semiclassical propagation ... - Steven D. Schwartz

THE JOURNAL OF CHEMICAL PHYSICS 126, 184107 共2007兲

New mixed quantum/semiclassical propagation method Dimitri Antoniou and David Gelman Department of Biophysics, Albert Einstein College of Medicine, New York 10461

Steven D. Schwartz Department of Biophysics, Albert Einstein College of Medicine, Bronx, New York 10461 and Department of Biochemistry, Albert Einstein College of Medicine, Bronx, New York 10461

共Received 9 November 2006; accepted 26 March 2007; published online 10 May 2007兲 The authors developed a new method for calculating the quantum evolution of multidimensional systems, for cases in which the system can be assumed to consist of a quantum subsystem and a bath subsystem of heavier atoms. The method combines two ideas: starting from a simple frozen Gaussian description of the bath subsystem, then calculate quantum corrections to the propagation of the quantum subsystem. This follows from recent work by one of them, showing how one can calculate corrections to approximate evolution schemes, even when the Hamiltonian that corresponds to these approximate schemes is unknown. Then, they take the limit in which the width of the frozen Gaussians approaches zero, which makes the corrections to the evolution of the quantum subsystem depend only on classical bath coordinates. The test calculations they present use low-dimensional systems, in which comparison to exact quantum dynamics is feasible. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2731779兴 I. INTRODUCTION

Many systems of current interest, such as chemical reactions in a solution and in biomolecules, consist of a large number of degrees of freedom. These chemical reactions are often of a quantum mechanical character, for example, proton or hydrogen transfer. However, the size of these systems precludes a full quantum-mechanical treatment. The last decade has witnessed the advent and development of a variety of methods for describing the quantum dynamics of these systems.1 One class of these methods treats a few reactive degrees of freedom fully quantum mechanically, and the rest are treated in some lower level of approximation. A set of these methods can be described as “mixed quantum classical:” a few degrees of freedom are propagated with the Schrödinger equation, while the rest are propagated with Newton’s equations of motion. Since the dynamics of the quantum subsystem allows for transitions between states with certain probabilities, and since the quantum and classical subsystems are coupled, one must invent an ansatz to describe the behavior of the classical subsystem when such quantum transitions take place. The simplest such ansatz is the mean-field approximation,2–4 and its extensions and variations include “surface-hopping” methods5–13 and multiconfiguration mean field.14–16 Remarkable agreement with exact calculations has been achieved with clever choices of the way the classical system responds to the quantum transitions.17,18 In this set of methods, other recent methodologies include Bohmian dynamics.19 In the last few years there has been a renewed interest in semiclassical methods. Among them, methods that are based on linearization of semiclassical initial value representations have already been successfully applied to systems with many 0021-9606/2007/126共18兲/184107/7/$23.00

degrees of freedom.20 It has been proved recently that approximations based on linearizing the forward-backward action in the exact path integral expression for the correlation function lead to expressions that are identical to those derived by linearizing the semiclassical initial-value representation result for the correlation function.21–23 This proof established the general relationship between semiclassical methodology and centroid-based methods.24,25 These methods share the common characteristic of treating all degrees of freedom on the same footing—no specific degrees of freedom are singled out as needing special quantum treatment over others. Finally, another successful method for studying quantum multidimensional systems is the multiconfigurational time-dependent Hartree method,14,26,27 which also treats all degrees of freedom on an equal footing. Since it uses as basis a product of one-dimensional functions, it has difficulties with certain forms of the coupling potentials. The method we propose has a philosophy similar to the mixed quantum-classical approaches: we propagate a reaction coordinate with the Schrödinger equation, and the bath degrees of freedom with an approximate method. Our method consists of three steps. In the first step, we describe the dynamics of the system using an exact propagator for the quantum subsystem, and frozen Gaussians28,29 for the bath subsystem. This means that the bath subsystem will evolve quantum mechanically, with a frozen Gaussian approximation. The second step, discussed in greater detail in the following paragraph, applies a recently developed method30 that allows a systematic evaluation of corrections to any propagating method. For example, it enables improving the frozen Gaussian propagating scheme up to potentially the exact result. In order to develop a tractable numerical approach from this exact formulation, further approximations are needed. In the third step, we take the limit in which the width of the

