Hartree–Fock and the correlation gap: where the orbital energies come from

In a previous post on water we drew a molecular-orbital diagram for \(\mathrm{H_2O}\) and read its levels off the page. Symmetry sorted the atomic orbitals into \(a_1\), \(b_2\), and \(b_1\) boxes; filling eight valence electrons from the bottom gave the configuration \((2a_1)^2(1b_2)^2(3a_1)^2(1b_1)^2\); and we checked the ordering against photoelectron spectroscopy, which knocks electrons out one at a time and reports the energy it takes: about 12.6 eV for \(1b_1\), 14.7 eV for \(3a_1\), 18.5 eV for \(1b_2\), and a deep 32 eV for \(2a_1\).¹ The diagram worked. But we placed those levels by hand — by symmetry, by chemical intuition, and by knowing the answer we were aiming for.

This post is about the machinery that actually computes them. It sits one floor up from the single-particle problems — particle in a box, harmonic oscillator, hydrogen atom — worked out in the fundamentals post, and one floor down from the excited-state, linear-response methods in the post on excited-state frameworks and basis sets. It is the ground-state self-consistent-field theory of Hartree and Fock: the set of equations whose eigenvalues are the levels on the water diagram. We will build it from the bottom — the Hamiltonian, the wavefunction we are allowed to write down, the variational principle that picks the best one, the mean-field equations that result, and the matrix form a computer actually solves — and then we will be honest about what it leaves out. Because the most important number in this whole story is the one Hartree–Fock cannot get: the correlation energy, the gap between the best mean-field answer and the truth.

1. The electronic Hamiltonian

Before any energy level means anything, we need a well-defined energy axis, and that means writing down the operator whose eigenvalues we are after. A molecule is a collection of nuclei and electrons interacting through Coulomb’s law, and the full Hamiltonian couples the motion of both. The first simplification — the one that makes “molecular structure” a meaningful idea at all — is the Born–Oppenheimer approximation: nuclei are thousands of times heavier than electrons, so on the timescale of electronic motion they are effectively frozen. We clamp the nuclei at fixed positions, solve for the electrons in that static field, and read the result as a function of where we put the nuclei.

With the nuclei clamped, the nuclear kinetic energy vanishes and the nuclear–nuclear repulsion becomes a constant we can add at the end. What remains is the electronic Hamiltonian (in atomic units, \(\hbar = m_e = e = 4\pi\varepsilon_0 = 1\)):

\[ \hat{H}_{\mathrm{elec}} = \underbrace{-\frac{1}{2}\sum_{i} \nabla_i^2}_{\text{kinetic}} \;\underbrace{-\sum_{i}\sum_{A} \frac{Z_A}{r_{iA}}}_{\text{electron–nuclear attraction}} \;+\;\underbrace{\sum_{i<j} \frac{1}{r_{ij}}}_{\text{electron–electron repulsion}} . \]

The first two sums are one-electron operators: each term involves a single electron \(i\) moving in the fixed external field of the nuclei. If those were the only terms, the problem would factor. Each electron would see the same fixed potential, the many-electron Schrödinger equation would separate into independent one-electron equations, and we would be back in the comfortable world of the hydrogen atom — orbitals, with energies, filled one at a time. The water diagram assumes exactly this picture: independent electrons in orbitals.

The third sum is what wrecks it. The term \(1/r_{ij}\) couples every pair of electrons: where electron \(i\) goes depends on where electron \(j\) is, instant by instant. It cannot be split into a sum of one-electron pieces, and it is the reason the electronic Schrödinger equation has no closed-form solution for any atom past hydrogen. Everything difficult about quantum chemistry — and everything in this post — is a strategy for coping with that one term. Hartree–Fock’s strategy is to approximate it; the correlation energy of §7 is the price of the approximation.

2. The Slater determinant

Even before we touch the repulsion, the form of the wavefunction is constrained. Electrons are fermions, and the wavefunction of a many-fermion system must be antisymmetric: swap the full coordinates (space and spin) of any two electrons and \(\Psi\) must change sign,

\[ \Psi(\mathbf{x}_1,\dots,\mathbf{x}_i,\dots,\mathbf{x}_j,\dots,\mathbf{x}_N) = -\,\Psi(\mathbf{x}_1,\dots,\mathbf{x}_j,\dots,\mathbf{x}_i,\dots,\mathbf{x}_N), \]

where \(\mathbf{x} = (\mathbf{r}, \omega)\) bundles position and spin. This is the Pauli principle in its deepest form, and it has an immediate consequence: set \(\mathbf{x}_i = \mathbf{x}_j\) and antisymmetry forces \(\Psi = 0\). Two electrons cannot occupy the same spin-orbital — not as a rule bolted on afterward, but as arithmetic.

