  On the basis of the point charge model, we formulate the crystalline
electronic field (CEF) Hamiltonian $H_{\rm CEF}$ in the rare-earth based
quasicrystal (QC) and approximant crystal (AC) with ligand ions located at
pseudo 5-fold configurations by using the operator equivalent method. By
setting the total angular momentum $J=7/2$, the CEF in the quantum critical QC
Au$_{51}$Al$_{34}$Yb$_{15}$ and the 1/1 AC Au$_{51}$Al$_{35}$Yb$_{14}$ is
analyzed with consideration for the effect of Al/Au mixed sites. We find that
the ratio of the valences of ligand ions $x=Z_{\rm Al}/Z_{\rm Au}$ plays an
important role in characterizing the CEF ground state. As $x$ decreases from
$x=3$, the 4f wave function of the CEF ground state with the flat shape lying
in the mirror plane is deformed around $x\approx 0.8$ to the flat shape
perpendicular to the pseudo 5-fold axis at $x=0$. The formulated $H_{\rm CEF}$
by $J$ is generally applicable to rare-earth-based QCs and ACs, which is useful
to analyze the CEF.
