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

Abstract

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 approximation, 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 derivation 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. Moreover, much material that is found scattered in the literature has been put together here to facilitate comprehension and to serve as a handy reference.

Assessment of theoretical methods for the study of hydrogen abstraction kinetics of global warming gas species during their degradation and byproduct formation (IUPAC Technical Report)

“…The simplest ab initio method is the application of the Hartree-Fock (HF) theory [116]. In the HF method, a system is reduced to a series of one-electron wave functions where each electron moves in an average field due to the other electrons.…”

Developing a New Storage Format and a Warp-Based SpMV Kernel for Configuration Interaction Sparse Matrices on the GPU

“…The Hartree-Fock (HF) method [9], also known as the self-consistent field (SCF) method, is an approximation method that is used to determine the wavefunction and the energy of a quantum system. In the HF method, there is an assumption that the wavefunction can be approximated using a single Slater determinant (a representation of the wavefunction).…”

The canonical equilibrium of constrained molecular models

“…(47) of such an energy function, we see that the quantities, say, l 0 α and θ 0 α , appearing in the harmonic energy terms associated to bond lengths and bond angles, are readily available to be elected as the candidate constant numbers to which equate this coordinates should we want to consider them as 'stiff' and constrain them; even if they are not, as mentioned, the actual minimum-energy values of the bond lengths and bond angles in any given conformation of the molecule. If, instead of a force field, we consider a less explicit energy function, such as the one produced by the ground-state Born-Oppenheimer approximation for the electrons in quantum chemistry [23,136], then no candidate number appears before our eyes as the constants to be used in the constrained model, and the flexible way of proceeding is even more natural.…”

“…The fact that all these issues are intricately coupled is easily seen if we consider examples such as the comparison between ab initio MD based on the ground-state Born-Oppenheimer approximation [1,12,13,14,15] and MD simulations using as energy functions the ones known as classical force fields [16,17,18,19,20,21]. On the one hand, it is clear that the accuracy of the two physical models is not the same; on the other hand, the size of the systems that can be practically tackled is also different, since quantum chemical methods present a cost that scales typically faster with the number of atoms than that of force fields [22,23]. Another example of this interplay between accuracy and cost is the use of coarse-grained descriptions, which, by selecting as interaction centers larger entities than individual atoms, and thus changing the physical model, allow to reach larger systems and longer time-scales than atomistic simulations using either force fields or ab initio methods [24,25,26,27,28].…”