Department of Chemistry, Moscow State University,

Moscow 119899, Russian Federation, C. I. S.

Quantum mechanical harmonic force fields are widely used at present for the calculation of frequencies and modes of normal vibrations. However, they require a certain empirical correction. This is due to the more or less systematic overestimation of the force constants in the Hartree-Fock method, which depends on the basis set employed (see, for example, Ref. 1), and to not so regular discrepancies in the Møller-Plesset pertubation theory [2]. The necessity of the empirical correction is also quite obvious when the approximate theory of the density functional [3] is used in the calculation vibrational spectroscopy.

In order to retain the characteristic features of quantum mechanical force fields, for example, a particular distribution of the signs of the off-diagonal elements and an approximate ratio between the force constants, a smaller number of adjusting parameters is used in the empirical correction than the number of parameters in the traditional solution of the inverse vibrational problem in which the role of the latter was played by the varying force constants themselves. Therefore, several procedures for scaling quantum mechanical force fields have been suggested.

Comparison analysis of different methods [4-9] of empirical scaling of quantum mechanical molecular force fields has been performed. The possibility of using each concrete scaling technique was shown to depend on the theoretical level of the quantum mechanical calculation [10]. Pulay's scaling method [8] (congruent transfomation of the force constant matrix, i.e.

**F**s** = D**^{1/2} **F**qm **D**^{1/2}** **
(1)

where **F**qm is the matrix of force constants, calculated, for example, in
the Hartree-Fock approximation, and **D** is the diagonal matrix of the
scaling factors) was found to be applicable when the relative accuracy of
determination of both the diagonal and the off-
diagonal
quantum mechanical force constants is approximately equal. This constraint is
fulfilled for the quantum mechanical force field as determined quite close to
the Hartee-Fock limit [10, 11]. This makes it possible to carry out its
correction with maximal retention of peculiarities inherent in the molecule
under investigation [12-14].

Solution of the inverse vibrational problem on the basis of a quantum mechanical force field may be considered as a limiting case of scaling procedure where the maximal number of scale factors was used. However, this approach does not maintain the distribution of signs and the approximate ratio of the absolute values of the quantum mechanical force constants.

This is true, in particular, for a "regularized force field" [15], since search
for the solution, nearest to **F** in the euclidean norm, ensures [16]
retention of the signs of the force constants and of the approximate ratio
between their absolute values only in the ideal case, i.e., in the case where
the desired matrix is close to **F**qm. However, in practical calculations,
the norm of the difference between these matrices is large, because the
calculated frequencies deviate from the experimental values by 10-20% on the
average. Therefore, substantial changes in the force matrix and, consequently,
in the vibrational modes, may arise during the iteration procedure (see, for
example, Ref. 15). Then the formal solution may be inconsistent with the
physical requirements to the assignment of the vibration frequencies that
follow from quantum mechanical calculation. The introduction of the
corresponding limitations on the assignment into the functional (2) being
minimized (see Ref. 15) is rather difficult, which dramatically restricts the
possibilities of the regularization method.

Thus, in the Pulay method [8], the attention is concentrated on the manner in
which the correction of the quantum mechanical force field **F**qm should be
carried out with account for the physical requirements (retention of the modes
of normal vibrations, i.e., the distribution of signs and the ratio of the
force constants), whereas in the regularization method [15], the attention is
focused on how much **F**qm should be minimally modified from the viewpoint
of the matrix norm, in order to achieve the best fit with the experiment. In
the former case, qualitative agreement with **F**qm is attained (the
vibrational modes are retained), while in the latter case, the formal
quantitative agreement, corresponding to the least deviation of the resulting
matrix **F**r from the initial matrix **F**qm in the matrix norm, is
obtained.

