# Determining Kohn-Sham and Hartree Fock virtual orbitals: The underlying field

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That is true: in the Kohn–Sham model electrons both in occupied and in virtual orbitals are "moving" in the field $n-1$ electrons, while in the Hartee-Fock model electrons in occupied orbitals are "moving" in the field $n-1$ electrons, but electrons in virtual orbitals are "moving" in the field all $n$ electrons. And this is simply "by construction": you just have to look at the very definitions of these models. So, in short:

In the Kohn–Sham model, it is the exchange-correlation hole that by definition "excludes" a single electron from equations describing each and every orbital.

If you look at the Kohn–Sham equations $$\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+v_{{{\rm {eff}}}}({\mathbf r})\right)\phi _{{i}}({\mathbf r})=\varepsilon _{{i}}\phi _{{i}}({\mathbf r}) \, ,$$ where $$v_{\rm {eff}}(\mathbf {r} )=v_{\rm {ext}}(\mathbf {r} )+e^{2}\int {\rho (\mathbf {r} ') \over |\mathbf {r} -\mathbf {r} '|}d\mathbf {r} '+v_{{{\rm {xc}}}} \, ,$$ it is last term of the Kohn–Sham effective potential, the exchange-correlation potential, $$v_{{{\rm {xc}}}}({\mathbf r})\equiv {\delta E_{{{\rm {xc}}}}[\rho ] \over \delta \rho ({\mathbf r})} \, ,$$ that causes the above mentioned feature of Kohn–Sham orbitals.

For the case of the Hartree-Fock equations $$\hat{F} \phi_i({\mathbf r}) = \epsilon_i \phi_i({\mathbf r}) \, ,$$ the summation in the Fock operator by the very definition of the model runs over all occupied orbitals only, $$\hat{F} = \hat{H}^{\text{core}} + \sum_{j=1}^{n}[\hat{J}_{j} - \hat{K}_{j}] \, .$$ For occupied orbitals, in the Hartree–Fock equation defining some particular (occupied) $i$-th orbital the Coulomb and exchange contribution from this orbital itself perfectly cancel each other, i.e. $\hat{J}_{j} \phi_i = \hat{K}_{j} \phi_i$ when $j=i$. Thus, in the Hartree-Fock model an electron from any occupied orbital is also moving in the field $n-1$ electrons. And this is of course physically more than reasonable: at the end of the day electron should not interact with itself.

But for virtual orbitals the situation in the Hartree-Fock model is different from that in the Kohn-Sham one: for any virtual orbital $\phi_k$ the sum in the Hartree-Fock equation defining it still runs over occupied orbitals only, but none of the terms $\hat{J}_{i} \phi_k$ and $\hat{K}_{i} \phi_k$ this time cancel each other since $k$ is simply grater than $n$.

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### Jes Aasbæk

Updated on August 01, 2022

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