Reduced projection operators and algebraic expressions for the symmetry-adapted functions of the icosahedral group

Abstract

The algebraic expressions for the reduced projection operators \wp^{(\lambda)\bar{\mu}}_{\mu} = \sum_{i = 1}^4 u_i \hat{\beta_i} for the irreducible representation (irrep) \lambda of the icosahedral group I are found by using the double-induced technique and eigenfunction method, where \hat{\beta}_i are the double-coset generators of I with respect to the cyclic subgroup C_5. Simple algebraic expressions are derived for the symmetry-adapted functions (SAF's) by applying the reduced projection operators \wp^{(\lambda) \bar{\mu}}_\mu to Y_{l \bar m}. The SAF's are functions of the angular momentum l, the quantum numbers \lambda, \mu of the group chain I \supset C_5 and the multiplicity label \bar m. In this way, the SAF problem of the group I is solved once for all instead of for one angular momentum l each time.