To find an alternative for the prediction of excited state properties of large systems at a moderate cost, time-dependent density functional theory (TD-DFT) appears to be a promising candidate. TD-DFT is an extension of Kohn-Sham DFT and based on almost 35-years old Runge-Gross theorem which has been thoroughly reviewed in the literature . Almost 25 years ago, Casida developed a constructive linear-response formalism for TD-DFT, known as Casida equations but which we will refer to as the random-phase approximations (RPA), allowing to efficiently determine the solution of the TD-DFT equations, which are formulated in matrix equation involving the excitation and de-excitation matrices.