Maxwell-Cartesian spherical harmonics in multipole potentials and atomic orbitals

The nature of the Maxwell-Cartesian spherical harmonics S-K((n)) and their relation to tesseral harmonics Y-nm is examined with the help of tricorn arrays that display the components of a totally symmetric Cartesian tensor of any rank in a systematic way. The arrays show the symmetries of the Maxwell-Cartesian harmonic tensors with respect to permutation of axes, the traceless properties of the tensors, the linearly independent subsets., the nonorthogonal subsets, and the subsets whose linear combinations produce the tesseral harmonics. The two families of harmonics are related by their connection with the gradients of 1/r, and explicit formulas for the transformation coefficients are derived. The rotational transformation of S-K((n)) functions is described by a relatively simple Cartesian tensor method. The utility of the Maxwell-Cartesian harmonics in the theory of multipole potentials, where these functions originated in the work of Maxwell, is illustrated with some newer applications which employ a detracer exchange theorem and make use of the partial linear independence of the functions. The properties of atomic orbitals whose angular part is described by Maxwell-Cartesian harmonics are explored, including their angular momenta, adherence to an Unsold-type spherical symmetry relation, and potential for eliminating an angular momentum contamination problem in Cartesian Gaussian basis sets.