Numerical simulation of charged fullerene spectrum
Abstract
The mathematical model of the ground state electron spectrum of a charged fullerene is constructed on the basis of the potential of a charged sphere and the spherically symmetric potential of a neutral fullerene, derived in a single-electron self-consistent field model approach. The electron spectrum is defined as the solution of the spectral problem for the one-dimensional Schrödinger equation. For the numerical solution of the spectral problem, piecewise-linear finite elements are used. The computational algorithm was tested on the analytical solution of the problem of the spectrum of the hydrogen atom. For solution of matrix spectral problems, a free library for solving spectral problems of SLEPc is used. The results of calculations of the electron spectrum of a charged fullerene C60 are presented.
Keyword : fullerene, Schrödinger equation, spectral problem, finite element method
How to Cite
Arutyunyan, R., Obukhov, Y., & Vabishchevich, P. (2019). Numerical simulation of charged fullerene spectrum. Mathematical Modelling and Analysis, 24(2), 263-275. https://doi.org/10.3846/mma.2019.017
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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F. Langa. Fullerenes: principles and applications. Royal Society of Chemistry, Cambridge, U.K, 2011. ISBN 9781849732956.
V. Ledoux and M. Van Daele. Matslise 2.0: A Matlab toolbox for Sturm-Liouville computations. ACM Transactions on Mathematical Software (TOMS),42(4):29:1–29:18, June 2016. https://doi.org/10.1145/2839299
A. Logg, K. A. Mardal and G. Wells. Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book. Springer, Berlin NewYork, 2012. ISBN 9783642230981. https://doi.org/10.1007/978-3-642-23099-8
L. L. Lohr and M. Blinder. Electron photodetachment from a Dirac bubble potential. A model for the fullerene negative ionCC−60. Chemical Physics Letters, 198(1–2):100–108, 1992. https://doi.org/10.1016/0009-2614(92)90055-R
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R. G. Polozkov, V. K. Ivanov and A. V. Solov’yov. Photoionization of thefullerene ionC+60. Journal of Physics B: Atomic, Molecular and Optical Physics, 38(24):4341–4348, 2005. https://doi.org/10.1088/0953-4075/38/24/001
J. Pryce. Numerical solution of Sturm-Liouville problems. Clarendon Press, Oxford England New York, 1993. ISBN 0198534159.
J. D. Pryce. A test package for Sturm-Liouville solvers. ACM Transactionson Mathematical Software,25(1):21–57,1999. https://doi.org/10.1145/305658.287651
M. J. Puska and R.M. Nieminen. Photoabsorption of atoms inside C60. PhysicalReview A, 47:1181, 1993. https://doi.org/10.1103/PhysRevA.47.1181
A. Rüdel, R. Hentges, U. Becker, H. S. Chakraborty, M. E. Madjet and J. M. Rost. Imaging delocalized electron clouds: Photoionization of C60 in Fourier reciprocal space. Physical Review Letters, 89:125503, 2002. https://doi.org/10.1103/PhysRevLett.89.125503
Y. Saad. Numerical methods for large eigenvalue problems. Society for Industrial and Applied Mathematics, Philadelphia, 2011. ISBN 9781611970722. https://doi.org/10.1137/1.9781611970739
R. Sahnoun, K. Nakai, Y. Sato, H. Kono, Y. Fujimura and M. Tanaka. Theo-retical investigation of the stability of highly chargedC60molecules producedwith intense near-infrared laser pulses.The Journal of Chemical Physics,125(18):184306, 2006. https://doi.org/10.1063/1.2371109
S. Saito, A. Oshiyama and Miyamoto Y. Electronic structures of fullerenes and fullerides. Computational Approaches in Condensed-Matter Physics, pp. 22–26,1992. https://doi.org/10.1007/978-3-642-84821-64
G. Sánchez, S. Dıaz-Tendero, M. Alcamı and F. Martın. Size dependence ofionization potentials and dissociation energies for neutral and singly-charged Cnfullerenes (n= 40−70). Chemical Physics Letters, 416(1–3):14–17, 2005. https://doi.org/10.1016/j.cplett.2005.09.033
