PREFACTOR-FREE FORWARD-BACKWARD SEMICLASSICAL DYNAMICS

An efficient time-dependent (TD) Monte Carlo (MC) importance sampling method has recently been developed [G. Tao and W. H. Miller, J. Chem. Phys. 135, 024104 (2011)] for the evaluation of time correlation functions using the semiclassical (SC) initial value representation (IVR) methodology. In this TD-SC-IVR method, the MC sampling uses information from both time-evolved phase points as well as their initial values, and only the "important" trajectories are sampled frequently. Even though the TD-SC-IVR was shown in some benchmark examples to be much more efficient than the traditional time-independent sampling method (which uses only initial conditions), the calculation of the SC prefactor-which is computationally expensive, especially for large systems-is still required for accepted trajectories. In the present work, we present an approximate implementation of the TD-SC-IVR method that is completely prefactor-free; it gives the time correlation function as a classical-like magnitude function multiplied by a phase function. Application of this approach to flux-flux correlation functions (which yield reaction rate constants) for the benchmark H + H(2) system shows very good agreement with exact quantum results. Limitations of the approximate approach are also discussed.

The Herman–Kluk (HK) semiclassical propagator is applied to two-dimensional (2D) and four-dimensional (4D) activated reactive scattering of H2 from a Cu(100) surface. Flux analysis is used to calculate collision-energy-dependent reaction probabilities from the time dependence of the semiclassical wave function; several different variations of this analysis method are implemented. Direct comparison is made between the reaction probabilities calculated using the HK method, and numerically exact quantum results, as well as quasiclassical (QC) trajectory calculations. Although aspects of results obtained using the HK propagator are an improvement over the QC results—for example, reproduction of the quantum reaction threshold and quantum resonances—overall the HK reaction probabilities are no more accurate than the QC probabilities. Furthermore, the HK method is relatively expensive to apply, and the computational expense does not scale with dimension as favorably as might be expected. The reasons for this are discussed.

The semiclassical (SC) initial value representation (IVR) for the time evolution operator e−iĤt/ℏ involves a phase space integral over the initial conditions of classical trajectories. It is shown in this paper how an IVR for the two time evolution operators in a typical quantum mechanical time correlation function, CAB(t)≡tr[ÂeiĤt/ℏB̂e−iĤt/ℏ], can be combined into one such phase space integral; i.e., time evolution from 0 to t and from t to 0 is combined into one overall SC-IVR propagation. This not only reduces the dimensionality of the phase space average, but the forward–backward (FB) nature of the net trajectory has a partial self-cancellation that reduces the oscillatory behavior of the integrand. Several applications of this FB-IVR to reactive flux correlation functions are presented to illustrate its possibilities.

We numerically compare the semiclassical "frozen Gaussian" Herman-Kluk propagator [Chem. Phys. 91, 27 (1984)] and the "thawed Gaussian" propagator put forward recently by Baranger et al. [J. Phys. A 34, 7227 (2001)] by studying the quantum dynamics in some nonlinear one-dimensional potentials. The reasons for the lack of long-time accuracy and norm conservation in the latter method are uncovered. We amend the thawed Gaussian propagator with a global harmonic approximation for the stability of the trajectories and demonstrate that this revised propagator is a true alternative to the Herman-Kluk propagator with similar accuracy.

The Herman---Kluk propagator is a well-known semi-classical approximation of the unitary evolution operator in quantum molecular dynamics. In this paper we formulate the Herman---Kluk propagator as a phase space integral and discretise it by Monte Carlo and quasi-Monte Carlo quadrature. Then, we investigate the accuracy of a symplectic time discretisation by combining backward error analysis with Fourier integral operator calculus. Numerical experiments for two- and six-dimensional model systems support our theoretical results.

The study of quantum stochastic processes presents severe difficulties, both on the theory level as well as on technical grounds. The numerically exact solution remains prohibitive even today. In this paper we review and present new results for three different methods used for the modelling of quantum stochastic processes. These include a mixed quantum classical approach, semiclassical initial value representations of the quantum propagator and the reduced density matrix approach as typified by the quantum Wigner-Fokker-Planck equation. A new semiclassical initial value representation that does away with cumbersome prefactors which depend on the monodromy matrix elements but is exact for a harmonic oscillator is presented and its properties analysed. A recently proposed systematic method for improving semiclassical initial value representations is reviewed. The generalization of the Wigner-Fokker-Planck equation to stochastic processes with memory is obtained by using a novel integral equation representation.

An efficient method, the self-consistent hybrid method, is proposed for accurately simulating time-dependent quantum dynamics in complex systems. The method is based on an iterative convergence procedure for a dynamical hybrid approach. In this approach, the overall system is first partitioned into a “core” and a “reservoir” (an initial guess). The former is treated via an accurate quantum mechanical method, namely, the time-dependent multiconfiguration self-consistent field or multiconfiguration time-dependent Hartree approach, and the latter is treated via a more approximate method, e.g., classical mechanics, semiclassical initial value representations, quantum perturbation theories, etc. Next, the number of “core” degrees of freedom, as well as other variational parameters, is systematically increased to achieve numerical convergence for the overall quantum dynamics. The method is applied to two examples of quantum dissipative dynamics in the condensed phase: the spin-boson problem and the electronic resonance decay in the presence of a vibrational bath. It is demonstrated that the method provides a practical way of obtaining accurate quantum dynamical results for complex systems.

