Construction of basis functions for the spin-cluster expansion of the magnetic energy on the atomic scale in rotationally invariant systems

The recently introduced spin-cluster expansion (SCE) is a powerful tool to represent on the atomic scale the adiabatic magnetic energy for each magnetic configuration of a system with N sites. In the present paper the theory is worked out for the very important case of rotationally invariant magnets. Appropriate basis functions for this SCE are rotationally invariant and exhibit time-reversal symmetry, are real, and constitute a complete orthonormal set for the representation of any rotationally invariant observable. It is also shown how generalized Heisenberg-type models of the magnetic energy of an isotropic magnet are represented in this symmetry-adapted SCE basis.