Quantum Symmetry of the Vibrational States and Electronic π-Orbitals in a Benzene Molecule C6H6. The Fine Structure of Spin-Dependent Splitting

Authors

  • V.O. Gubanov Taras Shevchenko National University of Kyiv
  • A.P. Naumenko Taras Shevchenko National University of Kyiv
  • D.V. Gryn Taras Shevchenko National University of Kyiv
  • L.A. Bulavin Taras Shevchenko National University of Kyiv

DOI:

https://doi.org/10.15407/ujpe66.1.28

Keywords:

benzene, vector and spinor representations of symmetry groups, normal-vibration patterns, classes of symmetry-group projective representations, electronic states, spin-dependent splitting

Abstract

Analytical expressions and vector images have been constructed for all patterns of the normal vibrations, including doubly degenerate ones, of a benzene molecule C6H6 using the projection operator on the matrix elements of irreducible representations of the point symmetry group 6/mmm (D6ℎ). The characters of representations corresponding to the symmetry of both the electronic п-orbitals in a benzene molecule (without taking the electron spin into account) and the projective representations of its spinor п′-orbitals are found. The representations of the spinor п′-orbitals of a benzene molecule C6H6 belong to the projective class K1 and describe the fine structure of spin-dependent splitting of the degenerate spinless п-orbitals, which are revealed for the first time.

References

Quantum Chemistry and Spectroscopy. Edited by T. Engel, P. Reid (Prentice Hall, 2006).

E.B. Wilson, Jr. The normal modes and frequencies of vibration of the regular plane hexagon model of the benzene molecule. Phys. Rev. 45, 706 (1934). https://doi.org/10.1103/PhysRev.45.706

F. Hamdache, G. Vergoten, P. Lagant, A. Benosman. Normal modes calculation for benzene in a local symmetry force field. J. Raman Spectrosc. 20, 297 (1989). https://doi.org/10.1002/jrs.1250200505

M. Preuss, F. Bechstedt. Vibrational spectra of ammonia benzene, and benzene adsorbed on Si (001) by first principles calculations with periodic boundary conditions. Phys. Rev. B 73, 155413 (2006). https://doi.org/10.1103/PhysRevB.73.155413

A.M. Gardner, T.G. Wright. Consistent assignment of the vibrations of monosubstituted benzenes. J. Chem. Phys. 135, 114305 (2011). https://doi.org/10.1063/1.3638266

H. Poulet, J.P. Mathieu. Vibration Spectra and Symmetry of Crystals (Gordon and Breach, 1976).

V.O. Gubanov, A.P. Naumenko, M.M. Bilyi, I.S. Dotsenko, M.M. Sabov, M.S. Iakhnenko, L.A. Bulavin. Energy spectra of electron excitations in graphite and graphene and their dispersion making allowance for the electron spin and the time-reversal symmetry. Ukr. J. Phys. 65, 342 (2020). https://doi.org/10.15407/ujpe65.4.342

V.O. Gubanov, A.P. Naumenko, M.M. Bilyi, I.S. Dotsenko, O.M. Navozenko, M.M. Sabov, L.A. Bulavin. Energy spectra correlation of vibrational and electronic excitations and their dispersion in graphite and graphene. Ukr. J. Phys. 63, 431 (2018). https://doi.org/10.15407/ujpe63.5.431

B.R. Sohnlein, S. Li, D.-S. Yang. Electron-spin multiplicities and molecular structures of neutral and ionic scandiumbenzene complexes. J. Chem. Phys. 123, 214306 (2005). https://doi.org/10.1063/1.2131867

M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G.A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H.P. Hratchian, A.F. Izmaylov et al. Gaussian 09, Revision D.01 (Gaussian, Inc., 2013).

M.I. Katsnelson. Graphene: Carbon in Two Dimensions (Cambridge Univ. Press, 2012). https://doi.org/10.1017/CBO9781139031080

V.V. Strelchuk, A.S. Nikolenko, V.O. Gubanov, M.M. Biliy, L.A. Bulavin. Dispersion of electron-phonon resonances in one-layer graphene and its demonstration in micro-Raman scattering. J. Nanosci. Nanotechnol. 12, 8671 (2012). https://doi.org/10.1166/jnn.2012.6815

Published

2021-01-29

How to Cite

Gubanov, V., Naumenko, A., Gryn, D., & Bulavin, L. (2021). Quantum Symmetry of the Vibrational States and Electronic π-Orbitals in a Benzene Molecule C6H6. The Fine Structure of Spin-Dependent Splitting. Ukrainian Journal of Physics, 66(1), 28. https://doi.org/10.15407/ujpe66.1.28

Issue

Section

Optics, atoms and molecules

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