An alternative to Wigner d-matrices for rotating real spherical harmonics

Abstract

Transformation of spherical harmonics under rotation is a major problem in many areas of theoretical and applied science. While elegantly and efficiently solved for complex spherical harmonics with Wigner D- and d-matrices, extending this method to real spherical harmonics (RSH) faces serious difficulties not yet overcome. This work presents novel explicit formulas and recurrence relations for building RSH rotation matrices with lesser complexity and better computational efficiency. It also gives general closed forms of Wigner d-matrix elements in terms of the rotation angle instead of half this angle as is usual.