A. Daniel Boese and Nicholas C. Handy a) Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 22 8 JUNE 2002 ARTICLES New exchange-correlation density functionals: The role of the kinetic-energy density A. Daniel Boese and Nicholas C. Handy a) Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom Received 14 December 2001; accepted 15 March 2002 New density functionals, using the kinetic-energy density Tau are reported. The newly introduced variable enhances the performance of previous functionals, leading to highly accurate functionals with and without the use of exact exchange. All these functionals are compared to commonly used functionals for a large test set, looking also at reactions and hydrogen bonded systems. Furthermore, their physical plausibility is discussed American Institute of Physics. DOI: / I. INTRODUCTION Density functional theory DFT is now a very important approach for computational quantum chemistry. The Kohn Sham implementation of DFT critically depends, for its success, on the quality of the exchange-correlation functional. It has long been clear that the local density approximation LDA is inadequate for applications in chemistry for LDA we refer to the exchange functional of Dirac, LDAX, and the correlation functional of either Vosko, Wilk, and Nusair, VWN, 1 or Perdew and Wang, PW91 2. The great step forward was due to Becke, who introduced a particular functional involving, B88X, 3 as a gradient correction to LDAX. A key to the success of B88X was the introduction of one semiempirical parameter to fit the Hartree Fock exchange energies of some atoms. Likewise gradient-corrected correlation functionals involving no semiempirical parameters have been introduced by Perdew, P86 4 and P91C. 5 Another correlation functional, ascribed to Lee, Yang, and Parr, LYP, was introduced and based on a correlated wave function, and parametrized for He. 6 As a result the generalized gradient approximation, GGA, functionals such as B88XLYP BLYP and B88XP91C BP91 are in common usage. Becke showed that if a fraction of exact exchange was reintroduced, there was a significant improvement in the performance of DFT, in particular the prediction of dissociation energies was improved. Becke used the theory of the adiabatic connection to underline his approach. 7 His hybrid functional, B3P91 and later equivalent forms B3LYP and B3P86 involved three parameters which were obtained through a least-squares refinement to the energetic data of a training set of atoms and molecules taken from Pople s G1 8 and G2 9,10 sets which later evolved into the G3 11 set. Becke s most successful hybrid functional B97 contains nine GGA parameters and one exact exchange parameter. 12 a Author to whom correspondence should be addressed. Electronic mail: nch1@cam.ac.uk We have exhaustively investigated GGA functionals. In particular we have used the GGA expansion forms for exchange and correlation introduced by Becke in his B97 functional. 13 We have given these functionals the acronym HCTH, training them on 93, 120, 147, and 407 atomic and molecular systems 14 hence HCTH/93, 13 HCTH/120, HCTH/147, 15 and HCTH/407; 16 each of these functionals contains 15 parameters. We demonstrated that the introduction of H-bonded dimers into the training set greatly increased the applicability of the functionals. 15,17 In another direction we have refined the Becke exchange functional B88X to all the first and second row atoms, giving our two parameter GGA functional OPTX. 18 The functional OPTX- LYP OLYP is significantly better than BLYP, although the HCTH functionals are better than OLYP, primarily because of the greater number of parameters which refine the correlation part. It may now be that optimum GGA functionals have been obtained. The next stage in the development would seem to be a study of the so-called meta-gga functionals. The success of the hybrid functional implies that exact exchange contains important information which is not present in a GGA. To consider this further, the expansion of the exact spherically averaged exchange hole with the spherical coordinate s is examined, 19 x r,s r s 2, 1 where i 2. The exchange hole arises in the following expression for the exchange energy 2 E x dr r 0 dsx r,s*4s /2002/116(22)/9559/11/$ American Institute of Physics