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© 2007 American Institute of Physics

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Antoniou, Gelman, and Schwartz

frozen Gaussians approaches zero, which allows us to replace the location of the centers of the frozen Gaussians with a classical position variable. What we gain with respect to standard mixed quantum-classical methods is that the interaction between the quantum and bath subsystems is calculated at a stage when they are both described quantum mechanically. The limit of frozen Gaussians towards classical position variables is taken only after an expression for the coupled evolution is derived. This avoids the need for an ansatz that describes the reaction of the classical subsystem to transitions of the quantum subsystem, but eventually preserves the simplicity of a mixed quantum/classical approach. In addition, the method described in this paper will be shown to result in a formalism which combines aspects of the timedependent Hartree methodology and mixed quantumclassical theories. Of course, earlier investigations have also used a frozen Gaussian approximation for multidimensional systems.31–41 What makes it novel is the second step above: after starting with a frozen Gaussian approximation for the bath subsystem, we use a systematic way to calculate corrections to the approximate propagation. In the perturbation theory one writes the Hamiltonian as H0 + H⬘ and uses a systematic way to correct the evolution propagator U0 = e−iH0t. However, in the present case we have a different situation: we are given the zeroth-order propagator U0 not through a Hamiltonian H0, but through a propagation methodology, e.g., a product of classically evolving frozen Gaussians for the bath subsystem. The question is, given this ansatz for U0 and the full Hamiltonian H, can we find a systematic way to calculate corrections to U0? In a recent publication30 we derived such a procedure, and we will use it for calculating quantum corrections to U0. The structure of the paper is as follows. In the first part of Sec. II we will show how one can systematically calculate quantum corrections given a guess for the zeroth-order propagator U0. Then we will calculate these corrections for the case in which the bath part of U0 is described by sharp Gaussians. Once this correction to the propagator is calculated, three steps remain: evaluating this correction for the evolution of the bath subsystem, taking the limit in which the width of Gaussians approaches zero, and finally propagating the quantum subsystem. The details of the derivation are shown in appendixes A and B. Section II C gives a brief review of the quantum/classical time dependent selfconsistent field approximation 共Q/C TDSCF兲. In Sec. III we will apply our formalism to lowdimensional problems that admit an exact solution: a hydrogen atom in a symmetric double well coupled symmetrically and antisymmetrically to a Morse oscillator with a mass of a carbon atom. The proposed method will be compared to numerically exact quantum mechanical calculations as well as to an alternative mixed quantum/classical method 共Q/C TDSCF兲. It should be noted that a.u. are used throughout the paper 共ប = 1兲.

II. THEORY

We will study a system described by a quantummechanical Hamiltonian, H = Hs共s兲 + Hb共q兲 + f共s,q兲,

共1兲

where s is the reaction coordinate, q is a multidimensional “bath” coordinate, and f共s , q兲 is the coupling term. We are interested in the dynamics of the quantum degree of freedom s, while the degrees of freedom q are also quantum, but because they are, e.g., heavy atoms, they will eventually be treated with some classical-like approximation. We emphasize though that at this stage of the calculation q is treated quantum mechanically. Let us assume that we have a guess or approximation U0 for the exact propagator e−iHt. We can write it as a product U0 = UsUb, where Us = e−i共Hs+f兲t, but Ub is not necessarily the operator corresponding to evolution according to Hb 共i.e., Ub is not e−iHbt, which would make U0 a simple Trotter product of two evolution operators兲, but could be some ansatz for the propagation of q. If we write the exact propagator as a product of the guess U0 times an unknown correction operator e−iHt ⬅ U0e−i兰 dt⬘H⬘ , t

共2兲

then after some algebra 共as shown in Appendix A兲 we derive the exact expression for the correction propagator e−i兰dtH⬘,

再冕冉冋册

exp − i

dt U−1 b Hb − i

d −1 df tUsUb Ub + U−1 b Us dt dt

冊冎

.