The simplest wavefunction that builds in antisymmetry automatically is a determinant. Take \(N\) one-electron spin-orbitals \(\chi_1,\dots,\chi_N\) (each a spatial orbital times a spin function) and assemble the Slater determinant

\[ \Psi_{\mathrm{HF}}(\mathbf{x}_1,\dots,\mathbf{x}_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{vmatrix}. \]

A determinant changes sign when two rows are swapped — that is exchanging two electrons, so antisymmetry is automatic — and it vanishes when two columns are equal — that is two electrons in the same spin-orbital, so Pauli is automatic. The determinant is not just a convenient antisymmetric function; it is the minimal one, the single-configuration ansatz, and it is the precise mathematical content of the phrase “each electron is in its own orbital.” The water configuration \((2a_1)^2(1b_2)^2(3a_1)^2(1b_1)^2\) names exactly which spin-orbitals fill the columns. Hartree–Fock theory is what you get when you insist the wavefunction is a single such determinant and then make it as good as it can be.

3. The variational principle

“As good as it can be” needs a yardstick, and the variational principle is the yardstick. For any normalized trial wavefunction \(\Psi\), the expectation value of the Hamiltonian is an upper bound on the true ground-state energy \(E_0\):

\[ E[\Psi] = \frac{\langle \Psi | \hat{H} | \Psi \rangle}{\langle \Psi | \Psi \rangle} \;\geq\; E_0 . \]

The proof is one line: expand \(\Psi\) in the exact (unknown) eigenstates \(\hat{H}|\Psi_n\rangle = E_n|\Psi_n\rangle\), and since every \(E_n \geq E_0\), the weighted average \(E[\Psi]\) cannot dip below \(E_0\). Equality holds only when \(\Psi\) is the ground state. This turns “solve the Schrödinger equation” — a differential equation we cannot solve — into “minimize a number,” which we can attack.

Hartree–Fock applies the principle inside the restricted arena of single determinants. Among all antisymmetric wavefunctions, the true ground state is some hopelessly complicated object; we are not searching there. We are searching only over single Slater determinants, asking: of all the ways to choose the spin-orbitals \(\{\chi_i\}\), which determinant has the lowest energy? That lowest single-determinant energy is the Hartree–Fock energy \(E_{\mathrm{HF}}\), and because the determinant is a legitimate trial function, \(E_{\mathrm{HF}} \geq E_0\) — always above the truth, by an amount we will name in §7. The minimization is a variational problem in the orbitals themselves, and carrying it out is what produces the equations that bear Hartree’s and Fock’s names.

4. The Hartree–Fock equations

Minimizing \(E[\Psi_{\mathrm{HF}}]\) with respect to the spin-orbitals, subject to keeping them orthonormal, yields a set of coupled one-electron eigenvalue equations²:

\[ \hat{f}\,\chi_i = \varepsilon_i\,\chi_i . \]

This looks like an ordinary single-particle Schrödinger equation, and that resemblance is the whole achievement: the intractable many-electron problem has been recast as a one-electron problem in an effective potential. The operator \(\hat{f}\) is the Fock operator,

\[ \hat{f} = \hat{h} + \sum_{j}^{\text{occ}} \big( \hat{J}_j - \hat{K}_j \big), \]

where \(\hat{h} = -\tfrac{1}{2}\nabla^2 - \sum_A Z_A/r_A\) is the bare one-electron core Hamiltonian — kinetic energy plus attraction to the nuclei — and the sum runs over all occupied orbitals. The two new operators are where the electron–electron repulsion has gone, and they are worth dwelling on because one of them is classical and the other is not.

