

RP

de(t)

dp(t)

dep(t)

qe

(a)

(b)

(c)

OU

DS

RP

OU

DS

RP

de(t)

dp(t)

dep(t)

qs

(a)

(b)

(c)

OU

DS

RP

OU

DS

RP

de(t)

dp(t)

dep(t)

-0.005

0

0.005

-300 -200 -100 0 100 200 300

d(t)

/ a.

u.

t / fs

(a)

(b)

(c)

OU

DS

RP

OU

DS

RP

de(t)

dp(t)

dep(t)

FIG. 4. (a) Reactive RPMD trajectories (red) and the average over the en-semble of reactive trajectories (yellow) for the concerted PCET reaction inSystem 1 reveal a Marcus-type solvent-gating mechanism indicated by theblack arrows. The trajectories are projected onto the FE surface in the electronbead-count coordinate, fb, and the solvent position coordinate, qs, with con-tour lines indicating FE increments of 2 kcal/mol. The regions correspondingto the concerted PCET reactant (OU), product (RP), and dividing surface(DS) are indicated. (b) Illustration of the mechanism for concerted PCET.The left panels present the vibronic diabatic free energy surfaces along thesolvent coordinate; the red dot indicates the solvent configuration associatedwith the OU, RP, and DS regions indicated in (a). The right panels present thedouble-well potential that is experienced by the electron in the OU, RP, andDS regions, as well as the ring-polymer configuration in the electron posi-tion coordinate at the corresponding points along a typical reactive trajectory.(c) The combined dipole for the transferring particles in the ensemble of reac-tive RPMD trajectories, dep(t) (black), as well as the individual componentsfrom the transferring electron, de (red), and the transferring proton, dp (blue),for the concerted PCET reaction in System 1.

ensemble of reactive trajectories (yellow), as described inSec. V A 5. As was seen for ET,33 the reactive RPMDtrajectories for concerted PCET follow a Marcus-typesolvent-gating mechanism (black arrows), in which solventreorganization precedes the sudden transfer of both the elec-tron and proton between wells that are nearly degenerate withrespect to solvent polarization.

Figure 4(b) elaborates on this mechanism, schematicallyillustrating the ring-polymer configurations that accompanythe various stages of the concerted PCET reaction. In the re-actant OU basin, the system rests at the bottom of the solventpotential well for the reactant vibronic diabat (indicated by ared point in the left panel); for this polarized solvent config-uration, the transferring electron and proton experience a po-tential energy surface that favors occupation of the donor sites(shown for the electron position in the right panel). In the di-viding surface (DS) region of the concerted PCET reaction,the solvent fluctuation brings the system to configurations atwhich the vibronic diabats for the transferring electron andproton are nearly degenerate (shown at left), and the transfer-ring particles undergo tunneling between nearly degeneratewells for the donor and acceptor sites (shown at right); alsoseen in the panel at right is the extended “kink-pair” configu-ration for the ring-polymer in which the electron spans the tworedox sites during the tunneling event. Finally, the figure pan-els associated with the product RP basin illustrate that as thesolvent relaxes to the minimum of the solvent potential wellfor the product vibronic diabat (left), the transferring electronand proton experience a potential energy surface that favorsoccupation of the product sites (right). This mechanism ob-served in the RPMD trajectories is consistent with the mech-anisms that are assumed by PCET rate theories in the fullynon-adiabatic regime.1, 17–19

Figure 4(c) illustrates part of the mechanistic basis forthe favorability of the concerted PCET reaction in this sys-tem. The figure presents the combined dipole for the transfer-ring particles in the ensemble of reactive RPMD trajectories,dep(t), as well as the individual components from the transfer-ring electron and proton, de(t) and dp(t), respectively. Theseterms are computed using

de(t) = −μes〈q̄e(t)〉traj, (47)

dp(t) = −μps〈q̄p(t)〉traj, (48)and

dep(t) = de(t) + dp(t), (49)where q̄e and q̄p are the ring-polymer centroids for the trans-ferring electron and proton, respectively, and 〈. . . 〉traj denotesthe non-equilibrium ensemble average over the time-evolvedreactive RPMD trajectories for concerted PCET (Sec. V A 5).Figure 4(c) shows that the orientation of de(t) and dp(t) switchduring the reaction on similar timescales, which follows fromthe fact that the two particles are moving both co-linearly andin concert. However, the figure also shows that the magnitudeof dep(t) is at all times smaller than the larger magnitude of thetwo component dipoles (i.e., |dep(t)| < max(|de(t)|, |dp(t)|)),due to the opposite charge of the two transferring particles.It is thus clear that throughout reactive trajectories for con-certed PCET, the degree to which the polar solvent couples tothe transferring particles is reduced by the opposing sign ofthe electron and proton charge. In this sense, the polar solventcreates a driving force for the co-localization of the electronand proton.

