Spatial degeneracy

The consideration of a simple example, the configuration ls22p of lithium, may make clearer what the different unperturbed functions are. Table 30-1 gives the sets of quantum num- 

1 The treatment of atoms which we are giving is due to J. C. Slater, Phys. Rev. 34, 1293 (1929), who showed that this method was very much simpler and more powerful than the complicated group-theory methods previously used. 

This simple case illustrates the idea of completed shells of electrons. The first two sets of quantum numbers remain the 

Problem 30-1. Construct tables similar to Table 30-2 for the configurations np3 and nd2. 

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Firstly, consider the spatial degeneracies of spin-parallel electronic arrangements within the configurations tig ej. Parallel spins must be placed in different... 

Now take the case for an octahedral vanadium(iii) ion. For d, the ground term is Tig. The spatial degeneracy of a 7 term is three-fold and we describe this with Leff = 1. Using (5.10) we find eff = VlO. So for this Tig term, the crystal field has quenched some, but not all, of the angular momentum of the parent free ion F term. 

For the cubic-field (octahedral or tetrahedral) subshells g(g), there is spatial degeneracy for e(g) but not for eQy Nevertheless, neither of these configurations give rise to an orbital contribution to the magnetic moment. The conditions for orbital contributions to arise in strong-field configurations are that the orbitals must be... 

It remains to consider the isotopically heteronuclear systems to complete the symmetry analysis of this system. Because the experiments are performed under natural abundance conditions, only systems containing a single rare isotope ( O or 0) need be considered. However, because the spatial degeneracy of the electronic state of the ion and the neutral differ, the case where either the neutral or the ion is isotopically heteronuclear must be considered separately. The results in Table 4 show that when the neutral is made isotopically heteronuclear the /-based restriction is removed, while that based on is preserved. Conversely, when... 

For degenerate states a problem arises with the definition of cumulants. We consider here only spin degeneracy. Spatial degeneracy can be discussed on similar lines. For S 0 there are (2S + 1) different Afs-values for one S. The n-particle density matrix p Ms) = of a single one of these states does not... 

As an example, consider a tetrahedral molecule in T symmetry, with two singly-occupied t2 symmetry orbitals, say tfy1. The direct product T2 (8) T2 reduces to A E Ti T2, so we obtain singlet states Mi, 1E, 1Ti, and 1T2, and triplet states Mi, 3E, 37), and 3T2. A handy check on the correctness of this sort of analysis is to add up the toted spin and spatial degeneracies of all the states and verify that it equals the spin and spatial degeneracy of the original orbital product (36 in this case). 

The operator M so couples states of different spin and space symmetries in second order, independent of spatial degeneracies. 

Focusing on a possible ground state, we note that interactions with the matrix would remove the spatial degeneracy, yielding two states. We therefore hazard to guess that the ground... 

We now suppose we have two ions with respective spins Si, S2 and no spatial degeneracy. The total spin for the system is... 

An additional contribution to equation 23), Rln gl/gs), arises from any difference in the spatial degeneracy of the ground and transition state for a self-exchange reaction this contributes aboutO.6 J M L Differences in the densities of vibrational states will also contribute in cross electron-transfer reactions. The activation enthalpy is... 

Thus, as a result of spin coupling (two 5= 1/2 = >5=1, 0) and spatial degeneracy (+/— combinations of local excitations), each CT state of a monomeric subunit (one Cu° center) corresponds to 2x2 = 4 states in the Cu -Cu dimer. Out of the four CT states given in Equation (4), two have the same spin and spatial symmetry of the singlet and triplet ground states (Equation (1)), respectively, which leads to configuration interaction (Cl)... 

Although an octahedron has a much lower symmetry than a sphere it would be reasonable to expect that many-electron wavefunctions would be handled similarly. This is so—symbols such as Eg and A g, like t g, g and aig orbitals, imply, respectively, triple, double and single orbital degeneracy. In each case they are associated with a spin degeneracy which, in each of these three examples, is identical to the spatial degeneracy. However the two vary independently and so symbols such as T g, Eg and A g are perfectly reasonable. 

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