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Quantum time correlation functions are often the principal objects of interest in experimental investigations of the dynamics of quantum systems. For instance, transport properties, such as diffusion and reaction rate coefficients, can be obtained by integrating these functions. The evaluation of such correlation functions entails sampling from quantum equilibrium density operators and quantum time evolution of operators. For condensed phase and complex systems, where quantum dynamics is difficult to carry out, approximations must often be made to compute these functions. We present a general scheme for the computation of correlation functions, which preserves the full quantum equilibrium structure of the system and approximates the time evolution with quantum-classical Liouville dynamics. Several aspects of the scheme are discussed, including a practical and general approach to sample the quantum equilibrium density, the properties of the quantum-classical Liouville equation in the context of correlation function computations, simulation schemes for the approximate dynamics and their interpretation and connections to other approximate quantum dynamical methods.

The dynamical properties of condensed-phase or complex systems are often investigated experimentally by applying external fields to weakly perturb a system and observe its relaxation back to the thermal equilibrium state. In such experiments, measurable quantities can be related to equilibrium time correlation functions via linear response theory [_{Q}

Exact numerical evaluation of _{b}_{s}_{c}_{s}

Several methods based on various master equations [

In this paper, we formulate MC-MD schemes to evaluate _{j}_{B}

Two main tasks are involved in evaluating _{eq}^{−βĤ}/Z_{Q}

The goals and outline of the paper are as follows: We first consider how the two ingredients, quantum equilibrium sampling and evolution of quantum operators, which are needed to compute quantum correlation functions, may be carried out. In Section 2, we describe a path-integral scheme to perform MC sampling from the partially Wigner transformed quantum density. In the

In general, analytical expressions for the Wigner transform of the density matrix cannot be determined easily. In this section, we present a path-integral-based scheme to perform MC sampling from the Wigner-transformed density,

First, we recall the definition of partial Wigner transform:
_{s}_{c}_{L}_{L}^{−βLĤ}_{W}_{i}_{W}_{i,i+1}(_{i}^{−βLVb(Ri)}^{−π𝒢(Ri−Ri+1)2}, respectively.

In this section, we discuss how one can simulate the time-evolved matrix elements,
_{W}_{W}

The QCLE has many desirable features, such as the conservation of energy, momentum and phase space volumes. Furthermore, the QCLE is equivalent to full quantum dynamics for arbitrary quantum subsystems, which are bilinearly coupled to a harmonic bath. For instance, commonly used spin boson models are of this type. In this circumstance, the combination of quantum and classical brackets in the QCLE does have a Lie algebraic structure. For the more general bath and coupling potentials, the QCLE provides an approximate description of the quantum dynamics. In this case, comparisons of simulations of QCL dynamics with exact quantum results have indicated that it is quantitatively accurate for a wide range of systems [

The QCLE equation can be simulated using ensembles of trajectories, which, in combination with the quantum initial condition sampling discussed above, provides a way to compute quantum correlation functions. As we shall see, the nature of the trajectories that enter in the simulations depends on the algorithm and should not be ascribed physical significance. It is only the observable, in this case, the correlation function, that has physical meaning and is independent of the manner in which it is simulated, provided the simulation algorithm is capable of exactly solving the QCLE, which is not always the case. One of the goals of this paper is to illustrate how a recently-developed FBTS [

In order to discuss the nature of the trajectory description involved in the TBSH algorithm, we briefly describe how it is implemented and, in particular, present the explicit generalization to an _{W}_{α}_{W}_{W}^{2}_{αα′} = (_{α}_{α′})_{αα′,ββ′} is responsible for nonadiabatic transitions and associated momentum changes in the bath. For an _{λλ′}, which introduces transitions only between the specific pair of _{αβ}_{λλ′}. We remark that it is difficult to exactly simulate the term, _{αβ}_{αβ}_{αβ}^{2}, in the above equation. Decomposing
_{‖} components by Δ_{c}

