Term symbols, quantum numbers

Finally, for linear molecules in Z states, the wavefunctions can be labeled by one additional quantum number that relates to their symmetry under reflection of all electrons through a ay plane passing through the molecule s Coo axis. If F is even, a + sign is appended as a superscript to the term symbol if F is odd, a - sign is added. 

Clearly, these coefficients contain no more or less information than do the CG coefficients. However, both sets of symbols have symmetries under interchange of the j and m quantum number that are more easily expressed in terms of the 3-j symbols. In particular, odd permutations of the columns of the 3-j symbol leave the magnitude unchanged and change... 

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. 

For linear molecules or ions the symbols are usually those derived from the term symbols for the electronic states of diatomic and other linear molecules. A capital Greek letter E, n, A, O,... is used, corresponding to k — 0,1,2,3,..., where A. is the quantum number for rotation about the molecular axis. For E species a superscript + or - is added to indicate the symmetry with respect to a plane that contains the molecular axis. 

In an atomic term symbol the value of L = 1,2,..., etc. is coded according to spectroscopic notation as 5, P, D,F,G,... etc. (alphabetically for L > 3). Likewise, each atomic state can be characterized by a total electronic spin quantum number S,... 

The problem is not simplified by Eq. (15), since there exists a closed-form expression for the multi-scattering matrix for n spheres in terms of spherical Bessel and Hankel functions, spherical harmonics and 3j-symbols, where l, l and to, m are total angular momentum and z-projection quantum numbers, respectively (Henseler, Wirzba and Guhr, 1997) ... 

A.12-1 for a brief review of atomic-spectroscopic notation). In a given term-symbol, T will be S, P, D, F, or G etc. depending on whether the total electronic orbital angular-momentum quantum number L is 0, 1,... 

As the electronic energy of an atom or ion will depend on the quantum numbers L, 5, and J, we designate the various energy states which may arise from a given electronic configuration by what is known as a spectro-scopic term symbol ... 

For equivalent electrons (i.e., electrons with the same principal quantum numbers) the term symbols given in Table 3.5 are possible. 

Table 3.4 Quantum Numbers n i and ms for Two Electrons in Configuration p Term Symbol Assignment9 l, Microstate Number, and...

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