10.6: Inversion Symmetry

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One more quantum number, that relating to the inversion (i) symmetry operator can be used in atomic cases because the total potential energy \(V\) is unchanged when all of the electrons have their position vectors subjected to inversion (i.e., \(i\textbf{r} = \textbf{-r}\)). This quantum number is straightforward to determine. Because each \(L, S, M_L, M_S, H\) state discussed previously consists of a few (or, in the case of configuration interaction several) symmetry adapted combinations of Slater determinant functions, the effect of the inversion operator on such a wavefunction \(\Psi\) can be determined by:

applying i to each orbital occupied in \(\Psi\) thereby generating a ± 1 factor for each orbital (+1 for s, d, g, i, etc orbitals; -1 for p, f, h, j, etc orbitals),
multiplying these \(\pm 1\) factors to produce an overall sign for the character of \(\Psi\) under \(\hat{i}\).

When this overall sign is positive, the function \(\Psi\) is termed "even" and its term symbol is appended with an "e" superscript (e.g., the \(^3P\) level of the \(O\) atom, which has \(1s^22s^22p^4\) occupancy is labeled \(^3P^e\)); if the sign is negative \(\Psi\) is called "odd" and the term symbol is so amended (e.g., the \(^3P\) level of \(1s^22s^12p^1\) \(B^+\) ion is labeled \(^3P_o\)).