Density matrix based time-dependent density functional theory and the solution of its linear response in real time domain

A density matrix based time-dependent density functional theory is extended in the present work. Chebyshev expansion is introduced to propagate the linear response of the reduced single-electron density matrix upon the application of a time-domain delta-type external potential. The Chebyshev expansion method is more efficient and accurate than the previous fourth-order Runge-Kutta method and removes a numerical divergence problem. The discrete Fourier transformation and filter diagonalization of the first-order dipole moment are implemented to determine the excited state energies. It is found that the filter diagonalization leads to highly accurate values for the excited state energies. Finally, the density matrix based time-dependent density functional is generalized to calculate the energies of singlet-triplet excitations.