which are directly computed with the forward and backward coefficients and then used to compute C AB (t) based on Eq. 51. For an arbitrary correlation function, the C AB (t) still have a full sum over the indices {j, k, a, b}, which can, in principle, be further reduced if Â and B exhibit the same type of sparsity. This is indeed the case for the linear response function (dipole-dipole autocorrelation function) with Â = μρ 0 and B = μ, as we showed in Eq. 49.