1. Crystal Field expanded on renormalized spherical Harmonics Clm
Maurits W. Haverkort
Institute for theoretical physics – Heidelberg University

2. From potentials to matrix elements of the Hamiltonian
Crystal-field theory
Approximate the solid by a single atom in an effective potential
And matrix elements of the Hamiltonian:
!!!! WARNING !!!!
Crystal-field potentials do not exist. They are effective Hamiltonians introduced to mimic covalent bonding.
(crystals bind due to a gain in kinetic energy and a smaller loss in potential energy, i.e. the contribution of the crystal-field potential to binding is negative)
Covalent bonds are stronger than ionic bonds

3. From potentials to matrix elements of the Hamiltonian
Atomic basis sets: (also Gaussian, muffin-tin orbitals, …)
Expand potential on spherical harmonics.

4. From potentials to matrix elements of the Hamiltonian
Atomic basis sets: (also Gaussian, muffin-tin orbitals, …)

5. Not all values of Akm are allowed (many are zero)
Selection rules on angular momentum conservation
Hamiltonian is Hermitian

6. Not all values of Akm are allowed (many are zero)
Symmetry related: For all “rotations” of the point-group Ci we have
Cubic: (Oh) eg d
t2g

7. Not all values of Akm are allowed (many are zero)
Tetragonal: (D4h)
a1g eg
b1g d eg
t2g
All other point-groups available several exported in Mathematica Help files and function PotentialExpandedOnClm() for others, just sent an
b2g