Mixed quantum-classical Hamiltonian in an optical field

1 Mixed quantum-classical Hamiltonian in an optical field 

We resume with a general theory of mixed quantum-classical representation of dynamics of a system composed of fast and slow subsystems, in which the former and latter are, respectively, treated quantum and classical mechanically. In the present case, the fast and slow subsystems happen to be electronic and nuclear parts, respectively. 

We start from the following full quantum mechanical molecular Hamiltonian in an optical field having the vector potential A, 

In the present theoretical scheme, the Hamiltonian of Eq. (8.40), which is represented in configuration space r, R, is re-expressed in the electronic Hilbert space and nuclear configuration space, whose basis is h/(R)) R) such that [425, 492] 

The tilde over H(R, P, elec. A) indicates the mixed quantum-classical representation. 

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