We examine the problem of choosing efficient basis sets for the calculation of vibrational states of molecules. An exact quantum functional is derived for optimizing the parameters of distributed Gaussian basis sets (DGBs). For a given Hamiltonian and energy range, the basis is optimized with respect to the accuracy of the computed eigenvalues. This procedure demonstrates that optimized DGBs are remarkably efficient, being essentially exact for the one-dimensional harmonic oscillator, and orders of magnitude more accurate for the 23-state Morse oscillator than previous DGB calculations of comparable size. Contrary to expectations however, the quantum optimized DGBs have large overlaps, resulting in nearly singular overlap matrices that may cause numerical instabilities in larger calculations. On the other hand, the optimized eigenvalue calculation is shown to be fairly robust with respect to DGB parameter variations, implying that accurate results are possible using more numerically stable DGBs.