共3兲 We emphasize that no approximation has been made up to this point: the exact propagator is UsUbe−i兰dt⬘H⬘. The physical interpretation of the terms in Eq. 共3兲 is as follows: 共1兲 The term in square brackets is similar to the timedependent Hartree term. 共2兲 The last term in Eq. 共3兲 is a “mixing” term, since as we will later show, it couples s eigenstates. This term can describe nonadiabatic effects. 共3兲 It is explained in Appendix A that this exact expression includes not only “backscattering” effects 共change in the time evolution of q due to its coupling to s兲, but also “correlation” effects 共change in the time evolution of s due to the backscattering correction兲. We will now evaluate the quantum correction Eq. 共3兲 for the particular case in which the dynamics of q is described by the evolution of frozen Gaussians. We assume that q are the coordinates of heavy atoms and we take Ub to be a frozen Gaussian propagator, i.e., Ub evolves the bath wave function 兩␾共q , t兲典 as a sharply peaked Gaussian, whose center is propagated according to the classical equations of motion for q. The exact expression for the wave function is UsUbe−itH⬘兩␺sq典兩␾q典.

共4兲

Equation 共4兲 merely expresses the evolution of an initial condition. Note that the s wave function ␺s depends parametri-


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Mixed quantum/semiclassical propagation method

cally on q. No approximation has been made 共as is done in time-dependent Hartree methods兲 that the wave function can be written as a separable product. We insert in Eq. 共4兲 a complete set of Gaussian bath states and obtain UsUb兩␺sq典兺兩␾q⬘,p⬘典具␾q⬘,p⬘兩e−itH⬘兩␾q典.

共5兲

q⬘,p⬘

We now begin to approximate the exact expression Eq. 共5兲, to yield an easily computable evolution methodology.

再冕冉冋

具␾q⬘兩e−i兰dtH⬘兩␾q典 ⯝ exp − i

dt

冓

A. Propagation of q

The coupled differential equation defined by Eq. 共5兲 will be solved numerically iteratively: qt will be propagated by one time step to qt+⌬t, then st will be propagated by one time step according to the updated coupling f共s , qt+⌬t兲. Since the time step ⌬t is small, we are justified in keeping a few terms in the expansions of the exponential e−itH⬘. After a cumulant expansion, and using the fact that Ub兩␾典 = 兩␾共t兲典, the matrix element of e−itH⬘ becomes

d␾q共t兲具␾q⬘共t兲兩Hb兩␾q共t兲典 − i ␾q⬘共t兲兩 dt

The term in the square brackets in Eq. 共6兲共which is just the usual time-dependent Hartree term兲 is a number, and we will group it with the s propagator Us. The last term is an operator acting on both s and q: we will have it act first on q and then on s. Up to this point, the method was general except for the specific rule for propagating q as a frozen Gaussian. To generate a concrete result we need to specify a form for the coupling f共s , q兲. To demonstrate that our method, based on the correction term Eq. 共3兲, is practical, we will first use as an example the “rate-promoting” coupling f = c共s2 − s20兲q. We have used this coupling in a series of papers that studied quantum proton transfer in condensed phases, because we and others42–44 have argued that it is the simplest realistic model for distance-modulated proton transfer. This coupling captures the fact that when the proton donor-acceptor distance oscillates, it modulates the height of the energetic barrier for the proton transfer. 1. Momentum shift for q

