An ion with the configuration \(\mathrm{f}^{2}\) enters an environment of octahedral symmetry. What terms arise in the free ion, and with which terms do they correlate in the complex? Hint. Follow the discussion of Section 8.9.

To start, consider an ion with the configuration \(\mathrm{f}^{2}\). This means there are two electrons in the 4f subshell. Using Hund’s rules, the term symbols for the two electron system can be found. Hund's rule states that: (1) the term with the largest multiplicity has the lowest energy. (2) For terms with the same multiplicity, the term with the largest value of L has the lowest energy. (3) For atoms with less than half-filled shells, the level with the smallest value of J lies lowest. For our \(\mathrm{f}^{2}\) configuration, only two electrons involved so we can easily identify the term symbols are \(\mathrm{{^3F}}_{0}\), \(\mathrm{{^3F}}_{2}\), \(\mathrm{{^3F}}_{4}\), \(\mathrm{{^1G}}_{4}\), and \(\mathrm{{^1S}}_{0}\). The superscript number indicates the spin multiplicity (2S+1), the letter denotes the total orbital momentum quantum number L, and the subscript value is the total angular momentum quantum number J.

Upon transition to an octahedral complex, the d orbitals split under octahedral symmetry. Because the f orbitals are radially further from the nucleus, they interact weakly with the field produced by ligands in an octahedral symmetry. Hence, there is no specific ligand field splitting diagram for f orbitals. However, in the octahedral crystal field, these terms are expected to split into more terms. The \(\mathrm{{^3F}}\) term in the free ion will correlate with the \(\mathrm{{^3F}}\) term in the complex, same for \(\mathrm{{^1G}}\) and \(\mathrm{{^1S}}\) terms.