Regional Self-Interaction Correction of Density Functional Theory

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1 Regional Self-Interaction Correction of Density Functional Theory TAKAO TSUNEDA, MUNEAKI KAMIYA, KIMIHIKO HIRAO Department of Applied Chemistry, Graduate School of Engineering, The University of Tokyo, Tokyo, Japan Received 21 October 2002; Accepted 27 January 2003 Abstract: We propose a new simple scheme for self-interaction correction (SIC) of exchange functionals in the density functional theory. In the new scheme, exchange energies are corrected by substituting exchange self-interactions for exchange functionals in regions of self-interaction. To classify the regions of self-interaction, we take advantage of the property of the total kinetic energy density approaching the Weizsäcker density in the case of electrons in isolated orbitals. The scheme differs from conventional SIC methods in that it produces optimized molecular structures. Applying the scheme to the calculation of reaction energy barriers showed that it provides a clear improvement in cases where the barriers are underestimated by conventional pure functionals. In particular, we found that this scheme even reproduces a transition state that is not given by pure functionals Wiley Periodicals, Inc. J Comput Chem 24: , 2003 Key words: density functional theory; self-interaction correction; exchange functional; reaction barrier energies; kinetic energy density Introduction Density functional theory (DFT) has become a major methodology in the field of computational chemistry. In DFT, a wide variety of electronic states are determined by solving the nonlinear Kohn Sham equation. 1 3 Determining a given set of physico-chemical molecular properties by DFT calculations requires far less storage capacity and time in computation than high-level ab initio molecular orbital methods. DFT has thus come to be applied in a wide range of investigations and particularly to the investigations of large molecular systems. In DFT, exchange-correlation functional plays a major role in the accuracy of the calculated results, because this is usually the only approximated and adjustable element in the process of calculation. Rather sophisticated exchange-correlation functionals have been developed over the last few decades. However, most conventional functionals still include certain common unsettled problems. Inherent self-interaction error (SIE) is one of the most serious problems with exchange-correlation functionals. 4 With the Hartree Fock (HF) method, the self-interaction of electrons should essentially be removed in such a way that the self-interaction components of the Coulomb and exchange potentials cancel each other out for each spin-orbital. Because exchange functionals have, in general, been developed as a single-electron approximation, SIE inevitably remains in a two-electron Coulomb interaction. 5 Past DFT studies have reported that SIE may be responsible for the finding of overly low barrier energies of chemical reactions, 5 overly narrow band-gaps of semiconductors and insulators, 4 and so forth. Various schemes for self-interaction correction (SIC) have been suggested as ways of ridding exchange functionals of the SIE. 4,6,7 Perdew and Zunger were the first to tackle the elimination of the SIE from exchange functionals. 4 They suggested a scheme for the application of SIC to occupied orbitals where the selfinteraction components of the Coulomb and exchange energies are simply subtracted from the total exchange-correlation energy, E SIC xc, E xc, i, and potential, 2 1 i R i R d R R 3 Rd 3 R E xc i,0, (1) Correspondence to: T. Tsuneda; tune@qcl.t.u-tokyo.ac.jp Contract/grant sponsors: Grant-in-Aid for Scientific Research on a Priority Area; Genesis Research Institute, Inc.; and Grant-in-Aid from the Japan Atomic Energy Research Institute 2003 Wiley Periodicals, Inc.

