only). Roughly at the same time [66], an extended database of FF SFs for 11- and
9-parameter frameworks, with a variety of density functionals and a variety of
Pople’s and Dunning’s basis sets (overall 370 computational levels) was presented.
All the FF SFs reported in the previous paragraph (except those mentioned in
Ref. [70]) were determined to reproduce spectra obtained in gas-phase experiments.
It was shown, however, that SFs are well transferable to molecules in condensed
phases unless hydrogen bonding strongly affects the bands’ position (see e.g., [71]).
The problem of gas phase versus Ar-matrix SFs was thoroughly considered in
reference [72]. A set of 33 molecules (347 vibrational modes), whose vibrational
spectra in both the gas phase and argon matrix are known was used. The authors
carried out calculations using four popular density functionals (PBE, B3LYP,
B3PW91, and M06-2X), two basis sets (6-31++G** and aug-cc-pVTZ), and
11-parameter scaling frame (with two additional types, i.e., SH and {H,X}XXH)
within the redundant PICs formalism. Surprisingly, the largest differences between
gas phase and Ar-matrix FF SFs were observed for the XXXX type (up to 3%)
rather than types involved in hydrogen bonding, i.e., OH and NH (<1%). Matrix
shifts were also reasonably reproduced with the new set of FF SFs. Finally, the
authors declared to report more FF SFs for recent density functionals (including
those accounting for the dispersion interactions) but we did not find works on that
in the post-2011 literature.
2.3.5 Effective Scaling Frequency Factor Method
2.3.5.1 Fundamentals of the Method
Effective Scaling Frequency Factor (ESFF) method is a multi-parameter frequency
rather than FF scaling method, in which SFs are applied directly to frequencies after
solving the vibrational problem (26). Here again the outline of the up-to-date theory
will be presented first, deferring a brief literature review on the historical background of methodology development.
Let us first consider the original formulation of the method, in which potential
energy U of a molecule is expressed in terms of non-redundant ICs, chosen to be
NICs. The potential energy in harmonic approximation, Eq. (2.28), reads
U s
ð Þ ¼
1
2
X K
l;m¼1
F lm s l s m :
ð2:51Þ
As in Sect. 2.2.3, s denotes here deviations of internal coordinates from their
equilibrium values. Introducing expansion (2.27) of ICs in terms of normal coordinates, i.e., s l ¼
P K
j¼1 a lj Q j , we obtain
2 Scaling Procedures in Vibrational Spectroscopy
81
9-parameter frameworks, with a variety of density functionals and a variety of
Pople’s and Dunning’s basis sets (overall 370 computational levels) was presented.
All the FF SFs reported in the previous paragraph (except those mentioned in
Ref. [70]) were determined to reproduce spectra obtained in gas-phase experiments.
It was shown, however, that SFs are well transferable to molecules in condensed
phases unless hydrogen bonding strongly affects the bands’ position (see e.g., [71]).
The problem of gas phase versus Ar-matrix SFs was thoroughly considered in
reference [72]. A set of 33 molecules (347 vibrational modes), whose vibrational
spectra in both the gas phase and argon matrix are known was used. The authors
carried out calculations using four popular density functionals (PBE, B3LYP,
B3PW91, and M06-2X), two basis sets (6-31++G** and aug-cc-pVTZ), and
11-parameter scaling frame (with two additional types, i.e., SH and {H,X}XXH)
within the redundant PICs formalism. Surprisingly, the largest differences between
gas phase and Ar-matrix FF SFs were observed for the XXXX type (up to 3%)
rather than types involved in hydrogen bonding, i.e., OH and NH (<1%). Matrix
shifts were also reasonably reproduced with the new set of FF SFs. Finally, the
authors declared to report more FF SFs for recent density functionals (including
those accounting for the dispersion interactions) but we did not find works on that
in the post-2011 literature.
2.3.5 Effective Scaling Frequency Factor Method
2.3.5.1 Fundamentals of the Method
Effective Scaling Frequency Factor (ESFF) method is a multi-parameter frequency
rather than FF scaling method, in which SFs are applied directly to frequencies after
solving the vibrational problem (26). Here again the outline of the up-to-date theory
will be presented first, deferring a brief literature review on the historical background of methodology development.
Let us first consider the original formulation of the method, in which potential
energy U of a molecule is expressed in terms of non-redundant ICs, chosen to be
NICs. The potential energy in harmonic approximation, Eq. (2.28), reads
U s
ð Þ ¼
1
2
X K
l;m¼1
F lm s l s m :
ð2:51Þ
As in Sect. 2.2.3, s denotes here deviations of internal coordinates from their
equilibrium values. Introducing expansion (2.27) of ICs in terms of normal coordinates, i.e., s l ¼
P K
j¼1 a lj Q j , we obtain
2 Scaling Procedures in Vibrational Spectroscopy
81