Calculating the linear response functions of noninteracting electrons with a time-dependent Schrödinger equation

An $O\left(N\right)\mathrm{}$ algorithm is proposed for calculating linear response functions of noninteracting electrons. This algorithm is simple and suitable to parallel and vector computation. Since it avoids ${O(N}^3)$ computational effort of matrix diagonalization, it requires only $O\left(N\right)\mathrm{}$ computational efforts, where $N$ is the dimension of the state vector. The use of this $O\left(N\right)\mathrm{}$ algorithm is very effective since, otherwise, we have to calculate a large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang and Zunger [L. W. Wang, Phys. Rev. B 49, 10 154 (1994); L. W. Wang and A. Zunger, Phys. Rev. Lett. 73, 1039 (1994)] is that our method can calculate linear response functions without any storage of huge state vectors on external storage.