Fig. 1

This situation can be illustrated by Fig. 1 in which curve 1 conventionally
describes the initial quantum mechanical force field **F**qm, and curves 2
and 3 describe the final force fields, i.e., the scaled field **F**s (see
Eq. (1) and Ref. 8) and the "regularized" field **F**r [15], respectively.
The points denote the experimental parameters (for example, the vibrational
frequencies) that have been used for the correction of the force field
**F**qm. The correction of curve 1 is conventionally reduced to its
displacement and a variation of its form. When the scaling is carried out
according to Pulay [8] using only one factor (homogeneous scaling), curve 1 is
transformed unchanged into curve 2. In some cases this scaling makes it
possible to assign even experimental vibrational frequencies of rotation
isomers (see, for example, Ref. 17). If several similar scaling factors are
used, curve 2 corresponds to a displaced and somewhat distorted curve 1, and
the deviation of the calculated vibrational frequencies from the experimental
values does not exceed the generally admissible limit. An example is provided
by the correction of the quantum mechanical force fields of trans-1,3-butadiene
[1], in which the scattering of the scaling factors for the force field in the
MP2/6-
31G*//
MP2/6-
31G*
approximation was 0.07 (the minimum value was 0.88 and the maximum value was
0.95). It is clear that with the scaling factors being so similar, the ratios
between the elements of a force matrix virtually do not change, and the
vibrational modes for the frequencies calculated from **F**s (see Eq. (1)),
are retained almost completely.

In the case where the inverse vibrational problem is solved by the
least-squares method or by the regularization method [15], when the disturbance
of the distribution of signs of the off-diagonal elements and of the ratio
between the diagonal and off-diagonal elements of the final force matrix
**F**r with respect to the initial matrix **F**qm in the practical
calculations is admissible, curve 1 is transformed into a type 3 curve. This is
caused by the fact that in the regularization method, the minimum condition

(2)

is used (**A** is a nonlinear operator that assigns a set of
computable **Λ** values to each matrix of force constants **F**r
and α is an additional varying parameter, which is selected by a special
procedure), with the calculated vibration frequencies being 10-20 % higher than
the experimental values. Indeed, this leads to an inevitable considerable
decrease in the diagonal force constants and to the compensation of this
decrease by a variation of the absolute values and the signs of the
off-diagonal elements of the force matrix. The resulting agreement of the
calculated vibrational frequencies with their experimental analogs, due to the
large number of fitting parameters, must be better than that in the case of
curve 2. However, the significant properties of the *ab initio* force
matrix that correspond to curve 1 are lost. The latter indicates that the
setting of the problem in the regularization method is obviously incorrect from
the viewpoint of the molecular vibration physics.

2. C. Møller and M. S. Plesset, *Phys. Rev.*, **46** (1934)
618.

3. T. Ziegler, *Chem. Rev.*, 1991, **91**, 651.

4. P. Pulay, W. Meyer, *J. Mol. Spectrosc.*, **40** (1971) 59.

5. W. Bleicher, P. Botschwina, *Mol. Phys.*, **30** (1975) 1029.

6. C. E. Blom, C. Altona, *Mol. Phys.*, **31** (1976) 1377.

7. K. W. Hipps, R. D. Poshusta, *J. Phys. Chem.*, **86** (1982)
4112.

8. G. Fogarasi, P. Pulay, *Annu. Rev. Phys. Chem.*, **35** (1984)
191.

9. J. Florián, B. G. Johnson, *J. Phys. Chem.*, **98** (1994)
3681.

10. Yu. N. Panchenko, *Rus. Chem. Bull*., **45** (1996) *in
press*.

11. V. I. Pupyshev, Yu. N. Panchenko, Ch. W. Bock, G. Pongor, *J. Chem.
Phys.*, **94** (1991) 1247.

12. Yu. N. Panchenko, N. F. Stepanov, *Rus. J. Phys. Chem.*, **69**
(1995) 535.

13. Yu. N. Panchenko, G. R. De Maré, N. F. Stepanov, *J. Mol.
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14. Yu. N. Panchenko, G. R. De Maré, V. I. Pupyshev, *J. Phys.
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15. G. M. Kuramshina, F. Weinhold, I. V. Kochikov, A. G. Yagola, Yu. A. Pentin,
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16. T. Kato, *Perturbation Theory for Linear Operators*, Springer Verlag,
Berlin, Heidelberg, New York, 1966.

17. Ch. W. Bock, Yu. N. Panchenko, and S. V. Krasnoshchiokov,
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