Kohn-Sham Orbitals

The interacting many-electron problem is unsolvable, so Kohn-Sham DFT swaps it for one that isn't: $N$ electrons that ignore each other completely, moving in a single shared potential $v_s(\mathbf{r})$. The swap comes with one non-negotiable condition — $v_s$ is defined as whatever potential makes the non-interacting density equal the true interacting density, point by point.

The ${\varphi }_i$ are the Kohn-Sham orbitals. Nothing in the construction says they are wavefunctions of anything physical — the fictitious electrons don't repel, so their determinant is not the molecule's wavefunction. The density is the only quantity the theory certifies. Everything else about the orbitals is a bonus we have to argue for case by case, which is exactly what the rest of this page does.

A converged SCF hands you a ladder: one energy ${\epsilon }_i$ per orbital, filled from the bottom with two electrons per rung. For water in a 6-31G basis that is 13 rungs — 5 occupied, 8 empty. The highest filled rung is the HOMO, the lowest empty one the LUMO, and the empty rungs are called virtual orbitals: solutions of the same one-electron equation that no electron actually occupies.

The ladder is the molecule's cast of characters. Chemistry happens at the frontier — the HOMO is where the molecule is easiest to ionize or attack, the LUMO where an incoming electron or an excited one lands. And spectroscopy, on this site's pages, is literally bookkeeping on this ladder: an excitation is an electron leaving rung $i$ for rung $a$. The run below prints the actual ladder, twice — once per mean field — so you can see how much the rungs depend on who built them.

How literally can you read the ${\epsilon }_i$? There is exactly one rigorous statement: in exact Kohn-Sham theory the HOMO energy is minus the vertical ionization potential, ${\epsilon }_{\mathrm{HOMO}}=-I$. The asymptotic decay of the density is governed by $I$, and the KS system must reproduce that decay, so its top level has no freedom. Every other eigenvalue — each deeper ${\epsilon }_i$, the LUMO, all the virtuals — is scaffolding with no exact meaning.

And even the one rigorous statement holds only for the exact functional, which nobody has. With PBE, water's $-{\epsilon }_{\mathrm{HOMO}}$ comes out at 6.2 eV against an experimental IP of 12.6 eV — off by a factor of two, the signature self-interaction error of semilocal functionals. Yet the same PBE calculation, asked for the IP the legitimate way (energy of cation minus energy of neutral), gives 12.4 eV. The functional's energies are far better than its eigenvalues. Keep the three gaps straight, too: the KS HOMO-LUMO gap, the fundamental gap $I-A$ (ionize here, attach there), and the optical gap (lowest excitation) are three different numbers, and only the first one is on the ladder.

Here is the structural difference from Hartree-Fock that matters for everything downstream. In Kohn-Sham theory every orbital — occupied and virtual alike — solves the same equation with the same local potential $v_s(\mathbf{r})$. A KS virtual is the answer to "where could one of these $N$ electrons go?" In Hartree-Fock the exchange operator acts differently: an occupied orbital feels the other $N-1$ electrons, but a virtual orbital feels all $N$ — it describes an extra electron stapled onto the molecule, not an excited resident.

The consequence is visible in the run below: HF virtuals sit too high (water's HF gap is 19.2 eV, PBE's is 7.2 eV) and are too diffuse, because they are anion states in disguise. For building excitations — which move a resident electron, not attach a new one — the KS virtuals are the better-shaped raw material, even from a functional whose eigenvalues are individually unreliable. Differences ${\epsilon }_a-{\epsilon }_i$ within one consistent ladder inherit a cancellation that absolute eigenvalues don't get.

This page exists because of what comes next. Take the occupied rungs $i$ and the virtual rungs $a$; every pair $i\to a$ is a particle-hole excitation, and those pairs form the basis in which Casida's equation is written. The orbital-energy differences ${\epsilon }_a-{\epsilon }_i$ are the zeroth-order spectrum — the diagonal of the Casida matrix — and the orbitals themselves are the ingredients of every coupling element in it.

So the working attitude of linear-response TDDFT is: trust the set, not the individual numbers. The KS ladder is a well-conditioned starting point whose errors are systematic and correctable — and correcting them is precisely the job of the response machinery in the pages ahead.