A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

A semiclassical initial value representation approach for molecular systems in Cartesian coordinates is combined with a recently proposed time averaging technique [J. Chem. Phys. 118, 7174 (2003)]. It is shown that a single trajectory can yield the zero-point energy of the water dimer with good accuracy for the model chosen when compared to fully constrained Cartesian semiclassical calculations. The convergence with respect to the number of averaging time origins is discussed.

Semiclassical approximation of real-time quantum effects is analyzed with the aid of the semiclassical initial value representation (SC-IVR) and Wigner distribution functions. We utilize these two ingredients to propose a new version of the semiclassical correlation function that contains, in principle, all quantum-mechanical effects. The advantage of this formulation is that it allows for a stepwise approximation specifically for real-time quantum effects based on a gradual inclusion of more degrees of freedom into the integral responsible for interference. From numerical calculations, this procedure does not seem to depend significantly on the choice coordinates if all degrees of freedom are coupled. This freedom from the coordinate choice removes possible ambiguities in applying this method. Several example cases are presented to demonstrate the usefulness of this approach.

Internal conversion is an inherently quantum mechanical process. To date, "ab initio" computation of internal conversion rates was limited to harmonic based approximations. These are questionable since the typical transition to the ground electronic state occurs at energies which are far from the harmonic limit. It is thus of interest to study the applicability of the Semiclassical Initial Value Representation (SCIVR) approach which is in principle amenable to "on the fly" studies even with "many" degrees of freedom. In this work we apply the Herman-Kluk-SCIVR methodology to compute the internal conversion rates for formaldehyde for a variety of initial vibronic states. The SCIVR computation gives reasonable agreement with experiment, while the harmonic approximation typically gives rates that are too high.

In this paper we develop a criterion of the applicability for the trajectory‐based semiclassical propagation methods that are exact for Gaussian wavepackets in locally quadratic potentials. The error estimate takes into account a deviation of the wavefunction from the Gaussian form due to the anharmonicity of a potential. The derivation is based on the time‐dependent noncoherent solutions of the Schrödinger equation for a general quadratic potential that form a complete orthonormal time‐dependent basis. The anharmonicity is treated as a time‐dependent cubic perturbation, which is further related to the nonlinearity of a force acting on trajectories. The criterion is expressed in terms of scalar products of the average force, its first moments, and dispersion of the wavefunction and is efficient in the context of dynamics with the Bohmian trajectories. We give an application for the Bohmian dynamics with the approximate quantum potential, for the Gaussian wavepacket dynamics and for the semiclassical initial value representation propagator. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Imaginary-time evolution (ITE) on a quantum computer is a promising formalism for obtaining the ground state of a quantum system. The probabilistic ITE (PITE) exploits measurements to implement nonunitary operations, and it can avoid the restriction of dynamics to a low-dimensional subspace imposed by variational parameters unlike other types of ITE. In this paper, we propose a PITE approach that uses only one ancillary qubit. Unlike the existing PITE approaches, the one proposed here constructs, under a practical approximation, the circuit from forward and backward real-time evolution (RTE) gates as black boxes for the original Hamiltonian. Thus all efficient unitary algorithms for RTE can be transferred to the ITE without any modifications. Our approach can be used to obtain the Gibbs state at a finite temperature and partition function. We validate the approach via several illustrative systems where the trial states are found to converge rapidly to the ground states. In addition, we discuss its applicability to quantum chemistry by focusing on the scaling of computational cost; this leads to the development of a framework referred to as a first-quantized eigensolver. The nonvariational generic approach will expand the scope of practical quantum computation for versatile objectives.

The main contribution of this paper is to present the systematic and low-complexity translation techniques between a class of hybrid systems referred to as automaton-based DHA and piecewise affine (PWA) systems. As an starting point the general modeling framework of the automaton-based DHA is represented which models the controlled and uncontrolled switching phenomena between linear continuous dynamics including discrete and continuous states, inputs and outputs. The basic theoretical definitions on the state trajectories of the proposed DHA with forward and backward evolutions which yield forward and backward piecewise affine (FPWA and BPWA) systems are given. Next, the well-posedness and equivalency properties are proposed and the sufficient conditions under which the well-posedness property is achieved with the automaton-based DHA and PWA systems are given. It is shown that the graphical structure of the proposed automaton-based DHA makes it possible to obtain analytically the equivalent PWA system with a polynomial complexity in contrast to the existing numerical translation techniques via decomposed structure of the DHA with an exponential complexity. Examples are presented to confirm the effectiveness of the proposed translation techniques.

A generic quantum Markov semigroup [Formula: see text] of a d-level quantum open system with a faithful normal invariant state ρ admits a dual semigroup [Formula: see text] with respect to the scalar product induced by ρ. We show that the difference of the generators [Formula: see text] can be written as the sum of a derivation 2i[H, ⋅] and a weighted difference of automorphisms [Formula: see text] where [Formula: see text] is a family of cycles on the d levels of the system, wc are positive weights and [Formula: see text] are unitaries. This formula allows us to represent the deviation from equilibrium (in a "small" time interval) as the superposition of cycles of the system where the difference between the forward and backward evolution is written as the difference of a reversible evolution and its time reversal. Moreover, it generalises cycle decomposition of Markov jump processes. We also find a similar formula with partial isometries instead of unitaries.