An efficient and accurate quantum method for the calculations of many large amplitude vibrational states of polyatomic molecules is proposed and tested on three triatomic molecules; H2O, SO2, and HCN. In this approach we define zero-order reduced dimensional Hamiltonians (h) over cap (k) using minimum energy reduced dimensional potentials. The eigenfunctions and eigenvalues of (h) over cap (k), phi(n)((k)), and epsilon(n)((k)), are used to form an energy selected basis (ESB) for the full system including all the product functions Pi(k)phi(n)((k)) for which Sigmaepsilon((k))less than or equal toE(cut). We show that ESB can be used efficiently in an iterative solution of the Schrodinger equation by the transformation between the ESB and the direct product quadrature grid. Application of the ESB of one-dimensional basis functions is shown to be very efficient for vibrational states of H2O and SO2 up to 30 000 and 23 000 cm(-1), respectively. A combined two-dimensional/one-dimensional basis is used very effectively for HCN above the isomerization energy to HNC. The present approach is shown to be substantially more efficient than either the direct product discrete variable representation (DVR) bases or compact bases from the DVR with the sequential diagonalization/truncation method. (C) 2003 American Institute of Physics.