^{1}and William H. Miller

^{1,a)}

### Abstract

A simple model is presented for treating local imaginary frequencies that are important in the study of quantum effects in chemical reactions and various dynamical processes in molecular liquids. It significantly extends the range of accuracy of conventional local harmonic approximations (LHAs) used in the linearized semiclassical initial value representation/classical Wigner approximation for real time correlation functions. The key idea is realizing that a local Gaussian approximation (LGA) for the momentum distribution (from the Wigner function involving the Boltzmann operator) can be a good approximation even when a LHA for the potential energy surface fails. The model is applied here to two examples where imaginary frequencies play a significant role: the chemical reaction rate for a linear model of the reaction and an analogous asymmetric barrier—a case where the imaginary frequency of the barrier dominates the process—and for momentum autocorrelation functions in liquid para-hydrogen at two thermal state points (25 and 14 K under nearly zero external pressure). We also generalize the LGA model to the Feynman–Kleinert approximation.

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) and the Lawrencium computational cluster resource provided by the IT Division at the Lawrence Berkeley National Laboratory.

I. INTRODUCTION

II. THEORY AND METHODOLOGY

A. Linearized semiclassical initial value representation

B. Local Gaussian approximation

C. Generalization to multidimensional systems

III. APPLICATIONS

A. Thermal reaction rate from the flux-side correlation function

B. Thermal reaction rate for the 1D Eckart barrier for

C. Thermal reaction rate for the asymmetric Eckart barrier

D. Liquid para-hydrogen

IV. CONCLUSIONS

### Key Topics

- Correlation functions
- 38.0
- Chemical reactions
- 18.0
- Thermogravimetric analysis
- 18.0
- Tunneling
- 14.0
- Boltzmann equations
- 12.0

## Figures

(a) Eckart barrier [given by Eq. (2.4)]. (b) Second derivative of the Eckart barrier.

(a) Eckart barrier [given by Eq. (2.4)]. (b) Second derivative of the Eckart barrier.

Off-diagonal elements at the top of the Eckart barrier for the Eckart barrier at (a) and (b) . The Fourier transform of produces the local momentum distribution . (c) Normalized local momentum distribution at . [Solid line: Exact results. Dotted-dashed line: LGA results.] (d) Same as (c) but at a lower temperature .

Off-diagonal elements at the top of the Eckart barrier for the Eckart barrier at (a) and (b) . The Fourier transform of produces the local momentum distribution . (c) Normalized local momentum distribution at . [Solid line: Exact results. Dotted-dashed line: LGA results.] (d) Same as (c) but at a lower temperature .

Quantum correction factor . As in the conventional way, imaginary frequencies are shown as on the negative axis (i.e., shown as ). Solid line: LHA. Dashed line: LGA. Note that the imaginary frequency for is where the LHA breaks down.

Quantum correction factor . As in the conventional way, imaginary frequencies are shown as on the negative axis (i.e., shown as ). Solid line: LHA. Dashed line: LGA. Note that the imaginary frequency for is where the LHA breaks down.

Gaussian width parameters of the local momentum distribution generated from the Fourier transform of . Imaginary frequencies are plotted on the negative axis. Solid line: Exact results. Solid circles: LGA results. Dashed line with hollow squares: LHA results. Note that for the imaginary frequency is where the LHA breaks down.

Gaussian width parameters of the local momentum distribution generated from the Fourier transform of . Imaginary frequencies are plotted on the negative axis. Solid line: Exact results. Solid circles: LGA results. Dashed line with hollow squares: LHA results. Note that for the imaginary frequency is where the LHA breaks down.

(a) An Arrhenius plot of the thermal rate constant for the 1D Eckart barrier. Solid line: Exact quantum results. Dotted line with solid circles: LSC-IVR results using the LGA. Dashed line: Classical results. Hollow squares: LSC-IVR results using the LHA. (b) Tunneling correction factors for the 1D Eckart barrier. Solid line: Exact quantum results. Solid circles: LSC-IVR results with the LGA. Hollow squares: LSC-IVR results with the LHA. Solid triangles: LSC-IVR results with the exact Wigner function (from Ref. 8). (c) Relative errors of tunneling correction factors or thermal rate constants. Solid line with solid circles: LSC-IVR results with the LGA. Dotted line with hollow triangles: LSC-IVR results with the exact Wigner function (from Ref. 8). [Since most LSC-IVR results with the LHA deviate from the exact results by a few orders as shown in (a) and (b), their relative errors are not demonstrated here.]

