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I'm wondering, what is anexactdefinition of theexchangeenergy in atomic physics and/or quantum chemistry ?

For the best of my knowledge, the case is quite simple forcorrelationenergy, namely

[itex]E_{corr} = E_{exact} - E_{HF}[/itex], where [itex]E_{exact}[/itex] is the exact solution of the Schrodinger equation and

[itex]E_{HF}=\left(\Psi_{HF},\hat{H}\Psi_{HF}\right)[/itex] is the Hamiltonian expectation value for the (approximate) complete basis set Hartree-Fock wavefunction. In other words, [itex]E_{corr}[/itex] is "everything beyond HF approximation" (in non-relativistic case of course).

I believe that there is some similar definition for alsoexchangeenergy ([itex]E_x[/itex]). But what is it?

It is clear that [itex]E_x[/itex] originates from the Pauli exclusion principle, i.e., the wavefunction symmetry.

So, am I right that one can define [itex]E_x[/itex] as something like

[itex]E_x = E_{HF} - E_H[/itex] where

[itex]E_H[/itex] is the variational Schrodinger equation solution with a Hartree product trial wavefunction instead of Slatter-determinant Hartree-Fock one?

Thank you in advance for your answers!

P.S.

I've found a https://www.physicsforums.com/archive/index.php/t-178573.html" on this Forum with

the definition being [itex]E_{exchange} = E_{Hartree-Fock} -E_{Coulomb}[/itex], but it is not clear

for me, what is [itex]E_{Coulomb}[/itex] here???

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# A definition exchange energy?

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