PhysicsLettersAl7O(1992) 116—126 PHYSICS LETTERS A North-Holland Eigenvalues and eigenfunctions of the Hückel Hamiltonian for carbon-60 Yuefan Deng 1 and Chen Ning Yang 2 State University of New York, Stony Brook, NY 11794, USA Received 24 April 1992; revised manuscript received 21 August 1992; accepted for publication 23 August 1992 Communicated by U. Sham Exploiting the symmetry of the molecule C 60, we obtain the precise algebraic and numerical expressions for the eigenvalues and eigenfunctions of the Htickel problem for C60. 1. Introduction With the recent discovery of fullerene molecules and simple methods for their production in bulk, there was an explosion of literature on various aspects of these molecules. In particular, there were many papers on the HUckel problem for C60 [1]. We exploit in the present paper the symmetry of C60 to obtain precise algebraic expressions for the eigen- functions and eigenvalues of the Hückel problem for C60. The symmetry group of C60 is Ix Z2 where I is the 60 element icosahedral group and Z2 is the two-element group consisting of the inversion operator P and the identity. Using this symmetry the 60 x 60 Hückel matrix will be reduced to six 6 x 6 real matrices, whose ei- genfunctions and eigenvalues are easily obtained, both algebraically and numerically. Many of the earlier papers on this problem did not use the symmetry of C60. In those that do use this sym- metry, the methods employed are very different from that of the present work. In particular for the case of unequal couplings for the hp and hh bonds (i.e. single and double bonds) our method is very different from the only existing treatment given by Samuel [1]. 2. The symmetry group 1XZ2 The icosahedral group I consists of the classes of rotations that leave an icosahedron invariant, given in table 1. The C60 molecule is a truncated icosahedron. It is clearly invariant under the group I that leaves the un- truncated icosahedron invariant. In addition inversion P leaves the C60 molecule invariant. Thus the full sym- metry group is Ix Z2. The character table of I can be easily computed by standard methods. It is exhibited in table 2. The irreducible representations are R1, R3, R3, R4 and R5 with linear dimensions given by the sub- scripts. We observe 12+32+32+42+52=60. For Ix Z2 there are 10 irreducible representations: R1 +, R1 —, R3 ÷,R3_, R3 +, R3. —, R4~, R4_, R5+ and R5. Department of AppliedMathematics. Work was partially supported by the Applied Mathematics Subprogram of the U.S. Department of Energy DE-FGO2-90ER2 5084. 2 Institute for Theoretical Physics. Work was partially supported by NSF PHY9O-08936. 116 0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.