Using this form for the coupling f, the last term of Eq. 共6兲 has the form exp关−itc共s2 − s20兲q兴, which can be interpreted as an operator that shifts the momentum of q by ␦ p = −c共s2 − s20兲t. We should point out that if we had used a Newton’s equation description for q, the extra force on q, because of its coupling to s, would be dp / dt = −df / dq, which leads to dp = −c共s2 − s20兲dt, i.e., the same amount as the momentum shift operator. This momentum shift depends on s, so we need an approximation to proceed further. We make a mean-field approximation for the momentum shift and set it equal to −c共具s2典 − s20兲t. At each time step the effect of exp兵−itc共s2 − s20兲q其 on 兩␾ p,q典 cause the momentum p to shift by −c共具s2典 − s20兲t. This approximation for the momentum shift is similar to the mean-trajectory approximation that is often employed in mixed quantum-classical methods. We will later compare our method numerically with that approximation. Note that the usual caveat against the mean-trajectory approximation does

冔册

+ 具␾q⬘共t兲兩Us−1

df tUs兩␾q共t兲典 dt

冊冎

.

共6兲

not hold here. Its weakness, for example, is that when the s wave packet is split between reactants and products, in the bilinear case f = csq that is often studied, the mean-trajectory approximation predicts a zero force on q. However, in the present case the coupling f is quadratic on s; therefore when the s wave packet is split between reactants and products, the mean-trajectory approximation does not necessarily fail. After the momentum shift ␦ p is applied, the matrix element 具␾q⬘,p⬘ 兩 ␾q,p+␦p典 is formed. Since sharp Gaussians form an orthogonal basis, the only term that survives in 兺 p⬘ is the one with a momentum equals to the shifted momentum p + ␦ p of 兩␾q典. Then the wave function Eq. 共5兲 is written as −1

UsUb兩␾q,p+␦p典e−itCTDH−itUs

共7兲

CmixUs

where we have denoted the matrix element of the first two terms of Eq. 共6兲 by CTDH and the matrix element in the last term of Eq. 共6兲 by Cmix.

2. Propagation of q by one time step

After the preliminary steps described in the previous subsection, we can now propagate 兩␾q,p+␦p典 according to the evolution rule Ub, thus completing the time evolution of q by one time step. After we have propagated q共t兲 to q共t + ⌬t兲, we project out the q part of the wave function, −1

具␾q共t + ⌬t兲兩Use−itCTDH−itUs

CmixUs

兩␺s0典兩␾q共t + ⌬t兲典.

共8兲

Since we assumed that the frozen Gaussians 兩␾典 are sharp, this allows us to replace all operators q by the position of the center of the frozen Gaussian ␾q共t + ⌬t兲. In what follows, q will be a classical variable, equal to the center of the frozen Gaussians. Before we proceed, we should point out that we have achieved what we set out to do: instead of separating the system into quantum and classical variables, and then trying to compensate by inventing a clever rule for the interaction between the two kinds of coordinates, we treated all coordinates as quantum, and only after we evaluated the effect of


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the dynamics q on the evolution of s, we took the limit where q is a classical-like variable. There was no need for an ansatz for the interaction between s and q. B. Propagation of s

We will complete the calculation of the evolution of the wave packet by evaluating the effect of the correction Eq. 共3兲 on the evolution of s. Having traced out q in Eq. 共8兲, the s wave function, including the correction Eq. 共3兲, is written as −1

兩␺sq共t兲典 = Use−itCTDH−itUs

CmixUs

兩␺s0典.

共9兲

Since the time step t is small, we can do a Taylor expansion of the exponential in Eq. 共9兲, and obtain, as shown in Appendix B, 兩␺sq共t兲典 =

兺 e−iE ⬘t关e−i共C q n

TDH␦n,n⬘+具nq兩Cmix兩nq⬘典兲t

n,n⬘

⫻具nq⬘兩␺s0兩nq典.

