A Model Calculation for the Vibrational Modes in C 60
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A MODEL CALCULATION FOR THE VIBRATIONAL MODES IN C 60 R.A. JISHIt, R.M. MIRJE*, and M.S. DRESSELHAUSt t Department of Physics, California State University at Los Angeles, Los Angeles, CA 90032 Department of Mathematics, University of Massachusetts at Lowell, Lowell, MA 01854 tDepartment of Electrical Engineering and Computer Science and Department of Physics, MIT, Cambridge MA 02139
Abstract A force constant model for the vibrational modes in C60 , based on bond-stretching and anglebending interactions, is presented. The results of this model are compared with the experimental data available from Raman, infrared, and high resolution electron energy loss spectroscopies, as well as neutron inelastic scattering measurements. Excellent agreement is obtained between the calculated and experimental mode frequencies. The pressure dependence of the Ramanand infrared-active mode frequencies is calculated within a simple model and is compared to available experimental data.
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Introduction
In a relatively short time, a great wealth of data has been accumulated regarding the vibrational modes in C6 0. Raman scattering measurements have been carried out [1, 2, 3] to identify the frequencies of the ten Raman-active modes in C 60 . Infrared absorption measurements [3, 4, 5] give four strong peaks which are identified with the frequencies of the infrared-active modes in C60 . High resolution electron energy loss spectroscopy has been employed [6] to identify 11 of the 46 distinct vibrational frequencies in C 60 , and neutron inelastic scattering measurements [7, 8] provide the frequencies of many more modes. Recently, the dependence on pressure of the Raman- and infrared-active modes has been reported [9, 10, 11]. All these measurements, taken together, provide a comprehensive picture of the vibrational modes of the C60 molecule. Several calculations of the frequencies of the vibrational modes in C60 have been carried out using various theories: MNDO [12], QFCC/PI [13], STO-3G/SCF [14], DZP/MP2 [15], readspace QMD [16], and Car-Parrinello molecular dynamics [17]. Of these, QCFF/PI gives the best results for the vibrational frequencies because this method has been parameterized with respect to the vibrational frequencies of conjugated aromatic hydrocarbons [18]. In addition, a number of force-constant models have been employed [19, 20, 21] to calculate the frequencies of the normal modes of vibration in C 60 , but none of these models yield results in satisfactory agreement with experimental data, because only nearest neighbor interactions are assumed. In this paper we present a force-constant model, formulated in terms of bond stretching and angle-bending forces, and the model is shown to be in excellent agreement with all the reported experimental data [22]. The model is then used to evaluate the bulk modulus of the C60 molecule. The effect of pressure on the force constants is also estimated within a simple model, and the pressure dependence of the vibrational mode frequencies is then calculated and found to be in reasonably
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