Chemical Physics 319 (2005) 93–100
www.elsevier.com/locate/chemphys
Comparison of dynamical aspects of nonadiabatic electron,
proton, and proton-coupled electron transfer reactions
Elizabeth Hatcher, Alexander Soudackov, Sharon Hammes-Schiffer
*
Department of Chemistry, 104 Chemistry Building, Pennsylvania State University, University Park, PA 16802, USA
Received 9 February 2005; accepted 20 May 2005
Available online 15 July 2005
Abstract
The dynamical aspects of a model proton-coupled electron transfer (PCET) reaction in solution are analyzed with molecular
dynamics simulations. The rate for nonadiabatic PCET is expressed in terms of a time-dependent probability flux correlation function. The impact of the proton donor–acceptor and solvent dynamics on the probability flux is examined. The dynamical behavior of
the probability flux correlation function is dominated by a solvent damping term that depends on the energy gap correlation function. The proton donor–acceptor motion does not impact the dynamical behavior of the probability flux correlation function but
does influence the magnitude of the rate. The approximations previously invoked for the calculation of PCET rates are tested. The
effects of solvent damping on the proton donor–acceptor vibrational motion are found to be negligible, and the short-time solvent
approximation, in which only equilibrium fluctuations of the solvent are considered, is determined to be valid for these types of
reactions. The analysis of PCET reactions is compared to previous analyses of single electron and proton transfer reactions. The
dynamical behavior is qualitatively similar for all three types of reactions, but the time scale of the decay of the probability flux
correlation function is significantly longer for single proton transfer than for PCET and single electron transfer due to a smaller
solvent reorganization energy for proton transfer.
Ó 2005 Elsevier B.V. All rights reserved.
Keywords: Proton transfer; Electron transfer; Dynamics
1. Introduction
Proton-coupled electron transfer (PCET) reactions
involve the simultaneous transfer of an electron and a
proton. These types of reactions play an important role
in a wide range of chemical and biological processes [1–
17]. A variety of theoretical approaches have been developed and applied to PCET reactions in solution and
proteins [18–34]. PCET reactions are typically nonadiabatic due to the relatively large separation between the
electron donor and acceptor and small overlap between
the localized proton vibrational wavefunctions for the
reactant and product vibronic states. While the dynam*
Corresponding author.
E-mail address: [email protected] (S. Hammes-Schiffer).
0301-0104/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.chemphys.2005.05.043
ical aspects of single electron transfer (ET) and single
proton transfer (PT) reactions have been studied extensively [35–44], the dynamical aspects of PCET reactions
have been neglected in most previous theoretical
treatments.
In general, nonadiabatic ET, PT, and PCET reactions
can be described in terms of nonadiabatic transitions between reactant and product quantum states. Despite the
similarities among nonadiabatic ET, PT, and PCET
reactions, each type of reaction has unique features that
deserve careful consideration. Solvent reorganization
plays an important role in all three types of reactions,
but the magnitude of the solvent reorganization energy
and the relevant time scales of the solvent dynamics
vary. An important feature of PT and PCET reactions
is the role of the proton donor–acceptor vibrational
94
E. Hatcher et al. / Chemical Physics 319 (2005) 93–100
motion, which strongly impacts the nonadiabatic couplings and thereby the rates. Understanding the similarities and differences among these types of reactions will
aid in the elucidation of the underlying physical principles of charge transfer reactions in condensed phases.
In this paper, we analyze the dynamical behavior of a
model PCET system in solution and compare the results
to previous analyses of the dynamical aspects of nonadiabatic ET and PT reactions [37,39]. Our analysis is
based on a general dynamical rate expression for nonadiabatic PCET reactions [27]. The overall rate is expressed in terms of a time-dependent probability flux
correlation function. The effects of the proton donor–
acceptor and solvent dynamics on this probability flux
correlation function are analyzed with molecular
dynamics simulations. In addition, the approximations
invoked for previously derived analytical rate expressions are tested.
The paper is organized as follows. Section 2 summarizes the theoretical formulation for PCET reactions and
describes the model system. Section 3 presents the application of the dynamical theory to this model system and
provides a detailed analysis of the dynamical aspects of
PCET reactions, as well as a comparison to single ET
and PT reactions. The conclusions and future directions
are given in Section 4.
2. Methodology
2.1. Theoretical formulation
PCET systems may be defined in terms of the proton
coordinate rp, the proton donor–acceptor coordinate R,
and the solvent coordinates n. In our theoretical formulation for nonadiabatic PCET reactions [23,24], the active electrons and transferring proton are treated
quantum mechanically. The PCET reaction is described
in terms of nonadiabatic transitions between pairs of
reactant and product electron–proton vibronic states.