2 Regional SIC of Density Functional Theory 1593 v i,sic xc R E SIC xc i R E xc, i R i R R R d3 R E xc i,0, i R where is the density of the -spin electron, and i is the i-th orbital component of. Atomic units are used in this paper ( e 2 m 1, energies are in hartree and distances in bohr). However, the self-interaction energy is not identically determined in this scheme because of the degrees of freedom in the unitary transformation of orbitals. To determine the self-interaction energy, past DFT studies have chiefly employed two techniques that are respectively based on orbital localization and on the optimized effective potential method; SIC is perturbatively applied after the localization of the occupied orbitals in the former scheme, 6 while the SIC is calculated after the transformation of functionals to an orbital-dependent form in the latter scheme. 7 However, either of these schemes may lead to some serious problems. The orbital localization may incorporate an obscure effect in the calculated properties, and the orbital-dependent transformation may greatly increase the time taken by computation. Above all, however, both schemes generally make the determination of transition-state structures impossible. In this article, we propose a new SIC scheme that operates by selectively improving exchange functionals in regions of selfinteraction. In a previous study, 8 the region of self-interaction was defined as the area where the density matrix is well-approximated by the product of the square roots of spin-density. On the basis of this definition, we have developed a new technique for the classification of self-interaction regions to identify those where the exchange functionals should be improved; this technique is described in the next section. On this basis, in turn, the exchange functionals in regions of self-interaction are substituted for the exact exchange self-interactions of hydrogenic wavefunctions. This is presented in third section. Then, this new SIC scheme is examined by applying it in the calculation of reaction energy barriers that are underestimated by conventional DFTs. Physical Connections in Regions of Self-Interaction Although the self-interaction error in conventional exchange functionals has been the source of some controversy, 4 7 little attention has thus far been given to the physical background of this error. In this section, we briefly examine physical connections that are based on the self-interaction of electrons. Conventional exchange functionals generally produce incorrect results for long-range asymptotic behavior. 9,10 In the far-field with respect to the nuclei of a molecule, the exchange energy per -spin electron, e x, should reproduce this asymptotic behavior 9,10 : (2) R 3 e x O 1 2 R, (3) where R is the radius from a nucleus and e x is defined as E x e x d 3 R. The important point to note here is that this asymptotic relation is derived from the long-range limiting case of the density matrix 2,11 : P 1 R r 2, R r 2 1/ 2 R r 2 1/ 2 R r 2, (4) where R is the radius vector from a nucleus to a center of paired electrons, and r is the position vector from one electron to another paired electron. This density matrix leads to the exact formula for the distribution of the density of exchange self-interaction energy for a well-separated orbital, R 3 x R O 2 1 r R 2 R r 2 r d 3 r, (5) where x is defined by E x x d 3 R. Because selfinteraction may play an important role, especially in the case of a well-separated orbital, we may say that eq. (4) is essentially the central relation for an electron that is subjected to self-interaction. By using the density matrix of eq. (4), it is easily confirmed that the kinetic energy density,, approaches the Weizsäcker density, W2 : 2 P 1 R r 2, R r 2 r 0 3 W 2 4. (6) We can also make sure that the diagonal term of the second-order density matrix for parallel-spin pairs, 12 P 2, reduces to zero in region of self-interaction, i.e., that P 2 R r 2 2, R r 1 2 R r 2 R 2 r P 1 R r 2 2, R r (7) This means that parallel-spin correlation energy density c also vanishes in this region, 12 c 3 0, (8) where c is defined by the correlation energy for parallel-spin pairs, E c c d 3 R. Let us now look at the energy of exchange self-interaction in detail. The electron density of molecule,, has been shown to be approximable in the far-field from nuclei (the tail of the density distribution) and near a nucleus (1s orbital is dominant), 13 by this exponential: 3 2 e 2 R. (9) Tao recently assumed that electrons in these regions have hydrogenic wave functions,

3 1594 Tsuneda, Kamiya, and Hirao Vol. 24, No. 13 Journal of Computational Chemistry The first question that we must consider is how we should distinguish the regions of self-interaction in molecule. It is interesting to note that the Weizsäcker kinetic energy density, W, gives the total density, total, for the density matrices for electrons that are subject to self-interaction, eq. (4). Moreover, W has been rigorously proven to provide a lower bound to total. 2 We can therefore clarify the regions where self-interaction applies by using the ratio of W to total : t W total 0 t 1. (13) Figure 1 displays the contour map of t for the formaldehyde molecule. Because t approaches 1 in the regions of self-interaction, the figure indicates that the regions of self-interaction may be concentrated in the far- and near-fields with respect to the nuclei. To employ t in eq. (13) as an indicator of self-interaction, we introduce a partition function f that has a parameter a: Figure 1. The contour map of t W / (0 t 1) for the formaldehyde molecule. Contour lines are used with t 0.1, 0.2,..., 0.8, 0.9, and The self-interaction regions correspond to the white ones, because t approaches one as the regions become bright. 3 1/ 2 e R (10) to arrive at this expression for the per electron exchange energy 14 : e x R 1 2R 1 1 R e 2 R, (11) where e x is defined as E x e x d 3 R. It is easy to prove that eq. (11) has the following long-range limit (R 3 ): R 3 e x O 1 2 R, (12) which is identical with the actual long-range asymptotic behavior as given in eq. (3). This correct behavior in the limiting case of R 3 may support the assumption used to obtain eq. (10). It also confirms that the Weizsäcker kinetic energy only becomes exact when the system is well described by orbitals that are strongly localized in separate regions, as is the case for the 1s orbital. 2 We can therefore expect that the regions of self-interaction may be limited to 1s-core orbitals and to those density-distribution tails that are closely approximated by eq. (10). f erf 5 t a 1 a 0 a 1. (14) This f allows us to construct the SIC exchange energy per -spin electron, e x, as a combination of self-interaction (SI) and exchange functional (DFT) parts: e x R f e SI x R 1 f e DFT x R. (15) Figure 2 shows the dependence of f on t. As can be seen in the figure, f bisects the region by t a; t a and t a correspond to the DFT and SI parts, respectively. We should note that parameter a must be kept at a certain large value (e.g., a 0.95), because it is easily confirmed that e SI x only becomes exact near t 1. According to Tao s argument, which was mentioned in the SI previous section, we use the spin-polarized form of eq. (11) for x in eq. (15): e SI x R 1 2R 1 1 R e 2 R. (16) A New Self-Interaction Correction Scheme We now develop a new self-interaction correction (SIC) scheme for exchange functionals on the basis of the physical connections in the previous section. Figure 2. The form of f (1/2)(1 erf[5(t a)/(1 a)]) in terms of t (0 t 1).

