The calculation of transport properties in quantum liquids using the maximum entropy numerical analytic continuation method: Application to liquid para -hydrogen

Abstract

We present a method based on augmenting an exact relation between a frequency-dependent diffusion constant and the imaginary time velocity autocorrelation function, combined with the maximum entropy numerical analytic continuation approach to study transport properties in quantum liquids. The method is applied to the case of liquid para-hydrogen at two thermodynamic state points: a liquid near the triple point and a high-temperature liquid. Good agreement for the self-diffusion constant and for the real-time velocity autocorrelation function is obtained in comparison to experimental measurements and other theoretical predictions. Improvement of the methodology and future applications are discussed.O ne of the major goals of, and perhaps the most challenging problem in, computational statistical mechanics is the simulation of quantum dynamics in condensed phases. In principle, the density matrix formalism provides all the tools necessary to study equilibrium and time-dependent properties of any chemical system. In practice, however, the exact solution of the time-dependent quantum Wigner-Liouville equation is possible for a very limited class of simple systems, and the numerical solution for a general many-body system is not possible because of the well known phase cancellation problem (the sign problem).This problem has led to a variety of different techniques to include the effects of quantum fluctuations on the dynamic response of the system. One of the viable alternatives to the exact quantum mechanical solution is the use of techniques that are ''semiclassical'' in nature; namely, the dynamic response is obtained with the aid of classical trajectories (1). Although such techniques appear promising, technical issues have prevented their use in describing dynamics in realistic quantum liquids.Another class of methods that has been used with success in a variety of problems involves sophisticated numerical analytical continuation of exact imaginary-time path-integral Monte Carlo (PIMC) data (2, 3). These methods have been applied to a variety of condensed phase problems, including the dynamics of an excess electron solvated in water (4), helium and xenon (5), vibrational relaxation (6, 7), optical spectroscopy (6-8), adiabatic reaction dynamics (9, 10), dynamics in various quantum lattice models (11,12), and density fluctuations in superfluid helium (13). However, the application of these approaches to study density fluctuations (13) and transport properties (4) in quantum liquids has not been completely successful.In this paper, we show that analytic continuation methods can be used successfully to study the transport properties of a ''realistic'' liquid. We express the imaginary time velocity autocorrelation function, which is obtained from a suitable PIMC method (14), in terms of a frequency-dependent diffusion constant and use the maximum entropy method to analytically continue the imaginary time data to real time and thus obtain the self-diffusion constant and the velocity autocorrelation function. We use ...

Static structure factor of liquid parahydrogen

“…23 As recently shown, the centroid correlation functions can be related to semiclassical approximations. 26 While analytic continuation of exact imaginary-time path integral Monte Carlo techniques 27 and quantum versions of the mode-coupling theory 28 are at present under development, PICMD provides a computationally convenient framework to calculate real-time properties of the condensed phases of many-body systems.…”

Linearized semiclassical initial value time correlation functions with maximum entropy analytic continuation

“…Section II briefly reviews the analytic relation between real time and imaginary time correlation functions 109 , the MEAC procedure [68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84] to utilize this relation efficiently, and the LSC-IVR methodology 6,24,25,34,[39][40][41][42][43][44] . Section III then presents several applications to demonstrate how the MEAC+LSC/IVR approach performs: Sec IIIA demonstrates how the LSC-IVR prior is superior to the conventional flat prior in the harmonic limit and the high temperature regime, and Sec IIIB shows how the MEAC procedure improves the LSC-IVR for a strongly anharmonic one-dimensional model at very low temperature; and finally, Sec IIIC describes application of the MEAC+LSC/IVR to a complex system (liquid para-hydrogen) and compares the results with those using priors from other trajectory-based methods (RPMD, CMD, purely classical dynamics itself) with their MEAC corrections.…”

“…Even for large systems, the imaginary time correlation function ( ) [68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84] based on the Bayesian approach 118,119 . Followed is a brief summary of the MEAC procedure, while more details are described in above literature.…”

Mode-coupling theory for reaction dynamics in liquids

“…A quantum mechanical generalization of this approach has recently been developed and applied successfully to study the dynamic response in quantum liquids. 34,35,36,37,38,39,40 The Grote-Hynes approach requires that the memory friction be approximated in the barrier region. We therefore replace the full memory friction of the GLE (ζ(t)) with that approximated at the saddle point (ζ † (t)), where (as before) the symbol " †" indicates that the position variable of the system is taken at the saddle point.…”