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Group theoretic method for the systematic study of multi-quark states isdeveloped. The calculation of matrix elements of many body Hamiltonian issimplified by transforming the physical bases (quark cluster bases) to symmetrybases (group chain classified bases), where the fractional parentage expansionmethod can be used. Five quark system is taken as example in this study. TheJaffe-Wilczek $qq-qq-\bar{q}$ configuration is chosen as one of examples toconstruct the physical bases and the transformation coefficients betweenphysical bases and symmetry ones are shown to be related to the${SU}_{mn}\supset{SU}_m\times{SU}_n$ isoscalar factors and a completetransformation coefficients table is given. The needed isoscalar factors andfractional parentage coefficients had been calculated with our new grouprepresentation theory and published before. Three quark models, the naiveGlashow-Isgur, Salamanca and quark delocalization color screening, are used toshow the general applicability of the new multi-quark calculation method andgeneral results of constituent quark models for five-quark states are given.

Systematic study of multi-quark states I. qq-qq-\bar{q} configuration