III. RESULTS

− 1 + ␦n,n⬘兴共10兲

The set of wave functions 兩nq典 and the associated are the adiabatic eigenfunctions and eigenvalues 共respectively兲 of the Hamiltonian Hs + f共s , q兲, defined for each configuration of q. Equation 共10兲 is our final result for the propagation of the quantum degree of freedom s. If we set Cmix = 0 we recover the time-dependent Hartree approximation. The term Cmix takes into account mixing between different s eigenstates induced by the interaction with the classical-like degrees of freedom q. Eqn

C. Q/C TDSCF method

We will compare our method to one of the mixed quantum/classical approximations—Q/C TDSCF.2,4 The latter is applied for the systems, in which one degree of freedom is treated quantum mechanically, while the rest are treated classically. In the following we briefly outline the main points of this method. For the given system 关Eq. 共1兲兴 the dynamics are determined by mixed quantum/classical TDSCF equations:

⳵ i 兩␺s共t兲典 = 关Hs共s兲 + f共s,qt兲兴兩␺s共t兲典, ⳵t

共11兲

We have applied the proposed method to a model system consisting of a double-well potential coupled either to a Morse or a harmonic oscillator. Two forms of coupling— linear and quadratic—have been considered. This kind of system has been used frequently to model proton transfer processes. A. Quadratic coupling

The s subsystem representing the reaction coordinate is described by a symmetric double-well potential, 1 1 V共s兲 = − as2 + bs4 , 2 4

共12兲 q˙ = pq/M . The dynamics of the quantum coordinate s are described by the Schrödinger equation 关Eq. 共11兲, which includes the coupling term f共s , qt兲 with q fixed at its classical value 共the TDSCF potential兲兴. The classical equations of motion 关Eq. 共12兲兴 contain the additional TDCSF force 共HellmannFeynman兲 defined as 共13兲

The additional force applied on the classical degree of freedom and the TDSCF potential f共s , qt兲 describe the effec-

共14兲

with parameters chosen such that the barrier height is 6 kcal/ mol and the distance between minima is equal to 2 bohrs. Both values are typical for proton/hydrogen transfer reactions. We coupled the proton to one q degree of freedom that has a mass equal to the mass of a carbon atom. The potential for the q degree of freedom was a Morse potential with a basic frequency of 440 cm−1. First, the coupling term was taken to be of a “ratepromoting” form, f = c共s2 − s20兲q, where s0 is the location of the double-well minimum. The calculations were performed for several values of the coupling constant c. We have calculated the decay time of the survival probability, P共t兲 = 円具␺s0兩␺sq共t兲典円2 .

⳵Hb + Fq共t兲, p˙q共t兲 = − ⳵q

Fq共t兲 = − 具共␺s共t兲兲兩ⵜq f共s,q兲兩共␺s共t兲兲典.

tive mean-field interaction between the quantum and classical subsystems. The equations are solved by assuming constant TDSCF potentials within small time step ⌬t, during which the modes are propagated independently. After updating the TDSCF values Fq and f共s , qt兲, the equations are then solved for the next time step. The momentum shift applied on q in our method 共Sec. II A 1兲 is completely equivalent to the additional TDCSF force described above 关Eq. 共13兲兴. The main difference, however, is in the evolution of the quantum subsystem. The corrected propagator 关Eq. 共9兲兴 includes correlations between the quantum and classical subsystems 共given by Cmix term兲, which are neglected in a simple mean-field approximation such as Q/C TDSCF.45

共15兲

The initial wave packet ␺s0 was localized at the reactant side of the double well 共it was taken to be a superposition of the two lowest eigenstates of the symmetric double-well potential兲. The q Gaussian was initially centered about q0 = 0.1 bohr, corresponding to energy, equals kBT / 2 at room temperature. The results obtained by the proposed method are compared to numerically exact quantum-mechanical calculations as well as to an alternative mixed quantum/classical method 共Q/C TDSCF兲.2,4 In the Q/C TDSCF one classical trajectory was propagated together with one wave packet, with the same initial conditions as above. The initial state for the full quantum calculations was chosen to be a product of two one-dimensional Gaussians. For the wave packet propagation


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FIG. 1. Tunneling splitting between the two lowest levels of the double-well potential as a function of the square of the coupling constant. The results obtained by the method proposed in the paper are given by full and empty circles 共the mean-field approximation and the full correction, respectively兲. The squares refer to the numerically exact quantum-mechanical results and the triangles are results of the application of the Q/C TDSCF method. The barrier height of the double-well potential is 6 kcal/ mode and the distance between the minima is equal to 2 bohrs. The frequency of the Morse potential is 440 cm−1.

