Coupled Cluster Methods
The Correlation Energy Revisited
Now that the CI method for determining the correlation energy has been
fully introduced, it is desirable to conclude the discussion of this method
with a look at the correlation energy itself. Consider the following table
in which the correlation energies of MgH2 and H2O
are provided at a number of geometries.
An initial examination of the table reveals the unusual conclusion that as these molecules are stretched away from their equilibrium geometries, the magnitude of the correlation energy is found to increase. It is an unusual conclusion because the correlation energy was described as the amount of energy recovered over the HF limit energy due to the true wave function allowing electrons with opposite spins to avoid one another; thus, it should be found to decrease,not increase, as the molecule is stretched and electrons are given more room to avoid one another. This conclusion indicates that the correlation energy must possess at least two physical origins: the ``dynamical'' correlation energy gained by allowing electrons to avoid one another and the remainder due to inadequacies inherent in the full HF wave function. Moreover, the table makes it clear that the magnitude of this non-dynamical, or static, component of the correlation energy grows very rapidly as one moves away from the equilibrium geometry where a single configuration dominates. The clearest description of the correlation energy found in the literature is the following quotation taken from Siegbahn [16]:
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.