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
(
) and the exact nonrelativistic energy of a system
(
)
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 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.
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 H
O at three different geometries.
Table 1: Electron Correlation in HO with a DZ Basis.
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.''