Perturbation degenerate

Let us consider an intermolecular perturbation where an orbital of the fragment A is degenerate with an orbital xjr°g of the fragment 6. Two very simple examples of this might be the assembly of the H2 molecule from two H atoms or that of ethane 

The second-order energy corrections resulting from these orbitals are given by equation 3.33. 

Finally, we briefly comment on those cases in which neither the degenerate nor the nondegenerate perturbation treatment is quite satisfectory [5]. To simplify our discussion, let us consider only the lower lying orbital and its ener gy e, that result from two interacting orbitals and For a nondegenerate interaction with 

These expressions are close to the exact ones if equation 3.36 is satisfied, that is, 

they are appropriate when the extent of perturbation is small in magnitude compared with the energy difference between the unperturbed levels. For a degenerate interaction with = e, the expressions for e, and are given by 

---

In this chapter, recent advances in the theory of conical intersections for molecules with an odd number of electrons are reviewed. Section II presents the mathematical basis for these developments, which exploits a degenerate perturbation theory previously used to describe conical intersections in nonrelativistic systems [11,12] and Mead s analysis of the noncrossing rule in molecules with an odd number of electrons [2], Section III presents numerical illustrations of the ideas developed in Section n. Section IV summarizes and discusses directions for future work. 

This analysis is heuristic in the sense that the Hilbert spaces in question are in general of large, if not infinite, dimension while we have focused on spaces of dimension four or two. A form of degenerate perturbation theory [3] can be used to demonstrate that the preceding analysis is essentially correct and, to provide the means for locating and characterizing conical intersections. 

Electron correlation was treated by the CIPSI multi-reference perturbation algorithm ([24,25] and refs, therein). The Quasi Degenerate Perturbation Theory (QDPT) version of the method was employed, with symmetrisation of the effective hamiltonian [26], and the Maller-Plesset baricentric (MPB) partition of the C.I. hamiltonian. 

The next lowest unperturbed energy level however, is four-fold degenerate and, consequently, degenerate perturbation theory must be used to determine its perturbation corrections. For simplicity of notation, in the quantities and we drop the index n, which has the value... 

Note that the position of the juLCR depends on the sign of the nuclear hyperfine parameter relative to that of the muon. Using degenerate perturbation theory one can calculate the effects of the level crossing on the... 

Selected entries from Methods in Enzymology [vol, page(s)] Degenerate perturbation treatment, 246, 84 ligand position functions, 246, 81-82 matrix elements, 246, 81-83 octahedral field potential, 246, 80 potential energy term, 246, 78-79 resource materials, 246, 16 tetrahedral field, 246, 81... 

Fig. 9. Comparison between the second-order perturbation theory and quasi-degenerate perturbation theory for the D value of ( 20) +.

There is another way of looking at this coupled ion system, namely, in terms of stationary states. From this point of view, one considers that the excitation belongs to both ions simultaneously. To determine the wave functions of the two-ion system, one resorts to degenerate perturbation theory. The coupling H can be shown to remove the degeneracy, and two new states that are mixtures of X20 and X11 are formed. For each the excita-... 

The derivations of the models as outlined here should be amenable to improvement. Basically the contractions to smaller subspaces were developed here within a first-order degenerate perturbation-theoretic framework, so that further improvements might be obtained in proceeding to higher orders. This then further renormalizes the parameters appearing in these models. Work in this area has been little developed. 

Here we will skip the notation details, as the relation established to the Coupled Perturbed frame allow us the shortcut of passing the references to the comprehensive works devoted to the analytic derivatives of molecular energy [9]. The recent advances in the analytic derivatives and Coupled Perturbed equations into multiconfigurational second order quasi-degenerate perturbation theory is the premise of further development in the ab initio approach of vibronic constants of JT effects [10]. 

---