4 Regional SIC of Density Functional Theory 1595 Figure 3. Exchange self-interaction energies of atoms, H through Ar, for three types of parameter a in eq. (14) in hartree. where R is the distance from each atomic nucleus. A second question remains: how should we determine the value of in eq. (16)? This question may be answered by assuming that the electron density is always well approximated in regions of self-interaction by eq. (9). This assumption gives us the following expression for : 2. (17) SI This is expected to provide a reliable e x for the linear combination of Gaussian-type functions that adequately describes the corresponding Slater-type function, i i exp( i R 2 ) as well as for the Slater-type function itself, exp( R). Calculations In this section, we numerically examine the new SIC scheme for various exchange functionals by calculating the exchange energies of atoms, the barrier energies in H H 2 3 H 2 H, and some other noteworthy reactions. The Effect of Self-Interaction Correction on Atomic Exchange Energies Firstly, the new SIC scheme is applied to the calculation of the exchange energies of the atoms from H through Ar by using the Clementi HF Slater-type orbitals. 16 For numerical integration, we use the Euler Maclaurin quadrature 17,18 with 50-point radial grids and the Lebedev quadrature 19 with 194-point angular grids. As exact values, we adopted the exchange energies as calculated by the numerical HF method. 21 In Figure 3, calculated self-interaction components of atomic exchange energies are plotted for three types of parameter a in eq. (14); a 0.90, 0.95, and For H and He atoms, the SIC energy corresponds to the total exchange energy, because the electron distributions of these atoms are dominated by the regions of self-interaction (i.e., t 1). The new SIC scheme correctly provides the exact exchange energy of the H atom, 5/16 hartree. For all three values of a, the SIC energy monotonically increases to the Be atom. As we continue through heavier atoms, however, the energy rapidly disappears in the case of a 0.99, gradually decreases in the case of a 0.95, and approaches to a constant in the case of a In all three cases, the proportion of the self-interaction component rapidly decreases as the atomic weight increases. This indicates that the regions of self-interaction in the 1s core orbital and in the tails of the density distribution, may be encroached upon in the case of heavier atoms. This effect may be responsible for the accuracy of DFT results in calculations for molecules that contains heavy metal atoms. Because, conversely, very light atoms like the H atom are dominated by the regions of self-interaction, we can expect a SIE to cause problems in the application of DFT to very light atoms. The H H 2 3 H 2 H Reaction Past DFT-based studies have included reports that self-interaction errors in exchange functionals may give rise to the underestimation of reaction energy barriers. This argument is, however, not well established, because conventional SIC schemes have not yet given the structures of the transition states (TS) for typical systems. We thus also applied the new scheme to the determination of TS structures. Before calculating the results for reactions of various types, we applied the SIC scheme to the hydrogen-abstraction reaction, that is, to H H 2 3 H 2 H. This reaction has been discussed as a challenging subject in computational chemistry, 6,26,27 because it is hard to obtain an accurate figure for the barrier energy, despite the simplicity of this system. Actually, this barrier is significantly underestimated by most DFT functionals. 6 Because eq. (4) holds throughout the H atom and the H 2 molecule, 2 this underestimation is obviously because of self-interaction errors in the functionals that have been used. The calculations were carried out by using the self-consistent Kohn Sham method 1 with the cc-pvdz, G (3df,3pd), 23 Roos augmented triple-zeta ANO (TZ-ANO), 24 and pv6z 25 Gaussian basis functions; point prune grids are employed in the numerical integration. Zero-point vibrational frequencies are taken into account in the calculations of barrier energy. We employed the Becke 1988 (B88) exchange 28 one-parameter progressive (OP) correlation 12,29 (BOP), B88 exchange Lee Yang Parr (LYP) correlation 30 (BLYP), and hybrid B3LYP functionals with self-interaction-corrected BOP functional (SIC-BOP) as the exchange-correlation functionals. Parameter a in eq. (14) was empirically determined as 0.95 by taking advantage of atomic kinetic energies that were calculated by substituting x and DFT x in eq. (15) with total and W, SI respectively. Table 1 shows the calculated energy barriers of the H H 2 3 H 2 H reaction with the optimized geometries of the TS structure (H 3 ) and the reactant (H 2 ). A highly accurate quantum Monte Carlo (QMC) estimate of the reaction energy barrier 32 is also given in the table. The table shows that a new SIC scheme is strongly dependent on the quality of the basis sets. Especially, the scheme produces a reaction barrier that is lower by 13.7 kcal/mol with cc-pvdz, although it conversely produces barriers that are higher by 3.0, 4.5, and 5.2 kcal/mol with 6-311G (3df,3pd), TZ-ANO and pv6z, respectively. The calculated bond lengths are also

