An adiabatic linearized path integral approach for quantum time-correlation functions II: a cumulant expansion method for improving convergence

Linearized mixed quantum-classical simulations are a promising approach for calculating time-correlation functions. At the moment, however, they suffer from some numerical problems that may compromise their efficiency and reliability in applications to realistic condensed-phase systems. In this paper, we present a method that improves upon the convergence properties of the standard algorithm for linearized calculations by implementing a cumulant expansion of the relevant averages. The effectiveness of the new approach is tested by applying it to the challenging computation of the diffusion of an excess electron in a metal-molten salt solution.     

Nonequilibrium transport in quantum impurity models: exact path integral simulations

We simulate the nonequilibrium dynamics of two generic many-body quantum impurity models by employing the recently developed iterative influence-functional path integral method [Phys. Rev. B: Condens. Matter, 2010, 82, 205323]. This general approach is presented here in the context of quantum transport in molecular electronic junctions. Models of particular interest include the single impurity Anderson model and the related spinless two-state Anderson dot. In both cases we study the time evolution of the dot occupation and the current characteristics at finite temperature. A comparison to mean-field results is presented, when applicable.     

Effective potential analytic continuation approach for real time quantum correlation functions involving nonlinear operators

We apply the effective potential analytic continuation (EPAC) method to the calculation of real time quantum correlation functions involving operators nonlinear in the position operator q. For a harmonic system the EPAC method provides the exact correlation function at all temperature ranges, while the other quantum dynamics methods, the centroid molecular dynamics and the ring polymer molecular dynamics, become worse at lower temperature. For an asymmetric anharmonic system, the EPAC correlation function is in very good agreement with the exact one at t = 0. When the time increases from zero, the EPAC method gives good coincidence with the exact result at lower temperature. Finally, we propose a simplified version of the EPAC method to reduce the computational cost required for the calculation of the standard effective potential.     

Forward-backward semiclassical and quantum trajectory methods for time correlation functions

Forward-backward trajectory formulations of time correlation functions are reviewed. Combination of the forward and reverse time evolution operators within the time-dependent semiclassical approximation minimizes phase cancellation, giving rise to an efficient methodology for simulating the dynamics of low-temperature fluids. A quantum mechanical version of the forward-backward formulation, based on the hydrodynamic formulation of time-dependent quantum mechanics, is also available but is practical only for small systems.     

A comparative study of imaginary time path integral based methods for quantum dynamics

The recently introduced approximate many-body quantum simulation method, ring polymer molecular dynamics (RPMD), is compared to the centroid molecular dynamics method (CMD). Comparisons of simulation results for liquid para-hydrogen at two state points and liquid ortho-deuterium at one state point are presented. The calculated quantum correlation functions for the two methods are shown to be in good agreement with one another for a large portion of the time spectrum. However, as the quantum mechanical nature of the system increases, RPMD is less accurate in predicting the kinetic energy of the system than is CMD. A simplified and highly efficient algorithm is proposed which largely corrects this deficiency.     

Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes

A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.     

An efficient ring polymer contraction scheme for imaginary time path integral simulations

A quantum simulation of an imaginary time path integral typically requires around n times more computational effort than the corresponding classical simulation, where n is the number of ring polymer beads (or imaginary time slices) used in the calculation. However, this estimate neglects the fact that the potential energies of many systems can be decomposed into a sum of rapidly varying short-range and slowly varying long-range contributions. For such systems, the computational effort of the path integral simulation can be reduced considerably by evaluating the long-range forces on a contracted ring polymer with fewer beads than are needed to evaluate the short-range forces. This idea is developed and then illustrated with an application to a flexible model of liquid water in which the intramolecular forces are evaluated with 32 beads, the oxygen-oxygen Lennard-Jones forces with seven, and the intermolecular electrostatic forces with just five. The resulting static and dynamic properties are within a few percent of those of a full 32-bead calculation, and yet they are obtained with a computational effort less than six times (rather than 32 times) that of a classical simulation. We hope that this development will encourage future studies of quantum mechanical fluctuations in liquid water and aqueous solutions and in many other systems with similar interaction potentials.     

