Perturbation theory degenerate systems

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. 

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. 

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-... 

To cover the gap between them the Hubbard model Hamiltonian was quite generally accepted. This Hamiltonian apparently has the ability of mimicking the whole spectrum, from the free quasi-particle domain, at U=0, to the strongly correlated one, at U —> oo, where, for half-filled band systems, it renormalizes to the Heisenberg Hamiltonian, via Degenerate Perturbation Theory. Thence, the Heisenberg Hamiltonian was assumed to be acceptable only for rather small t/U values. 

That there is something unusual about this description may be seen from the following argument. The d and / -isomers share the same Hamiltonian it for which [it, P] = 0 is true, so that tpi) and rpi) are degenerate in energy. However, since they are distinct physical systems we have assigned them their own Hilbert spaces= t, d, and in the perturbation theory calculation for a given isomer we tacitly asserted that at T = 0 all the excitations of say, the space-inversion operator P as an operator with domain = , since is dense, P has an adjoint P+. The condition... 

We now return to first-order perturbation theory for degenerate orbitals (Section 4.3). As any linear combination of two degenerate orbitals i/q and xj/j is equally valid, we set up a trial wavefunction function xj/jc = akixl/i-F akj t//7 and we have to solve the secular Equation 4.18. The eigenvalues of the unperturbed system will be equal for all linear combinations, = = (0). 

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