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Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su) 281
UDC 532.6
NUMERICAL ASPECTS OF THE CALCULATION OF SCALING FACTORS
FROM EXPERIMENTAL DATA
I. V. Kochikov 1 , G.M. Kuramshina 2 , A. V. Stepanova 2 , A. G. Yagola 3
New algorithms for finding molecular force field parameters expressed in terms of scaling factors
are developed. A new formulation of the inverse vibrational problem (the so­called inverse scaling
problem) is given and stable numerical methods are proposed. Examples include calculations of
scaling factors for the molecules of methylsilane and perfluoroethane within different models.
1. Introduction. Many problems of physical chemistry belong to the class of inverse problems. In these
problems, the properties of an object are determined from experimental data on the basis of a certain model
connecting these properties with characteristics measured. Inverse problems typically lead to mathematical
models that are not well­posed in the sense of Hadamard. This means that they may not have a solution in
the strict sense; when solutions exist, they may not be unique and/or may not depend continuously on input
data. Mathematical problems possessing such properties are called ill­posed problems, mostly due to instability
of solutions with respect to data perturbations. Numerical methods that can cope with this problem are the
so­called regularization methods. The theory of ill­posed problems (founded by A. N. Tikhonov and his scientific
school in 1960s; see, for example, [1]) investigates and develops efficient stable numerical methods for the solution
of ill­posed problems. An understanding of undetermined character of ill­posed problems and the concept of
regularizing operators (algorithms) form the basis for this theory.
Molecular force fields provide important information on the molecular structure and dynamics. Molecular
properties may be determined from experimental data of vibrational (infrared and Raman) spectroscopy as a
result of solving the so­called inverse vibrational problems. The rapid progress in quantum mechanical calcu­
lations of theoretical harmonic force fields provides new ways for more accurate interpretation of experimental
data and opens new opportunities in empirical force field calculations. The latter are particularly important for
the large­size molecules when accurate ab initio calculations are impossible, so that empirical methods based
on solving the inverse vibrational problems still remain the main source of data on the force field parameters.
The rapid progress in the investigation of rather large nanomolecules requires the development of special
approaches for solving inverse vibrational problems lying far beyond the traditional methods based on the least­
squares procedures. The analysis of large molecular systems (when a force constant matrix F is constructed from
previously evaluated force constants of model compounds) runs across difficulties of possible incompatibility
of the results obtained by different authors and by means of different numerical methods within different
approximations (force field models). These difficulties are caused by nonuniqueness and instability of the
solution to a inverse vibrational problem as well as by incompatibility of available experimental data with the
harmonic model.
In this paper we demonstrate how a priori model assumptions and ab initio quantum mechanical calcu­
lations can be used for constructing the regularizing algorithms for the molecular force field calculations. We
have proposed a principally new formulation of the problem of searching for the molecular force field parameters
using all available experimental data and quantum mechanical calculation results and taking into account the a
priori constraints for force constants. The essence of our approach is that we suggest (using given experimental
data and its accuracy) to find the stable approximations to the so­called normal pseudosolution (i.e., to find
a matrix F such that it is the nearest (in the chosen Euclidean norm) to a given force constant matrix F 0 ,
satisfies a set of a priori constraints D, and is compatible with experimental data \Lambda ffi with regard to the possible
incompatibility of the problem [2]).
Within this approach, the results tend to be as close to quantum mechanical data as an experiment allows.
From the mathematical point of view, the corresponding algorithm provides approximations to the solution
1 Research Computing Center, Moscow State University, Moscow 119992, Russian Federation; e­mail:
kochikov@tm­net.ru
2 Department of Physical Chemistry, Chemical Faculty, Moscow State University, Moscow 119992, Russian
Federation; e­mail: kuramshi@phys.chem.msu.ru
3 Department of Mathematics, Physical Faculty, Moscow State University, Moscow 119992, Russian Federa­
tion; e­mail: yagola@inverse.phys.msu.ru
c
fl Research Computing Center of Moscow State University

282 Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su)
that tend to the exact solution when experimental data become more extensive and accurate. Imposing certain
restrictions on the matrix of force constants (in our case, it is the requirement on the closeness of the solution
to the matrix F 0 )) allows one to select a unique solution from the variety of possible choices.
