US is simple and straightforward, which makes it very attractive for routine
applications. The only thing one has to do after solving the vibrational problem in
harmonic approximation is to find the SF relevant for a given computational level in
the literature and scale frequencies. However, there are twofold disadvantages of
US as compared with the more sophisticated, multi-parameter treatments. First, the
quality of the scaled frequencies is not spectacular. The RMS values are typically
large. As will be discussed in the case of frequency scaling, they frequently exceed
30 cm
−1 even with high-quality FFs (see, e.g., [15]). This is partially due to the fact
that the training set used in the optimization procedure in the cited work was very
diversified. It contains various types of molecules, including both organic and
inorganic ones, closed- and open-shell molecules, etc. Probably lower RMS would
be obtained if the training set consisted of molecules with rather similar structural
motifs (e.g., organic molecules only). Second, probably more important disadvantage, is associated with the fact that US does not provide the frequency swaps,
which are frequently necessary, in particular in the overcrowded spectral ranges.
Consider, for example, two close-lying vibrations, one being a simple stretch of
some bond, the other one—some out-of-plane bending vibration, with the harmonic
frequency of the former only somewhat greater than that of the latter one, and with
similar theoretical intensities. Assume, that they correspond to two well-resolved
close-lying bands on IR spectrum, also of similar intensities. Multiplying both
frequencies by the same SF would preserve their order, which may lead to
misassignment. The reason is that it may be expected, that the frequency lowering
for stretch due to anharmonicity is much greater than that for the out-of-plane
bending (the out-of-plane motion, like H motion out of the Ar plane in aromatic
compounds, has symmetric potential, like parabola, and for this reason it is more
harmonic than stretch). Having two SFs, lower one for the stretch, and larger one
for out-of-plane bending, would solve the problem. This is, in fact, what
multi-parameter scaling is handling quite well.
2.3.2.2 Development of US
Considerations presented in Sect. 2.3.2.1 exhaust the theory of US. In the following, we briefly report the literature on historical development of US to provide
the reader with the relevant references. For more details, the reader is referred to the
original papers. Probably, the first report on single frequency SF was given by
Pople and co-workers back in 1981 [16]. A value of 0.89 was derived for a training
set of 38 molecules (nearly 500 frequencies) and recommended for subsequent use
with HF/3-21G frequencies. This work initialized relentless progress in US procedure due to continuous QC methodology and basis set development as well as
advances in software and hardware manufacturing, which enable increasingly
accurate QC calculations on increasingly large molecules. Factors of 0.8929 and
0.9210 were derived one year later at more sophisticated HF/6-31G* and MP2-fu/
68
O. Bąk and P. Borowski
applications. The only thing one has to do after solving the vibrational problem in
harmonic approximation is to find the SF relevant for a given computational level in
the literature and scale frequencies. However, there are twofold disadvantages of
US as compared with the more sophisticated, multi-parameter treatments. First, the
quality of the scaled frequencies is not spectacular. The RMS values are typically
large. As will be discussed in the case of frequency scaling, they frequently exceed
30 cm
−1 even with high-quality FFs (see, e.g., [15]). This is partially due to the fact
that the training set used in the optimization procedure in the cited work was very
diversified. It contains various types of molecules, including both organic and
inorganic ones, closed- and open-shell molecules, etc. Probably lower RMS would
be obtained if the training set consisted of molecules with rather similar structural
motifs (e.g., organic molecules only). Second, probably more important disadvantage, is associated with the fact that US does not provide the frequency swaps,
which are frequently necessary, in particular in the overcrowded spectral ranges.
Consider, for example, two close-lying vibrations, one being a simple stretch of
some bond, the other one—some out-of-plane bending vibration, with the harmonic
frequency of the former only somewhat greater than that of the latter one, and with
similar theoretical intensities. Assume, that they correspond to two well-resolved
close-lying bands on IR spectrum, also of similar intensities. Multiplying both
frequencies by the same SF would preserve their order, which may lead to
misassignment. The reason is that it may be expected, that the frequency lowering
for stretch due to anharmonicity is much greater than that for the out-of-plane
bending (the out-of-plane motion, like H motion out of the Ar plane in aromatic
compounds, has symmetric potential, like parabola, and for this reason it is more
harmonic than stretch). Having two SFs, lower one for the stretch, and larger one
for out-of-plane bending, would solve the problem. This is, in fact, what
multi-parameter scaling is handling quite well.
2.3.2.2 Development of US
Considerations presented in Sect. 2.3.2.1 exhaust the theory of US. In the following, we briefly report the literature on historical development of US to provide
the reader with the relevant references. For more details, the reader is referred to the
original papers. Probably, the first report on single frequency SF was given by
Pople and co-workers back in 1981 [16]. A value of 0.89 was derived for a training
set of 38 molecules (nearly 500 frequencies) and recommended for subsequent use
with HF/3-21G frequencies. This work initialized relentless progress in US procedure due to continuous QC methodology and basis set development as well as
advances in software and hardware manufacturing, which enable increasingly
accurate QC calculations on increasingly large molecules. Factors of 0.8929 and
0.9210 were derived one year later at more sophisticated HF/6-31G* and MP2-fu/
68
O. Bąk and P. Borowski