Self-interaction error in density functional theory: a mean-field correction for molecules and large systems
Introduction
Density functional theory (DFT), combining good performances and low computational costs, has become an invaluable tool for chemists and physicists in understanding the electronic structure of atoms, molecules or solids and related properties [1]. In the framework of the Kohn–Sham (KS) approach to DFT, the quality of the results is strictly related to the functional used to evaluate the exchange and correlation energy, the only contribution that needs to be approximated in the expression of the total KS energy [2]. The research for improved approximations to this contribution has therefore become one of the main streams in theoretical DFT development (see for instance [3] and [4]). In such a quest for higher accuracy, some failures of the different models have been considered as “pathological”, that is intrinsic to the DFT approach itself. Among others, activation energies for SN2 and proton transfer reactions, dissociation energies of two center-three electron systems, ionization potentials and charge transfer systems, can be considered as representative examples [5], [6], [7], [8]. In many cases, these faults only depend on the approximate nature of the used functionals, which leads to the so-called self interaction (SI) error. This spurious effect arises from the interaction of an electron with itself, and it is related to Coulomb energy of the Kohn–Sham (KS) Hamiltonian which is not, in contrast to the Hartree–Fock approach, totally cancelled by the exchange contribution [2], [9]. One of the consequences is that the eigenvalue of the highest occupied KS orbital does not correspond to the ionization potential (IP), as it should. The SI problem has been clearly identified in the early DFT approaches and some solutions were proposed by Fermi and Amaldi [10] or by Slater [11]. More complex is, instead, the handling of SI in the framework of the KS approach. Some years ago Perdew and Zunger [12] have introduced a self-interaction correction (SIC), which is yet the only approach giving a correction for both exchange and correlation contributions. In fact, while some of the most recent correlation functionals are SI-free by construction, the handling of the exchange part is still troublesome [13], [14], [15].
The more effective recipe for the elimination of the SI error is the evaluation of an orbital dependent exchange-correlation potential, which is, in practical implementations, computationally excessively demanding, so that its use is so far restricted to small model systems or simplified Hamiltonians [16], [17]. For this reason, various attempts have been made to develop simplified SIC scheme, based mainly on mean field approximations, since they couple a reduced computational effort with an almost complete removing of the SI error. When they do not involve SIC pseudopotentials [18], [19], [20], [21], [22], the most successful approaches are those based on the optimized effective potential (OEP) [23], [24] and in particular that developed by Krieger, Li and Iafrate (KLI), which involves an integral equation for the averaged SIC field [25]. Even if such approach has been successfully applied in a number of different cases, including molecular or solid state problems [26], [27], [28], still it requires a significant amount of computer resources in order to evaluate the orbital-dependent Coulomb part [29].
We have recently implemented in a molecular code a simplified, yet effective, approach based on an average-density SIC (ADSIC) approximation, which takes into account the screening of both exchange-correlation and Coulomb contribution through a subtraction from the total density of a fraction proportional to 1/N, N being the total number of electron [30]. This simple, self consistent approach, first proposed by Legrand et al. [31] and applied to a jellium model is technically not expensive and, at the same time, still retains a number of theoretical features, like the correct behavior for the asymptotic potential and a variational formulation [31]. The first applications to molecular systems are quite promising, but some potentialities of ADSIC are still unexplored [30].
The aim of this paper is to investigate some of the limits of such an approach in the field of molecular applications. To this end, the evaluation of the vertical ionization potentials (IPs) for selected test cases, ranging from atoms to large conjugate systems, has been chosen as a difficult playground. Although the first IPs could be directly calculated using the DFT extension of the Koopmans theorem [25], [32], or more exactly of the Janak theorem [33], the negative of the KS HOMO energy is too small with respect to the experimental values [34], even if the shift, usually of several electronvolts, is rather constant [35], [36], [37], [38], [39]. It has been demonstrated that with the self interaction correction included, the HOMO energies are much closer to the first IPs, and that this effect is directly related to the SI error [40]. In this respect, IPs belong to key properties to validate any new SI corrections.
Section snippets
Theoretical background
In the Kohn–Sham (KS) approach to DFT the total exact energy can be written as [2]:where are the spin orbitals and ρσ are the total density of spin σ. The first term in Eq. (1) is the kinetic energy of a system of non-interacting particle, the second is the Coulomb interaction and the third is the interaction energy between the electron density ρ(r) and the external potential ν(r). These terms are all known exactly, which is not the case for
Computational details
We have implemented the ADSIC approach in one of the development versions of the Gaussian 03 code [64] for the generalized gradient approximation (GGA) for the exchange based on the Becke 88 functional [65]. This contribution has been next coupled with the GGA correlation of Lee, Yang and Parr [66], a functional by construction SI-free, so to have the BLYP approach.
All the molecules have been optimized using the 6-311G(d,p) basis set and energy evaluation have been done with the same basis,
Results and discussion
As first test, we have computed the IPs, from the highest orbital eigenvalue, for some atomic systems, even if we do not expect that the ADSIC approach performs particularly well on such systems, due to its average hypothesis. For this reason, only 10 spherical atoms, all belonging to the first and second group of the periodic table (configuration [core] n1 or [core] n2) have been chosen. The results are reported in Fig. 1 as function of the number of electrons, together with the experimental
Conclusions
In this paper, we have presented the validation of a simple approach to correct the self interaction error present in the common approximate exchange-correlation functionals used in density functional theory. This model rests on an average density self-interaction correction (ADSIC), so that the main advantages of the method with respect to other corrections are its simplicity and its favorable scaling with the size of the system. At the same time, it retains a number of theoretical features,
Note added in proofs
Eq. , of this paper were wrongly reported in Ref. [30] (one missing term), but correctly programmed. The correct form is given here.
Acknowledgements
Authors thank Thomas Heine (University of Dresden) help concerning fullerene structures. I.C. and C.A. also thank CNRS for a financial support from the ACI “Jeune Equipe 2002” project. This work has also been carried out within the framework of the Cost Action D26 “Integrative Computational Chemistry” (action n. D26/0013/02) and a support from the CNRS GdR “DFT”. The CINES is acknowledged for a grant of time (project cpt2130).
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