2. Mathematical formulation of inverse vibrational problems. The idea of the force field arises
from the treatment of a molecule as a mechanical system of nuclei where all the interactions due to electrons are
included into the effective potential function U (q 1 ; : : : ; q n ). Here by q 1 ; : : : ; q n we denote n = 3N \Gamma 6 generalized
coordinates of N­atomic nuclei of the molecule. The potential function minimum with respect to the nuclei
coordinates defines the equilibrium geometry of the molecule; the second derivatives of the potential at the
equilibrium
f ij = @ 2 U
@q i @q j
j eq (i; j = 1; : : : ; n)
constitute a positive definite matrix F determining all the molecular characteristics related to small vibrations.
The vibrational frequencies (obtained from the IR and Raman spectra) are the main type of experimental
information on molecular vibrations. They are connected with the matrix of force constants by the eigenvalue
equation
GFL = L\Lambda; (1)
where \Lambda is the diagonal matrix composed of the squares of the molecular normal vibration frequencies ! 1 ; : : : ; !n ,
\Lambda = diag
\Phi ! 2
1 ; : : : ; ! 2
n
\Psi , G is the kinetic energy matrix in the momentum representation, and L is the matrix of
normalized relative amplitudes.
If only the experimental frequencies of one molecular isotopomer are known, the inverse vibrational problem
of finding the force constant matrix F is reduced to an inverse eigenvalue problem; hence, when G is not singular,
the solution to equation (1) is represented by any matrix F such that
F = G \Gamma1=2 C \Lambda \Lambda C G \Gamma1=2 ; (2)
where C is an arbitrary orthogonal matrix (the asterisk indicates the transposed matrix).
Since equation (1) is the main source of data determining the force constants, it is evident that (except for
diatomic molecules) the n(n+1)=2 parameters of F cannot be found uniquely from the n frequencies ! 1 ; : : : ; !n .
This has led, on the one hand, to attempts of using certain model assumptions on the structure of the matrix F
and on the other hand, to introducing additional experimental data. Within the approximation considered, the
force field of a molecule does not depend on the nuclei masses; for the spectra of m isotopic species we have,
instead of equation (1), the system
(G i F ) L i = L i \Lambda i ; i = 1; 2; : : : ; m: (3)
Additional information may also be extracted from ro­vibrational spectra (Coriolis constants), gas electron
diffraction (mean­square amplitudes), and from other experimental evidence where the measured molecular
constants depend on the force constant matrix F . It is important to note, however, that even if the number
of available experimental measurements is equal to or exceeds the number of parameters to be determined, the
uniqueness of the solution is not guaranteed.
The mathematical relations between the vibrational properties of a molecule (equations (1), (3), etc.) and
their experimental representation can be summarized in the form of a single operator equation
AF = \Lambda ffi ; (4)
where F 2 Z = R n(n+1)=2 is an unknown force constant matrix (real and symmetrical), \Lambda 2 R m represents the
set of available experimental data (vibrational frequencies, etc.) determined within the ffi ­error level: k\Lambda \Gamma \Lambda ffi k 6
ffi , and A is a nonlinear operator that maps the matrix F on the set \Lambda.
This problem belongs to the class of nonlinear ill­posed problems, since it does not satisfy any of the three
well­posedness conditions according to Hadamard. In the general case (except for the diatomic molecules), it
may have a nonunique solution or no solutions at all (the incompatible problem); solutions may be unstable with
respect to errors in the operator A and in the set of experimental data \Lambda. Regularization methods for nonlinear
problems were developed in the last two decades [1] and were applied to the inverse problems of vibrational
spectroscopy [2].
The most important idea of the regularization theory is that experimental data are insufficient for the
unique and stable determination of some or all molecular parameters; hence, we should introduce some kind
of external knowledge or experience. In some cases, it is desirable to formulate these considerations as explicit

Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su) 283
additional restrictions on the set of possible solutions. When it is impossible, the more flexible approach is
to choose the solution nearest (in a certain sense) to some a priori defined parameter set. This set may not
necessarily conform to the experiment and should be based on data complementary to the experiment.
This external evidence may be derived from general ideas (for example, from molecular force field models or
data on similar molecular structures) or, preferably, be based on ab initio calculations. An inverse vibrational
problem is formulated as a problem of finding the so­called normal solution (or normal pseudo­(quasi­)solution
in the case of incompatibility of input data) to the nonlinear operator equation (4).
