Second-order perturbation transfer

Moriya (451) has extended the Anderson formalism for superexchange to include spin-orbit coupling in the perturbation Hamiltonian that appears in the transfer integral bij of equation 100. To first order, this leads to a second-order perturbation in the energy of the form... 

If the spins of the d and d electrons are antiparallel, the d electron can transfer to any level of Ui or in the excited states. The matrix element of the transition from to any one of is denoted by b, while that from to is /2b Thus the second-order perturbation of the energy is... 

In the first mechanism, a population transfer from LO phonon to acoustical phonons with half the frequency of the optical phonon and opposite wave vectors takes place. The problem is similar to the decay of a local mode. Thus, the lifetime of the optical mode should be similar to the lifetime of a local mode of the same frequency, because the strain field of a local mode has a similar character like the local mode. Estimations of this process can be made from second-order perturbation theory. In the case of population transfer, cubic terms are dominating in the vibrational potential. For the population transfer, energy and momentum conservation has to be fulfilled, which is the case for bulk material because of the overlap of the density of states of two acoustical phonons with that of the LO phonon. For pure population transfer, a linear increase of the LO half-width with the number n of the LO phonons is expected. 

The NBO method is particularly well suited to analyze charge transfer mechanisms between two interacting systems through the NBO second order perturbation analysis between the occupied orbitals of the base and the empty orbitals of the acid [36]. These localized hybrids are obtained as local block eigenvectors of the one-particle density matrix. 

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