We emphasize that although we have previously analyzedconcerted versus sequential PCET mechanisms in the context



of exact quantum simulations,89 the RPMD simulations pre-sented here constitute a trajectory-based simulation approachfor the detailed, side-by-side comparison of concerted and se-quential PCET mechanisms and thermal reaction rates, withboth reaction mechanisms treated on a consistent dynamicalfooting.

B. Reactions across multiple coupling regimes

In this section, we employ RPMD simulations to investi-gate concerted PCET in a range of physical regimes, includingthe fully non-adiabatic, partially adiabatic, and fully adiabaticregimes. We validate the accuracy of the RPMD method bycomparing thermal reaction rates obtained using the simula-tion method with those obtained using previously developedrate theories, and we investigate the variety of electron andproton tunneling processes that accompany concerted PCET.However, before delving into this analysis of PCET reac-tions, we first use RPMD to examine the crossover betweenelectronically non-adiabatic (i.e., weak electronic coupling)and electronically adiabatic (i.e., strong electronic coupling)regimes for pure ET; analysis of this more simple processwill provide useful context for the subsequent discussion ofPCET.

1. ET across electronic-coupling regimes

Figure 5(a) presents the reaction rates for Systems 4a-4g,computed using RPMD (red), the electronically adiabatic ETrate expression (Eq. (15), blue), and the electronically non-adiabatic ET rate expression (Eq. (22), black). The results areplotted as a function of the temperature-reduced electroniccoupling βVET. For the weak-coupling regime (βVET 1),the non-adiabatic rate expression constitutes the reference re-

sult, whereas for the strong-coupling regime (βVET � 1), theadiabatic rate expression is the reference. It is clear that theRPMD rate correctly transitions between agreement with thenon-adiabatic rate theory results at weak electronic couplingand the adiabatic rate theory results at strong electronic cou-pling. For systems with weak electronic coupling, we haveshown previously that RPMD accurately describes the ET re-action rate throughout the normal and activationless regimesfor the thermodynamic driving force,33 which follows fromthe method’s exact description of statistical fluctuations24–26

and its formal connection to semiclassical instanton theoryfor deep-tunneling processes.24, 90–93 Figure 5(a) shows thatfor symmetric systems, the accuracy of the method extendsfrom the weak-coupling to the strong-coupling limits.

It is important to note that the RPMD rates in Fig. 5(a)are obtained without prior knowledge or assumption of theelectronic coupling regime, and at no point in the RPMDrate calculation is VET required. A natural question, there-fore, is whether a posteriori analysis of the trajectories fromthe RPMD rate calculation can be used to determine the elec-tronic coupling regime for a given reaction. Figures 5(b)–5(d)demonstrate that this is indeed the case.

Figures 5(b) and 5(c) present snapshots of the electronposition in a ring-polymer configuration at the reaction di-viding surface, with the system either in the weak-couplingregime (b) or in the strong coupling regime (c). As describedin Sec. V, the dividing surface used for the RPMD ET ratecalculations is given by fb = 0, which corresponds to con-figurations for which the electron position evenly spans thetwo redox sites and for which the solvent is depolarizedto accommodate this symmetric charge distribution for theelectron.33 At left in Figs. 5(b) and 5(c), we plot the elec-tron position as a function of the ring-polymer bead index, α,where τ = β¯α/ne. At right, we schematically illustrate the

-19

-14

-9

-4

log(

k ET)

/ a.u

.

(a)

(b)

(c)

(d)

Non-Adiabatic

Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs 0

0.5

1

-3 -2 -1 0 1

Kin

k F

ract

ion

log(βVET)

(a)

(b)

(c)

(d)

Non-Adiabatic

Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

-5 0 5

q e

(a)

(b)

(c)

(d)

Non-Adiabatic

Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

Ene

rgy

(a)

(b)

(c)

(d)

Non-Adiabatic

Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

-5 0 5

0 0.5 1

q e

τ / β−h

(a)

(b)

(c)

(d)

Non-Adiabatic

Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

-5 0 5

Ene

rgy

qe

(a)

(b)

(c)

(d)

Non-Adiabatic

Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

FIG. 5. (a) ET reaction rates as a function of the temperature-reduced electronic coupling, obtained using RPMD (red), the non-adiabatic rate expression forET (Eq. (22), black), and the adiabatic rate expression for ET (Eq. (15), blue) for Systems 4a-4g. (b) and (c) At left, the electron position as a function of thering-polymer bead index for (b) System 4a (log (βVET) = −2.98) and (c) System 4g (log (βVET) = 1.32); at right, a schematic illustration of the correspondingdouble-well potentials that are experienced by the transferring electron at the dividing surface, as well as the ring-polymer configurations in the electronposition coordinate. The orange and purple stripes indicate the positions of the electron donor and acceptor sites, respectively. (d) The fraction of ring-polymerconfigurations at the dividing surface for ET that contain either a single kink-pair (black) or multiple kink-pairs (red) as a function of the temperature-reducedelectronic coupling.



