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Electron Correlation
The HF method gives "absolute" energies that are too high, because it overestimates electronelectron repulsion. There are two kinds of electron-electron repulsion: classical Coulomb (-->Coulomb
hole), arising from electric charge, and quantum mechanical or Fermi repulsion
(--> Fermi hole), which exists between electrons of like spin. The HF approximate treatment of
electron correlation arises from the use of just one determinant, and from the "smeared electron
cloud" integration used to get the J and K integrals.
In practice "electron correlation" is a term to describe the inadequacies of the Hartree-Fock
(single determinant) model.
In the Hartree-Fock model, the repulsion energy between two electrons is calculated between an
electron and the average electron density for the other electron. What is unphysical about this is
that it doesn't take into account the fact that the electron will push away the other electrons as
it moves around. This tendency for the electrons to stay apart diminishes the repulsion energy.
First, why is the Hartree-Fock method not capable of giving the correct solution to the Schrödinger
equation if a very large and flexible basis set is selected? In passing, we note that the very best
Hartree-Fock wave function, obtained with just such a large and flexible basis set, is called the
"Hartree-Fock limit". The problem is that electrons are not paired up in the way that the HartreeFock method supposes. It suggests that the two electrons have the same probability of being in the
same region of space as being in separate symmetry equivalent regions of space. For example, in H2
it would give the same probability of both electrons being near one atom as one being near one atom
and the other near the second atom. This is clearly wrong. The Hartree-Fock method also only
evaluates the repulsion energy as an average over the whole molecular orbital.
The two electrons in a molecular orbital are in reality moving in such a way that they keep more
apart from each other than being close. We call this effect "correlation". The difference in energy
between the exact result and the Hartree-Fock limit energy is called the "correlation energy".
The concept of electrons avoiding each other, is called the dynamical correlation, but there is also a
more subtle effect called nondynamical or static correlation energy. Nondynamical correlation energy
reflects the inadequacy of a single reference in describing a given molecular state, and is due to nearly
degenerate states or rearrangement of electrons within partially filled shells.
In many situations it is further convenient to subdivide the correlation energy into two parts with
different physical origins. For chemical reactions where bonds are broken and formed, and for most
excited states, the major part of the correlation energy can be obtained by adding only a few extra
configurations besides the Hartree-Fock configuration. This part of the correlation energy is due to
near degeneracy between different configurations and has its origin quite often in artifacts of the
Hartree-Fock approximation. The physical origin of the second part of the correlation energy is the
dynamical correlation of the motion of the electrons and is therefore sometimes called the
dynamical correlation energy. Since the Hamiltonian operator contains only one- and two-particle
operators this part of the correlation energy can be very well described by single and double
replacements from the leading, near degenerate, reference configurations.