_{j}_{j}_{j−1} = Δ_{αα′} (_{1}_{2}) = ^{iωαα′(t2−t1)}, and:
^{𝒥λλ′Δt} in the following block-diagonal matrix form:
_{i}_{λλ′} Δ_{λλ}_{′} and _{λ}_{→}_{λ}_{′} are the momentum-jump operators,
_{i}_{i}_{i}, λ_{i}, λ′^{λλ′}, of a size of (^{2}, and the associated null space is spanned by basis vectors, (_{1}_{2}), where _{i}^{(′)}. We remark that one has to permute the basis vectors in order to construct these block-diagonal matrices [

At this point, we have specified all the necessary details in order to simulate the QCL dynamics in the adiabatic basis:
_{j}α_{j−1}), one can determine the next pair at the time slice,

In this algorithm, we see that the trajectories in the ensemble that are used to simulate the time evolution are non-Newtonian in character, consisting of Newtonian segments where the system evolves on adiabatic surfaces, or the mean of two adiabatic surfaces, interspersed with quantum transitions and momentum changes.

This scheme is motivated by another way of writing the formally exact solution [_{W}

One approach to solve _{λ}_{1}, …, 1_{λ}_{N}_{λ}_{λ}_{λ}_{1} . . . 0_{N}

Next, we define the mapping version of operators,
_{W}_{m}_{b}

We now introduce the coherent states, |_{λ}|z_{λ}|z_{1}_{N}_{1}_{N}_{1}_{N}^{−(|z−z′|2)−i(z·z′*−z* ·z′)}. Finally, we remark that the coherent states provide the resolution of identity:
^{2}^{N}

Similar to the path integral approach for solving the quantum dynamics, we decompose the forward and backward evolution operators in _{i}_{i}_{i}_{0}(_{b}_{pseudo}_{e}^{2}

The main approximation introduced in the derivation of the FBTS, _{i}

In the simplest approach, one selects every (_{v}_{iv}_{iv−1 with ti0} = 0 and _{iK+1 = t}. According to this prescription, the continuous FB trajectories experience

The differences between the two QCLE simulation algorithms can be traced to the quantum basis that is used and the way that feedback between quantum and classical systems is treated. In the case of the TBSH algorithm, the trajectories are propagated through a Hellmann-Feynman force, or the mean of two Hellmann-Feynman forces [

For certain problems, such as proton transfer reactions, where the time scales of the bath and subsystem are well-separated, even during nonadiabatic transitions, the TBSH algorithm can yield quantitatively accurate results with a few hops. There are also dynamical problems in which distinct bath motions can be explicitly correlated with the subsystem’s quantum states. For instance, in the simple Tully I model [

Alternatively, there are also many examples where one would expect FBTS to be the preferred simulation method. In general, the TBSH algorithm has convergence issues, as the MC weights associated with nonadiabatic hops grows rapidly. Even for the simple spin boson model, one can identify parameter regimes where this numerical instability is clearly observed. In these cases, the FBTS and JFBTS are certainly the alternatives that one should adopt for efficient simulations.

As a specific example to illustrate the formalism outlined above, we consider a two-level system coupled to a quartic bistable oscillator with a single pair of phase space coordinates _{0} = (_{0}_{0}). The quartic oscillator is, in turn, coupled to an Ohmic heat bath of _{b}_{i}_{i}, P_{i}_{b}_{b}_{j}_{c}_{c}/δω_{j}_{j}^{1/2}_{j}_{max}/ω_{c}_{b}_{c}_{max}

The adiabatic states for the subsystem are:

We shall study the autocorrelation functions, _{LL}^{2}) = (2^{2})^{−1/2} ^{−x2/2σ2}, and:
_{0}_{b}, P_{b}_{0} from _{0}), respectively. The time-evolved matrix element,
_{eq}_{W}

In this study, we report numerical results in the energy unit, _{c}_{0} = 1.2, _{0} = 0.05 _{b}_{max}_{α}_{0}). The two diabatic surfaces, _{L,R}_{0}), remain close to each other, and the two adiabatic surfaces, _{±}_{0}), share essentially the same characteristics. In this case, a mean-field-based algorithm, like FBTS, should be accurate and efficient. This problem can also be handled easily in the adiabatic basis, since the surface-hopping trajectories will be initialized in both the adiabatic ground and excited states, because the system is in a thermal equilibrium state at _{b}_{LL}

