32
2 Computational Methods
For instance, for the MP2/VTZ value of r(CC) in benzene: r = r e − r(calc.) =
–0.13 pm, whereas the offset at the same level of theory is r = –0.65 pm for
NC–CN. This large difference cannot be explained by the variation of the r e value
because they are quite close. Constancy of the offset value in a given type of bond
implies that similar errors occur in the calculation of that value, i.e., the finite basis set
creates the same error, the partial neglect of electron correlation has the same effect on
the calculated bond length, etc. Consequently, it is not surprising that the magnitude
of the offset value is at least somewhat responsive to environmental perturbations
from the surroundings of the bond. The conclusion of this discussion is that the offset
method is useful but has to be used with caution. Nevertheless, in such a case, it is
advisable to verify that the bonding in the two molecules is similar by using, for
instance, the Atom In Molecule (AIM) theory; see Sect. 2.18.
It is possible to improve this method by using the linear regression approach
r e = a · r
MP2
e
+ b
(2.34)
where the coefficients a and b determined by least-squares fit.
For instance, the determination of the experimental structure of a fluorine derivative is a difficult problem because there is only one stable isotope for fluorine, making
studies of isotopic species impossible. The calculation of a reliable ab initio structure is further complicated by the fact that fluorine is a highly electronegative atom
which requires very large basis sets and highly correlated methods. On the other
hand, a least-squares fit of the C(sp
3 )-F single bond length of 15 molecules with the
MP2/6-311 + G(3df,2pd) method gives a = 0.99827(20) and b = 0 with a standard
deviation of 0.11 pm (Juanes et al. 2017). Likewise, it is possible to use this equation
to predict the length of CC bonds in phenyl rings using the MP2/VTZ method. For
30 molecules, it gives a = 0.998414(73) and b = 0 with a standard deviation of
0.055 pm (Demaison et al. 2019). Similar results were obtained for the CO bond
length. For a sample of 47 molecules with single and double bonds, the MP2/VQZ
level of theory gives a = 1.0215(30) and b = −3.03(27) with a standard deviation
of 0.2 pm (Demaison and Császár 2012).
2.13 Density Functional Theory (DFT)
The Density Functional Theory (DFT) is a way to treat electron correlation in a
much cheaper way than the correlated wavefunction methods like MP2, CCSD, or
CCSD(T). The basic idea is to use the electron density ρ(r) which only depends on
the three coordinates x, y, and z instead of the many-electron wavefunction which
depends on many variables (3n neglecting the spin, n being the number of electrons).
The theoretical basis is the Hohenberg and Kohn (1964) theorem that states that
the density of any system determines all the ground state properties of the system,
in particular the energy. The energy being variational with respect to the density, the
minimum of the energy defines the electron density.
2 Computational Methods
For instance, for the MP2/VTZ value of r(CC) in benzene: r = r e − r(calc.) =
–0.13 pm, whereas the offset at the same level of theory is r = –0.65 pm for
NC–CN. This large difference cannot be explained by the variation of the r e value
because they are quite close. Constancy of the offset value in a given type of bond
implies that similar errors occur in the calculation of that value, i.e., the finite basis set
creates the same error, the partial neglect of electron correlation has the same effect on
the calculated bond length, etc. Consequently, it is not surprising that the magnitude
of the offset value is at least somewhat responsive to environmental perturbations
from the surroundings of the bond. The conclusion of this discussion is that the offset
method is useful but has to be used with caution. Nevertheless, in such a case, it is
advisable to verify that the bonding in the two molecules is similar by using, for
instance, the Atom In Molecule (AIM) theory; see Sect. 2.18.
It is possible to improve this method by using the linear regression approach
r e = a · r
MP2
e
+ b
(2.34)
where the coefficients a and b determined by least-squares fit.
For instance, the determination of the experimental structure of a fluorine derivative is a difficult problem because there is only one stable isotope for fluorine, making
studies of isotopic species impossible. The calculation of a reliable ab initio structure is further complicated by the fact that fluorine is a highly electronegative atom
which requires very large basis sets and highly correlated methods. On the other
hand, a least-squares fit of the C(sp
3 )-F single bond length of 15 molecules with the
MP2/6-311 + G(3df,2pd) method gives a = 0.99827(20) and b = 0 with a standard
deviation of 0.11 pm (Juanes et al. 2017). Likewise, it is possible to use this equation
to predict the length of CC bonds in phenyl rings using the MP2/VTZ method. For
30 molecules, it gives a = 0.998414(73) and b = 0 with a standard deviation of
0.055 pm (Demaison et al. 2019). Similar results were obtained for the CO bond
length. For a sample of 47 molecules with single and double bonds, the MP2/VQZ
level of theory gives a = 1.0215(30) and b = −3.03(27) with a standard deviation
of 0.2 pm (Demaison and Császár 2012).
2.13 Density Functional Theory (DFT)
The Density Functional Theory (DFT) is a way to treat electron correlation in a
much cheaper way than the correlated wavefunction methods like MP2, CCSD, or
CCSD(T). The basic idea is to use the electron density ρ(r) which only depends on
the three coordinates x, y, and z instead of the many-electron wavefunction which
depends on many variables (3n neglecting the spin, n being the number of electrons).
The theoretical basis is the Hohenberg and Kohn (1964) theorem that states that
the density of any system determines all the ground state properties of the system,
in particular the energy. The energy being variational with respect to the density, the
minimum of the energy defines the electron density.