Coulomb repulsion and Slater Integrals

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Presentation transcript:

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

Coulomb interaction – Slater Integrals Triangular equations

Coulomb interaction – Slater Integrals Parity

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