1. Coulomb repulsion and Slater Integrals
Maurits W. Haverkort
Institute for theoretical physics – Heidelberg University

2. The Coulomb Integral is nasty: The integrant diverges at r1=r2
Coulomb Hamiltonian:
In order to create the Hamiltonian as a matrix we need to evaluate the following integral
Solution by Slater: Expand the operator on Spherical Harmonics. Solve the angular part analytical and the Radial integral numerical (Slater Integrals.)
Also works in solids. (Spherical Harmonics are not eigen-states, but still a valid basis set.

3. Coulomb interaction – Slater Integrals
Expansion on renormalized Spherical Harmonics
with
Useful expansion because our basis functions are (close to) spherical

4. Coulomb interaction – Slater Integrals
Integral to calculate
Expansion on renormalized Spherical Harmonics

5. Coulomb interaction – Slater Integrals
Radial part: Slater integrals
Angular part: Analytical solution

6. Coulomb interaction – Slater Integrals
Graphical representation

7. Coulomb interaction – Slater Integrals

8. Coulomb interaction – Slater Integrals
Triangular equations

9. Coulomb interaction – Slater Integrals
Parity

10. Coulomb interaction – Slater Integrals
d - electrons

11. Coulomb interaction – Slater Integrals
f - electrons

12. Coulomb interaction – Slater Integrals
Core (p) valence (d) interaction – direct term

13. Coulomb interaction – Slater Integrals
Core (p) valence (d) interaction – exchange term