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arXiv:1611.07290 · The Journal of Chemical Physics

Isomorphism between the multi-state Hamiltonian and the second-quantized many-electron Hamiltonian with only 1-electron interactions

Jian Liu

Published Jan 13, 2017• 23 citations
arXivHTMLPDF

Abstract

We introduce the isomorphism between an multi-state Hamiltonian and the second-quantized many-electron Hamiltonian (with only 1-electron interactions). This suggests that all methods developed for the former can be employed for the latter, and vice versa. The resonant level (Landauer) model for nonequilibrium quantum transport is used as a proof-of-concept example. Such as the classical mapping models for the multi-state Hamiltonian proposed in our previous work [J. Liu, J. Chem. Phys. 145, 204105 (2016)] lead to exact results for this model problem. We further demonstrate how these methods can also be applied to the second-quantized many-electron Hamiltonian even when 2-electron interactions are included.

DOI
10.1063/1.4973708

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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 U(F)/U(F−2)\mathrm{U}(F)/\mathrm{U}(F-2)U(F)/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.