double-well potential that is experienced by the transfer-ring electron at the dividing surface, as well as the ring-polymer configuration in the electron position coordinate.Note that for the weak-coupling regime (Fig. 5(b)), the con-figuration exhibits only a single kink-pair, in which the elec-tron position transits between the redox sites as a function ofthe ring-polymer bead index; for the strong-coupling regime(Fig. 5(c)), the configuration exhibits multiple kink-pairs.

It has long been recognized that the thermodynamicweight of ring-polymer kink-pair configurations is related tothe eigenstate splitting (i.e., coupling) in symmetric double-well systems.24, 75, 92–97 In particular, the weak-couplingregime corresponds to that for which the thermodynamicweight of ring-polymer configurations with multiple kink-pairs is small in comparison to the thermodynamic weightof ring-polymer configurations with only a single kink-pair;in the strong-coupling regime, configurations with multiplekink-pairs predominate. A straightforward approach to deter-mining the coupling regime from the RPMD reactive trajecto-ries is thus to simply count the fraction of ring-polymer con-figurations that exhibit multiple kink-pairs during the reactivetransition event.

For Systems 4a-4g, Fig. 5(d) presents the results of thisstrategy, in which RPMD results are used for the a posteri-ori determination of the regime of the electronic coupling.For each system, we calculate the fraction of ring-polymerconfigurations that exhibit either a single kink-pair (black)or multiple kink-pairs (red) in the ensemble from which theRPMD trajectories are initialized in the rate calculation (i.e.,the equilibrium ensemble constrained to the dividing surface).Here, a kink is defined as a segment of the ring-polymer forwhich the electron position spans from the donor region (qe< −0.7σ e) to the acceptor region (qe > 0.7σ e), where σ e isthe standard deviation of the ring-polymer bead position in the

dividing surface ensemble. We note that more sophisticatedstrategies for identifying the ring-polymer configurations inthe transition region may be needed for systems in which thetrajectories exhibit extensive recrossing through a given divid-ing surface,58, 84 although that is not the case for the systemsconsidered here. It is immediately clear from the comparisonof Figs. 5(a) and 5(d) that the onset of multiple kink-pair con-figurations coincides with the crossover between the adiabaticand non-adiabatic regimes for pure ET reactions at βVET ≈ 1.

We have thus shown that RPMD allows for the accuratecalculation of the ET reaction rate across multiple regimes,without prior assumption of the electronic coupling regime,and it also enables determination of the coupling regime viasimple analysis of the reactive trajectories.

2. Concerted PCET acrosselectronic-coupling regimes

We now shift our attention to Systems 2a-2f, whichexhibit weak vibrational coupling and which vary in elec-tronic coupling from the weak- to strong-coupling regimes.Figure 6(a) presents the thermal reaction rate for concertedPCET in these systems, calculated using the fully non-adiabatic rate theory (Eq. (19), black), the partially adia-batic rate theory (Eq. (16), blue), and the RPMD method(red). For the weak-coupling regime (βVET 1), the fullynon-adiabatic rate expression constitutes the reference result,whereas for the strong-coupling regime (βVET � 1), the par-tially adiabatic rate expression is the reference; the fully non-adiabatic results are discontinued (open-circle) at values ofthe electronic coupling for which the diabatic-state local-ization procedure becomes ill defined (Sec. V B). As ob-served for the pure ET reactions, the RPMD method tran-sitions correctly from the weak-coupling reference to the

-19

-15

-11

-7

log(

k CP

ET)

(a) (b)

(c)

(d)

(e)

Fully Non-Adiabatic

Partially Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs 0

0.5

1

-3 -2 -1 0 1 2 3

Kin

k F

ract

ion

log(βVET)

(a) (b)

(c)

(d)

(e)

Fully Non-Adiabatic

Partially Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

-5

0

5

q e

(a) (b)

(c)

(d)

(e)

Fully Non-Adiabatic

Partially Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

Ene

rgy

(a) (b)

(c)

(d)

(e)

Fully Non-Adiabatic

Partially Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

-5

0

5

q e

(a) (b)

(c)

(d)

(e)

Fully Non-Adiabatic

Partially Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

Ene

rgy

(a) (b)

(c)

(d)

(e)

Fully Non-Adiabatic

Partially Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs -5

0

5

0 0.5 1

q e

τ / β−h

(a) (b)

(c)

(d)

(e)

Fully Non-Adiabatic

Partially Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

-5 0 5

Ene

rgy

qe

(a) (b)

