Splitting of -levels in a weak crystal field

For the simplest case of a one-electron configuration dl the term functions are identical with the d-orbitals, and thus the formulae for the pertinent matrix elements listed in Table 8.10 are directly applicable. Then the 5 x 5 secular determinant is solved. For the case of an octahedral complex the matrix elements of the crystal field potential in the basis set of spherical harmonic functions Yi m form the matrix 

These roots span the irreducible representations and Eg of the octahedral group Oh. The same result is obtained when, from the basis set functions, the combinations transforming according to the irreducible representations of the octahedral group are formed 

In such a basis set the above Hamiltonian matrix is already diagonal. 

For a many-electron system the matrix elements between the term functions should be combined via the matrix elements over the one-electron functions. For example, the matrix elements of the crystal field potential over the functions of the 3 F term of the electron configuration d2 are 

The splitting of other atomic terms due to the crystal field potential of the given symmetry (group) can be evaluated in an analogous way [9]. The results for octahedral and tetrahedral fields are collected in Table 8.16. 

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