20
2 Computational Methods
It is important to have a look at the size of the basis sets because it determines the
computation time. For the VnZ basis sets, the number of contracted functions N V (n)
increases as the third power of the cardinal number n, see Table 2.4 which also shows
that the core–valence sets are considerably larger than the valence sets and that the
number of diffuse functions increase quadratically with the cardinal number.
The atomic natural orbital (ANO) basis sets provide another contraction approach
(Almlöf and Taylor 1987, 1990). The contraction coefficients are obtained by
optimizing atomic energies.
For atoms with many electrons, the standard basis sets become too large. Furthermore, relativistic effects, see Sect. 2.8, are no more negligible for the inner electrons.
For these reasons, the Effective Core Potentials (ECP) have been developed (Dolg
2000). Their goal is to reduce the basis set size and the number of electrons as
well as to include some relativistic effects (mass-velocity and Darwin, but the spin–
orbit effect are neglected, see Sect. 2.8). Their success is due to the fact that the
chemical bonding is mainly determined by the valence electrons, as for the frozen
core approximation. The core electrons are replaced by an approximate effective
pseudopotential.
Most basis sets may be found in the basis–set library of program packages. They
are also available on the EMSL basis set exchange web page: https://www.basissete
xchange.org (Pritchard et al. 2019).
2.7.2 Core Correlation
The (aug)-cc-pVnZ basis sets are not appropriate when all electrons are correlated
because they do not provide sufficient flexibility in the core region (Martin 1995;
Császár and Allen 1996). It is quite common in correlated calculations to use the
frozen core approximation, in which the orbitals of the inner-shell electrons are
constrained to remain doubly occupied in all configurations. Indeed, it significantly
reduces the computational effort and, fortunately, the error due to freezing the core
is nearly constant for molecules containing the same type of atoms; see Table 2.5.
Furthermore, it is small for bond angles, see Table 2.6.
Then, the core–core and core–valence correlations are calculated separately as
a correction, see (2.22) and (2.23c). From Table 2.7, it appears that an accurate
computation of the core correlation requires at least a CCSD(T)/wCVQZ level of
theory, the wCVTZ basis set giving a too small correction (as the wCVQZ basis
set is quite large, it may be advantageous to use a completely decontracted VTZ
basis set supplemented by an appropriate (1p3d2f) primitive set. This basis set called
Martin–Taylor basis set (denoted as MT) is significantly smaller without any loss in
accuracy (Martin 1995)). On the other hand, the MP2/wCVQZ level gives a correction
slightly too large for atoms of the row Li–F and is not accurate enough for heavier
atoms. For instance, for the C–Cl bond in ClCN, the MP2 method gives −0.43 pm,
whereas the CCSD(T) method gives −0.36 pm (Demaison et al. 2004). However,
it is still possible to achieve a high accuracy, starting from CCSD(T)/wCVTZ and
2 Computational Methods
It is important to have a look at the size of the basis sets because it determines the
computation time. For the VnZ basis sets, the number of contracted functions N V (n)
increases as the third power of the cardinal number n, see Table 2.4 which also shows
that the core–valence sets are considerably larger than the valence sets and that the
number of diffuse functions increase quadratically with the cardinal number.
The atomic natural orbital (ANO) basis sets provide another contraction approach
(Almlöf and Taylor 1987, 1990). The contraction coefficients are obtained by
optimizing atomic energies.
For atoms with many electrons, the standard basis sets become too large. Furthermore, relativistic effects, see Sect. 2.8, are no more negligible for the inner electrons.
For these reasons, the Effective Core Potentials (ECP) have been developed (Dolg
2000). Their goal is to reduce the basis set size and the number of electrons as
well as to include some relativistic effects (mass-velocity and Darwin, but the spin–
orbit effect are neglected, see Sect. 2.8). Their success is due to the fact that the
chemical bonding is mainly determined by the valence electrons, as for the frozen
core approximation. The core electrons are replaced by an approximate effective
pseudopotential.
Most basis sets may be found in the basis–set library of program packages. They
are also available on the EMSL basis set exchange web page: https://www.basissete
xchange.org (Pritchard et al. 2019).
2.7.2 Core Correlation
The (aug)-cc-pVnZ basis sets are not appropriate when all electrons are correlated
because they do not provide sufficient flexibility in the core region (Martin 1995;
Császár and Allen 1996). It is quite common in correlated calculations to use the
frozen core approximation, in which the orbitals of the inner-shell electrons are
constrained to remain doubly occupied in all configurations. Indeed, it significantly
reduces the computational effort and, fortunately, the error due to freezing the core
is nearly constant for molecules containing the same type of atoms; see Table 2.5.
Furthermore, it is small for bond angles, see Table 2.6.
Then, the core–core and core–valence correlations are calculated separately as
a correction, see (2.22) and (2.23c). From Table 2.7, it appears that an accurate
computation of the core correlation requires at least a CCSD(T)/wCVQZ level of
theory, the wCVTZ basis set giving a too small correction (as the wCVQZ basis
set is quite large, it may be advantageous to use a completely decontracted VTZ
basis set supplemented by an appropriate (1p3d2f) primitive set. This basis set called
Martin–Taylor basis set (denoted as MT) is significantly smaller without any loss in
accuracy (Martin 1995)). On the other hand, the MP2/wCVQZ level gives a correction
slightly too large for atoms of the row Li–F and is not accurate enough for heavier
atoms. For instance, for the C–Cl bond in ClCN, the MP2 method gives −0.43 pm,
whereas the CCSD(T) method gives −0.36 pm (Demaison et al. 2004). However,
it is still possible to achieve a high accuracy, starting from CCSD(T)/wCVTZ and