Effect of the self-interaction error for three-electron bonds: On the development of new exchange-correlation functionals Ju ¨rgen Gra ¨fenstein, Elfi Kraka and Dieter Cremer* Theoretical Chemistry, Go ¨ teborg University, Box 460, SE-405 30, Go ¨ teborg, Sweden Received 24th September 2003, Accepted 12th January 2004 F|rst published as an Advance Article on the web 26th February 2004 The dissociation behavior as well as the equilibrium properties of radical cations with three-electron bonds, namely He 2 þ ,N 2 H 6 þ ,O 2 H 4 þ ,F 2 H 2 þ , and Ne 2 þ are investigated using standard and self-interaction- corrected density functional theory (SIC-DFT) in connection with a variety of pure and hybrid exchange- correlation (XC) functionals. The impact of the self-interaction error (SIE) on the results of standard DFT is analyzed considering the individual orbital contributions to the SIE, the dependence of the SIE on the separation distance between the dissociation fragments, and its impact on the equilibrium properties of 2–6. A local analysis of the SIE in terms of exact and DFT exchange holes reveals that the SIE mimics not only non-dynamic but also an increasing amount of dynamic electron correlation effects as the number of valence electrons is enlarged. Standard DFT describes the dissociation of three-electron bonds qualitatively incorrectly. This can be traced back in the first instance to the SIE of the bonding b electron, which mimics a spurious long- range correlation with a non-existing delocalized a electron in the same bond. A comparison of the covalent (symmetric) and ionic (symmetry-broken) state of radical cations 2–6 at large interaction distances provides further insight in the inconsistencies of the DFT description: (i) Not only the SIE but also the approximate description of the interelectronic exchange contributes to the incorrect description of the dissociation. (ii) Dissociating three-electron bonds show a specific form of long-range correlation effects, which is neither accounted for by standard DFT, SIC-DFT nor Hartree–Fock theory. Indeed, SIC-DFT provides a qualitatively better description of the dissociation of radical cations, however in general a poor performance when describing equilibrium properties. There is no need for SIC-DFT methods. Instead, there is need for XC functionals with exact exchange and long-range correlation effects (e.g. mimicked by the exchange SIE) absorbed in the correlation functional. Implications of our findings for the construction of new density functionals are discussed. 1. Introduction The development of improved exchange-correlation (XC) functionals for Kohn–Sham (KS) density functional theory (DFT) 1,2 has to consider the self-interaction error (SIE) of approximate X- and C-functionals. 3,4 In recent work, we have described the consequences and the impact of the SIE on a DFT description of molecules both in their equilibrium geome- try and in the situation of bond breaking. 5–11 For this purpose, we have developed as descriptive tools an energy decomposi- tion analysis based on the Perdew–Zunger approach 3 for self-interaction corrected DFT (SIC-DFT), 5 the difference den- sity analysis reflecting the impact of the SIE, 5–7 and a decom- position analysis of the exchange hole, which makes it possible to relate the SIE to the correlation effects described in standard KS DFT. 8–10 Our work is in line with many investigations focusing on the SIE, 12–27 which had their forerunners already in the thirties in the work of Fermi and Amaldi 28 and in the fifties. 29 Our investigations revealed that the SIE of approxi- mate X functionals accounts for non-specified short-range and long-range electron correlation effects, which can improve results of a standard DFT calculation and in general increase the stability of the KS-DFT solutions. This was also found or anticipated by authors such as Slater 29 (in connection with the discussion of charge transfer complexes), Becke 30 (when deriving the hybrid functionals) and in particular Baerends and co-workers who presented the first in detail discussion on this aspect of the SIE. 23 A quantification of the extra-corre- lation effects mimicked especially by the X-functional was possible by the combination of SIC-DFT calculations with difference density and exchange hole studies carried out in our previous work. 5–10 The ambivalent role of the SIE, namely being an error and at the same time mimicking useful correlation effects, has to be considered when correcting standard DFT: SIC-DFT is not necessarily a better and more accurate method than stan- dard DFT. On the other hand, SIC-DFT seems to be unavoid- able if odd electron systems are investigated. 31–40 It is well known that standard DFT leads to an erroneous description of the dissociation of radical cations. For example, standard DFT predicts for the dissociation of H 2 þ an artificial transition state and a dissociation limit being just 14 kcal mol 1 above the equilibrium energy. Similar failures have been reported for the dissociation of other radical cations. 31–40 Consequently, much effort has been focused on the development of SIE-free DFT methods, which was either based on the Perdew–Zunger approach 3 or the idea to mix in exact exchange any time a large SIE is indicated by a suitable antenna as for example the von Weizsa ¨cker kinetic energy density. 41,42 In this work, we will show that SIE-free DFT leads to a number of new problems, which in general deteriorate the per- formance of DFT. For this purpose, we will investigate radical cations with three-electron bonds, which are known to present the most drastic SIE problems in DFT. We will describe these systems with different XC functionals ranging from the local density approximation (LDA) to modern hybrid functional theory and using both standard DFT and SIC-DFT. The accu- racy of the various methods will be tested against experimental data (if available) or high level ab initio calculations. The cal- culated SIEs will be first decomposed in a pure electronic effect PCCP www.rsc.org/pccp RESEARCH PAPER 1096 Phys. Chem. Chem. Phys. , 2004, 6, 1096–1112 This journal is Q The Owner Societies 2004 DOI: 10.1039/b311840a