#### Document Type

Journal Article

#### Department/Unit

Department of Physics

#### Title

Review of Feynman’s path integral in quantum statistics: From the molecular schrödinger equation to kleinert

#### Abstract

Feynman's path integral reformulates the quantum Schrodinger differential equation to be an integral equation. It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems. In this Review, the molecular Schrodinger equation will first be introduced, together with the Born-Oppenheimer approximation that decouples electronic and internuclear motions. Some effective semiclassical potentials, e.g., centroid potential, which are all formulated in terms of Feynman's path integral, will be discussed and compared. These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrodinger's energy eigenvalues. As a result, path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques. To complement these techniques, we will examine how Kleinert's variational perturbation (KP) theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential. To enable the powerful KP theory to be practical for many-body molecular systems, we have proposed a new path-integral method: automated integration-free path-integral (AIF-PI) method. Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method, we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions. The computational procedure of using our AIF-PI method, along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy (AMAZE), are also highlighted in this review.

#### Publication Year

2014

#### Journal Title

Communications in Computational Physics

#### Volume number

15

#### Issue number

4

#### Publisher

Global Science Press

#### First Page (page number)

853

#### Last Page (page number)

894

#### Referreed

1

#### DOI

10.4208/cicp.140313.070513s

#### ISSN (print)

18152406

#### Link to Publisher’s Edition

http://dx.doi.org/10.4208/cicp.140313.070513s

#### Copyright

© 2014 Global-Science Press

#### Keywords

Feynman's path integral, Kleinert's variational perturbation theory, Schrodinger's equation, quantum statistics, centroid potential energy, quantum tunneling, zero-point energy, isotope effects

#### ISSN (electronic)

19917120

#### Citation

Wong, Kin Yiu.
"Review of Feynman’s path integral in quantum statistics: From the molecular schrödinger equation to kleinert."
*Communications in Computational Physics*
15.4
(2014): 853-894.