The Hartree Approximation

Although the Born-Oppenheimer approximation considerably reduces the complexity of the Schrödinger equation, the resulting electronic Schrödinger equation is still extremely complex, due to the electron-electron interactions. It is possible to use wavefunctions which explicitly include inter-electronic distance [7,8,9], but this approach is computationally infeasible for all but the smallest systems.

A more satisfying solution is to introduce the molecular orbital approximation, the simplest of which is the independent-particle, or Hartree, approximation [10,11,12] wherein the total wavefunction is approximated by a product of orthonormal molecular orbitals (MOs). This idea closely follows the chemists' view of electrons occupying orbitals. The Hartree approximation assumes that each electron moves independently within its own orbital and sees only the average field generated by all the other electrons. The Hartree wavefunction (for an N electron system) is
$$\begin{align}\Psi = \chi_{1}({\bf x}_{1}) \chi_{2}({\bf x}_{2}) \dots \chi_{N-1}({\bf x}_{N-1}) \chi_{N}({\bf x}_{N}), \end{align}$$
where each $\chi_{i}$ is a spin orbital containing one electron. The $\chi_{i}$ are orthonormal, consisting of a spatial orbital, $\psi_{i}({\bf r})$, and one of two spin functions, $\alpha(s)$ and $\beta(s)$, representing spin up and spin down states. ${\bf x}$ is the space-spin coordinate, containing both the position, ${\bf r}$, and spin, s, of a particle.