5 1596 Tsuneda, Kamiya, and Hirao Vol. 24, No. 13 Journal of Computational Chemistry Table I. Calculated Reaction Energy Barriers of the H 2 H 3 H H 2 Reaction in kcal/mol and Optimized Bond Distances in Å for cc-pvdz, 6-311G (3df,3pd), TZ-ANO, and pv6z Gaussian Basis Sets. Barrier height Optimized geometry Method Classical ZPVC R(H 2 ) R(H 3 ) cc-pvdz basis set BOP SIC-BOP BLYP B3LYP G (3df,3pd) basis set BOP SIC-BOP BLYP B3LYP TZ-ANO basis set BOP SIC-BOP BLYP B3LYP pv6z basis set BOP SIC-BOP BLYP B3LYP Refs. 9.6 a Classical and ZPVC indicate reaction barriers before and after the zero-point vibrational correction, respectively. R(H 3 ) and R(H 2 ) are the optimized HOH bond distances of the transition state (H 3 ) and reactant (H 2 ). a This value corresponds to a quantum Monte Carlo result. greatly vary between cc-pvdz and other large basis sets. This may be attributable to the incomplete cusp behaviors of Gaussian-type basis sets. To investigate this possibility, we calculated the electron density and the exchange energy per electron in eq. (11) for the hydrogen atom on the basis of atomic orbitals that were determined by the Hartree Fock method. In Figure 4, these values are plotted against the distance from the nucleus to the electron. As can be seen in the figure, exchange energy diverges near the nucleus, especially in the case of small basis sets (Fig. 4b), despite the good descriptions of electron densities (Fig. 4a). We should note that such near-nucleus areas are generally classified as regions of self-interaction. We may therefore suppose that large Gaussiantype basis sets that have correct cusp behavior may be necessary to obtain accurate figures for exchange energies in regions of selfinteraction. However, it is presumed that the SIC scheme may not depend on the quality of basis set so seriously for other reactions. Because H and H 2 only contain self-interactions between orbitals, and self-interactions therefore have a direct effect on the bondings in the H H 2 reaction. In most reactions, self-interactions may indirectly take part in bondings. This is taken up in the next section. Figure 4. Calculated electron densities and exchange energies per electron, eq. (11), of hydrogen atom for three types of Gaussian basis set; cc-pvdz, TZ-ANO, pv6z. These values are calculated by using Hartree Fock atomic orbitals. Typical Reactions Finally, the new SIC scheme was applied to the calculation of typical reaction energy diagrams. We selected reaction systems where the energy barriers are rather underestimated by conventional exchange-correlation functionals. The unrestricted Kohn Sham calculations were carried out with the G(3df,3pd) basis set 23 and point prune grids. BOP, SIC-BOP, BLYP, and B3LYP were employed as exchange-correlation functionals. For SIC-BOP, the cc-pvdz and TZ-ANO basis sets were also applied to explore the basis-set dependency of the SIC scheme. The calculated barrier energies were corrected for zeropoint vibrational frequencies. Table 2 shows calculated and experimental reaction energy barriers. It can be seen from the table that the new SIC scheme obviously tends to improve underestimated reaction energy barriers. Despite the fact that no TS is reproduced by BOP, SIC-BOP gives a TS for the H 2 O OH 3 OH H 2 O reaction. SIC-BOP also provides a positive estimate of the barriers in the F HF 3 FH F and NH 3 OH 3 NH 2 H 2 O reactions, although BOP produced negative barriers for these reactions. As for the basis-set dependency of the new SIC scheme, although 6-311G (3df,3pd) gave seriously different barrier energies from the energies of cc-pvdz, TZ-ANO hardly changed the barrier energies from the energies of 6-311G (3df,3pd). Hence, if we use a high-level Gaussian basis set like 6-311G (3df,3pd), it may not be necessary to become overly sensitive about the basisset dependency of this SIC scheme. Actually, the new SIC scheme improve the calculated barrier energies of the present reactions even in the cases of cc-pvdz.

