A measure of the extent to which any particular ab initio calculation does not
deal perfectly with electron correlation is thecorrelation energy. In a canonical
exposition [ 79 ]L€owdin defined correlation energy thus: “The correlation energy for
a certain state with respect to a specified Hamiltonian is the difference between the
exact eigenvalue of the Hamiltonian and its expectation value in the Hartree–Fock
approximation for the state under consideration.” This is usually taken to be the
energy from a nonrelativistic but otherwise perfect quantum mechanical procedure,
minus the energy calculated by the Hartree–Fock method with the same nonrelativ-
istic Hamiltonian and a huge (“infinite”) basis set:
Ecorrel¼EðexactÞ#EðHF limitÞ
using the same Hamiltonian for both termsFrom this definition the correlation energy is negative, sinceE(exact) (actually a
nonrelativistic energy here) is more negative thanE(HF limit). The Hamiltonians of
Section 5.2.2, Eqs.5.4–5.6and associated discussion exclude relativistic effects,
which are significant only for heavy atoms. Unless qualified the term correlation
energy means nonrelativistic correlation energy. The correlation energy is essen-
tially the energy that the Hartree–Fock procedure fails to account for.Ifrelativistic
effects (and other, usually small, effects like spin-orbit coupling) are negligible then
Ecorrelis the difference between the experimental value (of the energy required to
dissociate the molecule or atom into infinitely separated nuclei and electrons) and
the limiting Hartree–Fock energy.
A distinction is sometimes made between dynamic (or dynamical), and non-
dynamic or static correlation energy. Dynamic correlation energy is the energy a
Hartree–Fock calculation does not account for because it fails to keep the electrons
sufficiently far apart; this is the usual meaning of “correlation energy”. Static
correlation energy is the energy a calculation (Hartree–Fock or otherwise) may
not account for because it uses a single determinant, or starts from a single
determinant (is based on a single-determinant reference –Section 5.4.3); this
problem arises with singlet diradicals, for example, where a closed-shell description
of the electronic structure is qualitatively wrong. This is because there are (two,
usually) highest-energy orbitals (frontier orbitals) of equal or nearly equal energy
and the Hartree–Fock method cannot unambiguously decide which of these should
receive an electron pair and which should be empty – which should be the HOMO
and which the LUMO. A singlet diradical actually has two essentially half-filled
orbitals. The term correlation energy is applied to the unaccounted-for energy in
such cases perhaps because as with dynamic correlation energy the problem can be
at least partly overcome by expressing the wavefunction with more than one
determinant. Dynamic correlation energy can be calculated (“recovered”) by the
Møller–Plesset method or by multiditerminant configuration interaction methods
(Sections 5.4.2and5.4.3) and static correlation energy can likewise be recovered by
basing the wavefunction on more than one determinant, as in a multireference
258 5 Ab initio Calculations