2 9560 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 A. D. Boese and N. C. Handy Two new terms arise in Eq. 1 not present in GGA functionals. The first is the Laplacian of the density 2. The second one is the kinetic-energy density. There are two other observations to be made regarding the use of meta-gga functionals: i The presence of 2 means that it is necessary to calculate second derivatives of the basis functions. It is known that 2 shows a much more erratic behavior than. 20 Therefore a larger quadrature grid must be used in the Kohn Sham calculations. This adds marginally to the cost in well-written programs. ii In meta-gga functionals, there is a specific dependence on the Kohn Sham orbitals. This means that the local multiplicative exchange-correlation potential v xc (r) v xc r F xc 4 cannot be defined. Indeed many practitioners regard this potential as fundamental to DFT, and construct parameterized expressions for it, which are not functional derivatives. 21,22 Such people cannot work with meta-ggas because of the above-mentioned. We have determined a class of functionals, HCTH, parameterizing them against energetic data, gradient data, and exchange-correlation potential data. We found that the inclusion of v xc data improved the accuracy of the functional. 23 Meta-GGAs cannot be refined using v xc data. However, we must also observe that a local multiplicative exchange potential does not exist in the Hartree Fock approximation. Before we give an analysis of the previous studies involving meta-ggas, we discuss how Kohn Sham calculations are performed if the functional involves. When we are working with the expression E x drf x,, 2, 5 for the exchange energy, Kohn Sham matrix elements arise from energy minimization, and therefore the particular -contribution to a matrix element may be evaluated as r F x s dr, 6 where r, s are basis functions. 24 The introduction of into exchange-correlation functionals therefore introduces no particular difficulty and it is unlikely to introduce numerical instability. In the next section we review previous studies involving meta-ggas. It was modeled on the exchange hole for a hydrogen-like one electron atom, with nuclear charge. For any value of r for this atom, the exact exchange hole is known. The BR model was to take any point r in a molecule and evaluate (r) and Q (r), where Q These are the first two terms in the expansion of the exact exchange hole; see Eq. 1. and r were determined such that the same first two terms in the expansion of the hydrogen-like exchange hole had the numerical values,q. Figures of this determined exchange hole showed, for Ne, that if r was inside a shell, the hole was centered at r, but if it was not, the hole was shifted from r. The exact exchange hole also has this property. The BR89X functional contains no parameters. The only other model we know for which the exchange hole is known and obeys Eq. 8 is the uniform electron gas. The BR89X functionals gave exchange energies for atoms which are in much better agreement with Hartree Fock values than LDAX, but they are inferior to those from LDAXB88X. 26 This is a disappointment, because BR89X is based on the H atom, which would appear to be much more reasonable than LDAXB88X, which is based on the uniform electron gas. In their development, Becke and Roussel observed that a part of Q, denoted D D was zero for one-electron systems. This enabled Becke to introduce a correlation functional which was zero for oneelectron systems. He simply multiplied all terms in the LDA expansion for the correlation energy, by D, and thus he presented the correlation functional Bc95, which was then combined with LDAXB88X. 27 The performance was not promising, especially to weakly bound systems, probably because of the restrictive form for the correlation functional. 12 The associated hybrid functional B1Bc95 was high quality for atomization energies only. However, it also offers no advantage over the later GGA hybrid functional B97, referred to earlier. In 1998, Becke introduced a more carefully considered form, B98, 28 which was based on his successful B97, but introduced a dependence on Q and D. His explicit form was E E x E c E E HF x, 10 II. PREVIOUS META-GGA FUNCTIONALS To our knowledge the first meta-gga functional was presented by Becke and Roussel in 1989; 25 It was an exchange functional, and we denote it BR89X. It is a remarkable functional, insofar as the spherically averaged exchange hole BR x (r,s) contains one electron for all r, i.e., E x e UEG x g x w x dr, E c e UEG c f SCC g c w c dr, E c e UEG c, g c w c dr, BR x r,ss 2 ds f SCC D /, 14

