Optical and Vibrational Spectra of Narrow Nanotubes: A Symmetry Based Approach I. Miloševi´ c, B. Nikoli´ c, E. Dobardži´ c and M. Damnjanovi´ c Faculty of Physics, University of Belgrade, P. O. Box 368, Belgrade 11001, Serbia and Montenegro Abstract. By use of the tight binding method for induced representations (based on the line group symmetry) polarized optical conductivity, radial breathing and high energy vibrational mode frequencies of the narrow single-wall carbon nanotubes are calculated. The absorption spectra features are assigned by the complete set of conserved quantum numbers and the results obtained are discussed in relation to the previously reported calculations and measurements. Recently, narrow single-wall carbon nanotubes (SWCNTs) have been synthesized in a porous zeolite [1]. By high-resolution transmission electron microscope diameter of these tubes is determined to be 0 42 0 02 nm [2, 3]. The polarized optical absorption spectra of the narrow SWCNTs has been measured [2] and modeled by first principle electronic band structure calculations based on local density function approximation and SWCNTs with chirality (5,0), (3,3) and (4,2) are suggested to be the only possibilities [2, 4, 5]. Apart from the dipole transitions assignment [2, 5], based, however, on the (more or less) incorrectly determined isogonal point groups, symmetry has not been used in the previously reported calculations. As for the vibrational properties, radial breathing mode (RBM) frequency for tubes (5,0), (3,3) and (4,2) has been calculated by H. J. Liu and C. T. Chan [4] and compared to the (by Z. K. Tang and X. D. Xiao) measured Raman spectra. In this Contribution we evaluate numerically, using the line group theoretical meth- ods [6], parallelly polarized (i.e. along the tube axis) optical conductivity for the nan- otubes (5,0), (4,2), (3,3) and (5,1) in the energy region 0 4eV. Also we carry out calculations of the RBM and high energy mode (HEM) frequencies in these tubes. The full line group symmetry [7] of the SWCNTs is used and the calculations are carried out by use of the POLSym package [6] which is based on the tight binding (TB) method for representations of the induced type [8]. In the calculations of the electronic band structure, the results of the density functional tight binding (DFTB) calculations [9] are taken as input data. By using a basis set consisting of one s and three p orbitals (per carbon atom) hybridization of the graphitic σ , π , σ and π states is taken into account. Within the dipole approximation, the optical transition matrix elements are calculated out of the completely symmetry adapted Bloch functions [10]. More details on the method of the optical absorption calculations can be found in Ref. [11]. As for the vibrational spectra calculations, we use the force-constant model, starting with the graphite constants [12] and adjusting them (kinematically and dynamically) to the geometry of a SWCNT [13]. The calculated (real part of) optical conductivity for the nanotubes (5,0), (3,3), (4,2) © 2003 American Institute of Physics 0-7354-0154-3/03/$20.00 Electronic Properties of Novel Materials, edited by H. Kuzmany, J. Fink, M. Mehring, and S. Roth CP685, Molecular Nanostructures: XVII Int'l. Winterschool/Euroconference on 431