Chemistry Reference
In-Depth Information
Primer on Electron Correlation
The first component of convergent approaches to quantum chemistry is
the computational procedure used to treat the electron correlation problem
(depicted along the horizontal axis of Figure 4). A tutorial on treating electron
correlation has been published earlier in this topic series.
112
References 113 and
114 provide two additional excellent overviews of the subject. In any system
with
n
interacting bodies (classical or quantum), the instantaneous motions of
the bodies are correlated. Except for the simplest cases (e.g., certain one-electron
systems), exact solutions to this
n
-particle (or many-body) problem cannot be
obtained. Mean-field approximations (such as Hartree-Fock theory) neglect
the instantaneous correlated motions of the bodies. The ''missing'' energy that
corresponds to these simultaneous and instantaneous interactions is the correla-
tion energy. In electronic structure theory, the correlation energy is typically
(although not unambiguously) defined as the difference between the exact (non-
relativistic) electronic energy and the Hartree-Fock energy.
115
Configuration interaction (CI) theory, coupled-cluster (CC) theory, and
many-body perturbation theory (MBPT), of which Møller-Plesset (MP) per-
turbation theory is a specific case, are three of the most popular and relevant
approaches that have been developed to systematically improve the computa-
tional description of electron correlation that is absent in Hartree-Fock theo-
ry. (Density functional methods are not mentioned here. Although DFT
provides a very cost-effective means of recovering part of the electron correla-
tion energy, the systematic improvement of individual functionals is proble-
matic.) The missing correlation energy is recovered by constructing the wave
function out of many different electron configurations (or Slater determinants)
that are generated by ''exciting'' electrons from the occupied orbitals of the
Hartree-Fock reference configuration to unoccupied (or virtual) orbitals.
These additional (or excited) configurations are typically classified by excita-
tion (or substitution) level: S for single excitations/substitutions, D for double,
T for triple, etc.
Approximate many-electron wave functions are then constructed from
the Hartree-Fock reference and the excited-state configurations via some
sort of expansion (e.g., a linear expansion in CI theory, an exponential ex-
pansion in CC theory, or a perturbative power series expansion in MBPT).
When all possible excitations have been incorporated (S, D, T,
n
for an
n
-electron system), one obtains the exact solution to the nonrelativistic electro-
nic Schr
¨
dinger equation for a given AO basis set. This
n
-particle limit is typi-
cally referred to as the full CI (FCI) limit (which is equivalent to the full CC
limit). As Figure 5 illustrates, several WFT methods can, at least in principle,
converge to the FCI limit by systematically increasing the excitation level (or
perturbation order) included in the expansion technique.
It is particularly important to note that while the linear CI expansion
necessarily converges and all evidence suggests the exponential CC expansion
always converges, the MBPT (or MP) series does diverge occasionally.
116,117
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