Next, we consider the following parameter set, _{0} = 0.6, _{0} = 0.3 _{b}_{max}_{α}_{0}), obtained from this set of parameters. In this case, the adiabatic, _{±}_{0}), and diabatic surfaces, _{L,R}_{0}), only differ markedly near the region of the barrier top, where an avoided crossing point indicates significant mixing of the two diabatic states. Nonadiabatic effects should be most prominent near this barrier top. A stronger coupling, _{0}, is also chosen in this case. _{LL}_{0}), has a bimodal distribution profile, as illustrated in _{0}) distribution profile (blue curve in _{0}) yields many _{0} values near the barrier top, where several hops immediately take place for this strong-coupling case, and the instability sets in early in the simulation. Lowering _{0}), but the quartic oscillator’s momentum, _{0}, will fluctuate with a larger variance in the presence of the heat bath in this case. Since nonadiabatic transitions depend non-trivially on _{0}· _{12}(_{0})Δ

The scheme for computing the quantum correlation function in

It is known that the quantum-classical bracket, defined in terms of the commutator and Poisson brackets in _{e}_{eq}^{−βĤ}/Z_{Q}_{e}

The dynamics described by the QCLE can be related to that prescribed by other methods. In [

Mixed quantum-classical methods are often the only feasible approach to explore the dynamics of large complex systems, such as condensed phase or biochemical systems, where only a few light-mass DOF need be treated quantum mechanically. In many rate processes of interest, such as electron transfer or proton transfer, the local polar solvent motions are responsible for important features of the reaction mechanism. As a result, it is essential that the dynamics of these environmental degrees of freedom be treated in detail. Open quantum system methods that trace out all bath details cannot capture important aspects of such dynamics.

Some recent work [

Finally, we provide comments that may help in choosing between the two algorithms for simulations. The TBSH algorithm, without filtering, provides a very accurate QCL dynamics before the onset of the sign problem associated with its heavy reliance on Monte Carlo sampling. While filtering can be used to extend simulations to much longer times, the problems related to Monte Carlo sampling limits its usefulness in performing long-time simulations, as vividly illustrated in Section 4. However, the TBSH is found to be the preferred simulation method (in comparison to the FBTS) when one investigates bath dynamical properties of systems in the vicinity of conical intersections and avoided crossings. For instance, the TBSH results accurately capture the intricate geometric phases [

Research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.

The authors declare no conflict of interest.

Many realistic chemical and biological processes take place at room temperature, in which case, it is often justified to apply a classical approximation to the bath. In this Appendix, we make two assumptions: As in most condensed phase models, we consider a pure subsystem observable, _{b}, X_{n}_{s}_{n}_{sn}_{b}_{bn}_{i}_{i}+ V̂_{i}_{i}_{i}_{W}_{n}_{s}_{sn}_{n}_{n}_{n}_{bn}_{n}_{b}_{bn}_{n}_{sn}_{W}_{n}_{n}_{c}_{c}

Under these assumptions, one needs to evaluate the partial Wigner transform of ^{−βĤ}^{−βĤsn}e^{−βĤbn}

We next apply a symmetric Trotter decomposition to the matrix element of

In this equation, the symmetric Trotter decomposition separates the subsystem potential in _{W}_{n}_{n}_{W}_{n}_{ho}_{n}_{ho}_{n}_{ho}

The anharmonic term in _{n}_{α} (_{n}_{W}_{n}_{α}_{n}_{α}_{n}_{ho}_{n}_{αα′} = 〈_{n}|_{Rn}_{n}

Substituting _{b}_{sn}_{b}_{sn}_{W}_{Q}

(_{α}_{0}), for the ground adiabatic state (black, dotted), excited adiabatic state (black, dotted) and for the diabatic states, L (green) and R (red). (_{LL}

(_{α}_{0}), for the ground adiabatic state (black, dotted), excited adiabatic state (black, dotted) and for the diabatic states, L (green) and R (red). The blue curve is a plot of the un-normalized distribution function, _{0}), _{LL}_{LL}