User:TungstenEinsteinium/sandbox

This is a collection of matrices that are used in the calculation of Tanabe–Sugano diagrams, which are relevant to octahedral coordination complexes. Configuration interaction mixes all states sharing a term symbol, meaning there is one matrix per term symbol relevant to ligand field transitions. This is accomplished through electron–electron repulsion, calculated using the Laplace expansion of Coulombic potential. All matrices below are real and Hermitian and therefore symmetric; thus, only the upper triangle is listed. The $$d^1$$ and $$d^9$$ diagrams are trivial but are still included below.

=Abridged Diagrams= Tanabe–Sugano diagrams generally do not show all ligand field states, instead highlighting states that are more likely to be observed. These diagrams are shown here plotted with C/B = 4.5.

=Matrices and Full Diagrams= The same matrices may be used for $$d^n$$ and $$d^{10-n}$$ ions. These matrices are Hermitian (and in fact symmetric), so only the upper triangle of entries are shown. For octahedral $$d^n$$ ions with $$n \leq 5$$ and for tetrahedral $$d^n$$ ions with $$n > 5$$, a positive value of $$Dq$$ should be used. For tetrahedral $$d^n$$ ions with $$n > 5$$ and for octahedral $$d^n$$ ions with $$n \leq 5$$, a negative value of $$Dq$$ should be used. For octahedral ions with up to five d electrons, the matrix is described by the electron configurations shown in the leftmost column; otherwise, the matrix is described be the electron configurations shown in the topmost row. Any contributions from the Racah $$A$$ parameter have been subtracted, but can be reintroduced by adding $$[n(n-1)/2]A$$ to all diagonal entries.