Review of Feynman's Path Integral in Quantum Statistics: From the Molecular Schrödinger Equation to Kleinert's Variational Perturbation Theory (Communications in Computational Physics, 2020)
Feynman's path integral reformulates the quantum Schrödinger differentialequation to be an integral equation. It has been being widely used to compute internuclearquantum-statistical effects on many-body molecular systems. In this Review,the molecular Schrödinger equation will first be introduced, together with the Born-Oppenheimerapproximation that decouples electronic and internuclear motions. Someeffective semiclassical potentials, e.g., centroid potential, which are all formulated interms of Feynman's path integral, will be discussed and compared. These semiclassicalpotentials can be used to directly calculate the quantum canonical partition functionwithout individual Schrödinger's energy eigenvalues. As a result, path integrationsare conventionally performed with Monte Carlo and molecular dynamics samplingtechniques. To complement these techniques, we will examine how Kleinert's variationalperturbation (KP) theory can provide a complete theoretical foundation for developingnon-sampling/non-stochastic methods to systematically calculate centroidpotential. To enable the powerful KP theory to be practical for many-body molecularsystems, we have proposed a new path-integral method: automated integration-freepath-integral (AIF-PI) method. Due to the integration-free and computationallyinexpensive characteristics of our AIF-PI method, we have used it to perform ab initiopath-integral calculations of kinetic isotope effects on proton-transfer and RNA-relatedphosphoryl-transfer chemical reactions. The computational procedure of using ourAIF-PI method, along with the features of our new centroid path-integral theory at theminimum of the absolute-zero energy (AMAZE), are also highlighted in this review.