1. TDDFT
Prof. Dr. E.K.U. Gross
Prof. Dr. Mark Casida

18. From these papers, you will learn that a TDDFT calculation of electronic excitation energies start first from a usual ground-state calculation, with a chosen XC functional. Such a calculation produces a spectrum of Kohn-Sham electronic energies. It is widely known that differences between occupied and unoccupied Kohn-Sham electronic energies resemble excitation energies (the difference in energy between an excited state and the ground state), although there is no real theoretical justification for this similarity.
These differences between Kohn-Sham electronic energies are the starting point of Casida's approach : in the framework of TDDFT, their square give the main contribution to the diagonal part of a matrix, whose eigenvalues will be the square of the seeked excitation energies. One has to add to the diagonal matrix made from the squares of Kohn-Sham energy differences, a coupling matrix, whose elements are four-wavefunction integrals of the Coulomb and Exchange-Correlation kernel. The exchange-correlation kernel contribution will differ in the spin-singlet and in the spin-triplet states, this being the only difference between spin-singlet and spin-triplet states. See Eqs.(1.3) and (1.4) of CasidaIJQC1998, and Eqs.(1-2) of Vasiliev1999
The construction of the coupling matrix can be done on the basis of an exchange-correlation kernel that is derived from the exchange-correlation functional used for the ground-state, but this is not a requirement of the theory, since such a correspondance only holds for the exact functional. In practice, the approximation to the XC potential and the one to the XC kernel are often different. See section III of CasidaJCP1998.
A big drawback of the currently known XC potentials and XC kernels is observed when the system is infinite in at least one direction (e.g. polymers, slabs, or solids). In this case, the addition of the coupling matrix is unable to shift the edges of the Kohn-Sham band structure (each four-wavefunction integral becomes too small). There is only a redistribution of the oscillator strengths. In particular, the DFT band gap problem is NOT solved by TDDFT. Also, the Casida's approach relies on the discreteness of the Kohn-Sham spectrum.
Thus, the TDDFT approach to electronic excitation energies in ABINIT is ONLY valid for finite systems (atoms, molecules, clusters). Actually, only one k-point can be used, and a "box center" must be defined, close to the center of gravity of the system.
The Casida formalism also gives access to the oscillator strengths, needed to obtain the frequency-dependent polarizability, and corresponding optical spectrum. In the ABINIT implementation, the oscillators strengths are given as a second-rank tensor, in cartesian coordinates, as well as the average over all directions usually used for molecules and clusters. It is left to the user to generate the polarisability spectrum, according to e.g. Eq.(1.2) of CasidaIJQC1998.
One can also combine the ground state total energy with the electronic excitation energies to obtain Born-Oppenheimer potential energy curves for excited states. This is illustrated for formaldehyde in CasidaIJQC1998.
Given its simplicity, and the relatively modest CPU cost of this type of calculation, Casida's approach enjoys a wide popularity. There has been hundreds of papers published on the basis of methodology. Still, its accuracy might be below the expectations, as you will see. As often, test this method to see if it suits your needs, and read the recent litterature ...