Real time correlation function in a single phase space integral beyond the linearized semiclassical initial value representation

It is shown how quantum mechanical time correlation functions [defined, e.g., in Eq. (1.1)] can be expressed, without approximation, in the same form as the linearized approximation of the semiclassical initial value representation (LSC-IVR), or classical Wigner model, for the correlation function [cf. Eq. (2.1)], i.e., as a phase space average (over initial conditions for trajectories) of the Wigner functions corresponding to the two operators. The difference is that the trajectories involved in the LSC-IVR evolve classically, i.e., according to the classical equations of motion, while in the exact theory they evolve according to generalized equations of motion that are derived here. Approximations to the exact equations of motion are then introduced to achieve practical methods that are applicable to complex (i.e., large) molecular systems. Four such methods are proposed in the paper--the full Wigner dynamics (full WD) and the second order WD based on "Wigner trajectories" [H. W. Lee and M. D. Scully, J. Chem. Phys. 77, 4604 (1982)] and the full Donoso-Martens dynamics (full DMD) and the second order DMD based on "Donoso-Martens trajectories" [A. Donoso and C. C. Martens, Phys. Rev. Lett. 8722, 223202 (2001)]--all of which can be viewed as generalizations of the original LSC-IVR method. Numerical tests of the four versions of this new approach are made for two anharmonic model problems, and for each the momentum autocorrelation function (i.e., operators linear in coordinate or momentum operators) and the force autocorrelation function (nonlinear operators) have been calculated. These four new approximate treatments are indeed seen to be significant improvements to the original LSC-IVR approximation.     

Molecular dynamics with quantum statistics: time correlation functions and weakly bound nano-clusters

An account of recent developments in the study of molecular dynamics with the inclusion of quantum exchange effects is presented. Approaches for quantum dynamical calculations are reviewed and the determination of time correlation functions is a special point of focus. It is shown that the exact basis set techniques can be used to perform highly accurate calculations but are restricted to relatively small systems since computational cost scales exponentially with system size. Alternate formulations can be introduced to circumvent this problem, and semi-classical initial value representation and Feynman path centroid approaches are considered. It is then showed that from a practical point of view, for complex bosonic systems such as doped helium clusters, Quantum Monte Carlo techniques can currently be used for the calculation of quantities of experimental interest. A perspective on future prospects for the calculation of real time correlation functions of bosonic nano-scale systems is presented     

Effective potential analytic continuation calculations of real time quantum correlation functions: asymmetric systems

We apply the effective potential analytic continuation (EPAC) method to one-dimensional asymmetric potential systems to obtain the real time quantum correlation functions at various temperatures. Comparing the EPAC results with the exact results, we find that for an asymmetric anharmonic oscillator the EPAC results are in very good agreement with the exact ones at low temperature, while this agreement becomes worse as the temperature increases. We also show that the EPAC calculation for a certain type of asymmetric potentials can be reduced to that for the corresponding symmetric potentials.     

Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach

A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantum master equation and the real-time diagrammatic formalism of Schon and co-workers can be readily obtained at well defined limits of the present theory. We also show that for a noninteracting electron system, the present hierarchical equations of motion formalism terminates at the second tier exactly, and the Landuer-Buttiker transport current expression is recovered. The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.     

Exciton transport in disordered crystals: velocity correlation functions

The two-particle coherent-potential approximation is used to calculate Frenkel-exciton group-velocity correlation functions for substitutionally disordered crystals. Most of the results can be described by a relaxation-time approximation, provided that k-dependent and complex relaxation times are allowed; however, some evidence for long-time tails, associated with the finite frequency range of the scattering potentials, is found. The probable accuracy of the approximation and its relationships with localization and kinetic theory are discussed, as is its relevance to experimental systems (triplet excitons in isotopic mixed naphthalene and anthracene crystals).     

Libcint: An efficient general integral library for Gaussian basis functions

An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian‐type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one‐electron overlap and nuclear attraction, two‐electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4‐component relativistic Hartree–Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth‐order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi‐CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20‐core node, we are able to achieve an average output 8.3 GB/s for C₆₀ molecule with cc‐pVTZ basis. © 2015 Wiley Periodicals, Inc.     

