Semiclassical Initial Value Representation Research Articles

Interference nature of quantum breather oscillation

In a system of coupled nonlinear oscillators, the breather (or local mode) solution is studied fully quantum mechanically, as well as by a semiclassical initial value representation of the propagator and classical Wigner dynamics. We show that the initial breather state is a superposition of almost degenerate eigenstates. From this simple observation it follows that the breather must decay and revive (i.e., oscillate with energy localization for extended times). Numerical results are shown for a two degree of freedom system. The fact that the semiclassical real-time result reproduces the full quantum one to a large degree, whereas the classical Wigner dynamics based on a similar set of trajectories does not, indicates that the breather oscillation can be viewed as an interference phenomenon.

Efficient importance sampling in semiclassical initial value representation calculations for time correlation functions

The semiclassical (SC) theory based on an initial value representation (IVR) methodology provides a practical way to describe quantum effects in complex molecular systems. The efficiency of SC–IVR calculations for time correlation functions depends heavily on how to perform the Monte Carlo sampling of initial conditions. Here, we compare a variety of possibilities of sampling initial conditions in the SC calculations by choosing the sampling function to be either time-dependent or time-independent (TI). The implementation of these importance sampling protocols to two benchmark system-bath models demonstrates its advantages over the standard sampling method. In particular, the recently developed TI importance sampling which incorporates path correlation in the bath degrees of freedom shows a great potential in describing many-body quantum dynamics efficiently and accurately.

Semiclassical initial-value representation of the transfer operator

The ability of semiclassical initial-value representation (IVR) methods to determine approximate energy levels for bound systems is limited due to problems associated with long classical trajectories. These difficulties become especially severe for large or classically chaotic systems. This work attempts to overcome such problems by developing an IVR expression that is classically equivalent to Bogomolny's formula for the transfer matrix [E. B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)] and can be used to determine semiclassical energy levels. The method is adapted to levels associated with states of desired symmetries and applied to two two-dimensional quartic oscillator systems, one integrable and one mostly chaotic. For both cases, the technique is found to resolve all energy levels in the ranges investigated. The IVR method does not require a search for special trajectories obeying boundary conditions on the Poincaré surface of section and leads to more rapid convergence of Monte Carlo phase space integrations than a previously developed IVR technique. It is found that semiclassical energies can be extracted from the eigenvalues of transfer matrices of dimension close to the theoretical minimum determined by Bogomolny's theory. The results support the assertion that the present IVR theory provides a different semiclassical approximation to the transfer matrix than that of Bogomolny for ℏ≠0. For the chaotic system investigated the IVR energies are found to be generally more accurate than those predicted by Bogomolny's theory.

On-the-fly semiclassical study of internal conversion rates of formaldehyde

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.

Generalized dephasing relation for fidelity and application as an efficient propagator

The dephasing relation (DR), a linearization of semiclassical fidelity, is generalized to include the overlap of "off-diagonal" elements. The accuracy of the formulation is tested in integrable and chaotic systems and its scaling with dimensionality is studied in a Caldeira-Leggett model with many degrees of freedom. It is shown that the DR is often in very good agreement with numerically analytic quantum results and frequently outperforms an alternative semiclassical treatment. Most importantly, since there is no computationally expensive prefactor, and Monte Carlo Metropolis sampling is used to facilitate the calculation, the DR is found to scale remarkably well with increasing dimension. We further demonstrate that a propagator based on the DR can include more quantum coherence and outperform other popular linearized semiclassical methods, such as forward-backward semiclassical dynamics (FBSD) and the linearized semiclassical initial value representation (LSC-IVR).

Quantum decoherence of I2 in liquid xenon: A classical Wigner approach

Vibrational decoherence of a "breathing sphere" oscillator in a thermal Lennard-Jones bath is examined using a classical analog approach. The equivalence between this approach and the linearized semiclassical initial value representation (IVR) is established and the method is exploited to produce a useful computational strategy that can efficiently evaluate the time dependence of the decoherence in these systems. A comparison between Harmonic and Morse "breathing sphere" models is presented and the rate of decoherence is found to depend on the choice of model, the initial state of the oscillator, the initial conditions of the bath (temperature, density), and the choice of quantity measuring the decoherence rate. The results are used to examine the utility of the Caldeira-Leggett model in this realistic system.

