Multi-State Dynamics

We have proposed a generic framework to construct mapping models in the Cartesian phase space for the multi-state Hamiltonian operator [J. Chem. Phys. 146, 024110 (2017)]. After mapping an F-dimensional Hamiltonian onto an F+1 dimensional space as suggested in Phys. Rev. Lett. 78, 578 (1997), creation and annihilation operators are defined such that the F+1 dimensional space is complete for any combined excitation. Commutation and anti-commutation relations are then naturally derived, which show that the underlying degrees of freedom are neither bosons nor fermions. The framework provides three mapping approaches: (a) from the analogy with the quantum-classical correspondence for two non-commutable operators; (b) from the analogy with the classical angular momentum; and (c) from the analogy with the classical vector. While approach (a) offers a novel means to derive the well-known Meyer-Miller Hamiltonian model [J. Chem. Phys. 70, 3214 (1979)], approaches (b) and (c) set the scene for developing other exact Hamiltonian formulations and their semiclassical/quasiclassical counterparts for multi-state dynamics/nonadiabatic dynamics.

We have further introduced the isomorphism between the multi-state Hamiltonian and the second-quantized many-electron Hamiltonian (with only 1-electron interactions) and demonstrated that all methods developed for the former can be used for the latter, and vice versa. It is shown that the recently proposed mapping model for the many-electron Hamiltonian [J. Chem. Phys. 137, 154107 (2012)] is not unique but one of a few exact mapping models naturally derived in the framework.

The relevant peer-reviewed papers are

1. "A unified theoretical framework for mapping models for the multi-state Hamiltonian"

   Jian Liu*

   Journal of Chemical Physics, 145, 204105 (2016)

   http://dx.doi.org/10.1063/1.4967815

   Note: In the article we formulate a generic framework to construct mapping models in the Cartesian phase space for the multi-state Hamiltonian operator, which sets the scene for developing a series of exact Hamiltonian formulations and their semiclassical/quasiclassical counterparts for multi-state dynamics/nonadiabatic dynamics.

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

   Jian Liu*

   Journal of Chemical Physics, 146, 024110 (2017)

   http://dx.doi.org/10.1063/1.497370

   Note: In the article we introduce the isomorphism between the multi-state Hamiltonian and the second-quantized many-electron Hamiltonian (with only 1-electron interactions) and demonstrated that all methods developed for the former can be used for the latter, and vice versa.