(a) An Arrhenius plot of the thermal rate constant for the 1D Eckart barrier. Solid line: Exact quantum results. Dotted line with solid circles: LSC-IVR results using the LGA. Dashed line: Classical results. Hollow squares: LSC-IVR results using the LHA. (b) Tunneling correction factors for the 1D Eckart barrier. Solid line: Exact quantum results. Solid circles: LSC-IVR results with the LGA. Hollow squares: LSC-IVR results with the LHA. Solid triangles: LSC-IVR results with the exact Wigner function (from Ref. 8). (c) Relative errors of tunneling correction factors or thermal rate constants. Solid line with solid circles: LSC-IVR results with the LGA. Dotted line with hollow triangles: LSC-IVR results with the exact Wigner function (from Ref. 8). [Since most LSC-IVR results with the LHA deviate from the exact results by a few orders as shown in (a) and (b), their relative errors are not demonstrated here.]

(a) Asymmetric Eckart barrier [given by Eq. (3.15)]. (b) Second derivative of the asymmetric Eckart barrier.

(a) Asymmetric Eckart barrier [given by Eq. (3.15)]. (b) Second derivative of the asymmetric Eckart barrier.

Tunneling correction factors for the 1D asymmetric Eckart barrier with respect to different dividing surfaces. (a) Temperature . (b) Temperature . (c) Temperature .

Tunneling correction factors for the 1D asymmetric Eckart barrier with respect to different dividing surfaces. (a) Temperature . (b) Temperature . (c) Temperature .

Comparison between LSC-IVR (LGA) and its QTST counterpart for tunneling correction factors for the 1D asymmetric Eckart barrier with respect to different dividing surfaces. (a) Temperature . (b) Temperature . (c) Temperature .

Comparison between LSC-IVR (LGA) and its QTST counterpart for tunneling correction factors for the 1D asymmetric Eckart barrier with respect to different dividing surfaces. (a) Temperature . (b) Temperature . (c) Temperature .

(a) Tunneling correction factors for the 1D asymmetric Eckart barrier at different temperatures. Solid line: Exact quantum results. Solid circles: LSC-IVR results with the LGA. (b) Relative errors of tunneling correction factors or thermal rate constants.

(a) Tunneling correction factors for the 1D asymmetric Eckart barrier at different temperatures. Solid line: Exact quantum results. Solid circles: LSC-IVR results with the LGA. (b) Relative errors of tunneling correction factors or thermal rate constants.

Normalized density of local frequencies from the path integral calculations for the liquid para- at the state points (a) and and (b) and . Dotted lines indicate the imaginary frequency .

Normalized density of local frequencies from the path integral calculations for the liquid para- at the state points (a) and and (b) and . Dotted lines indicate the imaginary frequency .

Kubo-transformed momentum autocorrelation functions (divided by ) based on the LSC-IVR formulation for the liquid para- at the state points (a) and and (b) and . Comparisons between the LHA and the LGA. LHA2 represents the LHA with only real frequencies.

Kubo-transformed momentum autocorrelation functions (divided by ) based on the LSC-IVR formulation for the liquid para- at the state points (a) and and (b) and . Comparisons between the LHA and the LGA. LHA2 represents the LHA with only real frequencies.

Kubo-transformed momentum autocorrelation functions (divided by ) based on the LSC-IVR formulation for the liquid para- at the state points (a) and and (b) and . Comparisons between the TGA (and its MEAC-corrected version) to the LGA (and its MEAC-corrected version).

Kubo-transformed momentum autocorrelation functions (divided by ) based on the LSC-IVR formulation for the liquid para- at the state points (a) and and (b) and . Comparisons between the TGA (and its MEAC-corrected version) to the LGA (and its MEAC-corrected version).

Probability distribution functions of path integral beads with the centroid fixed at the top of the 1D Eckart barrier . (a) and (b) . Comparisons between the FKA and the PIMC.

Probability distribution functions of path integral beads with the centroid fixed at the top of the 1D Eckart barrier . (a) and (b) . Comparisons between the FKA and the PIMC.

## Tables

Information entropies in the MEAC procedure for different priors for liquid para-hydrogen at and and and under nearly zero extent pressure.

Information entropies in the MEAC procedure for different priors for liquid para-hydrogen at and and and under nearly zero extent pressure.

Diffusion contants for liquid para-hydrogen at and under nearly zero extent pressure.

Diffusion contants for liquid para-hydrogen at and under nearly zero extent pressure.

Diffusion contants for liquid para-hydrogen at and under nearly zero extent pressure.

Diffusion contants for liquid para-hydrogen at and under nearly zero extent pressure.

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