  Computational crystal-field models have provided consistent models of both
electronic and Zeeman-hyperfine structure for several rare earth ions. These
techniques have not yet been applied to the Zeeman-hyperfine structure of
Eu$^{3+}$ because modeling the structure of the $J=0$ singlet levels in
Eu$^{3+}$ requires inclusion of the commonly omitted lattice electric
quadrupole and nuclear Zeeman interactions. Here, we include these terms in a
computational model to fit the crystal field levels and the Zeeman-hyperfine
structure of the $^7F_0$ and $^5D_0$ states in three Eu$^{3+}$ sites: the
C$_{4v}$ and C$_{3v}$ sites in CaF$_2$ and the C$_2$ site in EuCl$_3$.6H$_2$O.
Close fits are obtained for all three sites which are used to resolve
ambiguities in previously published parameters, including quantifying the
anomalously large crystal-field-induced state mixing in the C$_{3v}$ site and
determining the signs of Zeeman-hyperfine parameters in all three sites. We
show that this model allows accurate prediction of properties for Eu$^{3+}$
important for quantum information applications of these ions, such as relative
transition strengths. The model could be used to improve crystal field
calculations for other non-Kramers singlet states. We also present a spin
Hamiltonian formalism without the normal assumption of no $J$ mixing, suitable
for other rare earth ion energy levels where this effect is important.
