Wavefunctions degeneracy

The left superscript indicates that the arrangements are all spin triplets. The letter T refers to the three-fold degeneracy just discussed and it is in upper case because the symbol pertains to a many-electron (here two) wavefunction (we use lower-case letters for one-electron wavefunctions or orbitals, remember). The subscript g means the wavefunctions are even under inversion through the centre of symmetry possessed by the octahedron (since each d orbital is of g symmetry, so also is any product of them), and the right subscript 1 describes other symmetry properties we need not discuss here. More will be said about such term symbols in the next two sections. 

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

Kramers degeneracy theorem states that the energy levels of systems with an odd number of electrons remain at least doubly degenerate in the presence of purely electric fields (i.e. no magnetic fields). This is a consequence of the time-reversal invariance of electric fields, and follows from an application of the antiunitary T-operator to the wavefunction of an odd number of electrons [51]. 

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... 

The application of a magnetic field to the wavefunctions obtained by the procedure described in the previous sections results in the complete removal of the degeneracy of the / multiplet, either pertaining to Kramers or non-Kramers ions, and yields a temperature-dependent population of the different 2/ + 1 components (Figure 1.2) Thus, at low temperatures, large deviations from the Curie law are observed. The effect of the magnetic field is described by the Zeeman Hamiltonian ... 

We then discover an extremely important fact each normal coordinate belongs to one of the irreducible representations of the point group of the molecule concerned and is a part of a basis which can be used to produce that representation. Because of their relationship with the normal coordinates, the vibrational wavefunctions associated with the fundamental vibrational energy levels also behave in the same way. We are therefore able to classify both the normal coordinates and fundamental vibrational wavefunctions according to their symmetry species and to predict from the character tables the degeneracies and symmetry types which can, in principle, exist. 

This result is tremendously useful, it not only leads to selection rules for vibrational spectroscopy but also, as was the case with electronic wavefunctions (see 8-2), allows us to predict from inspection of the character table the degeneracies and symmetries which are allowed for the fundamental vibrational wavefunctions of any particular molecule. 

The fact that these HOMOs and LUMOs have a two-fold degeneracy implies that there are four isoenergetic one-electron transitions to yield the first excited states this complication is however resolved by the interaction of these one-electron excitations, and this is known as configuration interaction. The concept of configuration interaction (Cl) is somewhat similar to that of the interaction of atomic orbitals to form molecular orbitals. An electron configuration defines the distribution of electrons in the available orbitals, and an actual state is a combination of any number of such electron configurations, the state wavefunction being... 

Although most resonances in the F+HCl system are clearly of the reagent type or product type, there are some mixed" cases. Because the resonance manifold is dense, degeneracies can occur between zero-order reagent-type or product-type states. Thus, the resonance wavefunctions for these states are linear combinations of entrance and exit channel expressions. 

The first term is the initial state with energy Ei before the light beam is switched on and the second term represents the total wavefunction in the upper electronic state. The initial conditions for the corresponding time-dependent coefficients are aj(0) = 1 and af(0 Ef,n) = 0 for all energies Ef and all vibrational channels n. The sum over a in (2.9) is replaced in (2.64) by an integral over Ef and a sum over n. The integral over Ef reflects the fact that the spectrum in the upper electronic state is continuous and the summation over all open vibrational channels n accounts for the degeneracy of the continuum wavefunctions. 

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