Crystal Field Potentials

Let us assume that the crystal field arises from point charges lying outside the region that we want to describe. The crystal field potential satisfies Laplace's equation, the solution of which can be expanded in terms of spherical harmonics

must transform according to the identical representation of the point group. It would seem that we have to know the basis function of the identical representation for each I but for practical purposes we usually need just a few of them.

Usually, we are interested in howacts within a subspace spanned by the d-states (or f-states) of an unfilled shell. To find the crystal-field-split states of a single d-electron, we have to evaluate matrix elements of the form , where and are d-states. Because of the addition properties of the angular momentum, the expansion of cannot contain higher than = 4 components. Because of the orthogonality properties of the spherical harmonics, the matrix element vanishes for the > 4 components of ; hence for this particular problem, it is sufficient to expand to fourth order in . By similar arguments, for f-states it is sufficient to consider the terms with 6.

In what follows, we construct the Hamiltonian of a cubic crystal field. The largest term is the spherically symmetrical = 0 term . It is an important contribution to the total energy of the crystal but it does not influence the spectrum. Neither does the = 2 term which hasthe same symmetry (cf. Table 3.1). The first non-trivial contribution comes from = 4, and we do not have to go beyond sixth-order, thus

Source: Fazekas, Patrik.1999.Lecture Notes on Electron Correlation and Magnetism. USA:World Scientific.

Crystal Field Model

The Coulomb interactions between an electron i and a set of N nuclei g may expanded in a series of normalized spherical harmonics.

(1)

Approximate wave functions for this crystal field Hamiltonian will be obtained in two steps. First we solve the one-electron problem retaining only the first, spherically symmetric term of of the Laplace expansion. Thus the one electron zeroth-order Hamiltonian is

(2)

From this Hamiltonian we obtain a set of zeroth-order energy levels and molecular orbitals which will the form . The second step of our treatment introduces the higher spherical harmonics given in Table I as a first-order perturbation that lifts the degeneracy in quantum number l for functions of a given n. In the case of large (8NRZ)1/2 the eigenvalues of the Hamiltonian (2) are approximate by

(3)

Where n and l are the usual spherical quantum numbers.

Source: Hoffmann, Roald and Martin Gouterman. Theory of Polyhedral Molecules. II. A Crystal Field Model. Journal of the Chemical Physics 36, No. 8, pp. 2189-2195 (1962)