Icosahedron symmetry and optical properties of the fullerene C$_{60}$

The electron transitions property of the fullerene molecule is considered. Because of electron jumps between nearest neighbor is possible, the interaction of the $\pi$- electrons produces the appearance of the collective states, having truncated icosahedron symmetry. The irreducible representations of the icosahedral group allow us to subdivide the state functions into groups with separate energy sublevel. The good approximation of the wave functions for the irreducible representations developed by means of spherical functions superposition. The optical properties is presented by formula of the excitation cross-section for electronic levels and by formula of the spontaneous relaxation time, received with help of matrix element calculation. Because of Jahn–Teller effect, a time of non-radiative transition is much less than radiative relaxation time. The interaction with phonon spectrum produces the spontaneously broken symmetry and restricts electronic-emission spectrum, while the absorption in optical spectrum exists. Transfer of energy from electron states into phonon oscillations produces the thermal radiation. However, it is possible to solve problem of electron state decay.