Vibrational IR Active Frequencies of C36: an Algebraic Approach

  • M. D. Choudhury Department of Physics, Assam University
  • R. Sen Department of Physics, Assam University
  • B. I. Sharma Department of Physics, Assam University
Keywords: Lie algebra, Hamiltonian, C36, dynamic symmetry


The one-dimensional U(2) Lie algebra is employed to calculate the structural and vibrational properties of C36. The lowest energy configuration of the C36 cage is confirmed to have D6ℎ symmetry. The Lie algebraic method is based on the idea of dynamic symmetry, which can be expressed in terms of U(2) Lie algebra. By applying the algebraic techniques, a local Hamiltonian, which conveniently describes the rovibrational degrees of freedom of the physical system, can be obtained. In this technique, the Hamiltonian is constructed, by considering the invariant Casimir and Majorana operators replacing every bond of the molecule by a corresponding Lie algebra. At the same time, the fundamental stretching vibrational energy levels of the molecule C36 are calculated. Finally, the calculated results are compared with other theoretical findings.


T.A. Beu, J. Onoe, K. Takeuchi. Structural and vibrational properties of C36 and its oligomers (C36) =2,3,4 by tight-binding molecular dynamics. Eur. Phys. J. D 17, 205 (2001).

R.A. Jishi, M.S. Dresselhaus. Vibrational frequencies in C36. Chem. Phys. Lett. 302, 533 (1999).

S.J. Cyvin, E. Brendsdal, B.N. Cyvin, J. Brunvoll. Molecular vibrations of footballene. Chem. Phys. Lett. 143, 377 (1988);

R.A. Jishi, R.M. Mirie, M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund. Force-constant model for the vibrational modes in C70. Phys. Rev. B 48, 5634 (1993);

Z.C. Wu, D.A. Jelski, T.F. George. Vibrational motions of buckminsterfullerene. Chem. Phys. Lett. 137, 291 (1987).

R.E. Stanton, M.D. Newton. Normal vibrational modes of buckminsterfullerene. J. Phys. Chem. 92, 2141 (1988).

F. Negri, G. Orlandi, F. Zerbetto. Quantum-chemical investigation of Franck–Condon and Jahn–Teller activity in the electronic spectra of Buckminsterfullerene. Chem. Phys. Lett. 144, 31 (1988).

Z. Slanina, J.M. Rudzinski, M. Togasi, E. Osawa. Quantum-chemically supported vibrational analysis of giant molecules: the C60 and C70 clusters. J. Mol. Struc. (Theochem.) 202, 169 (1989).

G.B. Adams, J.B. Page, O.F. Sankey, K. Sinha, J. Menendez. First-principles quantum molecular-dynamics study of the vibrations of icosahedral C60. Phys. Rev. B 44, 4052 (1991);

C.H. Choi, M. Kertesz, L. Mihaly. Vibrational assignment of all 46 fundamentals of C60 and C6−60 : Scaled quantum mechanical results performed in redundant internal coordinates and compared to experiments. J. Phys. Chem. A 104, 102 (2000);

V. Schettino, M. Pagliai, L. Ciabini, G. Cardini. The vibrational spectrum of fullerene C60. J. Phys. Chem. A 105, 11192 (2001).

F. Iachello, S. Oss. Algebraic methods in quantum mechanics: From molecules to polymers. Euro. Phys. J. D 19, 307 (2002).

F. Iachello. Algebraic methods for molecular rotationvibration spectra. Chem. Phys. Lett. 78, 581 (1981).

P.W. Fowler, T. Heine, K.M. Rogers, J.P.B. Sandall, G. Seifert, F. Zerbetto. C36, a hexavalent building block for fullerene compounds and solids. Chem. Phys. Lett. 300, 369 (1999).

S.C. O'Brien, J.R. Heath, R.F. Curl, R.E. Smalley. Photophysics of buckminsterfullerene and other carbon cluster ions. J. Chem. Phys. 88, 220 (1988).

G. von Helden, M.T. Hsu, N.G. Potts, P.R. Kemperer, M.T. Bowers. Do small fullerenes exist only on the computer? Experimental results on C=/− 20 and C+/− 24 . Chem. Phys. Lett. 204, 15 (1993).

K.B. Shelimov, J.M. Hunter, M.F. Jarrold. Small carbon rings: dissociation, isomerization, and a simple model based on strain. Int. J. Mass Spectrom. Ion Process. 138, 17 (1994).

L.D. Book, C. Xu, G.E. Scuseria. Carbon cluster ion drift mobilities. The importance of geometry and vibrational effects. Chem. Phys. Lett. 222, 281 (1994).

C. Piskoti, J. Yarger, A. Zettl. C36, a new carbon solid. Nature (London) 393, 771 (1998).

N.K. Sarkar, J. Choudhury, R. Bhattacharjee. An algebraic approach to the study of the vibrational spectra of HCN. Mol. Phys. 104, 3051 (2006);

N.K. Sarkar, J. Choudhury, R. Bhattacharjee. An algebraic approach to the comparative study of the vibrational spectra of monofluoroacetylene (HCCF) and deuterated acetylene (HCCD). Mol. Phys. 106, 693 (2008);

N.K. Sarkar, J. Choudhury, R. Bhattacharjee. A comparative study of the vibrational spectra of OCS and HCP using the Lie algebraic method. Eur. Phys. J. D. 53, 163 (2009).

R. Sen, A. Kalyan, R.S. Paul, N.K. Sarkar, R. Bhattacharjee. A study of vibrational spectra of fullerene C70 and C80: An algebraic approach. Acta Phys. Polonica A 120(3), 407 (2011);

R. Sen, A. Kalyan, R. Das, N.K. Sarkar, R. Bhattacharjee. Vibrational frequencies of buckminsterfullerene: An algebraic study. Spectrosc. Lett. 45, 273 (2012).

R. Sen, A. Kalyan, R.S. Paul, J. Choudhury, N.K. Sarkar, R. Bhattacharjee. (2) Lie algebraic study of vibrational spectra of fullerene C80 and its epoxide C80–O. Ukr. J. Phys. 57(5), 500 (2012).

R. Sen, A. Kalyan, N.K. Sarkar, R. Bhattacharjee. Spectroscopic analysis of C20 isomers by the (2) algebraic model. Fuller. Nanotub. Car. No. 21, 403 (2013).

F. Iachello, R.D. Levine. Algebraic Theory of Molecules (Oxford Univ. Press, 1996).

S. Oss. Algebraic models in molecular spectroscopy. Adv. Chem. Phys. 93, 455 (1996).

K. Nakamoto. Infrared and Raman spectra of inorganic and coordination compounds: Part A: Theory and applications in inorganic chemistry (Wiley, 1997).

K.P. Huber, G. Herzberg. Molecular Spectra and Molecular Structure IV: Constants of Diatomic Molecules (Van Nostrand, 1979).

How to Cite
Choudhury, M., Sen, R., & Sharma, B. (2018). Vibrational IR Active Frequencies of C36: an Algebraic Approach. Ukrainian Journal of Physics, 62(8), 661.
Atoms and molecules