Molecular Hamiltonian, complex scaling

In order to appreciate the fine points in this analysis, we therefore return to the domain issues, i.e. how to define the operator and the basis functions so that the scaling operation above becomes meaningful. Following Balslev and Combes [3], we introduce the N-body (molecular) Hamiltonian as H = T + V, where T is the kinetic energy operator and V is the (dilatation analytic) interaction potential (expressed as sum of two-body potentials Vy bounded relative Ty = Ay, where the indices i and j refers to particles i and j respectively). As a first crucial point we realize that the complex scaling transformation is unbounded, which necessitates a restriction of the domain of H note that H is normally bounded from below. Hence we need to specify the domain of H as... 

A complete description of the dynamics of any molecular system is contained in the Hamiltonian H, which is the energy operator in quantum mechanics or the energy function in classical mechanics. In general, the Hamiltonian is a function of the electronic and nuclear degrees of freedom, as is the description of the system dynamics. This complex problem simplifies through the adoption of the Born-Oppenheimer approximation, which is the assumption that nuclear and electronic motion are independent due to their substantially different time scales and masses. This assumption allows one to first solve for the dynamics of the electrons and then obtain the forces experienced by the nuclei as determined by this fixed-electron configuration. 

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