We investigate the existence of holomorphic Hartree–Fock solutions using a revised SCF algorithm. We use this algorithm to study the Hartree–Fock solutions for H2 and H42+ and report the emergence of holomorphic solutions at points of symmetry breaking. Finally, we find these holomorphic solutions for H4 and use them as a basis for Non-Orthogonal Configuration Interaction at a range of rectangular geometries and show them to produce energies in good agreement with Full Configuration Interaction.