What Determines Hartree–Fock and Kohn–Sham Virtual Orbitals?

A virtual orbital is an unoccupied solution of the one-electron equations used in an electronic-structure calculation. It is not calculated from a separate cloud of electrons occupying that orbital. Instead, it belongs to the effective operator established by the method’s self-consistent ground-state calculation.

The main difference is what that operator contains. Hartree–Fock (HF) includes a nonlocal exchange operator built from occupied orbitals. Kohn–Sham (KS) density-functional theory uses an effective potential built from the electron density and an exchange-correlation approximation (or, in principle, the exact exchange-correlation potential).

The operators that determine the orbitals

In canonical Hartree–Fock theory, molecular orbitals satisfy the Fock equation

$$\hat{F}\phi_i = \varepsilon_i\phi_i.$$

The Fock operator includes the kinetic-energy operator, the electron–nuclear attraction, the Coulomb repulsion from the electron density, and exchange. Its Coulomb and exchange terms depend on the occupied orbitals, so the orbitals and operator are found iteratively. Once the occupied orbitals have converged, diagonalizing the Fock operator also gives unoccupied eigenfunctions: the HF virtual orbitals.

In Kohn–Sham theory, the corresponding equation is

$$\left[-\frac{1}{2}\nabla^2 + v_{\mathrm{ext}}(\mathbf{r}) + v_{\mathrm{H}}(\mathbf{r}) + v_{\mathrm{xc}}(\mathbf{r})\right]\phi_i(\mathbf{r}) = \varepsilon_i\phi_i(\mathbf{r}).$$

Here, $v_{\mathrm{ext}}$ is usually the potential from the nuclei, $v_{\mathrm{H}}$ is the electrostatic potential from the electron density, and $v_{\mathrm{xc}}$ is the exchange-correlation potential. For ordinary KS calculations, these are local, multiplicative potentials. The converged ground-state density determines the effective potential, and solving the KS equation in that potential gives both occupied and virtual orbitals.

Thus, “the underlying field” is best understood as the effective one-electron operator—not simply the electric field from the nuclei. If an external electric field is applied, its contribution is included in the external potential; it can change the density and, after self-consistent convergence, the orbitals as well.

A concrete example: H₂ in a minimal basis

Represent each hydrogen atom with one basis function, giving a two-function basis for H₂. For a closed-shell calculation, the two resulting molecular orbitals are a bonding combination and an antibonding combination. The bonding orbital is occupied by the two electrons; the antibonding orbital is virtual.

In HF, those two orbitals are eigenfunctions of the Fock operator constructed from the occupied bonding orbital. In KS, they are eigenfunctions of the KS operator constructed from the two-electron density. The basis and nuclear positions are the same, but the effective operators differ, so their orbital energies are not required to match; in this symmetric two-function basis, the bonding and antibonding shapes are fixed by symmetry. This example also shows why a virtual orbital is not made by putting an electron into it during the ground-state calculation: the antibonding orbital is determined by the operator even though it is unoccupied.

What virtual orbitals do—and do not—mean

Virtual orbitals provide a useful set of directions in which the wavefunction or density can change. For example, configuration-interaction and coupled-cluster methods use occupied-to-virtual orbital substitutions to build correlated states. Time-dependent DFT also uses KS orbital transitions as part of its response calculation.

But a virtual orbital is not automatically an observable excited state. In particular, a KS virtual orbital energy is generally not an exact electron-addition energy, and the difference between the KS LUMO and HOMO is generally not the exact fundamental gap. Approximate functionals can make this gap especially method-dependent. HF virtual orbital energies likewise do not generally equal measured electron affinities, since adding an electron allows orbital relaxation and introduces correlation effects.

The orbitals also depend on practical choices: the molecular geometry, basis set, charge and spin state, and computational method. In a finite basis, “virtual orbitals” are the remaining basis-space eigenvectors orthogonal to the occupied subspace. Adding diffuse basis functions, for example, can substantially alter orbitals intended to describe diffuse or weakly bound states.

Common mistakes

• Calling virtual orbitals empty physical states. They are solutions of the chosen effective equations; whether a corresponding excited or attached-electron state is bound requires separate analysis.
• Assuming HF and KS virtual orbitals must agree. Their operators have different mathematical forms, so agreement is not guaranteed even with the same geometry and basis.
• Reading every virtual orbital energy as an excitation energy. Excitation energies require a method that accounts for the relevant response, electron interaction, and often orbital relaxation.
• Ignoring the basis set. A small basis may not contain functions flexible or diffuse enough to represent the virtual orbital of interest.

For a particular calculation, inspect the method, basis set, orbital coefficients, and orbital visualization together. If the goal is an excitation energy or electron affinity, use a method designed to calculate that property rather than interpreting a virtual orbital energy by itself.