The use of the linear combination of atomic orbitals method forexcited states involves products of orbitals that are known to be linearly dependent. We identify a basis in the space of orbital products that is local for orbitals of finite support and with a residual error that vanishes exponentially with its dimension. As an application of our previously reported technique we compute the Kohn–Sham density response function x0 or a molecule consisting ofNatoms inN2Nv operations, withNv the number of frequency points. We test our construction of x0 by computing molecular spectra directly from the equations of Petersilka–Gossmann–Gross in N2Nv operations rather than from Casida's equations which takes N3 operations. We consider the good agreement with previously calculated molecular spectra as a validation of our construction of x0. Ongoing work indicates that our method is well suited for the computation of the GW self-energy S ¼ iGW and we expect it to be useful in the analysis of exitonic effects in molecules.