  Pauli-projected random gaussians are used as a representation to solve the
shell model equations. The elements of the representation are chosen by a
variational procedure. This scheme is particularly suited to describe cluster
formation and cluster decay in nuclei. It overcomes the basis-size problem of
the ordinary shell model and the technical difficulties of the
cluster-configuration shell model. The model reproduces the $\alpha$-decay
width of $^{212}$Po satisfactorily.
