The simplest approximation is the Hartree approximation [42]. The initial *ansatz* is that we may write the many-body wavefunction as

from which it follows that the electrons are *independent*, and interact only via the mean-field Coulomb potential. This yields one-electron Schrödinger equations of the form

where is the potential in which the electron moves; this includes both the nuclear-electron interaction

(2.5) |

and the mean field arising from the other electrons. We smear the other electrons out into a smooth negative charge density leading to a potential of the form

(2.6) |

where .

Although these *Hartree* equations are numerically tractable via the self-consistent field method, it is unsurprising that such a crude approximation fails to capture elements of the essential physics. The Pauli exclusion principle demands that the many-body wavefunction be antisymmetric with respect to interchange of *any* two electron coordinates, *e.g.*

(2.7) |

which clearly cannot be satisfied by a non-trivial wavefunction of the form 2.3. This *exchange* condition can be satisfied by forming a Slater determinant [43] of single-particle orbitals

(2.8) |

where is an anti-symmetrising operator; *i.e.* it ensures that all possible anti-symmetric combinations of orbitals are taken. Again, this decouples the electrons, leading to the single-particle *Hartree-Fock* equations [44] of the form

(2.9) |

The last term on the left-hand side is the *exchange* term; this looks similar to the direct Coulomb term, but for the exchanged indices. It is a manifestation of the Pauli exclusion principle, and acts so as to separate electrons of the same spin; the consequent depletion of the charge density in the immediate vicinity of a given electron due to this effect is called the *exchange hole*. The exchange term adds considerably to the complexity of these equations.

The Hartree-Fock equations deal with exchange exactly; however, the equations neglect more detailed correlations due to many-body interactions. The effects of electronic correlations are not negligible; indeed the failure of Hartree-Fock theory to successfully incorporate correlation leads to one of its most celebrated failures: its prediction that jellium is an insulating rather than a metallic system. The requirement for a computationally practicable scheme that successfully incorporates the effects of both exchange and correlation leads us to consider the conceptually disarmingly simple and elegant density functional theory.