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

^{1,b)}

### Abstract

The linearized approximation to the semiclassical initial value representation (LSC-IVR) is used to calculate time correlation functions relevant to the incoherent dynamic structure factor for inelastic neutron scattering from liquid para-hydrogen at 14 K. Various time correlations functions were used which, if evaluated exactly, would give identical results, but they do not because the LSC-IVR is approximate. Some of the correlation functions involve only linear operators, and others involve nonlinear operators. The consistency of the results obtained with the various time correlation functions thus provides a useful test of the accuracy of the LSC-IVR approximation and its ability to treat correlation functions involving both linear and nonlinear operators in realistic anharmonic systems. The good agreement of the results obtained from different correlation functions, their excellent behavior in the spectral moment tests based on the exact moment constraints, and their semiquantitative agreement with the inelastic neutron scattering experimental data all suggest that the LSC-IVR is indeed a good short-time approximation for quantum mechanical correlation functions.

We thank D. Manolopoulos for providing the RPMD results^{78} and the experiment data^{101} and for some useful discussions. This work was supported by the Office of Naval Research Grant No. N00014-05-1-0457 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).

I. INTRODUCTION

II. THEORY AND METHODOLOGY

A. Inelastic neutron scattering

B. LSC-IVR correlation functions using the TGA

C. Spectral moment tests

III. RESULTS AND DISCUSSIONS

A. Simulation details

B. Incoherent dynamic structure factors

C. Spectral moment test

D. Comparison with experimental data

IV. CONCLUSIONS

### Key Topics

- Correlation functions
- 75.0
- Thermogravimetric analysis
- 33.0
- Inelastic neutron scattering
- 15.0
- Quantum effects
- 8.0
- Boltzmann equations
- 6.0

## Figures

Self-parts of the intermediate scattering functions for liquid para-hydrogen at the state point ; . Dashed line: . Dotted line: . Dot-dashed line: . Solid line: .

Self-parts of the intermediate scattering functions for liquid para-hydrogen at the state point ; . Dashed line: . Dotted line: . Dot-dashed line: . Solid line: .

Incoherent dynamic structure factors for liquid para-hydrogen at the state point ; . Solid line: From the Kubo-transformed velocity correlation function (vv-kubo). Dot-dashed line: From the standard velocity correlation function (vv-std). Dashed line: From the self-part of the intermediate scattering function (inelastic-std).

Incoherent dynamic structure factors for liquid para-hydrogen at the state point ; . Solid line: From the Kubo-transformed velocity correlation function (vv-kubo). Dot-dashed line: From the standard velocity correlation function (vv-std). Dashed line: From the self-part of the intermediate scattering function (inelastic-std).

The first three moments of the incoherent dynamic structure factors shown in Fig. 2. Solid line: Exact result. Dashed line with solid circles: From the self-part of intermediate scattering function (inelastic-std). Hollow circles: From the Kubo-transformed velocity correlation function (vv-kubo). Crosses: From the standard velocity correlation function (vv-std). Dot-dashed line: From the Kubo-transform of [the self relaxation function ] by the RPMD method (RPMD-kubo). Hollow squares: From the Kubo-transformed velocity correlation function by the RPMD method (RPD-vv-kubo).

The first three moments of the incoherent dynamic structure factors shown in Fig. 2. Solid line: Exact result. Dashed line with solid circles: From the self-part of intermediate scattering function (inelastic-std). Hollow circles: From the Kubo-transformed velocity correlation function (vv-kubo). Crosses: From the standard velocity correlation function (vv-std). Dot-dashed line: From the Kubo-transform of [the self relaxation function ] by the RPMD method (RPMD-kubo). Hollow squares: From the Kubo-transformed velocity correlation function by the RPMD method (RPD-vv-kubo).

The relative error of the moment the incoherent dynamic structure factors shown in Fig. 2. Solid line with solid triangles: From the self-part of intermediate scattering function (inelastic-std). Dashed line with solid circles: From the Kubo-transform of [the self-relaxation function ] by the RPMD method (RPMD-kubo).

The relative error of the moment the incoherent dynamic structure factors shown in Fig. 2. Solid line with solid triangles: From the self-part of intermediate scattering function (inelastic-std). Dashed line with solid circles: From the Kubo-transform of [the self-relaxation function ] by the RPMD method (RPMD-kubo).

The first two even moments of the incoherent relaxation function based on the incoherent dynamic structure factors shown in Fig. 2. Solid line: Exact result. Dashed line with solid circles: From the self-part of intermediate scattering function (inelastic-std). Hollow circles: From the Kubo-transformed velocity correlation function (vv-kubo). Crosses: From the standard velocity correlation function (vv-std). Dot-dashed line: From the Kubo-transform of [the self-relaxation function ] by the RPMD method (RPMD-kubo). Hollow squares: From the Kubo-transformed velocity correlation function by the RPMD method (RPMD-vv-kubo).

The first two even moments of the incoherent relaxation function based on the incoherent dynamic structure factors shown in Fig. 2. Solid line: Exact result. Dashed line with solid circles: From the self-part of intermediate scattering function (inelastic-std). Hollow circles: From the Kubo-transformed velocity correlation function (vv-kubo). Crosses: From the standard velocity correlation function (vv-std). Dot-dashed line: From the Kubo-transform of [the self-relaxation function ] by the RPMD method (RPMD-kubo). Hollow squares: From the Kubo-transformed velocity correlation function by the RPMD method (RPMD-vv-kubo).

Wave-vector transfer accessible by the TOSCA-II experiment in backward scattering (dashed line) and forward scattering (solid line) as a function of the energy transfer parameter based on the conservation laws, Eqs. (2.3) and (D1).

Wave-vector transfer accessible by the TOSCA-II experiment in backward scattering (dashed line) and forward scattering (solid line) as a function of the energy transfer parameter based on the conservation laws, Eqs. (2.3) and (D1).

Comparison of the LSC-IVR simulations with the inelastic neutron scattering experiment results along two different kinematic lines in the plane: (a) Forward scattering and (b) backward scattering . Solid line: Experiment results. Solid squares: From the self-part of intermediate scattering function (inelastic-std). Hollow circles: From the Kubo-transformed velocity correlation function (vv-kubo). Crosses: From the standard velocity correlation function (vv-std).

Comparison of the LSC-IVR simulations with the inelastic neutron scattering experiment results along two different kinematic lines in the plane: (a) Forward scattering and (b) backward scattering . Solid line: Experiment results. Solid squares: From the self-part of intermediate scattering function (inelastic-std). Hollow circles: From the Kubo-transformed velocity correlation function (vv-kubo). Crosses: From the standard velocity correlation function (vv-std).

## Tables

Two sets of moments given by the three methods based on the LSC-IVR using the TGA as discussed in Sec. III. Those that can be analytically exact are marked with “✓”.

Two sets of moments given by the three methods based on the LSC-IVR using the TGA as discussed in Sec. III. Those that can be analytically exact are marked with “✓”.

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