Rotation matrices for real spherical harmonics: general rotations of atomic orbitals in space-fixed axes

Abstract

The angular factors of atomic orbitals are real spherical harmonics. This is independent of the choice of basis function. In the course of molecular electronic structure calculations, numerous rotations of real spherical harmonics are required in a suitably defined space-fixed coordinate system. The origin and axes are space-fixed and rotation matrices defined on the basis of spherical harmonics. Typically, rotations are used to align the local axes for atoms with a unique interatomic direction in a fixed frame for integral evaluations. In this work, a highly compact expression and efficient evaluation of the rotation matrices are given for a real spherical harmonic basis. Relations to Gaunt coefficients are shown explicitly as are recurrence formulae for rotation matrices. This leads to extremely rapid and precise rotation algorithms. The Wigner rotation matrices which are still often used in orbital rotations are shown to be completely surpassed by this approach. The present work is related to a method described by Kautz in the field of image processing but significant improvements have been made, especially in the study of structure and storage of the rotation matrices. After complete testing using computer algebra, a numerical program was written in C. Numerical tests are cited in the closing sections of this work.