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Approximate quantum dynamics methods for time correlation functions

Kyle K. G. Smith

Published May 1, 2014• 0 citations
OpenAlex

Abstract

The dynamic structure factor of liquid para-hydrogen and ortho-deuterium in corresponding thermodynamic states, (T = 20.0 K, n = 21.24 nm⁻³) and (T = 23.0 K, n = 24.61 nm⁻³) respectively, has been computed by both the Feynman-Kleinert linearized path-integral (FK-LPI) and Ring-Polymer Molecular Dynamics (RPMD) methods and compared with Inelastic X-ray Scattering spectra. The combined use of computational and experimental methods enables a reduction in experimental uncertainties for the determination of the true sample spectrum. Furthermore, the refined experimental spectrum of para-hydrogen and ortho-deuterium is consistently reproduced by both FK-LPI and RPMD at momentum transfers lower than 12.8nm⁻¹. At larger momentum transfers the F K - LP I results agree with experiment much better for ortho-deuterium than for para-hydrogen. More specifically we found that for k ~ 20.0 nm⁻¹ para-hydrogen provides a test case for improved approximations to quantum dynamics. We meet this demand for an improved approximate quantum dynamics method by developing two classes of quasi-classical dynamics that are shown to conserve the initial quantum ensemble when used in conjunction with the Feynman-Kleinert approximation of the density operator. As shown, both classes of dynamics are able to recover the exact classical and high temperature limits of the quantum time correlation function, while a subset is able to recover the exact harmonic limit. A comparison of the approximate quantum time correlation functions obtained from both classes of dynamics are made with the exact results for the challenging model problems of the quartic and double-well potentials. It is found that this new Feynman-Kleinert Quasi-Classical Wigner (FK-QCW) method provides a great improvement over the Feynman-Kleinert implementation of the classical Wigner approximation, also known as FK-LPI, in which purely classical dynamics are used. Furthermore, it is shown that the first class of dynamics reduces to Centroid Molecular Dynamics (CMD) when used within the framework of the classical Wigner approximation for the Kubo transformed time correlation function. Finally, we apply the Feynman-Kleinert Quasi-Classical Wigner (FK- QCW) method to the same liquid para-hydrogen and ortho-deuterium system, previously studied using FK-LPI and RPMD. When applied to this challenging system, it is shown that this new FK-QCW method consistently reproduces the experimental dynamic structure factor for all momentum transfers considered. This shows that FK-QCW provides a great improvement over FK-LPI for not only model problems, but also realistic systems. Furthermore, for small momentum transfers, where RPMD is applicable, it is shown that FK-QCW provides nearly the same results as RPMD, thus suggesting that FK-QCW provides a potentially more appealing algorithm than RPMD since one is not limited to correlation functions involving linear operators. This then suggests that the FK-QCW method is a top contender in the realm of approximate quantum dynamics methods which allow for the practical evaluation of time correlation functions.

