Perturbation theory, degenerate

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. 

As is well known, perturbation theory for a single state is different from that for degenerate states. The former leads to the traditional adiabatic... 

The first-order perturbation theory of the quantum mechanics (4, III) is very simple when applied to a non-degenerate state of a system that is, a state for which only one eigenfunction exists. The energy change W1 resulting from a perturbation function / is just the quantum mechanics average of / for the state in question i.e., it is... 

If the unperturbed system is degenerate, so that several linearly independent eigenfunctions correspond to the same energy value, then a more complicated procedure must be followed. There can always be found a set of eigenfunctions (the zeroth order eigenfunctions) such that for each the perturbation energy is given by equation 9 and the perturbation theory provides the... 

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. 

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... 

B) How many lines are expected from this model The total number of nuclear spin states is (2 f + 1) x (2I2 + 1) x (2/3 + 1). Thus, if the model structure has six protons (I = 1/2), there should be (2 x 1/2 + l)6 = 26 = 64 nuclear spin states. If some of the nuclei are expected to be equivalent, then the number of lines will be less than the number of spin states, i.e., some of the spin states will be degenerate (to first-order in perturbation theory). Thus, if the six protons are in three groups of two, it is as if you had three spin-1 nuclei and you expect (2 x 1 + l)3 = 33 = 27 distinct lines. If there is one group of four equivalent protons and another group of two, then it is as if you had one spin-2 nucleus and one spin-1 nucleus and you expect (2x2+ 1)(2 x 1+1) =15 lines. 

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