Fast algorithm for calculating nonlinear response functions of large quantum systems and its application to TPA spectra

We review the progress we have been making in recent years in the application of the real-time real-space higher-order finite-difference method to the calculation of various linear/nonlinear response functions. The method was devised for numerical calculation of electronic properties of large quantum systems, and has so far been applied primarily to the calculation of dielectric functions. However, the introduction of a fast statistical algorithm for intermediate state averaging makes the method promising also for computing nonlinear response functions. With the use of random vector averaging for the intermediate states, the task of evaluating the multi-dimensional time integral is reduced to calculating a number of one-dimensional integrals. Then the CPU time necessary for computing a nonlinear response function scales only linearly both with the number of basis states and with the inverse of the required energy resolution, irrespective of the order of nonlinearity. The effectiveness of the algorithm is demonstrated in the calculation of the TPA spectra of silicon. We discuss future applications in such areas as the investigation of electronic properties of biomolecules and complex systems, and designing materials of large nonlinear optical properties.