Molecular symmetry with quaternions.

A new and relatively simple version of the quaternion calculus is offered which is especially suitable for applications in molecular symmetry and structure. After introducing the real quaternion algebra and its classical matrix representation in the group SO(4) the relations with vectors in 3-space and the connection with the rotation group SO(3) through automorphism properties of the algebra are discussed. The correlation of the unit quaternions with both the Cayley-Klein and the Euler parameters through the group SU(2) is presented. Besides rotations the extension of quaternions to other important symmetry operations, reflections and the spatial inversion, is given. Finally, the power of the quaternion calculus for molecular symmetry problems is revealed by treating some examples applied to icosahedral symmetry.