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We propose to use a basis function constructed based on the microscopiccluster model for an efficient description of multi-cluster systems with theorthogonality condition originating from the Pauli principle. The basisfunction is expressed analytically by a superposition of correlated Gaussianfunctions. We demonstrate the power of this approach by taking an example of a$3\alpha$ system, $^{12}$C. A comparison with the conventional pseudopotentialmethod using the projection operator is made. The present method offersefficient and numerically stable computations as the number of basis functionsis significantly reduced compared to the conventional method. We show that thepresent basis function includes reasonably small components of thePauli-forbidden states, allowing us to discuss simply the structure of thefirst excited $0^+$ state, Hoyle state.

Novel approach to the removal of the Pauli-forbidden states in the orthogonality condition model: A case of multi-$\alpha$ systems