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The dynamic structure factor of liquid para-hydrogen and ortho-deuterium in corresponding thermodynamic states, (T = 20.0 K, n = 21.24 nm⁻³) and (T = 23.0 K, n = 24.61 nm⁻³) respectively, has been computed by both the Feynman-Kleinert linearized path-integral (FK-LPI) and Ring-Polymer Molecular Dynamics (RPMD) methods and compared with Inelastic X-ray Scattering spectra. The combined use of computational and experimental methods enables a reduction in experimental uncertainties for the determination of the true sample spectrum. Furthermore, the refined experimental spectrum of para-hydrogen and ortho-deuterium is consistently reproduced by both FK-LPI and RPMD at momentum transfers lower than 12.8nm⁻¹. At larger momentum transfers the F K - LP I results agree with experiment much better for ortho-deuterium than for para-hydrogen. More specifically we found that for k ~ 20.0 nm⁻¹ para-hydrogen provides a test case for improved approximations to quantum dynamics. We meet this demand for an improved approximate quantum dynamics method by developing two classes of quasi-classical dynamics that are shown to conserve the initial quantum ensemble when used in conjunction with the Feynman-Kleinert approximation of the density operator. As shown, both classes of dynamics are able to recover the exact classical and high temperature limits of the quantum time correlation function, while a subset is able to recover the exact harmonic limit. A comparison of the approximate quantum time correlation functions obtained from both classes of dynamics are made with the exact results for the challenging model problems of the quartic and double-well potentials. It is found that this new Feynman-Kleinert Quasi-Classical Wigner (FK-QCW) method provides a great improvement over the Feynman-Kleinert implementation of the classical Wigner approximation, also known as FK-LPI, in which purely classical dynamics are used. Furthermore, it is shown that the first class of dynamics reduces to Centroid Molecular Dynamics (CMD) when used within the framework of the classical Wigner approximation for the Kubo transformed time correlation function. Finally, we apply the Feynman-Kleinert Quasi-Classical Wigner (FK- QCW) method to the same liquid para-hydrogen and ortho-deuterium system, previously studied using FK-LPI and RPMD. When applied to this challenging system, it is shown that this new FK-QCW method consistently reproduces the experimental dynamic structure factor for all momentum transfers considered. This shows that FK-QCW provides a great improvement over FK-LPI for not only model problems, but also realistic systems. Furthermore, for small momentum transfers, where RPMD is applicable, it is shown that FK-QCW provides nearly the same results as RPMD, thus suggesting that FK-QCW provides a potentially more appealing algorithm than RPMD since one is not limited to correlation functions involving linear operators. This then suggests that the FK-QCW method is a top contender in the realm of approximate quantum dynamics methods which allow for the practical evaluation of time correlation functions.
The maximum entropy analytic continuation (MEAC) and ring polymer molecular dynamics (RPMD) methods provide complementary approaches to the calculation of real time quantum correlation functions. RPMD becomes exact in the high temperature limit, where the thermal time betavariant Planck's over 2pi tends to zero and the ring polymer collapses to a single classical bead. MEAC becomes most reliable at low temperatures, where betavariant Planck's over 2pi exceeds the correlation time of interest and the numerical imaginary time correlation function contains essentially all of the information that is needed to recover the real time dynamics. We show here that this situation can be exploited by combining the two methods to give an improved approximation that is better than either of its parts. In particular, the MEAC method provides an ideal way to impose exact moment (or sum rule) constraints on a prior RPMD spectrum. The resulting scheme is shown to provide a practical solution to the "nonlinear operator problem" of RPMD, and to give good agreement with recent exact results for the short-time velocity autocorrelation function of liquid parahydrogen. Moreover these improvements are obtained with little extra effort, because the imaginary time correlation function that is used in the MEAC procedure can be computed at the same time as the RPMD approximation to the real time correlation function. However, there are still some problems involving long-time dynamics for which the RPMD+MEAC combination is inadequate, as we illustrate with an example application to the collective density fluctuations in liquid orthodeuterium.
The classical Wigner model is one way to approximate the quantum dynamics of atomic nuclei. Here, a new method is presented for sampling the initial quantum mechanical distribution that is required in the classical Wigner model. The new method is tested for the position, position-squared, momentum, and momentum-squared autocorrelation functions for a one-dimensional quartic oscillator and double well potential as well as a quartic oscillator coupled to harmonic baths of different sizes. Two versions of the new method are tested and shown to possibly be useful. Both versions always converge toward the classical Wigner limit. For the one-dimensional cases, some results that are essentially converged to the classical Wigner limit are acquired and others are not far off. For the multi-dimensional systems, the convergence is slower, but approximating the sampling of the harmonic bath with classical mechanics was found to greatly improve the numerical performance. For the double well, the new method is noticeably better than the Feynman-Kleinert linearized path integral method at reproducing the exact classical Wigner results, but they are equally good at reproducing exact quantum mechanics. The new method is suggested as being interesting for future tests on other correlation functions and systems.
