  We propose a new basis state, which satisfies the Pauli principle in the
nuclear cluster model. The basis state is defined as the generalized coherent
state of the harmonic oscillator wave function using a pair of the creation
operators and is orthogonal to the Pauli-forbidden states having smaller
quanta. In the coherent basis state, the range parameter is changeable and
controls the radial dilation. This property is utilized for the precise
description of the relative motion between nuclear clusters. We show the
reliability of this framework for the $2\alpha$ system of $^8$Be in the
semi-microscopic orthogonality condition model. We obtain the resonances and
non-resonant continuum states of $2\alpha$ with complex scaling. The resonance
solutions and the phase shifts of the $\alpha$-$\alpha$ scattering agree with
those using the conventional projection operator method to remove the
Pauli-forbidden states. We further discuss the extension of the present
framework to the multi-$\alpha$ cluster systems using the SU(3) wave functions.