共used in both the exact and the Q/C TDSCF schemes兲 the kinetic energy operator was applied in Fourier space employing fast Fourier transform,46 and the Chebychev method47,48 was used to compute the evolution operator. Although in the Q/C TDSCF method the Hamiltonian is time dependent, the Chebychev polynomial expansion can be used with small time steps. Figure 1 shows the tunnel splitting in the ground state as a function of the square of the coupling constant. Tunnel splitting between the two lowest levels of the double-well potential is defined as an inverse time at which the survival probability decays to zero ⌬E = ␲ / ␶. We compared the results of four different calculations: 共a兲 the numerically exact quantum mechanical, 共b兲 the Q/C TDSCF, 共c兲 the results obtained by the proposed method, and 共d兲 a calculation that used Us for propagating s and a meantrajectory approximation 共MTA兲 for propagating q. The results obtained by the proposed method are in good agreement with the numerically exact results 共except for the strongest coupling兲 and far more accurate then either Q/C TDSCF and MTA, which show much larger errors for the weak couplings and fail completely for the stronger couplings. Figure 2 shows the survival probability for the same system, while the frequency of the q degree of freedom was chosen to be 900 cm−1 and the coupling constant equals 0.015 hartree bohr−2. Since the frequency of oscillation of q is fast, there is no separation of time scales of motion between s and q. This makes the calculation more challenging since one cannot assume that the motion of s takes place on an adiabatic surface of q. The results obtained by the proposed method are compared to the exact numerical calculations. While the simple mean-field approximation performs only at short times, the full correction result is accurate as the exact one. B. Linear coupling

The second problem we examined is a proton in a symmetric double well with a barrier height of 4 kcal/ mol and distance between minima equals to 2 bohrs. As before, we

J. Chem. Phys. 126, 184107 共2007兲

FIG. 2. Survival probability as a function of time. The calculations are performed for the double-well potential coupled to a Morse potential with frequency equal to 900 cm−1. The coupling strength c is equal to 0.015 hartree bohr−2. Thin solid line: the numerically exact quantum calculations. The dashed and thick solid line lines refer to the mean-field approximation and the full correction, respectively.

coupled the proton to one q degree of freedom that has a mass equal to the mass of a carbon atom, in a harmonic potential of frequency of 447 cm−1. The coupling between s and q was of the bilinear form csq with c = 0.01 hartree bohr−1. As it is well known, a mean-field approximation for the bilinear coupling, i.e., replacing s by 具s典, fails. The reason is that when the s wave packet starts to tunnel and is split between reactant and product wells, the average 具s典 approaches zero, while the effect of the true coupling csq is clearly nonzero. To proceed we need a propagation rule for q that is better than the mean-field approximation. A first attempt to correct the failure of the mean-field approximation for bilinear coupling is to approximate s in the coupling by sgn共s兲冑具s2典. This form does not go to zero when part of the s wave packet has tunneled and is split on either side of the barrier. The extra term sgn共s兲 gives it the correct sign depending on where the majority of the wave packet is, with respect to the barrier. Figure 3 shows the survival probability calculated by our method and by the simple mixed quantum-classical method. The exact decay time for the survival probability is 2828 fs.

