A new all-round density functional based on spin states and SN 2 barriers

Marcel Swart, Miquel Solà, and F. Matthias Bickelhaupt Departament de Química and Institut de Química Computacional, Universitat de Girona, Campus Montilivi, 17071 Girona, Catalunya, Spain Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluís Companys 23, 08010 Barcelona, Catalunya, Spain Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands


I. INTRODUCTION
Over the past 20 years, density functional theory ͑DFT͒ [1][2][3] has become the method of choice for computational studies in organic chemistry, organometallics, and biochemistry.This is mainly due to the combination of general accuracy and high efficiency, which enables one to treat real molecular systems at relatively low computational cost.Within Kohn-Sham DFT, only the exchange-correlation energy ͑E XC ͒ needs to be approximated, for which many formulations have been proposed.The earliest DFT functionals took only the density ͑r͒ into consideration ͓local density approximations ͑LDAs͔͒, which was observed to be a big improvement over Hartree-Fock.Further improvements were obtained by taking into account more and more density ingredients ͓see Eq. ͑1͔͒, such as the density gradient ٌ ͓in generalized gradient approximations, ͑GGAs͔͒, 4 and its Laplacian ٌ 2 and/or the kinetic-energy density ͑in meta-GGA or MGGA functionals; Perdew and Constantin recently showed 5 that these latter two ingredients carry essentially the same information beyond the density and density gradient͒. 6,7XC

͑1͒
Perdew et al. 7,8 used these ingredients in order to build Jacob's ladder of density functional approximations, where each rung is built around those prior to it and with increasing proximity to the heaven of chemical accuracy.Although the systematic and nonempirical approach taken by Perdew indeed seems to lead to constant improvement, there does not seem to be a systematic improvement for all properties.This can be directly traced back to the process of solving the Schrödinger equation H⌽ = E⌽.In contrast with wave function theory that systematically improves the wave function ⌽ ͑but not necessarily all molecular properties͒, density functional methods tinker with the Hamiltonian H.As a result, no systematic improvement can be expected within DFT ͑although Perdew's nonempirical series of functionals comes very close͒ and every new density functional has to be validated for its appropriateness for different properties and characteristics.
5][16][17] From these studies, it emerged that Perdew's GGA functional ͑PBE͒ 18,19 performs very well in many aspects, but also fails dramatically for others.Interestingly enough, for those interactions for which PBE fails ͑spin states, S N 2 barriers͒, 9,15 it helps to replace the PBE exchange part with Handy and Cohen's OPTX ͑Ref.20͒ functional ͓to give OPBE ͑Ref.14͔͒.Instead, for those interactions that PBE works very well ͑hydrogen bonding͒ or reasonably well ͑-stacking͒, OPBE fails. 11,12We attempted to rationalize what causes this different behavior, which was initially attributed to the inclusion of s 4 terms ͓where s is a reduced density gradient, s = ٌ͉͉ / ͑2͑3 2 ͒ 1/3 4/3 ͔͒ in the OPTX functional. 9We previously studied what was the influence of the constraints put on the PBE exchange functional by Perdew, but did not find any clear correlation with the performance for different interactions. 21n important step forward was made recently 22 when we investigated which part of the ͑exchange͒ functional is responsible for which interactions.In that study we exploited the fact that the only difference between PBE and OPBE is in the formulation of the exchange functional.This allowed for the design of a series of functionals that switch smoothly from OPBE to PBE at a predefined point P of the reduced density gradient s, where P runs from s = 0.1 to s = 10.0: The enhancement factor for the switched exchange functional then has the form: We observed in that study that for a number of properties the different behaviors of OPBE versus PBE is completely determined by one particular region, i.e., the region of s Ͻ 1.
That is, the excellent behavior of OPBE for spin states and the barriers of S N 2 reactions can be obtained with a P value of ca.0.7-0.9,without any deterioration of the PBE results for hydrogen bonding orstacking.Based on these data, we designed a small modification of the PBE exchange functional in order to mimic these findings: In obtaining these parameters, the A-value was constrained to the OPTX value, the C-value determined by the PBE constraint for s → ϱ ͑A + B / C = 1.804͒, 18 while the other three parameters had been fitted to minimize the difference be-tween F SSB-sw and F OPTX for s Ͻ 0.7 and between F SSB-sw and F PBEx for s Ͼ 0.9.It was shown 22 that this design of the exchange functional reproduces the results of the switched functional and thus combines the best of OPBE with the best of PBE.
Here we report an improvement of the switched functional, where the parameters of the functional have been optimized for a number of different interactions.Moreover, we included Grimme's dispersion correction 23,24 in order to give good behavior for all weak interactions ͑especially needed for -stacking͒.

