^{1,a)}

### Abstract

We show two more approaches for generating trajectory-based dynamics in the phase space formulation of quantum mechanics: “equilibrium continuity dynamics” (ECD) in the spirit of the phase space continuity equation in classical mechanics, and “equilibrium Hamiltonian dynamics” (EHD) in the spirit of the Hamilton equations of motion in classical mechanics. Both ECD and EHD can recover 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. Both ECD and EHD conserve the quasi-probability within the infinitesimal volume *dx* _{ t } *dp* _{ t } around the phase point (*x* _{ t }, *p* _{ t }) along the trajectory. Numerical tests of both approaches in the Wigner phase space have been made for two strongly anharmonic model problems and a double well system, for each potential auto-correlation functions of both linear and nonlinear operators have been calculated. The results suggest EHD and ECD are two additional potential useful approaches for describing quantum effects for complex systems in condense phase.

J.L. thanks Prof. William H. Miller for encouragements and stimulating discussions and Dr. Shervin Fatehi for reading the manuscript and giving useful comments. J.L. also thanks Dr. Ionut Georgescu and Prof. Vladimir A. Mandelshtam for sending their manuscript of Ref. 32 after the original version of this paper was done. This work was supported by the National Science Foundation (NSF) 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. TRAJECTORY-BASED DYNAMICS GENERATED IN THE SPIRIT OF THE PHASE SPACE CONTINUITY EQUATION IN CLASSICAL MECHANICS

A. Phase space continuity equation in classical mechanics

B. A heuristic viewpoint for deriving the force from the canonical distribution function for the thermal equilibrium system

C. Equilibrium continuity dynamics

D. ECD trajectories and canonical ensemble averages

E. Choice of the phase space distribution function

F. Application to the thermal correlation function

III. TRAJECTORY-BASED DYNAMICS GENERATED IN THE SPIRIT OF THE HAMILTON EQUATIONS OF MOTION

A. Hamilton equations of motion in classical mechanics

B. Equilibrium Hamiltonian dynamics

C. EHD trajectories

D. Relation between ECD and EHD

E. Application to the canonical ensemble average and the thermal correlation function

IV. NUMERICAL IMPLEMENTATION AND EXAMPLES

A. Methods for ECD and EHD

1. W-ECD with LGA-TGA

2. W-EHD with LGA-TGA

3. Relation between W-EHD with LGA-TGA and Gaussian molecular dynamics

B. Numerical examples

1. Harmonic potential

2. Asymmetric anharmonic potential

3. Quartic potential

4. Double-well potential

V. CONCLUSION REMARKS

### Key Topics

- Correlation functions
- 49.0
- Cumulative distribution functions
- 48.0
- Equations of motion
- 31.0
- Quantum mechanics
- 22.0
- Classical mechanics
- 19.0

## 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; dotted line: LSC-IVR; solid circles: ELD; solid triangles: ECD; solid squares: EHD. Panel (c) Kubo-transformed momentum autocorrelation function. Solid line: Exact quantum result; solid circles: RPMD; solid triangles: CMD with classical operator; solid squares: CMD with effective classical operator. (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; dotted line: LSC-IVR; solid circles: ELD; solid triangles: ECD; solid squares: EHD. Panel (c) Kubo-transformed momentum autocorrelation function. Solid line: Exact quantum result; solid circles: RPMD; solid triangles: CMD with classical operator; solid squares: CMD with effective classical operator. (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 TGA. Dotted line: LSC-IVR with full TGA. Solid circles: W-ELD with LGA-TGA. Solid triangles: W-ECD with LGA-TGA. Solid squares: W-EHD with LGA-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 TGA. Dotted line: LSC-IVR with full TGA. Solid circles: W-ELD with LGA-TGA. Solid triangles: W-ECD with LGA-TGA. Solid squares: W-EHD with LGA-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. 1, but for a much lower temperature β = 8.

As in Fig. 1, but for a much lower temperature β = 8.

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: LSC-IVR with full TGA. Solid circles: W-ELD with LGA-TGA. Solid triangles: W-ECD with LGA-TGA. Solid squares: W-EHD with LGA-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: LSC-IVR with full TGA. Solid circles: W-ELD with LGA-TGA. Solid triangles: W-ECD with LGA-TGA. Solid squares: W-EHD with LGA-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. 3, but for a much lower temperature β = 8.

As in Fig. 3, but for a much lower temperature β = 8.

The autocorrelation functions for the one-dimensional double well potential for β = 0.1 and for β = 8. Panels (a), (c), and (e): Kubo-transformed momentum autocorrelation function. Panels (b), (d), and (f): Real part of standard *x* ^{2} autocorrelation function. Panels (a)–(d)—Solid line: Exact quantum result; dotted line: LSC-IVR with LGA-TGA; solid circles: W-ELD with LGA-TGA; solid triangles: W-ECD with LGA-TGA; solid squares: W-EHD with LGA-TGA. Panels (e)–(f)—Solid line: Exact quantum result; Dotted line: LSC-IVR with LGA-TGA; Solid circles: W-ELD with full-TGA; Solid triangles: W-ECD with full-TGA.

The autocorrelation functions for the one-dimensional double well potential for β = 0.1 and for β = 8. Panels (a), (c), and (e): Kubo-transformed momentum autocorrelation function. Panels (b), (d), and (f): Real part of standard *x* ^{2} autocorrelation function. Panels (a)–(d)—Solid line: Exact quantum result; dotted line: LSC-IVR with LGA-TGA; solid circles: W-ELD with LGA-TGA; solid triangles: W-ECD with LGA-TGA; solid squares: W-EHD with LGA-TGA. Panels (e)–(f)—Solid line: Exact quantum result; Dotted line: LSC-IVR with LGA-TGA; Solid circles: W-ELD with full-TGA; Solid triangles: W-ECD with full-TGA.

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