FIG. 3. Survival probabilities for our method 共thin solid line兲, and for the simple mixed quantum-classical method with a mean-trajectory approximation共dashed line兲 for the bilinear coupling case that is discussed in the text. Thick solid line: the numerically exact quantum calculations. The barrier height of the double-well potential is chosen to be 4 kcal/ mole and the distance between the minima is 2 bohrs. The harmonic oscillator has a frequency of 447 cm−1 and the mass of a carbon atom. The coupling strength c is equal to 0.01 hartree bohr−1. In the simple mixed quantum-classical method the wave packet is reflected at the barrier instead of tunneling.


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While our method produces results in reasonably good agreement with the numerically exact calculations, the meanfield approximation fails completely. We have also compared the evolution of wave packets using the mean-field approximation for two cases: s was approximated by the mean-field result 具s典, and by sgn共s兲冑具s2典. In the former case, the wave packet remained in the reactant well for all times. In the latter case 共see MTA results in Fig. 3兲 a part of the wave packet was tunneled, but then it remained mostly in the product well. In this case the extra term sgn共s兲 corrects the unphysical prediction of the mean-field approximation that the coupling approaches zero as the wave packet starts to tunnel. Our method of calculating corrections only needs an approximate zeroth-order propagation rule; if we can guess an appropriate rule, the corrections may capture the essential physics, as this example shows.

ACKNOWLEDGMENTS

The authors acknowledge support of the NIH through Grant No. GM41916 and GM068036.

APPENDIX A: DERIVATION OF EQUATION „3…

The correction propagator H⬘ was defined by e−iHt ⬅ U0e−i兰 dt⬘H⬘ . t

共A1兲

After taking the time derivative of both sides of Eq. 共A1兲, we obtain −1 H⬘ = U−1 0 HU0 − iU0

dU0 . dt

共A2兲

We assume that U0 has a product form UsUb. Then the first term in the right-hand side of Eq. 共A2兲 is given by −1 U−1 b Us 共Hs + f + Hb兲UsUb

IV. CONCLUSIONS

In this paper we proposed a method for studying the quantum dynamics of multidimensional systems. The method consists of two main ideas: First, we used sharp Gaussians for the description of the bath subsystem, in order to avoid the difficulties that mixed quantum-classical methods have in treating the effect of the quantum on the classical subsystem. For actual dynamics we took the limit in which the width of the Gaussians approaches zero. Secondly, we calculated a correction, Eq. 共6兲, to the zeroth-order propagator, which accounts for improvements in the description of the coupled dynamics of the quantum and classical subsystems. The rest of the paper was a proof of concept. We showed that Eq. 共6兲 can be calculated for model problems 共including two different kinds of coupling—bilinear and quadratic兲. The corrections to the evolution of the bath subsystem were reduced to simple momentum shifts at each time step; the evolution of the quantum subsystem was described by Eq. 共10兲, which corrects the time-dependent Hartree result with a state-mixing term. The correction led to a good agreement between the proposed method and the exact results, even for relatively strong couplings. The method proved equally effective for linear coupling, while Q/C TDSCF and simple mean-trajectory approximations utterly fail. The advantage of the proposed methodology is that it starts from exact quantum mechanics. The coupling between the quantum and classical subsystems arises from the quantum correction operator. Then these corrections were taken to a classical limit. Although we used sharp frozen Gaussians, the other approximations are possible. We are currently evaluating methods for going beyond the mean-field approximation for the q evolution, and beyond the first-order cumulant approximation. The latter makes the method applicable only at relatively short times. There is a clear need for a more efficient numerical algorithm to expand the application of the method to truly multidimensional systems.

−1 −1 −1 = U−1 b Us 共Hs + f兲UsUb + Ub Us HbUsUb ,

共A3兲

and, after using dUs / dt = −i共Hs + f兲Us, the second term can be written as −1 − i共U−1 b Us 兲

冉

dUb dUs Ub + Us dt dt

冊

−1 −1 = − U−1 b Us 共Hs + f兲UsUb − iUb

dUb . dt

共A4兲

When we sum Eqs. 共A3兲 and 共A4兲 the first terms cancel and we are left with H⬘ = − iU−1 b

dUb −1 + U−1 b U s H bU sU b . dt

共A5兲

The second term in Eq. 共A5兲 is written as −1 −1 −1 U−1 b Us HbUsUb = Ub Us 共UsHb + 关Hb,Us兴兲Ub .

