Hartree-Fock Theory

A full understanding of Hartree-Fock theory would require a few weeks of study. Here, we will aim for only a rough idea of what is going on. Actually, you already know a few things about Hartree-Fock theory, even if you haven't heard the term before. You almost certainly have heard about molecular orbital (MO) theory, or the linear combination of atomic orbitals molecular orbital (LCAO-MO) theory. This is actually the same as Hartree-Fock theory, which is named after the two physicists who invented it. The basic idea is that we will describe the motion of each electron by a molecular orbital. The mathematics behind it is that each MO is made of a linear combination of atom-centered basis functions. The Hartree-Fock procedure simply solves for what the linear expansion coefficients actually are.

The variables in the Hartree-Fock equations unfortunately depend on themselves, so they must be solved in an iterative manner. You will see these iterations in the outputs you run in the lab. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method.

Unfortunately, you are probably under the illusion that molecular orbitals are somehow ``real'' or ``true.'' Except for the special case of the hydrogen atom, this is completely false. Molecular orbitals are the product of Hartree-Fock theory, and Hartree-Fock is not an exact theory: it is an approximation to the electronic Schrödinger equation. The approximation is that we pretend that each electron feels only the average Coulomb repulsion of all the other electrons. This approximation makes Hartree-Fock theory much simpler than the real problem, which is an N-body problem. Unfortunately, in many cases this approximation is rather serious and can give bad answers. It can be corrected by explicitly accounting for electron correlation by density functional theory (DFT), many-body perturbation theory (MBPT), configuration interaction (CI), and other means.