K. Sattler. Handbook of nanophysics. CRC Press, Boca Raton, 2011. ISBN9781420075540.
G. Schrange-Kashenock. 4d→4fresonance in photoabsorption of ceriumion Ce3+ and endohedral cerium in fullerene complexCe@C+82. Journal of Physics B: Atomic, Molecular and Optical Physics, 49(19):185002, 2016. https://doi.org/10.1088/0953-4075/49/18/185002
G. W. Stewart. A Krylov-Schur algorithm for large eigen problems. SIAM Journal on Matrix Analysis and Applications, 23(3):601–614, 2001. https://doi.org/10.1137/S0895479800371529
N. Troullier and J.L. Martins. Structural and electronic properties of C60. Physical Review B, 46(3):1754–1765,1992. https://doi.org/10.1103/PhysRevB.46.1754
A. V. Verkhovtsev, A. V. Korol and Solov’yov A. V. Quantum and classical features of the photoionization spectrum of C60. Physical Review A, 88:043201, 2013. https://doi.org/10.1103/PhysRevA.88.043201
Y. B. Xu, M.Q. Tan and U. Becker. Oscillations in the photoionization cross section of C60. Physical Review Letters, 76:3538, 1996. https://doi.org/10.1103/PhysRevLett.76.3538
K. Yabana and G. F. Bertsch. Electronic structure of C60 in a sphericalbasis. Physica Scripta, 48(5):633–637, 1993. https://doi.org/10.1088/0031-8949/48/5/022
H. Zettergren, J. Jensen, H. T. Schmidt and H. Cederquist. Electrostatic modelcal culations of fission barriers for fullerene ions. European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics, 29(1):63–68, 2004. https://doi.org/10.1140/epjd/e2004-00006-6
H. Zettergren, G. Sánchez, S.Dıaz-Tendero, M. Alcamı and F. Martın. Theoretical study of the stability of multiply charged C70 fullerenes. The Journal of Chemical Physics,127(10):104308, 2007. https://doi.org/10.1063/1.2768361
A. Zettl. Sturm-Liouville theory. American Mathematical Society, Providence, R. I, 2005. ISBN 9780821839058.
R. V. Arutyunyan and A. V. Osadchy. The systems of volume-localized electron quantum levels of charged fullerenes. Journal of Nanomaterials, 2018:7526869, 2018. https://doi.org/10.1155/2018/7526869
R. V. Arutyunyan, P. N. Vabishchevich and Y. N. Obukhov. Existence of a system of discrete volume-localized quantum levels for charged fullerenes. Physical Review B, 98(10):155427, 2018. https://doi.org/10.1103/PhysRevB.98.155427
A. S. Baltenkov, S. T. Manson and A. Z. Msezane. Jellium model potentials forthe C60 molecule and the photoionization of endohedral atoms, A@C60. Journal of Physics B: Atomic, Molecular and Optical Physics, 48(18):185103, 2015. https://doi.org/10.1088/0953-4075/48/18/185103
A. K. Belyaev, A. S. Tiukanov, A. I. Toropkin, V. K. Ivanov, R. G. Polozkovand A. V. Solov’yov. Photoabsorption of the fullerene C60 and its positiveions. Physica Scripta, 80(4):048121, 2009. https://doi.org/10.1088/0031-8949/80/04/048121
V. R. Bhardwaj, P. B. Corkum and D. M. Rayner. Internal laser-induced dipoleforce at work in C60 molecule. Physical Review Letters, 91(20):203004, 2003. https://doi.org/10.1103/PhysRevLett.91.203004
A. Brenac, F. Chandezon, H. Lebius, A. Pesnelle, S. Tomita and B. A. Huber. Multifragmentation of highly charged C60 ions: Chargestates and fragment energies. Physica Scripta, 1999(T80B):195,1999. https://doi.org/10.1238/Physica.Topical.080a00195
S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer, New York, 2008. ISBN 9780387759333. https://doi.org/10.1007/978-0-387-75934-0