II. COMPUTATIONAL DETAILS
All DFT calculations were performed with a locally adapted version of the Amsterdam Density Functional 25,26 program developed by Baerends et al.The MOs were expanded in a large uncontracted set of Slater-type orbitals ͑STOs͒ ͑TZ2P, QZ4P͒ ͑Ref.27͒, which are respectively of triple-quality augmented by two sets of polarization functions and of quadruple-quality augmented by four sets of polarization functions.An auxiliary set of s, p, d, f, and g STOs was used to fit the molecular density and to represent the Coulomb and exchange potentials accurately in each SCF cycle.
Energies and gradients were calculated using the LDA ͓Slater exchange and VWN ͑Ref.28͒ correlation͔ with gradient corrections ͑GGA͒ for exchange and correlation included self-consistently for each of the functionals used in this study, i.e., the OPBE functional uses the GGA parts of Handy and Cohen's OPTX ͑Ref.20͒ exchange functional with the PBEc ͑Ref.18͒ correlation functional, PBE ͑Ref.18͒ uses both the exchange PBEx and correlation PBEc GGA functionals, while the SSBx functionals combine the GGA part of Eq. ͑9͒ with that of the simplified PBE correlation ͑sPBEc͒ functional. 21We used here the sPBEc and not the original PBEc because the former gives better results for G2-set atomization energies and hydrogen-bonding interactions. 21In the case of the newly developed functionals that include Grimme's dispersion correction, we adapted only the s 6 factor with which this correction is included and kept all other parameters to their default values.
Geometries were optimized with the QUILD program 29 using adapted delocalized coordinates 30 because of its superior geometry optimization performance.In the optimization process of the different functionals, we used the Amoeba routines from Numerical Recipes 31 for minimization of the sum of mean absolute deviation ͑MAD͒ values ͑see below͒.