3 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 Exchange-correlation functionals 9561 q Q Q UEG /Q UEG, q avg q q /2, w x x q /1 2 x q 2 ) 1/2, 17 2 w c c q /1 c q 2 1/2, 18 2 w c c q avg /1 c q 2 avg 1/2. 19 The uniform electron is denoted in these equations as UEG. e UEG x, e UEG c, and e UEG c are the UEG values for exchange and the Stoll separation for the correlation energy with like and unlike-spin components. The g functions are polynomials whose coefficients are determined by fitting to a training set. All important definitions are here; the remainder are available in the reference. B98 is a hybrid functional, obtained by adding some exact exchange. In this functional Becke has introduced Q. He has also introduced D which ensures that E c is zero for one-electron systems. Becke considered that B98 was a theoretical improvement on B97, for which q had the GGA form x / 4/3. Becke claimed that B98 was superior to B97 because the mean absolute atomization error of the G data set drops from 1.79 to 1.54 kcal mol 1. We must ask whether such a drop is significant, and if it is sustainable over a larger set of molecules and geometries. 29,30 Also in 1998, Van Voorhis and Scuseria used the density matrix expansion to introduce a meta-gga. 31 Like Becke and Roussel, they commenced with an expansion of the spherically averaged exchange hole in terms of the Bessel functions, j 1 ks, j 3 ks,,, 2,. They were able to integrate the first few terms in their expansion with respect to s to determine a specific form E x DME F,x,z dr, 20 z. 21 5/3 Introduction of adjustable parameters and further arguments led VS to propose a functional of the B98 form, denoted VSXC, with the functions w x, w c, w c now being dependent on x, z. In distinction to B98, VSXC is not a hybrid functional. VSXC has 21 adjustable parameters. Their conclusion is the standard deviation in the atomization energies is lower than for the hybrid functional B1B95 and B3LYP. But they make the important point that the introduction of the kinetic energy density gives a more accurate exchange-correlation functional, and which therefore reduces the need for the introduction of exact exchange. However, one must ask the question whether there are too many parameters in this functional to draw any valid conclusions for the small set of molecules to which it has been fit. There are other meta-gga functionals which we must reference, but which we have not as closely examined: the meta-gga functional which has only one semiempirical parameter, PKZB, 32 has been mainly developed for solids and its performance for atomization energies is not comparable to functionals designed to do well for molecules. 29 Also, Salahub and co-workers have examined several meta-gga functionals, whose accuracy does not surpass Becke s three-parameter hybrid functional B3LYP. Now we pose some questions on the meta-gga functionals specifically BR89, Bc95, B98, VSXC: i ii iii iv All of these functionals are based on the expansion of the exchange hole; should such a form also be used for the correlation hole? The functionals are derived from the short-range expansion of the hole. Is this sufficient to study intermediate regions, where bonds occur? B98 and VSXC were parameterized using training sets, and claims on their performance were then made. Do these claims stand when their performance is compared with other GGA and hybrid functionals, using a larger molecular set? The training sets largely contain molecules with covalent bonds. How do they perform on other important classes of molecule, such as those which are hydrogen-bonded? As an answer to some of these questions, we now compare the performance of these functionals with other functionals using a large molecular set, some reactions and some hydrogen-bonded species. We use our recently published set of 407 atomic and molecular systems, which includes 31 total energies of atoms, 199 atomization energies of neutral molecules, 60 atomization energies of cations, 25 atomization energies of anions, 20 ionization potentials, 23 electron affinities and 7 dissociation energies of transition metal complexes and hydrogen bonds. This set includes the large G2-2 set, and it includes many more difficult systems such as hypervalent molecules instead of including larger hydrocarbons, as in the G3 Set of molecules. In Table I and Fig. 1 we give details of our comparisons, calculated using a TZ2P basis set, which we have used throughout the paper. We compare with a b c the hybrid functional B97-1, which is our own reparameterization of B97 using our 93 system training set, 13 the GGA functional HCTH/407. The form of HCTH/ 407 is identical to the GGA part of B97. It contains 15 parameters, refined