The ring polymer molecular dynamics (RPMD) method has gained popularity in recent years as a simple approximation for calculating real time quantum correlation functions in condensed media. However, the extent to which RPMD captures real dynamical quantum effects and why it fails under certain situations have not been clearly understood. Addressing this issue has been difficult in the absence of a genuine justification for the RPMD algorithm starting from the quantum Liouville equation. To this end, a new and exact path integral formalism for the calculation of real time quantum correlation functions is presented in this work, which can serve as a rigorous foundation for the analysis of the RPMD method as well as providing an alternative derivation of the well established centroid molecular dynamics method. The new formalism utilizes the cyclic symmetry of the imaginary time path integral in the most general sense and enables the expression of Kubo-transformed quantum time correlation functions as that of physical observables pre-averaged over the imaginary time path. Upon filtering with a centroid constraint function, the formulation results in the centroid dynamics formalism. Upon filtering with the position representation of the imaginary time path integral, we obtain an exact quantum dynamics formalism involving the same variables as the RPMD method. The analysis of the RPMD approximation based on this approach clarifies that an explicit quantum dynamical justification does not exist for the use of the ring polymer harmonic potential term (imaginary time kinetic energy) as implemented in the RPMD method. It is analyzed why this can cause substantial errors in nonlinear correlation functions of harmonic oscillators. Such errors can be significant for general correlation functions of anharmonic systems. We also demonstrate that the short time accuracy of the exact path integral limit of RPMD is of lower order than those for finite discretization of path. The present quantum dynamics formulation also serves as the basis for developing new quantum dynamical methods that utilize the cyclic nature of the imaginary time path integral.
A new numerical approach for the calculation of quantum time correlation functions in the canonical and micro-canonical ensembles has been developed within the Wigner formulation of quantum statistical mechanics. For electrons in disordered systems of scatterers the numerical results for the tensor of static electrical conductivity have been obtained according to the quantum Kubo formula in the one-electron approximation in the canonical ensemble. The second quantity considered is the flux position time correlation function characterizing the momentum flow for tunnelling particles at the barrier crossing from one potential well to another. The difficulties with studies of exponentially small effects in the temperature averaging for the canonical ensemble has been overcome by a transition to the micro-canonical ensemble with a fixed total energy. The related integral Wigner-Liouville type equations have been obtained and a numerical approach combining both molecular dynamics and Monte Carlo methods for solving this equation has also been developed. Numerical results have been obtained for the momentum flow of a particle tunnelling through the barrier dividing two potential wells in two- and three-dimensional cases.
Numerical algorithms are developed for the centroid molecular dynamics (centroid MD) method to calculate dynamical time correlation functions for general many-body quantum systems. Approaches based on the normal mode path integral molecular dynamics and staging path integral Monte Carlo methods are described to carry out a direct calculation of the force on the centroid variables in the centroid MD algorithm. A more efficient, but approximate, scheme to compute the centroid force is devised which is based on the locally optimized harmonic approximation for the centroid potential. The centroid MD equations in the latter method can be solved with the help of an iterative procedure or through extended Lagrangian dynamics. A third algorithm introduces an effective centroid pseudopotential to approximate the full many-body centroid mean force potential by effective pairwise centroid interactions. Numerical simulations for both prototype models and more realistic many-body systems are performed to explore the feasibility and limitations of each algorithm.
We present a method for approximating real-time correlation functions and quantum transport coefficients of bosonic condensed phases. The direct evaluation of quantum real-time correlation functions in the path integral formulation is impossible due to a severe dynamical sign problem. Evaluating imaginary-time correlation functions and inverting them using analytic continuation is famously ill-posed. Ring polymer molecular dynamics (RPMD) provides an alternative approach resulting in accurate approximations for real-time correlation functions of condensed phases but neglects exchange effects. In this work, we develop a bosonic RPMD method, which is exact in the harmonic, short time, and high temperature limits. A critical enabling observation is that the Kubo-transformed correlation function for bosons is real only when the correlated observables are symmetric under exchange. We benchmark the method on harmonic and anharmonic model systems and then use it to directly obtain the Lieb-Liniger gas density-density real-time correlation functions at various temperatures and momentum transfers. This work enables the direct simulation of finite-temperature, real-time dynamics of large bosonic condensed phases using RPMD for the first time.