III. CONSTRUCTION OF NEW FUNCTIONALS
We previously 22 designed a new exchange-functional formulation ͑SSB-sw͒ that reproduced the results obtained when switching between OPBE and PBE.That study showed that by a small modification of PBEx its poor performance for spin states and reaction barriers can be improved drastically.However, constraining the parameters to those of OPTX may not be optimal for the newly designed functional so we decided to try to optimize them.In one of our previous studies, 21 we already discovered that by optimizing the pa- rameters for atomic exchange energies, the performance for molecular properties is not optimal.On the other hand, by optimizing the parameters to the atomization energies of the G2-1 set, neither other molecular properties nor atomic properties are satisfactory.Therefore, we decided to optimize the parameters for a number of different interactions such as stacking, hydrogen bonding, S N 2 barriers, and accuracy of geometries of organic and organometallic systems.In particular, we determined the MAD from reference values ͑taken from the literature and usually obtained at high-level CCSD͑T͒ with large basis sets; see Ref. 21 for details͒ and multiplied it with a weight specific to each interaction ͑see Table I͒.The sum of these weighted MAD values was then minimized for obtaining the parameters of the new density functionals.
At first, we optimized simply the parameters in Eq. ͑4͒, which are in fact only three independent ones since we applied both the constraint that A + B / C = 1.804 ͑see above͒, and that D = B in order to reduce the number of parameters to be optimized.This led to good behavior for most interactions but showed an elevated MAD value forstacking and the G2-1 atomization energies.Becausestacking was found 12 to be a complicated issue for density functional methods and can easily be circumvented by including Grimme's empirical dispersion correction, 23,24 we decided to try this first.Indeed, this brought the MAD value for stacking down considerably, from ca. 7 kcal mol −1 to ca. 0.5 kcal mol −1 .The MAD value for the G2-set however remained quite elevated ͑ca.14-16 kcal mol −1 ͒.
We then realized that this is reminiscent of the situation of Keal and Tozer's 32 KT1 functional, which worked very well for NMR parameters, but gave sizeable MAD values for molecular properties and atomization energies.Keal and Tozer thereafter experimented with separate factors for LDA exchange and for LDA correlation contributions ͑to give their KT2 functional͒, but here we simply explored mixing in a portion of their gradient correction that is given in Eq. ͑5͒: In this equation runs over the spin-up and spin-down densities and ␥ and ␦ are parameters; in the KT1 functional, the values are, respectively, Ϫ0.006 ͑␥͒ and 0.1 ͑␦͒.Note that Eq. ͑5͒ does not satisfy the uniform density exchange scaling condition and for finite densities may be viewed as a combined exchange-correlation term.It resembles the leading term in the exchange gradient expansion, with the additional constant ␦ introduced to prevent divergence of the potential in asymptotic regions.Now, Eq. ͑5͒ can be easily rewritten in a form that resembles our −Ds 2 / ͑1+Es 4 ͒ correction in Eq. ͑4͒ ͓where u is a constant that remains to be determined and C x is the coefficient for LDA exchange in spin-polarized form, see Eqs. ͑7͒ and ͑8͔͒: In order to compare it with our formulation in Eq. ͑4͒, it should be kept in mind that Eq. ͑4͒ shows only the enhancement factor F͑s͒ over LDA exchange, i.e., the total energy is for LDA and GGA exchange functionals given by Therefore, the correction term in Eq. ͑4͒ for the exchange energy is ͑note that here it has been rewritten in spinpolarized form because that is how it is usually calculated͒

͑8͒
If we ignore for the moment the constants u and v, the only difference appears in the denominator that reads 1 + Es 4 for our SSB formula and 1 + ␦ / 4/3 for KT.It is interesting to note that both our SSB formula and the KT formula correct the enhancement factor F͑s͒ at low s values.However, in case of KT, the situation is a bit more complicated because the enhancement factor cannot be expressed in terms of s only ͓see Eq. ͑6͔͒, contrary to what is the case for GGA exchange functionals.So far, for our newly constructed functionals we constrained the D parameter to be equal to B ͓see Eq. ͑4͔͒, which we can now expand to include the KT formula.That is, we take the combination of the SSB and KT corrections and mix them with u + v = 1 such that the total effect at low s remains the same.By coincidence, we noticed in our optimization procedure of the parameters that an even better performance was observed when multiplying the KT correction with an additional factor F. The final formulation of our SSB functionals is therefore given in Eq. ͑9͒: 33 2 , We have no reason to adjust the values of ␣, ␤, ␥, and ␦, nor the high s limit of PBE, so we therefore keep these at their original values of 20, ϳ0.066 725, ϳ0.031 091, 0.1, and 1.804, respectively.The other parameter values for the SSB and SSB-D functionals are given in Table II.It should be noted that for the construction of the SSB-D functional that includes Grimme's dispersion correction, the s 6 factor for dispersion correction was included in the parameter set to be optimized and was thus optimized simultaneously with the other ones.Similar to OPTX and Grimme's B97-D functional, 24 the SSB functionals do not satisfy the uniform electron gas ͑UEG͒ limit at s = 0. Instead they have elevated values of 1.051 51 ͑OPTX͒, 1.086 62 ͑B97-D͒, and 1.079 966 ͑SSB-D͒.It seems that for reliable results in chemistry, factors different from one are needed as also evidenced by, e.g., the hybrid functionals B3LYP and TPSSh that have factors of 0.8 and 0.9, respectively ͑plus a wildly complicated functional, i.e., portion of HF exchange, that does not depend explicitly on s͒.In the design of our SSB-D functional, we aim at reliable results for biological and chemical applications, which seem to dictate that we have to sacrifice the UEG limit.

