Detailed Calculation: take the point group , the Stevens Operators taken as a vector form a representation of this point group, which is reducible. Follow the procedure outlined in the book  to split this representation into irreducible parts. The basis vectors of the unit representation may then be linear combined with some arbitrary crystal field parameters to give the most general crystal field. Note that the basis vectors of the unit representation can be obtained efficiently by constructing the projection operator (eq. 4.51 in ) into the subspace transforming according to the irreducible unit representation.
In general the bilinear two ion interaction has the form . and number the different positions of the magnetic ions in the lattice. Without loss of generality the interaction constants can be chosen such that (because the expression is symmetric in angular momentum components any anisotropic part of the interaction tensor does not contribute to the interaction energy).
If and are nearest neighbours on a orthorhombic lattice, they are situated on one of the
crystallographic axes, for example at . The off-diagonal components of the corresponding
vanish, because spin-configurations such as
a moment (m00) on  and (0m0) on  must have the same magnetic energy as (m00) on  and
(0-m0) on . Furthermore the spin-configuration with (m00) on  and (m'00) on  must
have the same magnetic energy as (m00) on  and (m'00) on [-100]. This and similar
considerations lead to the conclusion that the interaction tensor must be the same for [-100] and
[+100]. Therefore the most general form of the interaction
Martin Rotter 2017-01-10