Nonadiabatic dynamics equation derivation

Presumably the most straightforward approach to chemical dynamics in intense laser fields is to use the time-independent or time-dependent adiabatic states [352], which are the eigenstates of field-free or field-dependent Hamiltonian at given time points respectively, and solve the Schrodinger equation in a stepwise manner. However, when the laser field is approximately periodic, one can also use a set of field-dressed periodic states as an expansion basis. The set of quasi-static states in a periodic Hamiltonian is derived by a Floquet type analysis and is often referred to as the Floquet states [370]. Provided that the laser field is approximately periodic, advantages of using the latter basis set include (1) analysis and interpretation of the electron dynamics is clearer since the Floquet state population often vary slowly with the timescale of the pulse envelope and each Floquet state is characterized as a field-dressed quasi-stationary state, (2) under some moderate conditions, the nuclear dynamics can be approximated by mixed quantum-classical (MQC) nonadiabatic dynamics on the field-dressed PES. The latter point not only provides a powerful clue for interpretation of nuclear dynamics but also implies possible MQC formulation of intense field molecular dynamics. 

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be Just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrddinger equation are derived by employing the time-dependent variational principle (TDVP). 

With increasing system size, the implementation of ab initio electron wavepacket dynamics, such as the semiclassical Ehrenfest theory, using nuclear derivative coupling tends to be computationally more demanding because of the necessity of solving coupled perturbed equations. We therefore propose a useful treatment of nonadiabatic coupling, in which one can avoid the tedious coupled perturbed equations for the nuclear derivative of molecular orbitals and CSFs. 

The adiabatic states, hereinafter labeled by the superscript (A), are uniquely defined through equation (10), apart from arbitrary phase factors. The Bom-Oppenheimer approximation now consists in neglecting the dynamic couplings and T kl - eigenvalues Ek take the place of the matrix elements H kk as potential energy functions. The derivative matrix elements between adiabatic wavefunctions (the nonadiabatic couplings) obey the Hellman-Feynman-like formula ... 

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