The sought­for solution is a matrix F ff 2 Z that reproduces experimental data within a given error level and
is the nearest (in the Euclidean metrics) to some given matrix F 0 . All necessary model assumptions (explicit
and implicit) concerning the form of the force field may be taken into account by the choice of some given
a priori matrix of force constants F 0 and a pre­assigned set D of a priori constraints on the values of force
constants. This set defines a form of the matrix F in the framework of the desired force field model (i.e., with
specified zero elements, equality of some force constants, etc.). If no a priori data constrain the form of a
solution, then D coincides with the set Z.
3. Constraints on the values of force constants based on ab initio calculations. The inverse
scaling problem. It was stated earlier [3, 4] that in Tikhonov's regularizing procedure it is possible to increase
the stability and accuracy of the calculated solution F ff by using
a) an extended set of experimental data (including, for example, the Coriolis constants, mean­square
amplitudes, frequencies of isotopomers or related compounds, etc.);
b) an improved choice of the stabilizer matrix F 0 ;
c) an improved choice of the constraint set D.
As a particularly effective choice of a stabilizer, we have proposed [5] to use the ab initio quantum mechanical
matrix F 0 in the regularizing procedure. This leads to the concept of the regularized quantum mechanical force
field (RQM FF) defined as the force constant matrix which is nearest to the corresponding quantum mechanical
matrix F 0 and reproduces experimental frequencies within a given error level.
A correct choice of the constraint set D is also extremely important. Physically stipulated limitations
may either decrease the range of acceptable matrices F or may provide criteria for selecting a unique solution
from the set of tolerable ones. An incorrect choice of constraints may lead to an increased incompatibility of an
inverse problem, eventually resulting in a pseudosolution with no physical meaning. A set of a priori constraints
may arise from several types of limitations on force constant values (see, for example, [3, 4]):
1. some force constants may be assumed a priori to be zero;
2. some force constants may be assumed to satisfy the inequalities a ij 6 f ij 6 b ij , where a ij and b ij are
certain known values;
3. some force constants may be assumed to be equal in a series of related molecules (or conformers).
During the last 30 years, numerous attempts were undertaken in the development of quantum mechanical
methods for evaluation of molecular structure and molecular dynamics of polyatomic molecules directly from
Schr¨odinger's equation. At present, direct calculations of vibrational spectra by quantum mechanical calcula­
tions at different levels of theory are very important and widely used to interpret experimental data, especially
in the case of large molecules. However, the use of very restrictive assumptions was (and still remains) necessary
to make such calculations computationally feasible.
Most ab initio methods are based on the relatively simple Hartree--Fock approach (HF). At more advanced
levels of theory, electron correlation neglected by the HF­approach is taken into account by means of the
perturbation theory (the Moller--Plesset method (MP)) or by using the density functional theory (for example,
the Becke method referred to as B3LYP). Account for electron correlation, though being computationally
expensive, is proved to be very important for obtaining the correct equilibrium structure of many compounds.
Another important feature of ab initio calculations is the size of the used basis set. The standard basis sets
are composed of Gaussian functions and differ by the number of such functions (among the most popular sets,
the basis set 6­31G* is relatively scarce, while 6­311++G** is more rich). Under certain circumstances, the
standard basis sets have to be complemented by special functions; such extended sets have special notations
(for example, AUG­cc­pVDZ, etc.). The basis extension is also computationally costly, so the usual practice
is to find a reasonable compromise between the size of the basis set and the level of the theory. Ab initio
calculations are usually denoted by the specifying level of a theory and a basis set, for example, the notation
B3LYP/6­311++G** means that the DFT method with one of the Gaussian basis sets is used.
Among different approaches, the Hartree--Fock level calculations are routine and available even for very
large systems consisting of up to hundreds of atoms. However, the quality of these calculations is insufficient
for direct comparison of theoretical and experimental vibrational frequencies. As a rule, the HF frequencies are

284 Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su)
overestimated (up to 15%; these errors have a systematic character for the related compounds. The wide use
of quantum mechanical calculations of vibrational spectra and harmonic force fields of polyatomic molecules
induced the necessity of making empirical corrections to theoretical data for compatibility with an experiment.
The most popular approach is the so­called scaling procedure proposed by P. Pulay [6], where the disagreement
between experimental and theoretical frequencies is eliminated by introducing a finite (not very large) number
of scaling factors.
This approach can also be formulated in the following form of a priori constraints: the final solution may
be assumed to conform to Pulay's scaled force constant matrix [6], which may also be considered as a kind of
constraint.
In this approach we can specify [3, 4] the set D as: D = F : F= BF 0 Bg, B= diagffi 1 ,. . . ,fi ng, where fi i
are the scaling parameters.
