# Understanding Coulomb Repulsion and Slater Integrals

Dive into the intricate world of Coulomb repulsion and Slater integrals, essential concepts in quantum physics. Explore the challenges posed by the diverging Coulomb integral and the complex calculations required to evaluate these interactions. Discover how Slater integrals play a crucial role in creating Hamiltonian matrices and expanding operators on spherical harmonics, providing insights into the behavior of particles in solids.

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**Coulomb repulsion and Slater Integrals**Maurits W. Haverkort Institute for theoretical physics Heidelberg University M.W.Haverkort@thphys.uni-heidelberg.de**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.**Coulomb interaction Slater Integrals**Expansion on renormalized Spherical Harmonics with Useful expansion because our basis functions are (close to) spherical**Coulomb interaction Slater Integrals**Integral to calculate Expansion on renormalized Spherical Harmonics**Coulomb interaction Slater Integrals**Radial part: Slater integrals Angular part: Analytical solution**Coulomb interaction Slater Integrals**Graphical representation**Coulomb interaction Slater Integrals**Triangular equations**Coulomb interaction Slater Integrals**d - electrons**Coulomb interaction Slater Integrals**f - electrons**Coulomb interaction Slater Integrals**Core (p) valence (d) interaction direct term**Coulomb interaction Slater Integrals**Core (p) valence (d) interaction exchange term