On the advantages of a rectangular matrix collocation equation for computing vibrational spectra from small basis sets

We demonstrate that for calculations of vibrational spectra from small basis sets, a rectangular collocation matrix equation combined with fitting [S. Manzhos and T. Carrington Jr., Can. J. Chem. 87 (2009) 864] is advantageous compared to both a traditional approach where the energies and wavefunctions are found from eigenvectors and eigenvalues of a square matrix and to the method of Boutry et al. [G. Boutry, M. Elad, G.H. Golub, P. Milanfar, SIAM J. Matrix Anal. Appl. 27 (2005) 582] for rectangular matrix pencils. A rectangular matrix equation allows for the use of a small basis set without the loss of information induced by squaring, while fitting the energy insures stability.