Mathematically, the scaling procedure imposes rather strict limitations on the molecular force field [3] and
often does not provide enough freedom to eliminate all discrepancies between the calculated and observed data.
However, it has certain advantages that follow from the comparatively small number of adjustable parameters
and, consequently, moderate computational resources required to perform force­field refinement. Indeed, it
is very attractive to find a limited number of scaling factors for a series of model molecules and (assuming
their transferability) to use them to correct the quantum mechanical force constants of more complicated
molecular systems. The most popular numerical procedure for calculation (optimization) of scaling factors is
the least­squares procedure, but there are a few publications indicating the nonconvergence and instability of
this numerical procedure while solving an inverse scaling problem. This is explained by the impossibility of
using the traditional numerical methods for solving nonlinear ill­posed problems [1, 2, 7, 8].
In our works the following strict mathematical formulation of the inverse scaling problem has been pro­
posed [9, 10]: the problem of finding scaling factors on the basis of experimental data is treated as an operator
equation similar to (4):
AF (fi) = \Lambda ffi ; (5)
where fi are scaling factors. Let the following norms in the Euclidean space be introduced:
jjF jj =
Ÿ n
X
ij
f 2
ij
– 1=2
; jj\Lambdajj =
Ÿ l
X
k=1
– 2
k ae k
– 1=2
;
where ae k ? 0 are certain positive weights, f ij are the elements of the matrix F ; – k (k = 1; : : : ; l) are the
components of \Lambda.
Since problem (5) is also ill­posed, we have to regularize it. We formulate the problem as a requirement
to find an approximation to such a solution F n;ffi to equation (5) that it is nearest (in Euclidean norm) to the
quantum mechanical matrix F 0 , satisfies experimental data within a given error level ffi (jjA(F (fi)) \Gamma \Lambda ffi jj 6
ffi ), and has a special form proposed by Pulay. If we consider this problem taking into account its possible
incompatibility, we come to the following formulation:
to find
F n;ffi = arg min
fl fl F \Gamma F 0
fl fl
where
F 2
n
F : F 2 D =
n
F : F = B 1/2 F 0 B 1/2
o
; kAF \Gamma \Lambda ffi k 6 ¯ + ffi
o
:
Here B is a diagonal matrix of scaling factors fi i and ¯ is a measure of incompatibility of the problem [2].
The latter may appear due to the possible anharmonicity of experimental frequencies or the crudeness of the
chosen model.
Finding the matrix F n;ffi may be performed by minimization of the Tikhonov functional
M ff (fi) = M ff [F ] = jjA h F \Gamma \Lambdajj 2 + ffjjF \Gamma F 0 jj 2
;
where F = F (fi) and the regularization parameter ff is chosen in accordance with the generalized discrepancy
principle [1].
As a rule, the assumed limitations on the values of force constants of polyatomic molecules cannot be
strictly proved. Nevertheless, numerical quantum mechanical results on molecular force fields can provide
useful guidance in choosing realistic force field models for different types of molecules.

Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su) 285
The least­squares procedure applied to solution of a inverse vibrational problem has been described in
numerous publications. Very often, as a criterion of minimization, the authors choose the ``best'' agreement
between experimental and fitted vibrational frequencies. However, there are situations when the ``best'' agree­
ment criterion is of many meanings. We should like to note that this criterion is insufficient due to the ill­posed
nature of the inverse vibrational problem. Even in the case of a single molecule, it is a well­known fact that
there exist an infinite number of solutions which exactly satisfy any given set of the experimental frequencies.
Multiple solutions also have been shown to exist when scaling procedure is applied. Addition of the expanded
experimental information on frequencies of the isotopomers or related molecules may lead to incompatibility of
the mathematical problem and may result in no solution at all within the conventional harmonic model. This
means that, using any minimization procedure for solving the inverse vibrational problem, it is necessary to
apply some additional criteria (that can be mathematically formulated) in the minimization procedure to select
the unique solution.
4. Generalized inverse structural problem. A similar regularized procedure was proposed for solving
the generalized inverse structural problem (GISP) [11] in the case of joint treatment of the experimental data
obtained by different physical methods (the vibrational spectroscopy and the electron diffraction (ED) data
and microwave (MW) spectroscopy). In general, solving the generalized inverse problem should lead to the
simultaneous determination of molecular geometry (R) and the force field (F ), each described by a finite set of
certain parameters.