E. Campbell. Fullerene collision reactions. Kluwer Academic, Dordrecht London,2003. ISBN 9781402025242.
J. P. Connerade, V. K. Dolmatov, P. A. Lakshmi and Manson S. T. Electron structure of endohedrally confined atoms: atomic hydrogen in an attractive shell. Journal of Physics B: Atomic, Molecular and Optical Physics, 32(10):L239–L245, 1999. https://doi.org/10.1088/0953-4075/32/10/101
J. P. Connerade, V.K. Dolmatov and Manson S.T.A unique situation foran endohedral metallofullerene. Journal of Physics B: Atomic, Molecularand Optical Physics, 32(14):L395–L403, 1999. https://doi.org/10.1088/0953-4075/32/14/108
S. Dıaz-Tendero, M. Alcamı and F. Martın. Structure and electronic properties of highly charged C60 and C58 fullerenes. The Journal of Chemical Physics, 123(18):184306, 2005.https://doi.org/10.1063/1.2104467
Z. Felfli and A. Z. Msezane. Simple method for determining binding energies of fullerene negative ions. European Physical Journal B,72:78, 2018. https://doi.org/10.1140/epjd/e2018-80420-9
V. Hernandez, J. E. Roman and V. Vidal. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Transactions on Mathematical Soft-ware (TOMS),31(3):351–362, 2005. https://doi.org/10.1145/1089014.1089019
V. K. Ivanov, G. Y. Kashenock, R. G. Polozkov and A. V. Solov’yov. Photoionization cross sections of the fullerenes C20 and C60 calculated in a simple spherical model. Journal of Physics B: Atomic, Molecular and Optical Physics, 34(21):L669–L677, 2001. https://doi.org/10.1088/0953-4075/34/21/101
W. Jaskólski. Confined many-electron systems. Physics Reports, 271(1):1–66,1996. https://doi.org/10.1016/0370-1573(95)00070-4
J. Jensen, H. Zettergren, H. T. Schmidt, H. Cederquist, S. Tomita, S. B. Nielsen, J. Rangama, P. Hvelplund, B. Manil and B. A. Huber. Ionization of C70 and C60 molecules by slow highly charged ions: A comparison. Physical Review A, 69:053203, 2004. https://doi.org/10.1103/PhysRevA.69.053203
W. Krätschmer, L. D. Lamb, K. Fostiropoulos and D. R. Huffman. Solid C60 : a new form of carbon. Nature, 347(6291):354–358, 1990. https://doi.org/10.1038/347354a0
H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley. C60: Buckminsterfullerene. Nature, 318(6042):162–163, 1985. https://doi.org/10.1038/318162a0
L. D. Landau and E. M. Lifshitz. Quantum Mechanics : Non-Relativistic Theory. Elsevier Science, Burlington, 1977. ISBN 9781483149127.
F. Langa. Fullerenes: principles and applications. Royal Society of Chemistry, Cambridge, U.K, 2011. ISBN 9781849732956.
V. Ledoux and M. Van Daele. Matslise 2.0: A Matlab toolbox for Sturm-Liouville computations. ACM Transactions on Mathematical Software (TOMS),42(4):29:1–29:18, June 2016. https://doi.org/10.1145/2839299
A. Logg, K. A. Mardal and G. Wells. Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book. Springer, Berlin NewYork, 2012. ISBN 9783642230981. https://doi.org/10.1007/978-3-642-23099-8
L. L. Lohr and M. Blinder. Electron photodetachment from a Dirac bubble potential. A model for the fullerene negative ionCC−60. Chemical Physics Letters, 198(1–2):100–108, 1992. https://doi.org/10.1016/0009-2614(92)90055-R
M. E. Madjet, H. S. Chakraborty, J. M. Rost and S. T. Manson. Photoionization of C60: a model study. Journal of Physics B: Atomic, Molecular and Optical Physics, 41(10):105101, 2008. https://doi.org/10.1088/0953-4075/41/10/105101
R. G. Polozkov, V. K. Ivanov and A. V. Solov’yov. Photoionization of thefullerene ionC+60. Journal of Physics B: Atomic, Molecular and Optical Physics, 38(24):4341–4348, 2005. https://doi.org/10.1088/0953-4075/38/24/001
J. Pryce. Numerical solution of Sturm-Liouville problems. Clarendon Press, Oxford England New York, 1993. ISBN 0198534159.
