Chemistry Reference
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state of the system. The HF method predicts molecular geometries (distances,
angles) within a few percent of experiment. Even vibrational frequencies, derived
from the curvature of the total energy as a function of nuclear separations, were
found to be within about 10% of experiment. The binding energies of molecules
are less satisfactory. Some systems like F
2
caused serious problems. The F
2
mole-
cule was predicted to be less stable than two isolated F atoms. The calculation of the
geometry and vibrational properties of O
3
also turned out to be quite difficult.
The HF model has also been used for solid state applications. To obtain more
exact results, one has to calculate the correlation energy which is defined as the
difference between the HF energy and the exact energy:
E
corr
¼
E
exact
E
HF
;
(7)
where
E
exact
is the energy of the system obtained from solving exactly the non-
relativistic Schr
odinger equation. In the HF approach the instantaneous position of
an electron is not influenced by the presence of other electrons. In fact, the motions
of electrons are correlated and they tend to “avoid” each other more than HF
predicts, giving rise to a lower energy. There are a number of techniques to improve
the HF approach, like the configuration interaction (CI) method, Møller-Plesset
perturbation theory (MP) or CC approach, and many others [
5
]. The CI approach
describes the total wave function as a linear combination of the ground- and
excited-state wave functions. A CI calculation is variational and, therefore, gives
an upper bound of the true energy. Of course, a full CI calculation is expensive,
such that only a few excitations are taken into account. But those calculations are
not size consistent. That means that the energy of a number N of non-interacting
atoms or molecules is not equal to N times the energy of a single atom or molecule.
To overcome this problem the quadratic configuration interaction method (QCISD)
was introduced to try to deal with this. It can be considered a size consistent CISD
theory. The procedure involves the addition of higher excitation terms which are
quadratic in their expansion coefficients (see [
5
]).
The MP perturbation theory, basically the Rayleigh-Schrodinger perturbation
theory, is size-independent. The idea is a partitioning of the Hamiltonian into a HF
part and a perturbation
V
:
€
H
¼
H
HF
þ l
V
;
(8)
where
is a parameter that can vary between zero and one.
Further evaluation reveals that at least second-order perturbations have to be
included. This level of theory is referred to as MP2 and involves an evaluation of
E
ð
2
Þ
0
l
Higher order approximations are also possible. The MP perturbation theory is
not variational and can sometimes give energies that are lower than the “true”
energy. As any perturbation theory, MP perturbation theory depends on how close
the starting wave function is to the exact wave function. When this is the case,
convergence of the MP series is rapid. However, when bonds are stretched the MP
:
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