Constraint coordinate-momentum phase space formulations for finite-state quantum systems: The relation between...

We have recently developed the constraint coordinate-momentum 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 [J. Phys. Chem. A 2021, 125, 6845-6863, which followed J. Chem. Phys. 2016, 145, 204105 and J. Chem. Phys. 2019, 151, 024105], each connected component of the generalized CPS is the 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 formulation 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 representations 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 mapping Hamiltonian in phase space mapping model I of our previous work [J. Chem. Phys. 2016, 145, 204105] leads to a complex Stiefel manifold $U\left(F\right)/U(F-2)$ . The phase space expressions (of TCFs) proposed in this paper are extensively tested in our subsequent work on nonadiabatic dynamics [J. Chem. Theory Comput. 2025, 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.