fundamentals (and/or experimental harmonic frequencies, if available). In the case
of ethane, propane, and cyclopropane [57], they obtained scaled FFs that reproduced the experimental fundamentals of the molecules and their deuterated analogs
with remarkable accuracy (with RMS ca. 10 cm
−1 ) using merely 5 (or 6 in the case
of cyclopropane) SFs. This was the case of refinement carried out for each molecule
separately (SFs remained roughly the same, regardless of molecule) and for all
molecules simultaneously, indicating good transferability of SFs. Then the calculations of the scaled FF for water (to match both harmonic and fundamental
experimental frequencies) and methanol (to match experimental fundamentals)
were carried out [58]. The average error for harmonic frequencies of water was very
low (1.6 cm
−1 ) while for fundamentals much larger (14.8 cm
−1 ) indicating that the
significant part of the difference may be ascribed to anharmonicity. Similar calculations were performed on propene [59] using both precomputed [58] and optimized (for the set of experimental fundamentals of propene and its deuterated
analogs) SFs. The RMS turned out to be 9.3 and 8.3 cm
−1 , respectively, for the set
84 fundamentals using 7 SFs. The FF SFs were then obtained for ethylene and its
deuterated analogs [60] were the harmonic and fundamental frequencies were
reproduced with remarkable accuracy. In the case of 50 harmonic frequencies, the
RMS value turned out to be 3.9 cm
−1 . This value reflects the neglect of the correlation effects in QC treatment, which apparently cannot be accounted for by a few
SFs. In the case of 81 fundamental frequencies, the RMS was somewhat larger and
amounted to 7.1 cm
−1 . Further, calculations carried out on dimethylether [61]
proved, that the method can be generally applied to reproduce vibrational spectra
for more complex systems.
The research described above constitutes an embryo of the modern SQM
method. The problematic thing in Blom and Altona approach was the treatment of
off-diagonal (interaction or coupling) FCs. Apparently, they did not know how to
solve this problem, so they decided to attribute one of the SFs to all off-diagonal
FCs. In the pioneering work on SQM [11] Pulay et al. said: “It is difficult to see why
physically different off-diagonal elements should share the same scale factor…” and
decided to adopt the geometric mean of SFs as in Eq. (2.48). With this modification, SQM survived until now.
Further development of the SQM procedure
3 consisted in, inter alia, extending
its applicability to a wider range of systems. In 1983, Pulay et al. [12] considered a
series of molecules (glyoxal, acrolein, butadiene, formaldehyde, and ethylene) with
similar structural motifs treated at HF/4-21G computational level. They used six
SFs for 115 in-plane modes and three factors for 41 out-of-plane modes and
managed to reproduce the fundamentals with RMS of 12.3 and 6.7 cm
−1 , respectively. It was also concluded that, in contrast to SFs, the FCs themselves are not
well transferable between molecules. This means that common FF SFs can be used
for a large group of molecules. Twelve years later [62, 63] calculations of the
3
As mentioned in the introductory section in the following we will not cite papers, where only
applications are presented.
76
O. Bąk and P. Borowski
of ethane, propane, and cyclopropane [57], they obtained scaled FFs that reproduced the experimental fundamentals of the molecules and their deuterated analogs
with remarkable accuracy (with RMS ca. 10 cm
−1 ) using merely 5 (or 6 in the case
of cyclopropane) SFs. This was the case of refinement carried out for each molecule
separately (SFs remained roughly the same, regardless of molecule) and for all
molecules simultaneously, indicating good transferability of SFs. Then the calculations of the scaled FF for water (to match both harmonic and fundamental
experimental frequencies) and methanol (to match experimental fundamentals)
were carried out [58]. The average error for harmonic frequencies of water was very
low (1.6 cm
−1 ) while for fundamentals much larger (14.8 cm
−1 ) indicating that the
significant part of the difference may be ascribed to anharmonicity. Similar calculations were performed on propene [59] using both precomputed [58] and optimized (for the set of experimental fundamentals of propene and its deuterated
analogs) SFs. The RMS turned out to be 9.3 and 8.3 cm
−1 , respectively, for the set
84 fundamentals using 7 SFs. The FF SFs were then obtained for ethylene and its
deuterated analogs [60] were the harmonic and fundamental frequencies were
reproduced with remarkable accuracy. In the case of 50 harmonic frequencies, the
RMS value turned out to be 3.9 cm
−1 . This value reflects the neglect of the correlation effects in QC treatment, which apparently cannot be accounted for by a few
SFs. In the case of 81 fundamental frequencies, the RMS was somewhat larger and
amounted to 7.1 cm
−1 . Further, calculations carried out on dimethylether [61]
proved, that the method can be generally applied to reproduce vibrational spectra
for more complex systems.
The research described above constitutes an embryo of the modern SQM
method. The problematic thing in Blom and Altona approach was the treatment of
off-diagonal (interaction or coupling) FCs. Apparently, they did not know how to
solve this problem, so they decided to attribute one of the SFs to all off-diagonal
FCs. In the pioneering work on SQM [11] Pulay et al. said: “It is difficult to see why
physically different off-diagonal elements should share the same scale factor…” and
decided to adopt the geometric mean of SFs as in Eq. (2.48). With this modification, SQM survived until now.
Further development of the SQM procedure
3 consisted in, inter alia, extending
its applicability to a wider range of systems. In 1983, Pulay et al. [12] considered a
series of molecules (glyoxal, acrolein, butadiene, formaldehyde, and ethylene) with
similar structural motifs treated at HF/4-21G computational level. They used six
SFs for 115 in-plane modes and three factors for 41 out-of-plane modes and
managed to reproduce the fundamentals with RMS of 12.3 and 6.7 cm
−1 , respectively. It was also concluded that, in contrast to SFs, the FCs themselves are not
well transferable between molecules. This means that common FF SFs can be used
for a large group of molecules. Twelve years later [62, 63] calculations of the
3
As mentioned in the introductory section in the following we will not cite papers, where only
applications are presented.
76
O. Bąk and P. Borowski