^{1,a)}and William H. Miller

^{1,b)}

### Abstract

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.

J.L. thanks Dr. Shervin Fatehi for reading the manuscript and giving useful comments. This work was supported by the National Science Foundation Grant No. CHE-0809073 and by the Director, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, U.S. Department of Energy under Contract No. DE-AC02–05CH11231. We also acknowledge a generous allocation of supercomputing time from the National Energy Research Scientific Computing Center (NERSC) and the use of the Lawrencium computational cluster resource provided by the IT Division at the Lawrence Berkeley National Laboratory.

I. INTRODUCTION

II. THE EXACT FORMULATION OF THE CORRELATION FUNCTION IN THE PHASE SPACE FORMULATION OF QUANTUM MECHANICS

III. EQUILIBRIUM DISTRIBUTION APPROXIMATION

A. Evaluation of the correlation function in classical mechanics

B. Evaluation of the correlation function with the trajectory-based dynamics in the phase space formulation of quantum mechanics

IV. THERMAL GAUSSIAN APPROXIMATION (TGA) IN THE POSITION REPRESENTATION

A. W-ELD with full TGA

B. W-ELD with LGA–TGA

C. H-ELD with full TGA

V. NUMERICAL EXAMPLES

A. Harmonic potential

B. Asymmetric anharmonic oscillator

C. Quartic potential

VI. CONCLUSIONS

### Key Topics

- Correlation functions
- 62.0
- Thermogravimetric analysis
- 21.0
- Quantum effects
- 10.0
- Quantum mechanics
- 8.0
- Classical mechanics
- 7.0

## Figures

The autocorrelation functions for the one-dimensional harmonic oscillator for β = 8. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function. solid line: Exact quantum result; solid triangles: W-ELD; solid circles: H-ELD; hollow squares: LSC–IVR. Panel (c) Kubo-transformed momentum autocorrelation function. (d) Kubo-transformed *x* ^{2} autocorrelation function. Solid line: Exact quantum result; dashed line: RPMD; dotted line: CMD with classical operator; dot-dashed line: CMD with effective classical operator.

The autocorrelation functions for the one-dimensional harmonic oscillator for β = 8. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function. solid line: Exact quantum result; solid triangles: W-ELD; solid circles: H-ELD; hollow squares: LSC–IVR. Panel (c) Kubo-transformed momentum autocorrelation function. (d) Kubo-transformed *x* ^{2} autocorrelation function. Solid line: Exact quantum result; dashed line: RPMD; dotted line: CMD with classical operator; dot-dashed line: CMD with effective classical operator.

The autocorrelation functions for the one-dimensional anharmonic oscillator for β = 0.1. Solid line: Exact quantum result. In the following results, the Boltzmann operator is treated by the TGA. Dotted line: W-ELD with full TGA. Solid circles: W-ELD with LGA–TGA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Symmetrized force autocorrelation function. (c) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional anharmonic oscillator for β = 0.1. Solid line: Exact quantum result. In the following results, the Boltzmann operator is treated by the TGA. Dotted line: W-ELD with full TGA. Solid circles: W-ELD with LGA–TGA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Symmetrized force autocorrelation function. (c) Real part of standard *x* ^{2} autocorrelation function.

As in Fig. 2, but for a much lower temperature β = 8.

As in Fig. 2, but for a much lower temperature β = 8.

The autocorrelation functions for the one-dimensional anharmonic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: H-ELD with full TGA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Symmetrized force autocorrelation function. (c) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional anharmonic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: H-ELD with full TGA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Symmetrized force autocorrelation function. (c) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional quartic oscillator for β = 0.1. Solid line: Exact quantum result. In the following results, the Boltzmann operator is treated by the TGA. Dotted line: W-ELD with full TGA. Solid circles: W-ELD with LGA–TGA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Symmetrized force autocorrelation function. (c) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional quartic oscillator for β = 0.1. Solid line: Exact quantum result. In the following results, the Boltzmann operator is treated by the TGA. Dotted line: W-ELD with full TGA. Solid circles: W-ELD with LGA–TGA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Symmetrized force autocorrelation function. (c) Real part of standard *x* ^{2} autocorrelation function.

As in Fig. 5, but for a much lower temperature β = 8.

As in Fig. 5, but for a much lower temperature β = 8.

The autocorrelation functions for the one-dimensional quartic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: H-ELD with full TGA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Symmetrized force autocorrelation function. (c) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional quartic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: H-ELD with full TGA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Symmetrized force autocorrelation function. (c) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional anharmonic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: W-ELD with FKA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional anharmonic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: W-ELD with FKA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional quartic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: W-ELD with FKA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional quartic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: W-ELD with FKA. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional anharmonic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: H-ELD with TFG. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional anharmonic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: H-ELD with TFG. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional quartic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: H-ELD with TFG. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function.

The autocorrelation functions for the one-dimensional quartic oscillator for β = 8. Solid line: Exact quantum result. Solid triangles: H-ELD with TFG. Dashed line: LSC–IVR with full TGA. Panel (a) Kubo-transformed momentum autocorrelation function. (b) Real part of standard *x* ^{2} autocorrelation function.

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