Fast Algorithms for Estimating the Absorption Spectrum within the Linear Response Time-dependent Density Functional Theory

In this paper, we present two iterative algorithms for approximating the absorption spectrum of molecules within the linear response of time-dependent density functional theory (TDDFT) framework. These methods do not attempt to compute eigenvalues or eigenvectors of the linear response matrix. They are designed to approximate the absorption spectrum as a function directly. They take advantage of the special structure of the linear response matrix. Neither method requires the linear response matrix to be constructed explicitly. They only require a procedure that performs the multiplication of the linear response matrix with a vector. These methods can also be easily modified to efficiently estimate the density of states (DOS) of the linear response matrix without computing the eigenvalues of this matrix. We show by computational experiments that the methods proposed in this paper can be much more efficient than methods that are based on the exact diagonalization of the linear response matrix. We also show that they can also be more efficient than real-time TDDFT simulations. We compare the pros and cons of these methods in terms of their accuracy and computation and storage cost.