The reactant and product vibronic surfaces
eIl ðR; nÞ and eII
m ðR; nÞ, respectively, are obtained by calculating the proton vibrational states for the reactant
and product ET electronic states.
In our previous work [23,24], the electronic structure
of the solute is described in the framework of a fourstate valence bond (VB) model [45]. The most basic
PCET reaction involving the transfer of one electron
and one proton may be described in terms of the following four diabatic electronic basis states:
ð1aÞ
ð1bÞ
D
e
D
e
þ
ð2aÞ
ð2bÞ
De Dp Hþ Ap A
e
De Dp þ HAp A
e
Dp H Ap Ae
Dp þ HAp Ae
ð1Þ
where 1 and 2 denote the ET state, and a and b denote
the PT state. Typically, the intermediate states 1b and 2a
are much higher in energy than states 1a and 2b due to
differences in the electrostatic interactions. Thus, the
reactant ET state is predominantly 1a, and the product
ET state is predominantly 2b.
The coupling Vlm between two vibronic states Il and
IIm is defined as
D
E
;
ð2Þ
V lm ¼ /Il jV ðrp ; ny Þj/II
m
p
where the subscript of the angular brackets indicates
integration over rp, n represents the solvent coordinates
in the region of strong nonadiabatic coupling between
the reactant and product surfaces, V(rp, n) is the electronic coupling between reactant and product states,
and /Il and /II
m are the proton vibrational wavefunctions for the reactant and product vibronic states,
respectively. Here, I and II, respectively, denote reactant
and product ET states. For many PCET systems [46,47],
the coupling is approximately proportional to the overlap between the reactant and product proton vibrational
wavefunctions
V lm V el h/Il j/II
m ip ;
ð3Þ
where Vel is a constant effective electronic coupling.
Recently, we derived nonadiabatic rate expressions
for PCET that include the dynamical effects of both
the proton donor–acceptor mode (i.e., the R coordinate)
and the solvent [27]. One of the most important effects of
the R coordinate motion in PCET systems is the modulation of the proton tunneling distance and thereby the
nonadiabatic coupling between the reactant and product
vibronic states. The fluctuations of the nonadiabatic
coupling due to the R motion can be dynamically coupled to the fluctuations of the solvent degrees of freedom, which are responsible for bringing the system
into the degenerate state required for nonadiabatic transitions. Here, we consider the case in which the electron
and transferring proton are adiabatic with respect to the
R coordinate and the solvent within the reactant and
product states. The R mode is treated dynamically on
the same level as the solvent modes in order to include
the effects of the dynamical coupling between the R
mode and the solvent.
The nonadiabatic dynamical rate constant can be expressed as
k dyn ¼
fIg
X
l
P Il
fIIg
X
k dyn
lm ;
ð4Þ
m
where P Il is the Boltzmann weighting for the resistant
state Il. The partial rate constant dyn k dyn
lm describes nonadiabatic transitions between the quantum states for the
pair of electron–proton vibronic surfaces Il and IIm and
can be written as an integral of the time-dependent
probability flux correlation function jlm(t)
E. Hatcher et al. / Chemical Physics 319 (2005) 93–100
k dyn
lm ¼
1
h2
Z
1
jlm ðtÞ dt.
ð5Þ
1
We have used this formulation to derive rate expressions for both a dielectric continuum and a molecular
representation of the environment [27]. Here, we focus on only the molecular representation of the
environment.
To facilitate the derivation of an analytical expression
for the probability flux correlation function, the R
dependence of the overall coupling Vlm is approximated
by a single exponential
h
i
ð0Þ
Il Þ ;
V lm V lm
exp alm ðR R
ð6Þ
Il
is the equilibrium value of the R coordinate on
where R
ð0Þ
the reactant surface Il, V lm
is the value of the coupling
Il
, and alm can be estimated from the R depenat R ¼ R
dence of the coupling. The justification for Eq. (6) is that
the nonadiabatic coupling can be approximated as the
product of a constant electronic coupling and the
overlap of the reactant and product proton vibrational
wavefunctions, as given in Eq. (3). A more detailed
justification for Eq. (6) is provided in Appendix A.
For PCET reactions, typically this overlap depends only
weakly on the solvent coordinates but depends very
strongly on the proton donor–acceptor separation R.
The approximation in Eq. (6) has been shown to be
reasonable for model PCET systems and was also used
previously for nonadiabatic proton transfer systems
[36,38,48].
