The Correlation Energy

Approximate CI methods can be judged according to what fraction of the correlation energy they recover. The correlation energy is defined as the difference between the energy in the Hartree-Fock limit (E_HF) and the exact nonrelativistic energy of a system ( ${\cal E}_0$)

$$E_{\rm corr} = {\cal E}_0 - E_{HF}$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred10 E_corr = E_0 - E_HF )

This energy will always be negative because the Hartree-Fock energy is an upper bound to the exact energy (this is guaranteed by the variational theorem,  as explained in section 3). The exact nonrelativistic energy ${\cal E}_0$ could, in principle, be calculated by performing a full CI  in a complete one-electron basis set. If we have an incomplete one-electron basis set, then we can only compute the basis set correlation energy, which is the correlation energy for a given one-electron basis. For convenience, the basis set correlation energy is often simply referred to as the correlation energy.

The correlation energy is the energy recovered by fully allowing the electrons to avoid each other; Hartree-Fock improperly treats interelectron repulsions in an averaged way.² However, there is some inconsistency in this line of thinking. When a molecule is pulled apart, the electrons shouldn't need to avoid each other as much, so the magnitude of the correlation energy should decrease. In fact, the opposite is true, as shown by the basis set correlation energies given in Table 1 for H₂O at three different geometries.

 

 

Table 1: Electron Correlation in H₂O with a DZ Basis.

Geometry	E $_{\rm corr}$ (hartree)a
Re	-0.148028
1.5 Re	-0.210992
2.0 Re	-0.310067
aData from reference [6].

The correlation energy increases at stretched geometries, because our definition of the correlation energy in equation (2.10) includes not only the concept of electrons avoiding each other, which is called the ``dynamical''    correlation energy, but also a more subtle effect called the ``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. Shavitt [7] has pointed out this deficiency in the definition of the correlation energy, and has suggested that perhaps a multiconfigurational Hartree-Fock method may be more useful in the definition of correlation energy.

Siegbahn  [8] offers the following explanation of the difference between dynamical    and nondynamical correlation    energies:

``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.''