Maurits W Haverkort Institute for theoretical physics Heidelberg University MWHaverkortthphysuniheidelbergde The Coulomb Integral is nasty T he integrant diverges at r 1 r 2 ID: 620887
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Slide1
Coulomb repulsion and Slater Integrals
Maurits W. Haverkort
Institute for theoretical physics – Heidelberg UniversityM.W.Haverkort@thphys.uni-heidelberg.deSlide2
The Coulomb Integral is nasty:
T
he 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.Slide3
Coulomb interaction – Slater Integrals
Expansion on renormalized Spherical Harmonics
with
Useful expansion because our basis functions are (close to) spherical Slide4
Coulomb interaction – Slater Integrals
Expansion on renormalized Spherical Harmonics
Integral to calculateSlide5
Coulomb interaction – Slater Integrals
Radial part: Slater integrals
Angular part: Analytical solutionSlide6
Coulomb interaction – Slater Integrals
Graphical representationSlide7
Coulomb interaction – Slater IntegralsSlide8
Coulomb interaction – Slater Integrals
Triangular equationsSlide9
Coulomb interaction – Slater Integrals
ParitySlide10
Coulomb interaction – Slater Integrals
d - electronsSlide11
Coulomb interaction – Slater Integrals
f
- electronsSlide12
Coulomb interaction – Slater Integrals
Core (p) valence (d) interaction – direct termSlide13
Coulomb interaction – Slater Integrals
Core (p) valence (d) interaction – exchange termSlide14