Mixed quantum-classical Liouville molecular dynamics without momentum jump

Abstract

Exact and asymptotic solutions of the mixed quantum-classical Liouville equationAn alternative Liouville formulation of mixed quantum-classical dynamics outlined recently ͓K. Ando, Chem. Phys. Lett. 360, 240 ͑2002͔͒ is expanded in detail by taking an explicit account of the parametric dependence of the electronic ͑adiabatic͒ basis on the nuclear coordinates. As a consequence of the different operational order of the partial Wigner transformation for the nuclear coordinates and the calculation of the matrix elements in the adiabatic electronic basis, the present formula differs from the previously proposed one, slightly in the appearance but significantly in the treatment of nonadiabatic transitions in the trajectory implementation in that the former does not contain the ''off-diagonal Hellmann-Feynman forces'' representing the so-called ''momentum-jump'' associated with the nonadiabatic transitions. Because of this, the present formula is free from the numerical instability intrinsically coming from the momentum-jump operation at around the classical turning points of the nuclear motion. It is also shown that the density matrices from the two approaches coincide when the electronic basis is independent of the nuclear coordinates (R), and hence the momentum-jump approximation stems from the R-dependence of the adiabatic electronic basis. Improved stability and comparable to better reproduction of the quantum reference calculations are demonstrated by applications to one and three dimensional spin-boson models and a two-state three-mode model of the S 2 →S 1 internal conversion of pyrazine. Also discussed is the importance of electronic coherence for the proper treatment of nonadiabatic transition rates which is naturally described by the Liouville methods compared to the conventional independent trajectory approaches.

Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems

“…79,185,199,200) Equations in phase space have been formulated to treat such problems. [201][202][203][204][205][206] Here, we present the multi-state quantum Fokker-Planck equation, which can be applied to both optical and nonadiabatic transition problems taking into account the noise correlation and temperature effects. We consider the Hamiltonian expressed aŝ Note that the present equations hold even if U jk ðqÞ is timedependent, as is the case of optical spectroscopy.…”

Measuring nonadiabaticity of molecular quantum dynamics with quantum fidelity and with its efficient semiclassical approximation

“…Until now all operations have been exact and Eqs. (19) and (20) are equivalent to the original mixed quantum classical Liouville equation (13). Now we will make the locally mean field approximation: Specifically, once in the Lagrangian reference frame, all terms in both Eqs.…”

“…(20) corresponds to the right-hand side of Eq. (19) and is responsible for maintaining Tr e ρ e (X, t) = 1 during the (nonapproximated) MQC dynamics. Until now all operations have been exact and Eqs.…”

“…Now we will make the locally mean field approximation: Specifically, once in the Lagrangian reference frame, all terms in both Eqs. (19) and (20) containing phase space derivatives of ρ(X, t) or ρ e (X, t) are neglected to obtain the approximate locally mean-field equations of motion. The resulting equation for ρ(X, t) in the Lagrangian reference frame is simply…”

Evaluation of the importance of spin-orbit couplings in the nonadiabatic quantum dynamics with quantum fidelity and with its efficient “on-the-fly” ab initio semiclassical approximation

“…51). In terms of the computational cost per trajectory, the MSDR thus falls into the category of methods in which nuclei are treated classically such as surface hopping methods, [52][53][54] and other methods based on the mixed quantum-classical Liouville equation, [55][56][57][58][59][60][61][62][63] or methods obtained by linearization of the path integral expression for the quantum propagator. 64,65 (Costs of current implementations of the MSDR are essentially the costs of a mean field or surface hopping dynamics.…”