Does the Hartree-Fock energy of a virtual orbital satisfy the virial theorem?

Question

In calculating the ground state of atoms or molecules at the equilibrium geometry, the expectation values of the kinetic, $\langle T\rangle$, and potential, $\langle V\rangle$, energies relate to the total energy, $E$, according to the virial theorem: $$ E = -\langle T\rangle=\tfrac{1}{2}\langle V\rangle. $$

Since the solution of the Schrödinger equation at the Hartree Fock (HF) level is variational, the viral theorem holds for it. Also, the HF energy is the sum of the energies of occupied orbitals; therefore, these energies must also fulfill the virial conditions individually. Can the same be said about virtual orbitals?

Answer 1

As you can easily see the mean field potential potential acting on every electron in atoms and molecules do not have this shape $V=Ar^\alpha$ so the virial theorem do not hold because of the repulsive electron repulsion in atoms and the other atomic potential in molecules. If you perform a HF calculation you will see that the total energy is $E\approx -\langle T \rangle $. This means that the correlation effects do not deviate to much from this theorem because the atomic potential has a dominant contribution.

The total energy is not the sum of the occupied orbital energy. The total energy strictly is the negative value sum of the ionization energies of all the electrons reason why there is a correction taking into account electron-electron interaction affecting every ionization. The virtual and occupied orbitals are issued from the same eigenvalue equation and because of the potential, the virial theorem cannot be applied on them (or very approximately)