Quantum dynamics in condensed phases via extended modes and exact interaction propagator relations

Abstract

This paper presents a new approach to the study of quantum dynamics in condensed phases. The methodology is comprised of two main components. First, a formally exact method is described which allows the description of the liquid as a collection of coupled (through kinetic and potential coupling) harmonic modes. The modes are related to the Fourier modes of the component particle densities. Once the modes have been defined, a canonical transformation from the standard classical interparticle Hamilton function describes a new Hamilton function, which is exactly equivalent and defined on these harmonic coordinates. The final step in this section is the transformation of this Hamilton function into a quantum Hamiltonian operator. The second step in the process is the derivation of a new quantum mechanical evolution operator which is exact and allows the correction from a reference evolution operator, which is formed by adiabatic evolution on an approximate potential. A particular approximate potential which we suggest will be useful, is the collection of harmonic modes given in the Zwanzig Hamiltonian, weighted by the spectral density. Application of the reference interaction propagator methodology can then correct the approximate adiabatic evolution operator based on the approximate potential to the exact Hamiltonian of Fourier modes described above. A test problem of a double well nonlinearly coupled to a harmonic oscillator shows that the methodology obtains rapid numerical convergence. The paper closes with a description of how the methodology would be applied to a many-dimensional (hundreds of degrees of freedom) picture of reaction in a condensed phase.