A Mathematical and Computational Review of Hartree-Fock SCF
February 1, 2008 5:24 Molecular Physics introSCF Molecular Physics, Vol. 00, No. 00, DD Month 200x, 1–64 A mathematical and computational review of Hartree-Fock SCF methods in Quantum Chemistry Pablo Echenique∗ and J. L. Alonso †‡ †‡ Departamento de F´ısica Te´orica, Universidad de Zaragoza, † Pedro Cerbuna 12, 50009, Zaragoza, Spain. Instituto de Biocomputaci´on y F´ısica de los Sistemas Complejos (BIFI), ‡ Edificio Cervantes, Corona de Arag´on 42, 50009, Zaragoza, Spain. (May 2007 ) We present here a review of the fundamental topics of Hartree-Fock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discussing the Born-Oppenheimer ap- proximation, we arrive to the Hartree and Hartree-Fock equations for the electronic problem. Special emphasis is placed in the most relevant mathematical aspects of the theoretical deriva- tion of the final equations, as well as in 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. Then, the discretization of the one-electron orbitals space is reviewed and the Roothaan-Hall formalism introduced. This leads to a exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing in algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (specially those related to the design of linear-scaling methods) are commented and linked to the issues discussed. The whole work is intentionally introductory and rather self-contained, so that it may be useful for non experts that aim to use quantum chemical methods in interdisciplinary applications.
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