  This paper presents some absorbing boundary conditions (ABC) for simulations
based on the time-dependent density-functional theory (TDDFT). The boundary
conditions are expressed in terms of the elements of the density-matrix, and it
is derived from the full model over a much larger domain. To make the
implementation much more efficient, several approximations for the convolution
integral will be constructed with guaranteed stability. These approximations
lead to modified density-matrix equations at the boundary. The effectiveness is
examined via numerical tests.
