Relation between cluster and shell-model wave functions

A method is presented for the expansion of properly antisymmetrized cluster model (or resonating group method) wave functions in terms of translationally invariant shell-model wave functions. By coupling the relative motion (harmonic oscillator) wave functions and the internal cluster functions to resultant ${\mathrm{SU}}_3$,${\mathrm{SU}}_4$ strong coupled functions, the cluster model wave functions can be expanded in terms of standard shell-model wave functions by ${\mathrm{SU}}_3$, ${\mathrm{SU}}_4$ recoupling techniques, using readily available recoupling coefficients. As a specific example, the full shell-model expansions are given for ($\alpha +{}_{}{}^{12}{}_{}{}^{}\mathrm{C}_{}^{\mathrm{}\left(04\right)\mathrm{}}{}_{}{}^{}$) cluster functions for all possible ${\mathrm{SU}}_3$ symmetries and $\alpha -{}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ relative motion wave functions carrying from 4 to 8 oscillator quanta, hence ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ core excitations up to $4\hslash \omega$.