Degeneracy formal

Again the Ai term is split into three doublets each of which preserves Ms degeneracy. Formally the results are the same as those obtained previously. Here too we may define a zero field splitting parameter, D, given by... 

In Hartree-Fock (HF) theory, the energy of a given orbital is lower when the orbital is occupied than when it is vacant (due to the formal self-energy in the former case), and the degeneracy is broken. Thus, perturbative F1F expressions such as Eq. (2.7) often have wider numerical validity than would be anticipated in naive MO theory. 

Table B.l summarizes the ground-state electron configuration and formal APH indices (turn number t, angular number l-n) for each known element, together with atomic number (Z) and relative atomic mass). As shown by the asterisks in the Anal column, 20 elements exhibit anomalous electron configurations (including two that are doubly anomalous - Pd and Th), compared with idealized t/l-n APH descriptors. These are particularly concentrated in the first d-block series, as well as among the early actinides. Such anomalies are indicative of configurational near-degeneracies that may require sophisticated multi-reference approximation methods for accurate description.

The configurational entropy term, given by the degeneracy, gc, is included in AfG but not in AfGc. Let us assume the existence of two compounds with different formal oxidation states for the B atom, ABO3 and ABO2.5. The two compounds have the same (perovskite-type) structure and the non-stoichiometric phase... 

For instance it was shown that in the case of no degeneracy the expansion is correct up to the order k" if Rmfunction space), and it is quite natural to expect that in such a case the expansion of the eigenvalue be valid up to the order tc. Actually this was the case with Ex. 4 if we put < 01=1 const. (1 + ar)-1 (see (6. 5)). But the conditions hitherto given are not able to comprise such a case. 

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. 

Obviously, when severe near degeneracy is present in the reference spectrum, a full quasi-degenerate formulation of the perturbation series should be employed. However, there is considerable value in applying the much simpler non-degenerate formalism to as wide a range of problems as possible. 

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