The Hartree equations were derived by considering the electronic wavefunction as a product of many single particle wavefunctions, , each subject to its own external potential. Thus

The Schrödinger equation for each single particle state, ,is given by

(3) |

Hartree suggested that in a many electron system it was possible to
replace *V*(*r*) with *V*_{eff}(*r*), which included the mean
electrostatic potential from all of the electrons [13].
This leads to the Hartree self-consistent equations

The total many body wavefunction is then formed as the product of
these one-electron spin-orbitals. *V*(*r*) is the potential due to the
ions. This simple approximation leads to surprisingly accurate
results, but suffers from the lack of electronic exchange, *i.e.*
the many body wavefunction, is not antisymmetric with
respect to a switching of two of the electrons. This is in breach of
the Pauli principle, which in general imposes a change of sign of the
total wavefunction when two electrons are switched, and in the
independent electron approximation described above, does not allow
more than one electron to occupy a given state.