Real-time propagation of the reduced one-electron density matrix in atom-centered Gaussian orbitals: application to absorption spectra of silicon clusters.

We solve the time-dependent density functional theory equation by propagating the reduced one-electron density matrix in real-time domain. The efficiency of several standard solvers such as the short-iterative Krylov-subspace propagator, the low-order Magnus integration method with the matrix polynomial (MP) or Chebyshev matrix polynomial (CMP) expansion of the evolution operator, and Runge-Kutta algorithm are assessed. Fast methods for summing MP and CMP are implemented to speed the calculation of the matrix exponential. It is found that the exponential propagators can tolerate large time step size and retain the computational accuracy whereas the Krylov-subspace algorithm is a little inferior for a larger time step size compared with the second-order Magnus integration method with the MP/CMP expansion of the evolution operator in both weak and intense fields. As an application, we calculate the absorption spectra of hydrogen-passivated silicon nanoparticles Si(29)H(x). The popular hybrid and generalized gradient approximation exchange-correlation functionals are applied. We find that the experimental spectra can be reproduced by using B3LYP and that the silicon particles with sizes of 1 nm and the optical excitations at 3.7, 4.0, and 4.6 eV may consist of 29 Si atoms surrounded by 24 hydrogen atoms.