Article pubs.acs.org/JPCA
Calculation of Electron Transfer Rates Using Mixed Quantum Classical Approaches: Nonadiabatic Limit and Beyond Weiwei Xie, Shuming Bai, Lili Zhu, and Qiang Shi* Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China
ABSTRACT: We investigate the applicability of the Ehrenfest and surface hopping methods to calculate electron transfer rates using the spin−boson model with different parameters. Rate constants are obtained from short time dynamics performed in both the diabatic and adiabatic basis sets. Numerical results and theoretical analysis show that these two methods can be reasonably accurate in the nonadiabatic limit, by staying close to an approximate Fermi’s golden rule. Beyond the nonadiabatic limit, the calculated mixed quantum classical rates are compared with numerical exact results, and similar accuracy was found as in the nonadiabatic limit. The relation between the current finding and recent studies using the surface hopping method based on long time dynamics is also discussed. It is found that the short time dynamics could be more accurate in calculating rate constants using the mixed quantum classical methods.
1. INTRODUCTION
As approximations to the full quantum dynamics, both methods have been tested against benchmark problems to assess their accuracy and limits. For example, the surface hopping method has been benchmarked with a series of onedimensional two state model,4,14 and proton transfer reactions in a double well model coupled to a harmonic bath;17 the Ehrenfest and SH methods have also been tested in studies of dynamics at conical intersections and dynamics of spin−boson models.7,18 More recently, the question whether the long time dynamics from these two methods obeys the detailed balance is also investigated.19,20 In this work, we will apply the two methods to calculate electron transfer rates using the spin−boson type of models.21−23 There are recently several studies related to this topic. In calculating the electron transfer rates between a pair of molecules in an organic molecular crystal, we have studied the analytic rate of several mixed quantum classical methods in the nonadiabatic limit, and compared them with Fermi’s Golden rule.24 In two recent papers,25,26 Landry and Subotnik have applied the fewest switches surface hopping (FSSH) method to study the electron transfer rates in the nonadiabatic limit and
Quantum mechanic effects are important in a variety of chemical and physical phenomena, where typical examples include nonadiabatic dynamics in electron transfer and photochemistry reactions, proton and hydrogen transfer reactions, vibrational relaxation of high frequency modes, etc. Despite recent advances in rigorous methods to study both model and realistic systems,1−3 simulation of quantum dynamics in the condensed phase remains a significant challenge in theoretical chemistry. By treating only the most important degrees of freedom (DOFs) quantum mechanically while retaining a much simpler classical description for the other DOFs, the mixed quantum classical methods can reduce the computational costs dramatically4−8 and are still the method of choice in many problems involving quantum mechanic effects. The Ehrenfest5,9,10 and surface hopping (SH)4,11−13 methods are the two mixed quantum classical approaches proposed early in the literature. Despite some well-known deficiencies,7,14 they are still the most popular methods due to their simplicity and robustness. Recent examples of their applications include calculation of charge transfer rates, nonradiative decay, and photochemical reactions. (See, e.g., refs 12, 15, and 16 for some typical examples. More examples can be found in “Special topic: Nonadiabatic Dynamics”, in J. Chem. Phys. 2012, 137, 22.) © 2013 American Chemical Society
Special Issue: Prof. John C. Wright Festschrift Received: January 15, 2013 Revised: March 22, 2013 Published: March 27, 2013 6196
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the equilibrium nuclear configuration of donor state |1⟩. To account for the quantum effects in the initial sampling of the nuclear DOFs, they are sampled from the multidimensional Wigner distribution37 calculated using e−βH1/Z1,
compared them with the Marcus rate theory. They found that the FSSH may significantly overestimate the electron transfer rates, which can be avoided by adding decoherence to the SH algorithm with their augmented fewest switches surface hopping (A-FSSH) method. We will extend these studies by investigating these two methods in calculating the ET rates in a more systematic way, and analyze their successes and failures. The remaining parts of the article are organized as follows. In section 2, we present the model Hamiltonian and briefly outline several different schemes of the Ehrenfest and SH calculations in both the diabatic and adiabatic basis sets. In section 3, we first show examples in the nonadiabatic limit. Analysis of the numerical results and their connections to the abovementioned theoretical and numerical studies are made. Numerical results are then presented for problems beyond the nonadiabatic limit. Conclusions and discussions are made in section 4.
