Hartree–Fock and variational states

Expanding a normalized trial in exact energy eigenstates makes its expectation an average of eigenvalues, bounded below by the ground-state energy. A single Slater determinant automatically imposes fermionic antisymmetry but restricts the allowed many-electron states. Hartree–Fock seeks the lowest expectation among such determinants for a fixed Hamiltonian and one-particle basis. Enlarging the basis can improve that minimum; multiple determinants can capture correlations unavailable through orbital optimization alone.

The upper bound concerns the total ground-state energy of the specified Hamiltonian. It does not guarantee an upper bound for each orbital eigenvalue, an accurate excitation gap, or the correct density in every region. It also does not make a variational result for a different approximate Hamiltonian comparable without qualification. Nuclear positions are usually fixed under a Born–Oppenheimer electronic approximation, and basis truncation introduces an additional source of error. Keep those modeling choices separate from the approximation of using a single determinant.

For occupied spin-orbitals, the one-electron contribution is a sum of kinetic and external-potential expectations. Each pair also contributes a direct Coulomb integral and an exchange integral. The direct term resembles the electrostatic interaction between orbital densities. Exchange results from antisymmetrizing the state; it is not a new classical force added to Coulomb’s law. For a spin-independent interaction, spin-orthogonal orbitals have zero exchange integral because their spin overlap vanishes, although their direct repulsion remains.

The factor of one half in the double sum prevents counting every pair twice. Terms with identical orbital indices cancel between direct and exchange contributions, removing spurious self-interaction for that occupied spin-orbital in Hartree–Fock. This cancellation does not capture all correlation effects. For example, correlated electrons can avoid one another dynamically in ways unavailable to a single determinant. The exchange hole imposed by antisymmetry and the additional correlation hole are conceptually different even though both affect spatial probabilities.

Varying the occupied orbitals while enforcing orthonormality produces a one-particle Fock operator that itself depends on the occupied orbitals. A self-consistent iteration builds this operator from a guessed density, solves an orbital problem, and updates the density. Convergence means that the chosen update rule has reached a fixed point within tolerance. There can be multiple stationary solutions, including saddle points and metastable states; a converged iteration need not identify the lowest determinant. Energy comparisons, orbital-gradient residuals, and stability analysis answer different questions.

Unrestricted orbitals can lower a variational energy by breaking a symmetry that the exact finite-system ground state preserves. This can diagnose a tendency without proving symmetry breaking or the correct phase. Bond dissociation and near-degeneracy can expose the single-determinant restriction. The optimization lab illustrates only gradient descent on a scalar quadratic. Its stable step-size range makes a useful warning about iterative convergence, but it contains no orbital orthogonality, density matrix, exchange integrals, or self-consistent Fock operator. A lower numerical residual should never be confused with smaller model error.