Many Electrons, One Determinant: Hartree--Fock Theory, Basis Sets, and Molecular Integrals

Abstract

This chapter accompanies the lectures on Hartree--Fock theory and on basis sets and molecular integrals at the \href{https://www.esqc.org}{European Summer School of Quantum Chemistry (ESQC) 2026}. Intended for doctoral students, it develops the single-determinant description of electronic structure from antisymmetry and the variational principle to the orbital equations and their finite-basis matrix representation. Detailed derivations connect the energy functional, the Fock operator, excited determinants, and the interpretation of orbital energies. Restricted, unrestricted, and generalized formulations show how the choice of variational space affects spin symmetry, symmetry breaking, and the limitations of the Hartree--Fock approximation. The computational development covers orthogonalization, self-consistent-field iteration, convergence acceleration, overlap-based occupation selection, and orbital analysis, distinguishing orbital stationarity from stability. A connection to second-order Møller--Plesset perturbation theory explains the correlation-energy expression and the cost of transforming electron-repulsion integrals from atomic to molecular orbitals. The basis-set and integral development examines radial and angular flexibility, contraction, basis-set incompleteness, and superposition error. The Gaussian product theorem leads to one- and two-electron integrals, recurrence relations, integral screening, and low-rank representations. Worked examples and exercises with detailed solutions connect these derivations to practical calculations. Historical perspectives and explicit notation conventions provide context for the approximations and algorithms that underpin molecular electronic-structure theory.