Using a second-order cumulant expansion, the probability flux correlation function can be expressed as
i
MD
ð0Þ 2
jlm ðtÞ ¼ jV lm j exp helm it exp a2lm ½C R ð0Þ þ C R ðtÞ
h
Z t
2ialm ~
hDlm i
C R ðsÞ ds
h
0
Z t
Z s1
1
2
ds1
ds2 C e ðs1 s2 Þ
h 0
0
Z t
Z s1
1
2
ds1
ds2 C D ðs1 s2 ÞC R ðs1 s2 Þ ;
h 0
0
ð7Þ
Il
Il
the derivative of the energy gap
~ lm ðtÞ ¼ oelm D
;
oR R¼R Il
for the other variables are defined analogously. These
expressions are analogous to those derived previously
for vibrationally nonadiabatic PT reactions occurring
on a single adiabatic electronic surface [38,39].
The quantities in Eq. (7) can be evaluated with molecular dynamics simulations of the full solute–solvent
system on the reactant vibronic surface eIl ðR; nÞ. Specifically, the quantity CR(t) is calculated from a molecular
dynamics simulation with an unconstrained R coordi~ lm i, Ce(t), and CD(t)
nate, and the quantities helmi, hD
are calculated from a molecular dynamics simulation
of the full solute–solvent system with the R coordinate
Il . In the derivation of this expresconstrained to R ¼ R
Il ¼ hRi. The quantum effects of
sion, we assumed that R
the R coordinate may be included using the standard
analytical expression for the time correlation function
of an undamped quantum mechanical harmonic oscillator [49].
The transferring hydrogen may be represented as a
classical nucleus bound to the proton donor at a fixed
distance during the molecular dynamics simulation on
the reactant surface. For the calculation of the energy
gap, the proton is moved to the proton acceptor to obtain the product state energy. This approximation neglects the effects of delocalization of the hydrogen
nuclear wavefunction on the motion of the other nuclei.
We emphasize that the quantum behavior of the transferring hydrogen is included in the rate calculation
through the couplings between the vibronic states, which
depend on the overlap between the reactant and product
proton vibrational wavefunctions.
2.2. Model system
ð8Þ
The model system used for the simulations discussed
in this paper is depicted in Fig. 1. The De–Ae distance is
constrainted to be 7.54 Å, and the Dp–H distance is constrained to be 1.1 Å. Harmonic bond restraints with
force constants of 232 kcal/mol/Å2 and equilibrium values of 2.37 Å are applied to De–Dp and Ae–Ap. To maintain approximate linearity, harmonic restraints with
force constants of 100.0 kcal/mol/deg2 and equilibrium
values of 180° are applied to the DeDpAe angle, the
DeApAe angle, and the DpHAp angle. The masses of
the electron donor and acceptor are 55.8470 g/mol
(i.e., the atomic mass of iron), and the masses of the proton donor and acceptor are 14.0067 g/mol (i.e., the
atomic mass of nitrogen).
ð9Þ
R
where the time evolution on the reactant vibronic surface is described in terms of the energy gap
I elm ðtÞ ¼ eII
m ðR ; nÞ el ðR ; nÞ;
95
and the R mode. The time correlation function for the
energy gap is defined as Ce(t) = hdelm(0)delm(t)i, where
delm(t) = elm(t) helmi. The time correlation functions
De
Dp
H
Ap
Ae
Fig. 1. Schematic picture of the solute for the model PCET system.
E. Hatcher et al. / Chemical Physics 319 (2005) 93–100
a
Cε(t)/Cε(0)
0.5
0
–0.5
b
CR(t)/CR(0)
0.5
0
–0.5
0
1
2
3
time (ps)
Fig. 2. Normalized time correlation function C(t)/C(0) for (a) the
energy gap fluctuations de and (b) the R coordinate fluctuations dR.
a
1.5
1
~
The solute is solvated with 253 explicit water molecules in a periodic cubic box with sides of length 19.98
Å. The water molecules are represented with the TIP3P
flexible model [50,51]. The solvent–solute interactions
include both electrostatic and van der Waals interactions. In the reactant, the electron donor has a charge
of 0.5e and the electron acceptor has a charge of
+0.5e, and in the product these charges are reversed.
In the reactant, the proton donor has a charge of 0.0
and the proton acceptor has a charge of 0.41e, and
in the product these charges are reversed. The proton always has a charge of +0.41e. Thus, the overall solute is
neutral. The Ewald method is used to treat the electrostatic interactions [52]. The van der Waals pair parameters for the interactions of De and Ae with the oxygen
atoms of the water molecules are e = 0.153 kcal/mol
and r = 3.5 Å. The van der Waals pair parameters for
the interactions of Dp and Ap with the oxygen atoms
of the water molecules are e = 0.15595 kcal/mol and
r = 3.14155 Å. The classical equations of motion are
integrated with the Verlet algorithm [53] with a time step
of 0.5 fs. The solute constraints are applied with the
SHAKE algorithm [54]. A canonical ensemble with a
temperature of 300 K is maintained with a Nose-Hoover
thermostat [55]. The system is equilibrated for 70 ps,
and the data is collected for 200 ps. All of the simulations were performed with a modified version of
DL_POLY_2.14 [56]. In the calculations of the probability flux, a = 10 Å1 [27].