over the 407 molecule training set. In the discussion which follows, we must bear this fact in mind, and the very commonly used hybrid functional B3LYP. Table I gives the important numerical values. The first four rows, which are displayed in Fig. 1, give RMS errors for all the computed energetic properties. The fifth row shows the RMS errors for the dissociation energy of the five transition metal complexes. Likewise rows six and seven give the RMS errors in ev for the ionization potentials and electron affinities; therefore rows five to seven indicate only energy differences. Row eight evaluates the sum of the moduli of the energy gradients of 285 molecules; this should be zero

4 9562 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 A. D. Boese and N. C. Handy TABLE I. Errors in the predictions of five functionals for atoms and molecules in the 407 set, together with additional errors for reactions and H-bonded dimers. Class Evaluation of Error in B98 VSXC B97-1 HCTH/407 B3LYP All systems 366 kcal/mol Neutral systems 215 kcal/mol Anionic systems 58 kcal/mol Cationic systems 88 kcal/mol T-metal complexes 5 kcal/mol Ionization potentials 80 ev Electron affinities 58 ev Sum of gradient errors 3441 a.u H 2 OH 2 CO H 2 COH 2 % H 2 OSO 3 H 2 SO 4 % H-bonded dimers 9 Diss. energies % H-bonded dimers 9 Shift of H-bonds % because all calculations are at the geometrical equilibrium. Therefore, the smaller the entry, the more accurate the structures. Rows nine and ten give the percentage error in the reaction energies of the two cited reactions in absolute value 46.7, kj/mol, 37 respectively. Finally, rows eleven and twelve render the percentage RMS errors in the counterpoise-corrected dissociation energies and the shift of the hydrogen-bond lengths for nine hydrogen-bonded dimers, only two of which HF 2 and H 2 O 2 are in the 407 set. From these evaluations which include reactions, geometry errors via the error of the gradients, H-bonded species, and transition metal complexes, it is apparent that the -dependent functionals discussed previously are in fact inferior to hybrid and well-developed GGA functionals: their overall error for the 407 set is worse than the hybrid functional B97-1 and the GGA HCTH, albeit comparable to the hybrid functional B3LYP. From this evaluation, B97-1 clearly seems to be the best functional which has been published so far, having an RMS error for the energetic properties of 7.3 kcal/mol. B98, which seemed to be an improvement for a smaller set of molecules, has a larger RMS error of 8.4 kcal/mol. When only looking at neutral molecules, the error of the meta-gga functionals is slightly lower than for FIG. 1. The root-mean-square error in the predictions of the B98, VSXC, B97-1, HCTH/407, and B3LYP functionals for the energetic properties of the 407 set in kcal/mol. the HCTH/407 GGA functional, but this is counterbalanced by the errors of the anions and cations. The difference of performance between B97-1 and B98 is even more striking when looking at the sum of the errors of the gradients in Table I: it is largest for B98, thus we would expect the worst structures for this functional. VSXC, HCTH/407, and B3LYP are comparable, while the B97-1 clearly seems to render the best structures of the functionals tested. Examining the energy differences of molecules without taking their atomization energy into account rows six and seven, the GGA functional does very well, giving an RMS error of less than 0.3 ev for all tested species. Examining the errors of the transition metals complexes row five, itbecomes apparent why hybrid functionals are not as commonly used for these species. By introducing the Hartree Fock contribution, errors in the dissociation energies become large, especially when changing the coordination number on the transition metal atom. Now we look at the performance of functionals for hydrogen bonds and reactions rows nine to twelve. Reactions have to be looked at carefully: their results clearly rely on error cancellation. In our example, all molecules of the hydration of sulphur trioxide to sulphuric acid are in the 407 set. Thus, we would expect the HCTH/407 functional to do particularly well for that reaction. This is not the case; together with the VSXC functional it is performing with an error a little less than 50%. On the other hand, one molecule in the diol reaction has not been included in the 407 set, yet the HCTH/407 functional gives an error less than 2% compared to the respective MP4 value. Overall, the meta-gga functionals have quite large errors mostly larger than the GGA, whereas the hybrid GGA functionals rely exactly on the previously mentioned error compensation. For the hydrogen-bonded dimers a different picture emerges. Unlike the findings of others, 38 for our test set the meta-ggas are more accurate than the other hybrid GGA functionals tested. In particular B98 has very good energies and structures, although the errors of the structures have to be viewed with some caution: their reference value has been determined at the MP2 level. The HCTH/407 functional has been particularly developed for the reason of improving the