Let \Lambda be a set of all available experimental data and A(R; F ) be a procedure allowing to calculate these
data from a set of molecular parameters R and F . Here \Lambda is a finite­dimensional vector from the normalized
space R m ; the parameters (R; F ) may also be chosen so as to constitute a vector from R n ; hence, A is an
operator acting from R n to R m . We can write down the following nonlinear operator equation similar to (4):
A(R; F ) = \Lambda ffi : (6)
This equation has some properties similar to those of equation (4): the solution of equation (6) may be not
unique and/or may reveal instability with respect to experimental measurement errors. To solve this problem
and obtain the solution, we again need to involve external assumptions based on some preliminary knowledge,
analogous to those formulated for the inverse vibrational problem and, preferably, based on ab initio data.
In the case of GISP, this kind of ``soft'' constraints may be combined with a more rigid set of constraints
imposed on the possible solutions in order to obtain the unique solution. For example, when there are several
close (by magnitude) interatomic distances in a molecule, it is a common practice to determine only one of them
from ED experiment, fixing all differences between the distances in ab initio magnitude. When it is impossible
to find a solution within this model, a less restricting approach could be applied to find a solution that will
have the same properties, unless this approach does not contradict to experiment. Otherwise, we will look for
the solution nearest to the element possessing these properties.
Now let us give a strict formulation of the problem. Let (R 0 ; F 0 ) be an a priori given set of parameters
(obtained, for example, from ab initio calculations), jj\Lambda \Gamma \Lambda ffi jj 6 ffi , and ffi be an experimental error level.
Introduce a set of constraints D in R n , which our solution should belong to. The generalized inverse structural
problem (GISP) may be formulated as follows:
to find an approximation (R; F ) ffi to the exact (normal) parameters (R; F ) such that
(R; F ) ffi = arg min
(R;F )2Z ffi
fl fl (R; F ) \Gamma (R 0 ; F 0 )
fl fl ;
where the solution is compatible with experimental data within the accuracy range, taking into account the
possible measure of incompatibility ¯
(R; F ) ffi 2 Z ffi where Z ffi = f(R; F ) 2 D : kA(R; F ) \Gamma \Lambdak 6 ¯ + ffi g ;
with an increasing accuracy, we get more accurate approximations to the exact (normal) (pseudo)solution:
(R; F ) ffi ! (R; F ) when ffi ! 0:
Similar to (1), one of the possible implementation of this procedure is obtaining such approximations on
the basis of Tikhonov functional technique when we minimize
M ff (R; F ) = kA(R; F ) \Gamma \Lambdak 2 + ff
fl fl (R; F ) \Gamma (R 0 ; F 0 )
fl fl 2

286 Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su)
on the set D; the regularization parameter ff is chosen as a solution of the equation kA(R; F ) ff \Gamma \Lambdak = ¯ + ffi ,
where (R; F ) ff delivers a minimum to M ff (R; F ). Obviously, when (R 0 ; F 0 ) is compatible with experimental
data in itself, no further adjustment is necessary.
It is often the case that even in the absence of experimental errors an inverse problem remains incompatible.
This may be caused by the roughness of a molecular model (for example, by ignoring some minor effects that
cannot be properly accounted for at the selected level of model complexity). Under these circumstances, the
approach described above should be reformulated in terms of (normal) pseudosolutions rather than in terms of
exact (normal) solutions.
The details of mathematical properties of the above­mentioned equations, such as existence and uniqueness
of the solution, depend on the properties of the operator A(R; F ). It is possible to ensure the desired properties
for a wide range of inverse problems, including all the important problems under consideration.
The formulation given above is very general; in the implementation that follows we shall assume that R is
a set of independent equilibrium geometry parameters and F is a set of harmonic force constants.
Now it is appropriate to give a brief summary of the features distinguishing the given approach from the
various previously used attempts to solve a similar inverse problem [12].
Our approach is aimed at the simultaneous determination of the geometry and force field parameters of
a molecule. It combines the techniques previously used in the IR spectroscopy and the ED data analysis. In
particular, it allows one to use more flexible force field models when fitting ED data, far beyond the usually
employed scaling of the ab initio force field.
Ab initio data (or any other external data) are automatically ``weighed'' so as to serve as an additional
source of information when data supplied by an experiment are proved to be insufficient. There is no need to
supply ab initio data with some kind of assumed errors, etc.
Molecular geometry is defined in terms of equilibrium distances, thus allowing compatibility with spec­
troscopic models and ab initio calculations. In addition, the self­consistency of a geometric configuration is
automatically maintained at all stages of the analysis.