IV. PERFORMANCE OF NEW FUNCTIONALS
The inclusion of the KT correction indeed reduces the MAD value for the atomization energies considerably and is now similar to that of PBE ͑see Table III͒, although it remains higher than that of other functionals such as OPBE ͑see Table III͒, TPSSh 34,35 or B3LYP. 36This is observed both for the smaller G2-1 ͑55 molecules͒ and the larger G2/97 ͑148 molecules͒ set.It is probably related to its performance for atomic properties and not as much the molecular ones because for a wide range of molecular properties of biological interest ͑hydrogen bonding,stacking, spin-state energies, reaction barriers͒ our new SSB-D functional performs as well or better than these functionals ͑see below͒.It might be that the GGA formulation is simply not flexible enough for satisfying both the G2-set and other properties, as advocated by Perdew et al., 37 and one should go to the MGGA level for improvement of the atomization energies.Indeed, the PBE atomization energy error is greatly reduced when going to its MGGA counterpart ͑TPSS͒. 34,35We have not attempted to do that here and will leave it for future studies.

A. Spin-state splittings
One of the aims of our recent studies was to determine what makes the PBE not work well for spin-state splittings of transition-metal complexes and why OPBE does.Our preliminary study with our SSB-sw functional 22 indicated that a very small region in s space is responsible for it.Since we did not fit to any spin-state splittings ͑in contrast with B3LYP ‫ء‬ ͒, we were interested to see how the newly developed functional performs in this respect.Given in Table IV are the ⌬E HL splittings between the low-spin and high-spin states for a number of iron complexes.Note that no experimental or "exact" reference spin-state energies are available for these complexes, but previous studies by one of us have shown the reliability of OPBE. 9,10For the first complex Fe͑L 1 ͒, we calculated the vertical spin-state splitting and observed both with OPBE and the SSB functionals a high-spin ground-state, which is in accord with experimental data. 9,38he PBE functional on the other hand predicts a low-spin state, 9 already with the high-spin structure in which the ironligand distances are elongated ͑2.3-2.4Å͒, which disfavors the low-spin state considerably.
The other two iron complexes contain pyridylmethylamine ligands, which are structurally and chemically very similar, yet display different spin states.With the amine ͑amp͒ ligand a high-spin ground-state is observed experimentally, while with the diamine ͑dpa͒ ligand a low-spin is found.Previously, 10 it was shown that OPBE is one of very few DFT functionals that is able to distinguish between these two complexes in terms of their spin ground state.To that list we can now add the SSB-D functional, which shows even more favorable spin-state splittings than OPBE with values of 12.6 kcal mol −1 ͑SSB-D͒ versus 8.5 ͑OPBE͒ for Fe͑amp͒ 2 Cl 2 and −5.2 kcal mol −1 versus Ϫ2.3 ͑OPBE͒ for Fe͑dpa͒ 2 2+ ͑see Table IV͒.Finally, we included the nickel complex Ni͑edt͒ 2 2− , which is a model complex for the catalytic cycle of NiFe-hydrogenase. 39,40Previous studies showed that, in particular, B3LYP and, to a lesser degree B3LYP ‫ء‬ , were unable to predict the low-spin ground-state for this complex. 39However, the influence of the basis set is often very important 41 and also played a role in this case.With a poor Gaussian-type orbital ͑GTO͒ basis, B3LYP indeed predicts a ͑high-spin͒ triplet ground state, but both with a good GTO basis and with STO basis sets it narrowly ͑1.0 kcal mol −1 or less͒ predicts the singlet ground state. 40The situation is completely different for OPBE or SSB-D, which correctly predicts the singlet ground state ͑see Table IV͒ and with a substantial singlettriplet splitting ͑6 -10 kcal mol −1 ͒.In this case, PBE also correctly predicts the ground state but this is not surprising since it favors low-spin state like most early GGA functionals. 8

