Crystal-Field Splitting of Free-Ion D Terms

Term wavefunctions describe the behaviour of several electrons in a free ion coupled together by the electrostatic Coulomb interactions. The angular parts of term wavefunctions are determined by the theory of angular momentum as are the angular parts of one-electron wavefunctions. In particular, the angular distributions of the electron densities of many-electron wavefunctions are intimately related to those for orbitals with the same orbital angular momentum quantum number that is. 

The electron distributions in term wavefunctions and orbitals may be the same or complementary, as shown below 

The identity of (b) with (a) is obvious. In (c), a section of the density is shown to take the form of a spherical density from which a density of the form (b) has been subtracted. Alternatively, (c) may be viewed as a distribution of positive charge in the form of (b). Whether components of a D term take the form (b) or (c) depends upon the number of electrons described by the many-electron wavefunction. 

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. 

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