The effect of grid quality and weight derivatives in density functional calculations of harmonic vibrational frequencies

We investigate the accuracy of harmonic vibrational frequencies computed with and without the inclusion of quadrature weight derivatives in our recently completed initial implementation of density functional theory (DFT) analytical second derivatives. Unlike the situation with analytical DFT gradients, second derivatives are much more sensitive to the inclusion of weight derivatives, and omitting them can produce nonsensical results unless the numerical grid is of sufficiently high quality. Results are presented for the homonuclear diatomics F2, Cl2, Br2, and I2 and for a number of larger molecules. Errors introduced by excluding weight derivatives increase with increasing atomic number and increasing basis set size. The origin of the error is the difficulty of accurately integrating high-order derivatives of basis functions with large exponents around their own atomic center, and it is not the weight derivatives themselves that eliminate this error but the fact that proper allowance for atom-centered grids that “move” with the atom means that basis functions no longer directly contribute to derivative quantities evaluated on their own grid.