B. Nucleophilic substitution reactions
The other case where OPBE performs remarkably well and PBE does not 14 is for the reaction barriers of bimolecular nucleophilic substitution ͑S N 2͒ reactions ͑see Scheme 1͒.These barriers were included in the optimization process of the SSB-D functional and it is therefore no surprise that the functional works well for them.Indeed, it gives the lowest deviation for the central barrier ͑2.6 kcal mol −1 ͒ for any of the GGA functionals studied so far ͑see Table III͒.In order to get an independent check on the accuracy for these barriers, we also used our new functionals for a number of systems taken from a recent study by some of us. 16In that study, the anti-E2, syn-E2, and S N 2 pathways of X − +CH 3 CH 2 X ͑X =F,Cl͒ was investigated with DFT and a number of wave function methods ͑HF, MPx, coupled cluster͒.Good overall agreement with the ab initio benchmark data was observed for the OPBE functional, which gives a MAD value for the central barrier of 3.8 kcal mol −1 ͑see Table V͒.The MAD value for the PBE functional is almost twice as large ͑6.6 kcal mol −1 ͒, while the SSB-D gives the best performance with a MAD value of 3.5 kcal mol −1 ͑see Table V͒.Previously, we also investigated the geometries of stationary points and compared these with high-level CCSD͑T͒ geometries. 14The best agreement was observed for the OLYP and OPBE functionals, with P G values ͓P G = MAD͑R͒ • MAD͑͒, see Ref. 14͔ of 0.033 and 0.042 Å deg, respectively, which was already significantly better than the best MGGA and hybrid functionals, OLAP3 and mPBE0KCIS, which showed values 0.060 and 0.068 Å deg, respectively. 14Our new SSB-D functional improves upon all of them with a value of 0.014 Å deg ͑see Table V͒.

C. Accuracy of geometries of small molecules and metal compounds
The good performance of our new functional is corroborated by good performance regarding geometry parameters for two sets of molecules.The first one comprises a set of 19 small ͑first row͒ molecules that was previously studied by Helgaker and co-workers 42,43 with a number of high-level ab initio methods and correlation-consistent basis sets, and by one of us for DFT functionals. 13The best performance was observed with CCSD͑T͒ with an average absolute deviation from experimental data of less than 0.2 pm.In fact they predicted that one experimental value was incorrect, which was confirmed by a subsequent experimental study. 43Density functional methods showed in general good agreement as well, with MAD values of around 1 pm. 13The PBE and OPBE functionals were found to be one of the best DFT functionals for this set with MAD values of ca.0.9 pm. 13,14ur new SSB-D functional significantly improves upon them with a MAD value of 0.6 pm.͑see Table III͒.
Bühl and co-workers [44][45][46] recently made a number of benchmarks for transition-metal complexes.In their first paper, they studied a diverse set of 32 molecules and also a more compact set of metal halides. 44The performance of the DFT functionals for the latter compact set was found to be representative of the performance for the larger set.Therefore, we also studied this set with the new functional and found good behavior, with a MAD value of 0.8 pm that is between that of OPBE and PBE ͑see Table III͒.Note that the value of 1.1 pm for OPBE was already improving upon those of B3LYP and B3LYP ‫ء‬ , 10 while the SSB-D value is similar to the best functional ͑TPSS͒ in the study of Bühl and Kabrede. 44

