A calculation to obtain the properties of electrons in a solid starts with the interacting N-body Schrödinger equation to obtain the N-electron wavefunction (for simplicity, spin is not explicitly mentioned):
where the Hamiltonian describing the interaction of electrons and nuclei is
The first term is the many-body kinetic energy operator and the second describes the interaction of the electrons with the bare nuclei at fixed positions in the solid, R. The final term represents the interactions of electrons with each other. The total energy of the system will also include the Coulomb repulsion between the ions. In general, it is not possible to solve this equation. In the Hartree approximation, it is found that can be written as a product of N one-electron wavefunctions thus:
This condition follows from the assumption that the electrons interact only via the Coulomb force (see equation (3.6) below).
The one-electron Schrödinger equations are now
where the first term is now the one-electron kinetic energy and V( r) is the potential in which the electron is moving. The choice of is then made such that equation (3.4) is solvable. Clearly it should include the potential that the electron would feel from the ions
However the electron also feels the field from all other electrons. To incorporate this into V(r), the remaining electrons are assumed to be a smooth distribution of negative charge density . Therefore the potential energy of an electron in their field (often referred to as the direct term) would be
There are however several important features which cannot be represented in such a simple self-consistent field approximation. In particular the product of one electron wave functions is incompatible with the Pauli exclusion principle which requires the many-body wavefunction to be antisymmetric under the interchange of two electrons, that is
This cannot be satisfied by a non-zero wavefunction of the form given by Equation (3.3). The form of the wavefunction can be generalised to incorporate asymmetry by replacing the Hartree wavefunction by a Slater determinant of one electron wavefunctions. This is a linear combination of all possible Hartree wavefunctions obtainable from permutations of added together with weights 1 so as to give condition (3.7). The resultant ground state electron wavefunctions can the be found variationally by using this trial wavefunction to obtain the Hartree-Fock equations of the orthogonal one electron wavefunctions . These are
These equations differ from the Hartree equations by the exchange term on the left hand side. Similar to the direct term it is non-linear in , but it has the structure if an integral operator. As a result the complexity added to the Hartree calculation by the incorporating electron exchange is considerable.
Although the Hartree-Fock approximation treats electron exchange exactly, this is only a first order approximation to the total energy due to many body interactions. Electron correlations are introduced at the next level of approximation. This is discussed further in Section 3.4 where both exchange and correlation are described by a simple functional of the charge density.