5 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 Exchange-correlation functionals 9563 performance of HCTH to weak interactions. It gives extremely accurate structures and energetics for the systems tested. Overall, it is questionable when looking at this analysis, whether the inclusion of the kinetic-energy is worth the effort: The B98 meta-gga hybrid does not seem to do particularly better than B97-1 for the molecules tested, with the exception of the hydrogen-bonded dimers its errors are generally larger by a considerable amount. The same holds true for the VSXC functional, which is barely surpassing the accuracy of the hybrid functional B3LYP for atomization energies and gives generally worse geometries. In comparison, a well developed GGA functional in our case HCTH/407 is showing lower errors than all meta-gga functionals tested. III. THE -HCTH FUNCTIONAL Consideration of the questions i and ii posed for the meta-ggas introduced in the last section, it might be useful to develop a meta-gga functional that does not rigorously follow the short-range behavior of the exchange hole. Becke has done exactly this. 39 He claimed that local density functionals, including the kinetic-energy density, can simulate delocalized exchange and thus, will improve GGA functionals. He introduced the additional expansion: f X w w 2w 3 w 5, 22 with w being dependent on the kinetic-energy density : w t 1 t 1 ; t 3/5 62 2/3 5/3. 23 Equation 23 transforms the semiinfinite variable into a variable whose value is between 1 and 1, Eq. 22 enforces zero slope for f X (w ) at the exponential tails of a molecule. We have developed a new meta-gga functional, following the idea of Becke to introduce f X (w ). We consider a much more general representation for exchange: A good starting point for a functional seems to be a power series expansion of a GGA, like the HCTH or B97 functional. We did not use the Laplacian of the density, 2 in this functional because of the numerical instability reasons mentioned in the last section. The HCTH form has already yielded good performance, from which we deduce that it must map the exchange-correlation hole quite well. The new functional, which we will denote -HCTH incorporates this form, as well as Eq. 22 introduced by Becke. The functional is defined as E XC E X,l E X,nl E C, e LSDA X g X,l s 2 dr, E X,l M g X,l i0 E X,nl i c X,l,i u X, 26 e LSDA X g X,nl s 2 f X w dr, 27 M g X,nl i0 u X X s 2 1 X s 2 1, X 0.004, i c X,nl,i u X, E C E C E C, 31 E C e LSDA C g C s 2 dr, M g C i0 32 i c C,i u C, 33 u C C s 2 1 C s 2 1, C 0.2, E C e C, g C s 2 avg dr, 36 M g C i0 i c C,i u C, 37 u C C s 2 avg 1 C s 2 avg 1, 38 C 0.006, s 2 2 8/3, s 2 avg 1 2 s 2 s This form has been suggested by the following considerations; the exchange local form is directly taken from HCTH. The exchange nonlocal form multiplies the local exchange term by Eq. 22, but unlike Becke s original proposal with all coefficients adjustable. Local exchange and the nonlocal exchange correction should be of different origin, mainly because the incorporation of left right correlation into the normal exchange term. The correlation form is identical to HCTH. Note that no function has been introduced to ensure that Eq. 32 is zero for a one-electron system, indeed we investigated this with no success. In other words, the new functional is similar to the HCTH expansion including a term to simulate nonlocal exchange. In our present HCTH expansions only 15 coefficients had to be determined 3*(M1) with M4. Since this functional with that number of these coefficients appeared to be extremely successful, we determined -HCTH, with 16 coefficients 4*(M1), M3. The -HCTH functional will be shown and its results will be discussed in the following sections. Our functional is then obtained by minimizing the functional s expansion coefficients c X,local,i, c X,non-local,i, c C,i, c C,i in n e n g w mee ex m E ks m 2 m w lx lg E l ks 2 42 X over the 407 training set. In Eq. 42, E ex m, E ks m are the exact and Kohn Sham energies of molecule m and E ks l /X is the Kohn Sham energy gradient of molecule l which

