  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 H$_{2}$ and H$_{4}^{2+}$ and report the emergence of holomorphic
solutions at points of symmetry breaking. Finally, we find these holomorphic
solutions for H$_{4}$ 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.