(c)

(d)

(e)

Fully Non-Adiabatic

Partially Adiabatic

RPMD

Single Kink-Pair

Multiple Kink-Pairs

FIG. 6. (a) Concerted PCET reaction rates as a function of the temperature-reduced electronic coupling, obtained using RPMD (red), the fully non-adiabaticrate expression (Eq. (19), black) and the partially adiabatic rate expression (Eq. (16), blue) for Systems 2a-2f. (b)–(d) At left, the electron position as a functionof the ring-polymer bead index for (b) System 2a (log (βVET) = −2.28), (c) System 2d (log (βVET) = 0.72), and (d) System 2f (log (βVET) = 2.02); at right,a schematic illustration of the corresponding potentials that are experienced by the transferring electron at the dividing surface, as well as the ring-polymerconfigurations in the electron position coordinate. The orange, purple, and green stripes indicate the positions of the electron donor site, the electron acceptorsite, and transferring proton, respectively. (e) The fraction of ring-polymer configurations at the dividing surface for concerted PCET that contain either a singlekink-pair (black) or multiple kink-pairs (red) as a function of the temperature-reduced electronic coupling.



strong-coupling reference, while avoiding any assumptionsabout the coupling regime and while avoiding explicit calcu-lation of the electronic or vibrational coupling.

As for the pure ET reactions, we can analyze the en-semble of reactive RPMD trajectories for concerted PCETto elucidate the associated tunneling processes and to de-termine the electronic coupling regime for each system.Figures 6(b)–6(d) present snapshots of a typical electronring-polymer configuration at the concerted PCET reactiondividing surface, with the system either in the weak-couplingregime (b), the intermediate-coupling regime (c), or in thestrong coupling regime (d). In each case, the dividing surfacecorresponds to configurations for which the electron and pro-ton positions are distributed between the donor and acceptorsites; for such configurations the solvent is depolarized to ac-commodate this symmetric charge distribution. The left panelin Figs. 6(b)–6(d) presents the electron position as a functionof the ring-polymer bead index; the right panels schematicallyillustrate the potential that is experienced by the transferringelectron at the dividing surface, as well as the ring-polymerconfigurations in the electron position coordinate.

For the regime of weak electronic coupling (Fig. 6(b)),the electronic tunneling event that accompanies the PCET re-action is qualitatively similar to that observed for pure ET(Fig. 5(b)); the electron ring-polymer directly transitions be-tween the two redox sites, exhibiting a single kink-pair. Thecoincident transfer of the proton in this regime simply af-fects the electron tunneling event by increasing the effectiveelectronic coupling of the donor and acceptor redox sites,such that the concerted PCET mechanism may be describedas proton-mediated electron superexchange. However, for theregime of strong electronic coupling (Fig. 6(d)), the electrontransitions between the two redox sites via a mechanism thatis fundamentally different than that observed for the pureET reactions (Fig. 5(c)); in the PCET reaction, the electroncollapses to a localized configuration about the position ofthe transferring proton, such that it adiabatically “rafts” withthe proton between the donor and acceptor sites. This con-certed PCET mechanism is immediately recognized as hydro-gen atom transfer, or HAT.10, 98–100

In both limiting regimes for the electronic coupling(Figs. 6(b) and 6(d)), the RPMD trajectories reveal concertedPCET reaction mechanisms that are implicit in the associ-ated PCET rate theories (Eqs. (16) and (19)). However, theRPMD simulations additionally reveal a distinct – and toour knowledge, previously undiscussed – mechanism for con-certed PCET in the intermediate coupling regime, in whichthe tunneling electron partially localizes about three sites:the positions of the electron donor site, the electron acceptorsite, and the proton that is simultaneously undergoing trans-fer (Fig. 6(c)). This intermediate mechanism, which might becalled “transient-proton-bridge” PCET, exhibits hybrid fea-tures of the PCET mechanisms from both limiting regimes(Figs. 6(b) and 6(d)), and it reflects the changing parame-ters that are employed to modulate the electronic coupling inSystems 2a-2f (Table VII); in this sense, it appears to be aphysically reasonable mechanism for PCET in systems withintermediate electronic coupling, rather than an artifact ofthe approximate RPMD dynamics. Nonetheless, the transient-

proton-bridge mechanism is certainly one for which no pre-vious PCET rate theory has been derived, and it remains tobe seen whether an unambiguous kinetic signature of thisnew mechanism can be identified and observed in a physicalsystem.

Finally, Fig. 6(e) demonstrates that analysis of kink-pairformation in the reactive RPMD trajectories allows for the de-termination of the electronic coupling regime for the PCETreactions. As in Fig. 5(d), we present the calculated frac-tion of ring-polymer configurations that exhibit either a

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