Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons.
Within the Hartree–Fock method of quantum chemistry, the antisymmetric wave function is approximated by a single Slater determinant. Exact wave functions, however, cannot generally be expressed as single determinants. The single-determinant approximation does not take into account Coulomb correlation, leading to a total electronic energy different from the exact solution of the non-relativistic Schrödinger equation within the Born–Oppenheimer approximation. Therefore, the Hartree–Fock limit is always above this exact energy. The difference is called the correlation energy, a term coined by Löwdin. The concept of the correlation energy was studied earlier by Wigner.
A certain amount of electron correlation is already considered within the HF approximation, found in the electron exchange term describing the correlation between electrons with parallel spin. This basic correlation prevents two parallel-spin electrons from being found at the same point in space and is often called Fermi correlation. Coulomb correlation, on the other hand, describes the correlation between the spatial position of electrons due to their Coulomb repulsion, and is responsible for chemically important effects such as London dispersion. There is also a correlation related to the overall symmetry or total spin of the considered system.
The word correlation energy has to be used with caution. First it is usually defined as the energy difference of a correlated method relative to the Hartree–Fock energy. But this is not the full correlation energy because some correlation is already included in HF. Secondly the correlation energy is highly dependent on the basis set used. The “exact” energy is the energy with full correlation and complete basis set.
Electron correlation is sometimes divided into dynamical and non-dynamical (static) correlation. Dynamical correlation is the correlation of the movement of electrons and is described under electron correlation dynamics and also with the configuration interaction (CI) method. Static correlation is important for molecules where the ground state is well described only with more than one (nearly-)degenerate determinant. In this case the Hartree–Fock wavefunction (only one determinant) is qualitatively wrong. The multi-configurational self-consistent field (MCSCF) method takes account of this static correlation, but not dynamical correlation.
If one wants to calculate excitation energies (energy differences between the ground and excited states) one has to be careful that both states are equally balanced (e.g., Multireference configuration interaction).