Configuration Interaction

To correctly describe the instantaneous interaction of electrons, the inter-electron distance must be introduced. The most conceptually simple way of achieving this is via Configuration Interaction (CI) [33,34]. CI uses a wavefunction which is a linear combination of the HF determinant and determinants from excitations of electrons,
$$\begin{align}\Psi = C_{0}\Psi_{0} + \sum_{i}\sum_{a}C_{i}^{a}\Psi_{i}^{a} + \sum... ... + \sum_{i>j>k}\sum_{a>b>c}C_{ijk}^{abc}\Psi_{ijk}^{abc} + \, \ldots \end{align}$$
where $\Psi_{i}^{a}$ represents the determinant with an electron excited from the occupied orbital, $\chi_{i}$ to the virtual orbital, $\chi_{a}$.

The CI expansion is variational and, if the expansion is complete (Full CI), gives the exact correlation energy (within the basis set approximation). The number of determinants in Full CI grows exponentially with the system size, making the method impractical for all but the smallest systems. For this reason the CI expansion is usually truncated at some order, for example CISD, where only singly and doubly excited determinants are considered. Brillouin's Theorem states that singly excited determinants do not mix with the HF determinant [35]. Therefore CISD is the cheapest worthwhile form of CI, yet this method scales as O(N⁶) where N is the size of the system.

The other main problem with truncated CI is that it is not size consistent. For CISD, an approximate way to correct for these effects is to introduce the Davidson correction [36]
$$\begin{align}E_{corr} = E_{corr}(\mathrm{CISD}) + (1-c^{2}_{0})E_{corr}(\mathrm{CISD}) \end{align}$$
where c₀ is the coefficient of the Hartree-Fock wavefunction in the normalized CISD wavefunction.