Perturbation method degenerate case

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). 

The degenerate perturbation method described in Sec. 3 assumes/requires that the degeneracy be lifted at first order. In this case, first order means that displacements exclusively in the seam space, zeroth order in displacements in the branching space [Elqs. (58)-(59)] do not lift the degeneracy. In a triatomic system, the seam would locally be a straight line perpendicular to the g h plane, and globally the seam would be approximated by a... 

The perturbation method as described in the previous section does not apply if several wave functions correspond to the same zero-order energy (the degenerate case). For example, the zero-order orbital energies of the 2s and 2p hydrogen-like orbitals are all equal, so that all of the states of the (1 X2 ) and (U)(2/ ) helium configurations have the same energy in zero order. A version of the perturbation method has been developed to handle this case. We will describe this method only briefly and present some results for some excited states of the helium atom. There is additional information in Appendix G. 

In the degenerate case there is no guarantee that the wave functions that we obtain with the zero-order Schrodinger equation are in one-to-one correspondence with the correct wave functions. If not, the smooth dependence on the parameter A. depicted in Figure 19.2 will not occur. The first task of the degenerate perturbation method is to find the zero-order wave functions that are in one-to-one correspondence with the exact wave functions. We call them the correct zero-order wave functions. 

In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51], For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. 

Conditions in Strong Fields.—It will be necessary to discuss this case, though only our method of approach is new, while the results were known from Bohr s and Kramers work. If the electric perturbation dominates over the relativistic, our process of successive approximations must start from considering the electric terms. As we have pointed out, the most important one among them is the degenerate term (11). Our second step will be, therefore, the consideration of the Hamiltonian equation... 

As we know from theorem 3, due to Poincare, in the general case of a non-degenerate unperturbed Hamiltonian Ho(p) the construction of the normal form can not be completely performed in a consistent manner. However, we are still allowed to perfom a suitable construction in domains where some resonances may be excluded, provided we perform a Fourier cutoff on the perturbation. We shall discuss this method in connection with Nekhoroshev s theorem. 

Theory can now provide much valuable guidance and interpretive assistance to the mechanistic photochemist, and the evaluation of spin-orbit coupling matrix elements has become relatively routine. For the fairly large molecules of common interest, the level of calculation cannot be very high. In molecides composed of light atoms, the use of effective charges is, however, probably best avoided, and a case is pointed out in which its results are incorrect. It seems that the mean-field approximation is a superior way to simplify the computational effort. The use of at least a double zeta basis set with a method of wave function computation that includes electron correlation, such as CASSCF, appears to be imperative even for calculations that are meant to provide only semiquantitative results. The once-prevalent degenerate perturbation theory is now obsolete for quantitative work but will presumably remain in use for qualitative interpretations. 

This regular precession is, in general, possible only for certain initial conditions. We shall show later (by the method of secular perturbations, 18) that, for weak fields, the spatial quantisation holds in general for every motion the only exceptions to this arc certain cases of double degeneration (e.g. hydrogen atom in an electric field, rf. 35). 

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