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We propose an iterative method for computing vibrational spectra thatsignificantly reduces the memory cost of calculations. It uses a direct productprimitive basis, but does not require storing vectors with as many componentsas there are product basis functions. Wavefunctions are represented in a basiseach of whose functions is a sum of products (SOP) and the factorizablestructure of the Hamiltonian is exploited. If the factors of the SOP basisfunctions are properly chosen, wavefunctions are linear combinations of a smallnumber of SOP basis functions. The SOP basis functions are generated using ashifted block power method. The factors are refined with a rank reductionalgorithm to cap the number of terms in a SOP basis function. The ideas aretested on a 20-D model Hamiltonian and a realistic CH$_3$CN (12 dimensional)potential. For the 20-D problem, to use a standard direct product iterativeapproach one would need to store vectors with about $10^{20}$ components andwould hence require about $8 \times 10^{11}$ GB. With the approach of thispaper only 1 GB of memory is necessary. Results for CH$_3$CN agree well withthose of a previous calculation on the same potential.

Calculating vibrational spectra with sum of product basis functions without storing full-dimensional vectors or matrices