Coulomb Interaction in Many-Particle Hamiltonian

The following slides show you how to treat the Coulomb interaction in a many

particle Hamiltonian. As the Coulomb interaction diverges for the case where the

two electrons have the same position one needs to be careful on how to integrate.

As our basis functions are atom centered radial functions times spherical harmonics

(or linear combinations thereof) we can expand the Coulomb interaction on spherical

harmonics as well and treat the angular part analytical. The radial part are the Slater

integrals.

An excellent text book introducing this topic is Ballhausen

–

 Ligand field theory

chapter 1 and 2.

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 r

=r

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

Expansion on renormalized Spherical Harmonics

Integral to calculate

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

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Explore the treatment of Coulomb interaction in a many-particle Hamiltonian, where careful integration is crucial due to divergence issues. Learn about solving the Coulomb Hamiltonian with Slater integrals and expanding the operator on spherical harmonics for analytical solutions. Discover the significance of Coulomb repulsion, Slater integrals, and the radial and angular parts in this intricate process.

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The following slides show you how to treat the Coulomb interaction in a many particle Hamiltonian. As the Coulomb interaction diverges for the case where the two electrons have the same position one needs to be careful on how to integrate. As our basis functions are atom centered radial functions times spherical harmonics (or linear combinations thereof) we can expand the Coulomb interaction on spherical harmonics as well and treat the angular part analytical. The radial part are the Slater integrals. An excellent text book introducing this topic is Ballhausen Ligand field theory chapter 1 and 2.

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