共A6兲

The first term inside the parentheses in Eq. 共A6兲, along with the first term Eq. 共A5兲 gives

冉

U−1 b Hb − i

冊

d Ub . dt

共A7兲

Equation 共A7兲 is the first part of the correction. The re−1 maining part is U−1 b Us 关Hb , Us兴Ub. The commutator can be written as a Heisenberg derivative 关Hb,Us兴 = 关Hb,e−i共Hs+f兲t兴 = i共− i兲

¯兲 df共s,q tUs , dt

共A8兲

where the bar in ¯q indicates that the propagation is taken according to Hb, i.e., the uncoupled q Hamiltonian. Equation 共A8兲 can be written as Us times ¯ ⳵ f dq t. ⳵q dt

共A9兲

More correctly, since the term Eq. 共A9兲 will eventually appear in an integral e−i兰dt⬘. If we integrate by parts 兰t0dt⬘ the term in Eq. 共A8兲 we obtain


184107-7

J. Chem. Phys. 126, 184107 共2007兲


¯ t兲t − f共s,q

冕

0

¯ t⬘兲. dt⬘ f共s,q

共A10兲

This form 关Eq. 共A10兲兴 is similar to a result we obtained by a different method in Ref. 49, where we showed that the correction to the Trotter propagator e−i共Hs+f兲te−iHbt is equal to e−ifte+i兰dt⬘ f . An intuitive interpretation for this form of the correction is to consider it as a “correlation” term: s evolves according to the coupling f 共“direct” term兲, but this causes a correction 共“backscattering”兲 to the evolution of q, and we have to take into account the effect 共“correlation”兲 on s of this backscattering correction to q. Collecting terms, the final result for the correction propagator e−i兰dtH⬘ is

再冕冉冋册

exp − i

dt U−1 b Hb − i

d −1 df tUsUb Ub + U−1 b Us dt dt

冊冎

.

共A11兲 APPENDIX B: DERIVATION OF EQUATION „10…

In Eq. 共9兲 we found that the s wave function can be written as −1

兩␺sq共t兲典 = Use−itCTDH−itUs

CmixUs

兩␺s0典.

共B1兲

Because the time step t in Eq. 共B1兲 is small, we can make a Taylor expansion of the exponential and obtain Us共1 − itCTDH − itUs−1CmixUs兲兩␺s0典.

共B2兲

We insert complete sets of s states 兩nq典, which depend parametrically on the current value of q, because the potential V共s兲 + c共s2 − s20兲q depends on q. The first term is diagonal in s states and can be calculated as Us共1 − itCTDH兲兺兩nq典具nq兩␺s0典 n

兺 ␦n,n⬘共1 − itCTDH兲e−iE ⬘t具n兩␺s0典兩nq典. q n

=

共B3兲

n,n⬘

The CTDH term is calculated explicitly 关the first two terms in Eq. 共6兲兴 by taking the limit in which the width of the Gaussians approaches zero. The second part is nondiagonal in s states and can be calculated as UsUs−1共− itCmix兲Us兩␺s0典 =

兺兩nq典具nq兩共− itCmix兲e−iE ⬘t兩nq⬘典具nq⬘兩␺s0典. q n

共B4兲

n,n⬘

Adding Eqs. 共B3兲 and 共B4兲 and resumming the corrections back into an exponential, we obtain Eq. 共10兲 of the text 兩␺sq共t兲典 =

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t

兺 e−iE ⬘t关e−i共C q n

n,n⬘

⫻具nq⬘兩␺s0典兩nq典.

TDH␦n,n⬘+具nq兩Cmix兩nq⬘典兲t

− 1 + ␦n,n⬘兴共B5兲

2
