Perturbation theory degenerate-level

Section 12-8 Degenerate-Level Perturbation Theory by Inspection... 

These rules follow directly from the quantum-mechanical theory of perturbations and the resolution of the secular equations for the orbital interaction problem. The (small) interaction between orbitals of significantly different energ is the familiar second order type interaction, where the interaction energy is small relative to the difference between EA and EB. The (large) interaction between orbitals of same energy is the familiar first order type interaction between degenerate or nearly degenerate levels. 

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... 

H2 > Hi this is the j-j coupling. In this case only H2 is taken into account in first order perturbation theory. It commutes with the total angular momentum ji = If -f Sj. The configurations split into levels being (2ji -I- l)(2j2 -f 1) fold degenerate. 

Now consider perturbation theory for an unperturbed energy level that is n-fold degenerate, with the states... 

Since perturbation theory requires conservation of energy relative to the degenerate level, the r3 level is raised 6Dq and the r6 level is lowered 4Dq relative to the fivefold degenerate D level, as is indicated in Figure 10(a). In the four-, eight-, and twelve-coordinated structures, on the other hand, the rB orbitals are directed toward the... 

Tj g state yields two different four-fold degenerate Vq states which mix with Eg, and accordingly, Equation 9 yields two distinct coefficients, denoted cj and C2 At the level of first-order perturbation theory, these two states lie at +/2 and 4 2+/3 ith respect to the ground state, where is the effective spin-orbit coupling integral for the Co " 3d orbital (17). 

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