Orthogonality properties of states, configurations, and orbitals

This manuscript explores the orthogonality constraints on configurations and orbitals subject to the property that states are mutually orthogonal. The orthogonality constraints lead to properties that affect the description of chemical systems. When states are described as linear combinations of (orthogonal or non-orthogonal) configurations, the coefficient matrix (mapping configurations to states) diagonalises S⁻¹H. Therefore, single-configuration states are only possible in one-electron systems: non-orthogonal configurations yield single-configuration states only if S⁻¹H is diagonal, but this would violate the orthonormalisation constraint. Further, the coefficient matrix is not constrained to be square (the number of configurations may differ from the number of configurations). Similarly, the orbitals used to construct configurations may also be orthogonal or non-orthogonal; orbitals are only required to be mutually orthogonal at the one-electron limit. Orthogonal orbitals are generally preferred due to their mathematical and conceptual simplicity, leading to fictitious unoccupied orbitals. Since the Fock operator is orthogonality agnostic, non-orthogonal (occupied) orbitals can be generated by solving the Fock equation independently for each electron; the virtual orbitals produced by this conception are true excitation orbitals as they are eigensolutions of the Fock operator. Additionally, we show that the number of molecular orbitals generated is not restricted to the number of atomic orbitals (or basis functions) employed in the computation. This manuscript explores the mathematical relationships that need to be satisfied under the various orthogonality regimes. We also present mathematical relationships that provide results that are independent of the orthogonality approximation within a particular computational method.