This paper presents a rigorous state-specific multi-reference coupled cluster formulation of the method first proposed by Meller et al. [J. Chem. Phys. 104, 4068 (1996)]. Guess values of the amplitudes of the single and double excitations (the Tˆ operator) on the top of the references are extracted from the knowledge of the coefficients of the Multi-Reference Singles and Doubles Configuration Interaction (MR-CISD) matrix. The multiple parentage problem is solved by scaling these amplitudes from the interaction between the references and the singles and doubles. Then one proceeds to a dressing of the MR-CISD matrix under the effect of the triples and quadruples, the coefficients of which are estimated from the action of exp(Tˆ). This dressing follows the logic of the intermediate effective Hamiltonian formalism. The dressed MR-CISD matrix is diagonalized and the process is iterated to convergence. As a simplification, the coefficients of the triples and quadruples may in practice be calculated from the action of Tˆ(2) only, introducing 5th-order differences in the energies. The so-simplified method is tested on a series of benchmark systems from Complete Active Spaces (CASs) involving 2-6 active electrons up to bond breakings. The comparison with full configuration interaction results shows that the errors are of the order of a few millihartree, five times smaller than those of the CAS-CISD, and the deviation to strict separability is lower than 10 μ hartree. The method is totally uncontracted, parallelizable, and extremely flexible since it may be applied to selected MR and/or selected CISD. Some potential generalizations are briefly discussed.