The Coulomb operator \(\hat{J}_j\) acts on \(\chi_i\) as

\[ \hat{J}_j(\mathbf{x}_1)\,\chi_i(\mathbf{x}_1) = \left[ \int \frac{|\chi_j(\mathbf{x}_2)|^2}{r_{12}}\, d\mathbf{x}_2 \right] \chi_i(\mathbf{x}_1) , \]

and it is exactly what intuition expects: electron \(i\) feels the smeared-out electrostatic repulsion of the charge cloud \(|\chi_j|^2\) of every other electron. This is the mean field. Each electron no longer tracks the others moment by moment; it moves in the average potential they create. That is the central approximation of Hartree–Fock, and it is exactly where correlation will leak out — because real electrons do dodge each other instantaneously, and an average cannot capture a dodge.

The exchange operator \(\hat{K}_j\) has no classical analogue at all:

\[ \hat{K}_j(\mathbf{x}_1)\,\chi_i(\mathbf{x}_1) = \left[ \int \frac{\chi_j^*(\mathbf{x}_2)\,\chi_i(\mathbf{x}_2)}{r_{12}}\, d\mathbf{x}_2 \right] \chi_j(\mathbf{x}_1) . \]

Notice that \(\chi_i\) and \(\chi_j\) have traded places between the integrand and the result — the operator is nonlocal, mixing the value of the orbital at one point with its overlap somewhere else. It arises purely from antisymmetry: it is the determinant’s bookkeeping for the fact that you cannot tell electron \(i\) from electron \(j\). Its physical effect is real and measurable. Exchange keeps electrons of like spin apart — the “Fermi hole” — which lowers their mutual repulsion and is a genuine piece of the energy, not a correction. Hartree–Fock treats exchange exactly. What it misses is the analogous correlation between electrons of opposite spin, and the residual same-spin correlation beyond the Fermi hole. That distinction is the seed of §7.

The eigenvalues \(\varepsilon_i\) that come out of \(\hat{f}\chi_i = \varepsilon_i\chi_i\) are the orbital energies. These are not an analogy for the levels on the water diagram — they are those levels. When we wrote \(1b_1\) at \(-12.6\) eV and \(2a_1\) deep below, we were sketching the \(\varepsilon_i\) of the water Fock operator. The rest of this post is about how to compute them and how much to trust them.

5. Roothaan–Hall: the equations a computer solves

The Fock equations are still differential equations in continuous functions. The move that made quantum chemistry computational, due to Roothaan and independently Hall, is to expand each molecular orbital as a linear combination of atomic orbitals (LCAO) — a fixed set of \(K\) basis functions \(\{\phi_\mu\}\) centered on the atoms:

\[ \psi_i = \sum_{\mu=1}^{K} C_{\mu i}\,\phi_\mu . \]

Solving for the orbitals now means solving for the coefficients \(C_{\mu i}\) — finitely many numbers. Substituting this expansion into the Fock equation and projecting onto each basis function turns the differential equation into a matrix equation, the Roothaan–Hall equation³:

\[ \mathbf{F}\,\mathbf{C} = \mathbf{S}\,\mathbf{C}\,\boldsymbol{\varepsilon} . \]

Four objects appear, each with a clean meaning. The overlap matrix \(S_{\mu\nu} = \int \phi_\mu^* \phi_\nu \, d\mathbf{r}\) records how non-orthogonal the atomic orbitals are — atomic orbitals on different centers are not orthogonal, so this is not the identity, and that is why the equation is a generalized eigenproblem rather than a plain one. The Fock matrix \(F_{\mu\nu} = \int \phi_\mu^* \hat{f}\, \phi_\nu \, d\mathbf{r}\) is the Fock operator in the basis. The columns of \(\mathbf{C}\) are the orbital coefficients, and the diagonal matrix \(\boldsymbol{\varepsilon}\) holds the orbital energies. The Fock matrix splits into the one-electron core integrals plus a two-electron part built from the density matrix

\[ P_{\mu\nu} = \sum_{i}^{\text{occ}} n_i\, C_{\mu i}\, C_{\nu i}^{*} \qquad\Longrightarrow\qquad F_{\mu\nu} = H^{\text{core}}_{\mu\nu} + \sum_{\lambda\sigma} P_{\lambda\sigma} \left[ (\mu\nu\,|\,\lambda\sigma) - \tfrac{1}{2}(\mu\lambda\,|\,\nu\sigma) \right], \]

where \((\mu\nu\,|\,\lambda\sigma)\) are the two-electron repulsion integrals over the basis. (The two bracketed terms are the matrix incarnations of \(\hat{J}\) and \(\hat{K}\) from §4.) Here we have specialized to the closed-shell (restricted) case: water’s electrons are paired, so each occupied spatial orbital holds two electrons (\(n_i = 2\)). That factor of two lives in \(\mathbf{P}\), which is why exchange enters with a one-half — the spin-orbital Fock operator of §4 collapses to this spatial form once every orbital is doubly occupied.