The complexity of the molecular models used in the analysis strongly depends on the availability and quality
of experimental data. Since in most cases the vibrational spectra and ED patterns reveal the vibrational motion
in a molecule resulting from small deviations of the atoms from their equilibrium positions, the molecular models
are generally based on the assumption of small harmonic vibrations. In some cases of solving GISP within the
scaling approximation, it is necessary to include the cubic part of the force field [13]. Similarly, in order to
get a set of more reliable cubic force constants, it is undoubtedly beneficial to improve the ab initio values
empirically (for simplicity, for example, using the Pulay harmonic scale factors). It has been our experience
that two schemes of cubic constant scaling are generally feasible. Let the ab initio quadratic force constant f 0
ij
defined in natural internal coordinates be scaled as follows:
f ij (scaled) = f 0
ij fi 1/2
i fi 1/2
j ;
where fi i and fi j are the harmonic scale factors. Then, the cubic constant scaling mode can be formulated [14]
as
f ijk (scaled) = f 0
ijk fi 1/2
i fi 1/2
j fi 1/2
k
or, alternatively,
f ijk (scaled) = f 0
ijk fi 1/3
i fi 1/3
j fi 1/3
k ;
where f 0
ijk are the unscaled theoretical cubic constants. Both the scaling schemes reduce the vibrational problem
to the determination of a much smaller number of parameters. The examples of applying the last procedure to
different molecular systems including those with large amplitude motion are given in [15, 16].
The most important step in solving a inverse vibrational problem is formulating a priori constraints on
the solution, which are taken from quantum mechanical calculations. Plausible constraints for the force field
matrix followed recommendations discussed elsewhere [3,4].
5. Applications.
5.1. CH 3 SiH 3 . As an example of solving an inverse scaling problem within different scaling schemes, here
we consider the calculation of the scaled force field of the methylsilane molecule CH 3 SiH 3 at different theoretical
levels. The quantum mechanical DFT (B3LYP) calculation of optimized structures and harmonic force fields for
this molecule have been done using different standard basis sets (varying from 6­31G* to AUG­cc­pVDZ). The
corresponding ab initio calculations were also performed at the HF/6­31G* and HF/ AUG­cc­pVDZ levels of
the theory. Quantum mechanical calculations were carried out using the GAUSSIAN­94 program package [17].

Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su) 287
Optimized molecular structures of CH 3 SiH 3 obtained at all applied theoretical levels belong to the C 3v
symmetry point group. The theoretical and experimental parameters [18] are compared in Table 1. We intro­
duced a redundant system of 20 internal coordinates (all bond distances, valence bond angles, and C­C torsion).
Since the definition of the force field matrix in a redundant coordinate system is not unique, the convention was
introduced that the matrix off­diagonal norm be a minimum. This choice is reasonable in accordance with the
commonly used force field models [5].
Table 1. Theoretical and experimental structural parameters of methylsilane, CH3SiH3
(bond lengths in angstroms, valence angles in degrees)
Parameter HF
6­31G*
B3LYP
6­31G*
B3LYP
6­311++
G**
B3LYP
aug­cc­
pVDZ
experiment
[18]
R (C­Si)
R (C­H)
R (Si­H)
``Si­C­H
``C­Si­H
``H­C­H
``H­Si­H
OE H­C­Si­H
1.888
1.086
1.478
111.1
110.6
107.8
108.3
179.9
1.889
1.095
1.490
111.0
110.7
107.9
108.1
179.9
1.885
1.093
1.487
111.0
110.7
107.9
108.2
179.9
1.893
1.099
1.496
111.0
110.6
107.9
108.3
180.0
1.867
1.092
1.484
108.44
108.0
110.12
Using the B3LYP/6­31G* force constant matrix as F 0 and an expanded set of vibrational frequencies for
the CH 3 SiH 3 , CH 3 SiD 3 , CD 3 SiH 3 , CD 3 SiD 3 isotopomers [19--21], the inverse scaling problem ISP­I was solved
by means of the Spectrum software package [2, 16]. In the first model (referred to as ISP­I) we introduced eight
independent scaling factors (in accordance with the molecular symmetry). In the second model (ISP­II) only
the frequencies of the ``light'' isotopomer were used in the minimization procedure. During this minimization
procedure, the initial value of the regularization parameter was gradually reduced until the criterion of its choice
has been met.
The comparison of experimental and fitted frequencies for calculations of ISP­I and II is given in Table 2.