6 9564 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 A. D. Boese and N. C. Handy TABLE II. Expansion coefficients of the -HCTH and -HCTH hybrid functionals, compared to the coefficients of the HCTH/407 and B97-1 functionals. Functional -HCTH HCTH/407 -HCTH hyb. B97-1 c 1 c X,local, c 2 c X,non-local, c 3 c C, c 4 c C, c 5 c X,local, c 6 c X,non-local, c 7 c C, c 8 c C, c 9 c X,local, c 10 c X,non-local, c 11 c C, c 12 c C, c 13 c X,local, c 14 c X,non-local, c 15 c X, c 16 c X, c 17 c X,local, c 18 c X,non-local,4 c 19 c X, c 20 c X, should be zero because all our calculations are performed at experimental or theoretical equilibrium structures. The resulting functional is displayed in Table II, together with the HCTH/407 functional for comparison. IV. THE PHYSICAL PLAUSIBILITY OF THE -HCTH FUNCTIONAL To discuss the new meta-gga functional and its physical meaning, one possibility is to take a look at their respective coefficients in Table II. The coefficients of the -HCTH functional are quite similar to the ones of the HCTH/407 functional. One exception is the C coefficients; while the zeroth-order coefficient has become much smaller, the higher-order coefficients appear to counterbalance this effect and have even changed sign. More physical insight can be gained by looking at the enhancement factors of the exchange-correlation functionals. 40 Basically, the enhancement factors F XC (r s,s) show how a GGA functional behaves in relation to its uniform electron gas value. It is defined by writing the functionals as E XC C D 4/3 rf XC r s,sdr, where r s 3, s /3, 45 4/3 and C D is the Dirac constant. The uniform electron gas limit is achieved for s0. At r s 0, the enhancement factor should become unity to obey this limit. The Lieb Oxford bound describes the upper limit for the enhancement factor of F XC (r s,s) ,42 Furthermore, the scaling inequalities: FIG. 2. The enhancement factor F XC for the HCTH/407 functional. FIG. 3. The enhancement factor F XC for the -HCTH functional.

7 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 Exchange-correlation functionals 9565 FIG. 4. The root-mean-square error for the energetic properties of the 407 set in kcal/mol as a function of the exact exchange mixing factor for different functional forms. FIG. 5. The sum of all gradient errors for the energetic properties of the 407 set in atomic units as a function of the exact exchange mixing factor for different functional forms. F XC r s,sf XC r s,s;r s r s 46 have to be obeyed. These inequalities indicate that the enhancement factor should not have a crossing point. In order to display the enhancement factor for the -HCTH functional, the -dependent terms have to be transferred into a local form. 43 This is achieved by approximating by its von Weizsäcker form, 44 as suggested by Perdew et al., 43 W Figures 2 and 3 show the enhancement factor for the HCTH/ 407 and -HCTH functional, respectively. Both enhancement factors are plotted in the physically meaningful region of s 3. Of course all the resulting enhancement factors in the -HCTH form have to be considered with caution because of the usage of the von Weizsäcker formula. Still it is important to compare the new meta-gga form to known enhancement factors, such as the one of the HCTH functional. Comparing Figs. 2 and 3, both functionals appear very similar with the crossing point and the enhancement factor of the -HCTH functional shifted towards lower values of s. With the HCTH/93 and most other proposed GGA and meta- GGA functionals, 43 neither the HCTH/407 and -HCTH functional obey Eq. 46. Nevertheless, even the authors who develop their functionals primarily on the basis of these exact known conditions admit that a simple GGA form cannot reproduce all known properties of the exact functional. 45 The violation of the Lieb Oxford bound occurs in the highdensity limit of both functionals with r s 0) only in the physically unimportant regions of s3. These violations may indicate a bad performance of these functionals only in applications involving solids. 