frequency p contributes to the overall LSMF. It can be used to change the
importance of specific frequencies in the fit. It is recommended that for
well-established frequencies, w p is inversely proportional to the frequency itself
[12], although in many applications presetting all weights to unity is a standard
procedure. In the case of multi-parameter scaling procedures, some of the SFs may
be given preset values. This is easily accomplished using, e.g., Lagrange method of
undetermined multipliers in conjunction with Eq. (2.29). Having found f
opt , one
can calculate the root-mean-square (RMS) deviation as
RMS ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
N vib
X N vib
p¼1
m scl
p f
opt
ð ÞÀm
expt
p
2
v
u
u
t
½cm
À1
ð 2:31Þ
where we used w p = 1, p = 1,2,…,N vib , for simplicity, average relative percentage
error (ARPE) as
ARPE ¼
100
N vib
X N vib
p¼1
m
scl
p f
opt
ð ÞÀm
expt
p
m
expt
p
½%;
ð2:32Þ
and/or some other quantities determining the accuracy of a fit. In the above formulas, N vib can be replaced by N
mol
vib to obtain the corresponding values for a
particular molecule.
The main term used in all scaling procedures is transferability of Sfs. Sfs
meaning seems to be intuitively obvious. However, two aspects of transferability of
SFs: (i) transferability “within the molecule” and (ii) transferability “among (related) molecules” should be considered [13]. The first of these items can be formulated as follows: The SFs are transferable within the molecule if, after
calculating them from rather localized modes (group vibrations), they are capable of
reproducing frequencies of delocalized modes, i.e., normal modes, involving a
number of local vibrations. This kind of transferability is particularly applicable in
the case of multi-parameter scaling procedures. The transferability among related
molecules naturally means that the factors computed for a training set of molecules
can be successfully applied to obtain accurate spectra of other molecules.
2.3.2 Uniform Scaling
2.3.2.1 Fundamentals of the Method
The simplest scaling procedure is single-parameter frequency scaling, frequently
called uniform scaling (US) . Its basic idea reads as follows: Scale harmonic frequencies of a molecule obtained from solution of Eqs. (2.21) or (2.26) by a single
2 Scaling Procedures in Vibrational Spectroscopy
65
importance of specific frequencies in the fit. It is recommended that for
well-established frequencies, w p is inversely proportional to the frequency itself
[12], although in many applications presetting all weights to unity is a standard
procedure. In the case of multi-parameter scaling procedures, some of the SFs may
be given preset values. This is easily accomplished using, e.g., Lagrange method of
undetermined multipliers in conjunction with Eq. (2.29). Having found f
opt , one
can calculate the root-mean-square (RMS) deviation as
RMS ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
N vib
X N vib
p¼1
m scl
p f
opt
ð ÞÀm
expt
p
2
v
u
u
t
½cm
À1
ð 2:31Þ
where we used w p = 1, p = 1,2,…,N vib , for simplicity, average relative percentage
error (ARPE) as
ARPE ¼
100
N vib
X N vib
p¼1
m
scl
p f
opt
ð ÞÀm
expt
p
m
expt
p
½%;
ð2:32Þ
and/or some other quantities determining the accuracy of a fit. In the above formulas, N vib can be replaced by N
mol
vib to obtain the corresponding values for a
particular molecule.
The main term used in all scaling procedures is transferability of Sfs. Sfs
meaning seems to be intuitively obvious. However, two aspects of transferability of
SFs: (i) transferability “within the molecule” and (ii) transferability “among (related) molecules” should be considered [13]. The first of these items can be formulated as follows: The SFs are transferable within the molecule if, after
calculating them from rather localized modes (group vibrations), they are capable of
reproducing frequencies of delocalized modes, i.e., normal modes, involving a
number of local vibrations. This kind of transferability is particularly applicable in
the case of multi-parameter scaling procedures. The transferability among related
molecules naturally means that the factors computed for a training set of molecules
can be successfully applied to obtain accurate spectra of other molecules.
2.3.2 Uniform Scaling
2.3.2.1 Fundamentals of the Method
The simplest scaling procedure is single-parameter frequency scaling, frequently
called uniform scaling (US) . Its basic idea reads as follows: Scale harmonic frequencies of a molecule obtained from solution of Eqs. (2.21) or (2.26) by a single
2 Scaling Procedures in Vibrational Spectroscopy
65