^{1}, Bastiaan J. Braams

^{1,a)}and David E. Manolopoulos

^{1,b)}

### Abstract

The maximum entropy analytic continuation (MEAC) and ringpolymermolecular 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 tends to zero and the ringpolymer collapses to a single classical bead. MEAC becomes most reliable at low temperatures, where 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 combination is inadequate, as we illustrate with an example application to the collective density fluctuations in liquid orthodeuterium.

This work was supported by the U.S. Office of Naval Research under Contract No. N000140510460 and by the U.K. Engineering and Physical Sciences Research Council under Grant No. E01741X.

I. INTRODUCTION

II. QUANTUM MECHANICAL CORRELATION FUNCTIONS

III. MAXIMUM ENTROPY ANALYTIC CONTINUATION

IV. RINGPOLYMERMOLECULAR DYNAMICS

A. The nonlinear operator problem

B. Dynamics of liquid parahydrogen

C. Density fluctuations in liquid orthodeuterium

V. CONCLUDING REMARKS

### Key Topics

- Correlation functions
- 55.0
- Polymers
- 18.0
- Real functions
- 15.0
- Molecular dynamics
- 11.0
- Real time information delivery
- 11.0

## Figures

The functions in Eq. (17).

The functions in Eq. (17).

The functions in Eq. (23).

The functions in Eq. (23).

An example of the “L curve,” taken from an application to the velocity autocorrelation function of liquid parahydrogen at . The regularization parameter decreases by a factor of 0.9 between successive calculations (open circles).

An example of the “L curve,” taken from an application to the velocity autocorrelation function of liquid parahydrogen at . The regularization parameter decreases by a factor of 0.9 between successive calculations (open circles).

The curvature of the L curve and the optimum value of the regularization parameter for the example in Fig. 3.

The curvature of the L curve and the optimum value of the regularization parameter for the example in Fig. 3.

Canonical real time autocorrelation functions for the anharmonic oscillator in Eq. (35), with . Filled circles: Exact. Dotted lines: RPMD prior. Solid lines: .

Canonical real time autocorrelation functions for the anharmonic oscillator in Eq. (35), with . Filled circles: Exact. Dotted lines: RPMD prior. Solid lines: .

Imaginary time autocorrelation functions for the anharmonic oscillator in Eq. (35), with .

Imaginary time autocorrelation functions for the anharmonic oscillator in Eq. (35), with .

Canonical real time velocity autocorrelation functions for liquid parahydrogen at 25 and .

Canonical real time velocity autocorrelation functions for liquid parahydrogen at 25 and .

Imaginary time velocity autocorrelation functions for liquid parahydrogen at 25 and . The present imaginary time path integral results (filled circles) agree to graphical accuracy with the results in Fig. 1 of the paper by Rabani *et al.* (Ref. 21), which were obtained using a different estimator for the imaginary time velocity autocorrelation function (see Appendix A).

Imaginary time velocity autocorrelation functions for liquid parahydrogen at 25 and . The present imaginary time path integral results (filled circles) agree to graphical accuracy with the results in Fig. 1 of the paper by Rabani *et al.* (Ref. 21), which were obtained using a different estimator for the imaginary time velocity autocorrelation function (see Appendix A).

Canonical velocity autocorrelation spectra for liquid parahydrogen at 25 and .

Canonical velocity autocorrelation spectra for liquid parahydrogen at 25 and .

Thermally symmetrized velocity autocorrelation functions for liquid parahydrogen at . The filled circles are the complex time path integral results of Nakayama and Makri (Ref. 48), and MEAC denotes the MEAC procedure with a flat prior [Eq. (60)].

Thermally symmetrized velocity autocorrelation functions for liquid parahydrogen at . The filled circles are the complex time path integral results of Nakayama and Makri (Ref. 48), and MEAC denotes the MEAC procedure with a flat prior [Eq. (60)].

Kubo-transformed intermediate scattering functions for liquid orthodeuterium at , for three different momentum transfers .

Kubo-transformed intermediate scattering functions for liquid orthodeuterium at , for three different momentum transfers .

Normalized dynamic structure factors for liquid orthodeuterium at , for three different momentum transfers. The calculated structure factors have been normalized such that to facilitate the comparison with experiment. The open circles are the neutron scattering results of Mukherjee *et al.* (Ref. 56).

Normalized dynamic structure factors for liquid orthodeuterium at , for three different momentum transfers. The calculated structure factors have been normalized such that to facilitate the comparison with experiment. The open circles are the neutron scattering results of Mukherjee *et al.* (Ref. 56).

Canonical velocity autocorrelation spectra for liquid parahydrogen at 25 and , obtained using three different priors in the MEAC procedure.

Canonical velocity autocorrelation spectra for liquid parahydrogen at 25 and , obtained using three different priors in the MEAC procedure.

## Tables

Results of the L curve procedure using three different priors, for the canonical velocity autocorrelation spectrum of liquid parahydrogen.

Results of the L curve procedure using three different priors, for the canonical velocity autocorrelation spectrum of liquid parahydrogen.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content