In this table we also present the results of solving the direct problem for three other isotopomers with the scaling
factors obtained in calculation II.
Table 2. Comparison of theoretical, experimental, and fitted frequencies when solving the inverse scaling problem
for CH3SiH3 (B3LYP/6­31G*) with frequencies of the four isotopomers. Experimental frequencies are taken
from [20, 21]. Columns I and II show discrepancies between experimental values
and those calculated in the variants ISP­I and ISP­II
Sym. Assignment CH3SiH3 CH3SiD3
exp. theor. I II exp. theor. I II
A1 š(CH)
š(SiH)
ffi(CH3)
ffi(SiH 3 )
š(C­Si)
2920
2179
1264
933
701
3052
2235
1332
950
687
13
­11
1
12
4
­9
­7
­6
6
0
2924
1561
1264
652
741
3052
1834
1332
802
674
9
­17
0
­4
4
­12
­14
­6
­7
0
A2 Ü 183 197 3 0 169 185 0 ­3
E š(CH)
š(SiH)
ffi(CH3)
ffi(SiH 3 )
ae(CH3)
ae(SiH3)
2976
2168
1403
951
861
528
3129
2239
1497
961
904
522
30
4
16
­7
15
­13
9
7
3
­3
10
­19
2980
1577
1412
670
821
421
3129
1741
1497
872
822
439
26
­7
6
6
4
­9
5
­5
­5
10
3
­15
The resulting scaling factors obtained in these two calculations as well as the factors obtained by solving
the ISP­II for different theoretical levels with variation of a number of scaling factors are shown in Table 3.
We have also analyzed how the values of scaling factors depend on a priori constraints (the set D). In
our case, the set D is determined by the number of the introduced parameters (scaling factors). In different
calculations this number was taken equal to 8, 4, and 1. The results of solving the inverse scaling problems for
four theoretical levels are also presented in Table 3. In this table, ae=ffi is a ratio of the minimum mean­square

288 Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su)
Table 2 (continued)
Sym. Assignment CD3SiH3 CD3SiD3
exp. theor. I II exp. theor. I II
A1 š(CH)
š(SiH)
ffi(CH3)
ffi(SiH 3 )
š(C­Si)
2129
2181
1004
930
643
2199
2044
1045
874
630
39
­75
­1
9
0
43
­90
­6
0
­4
2133
1562
1001
717
617
2199
1597
1044
720
606
­27
­18
­2
0
­5
­42
­16
­7
­4
­8
A2 Ü 154 155 ­1 ­3 134 140 ­2 ­4
š(CH)
š(SiH)
ffi(CH3)
ffi(SiH 3 )
ae(CH3)
ae(SiH3)
2231
2170
1044
950
773
457
2323
2148
1085
852
767
440
­4
1
­19
­8
­1
­19
­20
4
­28
­3
­9
­21
2234
1577
1036
693
668
389
2323
1623
1085
712
687
376
­8
­7
­12
­4
4
­19
­24
­5
­21
­4
3
­23
discrepancy value obtained in the minimization procedure to the experimental error (assumed to be \Sigma6 cm \Gamma1
for each frequency).
Table 3. Comparison of scaling factors (fi i ) for methysilane obtained for different number of optimized parameters
(ae=ffi is a ratio of resulting mean­square discrepancy to the error level). Only experimental frequencies of CH3SiH3
[19, 20] (except for the results of the first column for the B3LYP/6­31G* level) were used in calculations
Coordinate HF 6­31G* B3LYP 6­31G* B3LYP 6­
311++G**
B3LYP
AUG­cc­
pVDZ
expanded
set \Lambda
8 scaling factors
C­Si
C­H
Si­C­H
H­C­H
Si­H
C­Si­H
H­Si­H
Ü
0.9405
0.8316
0.7639
0.7581
0.8426
0.8366
0.8169
0.8880
1.0545
0.9232
0.9002
0.8977
0.9414
1.0165
0.9627
0.8925
1.0444
0.9100
0.9037
0.8808
0.9443
0.9806
0.9731
0.8654
1.0376
0.9244
0.9429
0.9200
0.9555
0.9782
0.9729
0.9117
1.0633
0.9185
0.9782
0.9658
0.9820
1.0363
0.9981
0.9271
ae=ffi 1.10 2.23 1.10 0.90 1.17
4 scaling factors
C­Si
C­H
Si­H
Ü
0.9329
0.8166
0.8380
0.8860
1.0443
0.9059
0.9493
0.8650
1.0386
0.9249
0.9591
0.9102
1.0699
0.9273
0.9877
0.9255
ae=ffi 3.49 1.66 1.09 2.41
One scaling factor
All cords. 0.8253 0.9217 0.9377 0.9483
ae /ffi 4.06 3.38 2.57 3.49
These results show that the quality of the ISP solutions depends very strongly on the set D. The strong
narrowing of the set of minimized parameters leads to the poor reproduction of experimental data. The ill­posed
character of the inverse vibrational problem, as a rule, cannot be avoided by increasing the set of experimental
data because this could simultaneously lead to the incompatibility of input data.
