Towards a unified description of ground and excited state properties: \textit{GW} vs RPA and beyond

In the quest for finding an ``optimal'' first principles electronic structure method, that combines accuracy and tractability with transferability across different chemical environments and dimensionalities (e.g. molecules, wires/tubes, surfaces, solids), the treatment of exchange and correlation in terms of ``exact-exchange plus correlation in the random-phase approximation (EX+cRPA)'' offers a promising avenue. Likewise one can express the same level of theory in the Green's function context through the $GW$ approximation, which has the additional advantage that quasiparticle spectra as measured by direct and inverse photoemission become accessible. In this talk I will contrast both approaches and present the latest results from our continuous assessment. We find that self-consistent (sc) $GW$ provides excellent charge densities [1], which is particularly important for charge transfer systems [2]. Spectral properties for closed shell molecules are generally in good agreement with photoemission spectra, although a judicial choice of the starting point in perturbative $G_0W_0$ calculations can outperform scGW [1,3]. Other ground state properties do not improve over EX+cRPA calculations [1]. EX+cRPA, on the other hand, provides a good description of the ground state [4] even for challenging cases like chemical reaction barrier heights [5] and the $f$-electron metal cerium [6]. The notorious underbinding of EX+cRPA can be corrected by going beyond RPA to renormalised second order perturbation theory (rPT2) [7] that gives the overall most balanced performance. I will also discuss the associated rPT2 self-energy that goes beyond $GW$.\\[4pt] [1] F. Caruso, P. Rinke, X. Ren, M. Scheffler, and A. Rubio, Phys. Rev. B {\bf 86}, 081102(R) (2012), {\it ibid} Phys. Rev. B {\bf 88}, 075105 (2013).\\[0pt] [2] F. Caruso, V. Atalla, A. Rubio, M. Scheffler, and P. Rinke, Phys. Rev. B {\bf 90}, 085141 (2014).\\[0pt] [3] N. Marom, F. Caruso, X. Ren, O. Hofmann, T. K\"orzd\"orfer, J. R. Chelikowsky, A. Rubio, M. Scheffler, and P. Rinke, Phys. Rev. B {\bf 86}, 245127 (2012).\\[0pt] [4] X. Ren, P. Rinke, C. Joas, and M. Scheffler, J. Mat. Sci. {\bf 47}, 7447 (2012).\\[0pt] [5] J. Paier, X. Ren, P. Rinke, G. E. Scuseria, A. Gr\"uneis, G. Kresse, and M. Scheffler, New J.\ Phys.\ {\bf 14}, 043002 (2012).\\[0pt] [6] M. Casadei, X. Ren, P. Rinke, A. Rubio, and M. Scheffler, Phys. Rev. Lett. \textbf{109}, 14642 (2012).\\[0pt] [7] X. Ren, P. Rinke, G. E. Scuseria, and M. Scheffler, Phys. Rev. B {\bf 88}, 035120 (2013)