# Computation of Resonance Magnetic Fields of CW-EPR Spectra by Reversion of Power Series

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## Abstract

The linear relationship between the frequency of an EPR transition and the magnetic field, valid in the presence of only the Zeeman interaction, generally becomes nonlinear, when other interactions become operative. In such cases, obtaining accurate values of the resonance magnetic fields of a given spin system for simulating their EPR spectra, recorded at a fixed frequency, is not a trivial exercise. Because of its fundamental importance in the analysis of EPR spectra, there are several methods available in the literature to address this issue. These methods either use numerical techniques to compute the resonance fields from the resonance energies computed at various magnetic fields by diagonalization of the Hamiltonian matrix, or modify the Hamiltonian appropriately and resort to perturbation calculations. In this work, we have examined a method based on a mathematical technique of reversion of a power series, by which the resonance magnetic fields at a fixed frequency can be achieved in a relatively simple and straightforward manner. We have shown that, when the energy of an EPR transition can be expressed as a power series in powers of the magnetic field, obtained either from the analytical energy expression or by fitting the calculated energies to an empirical power series, a reversed power series in powers of the transition energy can be obtained to represent the resonance magnetic field. We have derived the necessary algebraic relationships between the coefficients of these two series. We have shown the success and usefulness of this method by applying it to calculate the resonance magnetic fields of well-studied EPR spectra of hydrogen atom, naphthalene triplet, and ^{6}*S* state of Fe^{3+} in an octahedral crystalline electric field, at different frequencies.

## Notes

## References

- 1.C.R. Byfleet, D.P. Chong, J.A. Hebden, C.A. McDowell, J. Magn. Reson.
**2**, 69 (1970)ADSGoogle Scholar - 2.G.G. Belford, R.L. Belford, J.F. Burkhalter, J. Magn. Reson.
**11**, 251 (1973)ADSGoogle Scholar - 3.K.T. McGregor, R.P. Scaringe, W.E. Hatfield, Mol. Phys.
**30**, 1925 (1975)ADSCrossRefGoogle Scholar - 4.M.I. Scullane, L.K. White, N.D. Chasteen, J. Magn. Reson.
**47**, 383 (1982)ADSGoogle Scholar - 5.A.S. Yang, B.J. Gaffney, Biophys. J.
**51**, 55 (1987)CrossRefGoogle Scholar - 6.G.C.M. Gribnau, J.L.C. van Tits, E.J. Reijerse, J. Magn. Reson.
**90**, 474 (1990)ADSGoogle Scholar - 7.D. Nettar, J.J. Villafranca, J. Magn. Reson.
**64**, 61 (1985)ADSGoogle Scholar - 8.B.J. Gaffney, H.J. Silverstone, J. Magn. Reson.
**134**, 57 (1998)ADSCrossRefGoogle Scholar - 9.D. Collison, F.E. Mabbs, J. Chem. Soc. Dalton Trans. 1565 (1982)Google Scholar
- 10.S. Stoll, A. Schweiger, Chem. Phys. Lett.
**380**, 464 (2003)ADSCrossRefGoogle Scholar - 11.S. Stoll, Computational modeling and least-squares fitting of EPR spectra, in
*Multifrequency electron paramagnetic resonance: data and techniques*, ed. by S.K. Misra (Wiley-VCH, Weinheim, 2014), pp. 69–138CrossRefGoogle Scholar - 12.J.O. Hirschfelder, W.B. Brown, S.T. Epstein, Recent developments in perturbation theory, in
*Advances in quantum chemistry*, vol. 1, ed. by P.-O. Löwdin (Academic Press, New York, 1964), pp. 255–374Google Scholar - 13.I. Mayer,
*Simple theorems, proofs, and derivations in quantum chemistry*(Kluwer Academic/Plenum Publishers, New York, 2003), pp. 323–324CrossRefGoogle Scholar - 14.A. Rockenbauer, P. Simon, J. Magn. Reson.
**11**, 217 (1973)ADSGoogle Scholar - 15.M. Iwasaki, J. Magn. Reson.
**16**, 417 (1974)ADSGoogle Scholar - 16.Y. Teki, T. Takui, K. Itoh, J. Chem. Phys.
**88**, 6134 (1988)ADSCrossRefGoogle Scholar - 17.Y. Teki, I. Fujita, T. Takui, T. Kinoshita, K. Itoh, J. Am. Chem. Soc.
**116**, 11499 (1994)CrossRefGoogle Scholar - 18.F.E. Mabbs, D. Collision,
*Electron paramagnetic resonance of d transition metal compounds*(Elsevier, Amsterdam, 1992), p. 134Google Scholar - 19.T. Yamane, K. Sugisaki, T. Nakagawa, H. Matsuoka, T. Nishio, S. Kinjyo, N. Mori, S. Yokoyama, C. Kawashima, N. Yokokura, K. Sato, Y. Kanzaki, D. Shiomi, K. Toyota, D.H. Dolphin, W.-C. Lin, C.A. McDowell, M. Tadokoroc, T. Takui, Phys. Chem. Chem. Phys.
**19**, 24769 (2017)CrossRefGoogle Scholar - 20.T. Yamane, K. Sugisaki, H. Matsuoka, K. Sato, K. Toyota, D. Shiomia, T. Takui, Dalton Trans.
**47**, 16429 (2018)CrossRefGoogle Scholar - 21.M. Abramowitz, I.A. Stegun (eds.),
*Handbook of mathematical functions with formulas, graphs and mathematical tables, 10th printing*(National Bureau of Standards, Washington, DC, 1972), p. 16.**(entry 3.6.25)**zbMATHGoogle Scholar - 22.S.M. Selby (ed.),
*CRC standard mathematical tables*, 18th edn. (The Chemical Rubber Company, Cleveland, 1970), p. 454Google Scholar - 23.G.B. Arfken, H.J. Weber,
*Mathematical methods for physicists*, 4th edn. (Prism Books, Bangalore, 1995), p. 326zbMATHGoogle Scholar - 24.H. Chernoff, Math. Comp.
**2**, 331 (1947)CrossRefGoogle Scholar - 25.J.A. Weil, J.R. Bolton,
*Electron paramagnetic resonance: elementary theory and practical applications*, 2nd edn. (Wiley-Interscience, New Jersey, 2007)Google Scholar - 26.C.A. Hutchison Jr., B.W. Mangum, J. Chem. Phys.
**34**, 908 (1961)ADSCrossRefGoogle Scholar - 27.M.S. de Groot, J.H. van der Waals, Mol. Phys.
**6**, 545 (1963)ADSCrossRefGoogle Scholar - 28.P. Debye, Ann. Phys.
**32**, 85 (1938)CrossRefGoogle Scholar - 29.R. de L. Kronig, C.J. Bouwkamp, Physica
**6**, 290 (1939)ADSCrossRefGoogle Scholar - 30.J.E. Wertz, J.R. Bolton,
*Electron spin resonance: elementary theory and practical applications*(McGraw-Hill, New York, 1972)Google Scholar - 31.W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery,
*Numerical recipes in C: the art of scientific computing*, 2nd edn. (Cambridge University Press, Cambridge, 1992), pp. 671–675zbMATHGoogle Scholar - 32.M.F. González-Cardel, R. Díaz-Uribe, Rev. Mex. Fís. E
**52**, 163 (2006)MathSciNetGoogle Scholar