Molecules with multiconfigurational wave functions play a key role across chemistry, including excited states, dissociating bonds, and reaction pathways. Recently, nonorthogonal configuration interaction (NOCI) using a basis of multiple Hartree-Fock (HF) states has been proposed to predict electronic energies in this type of system. However, NOCI has previously been hindered by the disappearance of HF states as the molecular structure changes, creating kinks and discontinuities in the energy. This thesis develops a new theory - holomorphic Hartree-Fock (h-HF) - to analytically extend HF states across all molecular structures. By removing the complex-conjugation of orbital coefficients from the conventional HF equations, h-HF theory forms the complex-analytic continuation of real HF theory. When real HF states disappear, their h-HF counterparts continue to exist with complex orbital coefficients and provide a continuous basis for NOCI.

To rigorously underpin h-HF theory, it is mathematically proved that every two-electron h-HF state must exist for all molecular structures. It is then shown that h-HF theory provides new insights into the nature of HF states in general. For example, enforcing particular symmetry conditions on the h-HF wave function is found to guarantee real h-HF energies even for a non-Hermitian effective Hamiltonian. Furthermore, by analytically continuing the electron-electron interaction itself into the complex plane, discrete h-HF states are shown to connect as multiple sheets of a continuous Riemann surface. These theoretical insights open entirely new avenues of research for electronic structure theory.

The guaranteed existence of h-HF states across all molecular structures now allows a general NOCI approach to be developed for computing potential energy surfaces. This NOCI approach is shown to provide a similar accuracy to state-of-the-art multireference methods for predicting static electron correlation, while requiring simpler computational optimisation. Finally, a rigorous second-order perturbative correction- NOCI-PT2 - is derived and shown to yield quantitative accuracy alongside equivalent multireference perturbation theories. Ultimately, the development of h-HF theory in this thesis allows NOCI and NOCI-PT2 to be established as quantitative approaches for predicting multireference potential energy surfaces in chemical processes.