  The computation of vibrational spectra of diatomic molecules through the
exact diagonalization of algebraically determined matrixes based on powers of
Morse coordinates is made substantially more efficient by choosing a properly
adapted quantum-mechanical basis, specifically tuned to the molecular
potential. A substantial improvement is achieved while still retaining the full
advantage of the simplicity and numerical light-weightedness of an algebraic
approach. In the scheme we propose, the basis is parameterized by two
quantities which can be adjusted to best suit the molecular potential through a
simple minimization procedure.