Time-dependent importance sampling in semi-classical initial value representation calculations for time correlation functions. III. A state-resolved implementation to electronically non-adiabatic dynamics

The recently developed time-dependent (TD) Monte Carlo (MC) importance sampling method [Tao and Miller; JCP 135, 024104 (2011)] provides an efficient implementation of the semi-classical (SC) initial value representation (IVR) methodology for the evaluation of time correlation functions. The key idea in this TD-SC-IVR method is to perform importance sampling of trajectories for the MC averages in the SC calculations with a sampling function that includes information about the final (time-evolved) values of the coordinates and momenta of trajectories in addition to (the usual) information of their initial values. This paper shows how this approach deals with electronically non-adiabatic dynamics, i.e. dynamics that involves coupled multiple electronic states. We suggest that a state-resolved sampling function may facilitate the efficient implementation of TD-SC-IVR for such processes. This is illustrated by application to the calculation of the nuclear momentum distribution function (i.e. the nuclear energy-loss spectrum) for a benchmark non-adiabatic scattering problem.

Communication: Importance sampling including path correlation in semiclassical initial value representation calculations for time correlation functions

Full semiclassical (SC) initial value representation (IVR) for time correlation functions involves a double phase space average over a set of two phase points, each of which evolves along a classical path. Conventionally, the two initial phase points are sampled independently for all degrees of freedom (DOF) in the Monte Carlo procedure. Here, we present an efficient importance sampling scheme by including the path correlation between the two initial phase points for the bath DOF, which greatly improves the performance of the SC-IVR calculations for large molecular systems. Satisfactory convergence in the study of quantum coherence in vibrational relaxation has been achieved for a benchmark system-bath model with up to 21 DOF.

Accelerated direct semiclassical molecular dynamics using a compact finite difference Hessian scheme

This paper shows how a compact finite difference Hessian approximation scheme can be proficiently implemented into semiclassical initial value representation molecular dynamics. Effects of the approximation on the monodromy matrix calculation are tested by propagating initial sampling distributions to determine power spectra for analytic potential energy surfaces and for "on the fly" carbon dioxide direct dynamics. With the approximation scheme the computational cost is significantly reduced, making ab initio direct semiclassical dynamics computationally more feasible and, at the same time, properly reproducing important quantum effects inherent in the monodromy matrix and the pre-exponential factor of the semiclassical propagator.

A Cartesian classical second-quantized many-electron Hamiltonian, for use with the semiclassical initial value representation

A new classical model for the general second-quantized many-electron Hamiltonian in Cartesian coordinates and momenta is presented; this makes semiclassical (SC) calculations using an initial value representation (IVR) more useful than the classical Hamiltonian in action-angle variables given earlier by Miller and White [J. Chem. Phys. 84, 5059-5066 (1986)]. If only 1-electron terms are included in this Hamiltonian, the classical equations of motion for the Cartesian variables are linear, and the SC-IVR gives exact results for the propagator (and thus for transition probabilities, the energy spectrum, etc.), as confirmed by analytic proof and numerical calculations. Though this new Hamiltonian is not exact when 2-electron interactions are included, we observe good results for the SC-IVR transition probabilities for times that are not too long. Test calculations, for example, show that the SC-IVR is accurate for times long enough to obtain good result for the eigenvalue spectrum (i.e., the energy levels of the electronic system).

Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions. II. A simplified implementation

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.

Representation of the Boltzmann operator via the chain of imaginary time coherent states and its application to calculation of real time correlation functions

Coherent states in imaginary time are used to represent the Boltzmann operator in terms of a classical chain of N coupled beads. The resulting representation is similar to the path integral molecular dynamics (PIMD) approach, except for additional nearest-neighbor coupling terms in the effective Hamiltonian. Each time step of the molecular dynamics evolution of the effective classical system provides appropriately thermalized Gaussians that can be used as distribution functions to sample initial conditions for real-time semiclassical initial value representation (IVR) methods. The approach is illustrated with numerical calculations of thermal expectation values and canonical time correlation functions for several one-dimensional quantum systems.