Test of the consistency of various linearized semiclassical initial value time correlation functions in application to inelastic neutron scattering from liquid para-hydrogen

The linearized approximation to the semiclassical initial value representation (LSC-IVR) is used to calculate time correlation functions relevant to the incoherent dynamic structure factor for inelastic neutron scattering from liquid para-hydrogen at 14 K. Various time correlations functions were used which, if evaluated exactly, would give identical results, but they do not because the LSC-IVR is approximate. Some of the correlation functions involve only linear operators, and others involve nonlinear operators. The consistency of the results obtained with the various time correlation functions thus provides a useful test of the accuracy of the LSC-IVR approximation and its ability to treat correlation functions involving both linear and nonlinear operators in realistic anharmonic systems. The good agreement of the results obtained from different correlation functions, their excellent behavior in the spectral moment tests based on the exact moment constraints, and their semiquantitative agreement with the inelastic neutron scattering experimental data all suggest that the LSC-IVR is indeed a good short-time approximation for quantum mechanical correlation functions.     

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. II. Thermal correlation functions

We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.     

Reduced density matrix hybrid approach: an efficient and accurate method for adiabatic and non-adiabatic quantum dynamics

We present a new approach to calculate real-time quantum dynamics in complex systems. The formalism is based on the partitioning of a system's environment into "core" and "reservoir" modes with the former to be treated quantum mechanically and the latter classically. The presented method only requires the calculation of the system's reduced density matrix averaged over the quantum core degrees of freedom which is then coupled to a classically evolved reservoir to treat the remaining modes. We demonstrate our approach by applying it to the spin-boson problem using the noninteracting blip approximation to treat the system and core, and Ehrenfest dynamics to treat the reservoir. The resulting hybrid methodology is accurate for both fast and slow baths, since it naturally reduces to its composite methods in their respective regimes of validity. In addition, our combined method is shown to yield good results in intermediate regimes where neither approximation alone is accurate and to perform equally well for both strong and weak system-bath coupling. Our approach therefore provides an accurate and efficient methodology for calculating quantum dynamics in complex systems.     

Coupled-channels quantum theory of electronic flux density in electronically adiabatic processes: application to the hydrogen molecule ion

This article presents the results of the first quantum simulations of the electronic flux density (j(e)) by the "coupled-channels" (CC) theory, the fundamentals of which are presented in the previous article [Diestler, D. J. J. Phys. Chem. A 2012, DOI: 10.1021/jp207843z]. The principal advantage of the CC scheme is that it employs exclusively standard methods of quantum chemistry and quantum dynamics within the framework of the Born-Oppenheimer approximation (BOA). The CC theory goes beyond the BOA in that it yields a nonzero j(e) for electronically adiabatic processes, in contradistinction to the BOA itself, which always gives j(e) = 0. The CC is applied to oriented H(2)(+) vibrating in the electronic ground state ((2)Σ(g)(+)), for which the nuclear and electronic flux densities evolve on a common time scale of about 22 fs per vibrational period. The system is chosen as a touchstone for the CC theory, because it is the only one for which highly accurate flux densities have been calculated numerically without invoking the BOA [Barth et al, Chem. Phys. Lett. 2009, 481, 118]. Good agreement between CC and accurate results supports the CC approach, another advantage of which is that it allows a transparent interpretation of the temporal and spatial properties of j(e).     

Dynamic correlation for MCSCF wave functions: An effective potential method

A method is suggested which allows the inclusion of dynamic correlation into CASSCF calculations. An effective Coulomb hole potential is added to the Hamiltonian. The potential has a simple form, which allows its implementation into existing LCAO programs using Gaussian integral packages. The parameters appearing in the potential are determined by fitting to empirical valence correlation energies for first row atoms. Calculations of ionization energies and electron affinities show considerable improvement compared to the MCSCF values. Test calculations on three molecules give the following results, H₂ r _e=0.745 (0.741) Å, D _e=4.62 (4.75) eV; N₂ r _e=1.099 (1.098) Å, D _e= 10.42 (9.91) eV; O₂ r _e=1.198 (1.207) Å, D _e=4.73 (5.21) eV. Experimental values within parenthesis. On leave from: Institute of Organic Chemistry, Polish Academy of Sciences, PL-01-224 Warszawa 42, ul. Kasprzaka 44, Poland.