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Direct simulation of proton-coupled electron transfer across
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J. Chem. Phys. 138, 134109 (2013); doi: 10.1063/1.4797462 View
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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
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134109-3 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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−1)nep+l)e , 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
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134109-4 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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−1)nep+l)e , 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
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134109-5 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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
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134109-6 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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
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134109-7 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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.
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134109-8 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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)
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134109-9 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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
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134109-10 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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
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134109-11 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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.
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134109-12 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
(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
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134109-13 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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.
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134109-14 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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.
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134109-15 J. S. Kretchmer and T. F. Miller III J. Chem. Phys.
138, 134109 (2013)
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