ρw (x 0,p0)
⎡ ⎛ ⎞2 ⎤ ⎢p 2 + ω 2⎜x + cj σ ⎟ ⎥ ∑⎢ j j ⎜ j z ωj 2 ⎟⎠ ⎥⎦ ⎝ j ⎣
∑ j
cj 2 ωj
=
∏ 1 tanh(βωj /2)
∫ dΔx e−ip ·Δx 0
x0 +
Δx e−βH1 Δx x0 − 2 Z1 2
π
2 ⎤⎫ ⎧ ⎡ ⎛ cj ⎞ ⎥⎪ ⎪ tanh(βωj /2) ⎢ 2 2⎜ ⎟ ⎬ p x ω × exp⎨− + + j ⎜ j0 ⎢ j0 ωj ωj 2 ⎟⎠ ⎥⎦⎪ ⎪ ⎝ ⎣ ⎩ ⎭
(3)
We note that in the nonadiabatic limit (i.e., weak electronic coupling case), it can be assumed that the nuclear DOFs are equilibrated on the donor surface. Beyond the nonadiabatic limit, as long as there is a well-defined separation of time scales (i.e., ET rate is much slower than any other relaxation processes), such a choice of initial condition will also give the correct ET rate by following the dynamics of electronic population (see eqs 17 and 18 below). The Ehrenfest and surface hopping calculations can be written in either the diabatic or adiabatic basis set. When using the diabatic basis set, we assume that the time-dependent wave function |ψ(t)⟩ can be written as |ψ (t )⟩ = a1(t )|1⟩ + a 2(t )|2⟩
(1)
(4)
In both the Ehrenfest and SH methods, dynamics of the quantum subsystem is governed by a time-varying Hamiltonian determined by the classical trajectory
where σx and σz are the Pauli matrices, σx = |1⟩⟨2| + |2⟩⟨1| and σz = |1⟩⟨1| − |2⟩⟨2|; Δ is the electronic coupling between the two diabatic donor and acceptor states |1⟩ and |2⟩; ε is the energy difference between them; xj, pj, and ωj are the massweighted coordinate, momentum, and frequency of the jth bath mode; the harmonic bath couples linearly to the electronic DOF and causes energy fluctuations on the two states, where cj is the coupling coefficient between the bath coordinate and the system operator σz. The system−bath coupling in the spin− boson model system is characterized by the spectral density J(ω) defined as π J(ω) = 2
1 (2π )N
j
2. THEORY The spin−boson Hamiltonian21,27 and its extensions have been widely used as microscopic models for ET processes in condensed phase, such as in solution and biological systems (see, e.g., ref 28 and various studies in ref 29) as well as in charge transfer in organic semiconductors.30 Its application to low temperature cases have also been reported.31,32 The total Hamiltonian of the system and bath is written as 1 ε H = Δσx + σz + 2 2
=
a1̇ (t ) = −i{[ε /2 + F(t )]a1(t ) + Δa 2(t )}
(5a)
a 2̇ (t ) = −i{Δa1(t ) − [ε /2 + F(t )]a 2(t )}
(5b)
where F = ∑jcjxj is defined as the collective bath coordinate. With the adiabatic basis set, the wave function is written as |ψ (t )⟩ = b1(t )|e1⟩ + b2(t )|e 2⟩
(6)
where |e1⟩ and |e2⟩ are the eigenstates of the time-dependent Hamiltonian Hel(t) = Δσx + [ε/2 + F(t)]σz, with energies
δ(ω − ωj) (2)
E1,2 = ∓ (ε /2 +
The spin−boson model adopts a diabatic basis set in describing the electronic states, rather than the commonly used adiabatic states from first principle calculations based on the Born−Oppenheimer approximation. Nevertheless, now it is possible to calculate both the coupling constant Δ and intramolecular part of electron−vibrational coupling ωj and cj using first principle methods.30,33,34 To obtain the intermolecular contributions to the electron−vibrational coupling, methods based on molecular dynamics simulations can be employed.35,36 In the Ehrenfest and surface hopping methods, the electronic DOF is treated quantum mechanically, and the nuclear DOFs are treated classically. It is assumed that the initial state is equilibrated on the donor state ρ0 = e−βH1/Z1 ⊗ |1⟩⟨1|, where H1 = (1/2)∑j(pj2 + ωj2xj2) + ∑jcjxj, and Z1 = Tr e−βH1. So the initial electronic wave function is |ψ(0)⟩ = |1⟩. The initial nuclear positions {xj0} and momenta {pj0} are sampled from
∑ cjxj)2 + Δ2 (7)
j
The equation of motion in the adiabatic basis set is given by b1̇ (t ) = −iE1b1(t ) − p·d12b2(t )
(8a)
b2̇ (t ) = −iE2b2(t ) − p·d 21b1(t )
(8b)
where the nonadiabatic coupling vector d21 = −d12, and d12j =
e1
∂ e2 ∂xj
=
Δcj 1 2 (ε /2 + ∑j cjxj)2 + Δ2
