  Recently the Eu$_{0.5}$Ba$_{0.5}$TiO$_{3}$ solid was suggested as a promising
candidate for experimental search of the electron electric dipole moment. To
interpret the results of this experiment one should calculate the effective
electric field acting on an unpaired (spin-polarized) electrons of europium
cation in the crystal because the value of this field cannot be measured
experimentally. The Eu$^{++}$ cation is considered in the paper in the uniform
external electric field $E_{\rm ext}$ as our first and simplest model
simulating the state of europium in the crystal. We have performed high-level
electronic structure correlation calculation using coupled clusters theory (and
scalar-relativistic approximation for valence and outer core electrons at the
molecular pseudopotential calculation stage that is followed by the
four-component spinor restoration of the core electronic structure) to evaluate
the enhancement coefficient $K= E_{\rm eff}/E_{\rm ext}$ (where $E_{\rm ext}$
is the applied external electric field and $E_{\rm eff}$ is the induced
effective electric field acting on an unpaired electron in Eu$^{++}$). A
detailed computation analysis is presented. The calculated value of $K$ is
-4.6.
