Perturbation nondegenerate

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). 

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... 

The approximate block diagonalization produced by the Van Vleck transformation may be understood by first illustrating how nondegenerate perturbation... 

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-... 

The molecule also chooses the physically appropriate basis set. Sometimes the molecule suggests to the experimentalist one basis set to describe an exper-imentally-realizable pluck and a different basis set to describe the post-pluck dynamics. No matter what basis set the molecule chooses, there always will be some pairs of basis states connected by an off-diagonal matrix element of H(b larger than their difference in zero-order energies. Under such circumstances, nondegenerate perturbation theory fails and a quasi-degenerate block of the Heff... 

There is a profound but subtle difference between an exact vibrational H and a polyad Heff fit model. Many of the off-diagonal matrix elements in the exact H couple states that satisfy the convergence criterion for nondegenerate perturbation theory... 

Nondegenerate Perturbation Theory. Let be the wave function of some particular unperturbed nondegenerate level with energy Let >j/ be the perturbed wave function into which is converted when the perturbation is applied. From (9.1) and (9.7), the Schrodinger equation for the perturbed state is... 

J26 True or false (a) Every linear combination of solutions of the time-dependent Schrodinger equation is a solution of this equation, (b) Every linear comlnnation of solutions of the time-independent Schrddinger equation is a solution of this equation, (c) Ihe nondegenerate perturbation-theory formula applies only to the ground state, (d) The exact... 

The above analysis is restricted to the framework of nondegenerate perturbation theory here in order to focus on the essentials of our discussion of the transition from four- to two-component formulations, but emphasizes that all findings hold equally for the degenerate case. 

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