First-order, degenerate mixing

Orbital mixing. Simple spherical orbitals are shown to illustrate the concept. A. The first-order, degenerate mixing. B. The second-order, nondegenerate mixing. 

A mixed-state d cription of a given system (see e.g. in [10] or [28]) is necessary in such situations, among others, when the GS is degenerate and/or when a non-integer average number of electrons is of interest. The first-order spinless ensemble DM and the ensemble electron density, characterizing this system, are in a form... 

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). 

The effect of spin-orbit coupling on mixing of singlet and triplet states can be seen by considering first-order non-degenerate perturbation theory [8). Let us consider the perturbed triplet wavefunction for the state where represents the unperturbed triplet wavefunction for the... 

Thus, for, 2s and 2po in Eqs. (13.71) and (13.72) should be replaced by 2s + 2p and 2s - 2po.] For molecules that dissociate into many-electron atoms, the separated-atoms 2s and 2p AOs are not degenerate but do lie close together in energy. Hence the first-order corrections to the wave functions will mix substantial 2s character into the alp MOs and substantial 2p character into the a-ls MOs. Thus the designation of an MO as als or alp should not be taken too literally. For H2 and H2, the united-atom designations of the MOs are preferable to the separated-atoms designations, but we shall use mostly the latter. 

Mixing of Degenerate Orbitals—First-Order Perturbations... 

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