Closure procedures in crystal field theory

The group symmetry and unitary invariance of certain projection operators are discussed, and it is pointed out that closure approximations impose particular group symmetries on the intermediate states projection operators of second-order perturbation theory. The unitary invariance property of projection operators enables evaluation of second-order crystal field effects for a term without detailed knowledge of the crystal field levels of the perturbing terms. The latter result is applied in some numerical calculations for the rare earth Pr 3+ .