  In order to promote our basic understanding on the Kondo behavior recently
observed in europium compounds, we analyze an impurity Anderson model with
seven $f$ electrons at an impurity site by employing a numerical
renormalization group method. The local part of the model consists of Coulomb
interactions among $f$ electrons, spin-orbit coupling $\lambda$, and
crystalline electric field (CEF) potentials, while we consider the
hybridization $V$ between local $f$ electrons and single-band conduction
electrons with $a_{\rm u}$ symmetry. For $\lambda=0$, we observe the
underscreening Kondo behavior for appropriate values of $V$, characterized by
the entropy change from $\ln 8$ to $\ln 7$, in which one of seven $f$ electrons
is screened by conduction electrons. When $\lambda$ is increased, we obtain two
types of behavior depending on the values of $V$. For large $V$, we find the
entropy release of $\ln 7$ at low temperatures, determined by the level
splitting energy due to the hybridization. For small $V$, we also observe the
entropy change from $\ln 8$ to $\ln 2$ by the level splitting due to the
hybridization, but at low temperatures, $\ln 2$ entropy is found to be
released, leading to the Kondo effect. We emphasize that the Kondo behavior for
small $V$ is observed for realistic values of $\lambda$ in the order of $0.1$
eV. We also discuss the effect of CEF potentials and the multipole properties
in the Kondo behavior found in this paper.
