Degeneracy molecular electronic

In molecular physics, the topological aspect has met its analogue in the Jahn-Teller effect [47,157] and, indeed, in any situation where a degeneracy of electronic states is encountered. The phase change was discussed from various viewpoints in [144,158-161] and [163]. 

Each energy level of a multiplet with A 0 is doubly degenerate, corresponding to the two values for M. Thus a A term has six different wave functions [Eqs. (13.87), (13.89), (11.57) to (11.59)] and therefore six different molecular electronic states. Spin-orbit interaction splits the A term into three levels, each doubly degenerate. The double degeneracy of the levels is removed by the A-type doubling mentioned previously. 

We now consider the orbital degeneracy of molecular electronic terms. This is degeneracy connected with the electrons spatial (orbital) motion, as distinguished from spin degeneracy. Thus 11 and 11 terms of linear molecules are orbitally degenerate, while and 2 terms are orbitally nondegenerate. Consider an operator P that commutes with the molecular electronic Hamiltonian and that does not involve spin we have... 

AI and Cq (see Chapter 2.14 for further details) are related to molecular electronic structure by Equations (2) and (3), where /) is the excited state of the corresponding MCD transition, df is the electronic degeneracy of the ground state A) and the summation is over all components of 1 4) and /). The first part of the /4-term expression is the difference between the excited and ground-state eman terms, while the first part of the C-term expression gives the Zeeman effect in the ground state. The second parts in both equations give the difference between the Icp (m+) and rep (m ) electric dipole moments ... 

In this section we focus on narrow, close-lying levels of varying nature in diatomic molecules. Such levels may come about due to a quasi-degeneracy of either hyper-fine and rotational levels [45], or between the fine and vibrational levels within the molecular electronic ground state [49] (see Figure 16.1). The transitions between the quasi-degenerate levels correspond to microwave frequencies, which are experimentally accessible, and have narrow linewidths, typically of the relative variation can exceed 10 in such cases. 

Divergent couplings ai e a nuisance for the computational treatment of the nuclear dynamics. In cases of exact or near degeneracy of electronic potential-energy surfaces it is therefore preferable to introduce an alternative electronic representation, the so-called diabatic (or quasi-diabatic) representation, which avoids singular coupling elements. The basic concept of diabatic states has been introduced in early descriptions of atomic collision processes and vibronic-coupling phenomena in molecular spectroscopy. ... 

The phase-change nale, also known as the Ben phase [101], the geometric phase effect [102,103] or the molecular Aharonov-Bohm effect [104-106], was used by several authors to verify that two near-by surfaces actually cross, and are not repelled apart. This point is of particular relevance for states of the same symmetry. The total electronic wave function and the total nuclear wave function of both the upper and the lower states change their phases upon being bansported in a closed loop around a point of conical intersection. Any one of them may be used in the search for degeneracies. 

In Chapter 10, we will make quantitative calculations of U- U0 and the other thermodynamic properties for a gas, based on the molecular parameters of the molecules such as mass, bond angles, bond lengths, fundamental vibrational frequencies, and electronic energy levels and degeneracies. 

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. 

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