Correlation

Hartree-Fock theory is not an exact theory simply because it only considers a single determinant for the electron wavefunction, and this is only a small subset of the total number of allowable wavefunctions. Consequently, it is highly unlikely that the true wavefunction is contained within this subset. The only case when a single determinant is exact is for a non-interacting system of electrons.

In real systems the motions of electrons are more correlated than the mean-field description provided by Hartree-Fock. The interaction energy missed by Hartree-Fock is commonly termed the correlation energy $E_{\hbox{{C}}}$ [17],

$$E_{\hbox{{C}}} = E_0 - E_{\hbox{{HF}}}$$ (1.27)

where $E_0$ is the exact groundstate energy. Since Hartree-Fock is a variational method, i.e. $E^{\hbox{{HF}}} \ge E_0$, the correlation energy is a negative quantity according to (1.29), the exception is for a one-electron system, where in this case Hartree-Fock theory is exact and $E_{\hbox{{C}}}=0$.

A natural way to incorporate correlation effects beyond the Hartree-Fock level is to mix a linear combination of Slater determinants corresponding to excited state configurations. These post Hartree-Fock methods, such as configuration interaction, coupled-cluster and Møller-Plesset theory have been extensively developed in quantum chemistry [18], and although the approach may be systematic, the computational cost increases dramatically with excitation level. As a result, the best correlated methods are currently limited to small systems such as atoms and small molecules.