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timntimn
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Dear Forum members!
I'm wondering, what is an exact definition of the exchange energy in atomic physics and/or quantum chemistry ?
For the best of my knowledge, the case is quite simple for correlation energy, 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 also exchange energy ([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?
I'm wondering, what is an exact definition of the exchange energy in atomic physics and/or quantum chemistry ?
For the best of my knowledge, the case is quite simple for correlation energy, 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 also exchange energy ([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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