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

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?

• Strictly speaking, virtual orbitals are not defined in HF. Additionally to the point that orbital energies so not sum to the total energy. Apr 11 at 19:28

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.