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.
Global solutions of restricted open-shell Hartree-Fock theory from semidefinite programming with applications to strongly correlated quantum systems
基于半定规划（SDP）给出了一种计算受限开壳层Hartree-Fock理论整体解的密度矩阵方法，该方法给出了量子系统Hartree-Fock能量的上下界.虽然波函数方法的Hartree-Fock理论产生一个上限的Hartree-Fock能量，我们推导出一个半定的Hartree-Fock理论松弛，产生一个严格的Hartree-Fock能量的下限。我们还开发了一个上限算法，其中Hartree-Fock理论被铸造为一个SDP与非凸约束的秩的矩阵变量。上下界能量相等保证了计算的解是Hartree-Fock理论的全局最优解。本文的工作推广了以前提出的闭壳系统的方法[S. Veeraraghavan和D. a. Mazziotti，Phys.Rev.A89，010502R（2014）]。对于强相关系统，SDP方法提供了局部优化Hartree-Fock能量和密度的替代方案，并获得了全局最优性证书。应用C2，CN，Cr2和NO2的势能曲线。