(9)
In the Ehrenfest method, the classical dynamics is calculated on the average potential surface of the two states, xj̇ (t ) = pj (t ) 6197
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the FGR rate in eq 13 reduces to the well-known Marcus’s rate equation38,39
pj̇ (t ) = − ψ (t )
∂Hel ψ (t ) ∂xj
= − ωj 2xj(t ) − cj⟨ψ (t )|σz|ψ (t )⟩
kM =
(10b)
λ=
(11)
⎧ × exp⎨− ⎩
∞
∫0
⎫ − i sin ωt ]⎬ ⎭
∞
dω J(ω) ω
(15)
(17)
where k is the forward rate constant from |1⟩ to |2⟩, and kb is the backward rate constant from |2⟩ to |1⟩. In the following calculations, we all have kb ≤ k, and P2(0) = 0. So at short time, we have P1̇(t ) ≈ −kP1(t )
(18)
The rate constant is then calculated using short time dynamics after the rate defined in eq 18 reaches a plateau. In numerical calculations using the mixed quantum classical methods, the spectral density is discretized into 900 bath modes. The simulations were performed with a time step of 2.5 au, and a sampling of 5 × 104 trajectories was used for the statistical average unless stated otherwise. In the Ehrenfest method, the populations on the |1⟩ and |2⟩ states can be calculated conveniently using the wave functions |ψ(t)⟩. In the standard SH dynamics, the population in the diabatic basis can be obtained by counting the number of trajectories on each diabatic surface. However, in the SH dynamics using the adiabatic basis, it is necessary to obtain the diabatic population P1(t) from the adiabatic trajectory. Two different methods are used to calculate P1(t): the first method is to calculate the diabatic population from the adiabatic wave functions; the second is by transforming the adiabatic populations on |e1⟩ and |e2⟩ into the diabatic basis, which is the same method as used in refs 25 and 26. We note that this ambiguity is due to the lack of a clear scheme to determine diabatic populations from the adiabatic trajectory, and both approaches have their problems. For the first approach, although the wave functions are used to determine the
dt e iεt
dω
∞
P1̇(t ) ≈ −kP1(t ) + k bP2(t )
3. RESULTS In this section, we apply the above methods to calculate ET rates. We start from the nonadiabatic limit, where the coupling between the two diabatic states, Δ, is small enough such that a perturbative treatment of the coupling strength is valid. The case of larger Δ beyond the weak coupling limit is then investigated. 3.1. Nonadiabatic Limit. In the nonadiabatic limit, the ET rate can be calculated using the Fermi’s Golden rule23,27 (FGR),
∫0
∫0
Here Ω denotes the frequency of the Brownian oscillator, and η is the friction coefficient. The two parameters Ω and η determine the relaxation dynamics of the nuclear DOFs. The FGR rates can thus be calculated with eqs 13 and 16. When there is a separation of time scales, i.e., when the nuclear DOFs relax on a much faster scale than the ET reaction, the ET rate can be calculated by assuming the following phenomenological rate dynamics after a short transition relaxation plateau time42
(12)
on surfaces |e1⟩ and |e2⟩. Different hopping algorithms have been proposed in the literature. In this study, the standard “fewest switches” surface hopping (FSSH) algorithm suggested in ref 4 has been used, along with the standard treatment of forbidden transitions, and the velocity adjustment to conserve the total energy in each trajectory.4,14 As is well-known, the SH approach gives different results in the diabatic and adiabatic basis sets.14 In the literature, the SH calculations are usually performed in the adiabatic representation, but a similar algorithm can certainly be applied in the diabatic basis. Moreover, because the spin−boson problem is presented in the diabatic basis when electron transfer reactions are studied, SH calculations in the diabatic basis is certainly justified, and we will also compare the results calculated in these two basis sets.
