actly in the framework of the time鄄dependent density functional theory (TDDFT) [14-15] , in which the central quantity is the exchange鄄correlation kernel, fxc, corresponding to the second order functional derivative of the xc energy functional with respect to electron density in a time鄄dependent framework. For many finite systems, even the simplest approximation to the xc kernel, the adiabatic local density approximation (ALDA), can already give very accurate optical excitation properties. But the ALDA鄄 TDDFT method, like its static counterpart, has serious limitations[16]. It significantly underestimates charge鄄transfer type excitation energies, and has difficulty in describing multi鄄electron excitations. For extended systems, the ALDA鄄TDDFT has more serious problems[17-18]. Recent years have seen promising progresses to overcome these difficulties in the TDDFT for solids [17-18], but substantial efforts are still needed. Furthermore, the TDDFT is formulated mainly to describe neutral (optical) excitation. Quasi鄄particle excitations, as probed by photoemission and inverse photoemission spectroscopy, can not be accessed straightforwardly in the TDDFT framework. Electronic quasi鄄particle excitations are best described by the many鄄body perturbation theory based on the one鄄body Green忆 s function [19-20] . The central quantity is the exchange鄄correlation self鄄energy, 撞xc, which is non鄄local, energy鄄dependent and non鄄 hermitian, and includes all non鄄classical electron鄄electron interaction effects. Exact 撞xc can be obtained by solving a set of complicated integro鄄differential equations, Hedin忆 s equations after Hedin [19], which is unfortunately out of reach even for the simplest systems like homogeneous electron gas (HEG). Therefore one has to resort to various approximations. The state鄄of鄄the鄄art approach is the GW approximation (GWA), which is the first order term in a systematic many鄄body perturbation expansion of 撞xc with respect to the screened Coulomb interaction W. In practice, a "best G best W" strategy is usually used, where the quasi鄄 particle energies are calculated as a first鄄order correction to some reference single鄄particle Hamiltonian H0, and both G and W are calculated using eigen鄄energies and eigen鄄functions of H0, hence called G0W0. The G0W0 method based on the LDA H0, denoted henceforth as G0W0@LDA, has become the method of choice for the description of quasi鄄particle band structures in weakly correlated solids [20]. However, its applications, and even its validity to d/ f鄄electron systems are still far from established. The difficulty first arises from the failure of the LDA for d/f鄄electron systems so that a perturbative treatment based on the LDA single鄄particle Hamiltonian is no longer adequate any more. Obviously the failure of G0W0 could be due to either the failure of LDA H0 as the starting point, or the GW approximation itself. On the other hand, the fully self鄄consistent GW, besides its formidable computational demand, is also problematic because a self鄄consistency without including higher order contributions beyond the GW approximation to the self鄄energy is internally inconsistent. The latter seems to be confirmed by recent investigations of homogeneous electron gas, where it was found that although the ground

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