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## Approximate Methods: the Hartree and Hartree-Fock Methods

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

 (2.3)

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

 (2.4)

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.

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