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In this work we derive closed expressions for the head of the frequency-dependent microscopic polarizability matrix in the projector-augmented wave ͑PAW͒ methodology. Contrary to previous applications, the longitudinal expression is utilized, resulting in dielectric properties that are largely independent of the applied potentials. The improved accuracy of the present approach is demonstrated by comparing the longitudinal and transversal expressions of the polarizability matrix for a number of cubic semiconductors and one insulator, i.e., Si, SiC, AlP, GaAs, and diamond ͑C͒, respectively. The methodology is readily extendable to more complicated nonlocal Hamiltonians or to the calculation of the macroscopic dielectric matrix including local field effects in the random phase or density functional approximation, which is demonstrated for the previously mentioned model systems. Furthermore, density functional perturbation theory is extended to the PAW method, and the respective results are compared to those obtained by summation over the conduction band states.

We present a comparative full-potential study of generalized Kohn-Sham schemes (gKS) with explicit focus on their suitability as starting point for the solution of the quasiparticle equation. We compare $G_0W_0$ quasiparticle band structures calculated upon LDA, sX, HSE03, PBE0, and HF functionals for exchange and correlation (XC) for Si, InN and ZnO. Furthermore, the HSE03 functional is studied and compared to the GGA for 15 non-metallic materials for its use as a starting point in the calculation of quasiparticle excitation energies. For this case, also the effects of selfconsistency in the $GW$ self-energy are analysed. It is shown that the use of a gKS scheme as a starting point for a perturbative QP correction can improve upon the deficiencies found for LDA or GGA staring points for compounds with shallow $d$ bands. For these solids, the order of the valence and conduction bands is often inverted using local or semi-local approximations for XC, which makes perturbative $G_0W_0$ calculations unreliable. The use of a gKS starting point allows for the calculation of fairly accurate band gaps even in these difficult cases, and generally single-shot $G_0W_0$ calculations following calculations using the HSE03 functional are very close to experiment

The phonon dispersion relations of diamond and graphite are calculated using an ab initio force constant method. The force constants are calculated via a self-consistent supercell approach in the local-density approximation in terms of the Hellmann-Feynman forces induced by the displacement of a single atom in the supercell. For diamond our ab initio results are in very good agreement with the neutron inelastic scattering and Raman data. For graphite we find good agreement with the neutron data for the low-energy modes as well as with the reflections electron energy loss spectroscopy (REELS) and optical data at higher energies. Significant differences to the predictions of semi-empirical models appear.