Particle-Hole Excitations

The ground state of a closed-shell molecule, at the mean-field level, is one Slater determinant $|{\Phi }_0⟩$: the $N/2$ lowest orbitals, each doubly filled. The simplest thing you can do to it is a single substitution — remove the electron in occupied orbital ${\varphi }_i$, place it in virtual orbital ${\varphi }_a$ — producing the singly-excited determinant

Read the operators right to left: ${\hat{a}}_i$ digs a hole in the occupied sea, ${\hat{a}}_a^{†}$ drops the particle into an empty level. That ordered pair $(i,a)$ — hole index, particle index — is the whole identity of the excitation. Every absorption spectrum on this site starts life as a list of these substitutions.

How many single excitations does a molecule have? Exactly $N_{\mathrm{occ}}\times N_{\mathrm{vir}}$ — a rectangular grid, one axis for the hole, one for the particle. Water in 6-31G: 13 basis functions, 5 occupied, 8 virtual, so the grid is $5\times 8=40$ pairs. Not "about forty" — exactly forty, fixed the moment you choose the basis, before any physics happens.

This grid is the vector space low-energy spectroscopy lives in. A general excited state, in linear response, is a vector $X_{ia}$ over the grid — so much of $5\to 6$, a little of $4\to 6$, and so on — and the Casida matrix for water/6-31G is a $40\times 40$ matrix indexed by pairs on both sides. When quantum chemists say "the singles space," this grid is what they mean; grow the basis and the grid grows with the virtual axis.

Why insist on "particle-hole" instead of just "excited electron"? Because the vacancy left behind is not passive bookkeeping — it is a positively-charged quasiparticle with its own dynamics. The promoted electron and its hole attract each other Coulombically, and that attraction is enormous: in the run below, the bare cost of the $5\to 6$ promotion is 19.2 eV, but the actual lowest excitation lands at 9.4 eV. The electron-hole attraction claws back almost 10 eV — half the zeroth-order energy.

A bound electron-hole pair is an exciton, and the language is deliberately shared with solid-state physics: same object, wildly different binding scale (eV in a small molecule, meV in a semiconductor, because the pair delocalizes). The interaction terms that the Casida coupling matrix adds on top of the bare grid are precisely this electron-hole physics — restoring what the frozen mean field left out.

If the electrons truly didn't interact, the excitation spectrum would cost nothing to compute: promoting $i\to a$ would cost exactly the orbital-energy difference,

and the full spectrum would be the sorted list of all forty differences. That list is the bare particle-hole ladder, and it is genuinely the zeroth order of the real theory: the Casida matrix has exactly these differences on its diagonal, with interaction supplying everything off it (and part of the diagonal too). For water's lowest pairs the bare ladder reads 19.18, 20.80, 21.80 eV — wrong by a factor of two, but wrong systematically, and in the right order for the first few states.

This is the same trust-the-differences logic as on the orbital ladder page: individual eigenvalues are unreliable, but differences within one consistent ladder are a well-conditioned starting point that the response machinery then corrects.

One spatial pair $(i,a)$ is secretly two excitations. The electron carries spin, and after the promotion the open-shell pair can couple to total spin 0 or 1:

The singlet is what light makes (photons don't flip spins); the triplet is dark but always lies lower, because parallel spins keep the electrons apart and reduce their repulsion — Hund's rule, relocated to excited states. At zeroth order the two are degenerate (both cost ${\epsilon }_a-{\epsilon }_i$); the interaction splits them. That is why the Casida problem for a closed-shell molecule falls into separate singlet and triplet blocks of size $N_{\mathrm{occ}}{N}_{\mathrm{vir}}$ each — same grid, two spin channels.