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.