Perturbation degeneracy

Eqs. (D.5)-(D.7). However, when perturbations occur due to anharmonicity, the wave functions in Eqs. (D.11)-(D.13) will provide the conect zeroth-order ones. The quantum numbers and v h are therefore not physically significant, while V2 arid or V2 and I2 = m, are. It should also be pointed out that the degeneracy in the vibrational levels will be split due to anharmonicity [28]. 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

If the gn roots are all different, then in first order the g -fold degenerate unperturbed eigenvalue Ef is split into g different perturbed eigenvalues. In this case, the degeneracy is removed in first order by the perturbation. We... 

However, in the first excited state the degree of degeneracy is equal to four. Hence, the first-order perturbation calculation requires the application of Eq. (62). The wavefunctions for the first excited state can be written in the form... 

As in the case of II electronic states of tetraatomic molecules, because of generally high degeneracy of zeroth-order vibronic leves only several particular (but important) coupling cases can be handled efficiently in the framework of the perturbation theory. We consider the following particular cases ... 

In Hartree-Fock (HF) theory, the energy of a given orbital is lower when the orbital is occupied than when it is vacant (due to the formal self-energy in the former case), and the degeneracy is broken. Thus, perturbative F1F expressions such as Eq. (2.7) often have wider numerical validity than would be anticipated in naive MO theory. 

In this expression p is a mass parameter associated to the electronic fields, i.e. it is a parameter that fixes the time scale of the response of the classical electronic fields to a perturbation. The factor 2 in front of the classical kinetic energy term is for spin degeneracy. The functional f [ i , ] plays the role of potential energy in the extended parameter space of nuclear and electronic degrees of freedom. It is given by. 

The ex model has been elaborated in a number of ways. An electrostatic perturbation was added (33) to account for band splittings in the d-d spectra of tetragonal copper(II) ammine complexes where the simple AOM predicted accidental degeneracy the merits of this refinement will be discussed in 2.5.1. Another development has been the introduction of d—s and d—p mixing, which is apparently necessary to account for the d-d spectra of chlorocuprates(II) (34). This requires the additional parameters e, edpa and edpv. 

The diagrams are best understood in terms of the apparent repulsion between the energy levels of combining systems, which can easily be related to a perturbation treatment of the secular equations. For example, two carbon atom ir electron levels (1) and (2) with energies ao would interact to remove the degeneracy... 

So far, we have discussed the crystalline field acting on the ion A due to an octahedral environment of six B ligand ions. In many optically ion activated crystals, such as Ti +rAlaOj, the local symmetry of the active ion A is slightly distorted from the perfect octahedral symmetry Oh symmetry). This distortion can be considered as a perturbation of the main octahedral field. In general, this perturbation lifts the orbital degeneracy of the tag and eg levels and then produces additional structure in the tag eg absorption/emission bands. 

Distortions need to have their symmetry contained in the direct product — ai + e + ti + t2 and we will consider only such displacements that leave the Cl-0 bonds invariant. This leaves only e-type and t2-type to be of concern. An e-deformation shortens or lengthens opposite edges of the tetrahedron and does little to split the ti degeneracy. The 2 modes will be classified as el-orbitals and the perturbing potential will have the form V — + rjV + t V y with the... 

The t degeneracy is cancelled, to first order, hy the perturbation and the orbital energy changes are determined by the eigenvalues of the matrix above. They are expressed as... 

Typical for cyclic jr—systems are d enerate orbitals degeneracy can disappear by do— nor-or acceptor-perturbation ... 

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