Energy Spectrum and Optical Absorption Spectrum of Fullerene C 28 within the Hubbard Model
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Energy Spectrum and Optical Absorption Spectrum of Fullerene C28 within the Hubbard Model A. V. Silant’ev* Mari State University, Yoshkar-Ola, Russia *e-mail: [email protected] Received March 2, 2020; revised March 27, 2020; accepted April 1, 2020
Abstract—Anticommutator Green’s functions and energy spectra of the C28 fullerene and the endohedral Zr@C28 fullerene with the symmetry group Td are obtained within the Hubbard model in the approximation of static fluctuations. Using the methods of group theory, the classification of energy states is carried out, and the allowed transitions in the energy spectra of C28 and Zr@C28 molecules are determined. Keywords: Hubbard model, Green’s functions, energy spectrum, nanosystems, fullerene C28 DOI: 10.1134/S1063783420110335
INTRODUCTION Currently, a large number of studies are devoted to the study of the physical and chemical properties of fullerenes [1, 2]. These studies have shown that the majority of Cn fullerenes are unstable molecules, many of which become stable when metal atoms are placed inside the fullerene and endrohedral fullerenes M@Cn are formed. One of these fullerenes is C28 fullerene. This fullerene was discovered in 1993 in the form of the endrohedral fullerene U@C28 [3]. Further studies have shown that fullerene C28 is an unstable molecule that is stabilized during the formation of endofullerenes M@C28 with elements Zr, W, Mo, Os, Ti, Th, U, Ce, which are capable of assuming electronic configurations M4+ [4]. Quite a lot of works have been devoted to the study of the properties of fullerene C28 [5–8]. As is known [9], it is possible to construct two isomers of fullerene C28 with symmetry groups Td and D2 from 28 carbon atoms. The study of these isomers showed that the C28 fullerene with the Td symmetry group is more stable than the C28 fullerene with the D2 symmetry group [10]. As seen from Fig. 1, the C28 fullerene with the Td symmetry group consists of twelve pentagons and four hexagons. It can be seen from the Schlegel diagram shown in Fig. 1, that the C28 fullerene with the Td symmetry group contains three nonequivalent bonds, which are denoted by a, b and c; and three groups of nonequivalent carbon atoms: G1 = {1, 3, 5, 9, 13, 17, 19, 20, 22, 23, 25, 26}, G2 = {2, 4, 6, 8, 10, 12, 14, 16, 18, 21, 24, 27}, G3 = {7, 11, 15, 28}. Bond a, connects two pentagons, bond b connects pentagon and hexagon, and bond c con-
nects two hexagons. The set G1 includes atoms that are at the vertices of the junction of two hexagons and one pentagon. The set G2 includes atoms that are at the vertices of the junction of one hexagon and two pentagons. The set G3 includes atoms that are at the vertices of the junction of the three pentagons. The Hubbard model [23] is widely used to describe the electronic and optical properties of carbon nanosystems [11–22]. For example, within the framework of the Hubbard model in the approximation of static fluctuations (ASF), the energy spectra and optical absorption spectra of the C60 fullerene [11, 21], C70 fullerene [13, 22],
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