Linear-response time-dependent density-functional theory (TDDFT)is nowadays one of the most widely used method to compute molecular excitation energies thanks to its good cost versus accuracy ratio. The key object in TDDFT is the Hartree-exchange-correlation kernel which must describe the effects of the electron-electron interaction on the excitation energies of the system. Unfortunately the form of this kernel is unknown and the design of approximations remains a major challenge. Within the usual adiabatic semi-local approximations, although it reproduces correctly valence excitations, TDDFT is not able to describe properly Rydberg, charge-transfer or multiple excitations. Range separation of the electron-electron interaction allows one to mix rigorously density-functional methods at short range and wave-function or Green’s function methods at long range. When applied to the exchange kernel, the inclusion of the long-range Hartree-Fock exchange kernel already corrects most of TDDFT deficiencies as in particular the correct asymptotic behavior of the potential at long range is recovered. However multiple excitations are still missed by such a kernel as they need a frequency-dependent kernel in order to be captured which is prevented by the adiabatic approximation. In this talk, I will present several developments in range-separated time-dependent and time-independent density-functional theory to improve the treatment of such excitations. The effects of range separation are first assessed on the excitation energies of a partially-interacting system in an analytic and numerical study in order to provide guidelines for future developments of range-separated methods for excitation energy calculations . It is then applied on the exchange and correlation TDDFT kernels in a single-determinant approximation in which the long-range part of the correlation kernel vanishes . A long-range frequency-dependent second-order correlation kernel is then derived from the Bethe-Salpeter equation and added perturbatively to the range-separated TDDFT kernel in order to take into account the effects of double excitations.
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