J. D. Pryce. A test package for Sturm-Liouville solvers. ACM Transactionson Mathematical Software,25(1):21–57,1999. https://doi.org/10.1145/305658.287651
M. J. Puska and R.M. Nieminen. Photoabsorption of atoms inside C60. PhysicalReview A, 47:1181, 1993. https://doi.org/10.1103/PhysRevA.47.1181
A. Rüdel, R. Hentges, U. Becker, H. S. Chakraborty, M. E. Madjet and J. M. Rost. Imaging delocalized electron clouds: Photoionization of C60 in Fourier reciprocal space. Physical Review Letters, 89:125503, 2002. https://doi.org/10.1103/PhysRevLett.89.125503
Y. Saad. Numerical methods for large eigenvalue problems. Society for Industrial and Applied Mathematics, Philadelphia, 2011. ISBN 9781611970722. https://doi.org/10.1137/1.9781611970739
R. Sahnoun, K. Nakai, Y. Sato, H. Kono, Y. Fujimura and M. Tanaka. Theo-retical investigation of the stability of highly chargedC60molecules producedwith intense near-infrared laser pulses.The Journal of Chemical Physics,125(18):184306, 2006. https://doi.org/10.1063/1.2371109
S. Saito, A. Oshiyama and Miyamoto Y. Electronic structures of fullerenes and fullerides. Computational Approaches in Condensed-Matter Physics, pp. 22–26,1992. https://doi.org/10.1007/978-3-642-84821-64
G. Sánchez, S. Dıaz-Tendero, M. Alcamı and F. Martın. Size dependence ofionization potentials and dissociation energies for neutral and singly-charged Cnfullerenes (n= 40−70). Chemical Physics Letters, 416(1–3):14–17, 2005. https://doi.org/10.1016/j.cplett.2005.09.033
K. Sattler. Handbook of nanophysics. CRC Press, Boca Raton, 2011. ISBN9781420075540.
G. Schrange-Kashenock. 4d→4fresonance in photoabsorption of ceriumion Ce3+ and endohedral cerium in fullerene complexCe@C+82. Journal of Physics B: Atomic, Molecular and Optical Physics, 49(19):185002, 2016. https://doi.org/10.1088/0953-4075/49/18/185002
G. W. Stewart. A Krylov-Schur algorithm for large eigen problems. SIAM Journal on Matrix Analysis and Applications, 23(3):601–614, 2001. https://doi.org/10.1137/S0895479800371529
N. Troullier and J.L. Martins. Structural and electronic properties of C60. Physical Review B, 46(3):1754–1765,1992. https://doi.org/10.1103/PhysRevB.46.1754
A. V. Verkhovtsev, A. V. Korol and Solov’yov A. V. Quantum and classical features of the photoionization spectrum of C60. Physical Review A, 88:043201, 2013. https://doi.org/10.1103/PhysRevA.88.043201
Y. B. Xu, M.Q. Tan and U. Becker. Oscillations in the photoionization cross section of C60. Physical Review Letters, 76:3538, 1996. https://doi.org/10.1103/PhysRevLett.76.3538
K. Yabana and G. F. Bertsch. Electronic structure of C60 in a sphericalbasis. Physica Scripta, 48(5):633–637, 1993. https://doi.org/10.1088/0031-8949/48/5/022
H. Zettergren, J. Jensen, H. T. Schmidt and H. Cederquist. Electrostatic modelcal culations of fission barriers for fullerene ions. European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics, 29(1):63–68, 2004. https://doi.org/10.1140/epjd/e2004-00006-6
H. Zettergren, G. Sánchez, S.Dıaz-Tendero, M. Alcamı and F. Martın. Theoretical study of the stability of multiply charged C70 fullerenes. The Journal of Chemical Physics,127(10):104308, 2007. https://doi.org/10.1063/1.2768361
A. Zettl. Sturm-Liouville theory. American Mathematical Society, Providence, R. I, 2005. ISBN 9780821839058.