5.2. C 2 F 6 . The molecule of perfluoroethane C 2 F 6 belongs to the D 3d symmetry group. Quantum­
mechanical calculations have been performed at the levels HF/6­31G*, MP2/6­31G*, and B3LYP/6­31G*.
The optimized molecular geometries for these three cases are compared in Table 4. Here again a redundant
system of 20 internal coordinates was introduced.

Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su) 289
Table 4. Theoretical and experimental structural parameters of perfluoroethane, C2F6
(bond lengths in angstroms, valence angles in degrees)
Parameter HF
6­31G*
B3LYP
6­31G*
MP2
6­31G*
Experiment[22]
R (C­C)
R (C­F)
``C­C­F
``F­C­F
1.526
1.311
109. 8
109.1
1.545
1.338
109.9
109.1
1.530
1.340
109.7
109.2
1.545
1.326
109.8
Due to symmetry considerations, five independent scale factors have been introduced for twenty internal
coordinates. In table 5 we present the vibrational frequencies obtained in three different quantum­mechanical
calculations. Initial values as well as those obtained after scaling are included.
Table 5. Experimental, theoretical and scaled values of vibrational frequencies for different
levels of ab initio calculations. Error corresponds to the mean­square error (in cm \Gamma1 )
Sym. Assign. Exper. [23] HF/6­31G* MP2/6­31G* B3LYP/6­31G*
Initial Scaled Initial Scaled Initial Scaled
A1g š(CC)
š(CF)
ffi(CF3 )
1417
807
348
1627
887
377
1426
781
335
1501
813
354
1428
783
340
1426
803
344
1421
797
345
A1u Ü 68 70 68 65 68 63 67
A2u š(CF)
ffi(CF3 )
1116
714
1242
773
1110
702
1150
706
1124
697
1130
700
1127
703
Eg š(CF)
ffi(CF3 )
ae(CF3)
1250
619
372
1426
673
415
1268
614
383
1292
621
378
1260
615
378
1261
611
371
1256
615
376
Eu š(CF)
ffi(CF3 )
ae(CF3)
1251
522
219
1433
564
230
1263
523
210
1305
516
213
1265
517
212
1269
512
206
1257
518
210
Error 116.1 12.3 34.0 11.3 10.5 6.9
Note that the B3LYP calculation has resulted in close correspondence of top experimental frequencies prior
to any scaling, while the HF level of the theory gives systematically overestimated values. The scale factors for
all the three variants of calculation are given in Table 6.
Table 6. Scale factors for perfluoroethane
HF/6­31G* MP2/6­31G* B3LYP/6­31G*
R(C­C)
R(C­F)
``CCF
``FCF
Ü
0.7130
0.7675
0.8375
0.8706
0.9459
0.8330
0.9327
0.9850
1.0135
1.0802
0.9793
0.9766
1.0309
1.0266
1.1594
6. CONCLUSIONS. The above discussion allows us to come to the following conclusions.
1. The strict mathematical formulation of the inverse scaling problem could provide a possibility of compar­
ison of results obtained by different investigators. Obviously, it is necessary to work within the same physical
and mathematical models and use the stable numerical methods for the unification of different calculations.
Otherwise, the situation may be the same as in the empirical force field calculations when for the same molecule
it is possible to find a lot of very different sets of force constants.
2. It is important that using the scaling scheme, as a rule, it is impossible to obtain a solution that
reproduces experimental data within a given error level. We may search only for the pseudosolution of Problem
I that satisfies the input frequencies (or the expanded set of experimental data) in the least­squares sense.
7. Acknowledgements. This work was partially supported by the RFBR­YUGRA (grant 03--07--96842)
and by the RFBR (grant 02--01--00044).

290 Numerical Methods and Programming, 2004, Vol. 5 (http://num­meth.srcc.msu.su)
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Received 08 October 2004