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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 and diagrams are trivial but are still included below.

Abridged Diagrams

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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.

d1 Tanabe–Sugano diagram
d1 electron configuration
d2 Tanabe–Sugano diagram
d2 electron configuration
d3 Tanabe–Sugano diagram
d3 electron configuration
d4 Tanabe–Sugano diagram
d4 electron configuration
d5 Tanabe–Sugano diagram
d5 electron configuration
d6 Tanabe–Sugano diagram
d6 electron configuration
d7 Tanabe–Sugano diagram
d7 electron configuration
d8 Tanabe–Sugano diagram
d8 electron configuration
d9 Tanabe–Sugano diagram
d9 electron configuration

Matrices and Full Diagrams

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The same matrices may be used for and ions. These matrices are Hermitian (and in fact symmetric), so only the upper triangle of entries are shown. For octahedral ions with and for tetrahedral ions with , a positive value of should be used. For tetrahedral ions with and for octahedral ions with , a negative value of 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 parameter have been subtracted, but can be reintroduced by adding to all diagonal entries.

d1 and d9 Ions

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Energy Matrix for States
Energy Matrix for States

d2 and d8 Ions

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Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States

d3 and d7 Ions

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Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States

d4 and d6 Ions

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Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States
Energy Matrix for States

d5 Ions

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Energy Matrix for States
Energy Matrix for States