Cite this article as:

Berezin K. V., Novoselova A. V., Chernavina M. L., Berezin V. I., Novoselov V. V. Calculation of Scale Factors for Quantum-mechanical Force Fields. Izvestiya of Saratov University. New series. Series Physics, 2015, vol. 15, iss. 4, pp. 41-44. DOI:

УДК 539.182/.184, 519.677

Calculation of Scale Factors for Quantum-mechanical Force Fields


A computer-assisted method for the calculation of scaling factors for refining the quantum-mechanical force fields of polyatomic molecules by the Pulay technique is suggested. The method is an iteration procedure and does not involve the calculation of derivatives of the frequencies of vibrations with respect to the scaling factors.


1. Кон В. Электронная структура вещества ‒ волновые функции и функционалы плотности // УФН. 2002. Т. 172, № 3. С. 336‒348.

2. Frisch M. J., Trucks G. W., Schlegel H. B. et al. Gaussian 03. Gaussian Inc., Pittsburgh PA, 2003.

3. Yoshida H., Ehara A., Matsuura H. Density functional vibrational analysis using wave number-linear scale factors // Chem. Phys. Lett. 2000. Vol. 325, № 4. P. 477–483.

4. Pulay P., Fogarasi G., Pongor G., Boggs J. E., Vargha A. Combination of theoretical ab initio and experimental information to obtain reliable harmonic force constants. Scaled quantum mechanical (SQM) force fi elds for glyoxal, acrolein, butadiene, formaldehyde and ethylene // J. Amer. Chem. Soc. 1983. Vol. 105. P. 7037‒7047.

5. Baker J., Jarzecki A. A., Pulay P. Direct Scaling of Primitive Valence Force Constants: An Alternative Approach to Scaled Quantum Mechanical Force Fields // J. Phys. Chem. А. 1998. Vol. 102, № 8. P. 1412–1424.

6. Pulay P., Fogarasi G., Zhou X., Taylor P.W. Ab initio prediction of vibrational spectra : a database approach // Vibr. Spectr. 1990. Vol. 1, № 2. P. 159–165.

7. Jarzecki A. A., Kozlowski P. M., Pulay P., Ye B. -H., Li X. -Y. Scaled quantum mechanical and experimental vibrational spectra of magnesium and zinc porphyrins // Spectrochim. Acta. 1997. Vol. 53, № 8. P. 1195–1209.

8. Kozlowski P. M., Jarzecki A. A., Pulay P. Vibrational Assignment and Definite Harmonic Force Field for Porphine. 1. Scale Quantum Mechanical Results and Comparison with Empirical Force Field // J. Phys. Chem. 1996. Vol. 100, № 17. P. 7007–7013.

9. Kozlowski P. M., Zgierski M. Z., Pulay P. An accurate in-plane force fi elds for porphine. A scaled quantum mechanical study // Chem. Phys. Lett. 1995. Vol. 247, № 4‒6. P. 379–385.

10. Kozlowski P. M., Rush T. S.III., Jarzecki A. A., Zgierski M. Z., Chase B., Piffat C., Ye B. -H., Li X. -Y., Pulay P., Spiro T. G. DFT-SQM force fi eld for nickel porphine : Intrinsic ruffl ing // J. Phys. Chem. A. 1999. Vol. 103, № 10. P. 1357–1366.

11. Краснощеков С. В., Абраменков А. В., Панченко Ю. Н. Определение масштабирующих множителей молекулярных силовых полей методом наименьших квадратов с использованием псевдообратной матрицы // Вестн. Моск. ун-та. Сер. 2. Химия. 1985. Т. 26, № 1. С. 29‒33.

12. Березин К. В. Матричный метод нахождения масштабирующих множителей для квантово-механических силовых полей // Оптика и спектр. 2003. Т. 94, № 3. С. 309‒314.

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