J(ω)
96
3. Results
and the normalized spectral density is defined as [37]
Z 1
J ðxÞ
J~ðxÞ ¼ J ðxÞ
dx;
ð11Þ
x
0
where b = 1/kBT. The decay of the energy gap correlation function is characterized by two distinct relaxation
times. The initial fast decay on the time scale of 40 fs
can be attributed to the relatively high frequency librational modes of water, and the slower exponential tail
with a relaxation time of 0.5 ps is attributable to the
low frequency rotational and translational modes of
the solvent. In the spectral density for the solvent, the
stretching motions of water are evident at 3600
cm1, the bending motions of water are evident at
b
1.5
~
J(ω)
The time correlation functions Ce(t) and CR(t) for the
energy gap and R coordinate, respectively, are depicted
in Fig. 2. The corresponding spectral densities are depicted in Fig. 3. The spectral density J(x) is defined in
terms of the cosine transform of the time correlation
function C(t) as
Z 1
8
J ðxÞ
CðtÞ ¼
cosðxtÞ dx;
ð10Þ
pb 0
x
–0.5
1
–0.5
0
1000
2000
3000
4000
-1
frequency (cm )
Fig. 3. Normalized spectral density J~ðxÞ defined in Eq. (11) for (a) the
energy gap fluctuations de and (b) the R coordinate fluctuations dR.
1500 cm1, and the librational, rotational, and translational solvent motions correspond to the broad band at
lower frequencies. The R coordinate correlation function decays on a slower time scale of 3 ps. The sharp
peak in the spectral density for the R coordinate indicates that the frequency of this mode is 400 cm1.
E. Hatcher et al. / Chemical Physics 319 (2005) 93–100
For this model, hRi = 2.85 Å and hdR2i = 0.005 Å2. If
the R mode were harmonic and uncoupled to the solvent, CR(t) would oscillate indefinitely with fixed amplitude, and the spectral density would be a delta function
centered at the corresponding frequency. The observed
decay of CR(t) and the broadening of the peak in the
spectral density for the R mode are due to damping effects from the solvent–solute interactions.
If the potential energy of the system is assumed to be
harmonic along the R coordinate, the energy gap derivative defined in Eq. (9) becomes the constant
~ lm ¼ K~lm MX2 DRlm ;
D
IIm
ð12Þ
Il
R
is the difference between the equiwhere DR ¼ R
librium R coordinates on the product and reactant surfaces and the R mode frequency X is given by
o2 eIl ðR; nÞ
2
MX ¼
.
ð13Þ
oR2 Il
R¼R
In this limit, the energy gap derivative is independent of
~ lm i ¼ K~lm and C D ðtÞ ¼ K~2 . For a symmetric
time, so hD
lm
~ lm i ¼
system, K~lm ¼ 0. For this model system, hD
3.67 kcal=mol=Å. The two terms in Eq. (7) that include
this factor are negligible relative to the other terms due
~ lm i.
to the relatively small magnitude of hD
Fig. 4 depicts the time dependence of the flux and the
dominant contributing factors. The highly oscillatory
term exp½hi helm it shown in Fig. 4(b) has a period of
0.005 ps and describes the coherent oscillations of
the quantum amplitudes between the reactant and product vibronic states. For a symmetric system, the average
energy gap helmi = 72.25 kcal/mol corresponds to the
solvent reorganization energy. The period of the coherent term increases as the solvent reorganization energy
decreases. The quantum coherent term is damped by
other terms in Eq. (7), thereby enabling the calculation
of a reaction rate.
The R coordinate term expfa2lm ½C R ð0Þ þ C R ðtÞg
shown in Fig. 4(b) oscillates with a period of 0.08 ps
and approaches the constant value expfa2lm C R ð0Þg at
long times. A comparison of Figs. 2(b) and 4(a) indicates that the damping of the R mode by the solvent occurs on a longer time scale than the decay of the
probability flux. If the R coordinate motion is assumed
to be harmonic, and the solvent damping effects on this
harmonic motion are neglected, then the standard analytical expression for the time correlation function of
an undamped quantum mechanical harmonic oscillator
can be used for CR(t) [49]
h
1
coth
bhX cos Xt þ i sin Xt .
