Semiclassical approximations to real-time quantum-mechanical effects in correlation functions of complex molecular systems

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.

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.

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.

Within a differential expression of the Heisenberg operator, the forward and backward evolution can be joined together along a closed time contour. This manipulation leads to a dramatic cancellation of oscillations due to the two individual propagators in the Heisenberg operator and the resulting forward-backward propagator is more tractable to semiclassical approximations. This article gives a detailed description of the forward-backward semiclassical dynamics (FBSD) formalism. The semiclassical propagators, especially those of the initial value representations (IVRs), are briefly discussed. The derivation of the FBSD based on the Herman–Kluk propagator is reviewed. Different FBSD formulations with other semiclassical IVRs are worked out and numerical calculations show that they are also capable of describing quantum dynamics semiquantitatively and all display accuracy similar to the classical Wigner method.

Semi-classical theories are approximations to quantum theory that treat some degrees of freedom classically and others quantum mechanically. In the usual approach, the quantum degrees of freedom are described by a wave function which evolves according to some Schrödinger equation with a Hamiltonian that depends on the classical degrees of freedom. The classical degrees of freedom satisfy classical equations that depend on the expectation values of quantum operators. In this paper, we study an alternative approach based on Bohmian mechanics. In Bohmian mechanics the quantum system is not only described by the wave function, but also with additional variables such as particle positions or fields. By letting the classical equations of motion depend on these variables, rather than the quantum expectation values, a semi-classical approximation is obtained that is closer to the exact quantum results than the usual approach. We discuss the Bohmian semi-classical approximation in various contexts, such as nonrelativistic quantum mechanics, quantum electrodynamics and quantum gravity. The main motivation comes from quantum gravity. The quest for a quantum theory for gravity is still going on. Therefore a semi-classical approach where gravity is treated classically may be an approximation that already captures some quantum gravitational aspects. The Bohmian semi-classical theories will be derived from the full Bohmian theories. In the case there are gauge symmetries, like in quantum electrodynamics or quantum gravity, special care is required. In order to derive a consistent semi-classical theory it will be necessary to isolate gauge-independent dependent degrees of freedom from gauge degrees of freedom and consider the approximation where some of the former are considered classical.

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.

We present the lowest order quantum correction to the semiclassical Boltzmann distribution function, and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation, and the quantum Wigner distribution function is expanded in powers of Planck constant, too. The negative quantum correlation in the Wigner distribution function which is just the quantum correction terms is naturally singled out, thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.

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.

In this work, we study two examples of frustrated magnetic systems whose degrees of freedom are spins (or pseudo-spins) on two-dimensional lattices. The first part presents the results obtained for the quantum compass model on a square lattice. This model is a minimal model for the orbital degrees of freedom of transition metal compounds. We use exact diagonalizations, quantum Monte Carlo simulations as well as a perturbative approach in order to study the ground state degeneracy in the thermodynamic limit and the type of order in the ground state. The second part of this work is devoted to the study of the magnetic properties of SrCu2(BO3)2, which is one of the few quasi two-dimensional quantum magnets whose magnetization curve as a function of the magnetic field shows plateaus. The magnetic degrees of freedom are described by a spin 1/2 Heisenberg model on the Shastry-Sutherland lattice in the presence of an external magnetic field. In order to study the magnetic properties at zero-temperature, we use perturbative continuous unitary transformations to derive an effective Hamiltonian in which the triplets Sz = 1 are particles moving on a background of singlets. This Hamiltonian is characterized by a kinetic energy dominated by correlated hopping, which let a particle hop only if there is another particle nearby. In order to better understand the effect of the correlated hopping, we start by studying a minimal model. By using exact diagonalizations, quantum Monte Carlo and a semi-classical approximation we show that the correlated hopping can stabilize a phase characterized by a condensation of pairs of particles and strongly favor supersolid phases. Next, we determine the zero-temperature phase diagram of the effective model by using a classical approximation. A comparison to previous theoretical works and experimental measurements shows that the Shastry-Sutherland model cannot reproduce the series of magnetization plateaus measured experimentally. This suggest that the residual interactions, which were neglected, could be crucial for SrCu2(BO3)2.

