the given NIC may not fully account for the contributions of PICs in a given mode
in some cases. This problem is of particular importance in the case of the
multi-parameter frequency scaling procedure discussed in the next section. Thus, it
appears that set of PICs is more flexible. Second, manual generation of NICs is
tedious. Although there are programs, e.g., [65] that automatically generate NICs,
they are not free of shortcomings, and—as stated in [54]—“… algorithms … may
fail occasionally for unusual topologies, e.g., cage compounds.” Third, NICs are
frequently delocalized, which is also a disadvantage. Fourth, some of the NICs are
linear combinations of PICs which should not share the same SF. For example, NIC
for the symmetric bending of the methyl group is a linear combination of XCH and
HCH primitive bends, which are now usually put into different types.
On the other hand, the choice of PICs that form a non-redundant set is not
unique. The only thing one has to take care of is their linear independence. As
frequencies rather than FCs are observables, they are the same for all possible
choices of ICs. It was thus decided [54] to modify the original SQM procedure and
transform the Cartesian FF to the set of redundant PICs. They are easily generated
based on atomic connectivities, and their number L is greater than K, and even
greater than 3N in an overwhelming majority of cases. Consider, e.g., ethylene
molecule (3N = 18, K = 12). The redundant set of PICs consists of 5 stretches, 4
CCH and 2 HCH bendings, and 4 HCCH torsions (15 primitives overall). Thus, the
B matrix augmented with translations and rotations is a 15 + 6 = 21 Â 18 matrix,
inversion of which can be accomplished using the generalized Moore–Penrose
inverse. Introduction of the generalized inverse whenever necessary is the main
difference between the “classical” and “modified” SQM procedure. The scaling
idea, i.e., division of ICs into chemically similar groups, is retained.
This time [54] the authors used a new training set of 30 small organic molecules,
the fundamentals of which were taken from the post-1993 literature. The set
includes a wide variety of organic molecules, including saturated and unsaturated
compounds, both having chain, cyclic and heterocyclic structures with hydroxyl,
carbonyl, amino, etc. substituents. They carefully selected as many as 663 vibrational fundamentals, eliminated misassignments and fundamentals of uncertain
positions. Again the B3LYP/6-31G* computational level was used. A new classification of ICs into chemically similar groups was also proposed (see Table 2.5).
In the following, it will be called “11-parameter scaling frame.” Optimization of
SFs for the set of 663 fundamentals gave the RMS value of merely 12.04 cm
−1 .
Comparison of both classifications along with the recommended values is given in
Table 2.5. As can be seen some factors, in particular for the CCl stretch, turned out
to have a value larger than unity. This is also the case of multi-parameter frequency
scaling procedure described in the next section. This is apparently a “geometric
effect”—CCl bonds predicted by B3LYP are too long and, consequently, too weak.
For this reason, the harmonic frequency drops below the fundamental one. SFs for
CCl calculated, say, at the MP2 computational level have values lower than unity
(see, e.g., [66]). As before the factors calculated for the training set were applied to
the test set, which also consisted of 30 molecules (843 fundamentals found in the
post-1993 literature). The calculations proved again excellent transferability of FF
78
O. Bąk and P. Borowski
in some cases. This problem is of particular importance in the case of the
multi-parameter frequency scaling procedure discussed in the next section. Thus, it
appears that set of PICs is more flexible. Second, manual generation of NICs is
tedious. Although there are programs, e.g., [65] that automatically generate NICs,
they are not free of shortcomings, and—as stated in [54]—“… algorithms … may
fail occasionally for unusual topologies, e.g., cage compounds.” Third, NICs are
frequently delocalized, which is also a disadvantage. Fourth, some of the NICs are
linear combinations of PICs which should not share the same SF. For example, NIC
for the symmetric bending of the methyl group is a linear combination of XCH and
HCH primitive bends, which are now usually put into different types.
On the other hand, the choice of PICs that form a non-redundant set is not
unique. The only thing one has to take care of is their linear independence. As
frequencies rather than FCs are observables, they are the same for all possible
choices of ICs. It was thus decided [54] to modify the original SQM procedure and
transform the Cartesian FF to the set of redundant PICs. They are easily generated
based on atomic connectivities, and their number L is greater than K, and even
greater than 3N in an overwhelming majority of cases. Consider, e.g., ethylene
molecule (3N = 18, K = 12). The redundant set of PICs consists of 5 stretches, 4
CCH and 2 HCH bendings, and 4 HCCH torsions (15 primitives overall). Thus, the
B matrix augmented with translations and rotations is a 15 + 6 = 21 Â 18 matrix,
inversion of which can be accomplished using the generalized Moore–Penrose
inverse. Introduction of the generalized inverse whenever necessary is the main
difference between the “classical” and “modified” SQM procedure. The scaling
idea, i.e., division of ICs into chemically similar groups, is retained.
This time [54] the authors used a new training set of 30 small organic molecules,
the fundamentals of which were taken from the post-1993 literature. The set
includes a wide variety of organic molecules, including saturated and unsaturated
compounds, both having chain, cyclic and heterocyclic structures with hydroxyl,
carbonyl, amino, etc. substituents. They carefully selected as many as 663 vibrational fundamentals, eliminated misassignments and fundamentals of uncertain
positions. Again the B3LYP/6-31G* computational level was used. A new classification of ICs into chemically similar groups was also proposed (see Table 2.5).
In the following, it will be called “11-parameter scaling frame.” Optimization of
SFs for the set of 663 fundamentals gave the RMS value of merely 12.04 cm
−1 .
Comparison of both classifications along with the recommended values is given in
Table 2.5. As can be seen some factors, in particular for the CCl stretch, turned out
to have a value larger than unity. This is also the case of multi-parameter frequency
scaling procedure described in the next section. This is apparently a “geometric
effect”—CCl bonds predicted by B3LYP are too long and, consequently, too weak.
For this reason, the harmonic frequency drops below the fundamental one. SFs for
CCl calculated, say, at the MP2 computational level have values lower than unity
(see, e.g., [66]). As before the factors calculated for the training set were applied to
the test set, which also consisted of 30 molecules (843 fundamentals found in the
post-1993 literature). The calculations proved again excellent transferability of FF
78
O. Bąk and P. Borowski