using redundant set of PICs was proposed [75]. It involved utilization of generalized Moore–Penrose inverse of a matrix. As in the case of SQM, it is necessary to
invert the transformation matrix B r between the redundant PICs and the Cartesian
displacements (L Â 3N matrix) which is then used in the transformation of the
Cartesian FFs f
x to obtain F r and generation of a mass tensor G
À1
r . It was also
decided to use the canonical FFs (see, e.g., [76]) in order to ensure that the solution
is unique. Since both matrices: F r and G
À1
r are now singular, Eq. (26) was projected
onto some set of non-redundant ICs s nr but the final a r matrix used to calculate
PEDs must refer to redundant PICs, i.e., to the transformation s r ¼ a r Q (indices “r”
and “nr” refer to redundant and non-redundant, respectively). It can be shown that
the A transformation matrix from s r to s nr ðs nr ¼ As r Þ can be obtained by diagonalizing B r B
T
r , and taking the transpose of O eigenvector matrix corresponding to
nonzero eigenvalues, i.e., A ¼ O
T . Then we have F nr ¼ B
À
nr
À Á T f
x B
À
nr and
a r ¼ B r B
À
nr a nr , where B nr ¼ AB r , a nr is a transformation matrix obtained from
solution of WDC equations (2.26) with F nr , and the superscript “
−
” denotes the
generalized Moore–Penrose inverse. Having unique F r and a r matrices the PEDs
can be calculated according to Eq. (2.54).
Finally, one more extension of ESFF, called ESFF2, was made [75]. Namely,
when calculating contributions of local modes to a given normal mode (note, the
term “contribution” is used instead of “PED”), one can skip the FCs in Eq. (2.54)
and use only amplitudes, i.e.,
p ll;j ¼
a
2
lj
P K
m¼1 a 2
mj
:
ð2:62Þ
This approach, which turned out to be equally efficient as PED-based ESFF, is
particularly well suited to calculations on really large systems. This problem will be
briefly discussed in Sect. 2.3.6.3.
2.3.5.2 Development of ESFF
First application of the ESFF procedure, application to toluene, was made in the
first, methodological paper [74]. As in the case of pioneering works of scaling
procedures, the least-squares fit was made to frequencies of toluene itself, i.e., no
training set was used. Six LSFs were optimized, and the obtained results were
highly encouraging. However, as already discussed, the applicability of the scaling
procedures is contingent upon the already mentioned transferability of the SFs
among the molecules. Thus, the determination of the LSFs using a very diversified
training set of molecules (Baker’s training set [54]) was made [68]. Using this set,
apart from obtaining the SFs for the routine applications, comparison of the ESFF
and the SQM benchmark calculations was possible. The determination of both
LSFs and FF SFs at the B3LYP/6-311G** level was based on 660 experimental
fundamentals observed in the gas-phase spectra. The FF SFs for the redundant
84
O. Bąk and P. Borowski
invert the transformation matrix B r between the redundant PICs and the Cartesian
displacements (L Â 3N matrix) which is then used in the transformation of the
Cartesian FFs f
x to obtain F r and generation of a mass tensor G
À1
r . It was also
decided to use the canonical FFs (see, e.g., [76]) in order to ensure that the solution
is unique. Since both matrices: F r and G
À1
r are now singular, Eq. (26) was projected
onto some set of non-redundant ICs s nr but the final a r matrix used to calculate
PEDs must refer to redundant PICs, i.e., to the transformation s r ¼ a r Q (indices “r”
and “nr” refer to redundant and non-redundant, respectively). It can be shown that
the A transformation matrix from s r to s nr ðs nr ¼ As r Þ can be obtained by diagonalizing B r B
T
r , and taking the transpose of O eigenvector matrix corresponding to
nonzero eigenvalues, i.e., A ¼ O
T . Then we have F nr ¼ B
À
nr
À Á T f
x B
À
nr and
a r ¼ B r B
À
nr a nr , where B nr ¼ AB r , a nr is a transformation matrix obtained from
solution of WDC equations (2.26) with F nr , and the superscript “
−
” denotes the
generalized Moore–Penrose inverse. Having unique F r and a r matrices the PEDs
can be calculated according to Eq. (2.54).
Finally, one more extension of ESFF, called ESFF2, was made [75]. Namely,
when calculating contributions of local modes to a given normal mode (note, the
term “contribution” is used instead of “PED”), one can skip the FCs in Eq. (2.54)
and use only amplitudes, i.e.,
p ll;j ¼
a
2
lj
P K
m¼1 a 2
mj
:
ð2:62Þ
This approach, which turned out to be equally efficient as PED-based ESFF, is
particularly well suited to calculations on really large systems. This problem will be
briefly discussed in Sect. 2.3.6.3.
2.3.5.2 Development of ESFF
First application of the ESFF procedure, application to toluene, was made in the
first, methodological paper [74]. As in the case of pioneering works of scaling
procedures, the least-squares fit was made to frequencies of toluene itself, i.e., no
training set was used. Six LSFs were optimized, and the obtained results were
highly encouraging. However, as already discussed, the applicability of the scaling
procedures is contingent upon the already mentioned transferability of the SFs
among the molecules. Thus, the determination of the LSFs using a very diversified
training set of molecules (Baker’s training set [54]) was made [68]. Using this set,
apart from obtaining the SFs for the routine applications, comparison of the ESFF
and the SQM benchmark calculations was possible. The determination of both
LSFs and FF SFs at the B3LYP/6-311G** level was based on 660 experimental
fundamentals observed in the gas-phase spectra. The FF SFs for the redundant
84
O. Bąk and P. Borowski