Nondegenerate perturbation theory

In this context equations (50) and (53) can be considered forming a completely general perturbation theory for nondegenerate systems, although a recent development permits to extend the formalism to degenerate states [lej. 

For the energy, one can make use of first-order perturbation theory for a nondegenerate state. Thus, if the state P is perturbed to the state P = P + P by the one-electron perturbation AV = JT<5v(r,-), the energy change to first order is [1]... 

In the absence of an external magnetic field, orbitally nondegenerate levels with spin multiplicity greater than 2 split due to direct electron spin—spin coupling (in first order) and spin—orbit coupling (in second and higher orders of perturbation theory). This phenomenon is called zero-field splitting (ZFS). The SH that describes this phenomenon can be formulated in... 

Consider a dn configuration present in a crystal field that leaves the ground state nondegenerate except for spin. The ground state then consists of (25+ l)-spin states and the effect of the spin-orbit interaction plus the magnetic field can be computed using first- and second-order perturbation theory. If we take as the perturbation operator... 

We shall restrict the discussion here to nondegenerate perturbation theory for the ground-state case. However, there is no formal difficulty extending the method to degenerate cases if we wish to treat excited states (Cizek and Vrscay, 1982 Adams et al, 1980). 

Quasi-degeneracy Effects.—The convergence properties of the non-degenerate formulation of the many-body perturbation theory deteriorate when quasidegeneracy is present in the reference spectrum. In view of its simplicity, however, there is considerable interest in exploring the range of applicability of the nondegenerate formalism. 

In the perturbation theory for degenerate states the resonant hyperpolarizability is determined by the tensor part of polarizability [9] and may be extracted out of the fourth-order terms self-consistently in the case of nondegenerate perturbation theory the resonant part appears for separate sublevels of an atomic multiplet. The numerical results are listed in Table 2. 

If there is no degeneracy in the MO energies of the unperturbed system, the nondegenerate perturbation theory leads to the results. 

The Van Vleck transformation is an approximate block diagonalization procedure. It allows one, in effect, to throw away an infinite number of unimportant (class 2) basis functions after taking into account, through second-order nondegenerate perturbation theory, the effect of these ignorable functions on the finite number of important (class 1) functions. The Van Vleck transformation, T, is defined by... 

Perturbation theory is an extremely useful analytic tool. It is almost always possible to treat a narrow range of. /-values in a multistate interaction problem by exactly diagonalizing a two-level problem after correcting, by nondegenerate perturbation theory or a Van Vleck transformation, for the effects of other nearby perturbers. Such a procedure can enable one to test for the sensitivity of the data set to the value of a specific unknown parameter. 

The mixing coefficient, a, is given by nondegenerate perturbation theory as... 

Treatment of Two-State Interaction by Nondegenerate Perturbation Theory.674... 

Nondegenerate perturbation theory enables algebraically transparent analysis of the interactions between any number of decaying quasi-eigenstates pro-... 

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