Here is the knot at the center of the whole method. The Fock matrix \(\mathbf{F}\) depends, through \(\mathbf{P}\), on the very coefficients \(\mathbf{C}\) we are trying to find. You cannot build the operator without already knowing its eigenvectors. The escape is iteration: guess a density, build \(\mathbf{F}\), solve the eigenproblem for a new \(\mathbf{C}\), build a new density, and repeat until the density that comes out matches the one that went in. This is the self-consistent field (SCF) procedure, and “self-consistent” is literal — the field and the orbitals it produces must agree. Figure 1 traces one turn of the cycle.

Figure 1. The self-consistent-field loop: from a trial density build the Fock matrix, solve the Roothaan–Hall eigenproblem, form a new density from the occupied orbitals, and repeat until the density that comes out matches the one that went in.

Symmetry block-diagonalizes the problem

This is where the water post and this one fuse. The secular problem \(\mathbf{F} \mathbf{C} = \mathbf{S}\mathbf{C}\boldsymbol{\varepsilon}\) is a \(K \times K\) eigenproblem, and in general every basis function can mix with every other. But the Fock operator commutes with every symmetry operation of the molecule: it has the full \(C_{2v}\) symmetry of bent water. A standard result then says that \(\mathbf{F}\) (and \(\mathbf{S}\)) cannot connect basis functions of different irreducible representations — those matrix elements are exactly zero. If we first combine the raw atomic orbitals into the symmetry-adapted combinations of §2 of the water post — the \(a_1\), \(b_2\), and \(b_1\) sets — the Fock matrix becomes block-diagonal (Figure 2):

Figure 2. In a symmetry-adapted basis the Fock matrix is block-diagonal: the \(a_1\) (\(3\times3\)), \(b_2\) (\(2\times2\)), and \(b_1\) (\(1\times1\)) blocks carry no matrix elements between them, so the secular problem splits into one independent eigenproblem per irrep.

The big eigenproblem factors into independent small ones — one per irrep — and the labels on those blocks are exactly the \(a_1\), \(b_2\), \(b_1\) labels from the water diagram. Orbitals of different symmetry do not mix because the off-block elements are zero; that is not a modeling choice but a consequence of the molecule’s shape.

The smallest block makes “solve the eigenproblem” concrete. The \(b_1\) block is \(1\times 1\): in a minimal valence description the only function of \(b_1\) symmetry is the oxygen \(2p_x\) orbital sticking out of the molecular plane, with no hydrogen partner to pair with. A \(1\times 1\) secular “matrix” is already its own eigenvalue, \(\varepsilon = F_{b_1 b_1}\), and the orbital is the bare atomic orbital unchanged. That is the algebraic reason \(1b_1\) is a pure, nonbonding oxygen lone pair — it has nothing to combine with, so the variational principle leaves it alone. The “lonely box” of the water post is a one-dimensional block here.

The \(a_1\) block is where mixing happens. Take a minimal \(a_1\) set — the oxygen \(2s\), the oxygen \(2p_z\) along the symmetry axis, and the symmetric hydrogen combination \(\phi_{\mathrm{H}^+} = (1s_A + 1s_B)/\sqrt{2}\). The orbital energies in this symmetry are the roots \(\varepsilon\) of the \(3\times 3\) secular determinant

\[ \begin{vmatrix} F_{ss} - \varepsilon S_{ss} & F_{sz} - \varepsilon S_{sz} & F_{sh} - \varepsilon S_{sh} \\ F_{zs} - \varepsilon S_{zs} & F_{zz} - \varepsilon S_{zz} & F_{zh} - \varepsilon S_{zh} \\ F_{hs} - \varepsilon S_{hs} & F_{hz} - \varepsilon S_{hz} & F_{hh} - \varepsilon S_{hh} \end{vmatrix} = 0 , \]

whose three roots are precisely the bonding \(2a_1\), the roughly nonbonding \(3a_1\), and the empty antibonding \(4a_1^{*}\) of the diagram. The \(b_2\) block is a \(2\times 2\) between the oxygen \(2p_y\) and the antisymmetric hydrogen combination, \((1s_A - 1s_B)/\sqrt{2}\), giving the bonding \(1b_2\) and its antibonding partner. The point is not the numbers — those come out of the SCF — but the structure: the hand-drawn correlation lines of the water MO diagram are the off-diagonal Fock elements \(F_{sz}\), \(F_{sh}\), \(F_{zh}\) inside one symmetry block, and “which orbitals mix” is “which functions share a block.”