ð14Þ
C R ðtÞ ¼
2MX
2
The classical (high temperature) limit of this expression
is
C R ðtÞ ¼
a
1
j(t)/j(0)
0.5
0
–0.5
–1
Flux Components
b
1
0.5
0
–0.5
–1
0
0.01
0.02
time (ps)
0.03
0.04
Fig. 4. (a) Normalized probability flux time correlation function j(t)/
j(0) defined in Eq. (7). (b) Time dependence of normalized terms in Eq.
(7): coherent term exp½hi helm it with solid line, R coordinate term
expfa2lm ½C R ð0Þ þ
R t C R ðtÞg
R s with dot-dashed line, and solvent damping
term expf h12 0 ds1 0 1 ds2 C e ðs1 s2 Þg with dashed line. Each term is
normalized by the value at t = 0.
97
1
cos Xt.
bMX2
ð15Þ
Fig. 5(a) compares the R coordinate term with CR(t) calculated from the simulations to this term with CR(t) calculated from the expressions in Eqs. (14) and (15). The
agreement between the calculated R coordinate term
and the classical harmonic oscillator results indicates
that solvent damping effects are not important on the
time scale of the probability flux. The classical and
quantum harmonic oscillator results are both fairly constant for the relevant time scale, suggesting that the
quantum effects of the R coordinate do not impact the
dynamical behavior of the probability flux. On the other
hand, inclusion of the quantum effects of the R coordinate increases the magnitude of the rates slightly.
Rt
Rs
The solvent damping term expf h12 0 ds1 0 1
ds2 C e ðs1 s2 Þg decays to zero on a time scale of
0.005 ps, as shown in Fig. 4(b). Since the solvent term
decays much faster than the R coordinate term, the time
scale of the decay of the flux is determined mainly by the
solvent relaxation time. Moreover, the decay time of the
probability flux j(t) shown in Fig. 4(a) is short with respect to the initial decay time of the energy gap correlation function Ce(t) shown in Fig. 2(a). As a result, only
the initial value of the energy gap correlation function
(i.e., Ce(0) = hde2i) impacts the rate. In this case, the
short-time solvent approximation, which considers
only equilibrium fluctuations of the solvent, is valid.
98
E. Hatcher et al. / Chemical Physics 319 (2005) 93–100
2
exp[α (CR (0)+CR(t))]
a
4.5
4
3.5
3
2.5
Solvent Terms
b
0.75
0.5
0.25
0
0
0.005
time (ps)
0.01
Fig. 5. (a) Comparison of R coordinate term expfa2lm ½C R ð0Þ þ C R ðtÞg
calculated with CR(t) from classical molecular dynamics simulations
(solid line) to this term with CR(t) for an undamped quantum
harmonic oscillator given in Eq. (14) (dotted line) and with CR(t) for
an undamped classical harmonic oscillator given in Eq. (15)R(dashed
Rs
t
line). (b) Comparison of solvent damping term expf h12 0 ds1 0 1
ds2 C e ðs1 s2 Þg calculated with molecular dynamics simulations (solid
line) and calculated with the short-time approximation given in Eq.
(16) (dashed line).
According to the short-time approximation, Ce(t) =
Ce(0) and the solvent damping term becomes a Gaussian
Z t
Z s1
1
ds1
ds2 C e ðs1 s2 Þ
exp 2
h 0
0
C e ð0Þt2
¼ exp .
ð16Þ
2
h2
Fig. 5(b) compares the exact solvent damping term with
Ce(t) calculated from the simulations to the approximate
expression in Eq. (16). These results are virtually indistinguishable, indicating that the short-time approximation is valid for these types of systems. Typically, the
solvent damping term decays faster as the solvent reorganization energy increases.
The probability flux correlation function for nonadiabatic single ET is analogous to Eq. (7) with the removal
of all terms involving the R coordinate. For single ET,
the dominant terms are the coherent term, which oscillates with a frequency dependent on the solvent reorganization energy, and the solvent damping term, which
decays on a time scale dependent on the solvent reorganization energy. The solvent reorganization energies for
PCET and single ET are typically similar for the same
De–Ae separation because the proton moves only a relatively small distance (0.5 Å) and therefore does not
significantly influence the difference in solute charge
density for the reactant and product states. Previously,
Ando [37] calculated the energy gap correlation functions and the corresponding spectral densities for a series of single ET reactions in water. The results are
qualitatively similar to those shown here, with similar
time scales for the decay of the energy gap correlation
functions.
