28
2 Computational Methods
In most cases, a Taylor series expansion of the energy is made about a reference
point R 0
E = E(R 0 ) + (R − R 0 )
T G +
1
2
(R − R 0 )
T H(R − R 0 )
(2.29)
where T means the transpose, G is the gradient and H the Hessian.
By differentiating to find the minimum, it gives
R = R 0 − H
−1 G
(2.30)
The final solution is obtained by iteration. This is a Newton–Raphson method.
The main difficulty is the calculation of the inverse of the Hessian. Variants, called
quasi-Newton methods have been devised to simplify the calculation of H
−1 .
2.11 Strategy
To determine the structure of a molecule, if it is not too large and if a high accuracy is
required, (2.22–2.23c) should be used together with the CBS extrapolation procedure,
(2.24–2.25). The CCSD(T) method may be advantageously replaced by an explicitly
correlated method such as CCSD(T)_F12 (Werner et al. 2010), see Sect. 2.7.4.2.
When the molecule is too large, or when a very high accuracy is not required, the
CCSD(T)_ae/wCVQZ level of theory gives satisfactory results, its mean error being
only 0.1 pm (Coriani et al 2005); see also Table 2.14. This good result is mainly
due to a compensation of errors because the introduction of connected quadruples (CCSDTQ) increases the bond length by 0.1–0.2 pm, whereas the basis set
extension from quadruple zeta to sextuple zeta shortens it by about 0.1 pm. Some
CCSD(T)_ae/CVQZ bond lengths are compared to the CCSD(T)_ae/CV5Z values
and to the best equilibrium structure in Table 2.14. The CCSD(T)_ae/CVQZ bond
length is larger than the equilibrium one by 0.04 pm with a maximum absolute deviation (MAD) of 0.04 pm (corresponding to a standard deviation of 0.06 pm). On the
other hand, The CCSD(T)_ae/CV5Z bond length is smaller than the equilibrium one
by 0.04 pm with a maximum absolute deviation (MAD) of 0.02 pm (corresponding
to a standard deviation of 0.03 pm). There is one outlier: HF. Fluorine is extremely
electronegative and, in such a case, diffuse functions are required or a basis set up
to CV6Z has to be used. For instance, in the case of HF, r[CCSD(T)_ae/ACVQZ] =
91.73 pm is close to the r e value.
For large molecules, it may happen that the CVQZ (or wCVQZ) basis set is
too large. In such a case, a compound method, similar to (2.22) may be used.
For instance, the Born–Oppenheimer (BO) equilibrium structure is optimized at
the CCSD(T)_ae/wCVTZ level of theory and the small effect of further basis set
enlargement (wCVTZ → wCVQZ) is then estimated at the MP2 level. The resulting
r
BO
e estimate is:
2 Computational Methods
In most cases, a Taylor series expansion of the energy is made about a reference
point R 0
E = E(R 0 ) + (R − R 0 )
T G +
1
2
(R − R 0 )
T H(R − R 0 )
(2.29)
where T means the transpose, G is the gradient and H the Hessian.
By differentiating to find the minimum, it gives
R = R 0 − H
−1 G
(2.30)
The final solution is obtained by iteration. This is a Newton–Raphson method.
The main difficulty is the calculation of the inverse of the Hessian. Variants, called
quasi-Newton methods have been devised to simplify the calculation of H
−1 .
2.11 Strategy
To determine the structure of a molecule, if it is not too large and if a high accuracy is
required, (2.22–2.23c) should be used together with the CBS extrapolation procedure,
(2.24–2.25). The CCSD(T) method may be advantageously replaced by an explicitly
correlated method such as CCSD(T)_F12 (Werner et al. 2010), see Sect. 2.7.4.2.
When the molecule is too large, or when a very high accuracy is not required, the
CCSD(T)_ae/wCVQZ level of theory gives satisfactory results, its mean error being
only 0.1 pm (Coriani et al 2005); see also Table 2.14. This good result is mainly
due to a compensation of errors because the introduction of connected quadruples (CCSDTQ) increases the bond length by 0.1–0.2 pm, whereas the basis set
extension from quadruple zeta to sextuple zeta shortens it by about 0.1 pm. Some
CCSD(T)_ae/CVQZ bond lengths are compared to the CCSD(T)_ae/CV5Z values
and to the best equilibrium structure in Table 2.14. The CCSD(T)_ae/CVQZ bond
length is larger than the equilibrium one by 0.04 pm with a maximum absolute deviation (MAD) of 0.04 pm (corresponding to a standard deviation of 0.06 pm). On the
other hand, The CCSD(T)_ae/CV5Z bond length is smaller than the equilibrium one
by 0.04 pm with a maximum absolute deviation (MAD) of 0.02 pm (corresponding
to a standard deviation of 0.03 pm). There is one outlier: HF. Fluorine is extremely
electronegative and, in such a case, diffuse functions are required or a basis set up
to CV6Z has to be used. For instance, in the case of HF, r[CCSD(T)_ae/ACVQZ] =
91.73 pm is close to the r e value.
For large molecules, it may happen that the CVQZ (or wCVQZ) basis set is
too large. In such a case, a compound method, similar to (2.22) may be used.
For instance, the Born–Oppenheimer (BO) equilibrium structure is optimized at
the CCSD(T)_ae/wCVTZ level of theory and the small effect of further basis set
enlargement (wCVTZ → wCVQZ) is then estimated at the MP2 level. The resulting
r
BO
e estimate is: