Phase-space formulation of Nonadiabatic Quantum Processes

William A. Wassam, Jack H. Freed

A quantum stochastic theory for nonadiabatic processes in condensed phases and on surfaces is given. The present theory is primarily concerned with processes involving the fragmentation of two-particle systems and/or the ’’collision’’ of two particles with internal structure, i.e., electronic, spin, and nuclear degrees of freedom. A fully quantum mechanical joint energy space–quantum phase space master equation for two-particle systems is presented. A simpler form of this equation emerges in the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion. (This limit is equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2.) The simpler master equation is converted into a nonlinear quantum Fokker–Planck equation for nonadiabatic systems. Introducing simplifying assumptions, we obtain a quantun analog of the stochastic Liouville equation augmented with irreversible kinetic terms, which describes nonadiabatic transitions and the interruption of coherence in energy space at different points in quantum phase space. The final results are appropriately modified to incorporate the influence of applied time-dependent electric and magnetic fields. The present work provides a formal theoretical framework for examining the role of quantum effects that are neglected in ’’semiclassical’’ kinetic and ’’diffusional’’ theories for nonadiabatic processes.

Charis Anastopoulos

Anatoli Polkovnikov

Youhao Shang, Xiangsong Cheng, Jian Liu

We have recently developed the \textit{constraint} coordinate-momentum \textit{phase space} (CPS) formulation for finite-state quantum systems. It has been implemented for the electronic subsystem in nonadiabatic transition dynamics to develop practical trajectory-based approaches. In the generalized CPS formulation for the mapping Hamiltonian of the classical mapping model with commutator variables (CMMcv) method [\textit{J. Phys. Chem. A} \textbf{2021}, 125, 6845-6863], each {connected} component of the generalized CPS is the \textit{complex Stiefel manifold} labeled by the eigenvalue set of the mapping kernel. Such a phase space structure allows for exact trajectory-based dynamics for pure discrete (electronic) degrees of freedom (DOFs), where the equations of motion of each trajectory are isomorphic to the time-dependent Schrödinger equation. We employ covariant kernels {within the generalized CPS framework} to develop two approaches that naturally yield exact evaluation of time correlation functions (TCFs) for pure discrete (electronic) DOFs. In addition, we briefly discuss the phase space mapping formalisms where the contribution of each trajectory to the integral expression of the {TCF} of population dynamics is strictly positive semi-definite. The generalized CPS formulation also indicates that the equations of motion in phase space mapping model I of our previous work [\textit{J. Chem. Phys.} \textbf{2016}, 145, 204105; \textbf{2017}, 146, 024110; \textbf{2019}, 151, 024105] lead to a complex Stiefel manifold $\mathrm{U}(F)/\mathrm{U}(F-2)$. It is expected that the generalized CPS formulation has implications for simulations of both nonadiabatic transition dynamics and many-body quantum dynamics for spins/bosons/fermions.

Youhao Shang, Xiangsong Cheng, Jian Liu

, 21, 3775-3813]. It is expected that the generalized CPS formulation will have more implications for studying nonadiabatic transition dynamics and many-body quantum dynamics.

Kazuma Nagao