As mentioned above, the decay of the solvent damping term in Eq. (7) dominates the flux because it decays
faster than all other terms, so the R-dependent terms in
Eq. (7) do not significantly impact the dynamical aspects
of the probability flux correlation function. Thus, the
dynamical behavior of the probability flux for PCET
and ET reactions is similar. On the other hand, the magnitude of the rates for PCET and ET often vary substantially because of differences in the nonadiabatic coupling
ð0Þ
. As given by Eq. (3), the coupling for PCET can be
V lm
approximated as the product of an electronic coupling
and the overlap between the reactant and product proton vibrational wavefunctions. The overlap factor tends
to decrease the rate of PCET relative to the corresponding ET reaction [46].
The probability flux expression in Eq. (7) is analogous to the probability flux expression given in [39] for
vibrationally nonadiabatic PT reactions occurring on a
single adiabatic electronic surface. In PCET reactions,
however, the energy gap and coupling are defined for
pairs of electron–proton vibronic surfaces corresponding to different electronic states. Thus, the coupling for
PCET reactions involves ET nonadiabatic coupling as
well as PT coupling. Moreover, the value of a (i.e., the
R dependence of the coupling) could differ for PCET
and PT reactions due to qualitatively different electrostatic interactions between the transferring proton and
the solute electrons. In addition, the solvent reorganization is typically much smaller for PT than for PCET because the proton is transferred a shorter distance than
the electron, so PCET reactions involve substantially
greater differences in solute charge density for the reactant and product states. As a result of this difference in
solvent reorganization energies, the period of the coherent term is smaller for PCET than for PT, and the solvent damping term decays faster for PCET than for PT.
Previously, Borgis and Hynes [39] calculated the time
correlation functions for the energy gap, R coordinate
frequency shift, and probability flux for a nonadiabatic
PT reaction in a model polar aprotic solvent. The results
are qualitatively similar to those shown here in that the
solvent decay dominates the probability flux. For PT,
however, the decay of the probability flux correlation
function is on a longer time scale (0.02 ps) than the
time scale of 0.005 ps observed for PCET. Borgis
and Hynes [39] also found that the damping of the R
mode is negligible and that the short-time approximation is valid for nonadiabatic PT reactions.
E. Hatcher et al. / Chemical Physics 319 (2005) 93–100
4. Conclusions
In this paper, we analyzed the dynamical aspects of a
model PCET reaction in solution and tested the approximations previously invoked for the analytical calculation of rates. The rate constant for nonadiabatic
PCET reactions is expressed as a time integral of a
time-dependent probability flux correlation function.
The dynamical behavior of the probability flux correlation function is dominated by a solvent damping term
that depends on the energy gap correlation function.
Only the initial value of the energy gap correlation function impacts the rate, so the short-time solvent approximation, in which only the equilibrium solvent
fluctuations are considered, is valid for these types of
systems. As the solvent reorganization energy increases,
the probability flux correlation function decays faster.
The time dependence of the correlation function for
the proton donor–acceptor mode does not impact the
dynamical behavior of the probability flux correlation
function, but the initial value of this correlation function
influences the magnitude of the rate. Solvent damping
effects on the proton donor–acceptor mode are negligible, and the harmonic approximation for this mode is
physically reasonable. The quantum effects of the proton donor–acceptor mode do not impact the dynamical
behavior of the probability flux but slightly increase the
magnitude of the rate. These results provide validation
for the use of the analytical dynamical rate expressions
discussed in [27].
We also compared the dynamical aspects of PCET to
those of single ET and single PT. For PCET and ET
reactions with similar solvent reorganization energies,
the dynamical behavior of the probability flux correlation function, including the time scale of the decay, is
similar. Typically, the magnitude of the rate of a PCET
reaction is significantly lower than that of the corresponding ET reaction because the coupling for PCET
is the product of an electronic coupling and the overlap
between the reactant and product proton vibrational
wavefunctions. The dynamical behavior of nonadiabatic
PT reactions is also qualitatively similar to that of PCET
reactions, but typically PT reactions have much smaller
solvent reorganization energies than PCET reactions. As
a result, the probability flux correlation function decays
on a significantly longer time scale for PT than for
PCET reactions. In addition, the magnitude of the rate
will be influenced by differences in the couplings for
PCET and PT. Specifically, PCET couplings are defined
for pairs of vibronic surfaces corresponding to different
electronic states, wheres PT reactions typically occur on
a single adiabatic electronic surface.
This paper provided a detailed analysis of the dynamical behavior for a single representative PCET model
system. Future work will examine a series of PCET
model systems to analyze the dependence of PCET rates
99
and kinetic isotope effects on the solvent reorganization
energy, the frequency of the proton donor–acceptor
mode, and the PCET vibronic coupling. The temperature dependence of PCET rates and kinetic isotope effects will also be investigated at various levels of
theory. We emphasize that the dynamical behavior studied in this paper occurs in the linear response regime.
