Troubleshooting

1. Unphysical polarisability at some frequency points: As explained above, linear response TDDFT is caught on the horns of a dilemma, between using a small enough regularisation parameter to make individual resonances visible, and not choosing it so small that the linear system defining the polarisability is singular. If a frequency point happens to lie too close to one of the resonances of the system, the matrix is numerically singular and GMRES may fail to converge, or return unphysical results such as a negative polarisability.

   Since such singularities point out an underlying resonance, it may be useful to increase the regularisation parameter $\epsilon$ (freq_eps), or to shift the bounds of the frequency interval, omega_min, omega_max, or to reduce the number of frequency points.

   The question of numerical instability is in part a question of computer word length. Considering that the fit procedure with $\epsilon=10^{-3}$Ry typically extracts resonances to an accuracy of $10^{-5}$Ry, way beyond the inherent accuracy attainable with any flavour of DFT, FAST is written in single precision arithmetic, because that saves a factor of about two on the execution time.