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We analyze the $\alpha$-cluster wave functions in cluster states of $^8$Beand $^{20}$Ne by comparing the exact relative wave function obtained by thegenerator coordinate method (GCM) with various types of trial functions. Forthe trial functions, we adopt the fixed range shifted Gaussian of theBrink-Bloch (BB) wave function, the spherical Gaussian with the adjustablerange parameter of the spherical Thosaki-Horiuchi-Schuck-R\"opke (sTHSR), thedeformed Gaussian of the deformed THSR (dTHSR), and a function with the Yukawatail (YT). The quality of the description of the exact wave function with atrial function is judged by the squared overlap between the trial function andthe GCM wave function. The better result is obtained with the sTHSR wavefunction than the BB wave function, and further improvement can be done withthe dTHSR wave function because these wave functions can describe the outertail better. The YT wave function gives almost the equal quality to or evenbetter quality than the dTHSR wave function indicating that the outer tail of$\alpha$ cluster states is characterized by the Yukawa-like tail rather thanthe Gaussian tail. In the weakly bound $\alpha$ cluster states with the small$\alpha$ separation energy and the low centrifugal and Coulomb barriers, theouter tail part is the slowly damping function described well by the quantumpenetration through the effective barrier. This outer tail characterizes thealmost zero-energy free $\alpha$ gas behavior, i.e., the delocalization ofcluster.

Description of $\alpha$-cluster tail in $^{8}$Be and $^{20}$Ne: Delocalization of $\alpha$ cluster because of quantum penetration