6 Regional SIC of Density Functional Theory 1597 Table 2. Calculated Reaction Energy Barriers of Some Reactions in kcal/mol, Including Zero-Point Vibrational Corrections. SIC-BOP Reactions BOP BLYP B3LYP B1 B2 B3 Expt. H 2 O OH 3 OH H 2 O F HF 3 FH F a NH 3 OH 3 NH 2 H 2 O H 2 CO 3 H 2 CO HC(OH)CHC(O)H 3 HC(O)CHC(OH)H ,2,4,5-C 2 N 4 H 2 3 N 2 2HCN The G(3df,3pd) basis set (B2) is used for all functionals, while the cc-pvdz (B1) and TZ-ANO (B3) basis sets are also applied to SIC-BOP functional. a This value corresponds to the calculated QCISD(T)/D95 (3df,2p) result. However, impossible to overlook is that the new SIC scheme does not sufficiently improve the calculated barriers in the cases of the intramolecular rearrangement in malonaldehyde and the decomposition of tetrazine. The failure of SIC in the former case may be because of poor optimization of the geometries of the TS and reactant. Actually, the overestimation of the bond lengths with SIC-BOP is probably due to the increase of exchange repulsions. The reaction barrier may be non-negligibly affected by the changes in geometry. The ineffectiveness of SIC in the latter case may be attributable to the very narrow self-interaction regions in the present case (i.e., a 0.95). Unlike the other reactions, carbon and nitrogen atoms play a major role in this reaction. In these atoms, the self-interaction regions are confined in much smaller regions that are very near to and very far from the nuclei in comparison with the case of the hydrogen atom. Although selfinteraction regions may be enlarged by reducing a in eq. (14), it then becomes hard to solve the Kohn Sham calculation, probably due to the inclusion of critical areas where the density matrix may not be well approximated by eq. (4), in self-interaction regions. There is room for further investigation. The conclusion is that the underestimation of the reaction energy barriers in DFT may be a product of poor results of calculation for exchange energies in self-interaction regions, that is, in the short- and long-range areas relative to the atomic nuclei. Conclusions In this article, we have developed a new form of self-interaction correction (SIC) for the exchange functionals of density functional theory (DFT). In this scheme, the exact exchange self-interaction energy of a hydrogenic orbital was substituted for the exchange functional energy of regions of self-interaction, after the regions had been partitioned from the electron distributions of molecules. As an indicator of regions of self-interaction, the scheme employs the ratio of the Weizsäcker kinetic energy density ( W ) to the total density ( total ), t W / total. The scheme takes advantage of the property that W becomes total in the case of electrons that are subject to self-interaction. The new SIC scheme was initially applied in the calculation of the atomic exchange energies of regions of self-interaction. Calculated exchange energies revealed that self-interaction regions (t 1) may be gradually encroached upon with increasing the atomic weight. This may account for the accuracy of the results of conventional DFTs for heavy molecules. We then used the SIC scheme to calculate reaction-barrier energies that are severely underestimated by conventional DFTs. Calculated barrier energies for the H H 2 3 H 2 H reaction showed that, because of the cusp problem in such sets, high-quality Gaussian-type basis sets are necessary for estimation of exchange energies in regions of self-interaction. Calculations of typical reaction energy diagrams showed that the new SIC scheme clearly reduces the degree of underestimation of reaction energy barriers. In particular, even the transition state of the H 2 O OH 3 OH H 2 O reaction is reproduced by the SIC scheme, although uncorrected pure functionals give no transition state for this reaction. Compared with the conventional SIC scheme, the advantages of the new scheme are as follows; 1. It requires much less computational time and relieves the user of computational trouble. 2. It has a simple form that smoothly corrects the exchange functional in the Kohn Sham equation for self-interaction regions. 3. Optimized molecular structures and the Kohn Sham orbitals are analytically obtained for SIC functionals. 4. It is easily implementable in existing DFT programs as long as a meta-generalized-gradient-approximation exchange functional is already available. In the development of the new scheme, SIC energies were assumed to be given correctly by the exact exchange self-interaction energies of hydrogenic atoms. This assumption may be appropriate for molecules where the regions of self-interaction are cut out by the border of t 1. Heavier molecules, however, may be assumed to contain further regions where t is not close to 1, while these regions may be intermediate between slowly varying density and self-interaction regions. We, therefore, expect that

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