kFGR = 2Δ2 Re
4 π
The parameters were taken from two recent studies,25,26 where the collective bath coordinate is described as a damped Brownian oscillator model with the reorganization energy λ = 2.39 × 10−2 au (atomic unit) (5245 cm−1), kBT = 9.5 × 10−4 au (300 K), and the harmonic oscillator frequency Ω = 3.5 × 10−4 au (77 cm−1). The other two parameters, the energy difference ε between the donor and acceptor states and the friction constant of the bath η are varied in the following calculations. It is well-known that the spectral density of a Brownian oscillator can be described as40,41 ηω 1 J(ω) = λ Ω2 2 2 (ω − Ω2)2 + η2ω 2 (16)
where −(+) is used for dynamics on the donor (|1⟩) and acceptor (|2⟩) surfaces; in the adiabatic basis, eq 10b should be replaced by pj̇ (t ) = −ωj 2xj(t ) − ∂E1,2 /∂xj
(14)
where the reorganization energy λ can be calculated as
The expectation values in eq 10b can be calculated in either the diabatic or adiabatic basis set. And it can be shown that, if started from the same initial wave function and nuclear positions and momenta, the Ehrenfest dynamics using the two different basis sets will give the same result. In the SH method, quantum transitions are realized by “hoppings” between different electronic surfaces. The forces felt by the nuclear DOFs are determined by which state the electronic DOF is on. In the diabatic basis, eq 10b should be replaced by
pj̇ (t ) = −ωj 2xj(t ) ∓ cj
⎡ (ε − λ)2 ⎤ 2π Δ2 exp⎢ − ⎥ ⎣ 4λkBT ⎦ 4πλkBT
4J(ω) [coth(βω /2)(1 − cos ωt ) πω 2 (13)
When assuming a classical limit coth(βω/2) ≈ 2/ βω, and the short time approximation 1 − cos ωt ≈ 1/2ω2t2, sin ωt ≈ ωt, 6198
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from a previous theoretical study,24 where it is found that the Ehrenfest rate at the nonadiabatic limit can be approximated as
transition probabilities in the SH method, they are usually not used to calculate the populations directly. For the second approach, the coherence between the adiabatic states are completely neglected in calculating the diabatic populations. Figure 1 shows the short time dynamics of P1(t) with the Ehrenfest and SH methods in the diabatic basis, using
kAFGR = 2Δ2 Re ⎧ × exp⎨− ⎩ ⎫ − iωt ]⎬ ⎭
∫0
∫0
∞
∞
dω
dt e iεt 4J(ω) [coth(βω /2)(1 − cos ωt ) πω 2 (19)
Here, the subscript AFGR stands for an approximated FGR rate. The only difference between the above equation and FGR rate is that the term sin ωjt in eq 13 is replaced by ωjt in eq 19. Derivation of this result is given in Appendix A for completeness. It was also found in ref 24 that the same results should also apply for the SH rates in the diabatic basis. However, the previous analysis of the SH method in the diabatic basis have made some strong assumptions on the hopping probability,24 and an improved analysis is provided in Appendix B. For the specific parameters, the AFGR rates turned out to be the same as the FGR/Marcus results, which explains the good agreements in Figure 2. We then compare the long time dynamics from the Ehrenfest and SH methods in the diabatic basis, with the rate dynamics predicted using the calculated rate constant. The results are shown in Figure 3, where the donor population is averaged over
Figure 1. Time evolution of the donor state population by two methods in diabatic basis. The parameters used are Δ = 5 × 10−5, ε = 0.015, and η = 1.875 × 10−5 (all in atomic units). An average over 5 × 105 trajectories was used in the population calculation.
parameters ε = 0.015 au, and η = 1.875 × 10−5 au. The inset of Figure 1 shows that the rate dynamics is reached in a rather short time ( tp, which is the plateau time. In practice, we can replace tp with ∞ in eq 28 and make use the stationary property of F(t) to obtain ∞
dt
We note that P1→2 and P2→1 always have different signs. If we define X(t) = Im a1(t) a2*(t), the total probability for the decay of P1(t), the population on state |1⟩, can be calculated as ⟨[P1P1→2H(X(t)) − P2P2→1H(−X(t))]⟩ dt, where the average is taken for all the trajectories and H is the Heaviside function [H(X) = 1 for X ≥ 0 and 0 for X < 0]. If we further assume P1 ≈ |a1(t)|2 and P2 ≈ |a2(t)|2, which is reasonable for the SH dynamics at short time, the rate of decay should be k ≈ limt→tp ⟨2Δ Im a1(t) a*2 (t)⟩. If we further neglect the constraints posted by the requirement of the energy conservation (the forbidden transitions and the adjustment of the momentum after the hop), the decay rate is then the same as the Ehrenfest dynamics presented in the previous section A.
The electronic dynamics can be obtained by averaging over mixed quantum classical trajectories. For a rate process, the change of the occupation probability per unit time for a certain quantum state |i⟩ is given by d ⟨|a 2(t )|2 ⟩ ≈ −k′⟨|a 2(t )|2 ⟩ + k⟨|a1(t )|2 ⟩ dt
|a 2(t )|2
dt (31)
where dt is the time step in the simulation. The transition probability P1→2 can be either positive or negative, and the hop to state |2⟩ is not allowed when P1→2 is negative. Similarly, the transition probability from state |2⟩ to |1⟩ can be calculated 6203
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