A mathematical and computational review of Hartree-Fock SCF methods in quantum chemistry

We present a review of the fundamental topics of Hartree-Fock theory in quantum chemistry. From the molecular Hamiltonian, using and discussing the Born-Oppenheimer approximation, we arrive at the Hartree and Hartree-Fock equations for the electronic problem. Special emphasis is placed on the most relevant mathematical aspects of the theoretical derivation of the final equations, and on the results regarding the existence and uniqueness of their solutions. All Hartree-Fock versions with different spin restrictions are systematically extracted from the general case, thus providing a unifying framework. The discretization of the one-electron orbital space is then reviewed and the Roothaan-Hall formalism introduced. This leads to an exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing on algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (especially those related to the design of linear-scaling methods) are commented on and linked to the issues discussed. The paper is intentionally introductory and rather self-contained, and may be useful for non-experts intending to use quantum chemical methods in interdisciplinary applications. Moreover, much material that can be found scattered in the literature has been put together to facilitate comprehension and to serve as a handy reference.