6-31G* computational levels, respectively [17]. The calculations were carried out
for a set of 36 one- and two-heavy-atom molecules comprising second-row elements (first-row using terminology of Ref. [17]). The calculated frequencies were
recommended to obtain molecular entropies, from which entropies of reactions that
are in good agreement with experimental ones can be calculated. Similar calculations, this time using much more extended training set of molecules (122 molecules,
1066 frequencies) were reported in 1993 [18]. The recommended scaling factors are
0.8929 and 0.9427 for HF/6-31G* and MP2/6-31G* fundamentals, respectively,
and 0.9135 and 0.9646 for HF/6-31G* and MP2/6-31G* ZPVE corrections,
respectively. More comprehensive study came in 1996 [19]. The authors determined SFs for vibrational frequencies, ZPVE corrections as well as thermal contributions to enthalpy and entropy for subsequent usage with 19 computational
levels. They include semiempirical AM1 and PM3 levels, ab initio HF, MP2, and
QCISD as well as a few DFT (BLYP, B3LYP, BP86, B3P86, and B3PW91)
computational levels with a few Pople basis sets. The calculations were based on
various training sets, including that of 122 molecules for frequencies, as before. It
was shown that the best DFT results were obtained with B3LYP and B3PW91
functionals. In addition, HF procedure turned out to provide better results as
compared with the post-HF MP2 method.
2 The recommended SFs for the
above-mentioned three computational levels are reported in Table 2.1.
A number of other contributions to the development of US between 1996 and
2007 appeared. The recommended literature includes works by Wong [20], Truhlar
[21–26], Curtiss [27], Schlegel [28], Wilson [29], Uvdal [30], Csonka [31], and
Tantirungrotechai [32]. A brief summary is reported in Table 2.2. These works
Table 2.1 Recommended SFs reported in [19] for HF, DFT/B3LYP and DFT/B3PW91 with
6-31G* basis set for various quantities (frequencies, ZPVE, DH vib (T), and S vib (T))
Method
Frequencies
Low
frequencies
ZPVE
DH vib (T)
d
S ib (T)
d
HF
0.8953
(50
a )
0.9061 (15
b
)
0.9135
(0.71
c )
0.8905
(0.044
e )
0.8978
(0.28
f
)
DFT/
B3LYP
0.9614
(34
a )
1.0013 (15
b
)
0.9806
(0.42
c )
0.9989
(0.036
e )
1.0015
(0.26
f
)
DFT/
B3PW91
0.9573
(34
a )
0.9930 (14
b
)
0.9772
(0.37
c )
0.9885
(0.036
e )
0.9920
(0.24
f
)
The RMS values for the considered quantities are reported in parentheses
a In [cm
−1
]
b
In units of 10
−5 cm
c In [kJ/mol]
d
At 25 °C
e In [kJ/mol]
f
In [J/mol K]
2
This behavior was also observed in the case of multi-parameter scaling procedures. Although
MP2 is known to predict very accurate molecular geometries, it does not provide accurate
vibrational spectra.
2 Scaling Procedures in Vibrational Spectroscopy
69
for a set of 36 one- and two-heavy-atom molecules comprising second-row elements (first-row using terminology of Ref. [17]). The calculated frequencies were
recommended to obtain molecular entropies, from which entropies of reactions that
are in good agreement with experimental ones can be calculated. Similar calculations, this time using much more extended training set of molecules (122 molecules,
1066 frequencies) were reported in 1993 [18]. The recommended scaling factors are
0.8929 and 0.9427 for HF/6-31G* and MP2/6-31G* fundamentals, respectively,
and 0.9135 and 0.9646 for HF/6-31G* and MP2/6-31G* ZPVE corrections,
respectively. More comprehensive study came in 1996 [19]. The authors determined SFs for vibrational frequencies, ZPVE corrections as well as thermal contributions to enthalpy and entropy for subsequent usage with 19 computational
levels. They include semiempirical AM1 and PM3 levels, ab initio HF, MP2, and
QCISD as well as a few DFT (BLYP, B3LYP, BP86, B3P86, and B3PW91)
computational levels with a few Pople basis sets. The calculations were based on
various training sets, including that of 122 molecules for frequencies, as before. It
was shown that the best DFT results were obtained with B3LYP and B3PW91
functionals. In addition, HF procedure turned out to provide better results as
compared with the post-HF MP2 method.
2 The recommended SFs for the
above-mentioned three computational levels are reported in Table 2.1.
A number of other contributions to the development of US between 1996 and
2007 appeared. The recommended literature includes works by Wong [20], Truhlar
[21–26], Curtiss [27], Schlegel [28], Wilson [29], Uvdal [30], Csonka [31], and
Tantirungrotechai [32]. A brief summary is reported in Table 2.2. These works
Table 2.1 Recommended SFs reported in [19] for HF, DFT/B3LYP and DFT/B3PW91 with
6-31G* basis set for various quantities (frequencies, ZPVE, DH vib (T), and S vib (T))
Method
Frequencies
Low
frequencies
ZPVE
DH vib (T)
d
S ib (T)
d
HF
0.8953
(50
a )
0.9061 (15
b
)
0.9135
(0.71
c )
0.8905
(0.044
e )
0.8978
(0.28
f
)
DFT/
B3LYP
0.9614
(34
a )
1.0013 (15
b
)
0.9806
(0.42
c )
0.9989
(0.036
e )
1.0015
(0.26
f
)
DFT/
B3PW91
0.9573
(34
a )
0.9930 (14
b
)
0.9772
(0.37
c )
0.9885
(0.036
e )
0.9920
(0.24
f
)
The RMS values for the considered quantities are reported in parentheses
a In [cm
−1
]
b
In units of 10
−5 cm
c In [kJ/mol]
d
At 25 °C
e In [kJ/mol]
f
In [J/mol K]
2
This behavior was also observed in the case of multi-parameter scaling procedures. Although
MP2 is known to predict very accurate molecular geometries, it does not provide accurate
vibrational spectra.
2 Scaling Procedures in Vibrational Spectroscopy
69