Hartree–Fock theory

We now have the pieces: an electronic Schrödinger equation (chapter 1), the variational principle, and a finite basis (chapter 2). Hartree–Fock (HF) puts them together with one decisive simplification of the wavefunction, and is the foundation on which almost every other method builds. This is exactly what mychk.scf(ref="r").run() computes.

One determinant: the Hartree–Fock ansatz

The exact wavefunction of \(N\) interacting electrons is hopelessly complicated. Hartree–Fock makes the simplest possible antisymmetric guess: put each electron in its own spin-orbital \(\chi_i\) (a spatial orbital times a spin function) and combine them into a single Slater determinant.

Definition 2 (Slater determinant)

Given \(N\) orthonormal spin-orbitals \(\{\chi_i\}\), the Slater determinant

\[\begin{split} \Psi(\mathbf x_1,\dots,\mathbf x_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_1(\mathbf x_1) & \chi_2(\mathbf x_1) & \cdots & \chi_N(\mathbf x_1) \\ \chi_1(\mathbf x_2) & \chi_2(\mathbf x_2) & \cdots & \chi_N(\mathbf x_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf x_N) & \chi_2(\mathbf x_N) & \cdots & \chi_N(\mathbf x_N) \end{vmatrix} \end{split}\]

is automatically antisymmetric (swapping two electrons swaps two rows, flipping the sign) and vanishes if two spin-orbitals are equal (two equal columns) — so it builds in the Pauli principle for free.

A single determinant means each electron feels only the average field of the others, not their instantaneous positions — a mean-field approximation. That is the one big simplification; everything else is exact.

The Hartree–Fock energy

Take the expectation value of the electronic Hamiltonian over this determinant. The result is a sum of a one-electron part and a two-electron part,

\[ E_{\text{HF}} = \sum_{i} \langle i|\hat h|i\rangle + \frac{1}{2}\sum_{i,j}\big( J_{ij} - K_{ij}\big), \]

where \(\hat h = -\tfrac12\nabla^2 - \sum_A Z_A/r_A\) is the one-electron (kinetic + nuclear-attraction) operator, and the two-electron terms are

\[ J_{ij} = \langle ij | ij \rangle = \iint \frac{|\chi_i(\mathbf x_1)|^2\,|\chi_j(\mathbf x_2)|^2}{r_{12}}\,d\mathbf x_1 d\mathbf x_2, \qquad K_{ij} = \langle ij | ji \rangle . \]

\(J_{ij}\) is the Coulomb repulsion between the charge clouds of orbitals \(i\) and \(j\) — classical and intuitive. \(K_{ij}\) is the exchange term: it has no classical analogue and arises purely from antisymmetry. It lowers the energy for same-spin electrons (they avoid each other automatically — the “Fermi hole”), and, importantly, the \(i=j\) terms of \(J\) and \(K\) cancel, so an electron does not spuriously repel itself.

The Fock operator and the self-consistent field

To find the best orbitals, apply the variational principle: minimize \(E_{\text{HF}}\) over the spin-orbitals (keeping them orthonormal). The condition that comes out is a one-electron eigenvalue equation — each orbital is an eigenfunction of an effective operator, the Fock operator,

\[ \hat f\,\chi_i = \varepsilon_i\,\chi_i, \qquad \hat f = \hat h + \sum_{j}\big(\hat J_j - \hat K_j\big), \]

where \(\hat J_j,\hat K_j\) are the Coulomb and exchange operators built from the occupied orbitals. The \(\varepsilon_i\) are the orbital energies. In the finite LCAO basis of chapter 2 this becomes the matrix form we previewed — the Roothaan equations,

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

with the Fock matrix \(F_{\mu\nu} = \langle \phi_\mu | \hat f | \phi_\nu\rangle\) and the overlap \(\mathbf S\).

Here is the catch that shapes every SCF program: \(\hat f\) depends on the orbitals it is supposed to produce (through \(\hat J\) and \(\hat K\), which are built from the occupied orbitals). The equation is nonlinear, so we solve it iteratively — the self-consistent field (SCF) procedure:

1. Start from a guess density (qc-rs’s default is sad).

2. Build the Fock matrix \(\mathbf F\) from the current density.

3. Solve \(\mathbf{F}\mathbf{C}=\mathbf{S}\mathbf{C}\boldsymbol\varepsilon\) for new orbitals; fill the lowest ones with electrons to get a new density.

4. Repeat until the density (and energy) stop changing — self-consistency.

This is exactly the loop you watched in the run(log=...) transcript: the cycle-by-cycle table of the energy and its gradient, converging in a handful of iterations, accelerated by DIIS. The SCF chapter covers the convergence toolkit in depth.

Closed and open shells: RHF, UHF, ROHF

How the spin-orbitals are constrained gives the three flavours you select with ref=:

• RHF (ref="r") — restricted: each spatial orbital holds a paired \(\alpha\) and \(\beta\) electron. For closed-shell molecules (the water in the quickstart).

• UHF (ref="u") — unrestricted: \(\alpha\) and \(\beta\) electrons use different spatial orbitals. The natural choice for radicals (the methyl-radical example), at the cost of some spin contamination.

• ROHF (ref="ro") — restricted-open: open-shell but keeping paired electrons in common spatial orbitals; a spin-pure compromise.

What Hartree–Fock gets right — and wrong

Hartree–Fock is remarkably good: for a molecule like water it already recovers about 99% of the total electronic energy, and it gives sensible structures and trends. But the mean-field ansatz has a built-in blind spot. Because each electron sees only the average field of the others, HF misses the way electrons instantaneously avoid one another. The energy this costs is, by definition, the correlation energy

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

It is a small fraction of the total (that leftover ~1%), but it is chemically decisive — bond energies, reaction barriers, and dispersion all live there. Recovering it is the job of the post-Hartree–Fock methods (MP2, coupled cluster, …; qc-rs’s RI-MP2 family, in the User guide) — and of the alternative route taken by density functional theory, next, which folds correlation into an approximate functional of the density instead.

Note

Why the water dipole was a little large The tutorial dipole and the basis-convergence table both hinted at a “basis and method” error. You have now met both halves: finite basis (chapter 2) and the missing correlation of the mean-field method (this chapter).