\, \sum_f
       |\phi_f({\bf r}_{\alpha x}) \rangle \langle
        \phi_f({\bf r}'_{\alpha x})| 
%(3.1)$$ which rules out the amplitude of the Pauli-forbidden $\alpha-x$ relative states $\phi_f({\bf r}_{\alpha x})$ from the four-body total wave function [@Kukulin84]. The forbidden states are $f=0S$ for $x=n (p)$, $f=\{0S, 0P\}$ for $x=d$, $f=\{0S,  1S, 0P, 0D\}$ for $x=t (^3$He) and $f=\{0S,  1S, 0D\}$ for $x=\alpha$. The Gaussian range parameter $b$ of the single-particle $0s$ orbit in the $\alpha$ particle is taken to be $b=1.358$ fm so as to reproduce the size of the $\alpha$ particle. The same size is assumed for clusters $x=d, t $ and $ ^3$He to manage the Pauli principle avoiding the calculational difficulty. In the actual calculations, the strength $\lambda$ for $V_{\rm Pauli}$ is taken to be 10$^5$ MeV which is large enough to push up away the unphysical forbidden states in the very high energy region with keeping the physical states unchanged. Usefulness of this Pauli operator method of OCM has been verified in many cluster-model calculations.

In some calculations [@Bodmer84; @Bodmer87; @Gal01; @Gal02; @c12] of three-body systems including two or three $\alpha$ clusters, use is made of an $\alpha \alpha$ potential with a strong repulsive core [@Bodmerxx] so as to describe the 