Accelerating the calculation of time-resolved electronic spectra with the cellular dephasing representation

Dephasing representation of fidelity, also known as the phase averaging method, can be considered as a special case of Miller's linearized semiclassical initial value representation and belongs among the most efficient approximate semiclassical approaches for the calculation of ultrafast time-resolved electronic spectra. Recently it has been shown that the number of trajectories required for convergence of this method is independent of the system's dimensionality. Here we propose a further accelerated version of the dephasing representation in the spirit of Heller's cellular dynamics. The basic idea of the ‘cellular dephasing representation’ is to decompose the Wigner transform of the initial state into a phase space Gaussian basis and then evaluate the contribution of each Gaussian to the relevant correlation function approximately analytically, using numerically acquired information only along the trajectory of the Gaussian's centre. The approximate nature of the DR classifies it among semiclassical perturbation approximations proposed by Miller and Smith, and suggests its limited accuracy. Yet, the proposed method turns out to be sufficiently accurate whenever the interaction with the environment diminishes the importance of recurrences in the correlation functions of interest. Numerical tests on a collinear NCO molecule indicate that even results based on a single classical trajectory are in a remarkable agreement with the fully converged DR requiring approximately 104 trajectories.

Semiclassical initial value representation with complex trajectories

Semiclassical time-dependent solutions to the Schroedinger equation can be obtained by use of integral formulas known as initial value representations (IVRs). Over the last few decades IVR formulas have established a sucessful reputation, most notably in the field of physical chemistry, for it is often found that the quantum behavior of molecular systems is very well approximated by semiclassical wavefunctions. The IVR input consists in a swarm of independent classical trajectories, each of which obeys the dynamics dictated by the system’s unquantized Hamiltonian. In a recent work [Aguiar et al, Chem. Phys. 370 (2010)] an IVR expression based on complex classical trajectories was formulated. The complex nature of the classical trajectories in this particular formula is a consequence of employing minimum uncertainty wavepackets (coherent states) to represent the time evolution operator upon which the formula is constructed. Here, an improved and simpler version of this complex IVR formula is presented. In order to test its accuracy, we apply it to a simple model system.

Measuring nonadiabaticity of molecular quantum dynamics with quantum fidelity and with its efficient semiclassical approximation

We propose to measure nonadiabaticity of molecular quantum dynamics rigorously with the quantum fidelity between the Born-Oppenheimer and fully nonadiabatic dynamics. It is shown that this measure of nonadiabaticity applies in situations where other criteria, such as the energy gap criterion or the extent of population transfer, fail. We further propose to estimate this quantum fidelity efficiently with a generalization of the dephasing representation to multiple surfaces. Two variants of the multiple-surface dephasing representation (MSDR) are introduced, in which the nuclei are propagated either with the fewest-switches surface hopping or with the locally mean field dynamics (LMFD). The LMFD can be interpreted as the Ehrenfest dynamics of an ensemble of nuclear trajectories, and has been used previously in the nonadiabatic semiclassical initial value representation. In addition to propagating an ensemble of classical trajectories, the MSDR requires evaluating nonadiabatic couplings and solving the Schrödinger (or more generally, the quantum Liouville-von Neumann) equation for a single discrete degree of freedom. The MSDR can be also used in the diabatic basis to measure the importance of the diabatic couplings. The method is tested on three model problems introduced by Tully and on a two-surface model of dissociation of NaI.