D. Weak interactions
An important interaction for the study of the chemistry of biological systems is apart from the spin-state splittings ͑when transition metals are present͒ also weak interactions such as hydrogen bonding andinteractions.Since these interactions are much weaker than the strong intramolecular interactions, it is more difficult to obtain them correctly.This holds not only for DFT but also for wave function-based methods where the basis set superposition error ͑BSSE͒ is often of the same order of magnitude as the interaction itself, especially when using small or medium-sized GTO basis sets.
In our design of the SSB-D functional, we included both hydrogen bonding andstacking systems in the optimization process, which helped to get performance for both.For the hydrogen bonding in four dimers ͑ammonia, water, formic acid, formamide͒, we showed previously 22 that PBE performs very well, while OPBE significantly underestimates the hydrogen-bonding energies.This is also reflected in the hydrogen-bond distances of the same four dimers, where SCHEME 1. Energy profile for S N 2 reactions.OPBE shows a deviation of 29.3 pm from reference CCSD͑T͒ data, 47 while PBE gives a deviation of only 1.7 pm ͑see Table III͒.SSB-D further improves upon them with a value of only 1.2 pm.
For the stacking interaction we included the antiparallel cytosine dimer, which has an interaction energy of −9.9 kcal mol −1 with CCSD͑T͒.PBE underestimates this interaction considerably ͑−2.8 kcal mol −1 and hence a MAD value of 7.1 kcal mol −1 , see Table III͒, while OPBE even shows repulsion instead of attraction ͑MAD value 15.6 kcal mol −1 ͒.Our new SSB-D again performs very well with a MAD value of only 0.1 kcal mol −1 .
Hobza and co-workers 47 designed a benchmark set of 22 weakly bound molecules ͑S22͒ with high-level reference data with which other methods can be compared.The poor performance of OPBE for weak interactions is well reflected in its MAD value for this S22 set ͑8.1 kcal mol −1 , see Table VI͒, while the MAD value of PBE ͑2.8 kcal mol −1 ͒ reflects both its better hydrogen-bonding capacities as well as its poorer performance forstacking.This is confirmed by the PBE-D data ͑including Grimme's dispersion correction to improve -stacking͒ for the S22 set that improves considerably ͑0.6 kcal mol −1 ͒, while the OPBE-D data ͑where we used a s 6 value of 1.0͒ remain poor.Our new SSB-D that works well for both H-bonding andstacking gives a very good performance for this S22 set, with a MAD value of only 0.5 kcal mol −1 .Note that our SSB-D energetic data have been corrected for BSSE ͑even though with STO basis sets, this correction is small, i.e., 0.5 kcal mol −1 or less͒ in spite of Grimme's proposal 24 to avoid doing so when sufficiently large ͑polarized triple-zeta valence or larger͒ basis sets are used.Since the dispersion correction does not involve basis sets, the BSSE will affect only the other energy terms just as in regular DFT functionals without the dispersion correction.
The good performance of SSB-D for weak interactions is further shown for the hydrogen-bonding patterns of DNA and RNA base pairs A:T and G:C.Recently, 11 we compared the performance of a number of density functionals with STO and GTO basis sets and found good performance for either BP86, PBE, or PW91.Note that PBE-D overbinds and results in a deterioration compared to the PBE data ͑see Table VI͒.Our new SSB-D functional performs very well, with a MAD value with respect to the CCSD͑T͒ energies 47 of only 0.1 kcal mol −1 ͑see Table VI͒.
Grimme reported recently a study 48 on the importance of intramolecular and intermolecular van der Waals interactions for organic reactions by looking at the dimerization of anthracene ͑see Scheme 2͒.High-level ab initio data at SCS-MP2, QCISD͑T͒, and diffusion Monte Carlo ͑DMC͒ indicated that this dissociation is endothermic, i.e., the dimer is more stable than two infinitely separated monomers.The en-   VI because of the different basis set͒.Note that all these three functionals resulted in exothermic dissociation.We also studied this process at the SSB-D/ TZ2P level and found in contrast an endothermic dissociation of 10.6 kcal mol −1 .Not only is the sign correct, the energy value is also in perfect agreement with the DMC and QCISD͑T͒ data. 48n another recent study, Grimme reported another failure of density functionals, this time for the stereoelectronic effects in alkane isomers. 49The difference in stability of branched versus linear alkanes was with density functionals found to be systematically in favor of the linear alkanes.In contrast, experiment, MP2, and SCS-MP2 indicated that the branched alkanes should be more favorable.In the case of octane, the difference in energy should be 1.9Ϯ 0.5 kcal mol −1 in favor of branched octane ͓3͑a͒, see Scheme 3͔, 49 which is well reproduced with SCS-MP2 that gives a value of 1.4 kcal mol −1 .Density functionals on the other hand showed a systematic preference in favor of linear octane ͓3͑b͒, see Scheme 3͔ with values ranging from Ϫ3.5 ͑B2PLYP͒ to Ϫ9.9 ͑BLYP͒ kcal mol −1 .Previous studies speculated that this might be due to the failure of density functionals for van der Waals interactions, but this was found to play only a secondary role ͑e.g., BLYP-D still gave a value of −5.5 kcal mol −1 ͒.Instead, Grimme attributed the failure of density functionals to the neglect of medium-range electron correlation effects and strongly suggested that the energy for the 3͑a͒→3͑b͒ reaction be mandatory for checking newly developed density functionals. 49We therefore studied these two molecules and found indeed for PBE and OPBE the anticipated preference for linear alkanes ͑see Table VI͒.However, our SSB-D functional does not fail at all, but instead indicates a clear preference for the branched octane structure as it should.Moreover, the energy for the range electron correlation effects and strongly suggested that the energy for the 3͑a͒→3͑b͒ reaction ͑2.1 kcal mol −1 at SSB-D/TZ2P, see Table VI͒ is in excellent agreement with the reference data of 1.9 kcal mol −1 .

