2.4 Hartee–Fock (HF) Method
13
basis set increases, it is particularly a problem when electron correlation is taken into
account; see Sect. 2.7.4.
For a molecule with n electrons, the n/2 orbitals of smallest energy (which is
negative) are called occupied orbitals and the orbitals of higher energy are called
virtual orbitals.
Typical errors are 1% in bond distances (which are underestimated), the energy is
too high and the dissociation energy is not correctly calculated. For instance, in the
case of the HF theory, the two electrons of dihydrogen, H 2 , spend half the time on
the same atom and half the time on both atoms, even when the distance between the
two atoms is infinite, which is obviously not possible. The main weakness of the HF
method is that it does not take into account the instantaneous interactions between
the electrons, called electronic correlation. The correlation of electrons of same spin
is partially accounted for because they cannot belong to the same spin orbital (the
determinant, (2.8) would be zero). On the other hand, electrons of opposite spin are
allowed to approach each other closely. The difference between the HF energy and
the exact non-relativistic energy is the correlation energy
E(exact) = E(HF) + E(correlation)
(2.15)
There are different ways to estimate this correlation energy. One of the simplest
ones is to use the second-order perturbation theory (the first order is the E(HF)
energy), often called Møller–Plesset perturbation theory (Møller and Plesset 1934),
abbreviated as MP2. As we will see in Sect. 2.6 and Table 2.1, this method significantly improves the situation but is not always accurate enough. Higher-order perturbation theory is possible (MP3, MP4) but not fully satisfactory. It is also possible to
use configuration interaction methods that will be described below, Sect. 2.5, but they
require large amounts of computer resources, even for small molecules. Although
they are powerful, they are difficult to use. Another recent method giving excellent
results is the coupled cluster theory. Finally, a completely different method, nonab initio, is the Density Functional Theory (DFT, see Sect. 2.13) which may give
satisfactory results, particularly for large molecules.
2.5 Post Hartree–Fock Methods (Bartlett and Stanton
1994; Helgaker et al. 2000, 2004)
For these methods the HF solution corresponds to a single Slater determinant. Assume
a molecule with n electrons, the HF solution is
ψ 0 =
1
√
n!
|χ 1 χ 2 · · · χ n |
(2.16)
where the determinant is written in abbreviated form, | |, and the χ i are spin orbitals,
i.e., the product of a molecular orbital and a spin function. The spin orbitals χ 1 , χ 2 ,
13
basis set increases, it is particularly a problem when electron correlation is taken into
account; see Sect. 2.7.4.
For a molecule with n electrons, the n/2 orbitals of smallest energy (which is
negative) are called occupied orbitals and the orbitals of higher energy are called
virtual orbitals.
Typical errors are 1% in bond distances (which are underestimated), the energy is
too high and the dissociation energy is not correctly calculated. For instance, in the
case of the HF theory, the two electrons of dihydrogen, H 2 , spend half the time on
the same atom and half the time on both atoms, even when the distance between the
two atoms is infinite, which is obviously not possible. The main weakness of the HF
method is that it does not take into account the instantaneous interactions between
the electrons, called electronic correlation. The correlation of electrons of same spin
is partially accounted for because they cannot belong to the same spin orbital (the
determinant, (2.8) would be zero). On the other hand, electrons of opposite spin are
allowed to approach each other closely. The difference between the HF energy and
the exact non-relativistic energy is the correlation energy
E(exact) = E(HF) + E(correlation)
(2.15)
There are different ways to estimate this correlation energy. One of the simplest
ones is to use the second-order perturbation theory (the first order is the E(HF)
energy), often called Møller–Plesset perturbation theory (Møller and Plesset 1934),
abbreviated as MP2. As we will see in Sect. 2.6 and Table 2.1, this method significantly improves the situation but is not always accurate enough. Higher-order perturbation theory is possible (MP3, MP4) but not fully satisfactory. It is also possible to
use configuration interaction methods that will be described below, Sect. 2.5, but they
require large amounts of computer resources, even for small molecules. Although
they are powerful, they are difficult to use. Another recent method giving excellent
results is the coupled cluster theory. Finally, a completely different method, nonab initio, is the Density Functional Theory (DFT, see Sect. 2.13) which may give
satisfactory results, particularly for large molecules.
2.5 Post Hartree–Fock Methods (Bartlett and Stanton
1994; Helgaker et al. 2000, 2004)
For these methods the HF solution corresponds to a single Slater determinant. Assume
a molecule with n electrons, the HF solution is
ψ 0 =
1
√
n!
|χ 1 χ 2 · · · χ n |
(2.16)
where the determinant is written in abbreviated form, | |, and the χ i are spin orbitals,
i.e., the product of a molecular orbital and a spin function. The spin orbitals χ 1 , χ 2 ,