Role of a discontinuous functional in linear response TDDFT

An interesting consequence for density functional theory (DFT) arises when considering ensembles with densities integrating to fractional particle number. The ground-state energy as a function of particle number consists of straight-line segments with cusps at the integers and, as a consequence, the corresponding exchange-correlation (XC) potential jumps discontinuously. This feature of the exact theory turns out to be important to incorporate in approximate functionals in order to obtain, e.g, accurate band-gaps of solids and correct molecular dissociation limits. How the same property is reflected in the XC kernel, defined as the functional derivative of the XC potential with respect to the density, has recently been investigated [Phys. Rev. A 85, 022514 (2012)]. The XC kernel is of fundamental importance in TDDFT as it gives access to the particle conserving excitation spectrum. The discontinuity of the XC kernel is shown to consist of a space -and frequency dependent function, which asymptotically diverges, a behavior which turns out to be crucial for the description of long-range charge-transfer excitations. Using many-body perturbation theory we derive density functionals which contain the discontinuity. Whereas the discontinuity at even integers enters already at the simplest level of approximation the one at odd integers remains a challenge.