6. Koopmans’ theorem and the water spectrum

We now have orbital energies \(\varepsilon_i\). The water post compared them to ionization energies — the cost of removing an electron — as if the two were the same thing. Koopmans’ theorem is the bridge, and it is almost suspiciously simple⁴:

\[ \mathrm{IE}_i \approx -\varepsilon_i . \]

The ionization energy of the electron in orbital \(i\) is minus its orbital energy. This is why the photoelectron spectrum can be read straight off an MO diagram, and it is the single fact that made the water diagram checkable against experiment.

But the approximation rests on two assumptions that should be stated out loud, because both are false in detail and the post would be dishonest to hide them. First, frozen orbitals: Koopmans assumes that when you eject an electron, the remaining \(N-1\) electrons do not move — they stay in exactly the orbitals they occupied in the neutral. In reality the remaining electrons relax inward toward the now-less-shielded nuclei, which lowers the energy of the cation and therefore lowers the true ionization energy below \(-\varepsilon_i\). Second, no correlation: \(\varepsilon_i\) is a Hartree–Fock quantity, so it inherits the mean-field error of §7. The neutral, with more electrons, has more correlation energy than the cation, and that difference works in the opposite direction, raising the true IE relative to \(-\varepsilon_i\).

The reason Koopmans is useful at all is that these two errors have opposite signs and, for outer-valence orbitals, they substantially cancel. For deep orbitals the cancellation fails, because ejecting a tightly bound electron triggers a large relaxation that the frozen-orbital picture cannot see. Water shows the pattern cleanly. Lining up representative near-Hartree–Fock orbital energies against the experimental ionization energies from the water post (Table 1):

Table 1. Koopmans estimates \(-\varepsilon_i\), from representative near-Hartree–Fock orbital energies, against the experimental vertical ionization energies of water. The overestimate runs a little over an electron-volt across the outer valence and widens to several eV for the deep \(2a_1\).

(The exact \(-\varepsilon\) values shift by a few tenths of an eV with the basis set; these are representative near-HF-limit numbers, and the point is the pattern, not the third digit.) For the three outer orbitals Koopmans lands a little over an electron-volt high — a consistent overestimate, exactly as the relaxation-versus- correlation story predicts, with the two errors nearly but not perfectly cancelling. For the deep \(2a_1\) the error blows out to several eV: removing an electron from an orbital built largely of oxygen \(2s\) leaves a compact hole that the other electrons relax around dramatically, and the frozen-orbital assumption is simply too crude. The lesson is the right amount of trust: Koopmans gets the ordering right and the outer-valence energies right to about a volt, which is plenty to assign a spectrum, but it is not a quantitative theory of ionization, and the deep levels are where it visibly frays.

7. The correlation gap

Everything so far has lived inside the single-determinant approximation, and now we pay for it. The exact nonrelativistic energy of the molecule, at fixed nuclei and in a given basis, is lower than the Hartree–Fock energy. The difference defines the correlation energy:

\[ E_{\mathrm{corr}} = E_{\mathrm{exact}} - E_{\mathrm{HF}} \;<\; 0 . \]

By construction it is the part of the energy that the mean field cannot reach. The variational principle of §3 guarantees its sign: since the Hartree–Fock determinant is just one trial function, \(E_{\mathrm{HF}} \geq E_{\mathrm{exact}}\), so \(E_{\mathrm{corr}}\) is never positive. (Strictly, both energies must be taken in the same one-particle basis; the limiting value uses the Hartree–Fock basis-set limit and the exact energy in a complete basis.)

Figure 3. The correlation gap. \(E_{\mathrm{HF}}\) sits above the exact nonrelativistic energy; enlarging the basis lowers \(E_{\mathrm{HF}}\) only as far as the Hartree–Fock limit, and the residual distance down to \(E_{\mathrm{exact}}\) is the correlation energy \(E_{\mathrm{corr}}\).

