2.13 Density Functional Theory (DFT)
35
Table 2.17 Mean, standard deviation (St. dev.), and mean absolute error (MAE) for the
B2PLYP/VTZ and B3LYP/SNSD deviations from CCSD(T) semiexperimental equilibrium
structures (distances in pm and angles in degree)
Bond
n# a
B2PLYP/VTZ
B3LYP/SNSD
Mean
St. dev.
MAE
Mean
St. dev.
MAE
All bonds
74
0.00
0.05
0.04
−0.01
0.09
0.07
CH
30
−0.02
0.04
0.03
−0.05
0.05
0.06
CC
21
0.02
0.05
0.04
0.05
0.09
0.09
CO
7
0.02
0.02
0.02
0.03
0.04
0.05
Angles
46
0.00
0.05
0.03
0.00
0.07
0.05
Source Penocchio et al. (2015)
a Number of data in the sample
as shown by Piccardo et al. (2015), a double-zeta basis set such as SNSD (downloadable on http://dreamslab.sns.it) already gives satisfactory results; see Table 2.17.
Finally, B3LYP is able to give reliable geometries as well as force fields (Bauschlicher
et al. 1997). B3LYP slightly overestimates single bond lengths and underestimates
multiple bond lengths. Nevertheless, it is possible to correct B3LYP results (as for
the MP2 ones, see Sect. 2.12) and obtain mean absolute errors of about 0.02 pm for
all types of bonds. For instance, for the CO bond, using the 6-311 + G(3df,2pd) basis
set, the median value of residuals is +0.018 pm for multiple bonds and −0.021 pm
for single bonds (Demaison and Császár 2012). However, the differences are not
constant, but there is a linear relationship between the residuals and the B3LYP
values. Still for the CO bond, a linear fit gives
r e (CO) = 0.9780(29)r [B3LYP/6 − 311 + G(3df, 2pd)] + 0.0283(37) (2.39)
with a correlation coefficient of ρ = 0.9996 and a standard deviation of σ = 0.21 pm.
This is of quality comparable to the MP2/VTZ level of theory.
Table 2.18 compares the results of the structures optimized with the B3LYP/SNSD
or B2PLYP/VTZ levels of theory with accurate semiexperimental equilibrium
structures. The agreement is satisfactory.
2.14 Calculation of the Force Field (Császár 2012)
The concept of potential energy (hyper)surface (PES) is fundamental in chemistry.
The equilibrium structure corresponds to its minimum. The PES governs the vibrations of the atoms. Its knowledge is also important for a better understanding of
reaction kinetics. It is also a useful model for the study of local mode behavior. For
most applications, it is enough to know the PES near its minimum. Therefore, it is
35
Table 2.17 Mean, standard deviation (St. dev.), and mean absolute error (MAE) for the
B2PLYP/VTZ and B3LYP/SNSD deviations from CCSD(T) semiexperimental equilibrium
structures (distances in pm and angles in degree)
Bond
n# a
B2PLYP/VTZ
B3LYP/SNSD
Mean
St. dev.
MAE
Mean
St. dev.
MAE
All bonds
74
0.00
0.05
0.04
−0.01
0.09
0.07
CH
30
−0.02
0.04
0.03
−0.05
0.05
0.06
CC
21
0.02
0.05
0.04
0.05
0.09
0.09
CO
7
0.02
0.02
0.02
0.03
0.04
0.05
Angles
46
0.00
0.05
0.03
0.00
0.07
0.05
Source Penocchio et al. (2015)
a Number of data in the sample
as shown by Piccardo et al. (2015), a double-zeta basis set such as SNSD (downloadable on http://dreamslab.sns.it) already gives satisfactory results; see Table 2.17.
Finally, B3LYP is able to give reliable geometries as well as force fields (Bauschlicher
et al. 1997). B3LYP slightly overestimates single bond lengths and underestimates
multiple bond lengths. Nevertheless, it is possible to correct B3LYP results (as for
the MP2 ones, see Sect. 2.12) and obtain mean absolute errors of about 0.02 pm for
all types of bonds. For instance, for the CO bond, using the 6-311 + G(3df,2pd) basis
set, the median value of residuals is +0.018 pm for multiple bonds and −0.021 pm
for single bonds (Demaison and Császár 2012). However, the differences are not
constant, but there is a linear relationship between the residuals and the B3LYP
values. Still for the CO bond, a linear fit gives
r e (CO) = 0.9780(29)r [B3LYP/6 − 311 + G(3df, 2pd)] + 0.0283(37) (2.39)
with a correlation coefficient of ρ = 0.9996 and a standard deviation of σ = 0.21 pm.
This is of quality comparable to the MP2/VTZ level of theory.
Table 2.18 compares the results of the structures optimized with the B3LYP/SNSD
or B2PLYP/VTZ levels of theory with accurate semiexperimental equilibrium
structures. The agreement is satisfactory.
2.14 Calculation of the Force Field (Császár 2012)
The concept of potential energy (hyper)surface (PES) is fundamental in chemistry.
The equilibrium structure corresponds to its minimum. The PES governs the vibrations of the atoms. Its knowledge is also important for a better understanding of
reaction kinetics. It is also a useful model for the study of local mode behavior. For
most applications, it is enough to know the PES near its minimum. Therefore, it is