Hartree-Fock and What a Basis Set Decides

Molecular mechanics models shapes with fitted forces. Hartree-Fock (HF) instead represents electrons with orbitals. It optimises these orbitals using nuclear attraction, average electron repulsion and exchange. Exchange follows from antisymmetry: swapping identical electrons changes the sign of the total wavefunction.

In the self-consistent field (SCF) cycle, guess orbitals, build the field, solve for new orbitals and repeat. Here, convergence requires changes in energy and electron density to meet chosen tolerances. This does not prove an exact energy or the lowest HF solution. More cycles can tighten convergence, but cannot add basis functions or missing electron correlation.

HF uses one Slater determinant, an antisymmetric product of orbitals. It includes exchange but misses electron correlation beyond this approximation. At fixed nuclei, correlation energy is the exact ground-state electronic energy minus the minimum complete-basis HF energy for the same Hamiltonian. It is zero or negative. Finite-basis error is separate.

Each molecular orbital combines basis functions. A minimal basis uses a small set of atomic functions: for carbon, one 1s, one 2s and three 2p functions. Split-valence sets use several radial functions for valence orbitals, allowing their radial sizes to vary.

Polarisation functions add higher angular momentum functions, such as p functions on hydrogen or d functions on carbon. These allow density to change direction in bonding. Split-valence adds radial flexibility; polarisation adds angular flexibility. Neither changes the electron count.

Diffuse functions decay slowly and describe density far from nuclei. They often matter for anions and weakly interacting systems. A compact basis can restrict this outer density. Adding diffuse functions can reduce that basis error and lower the energy, but does not add missing electron correlation. A lower HF energy alone does not prove accuracy.