The gap drawn in Figure 3 is small in absolute terms — typically under 1% of the total electronic energy — but it is decisive in chemistry, because the quantities we care about are energy differences of the same order: bond energies, reaction barriers, excitation energies. A method that recovers 99% of the energy can still be useless for a barrier height if the missing 1% varies between reactant and product. So the correlation energy is not a rounding error to wave away; it is frequently the whole answer.

It helps to split it in two. Dynamic correlation is the instantaneous dodging the mean field smears over: each electron carves out a “Coulomb hole” around itself that the average potential of §4 ignores. It is the small, ubiquitous, additive correlation present in every system, and it is the correlation contribution lurking inside the Koopmans errors of §6. Static (or near-degeneracy) correlation is different in kind: it appears when a single determinant is qualitatively wrong because two or more configurations are close in energy and contribute comparably — stretched bonds, diradicals, transition-metal centers. There, no amount of patching one determinant helps, because the starting picture itself is broken.

Two broad routes go past Hartree–Fock. The wavefunction route keeps the determinant as a reference and adds others on top. Møller–Plesset perturbation theory treats the fluctuation potential as a perturbation; its second-order correction, the workhorse MP2, is

\[ E^{(2)} = \sum_{i<j}^{\text{occ}} \sum_{a<b}^{\text{virt}} \frac{\big| \langle ij \,\|\, ab \rangle \big|^2} {\varepsilon_i + \varepsilon_j - \varepsilon_a - \varepsilon_b}, \]

a sum over excitations of electron pairs from occupied orbitals \(i,j\) into virtual orbitals \(a,b\) — note that it is built entirely from the same \(\varepsilon\)’s and two-electron integrals we already have. Configuration interaction diagonalizes the Hamiltonian in a space of many determinants; carried to its limit (full CI) in a given basis it is exact for that basis — and combinatorially unaffordable for all but the smallest systems, which is why it serves mostly as a benchmark. Coupled cluster reorganizes the same idea into an exponential ansatz that captures dynamic correlation compactly; CCSD(T) — singles, doubles, and a perturbative treatment of triples — is accurate and affordable enough to be the de facto “gold standard” for well-behaved molecules.⁵ The density-functional route is different in spirit: Kohn–Sham DFT is exact in principle, folding exchange and correlation into a single exchange–correlation functional of the electron density — but that functional is unknown, and in practice one chooses among approximations, trading the systematic improvability of the wavefunction hierarchy for far lower cost.

And that hands off cleanly to where this series goes next. Every method just named has a linear-response, excited-state sibling: TD-DFT is the response version of Kohn–Sham, EOM-CCSD the response version of coupled cluster, and CASSCF the way to handle the static correlation that defeats a single reference — and those are exactly the methods worked through in the post on excited-state frameworks and basis sets. Ground-state Hartree–Fock is the floor they all stand on.

The equations read as one piece

Step back and the chain is short and tight. The electronic Hamiltonian (§1) is honest but unsolvable, wrecked by a single pairwise term. Antisymmetry forces the wavefunction into a determinant (§2); the variational principle (§3) tells us to pick the best determinant; and “best” turns out to mean the orbitals satisfy the Fock equations (§4), in which the feared repulsion has been replaced by an average field plus an exchange term that antisymmetry demands. Cast in a basis, that becomes a matrix eigenproblem solved self-consistently (§5), and the molecule’s symmetry breaks it into the very blocks — \(a_1\), \(b_2\), \(b_1\) — we had labelled by hand on the water diagram. The eigenvalues are the orbital energies; Koopmans (§6) turns them into the ionization energies we measured, well enough for the outer valence and visibly worse for the deep core. And the gap between this whole construction and reality (§7) is the correlation energy, the exact price of having replaced “what the other electrons are doing right now” with “what they do on average.”

That is the one story. The variational principle says get as low as you can; the single determinant says but only within the mean field; and the correlation energy is the distance between those two commands. The water diagram we read off the page in the previous post was the output of this machine, run once and rounded. Knowing the machine, the same diagram reads differently: not a set of facts to memorize, but the converged solution of an eigenvalue problem that the molecule’s own symmetry was kind enough to make small.

References