Future work will explore the impact of nonlinear effects.
The predictions emerging from all of these studies will
be experimentally testable and will enhance our understanding of the underlying physical principles of PCET
reactions.
Acknowledgements
This work was supported by NSF Grant CHE0096357 and NIH Grant GM56207.
Appendix A
This appendix provides the justification for the form
of the coupling given in Eq. (6). The PCET nonadiabatic
coupling Vlm can be approximated as the product of a
constant ET coupling VET and the overlap S(R) between
the reactant and product proton vibrational
wavefunctions
V ðRÞ V ET SðRÞ.
ðA1Þ
For a simple model based on two ground state harmonic
oscillator wavefunctions with centers separated by R,
the overlap has the form
SðRÞ ¼ exp½a0 R2 .
ðA2Þ
Note that the coupling in Eq. (6) is given in terms of
where R
is the equilibrium value of the R
dR ¼ R R,
coordinate on the reactant surface. Eq. (A2) can be
rewritten as
h
i
þ RÞ
2
SðRÞ ¼ exp a0 ðR R
exp adR a0 dR2 ;
ðA3Þ
¼ SðRÞ
In the region around dR = 0, we can rewhere a ¼ 2a0 R.
tain only the first order term in the exponential, and the
overlap is expressed as
exp½adR;
SðRÞ SðRÞ
ðA4Þ
which leads to the form of the coupling given in Eq. (6).
For the more general case, the natural logarithm of
the coupling can be expanded in a Taylor series around
R¼R
¼ adR a0 dR2 þ ;
ln ½V ðRÞ=V ðRÞ
ðA5Þ
which leads to the form of the coupling given in Eq. (6)
when only linear terms are retained.
100
E. Hatcher et al. / Chemical Physics 319 (2005) 93–100
References
[1] G.T. Babcock, B.A. Barry, R.J. Debus, C.W. Hoganson, M.
Atamian, L. McIntosh, I. Sithole, C.F. Yocum, Biochemistry 28
(1989) 9557.
[2] M.Y. Okamura, G. Feher, Annual Reviews of Biochemistry 61
(1992) 861.
[3] C. Tommos, X.-S. Tang, K. Warncke, C.W. Hoganson, S.
Styring, J. McCracken, B.A. Diner, G.T. Babcock, Journal of
the American Chemical Society 117 (1995) 10325.
[4] C.W. Hoganson, G.T. Babcock, Science 277 (1997) 1953.
[5] C.W. Hoganson, N. Lydakis-Simantiris, X.-S. Tang, C. Tommos,
K. Warncke, G.T. Babcock, B.A. Diner, J. McCracken, S.
Styring, Photosynthesis Research 47 (1995) 177.
[6] M.R.A. Blomberg, P.E.M. Siegbahn, S. Styring, G.T. Babcock,
B. Akermark, P. Korall, Journal of the American Chemical
Society 119 (1997) 8285.
[7] B.A. Diner, G.T. Babcock, in: D.R. Ort, C.F. Yocum (Eds.),
Oxygenic Photosynthesis: The Light Reactions, Kluwer, Dordrecht, 1996, p. 213.
[8] G.T. Babcock, M. Wikstrom, Nature 356 (1992) 301.
[9] B.G. Malmstrom, Accounts of Chemical Research 26 (1993) 332.
[10] P.E.M. Siegbahn, L. Eriksson, F. Himo, M. Pavlov, Journal of
Physical Chemistry B 102 (1998) 10622.
[11] J.P. Roth, S. Lovel, J.M. Mayer, Journal of American Chemical
Society 122 (2000) 5486.
[12] R.A. Binstead, T.J. Meyer, Journal of the American Chemical
Society 109 (1987) 3287.
[13] B.T. Farrer, H.H. Thorp, Inorganic Chemistry 38 (1999) 2497.
[14] J.P. Kirby, J.A. Roberts, D.G. Nocera, Journal of the American
Chemical Society 119 (1997) 9230.
[15] M.H.V. Huynh, T.J. Meyer, Angewandte Chemie-International
Edition 41 (2002) 1395.
[16] M. Sjodin, S. Styring, B. Akermark, L. Sun, L. Hammarstrom,
Journal of the American Chemical Society 122 (2000) 3932.
[17] M.J. Knapp, K.W. Rickert, J.P. Klinman, Journal of the
American Chemical Society 124 (2002) 3865.
[18] R.I. Cukier, Journal of Physical Chemistry 98 (1994) 2377.
[19] R.I. Cukier, Journal of Physical Chemistry 100 (1996) 15428.