We report a new method for calculating the Wigner transform of the Boltzmann operator in the canonical ensemble. The transform is accomplished by writing the Boltzmann operator in a semiharmonic form, utilizing the variational centroid effective frequencies introduced by Feynman and Kleinert (FK). The approximate many-body Wigner transformed Boltzmann operator is then utilized with a linearized path integral (LPI) representation for correlation functions. It is shown that this new FK-LPI method is capable of calculating quite accurately the short time behavior of linear and highly nonlinear correlation functions for low temperature Lennard-Jones model systems and that it is vastly superior to classical dynamics. The feasibility of the FK-LPI method for large systems is illustrated by considering a model liquid composed of 32 oxygen molecules with periodic boundary conditions. Initial conditions for molecular dynamics are obtained from its Boltzmann Wigner transform and the FK-LPI method is shown to describe correctly the zero-point motion of the liquid. The effective frequency representation of the Wigner transformed thermal density operator provides an efficient way of sampling nonclassical initial conditions for molecular-dynamics simulations more generally. Applications to vibrational energy relaxation and rate constant calculations in complex molecular systems are discussed.

Introduction. The paper presents a description of the process of constructing a mathematical model for managing risks arising from the operation of complex technical unique systems for critical purposes under conditions of uncertainty of information about the parameters and phase state. The purpose of the study is to create an individual optimal strategy for managing risk events that occur during the operation of complex technical systems, which minimizes the cost of preventive work, as well as the amount of damage that may result from the occurrence of relevant events. The objectives of the study include the construction of a mathematical model for control risks arising from the operation of complex dynamic unique technical systems for critical purposes under conditions of interval uncertainty of information about the values of parameters and the phase state, since as well as an analysis of existing approaches to creating an individual forecast of changes in the state of the considered class of systems.
 Materials and Methods. The article gives a comparative description of the effectiveness of the application of the mathematical apparatus of the theory of outliers of random processes and the method of individual forecasting in solving the problem of managing risks arising from the operation of complex unique technical systems for critical purposes under conditions of uncertainty. Based on the statistical methods of interval data, a mathematical model of risk management was created, taking into account possible errors in measuring the values of the parameters of the considered class of complex systems at all control windows during the operation period.
 Results. The scientific novelty of the implemented approach lies in the use of interval data statistics, which allow the most correct consideration of possible errors associated with measuring the values of the characteristics of the studied technical systems at all stages of the control process.
 Discussion and Conclusions. The mathematical model of risk control developed in the course of the study, arising from the operation of complex technical systems for critical purposes, makes it possible to make the optimal choice of risk management strategy when using objects of this class. Along with the above, an algorithm has been developed for predicting changes in the state of a technical system during the entire period of its operation based on the mathematical apparatus of statistics of interval data, which makes it possible to take into account in the calculations the errors that occur when measuring the values of the main parameters of the system under consideration at all stages of the control process.

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.

We present both full quantum mechanical and semiclassical calculations for the photoionization of a hydrogen atom induced by a few-cycle linearly polarized infrared laser pulse. As a quantum treatment, we applied the direct integration of the time-dependent Schrödinger equation. In the three-dimensional semiclassical trajectory Monte Carlo approximation, it is assumed that the wavepacket propagation in the post-tunneling process can be well described within the classical framework. By comparing the results of two methods, we found good agreement in the final momentum distributions of the emitted electrons for longer exciting pulses. This allowed us to use the semiclassical approximation and clearly identify and separate the initial conditions that correspond to various regions in the final momentum distributions of the emitted electrons. We also illustrate the corresponding regions with typical electron trajectories.

Conventional gas kinetic scheme (GKS) has been successfully utilized to obtain the accurate solution of Navier–Stokes (NS) equations. However, when it comes to flows beyond NS level, the conventional GKS is not reliable because its initial gas distribution function is approximated by the first-order Chapman–Enskog expansion, which merely recovers the NS equations. In order to make an extension for flows beyond NS level, we propose the Grad's distribution functions-based GKS in this paper. This scheme retains the advantage of conventional GKS and constructs the numerical fluxes through a time-dependent gas distribution function, which is derived from the integral solution of Boltzmann equation. In the present scheme, the initial gas distribution function in the local solution of Boltzmann equation is approximated by Grad's 13 and 26 distribution functions. Furthermore, the high-order moments in the initial Grad's distribution function are calculated by moment relationship directly, and thus, the solution of complicated partial differential equations for these high-order moments is avoided. Four benchmark numerical examples are tested to validate the performance of the present scheme, and the results demonstrate that the present scheme can not only recover NS solutions in the continuum region but also predict reasonable results for flows in the slip and transition regimes.