V. CONCLUSIONS
Based on previous studies where we investigated what governs the behavior of OPBE and PBE for spin states and S N 2 reaction barriers, we designed a new DFT functional that mixes the best of both for these interactions.Together with Grimme's dispersion correction we have thus constructed a new density functional that is based on a small correction of the nonempirical PBE exchange functional.Moreover, our new SSB-D functional partly contains the gradient expansion functional by Keal and Tozer, which could easily be rewritten in a form similar to our small correction.The combination of our small correction and the gradient expansion functional affects only a small region in the reduced gradient s space, but has a large influence on the performance of the functional.The SSB-D functional is shown to work well for spin states of transition-metal complexes, S N 2 reaction barriers, accuracy of geometries, hydrogen bonding, andstacking interactions.We tested the new functional on a number of standard test sets such as Hobza's S22 set for weak interactions for which we find a MAD of 0.46 kcal mol −1 .Moreover, we investigated two systems that had been brought forward by Grimme as difficult cases and for which density functional methods were shown to fail dramatically.For both the dissociation of the anthracene dimer as for the branching energy of octane do we observe excellent behavior.Not only do we predict the correct sign for these interactions, our energies of +10.6 kcal mol −1 for the former and +2.1 kcal mol −1 for the latter are in excellent agreement with SCS-MP2 and DMC data and experiment.We therefore show that it is possible to get the correct energetics for these difficult cases also with a density functional method, at least if SSB-D is used.

TABLE I .
Different interaction types taken into account for parameter optimization with their specific weights ͑see also Ref. 21͒.P E is the average of four energy components involved in the S N 2 energy profile.
b Ten distances in total.c 28 bonds in total.d Seven reactions in total, e Ammonia dimer, water dimer, formic acid dimer, and formamide dimer.Downloaded 02 Dec 2010 to 84.88.138.106.Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

TABLE II .
Parameters of new functionals ͓see Eq. ͑8͔͒.

TABLE III .
MAD values for energies and geometry parameters.
a Taken from Ref. 15. b Performance for energetics, average of four energy terms above.c Performance for geometries, product of R all and all , divided by 100 ͑due to pm to Å conversion͒.d Benchmark central barriers for anti-E2, syn-E2, and S N 2 pathways of X − +CH 3 CH 2 X with X = F , Cl, six reactions in total.
b MAD from reference CCSD͑T͒ energies ͑Ref.47͒.Energies have been corrected for BSSE.c Values in parentheses refer to OPBE-D.d Values in parentheses refer to PBE-D.e MAD from reference MP2 distances ͑Ref.54͒ and CCSD͑T͒ energies ͑Ref.47͒ for A:T and G:C basepairs, see Ref. 11. f Dissociation energy for anthracene dimer, reference value 9 Ϯ 3 kcal mol −1 , see text and Ref. 48. g Single-point on SSB-D/TZ2P geometries.h Branching energy for octane, reference value 1.9Ϯ 0.5 kcal mol −1 , see text and Ref. 49 SCHEME 2. Dimerization of anthracene ͑1͒.