146
6 Equilibrium Structures from Spectroscopy
In this equation, the denominator ω
2
k − ω
2
l disappears. The calculation of this
rovibrational correction is a more robust numerical procedure than the calculation of
the individual α-constants.
In conclusion, equilibrium rotational constants can be estimated using experimental ground-state rotational constants and computed rovibrational corrections.
The first important question is the origin of the force field because it is not easier to
obtain an experimental cubic force field than a full set of α-constants. Fortunately, it
is possible to calculate ab initio anharmonic force fields as explained in Sect. 2.14.
The next question is the accuracy of the results. It obviously depends on the level of
theory. A statistical analysis of available data shows that the rovibrational correction
calculated at the CCSD(T)_ae/CVQZ level of theory is systematically smaller by
about 2% than the corresponding experimental value (Vogt et al. 2011). It remains
true for MP2/VTZ calculations, but the median absolute deviation at 4% is larger.
The semiexperimental and experimental equilibrium structures of several polyatomic molecules were compared by many authors, among them Botschwina (1988,
2005) and Bak et al. (2001). The results were found in excellent agreement. They
also compared the semiexperimental structures to high-level ab initio optimizations,
and Bak et al. (2001) concluded that the accuracy of the semiexperimental structures
surpasses that reported in most experimental determinations. Actually, this conclusion is probably not true for molecules containing very heavy atoms. Moreover, the
solution is less accurate for very light molecules because the γ -terms in (6.1) are not
taken into account whereas they may have a sizeable effect.
There is another way to confirm the conclusion that the semiexperimental method
is highly accurate. The semiexperimental structure of 1,1-difluoroethene was determined using for the calculation of the rovibrational correction a cubic force field
computed at five different levels of theory from HF/6-31G*to MP2/VQZ (Vogt et al.
2014). The results are given in Table 6.9. It is obvious that the rovibrational corrections are quite sensitive to the level of calculation. For instance, the range for the
correction of the A rotational constant of the parent species is 13 MHz for a mean value
of 43 MHz. Nevertheless, the structures derived from the five sets of rovibrational
corrections are almost identical.
It is possible to explain this behavior. The calculated rovibrational corrections
are affected by a (mainly) systematic error of a few %. These corrections are quite
small, smaller than 1% of the moments of inertia in most cases; see Table 6.1 and
Sect. 6.2. Therefore, the error on the derived semiexperimental equilibrium moments
of inertia is only about 0.01
0 / 0 (or less) and, furthermore, this error happens to be
mainly systematic. This effect is equivalent to multiplying the equilibrium moments
of inertia by a constant factor f , very close to one. Therefore, the solution is little
changed (multiplied by about
√
f ), provided f has exactly the same value for all y i .
The systematic errors are further discussed in Sect. 9.10.3.
In conclusion, a survey of the literature shows that the MP2 method with a basis set
of triple-zeta quality often gives results that are sufficiently accurate for computing
vibration–rotation constants. Some DFT methods are of comparable quality or even
better (Penocchio et al. 2015; Piccardo et al. 2015). In Table 6.10, the equilib-
6 Equilibrium Structures from Spectroscopy
In this equation, the denominator ω
2
k − ω
2
l disappears. The calculation of this
rovibrational correction is a more robust numerical procedure than the calculation of
the individual α-constants.
In conclusion, equilibrium rotational constants can be estimated using experimental ground-state rotational constants and computed rovibrational corrections.
The first important question is the origin of the force field because it is not easier to
obtain an experimental cubic force field than a full set of α-constants. Fortunately, it
is possible to calculate ab initio anharmonic force fields as explained in Sect. 2.14.
The next question is the accuracy of the results. It obviously depends on the level of
theory. A statistical analysis of available data shows that the rovibrational correction
calculated at the CCSD(T)_ae/CVQZ level of theory is systematically smaller by
about 2% than the corresponding experimental value (Vogt et al. 2011). It remains
true for MP2/VTZ calculations, but the median absolute deviation at 4% is larger.
The semiexperimental and experimental equilibrium structures of several polyatomic molecules were compared by many authors, among them Botschwina (1988,
2005) and Bak et al. (2001). The results were found in excellent agreement. They
also compared the semiexperimental structures to high-level ab initio optimizations,
and Bak et al. (2001) concluded that the accuracy of the semiexperimental structures
surpasses that reported in most experimental determinations. Actually, this conclusion is probably not true for molecules containing very heavy atoms. Moreover, the
solution is less accurate for very light molecules because the γ -terms in (6.1) are not
taken into account whereas they may have a sizeable effect.
There is another way to confirm the conclusion that the semiexperimental method
is highly accurate. The semiexperimental structure of 1,1-difluoroethene was determined using for the calculation of the rovibrational correction a cubic force field
computed at five different levels of theory from HF/6-31G*to MP2/VQZ (Vogt et al.
2014). The results are given in Table 6.9. It is obvious that the rovibrational corrections are quite sensitive to the level of calculation. For instance, the range for the
correction of the A rotational constant of the parent species is 13 MHz for a mean value
of 43 MHz. Nevertheless, the structures derived from the five sets of rovibrational
corrections are almost identical.
It is possible to explain this behavior. The calculated rovibrational corrections
are affected by a (mainly) systematic error of a few %. These corrections are quite
small, smaller than 1% of the moments of inertia in most cases; see Table 6.1 and
Sect. 6.2. Therefore, the error on the derived semiexperimental equilibrium moments
of inertia is only about 0.01
0 / 0 (or less) and, furthermore, this error happens to be
mainly systematic. This effect is equivalent to multiplying the equilibrium moments
of inertia by a constant factor f , very close to one. Therefore, the solution is little
changed (multiplied by about
√
f ), provided f has exactly the same value for all y i .
The systematic errors are further discussed in Sect. 9.10.3.
In conclusion, a survey of the literature shows that the MP2 method with a basis set
of triple-zeta quality often gives results that are sufficiently accurate for computing
vibration–rotation constants. Some DFT methods are of comparable quality or even
better (Penocchio et al. 2015; Piccardo et al. 2015). In Table 6.10, the equilib-