Insights in quantum dynamical effects in the infrared spectroscopy of liquid water from a semiclassical study with an ab initio-based flexible and polarizable force field

The dynamical properties of liquid water play an important role in many processes in nature. In this paper, we focus on the infrared (IR) absorption spectrum of liquid water based on the linearized semiclassical initial value representation (LSC-IVR) with the local Gaussian approximation (LGA) [J. Liu and W. H. Miller, J. Chem. Phys. 131, 074113 (2009)] and an ab initio based, flexible, polarizable Thole-type model (TTM3-F) [G. S. Fanourgakis and S. S. Xantheas, J. Chem. Phys. 128, 074506 (2008)]. Although the LSC-IVR (LGA) gives the exact result for the isolated three-dimensional shifted harmonic stretching model, it yields a blueshifted peak position for the more realistic anharmonic stretching potential. By using the short-time information of the LSC-IVR correlation function; however, it is shown how one can obtain more accurate results for the position of the stretching peak. Due to the physical decay in the condensed phase system, the LSC-IVR (LGA) is a good and practical approximate quantum approach for the IR spectrum of liquid water. The present results offer valuable insight into future attempts to improve the accuracy of the TTM3-F potential or other ab initio-based models in reproducing the IR spectrum of liquid water.

Fighting the curse of dimensionality in first-principles semiclassical calculations: Non-local reference states for large number of dimensions

Semiclassical methods face numerical challenges as the dimensionality of the system increases. In the general context of the theory of differential equations, this is known as the "curse of dimensionality." In the present manuscript, we apply the recently-introduced multi-coherent states semiclassical initial value representation (MC-SC-IVR) approach to extend the applicability of first-principles semiclassical calculations. The proposed strategy involves the use of non-local coherent states with the goal of increasing accuracy in the Fourier transforms, and on the other hand, allows for the selection of peaks of different frequencies. The ability to filter desired peaks is important for analyzing the power spectra of complex systems. The MC-SC-IVR approach allows us to solve a 19-dimensional test system and to resolve on-the-fly the power spectra of the formaldehyde molecule with very few classical trajectories.

Semiclassical study of quantum coherence and isotope effects in ultrafast electron transfer reactions coupled to a proton and a phonon bath

The linearized semiclassical initial value representation is employed to describe ultrafast electron transfer processes coupled to a phonon bath and weakly coupled to a proton mode. The goal of our theoretical investigation is to understand the influence of the proton on the electronic dynamics in various bath relaxation regimes. More specifically, we study the impact of the proton on coherences and analyze if the coupling to the proton is revealed in the form of an isotope effect. This will be important in distinguishing reactions in which the proton does not undergo significant rearrangement from those in which the electron transfer is accompanied by proton transfer. Unlike other methodologies widely employed to describe nonadiabatic electron transfer, this approach treats the electronic and nuclear degrees of freedom consistently. However, due to the linearized approximation, quantum interference effects are not captured accurately. Our study shows that at small phonon bath reorganization energies, coherent oscillations and isotope effect are observed in both slow and fast bath regimes. The coherences are more substantially damped by deuterium in comparison to the proton. Further, in contrast to the dynamics of the spin-boson model, the coherences are not long-lived. At large bath reorganization energies, the decay is incoherent in the slow and fast bath regimes. In this case, the extent of the isotope effect depends on the relative relaxation timescales of the proton mode and the phonon bath. The isotope effect is magnified for baths that relax on picosecond timescales in contrast to baths that relax in femtoseconds.

A semiclassical study of the thermal conductivity of low temperature liquids

The conventional classical energy current auto-correlation function has been extended into a quantum mechanical version and then approximated by the linearized semiclassical initial value representation approach. Comparison of the thermal conductivity to simulation results shows that about 15% quantum correction to the classical molecular dynamics results for liquid neon are quantitatively predicted. For liquid para-hydrogen the quantum effects are sufficiently large that the linearized semiclassical approach is only 20% accurate, while for both liquid He(4) and He(3) the thermal conductivity disagrees by a factor of 2, although exchange effects appear to play a minor role.

Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions

An efficient time-dependent importance sampling method is developed for the Monte Carlo calculation of time correlation functions via the initial value representation (IVR) of semiclassical (SC) theory. A prefactor-free time-dependent sampling function weights the importance of a trajectory based on the magnitude of its contribution to the time correlation function, and global trial moves are used to facilitate the efficient sampling the phase space of initial conditions. The method can be generally applied to sampling rare events efficiently while avoiding being trapped in a local region of the phase space. Results presented in the paper for two system-bath models demonstrate the efficiency of this new importance sampling method for full SC-IVR calculations.