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Coulomb integrals (Read 3123 times)
Gerrit-Jan Linker
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Coulomb integrals
27.08.09 at 13:03:10
Coulomb integrals
Two electron coulomb integrals in different notations:
Jij = <ΦiΦjiΦj> =  Φi(r1j(r2)|  r12-1i(r1j(r2)>  
The indices for r are 12 and 12!
Note that the scalar r12 is |r1 - r2|
Jij = (ΦiΦijΦj) = ∫|Φi(r1)|² r12-1  |Φj(r2)|² dr1dr2
The indices for r are 11 and 22!
We recognise |Φj(r)|² as the electron density. ρj(r) = |Φj(r)|²
In words: The electron density of electron j in position r is given by the square of the orbital of electron j in that position.
Therefore we can also write:
Jij = ∫|Φi(r1)|² r12-1  |Φj(r2)|² dr1dr2
     = ∫ ρi(r1) r12-1 ρj(r2) dr1dr2
We have written the Coulomb integral  as an integral over electron densities ρ
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« Last Edit: 10.09.09 at 08:31:57 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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