Quasiclassical Correlation Functions from the Wigner Density Using the Stability Matrix

Abstract

Accounting for zero-point energy in the initial conditions of classical trajectory calculations of time correlation functions requires sampling from a quantized phase space distribution, which is often chosen as the Weyl–Wigner transform of a thermalized operator. The numerical construction of the latter and its use as a sampling function can be challenging. We show that the operator dependence of the phase space distribution can be transferred to the dynamics, allowing sampling from the simpler Wigner phase space density. The method involves augmenting the classical equations of motion with additional differential equations for elements of the stability matrix. We also propose a local harmonic approximation for the dynamical derivatives, which significantly reduces the computational cost required to obtain correlation functions of nonlinear operators. We illustrate the method with application to linear and nonlinear correlation functions of model Hamiltonians. While the local harmonic approximation is not always successful in predicting nonlinear correlation functions of one degree of freedom, it quantitatively captures the full quasiclassical results for systems in contact with dissipative environments.

Vibrational spectrum of a 1D oscillator: The quantum, the Wigner, and the classical ways

“…Then, we propagate each of these sampled points using CM and calculate the SC dipole autocorrelation function. Recently Bose and Makri [ 42 ] have suggested using the Weyl transform [ 23 ] of the operator, as an SC distribution, . here, refers to the ground state density operator.…”

“…Then, we propagate each of these sampled points using CM and calculate the SC dipole autocorrelation function. Recently Bose and Makri [42] have suggested using the Weyl transform [23] of the b ρ 0 b μ operator, as an SC distribution, P Wig ρ 0 ,μ q 0 , p 0 ð Þ.…”

Quantum–Classical Path Integral Simulation of Excess Proton Dynamics in a Water Dimer Embedded in the Gramicidin Channel

“…Quantization of the phase-space distribution can be rigorously done using the Wigner transform of the quantum mechanical density. A simple, approximate procedure based on classical adiabatic switching (ASW) is available for this purpose, , and a fully quantum mechanical method based on the imaginary-time path integral representation combined with information-guided noise reduction (IGNoR) offers a highly accurate alternative . However, the motion of the water hydrogens is coupled to those of the gramicidin atoms, making the use of the ASW method a demanding task.…”

A Celebration of Women in Computational Chemistry

“…This is achieved by accounting for the first operator in the dynamics and using properties of the stability matrix to obtain the required derivatives. 12 All these findings undoubtedly have potential important implications for a variety of biomolecular modeling applications.…”

“…In another article, Bose and Makri describe a procedure for proper inclusion of zero-point energy effects of time-correlation functions in classical trajectory calculations without sacrificing accuracy, which involves the simple Wigner phase space distribution. This is achieved by accounting for the first operator in the dynamics and using properties of the stability matrix to obtain the required derivatives . All these findings undoubtedly have potential important implications for a variety of biomolecular modeling applications.…”