Computational Chemistry 4.17 - Fock Operator

Short lecture on the Fock operator in Hartree-Fock theory.
The Fock operator forms a pseudo-eigenvalue equation where the eigenfunction is a spin orbital and the eigenvalue is the orbital energy. The Fock operator consists of the one-electron core Hamiltonian operator plus the Coulomb potential and exchange potential, each of which is a sum of the Coulomb and exchange operators for all the other occupied spin orbitals, respectively. This is a pseudo-eigenvalue equation because the mean-field operator depends on the form of all the other spin orbitals in the molecule. A simplified representation of the Fock operator may be obtained by making use of the permutation operator, which exchanges the indices of electrons 1 and 2 wherever they are encountered when acted upon by the operator. For the current spin orbital, the Coulomb and exchange operators cancel out, thus we can include the current spin orbital as well in the sum over all other occupied orbitals.
Notes Slide: i.imgur.com/Bs...
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[@vitusberger1141]

You really are a great teacher =) thanks!

[@isaacleach1]

Great vid, as always. Something I'm a little confused about is whether or not the Fock operator is a 1 electron or 2 electron operator. I see it contains both 1 electron and 2 electron integrals but I've also heard it referred to as a single electron operator...

[@TMPChem]

In some regards it is a one-electron operator, and in others it is a two-electron operator. I believe technically it is a one-electron operator, though you could make an opposing argument. It is technically a one electron operator because the effects of the 2nd electron (i.e. all other electrons) is abstracted away into the mean field potential. The electron isn't actually feeling any specific electron, but is experiencing the net average effect of all electrons. However, this depends on which orbital it's in, as the exchange operator cancels out self-interaction, so while each electron is interacting with all others it isn't interacting with itself, therefore each orbital experiences the mean field operator differently. And the core Hamiltonian part is of course uncontroversially a one-electron component.

[@isaacleach1]

@@TMPChem hmmmmm okay that clears things up. Thanks!

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[@apburner1]

First.