1 P. Hohenberg and W. Kohn Phys. Rev. 136 , B864 (1964). 2 

W. Kohn and L. J. Sham Phys. Rev. 140 , A1133 (1965). 3 

M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34 , 5390 (1986). 4 

J. P. Perdew and A. Zunger, Phys. Rev. B 23 , 5048 (1981). 5 

J. Paier, M. Marsman, K. Hummer, G. Kresse, I. C. Ger- ber, and J. G. Angyan, J. Chem. Phys. 124 , 154709 (2006) 6 

J. Heyd, J. E. Peralta, G. E. Scuseria, and R. L. Martin, J. Chem. Phys. 123 , 174101 (2005). 7 

C. Adamo and V. Barone, J. Chem. Phys. 110 , 6158 (1999) 8 

A. D. Becke, J. Chem. Phys. 98 , 5648 (1993) 9 

C. Lee, W. Yang and R. G. Parr, Phys. Rev. B 37 , 785 (1988) 10 

J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118 , 8207 (2003) 11 

According to Janak’s theorem $^{12}$, for finite systems the po- sition of the HOMO in exact Kohn-Sham theory should coincide with the ionization potential. However, for practi- cal exchange-correlation functionals this is far from being the case $^{4,13}$. 12 

J. F. Jarnak, Phys. Rev. B 18 , 7165 (1978). 13 

C. D. Pemmaraju, T. Archer, D. Sanchez-Portal, and S. Sanvito, Phys. Rev. B 75 , 045101 (2007) 14 

J. Q. Sun and R. J. Bartlett, J. Chem. Phys. 104 , 8553 (1996) 15 

L. Dagens and F. Perrot, Phys. Rev. B 5 , 641 (1972) 16 

Solid State Physics , G. Grosso and G. P. Parravicini, Cam- bridge University Press, Cambridge 2000 17 

L. Hedin, Phys. Rev. 139 , A796 (1965) 18 

F. Aryasetiawan and O. Gunnarson, Rep. Prog. Phys. 61 , 237 (1998) 19 

W. G. Aulbur, L. Jonsson, and J. Wilkins, in Solid State Physics , edited by H. Ehrenreich and F. Saepen (Academic Press, New York 2000), Vol. 54, 1. 20 

G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74 , 601 (2002) 21 

J. C. Grossman, M. Rohlfing, L. Mitas, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 86 , 472 (2001). 22 

P. H. Hahn, W. G. Schmidt, and F. Bechstedt Phys. Rev. B 72 , 245425 (2005) 23 

T. Niehaus, M. Rohlfing, F. D. Sala, A. Di Carlo, and T. Frauenhaim, Phys. Rev. A 71 , 022508 (2005) 24 

A. Stan, N. E. Dahlen, and R. van Leeuwen, Europhys. Lett. 76 , 298 (2006) 25 

C. D. Spataru, S. Ismail-Beigi, X. L. Benedict, and S. G. Louie, Phys. Rev. Lett. 92 , 077402 (2004) 26 

P. E. Trevisanutto, C. Giorgetti, L. Reining, M. Ladisa, and V. Olevano, Phys. Rev. Lett. 101 , 226405 (2008) 27 

M. L. Tiago and J. R. Chelikowsky, Phys. Rev. B 73 , 205334 (2006) 28 

J. B. Neaton, M. S. Hybertsen, and S. G. Louie Phys. Rev. Lett. 97 , 216405 (2006) 29 

K. S. Thygesen and A. Rubio, Phys. Rev. Lett. 102 , 046802 (2009) 30 

C. Freysoldt and P. Rinke and M. Scheffler, Phys. Rev. Lett. 103 , 056803 (2009) 

31 J. M. Garcia-Lastra, C. Rostgaard, A. Rubio, and K. S. Thygesen Phys. Rev. B 80 , 245427 (2009) 32 

K. S. Thygesen and A. Rubio, J. Chem. Phys. 126 , 091101 (2007). 33 

K. S. Thygesen, Phys. Rev. Lett. 100 , 166804 (2008) 34 

P. My¨ ohanen and A. Stan and G. Stefanucci and R. van Leeuwen, Euro. Phys. Lett. 84 , 67001 (2008) 35 

C. D. Spataru and M. S. Hybertsen and S. G. Louie and A. J. Millis, Phys. Rev. B 79 , 155110 (2009) 36 

M. P. von Friesen and C. Verdozzi and C.-O. Almbladh, Phys. Rev. Lett. 103 , 176404 (2009) 37 

K. Kaasbjerg and K. S. Thygesen (submitted) 38 

B. Holm and U. von Barth Phys. Rev. B 57 , 2108 (1998) 39 

M. Usuda, N. Hamada, T. Kotani, and M. van Schifl- gaarde, Phys. Rev. B 66 , 125101 (2002) 40 

W. Ku and A. G. Eguiluz, Phys. Rev. Lett. 89 , 126401 (2002) 41 

M. van Schiflgaarde, T. Kotani, and S. V. Faleev, Phys. Rev. B 74 , 245125 (2006) 42 

M. Shishkin and G. Kresse, Phys. Rev. B 74 , 035101 (2006) 43 

P. Rinke, A. Qteish, J. Neugebauer, C. Freysoldt, and M. Scheffler New J. Phys. 7 126 (2005) 44 

X. Qian, J. Li, L. Qi, C.-Z. Wang, T.-L. Chan, Y.-X. Yao, K.-M. Ho, and S. Yip, Phys. Rev. B 78 , 245112 (2008) 45 

J. J. Mortensen, L. B. Hansen, K. W. Jacobsen, Phys. Rev. B 71 , 035109 (2005) 46 

A. H. Larsen, M. Vanin, J. J. Mortensen, K. S. Thygesen, and K. W. Jacobsen Phys. Rev. B 80 , 195112 (2009) 47 

K. S. Thygesen and A. Rubio, Phys. Rev. B 77 , 115333 (2008). 48 

K. S. Thygesen, Phys. Rev. B 73 , 035309 (2006) 49 

H. Haug and A. -P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors , Springer (1998) 50 

R.van Leeuwen, N.E.Dahlen, G.Stefanucci, C.- O.Almbladh, and U.von Barth, Lectures Notes in Physics vol. 706 (Springer) 51 

A. Stan, N. E. Dahlen, and R. van Leeuwen, J Chem. Phys. 130 , 114105 (2009) 52 

F. Aryasetiawan and O. Gunnarsson, Phys. Rev. B 49 , 16214 (1994) 53 

M. Walter, H. H¨ akkinen, L. Lehtovaara, M. Puska, J. Enkovaara, C. Rostgaard, and J. J. Mortensen, J Chem. Phys. 128 , 244101 (2009) 54 

P. E. Bl¨ ochl, Phys. Rev. B 50 , 17953 (1994) 55 

E. Engel, Phys. Rev. B 80 , 161205(R) (2009) 56 

L. A. Curtiss, K. Raghavachari, P. Redfern, and J. A. Pople, J. Chem. Phys. 106 , 1063 (1997). 57 

J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 , 3865 (1996) 58 

M. van Schilfgaarde, T. Kotani, and S. Faleev, Phys. Rev. Lett. 96 , 226402 (2006) 59 

NIST Computational Chemistry Comparison and Bench- mark Database, NIST Standard Reference Database Num- ber 101 Release 14, Sept 2006, Editor: Russell D. Johnson III http://srdata.nist.gov/cccbdb 