[20] R.I. Cukier, D.G. Nocera, Annual Reviews of Physical Chemistry
49 (1998) 337.
[21] R.I. Cukier, Journal of Physical Chemistry A 103 (1999) 5989.
[22] R.I. Cukier, Journal of Physical Chemistry B 106 (2002) 1746.
[23] A. Soudackov, S. Hammes-Schiffer, Journal of Chemical Physics
111 (1999) 4672.
[24] A. Soudackov, S. Hammes-Schiffer, Journal of Chemical Physics
113 (2000) 2385.
[25] S. Hammes-Schiffer, Accounts of Chemical Research 34 (2001)
273.
[26] S. Hammes-Schiffer, in: V. Balzani (Ed.), Electron Transfer in
Chemistry, Principles, Theories, Methods, and Techniques, vol. I,
Wiley–VCH, Weinheim, 2001, p. 189.
[27] A.V. Soudackov, E. Hatcher, S. Hammes-Schiffer, Journal of
Chemical Physics 122 (2005) 014505.
[28] J.M. Mayer, D.A. Hrovat, J.L. Thomas, W.T. Borden, Journal of
the American Chemical Society 124 (2002) 11142.
[29] J.S. Mincer, S.D. Schwartz, Journal of Chemical Physics 120
(2004) 7755.
[30] Y. Georgievskii, A.A. Stuchebrukhov, Journal of Chemical
Physics 113 (2000) 10438.
[31] D.B. Moore, T.J. Martinez, Journal of Physical Chemistry A 104
(2000) 2367.
[32] P.E.M. Siegbahn, M.R.A. Blomberg, R.H. Crabtree, Theoretical
Chemistry Accounts 97 (1997) 289.
[33] W. Siebrand, Z. Smedarchina, Journal of Physical Chemistry B
108 (2004) 4185.
[34] A.M. Kuznetsov, J. Ulstrup, Canadian Journal of Chemistry 77
(1999) 1085.
[35] A.V. Barzykin, P.A. Frantsuzov, K. Seki, M. Tachiya, in: I.
Prigogine, S.A. Rice (Eds.), Advances in Chemical Physics, vol.
123, John Wiley & Sons, Inc., New York, 2002, p. 511.
[36] A. Suarez, R. Silbey, Journal of Chemical Physics 94 (1991)
4809.
[37] K. Ando, Journal of Chemical Physics 106 (1997) 116.
[38] D. Borgis, S. Lee, J.T. Hynes, Chemical Physics Letters 162
(1989) 19.
[39] D. Borgis, J.T. Hynes, Journal of Chemical Physics 94 (1991)
3619.
[40] D. Borgis, J.T. Hynes, Chemical Physics 170 (1993) 315.
[41] A.M. Kuznetsov, Charge Transfer in Physics, Chemistry, and
Biology, Gordon & Breach, Reading, MA, 1995.
[42] A.M. Kuznetsov, J. Ulstrup, Electron Transfer in Chemistry and
Biology: An Introduction to the Theory, Wiley, Chichester, 1999.
[43] M.D. Newton, H.L. Friedman, Journal of Chemical Physics 88
(1988) 4460.
[44] L.W. Ungar, M.D. Newton, G.A. Voth, Journal of Physical
Chemistry B 103 (1999) 7367.
[45] A. Warshel, Computer Modeling of Chemical Reactions in
Enzymes and Solutions, Wiley, New York, 1991.
[46] N. Iordanova, H. Decornez, S. Hammes-Schiffer, Journal of the
American Chemical Society 123 (2001) 3723.
[47] N. Iordanova, S. Hammes-Schiffer, Journal of the American
Chemical Society 124 (2002) 4848.
[48] L.I. Trakhtenberg, V.L. Kochikhim, S.Y. Pshezhetsky, Chemical
Physics 69 (1982) 121.
[49] D. Chandler, Liquids Freezing and Glass Transition, Elsevier,
Amsterdam, 1991.
[50] W.L. Jorgensen, Journal of the American Chemical Society 103
(1981) 335.
[51] W.L. Jorgensen, J. Chandreskhar, J.D. Madura, R.W. Impey,
M.L. Klein, Journal of Chemical Physics 79 (1982) 926.
[52] P.P. Ewald, Annals of Physics 64 (1921) 253.
[53] L. Verlet, Physical Review 159 (1967) 98.
[54] J.P. Ryckaert, G. Ciccotti, H.J.C. Berendsen, Journal of Computational Physics 23 (1977) 327.
[55] W.G. Hoover, Physical Review A 31 (1985) 1695.
[56] W. Smith, T.R. Forester, DL_POLY_2.14. CCLRC, Daresbury
Laboratory, Warrington, England, 2003.