# 10.6: Inversion Symmetry

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- 60602

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 \(^3P_o\)).