  Eigenvalue problems and linear systems of equations involving large symmetric
matrices are commonly solved in quantum chemistry using Krylov space methods,
such as the Davidson algorithm. The preconditioner is a key component of Krylov
space methods that accelerates convergence by improving the quality of new
guesses at each iteration. We systematically design a new preconditioner for
time-dependent density functional theory (TDDFT) calculations based on the
recently introduced TDDFT-ris semiempirical model by re-tuning the empirical
scaling factor and the angular momenta of a minimal auxiliary basis. The final
preconditioner produced includes up to $d$-functions in the auxiliary basis and
is named "rid". The rid preconditioner converges excitation energies and
polarizabilities in 5-6 iterations on average, a factor of 2-3 faster than the
conventional diagonal preconditioner, without changing the converged results.
Thus, the rid preconditioner is a broadly applicable and efficient
preconditioner for TDDFT calculations.
