Schrodinger equation Hamiltonian selection

The infrared transitions obey the following selection rules Ap — 1, and A./ = 1 or 0. The wave functions for the initial and final states obtained by solving the Schrodinger equation with the Hamiltonian of Eq. (1-261) or (1-265) can be used to compute the infrared absorption intensities for the complex. The infrared absorption coefficient J(J" ->./ ) for the transition J" J is proportional to,... 

We shall start from methods similar to that previously described, characterized by the use of the apparent surface charge (ASC) description of the electrostatic interaction term Ve/, passing then to consider other continuum methods, which use a different description of Ve/.To complete the exposition we shall introduce, where appropriate, methods not based on the solution of a Schrodinger equation, and hence not belonging to the category of continuum effective Hamiltonian methods. We shall pass then to a selection of methods based on mixed continuum-discrete representation of the solvent, to end up with the indication of some approaches based on a full discrete representation of the solvent. 

As one can see, for each set of parameters Z, k, L and d, one can select a in such a way that Eqs. (21) and (22) are fulfilled. Then, the resulting expectation values correspond to the variational minima and are equal to the appropriate exact eigenvalues of either Dirac or Schrodinger (or rather Levy-Leblond) Hamiltonian. However the corresponding functions do not fulfil the pertinent eigenvalue equations they are not eigenfunctions of these Hamiltonians. This example demonstrates that the value of the variational energy cannot be taken as a measure of the quality of the wavefunction, unless the appropriate relation between the components of the wavefunction is fulfilled [2]. 

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