A new algorithm to solve the Time Dependent Density Functional Theory (TDDFT) equations in the
space of the density fitting auxiliary basis set has been developed and implemented. The method
extracts the spectrum from the imaginary part of the polarizability at any given photon energy,
avoiding the bottleneck of Davidson diagonalization. The original idea which made the present
scheme very efficient consists in the simplification of the double sum over occupied-virtual pairs
in the definition of the dielectric susceptibility, allowing an easy calculation of such matrix as a linear
combination of constant matrices with photon energy dependent coefficients. The method has been
applied to very different systems in nature and size (from H2 to [Au147]−). In all cases, the maximum
deviations found for the excitation energies with respect to the Amsterdam density functional code are
below 0.2 eV. The new algorithm has the merit not only to calculate the spectrum at whichever photon
energy but also to allow a deep analysis of the results, in terms of transition contribution maps, Jacob
plasmon scaling factor, and induced density analysis, which have been all implemented
