Here, M is the total number of time steps and t k = k∆t. In a TDKS calculation, one computes the time dependent density ρ(r, t) by solving either Eq. (2) or Eq. ( ), from which one can calculate µ λ (t), leading ultimately to an absorption spectrum. The frequency resolution of that absorption spectrum improves as the total simulation time (t max = M ∆t) is increased. Integration time steps on the order of ∆t = 0.02-0.20 a.u. are common, where the atomic unit of time is ≈ 0.0242 fs. Values on the smaller end of that range are necessary for XAS, due to the Fourier transform limit on the maximum excitation energy, Eq. (6). A total propagation time t max = 10-30 fs is typically required in order to obtain an adequately converged spectrum, which equates to 10,000 or more time steps. The cost of each step is roughly equivalent to that of one self-consistent field (SCF) iteration of a groundstate DFT calculation, when the modified-midpoint algorithm is used. 2.A.3. Padé-accelerated transforms. The requisite propagation time can be reduced through the application of a Padé-accelerated Fourier transform technique, which reduces the simulation time required to obtain a converged spectrum. This method is widely used in magnetic resonance spectroscopy to treat noisy spectra with low resolution, and Padé approximants have also been used in real-time electronic structure calculations. To use the Padé technique, the quantity µ λ (ω) in Eq. ( ) is viewed as a polynomial expansion in z = exp(-iω∆t):