Time-dependent density-functional theory (DFT) leads to a formally exact eigenvalue equation for determining excitation energies. In an adiabatic approximation, we have first calculated the lowest excitation energies for various neutral atoms and positively charged atomic ions, for comparison with experimental data. Then, to gain further insight, the time-dependent theory is reformulated by using the chemical potential equation of time-independent DFT. The central new quantity then appearing is a second functional derivative of the single-particle kinetic energy T_s. If chemical hardness can be treated as a correction to the term involving T_s, then further analytical progress is effected. Good numerical results testify to the usefulness of invoking the time-independent DFT within the present context.