The dimension of the variational problem in eq. ( ) is typically very large. Hence, to reduce dimensionality, but also to set up an effective ab initio crystal field splitting theory i.e a degenerate perturbation theory within the lanthanide ion's degenerate ground level, we apply the Löwdin partitioning procedure. This is achieved by partitioning the full space spanned by H, D and M in the two subspaces A (here the (2J + 1)-degenerate space of the lanthanide ground spin-orbit multiplet) and B (the complement excitation space) 23 :