46 We have not yet observed any problems inherent to this class of functionals in the application to molecules and their properties. V. A NEW HYBRID Now that we have a new functional form for the -HCTH functional, we might ask if the new term is really able to simulate Hartree Fock exchange. Even if it does, it might just enhance the current hybrid functional by adding and simulating variable nonlocal exchange. The variable nonlocal exchange comes from the observation that different types of molecules need different amounts of exact exchange. Thus, the amount of exact exchange should be variable rather than a constant amount added to the exchangecorrelation functional. In the case of simulating constant exact exchange, we should first and foremost see a shift of the minimum of the variable see 42 towards lower Hartree Fock values. Otherwise, if it simulates the variable nonlocal exchange, we would only see a further lowering of the error without observing a shift. In Figs. 4 and 5, the RMS energy and the sum of the gradient errors, respectively, for the 407 set are depicted as a function of the Hartree Fock exact-exchange mixing factor. Here, we test several hybrid functionals with different cutoffs for the power series expansion: i.e., HCTH2 corresponds to the B97-1 functional fit to 407 systems with M2 in Eqs. 26, 33, and 37. HCTH3 is the same functional with M3, while HCTH4 with no exact exchange is the HCTH/407 functional. The -HCTH3 functional with no exact exchange mixing is the actual -HCTH functional displayed in Table II. Although the -HCTH4 functional shows an improved error with respect to the properties in the 407 set, it displays erratic behavior towards various other properties, such as an enhancement factor with a double crossing and unreasonably low molecular frequencies. Apart from this functional, the -HCTH3 hybrid functional clearly shows the lowest errors at an exact exchange mixing coefficient of 15%. Interestingly, even the B97-1 form has its lowest error around 15% rather than the previously determined 21%. 13 This shift is a result of the different training set used. We do not see a shift of the minima towards a lower exact-exchange mixing factor going from the HCTH to -HCTH as evident from Figs. 4 and 5. Moreover, there is very little difference in the energy errors of the HCTH2, HCTH3, and -HCTH2 functionals. This justifies the use of M2 in case of the B97, B97-1, and B functionals since the next term in the expansion provides little improvement. In case of the -HCTH func-

8 9566 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 A. D. Boese and N. C. Handy FIG. 6. The RMS error of the -HCTH, HCTH/407, -HCTH hybrid and B97-1 functionals for all molecules, neutrals, cationic, and anionic species of the 407 set in kcal/mol. FIG. 7. The distribution curve for the error frequency kcal/mol of the -HCTH functional to the 407 set, shown as a fit to a Gaussian curve. tional, however, there is a more significant improvement of approximately 5% in the RMS energy error between M2 and M3. The coefficients of the newly obtained -HCTH hybrid functional are compared to B97-1 in Table II. Except for the very first coefficient, which indicates that less exact exchange is used for the -HCTH hybrid functional, all the coefficients are quite similar. In addition, the nonlocal coefficients are quite similar to the -HCTH functional without any exact exchange mixing with perhaps exception of the highest-order coefficient, c X,non-local,3. Hence, we would expect a quite similar, but somewhat improved performance for the -HCTH hybrid compared to the B97-1 hybrid functional. VI. COMPARISON OF THE FUNCTIONALS Now we will take a closer look at the new functionals by examining their performance to the 407 set, compared to the HCTH/407 and B97-1 functionals. This analysis is displayed in Fig. 6. The exact values from Figs. 1 and 6 are summarized in Tables I and III for comparison. From this data, a quite accurate picture emerges on how the different functionals perform: with the largest set of molecules 407 being the most important. The results for this set are quite surprising, because the new -dependent functionals show lower errors than any other functional tested so far. Their respective errors of the -HCTH and -HCTH hybrid functionals over the full set are 25% and 35% lower than the most commonly used functional, B3LYP. In the new form, even the hybrid functional B97-1 gives slightly larger errors than -HCTH. To ensure that we did not just shift the atomization energies, we compared the distribution curves of the functionals. Such a shift would render exactly the same reaction energies as another functional but a larger RMS error for the atomization energy. In Figs. 7 and 8, the error distribution curves of the -HCTH meta-gga and hybrid functionals are presented. Their respective maxima are at 57 molecules and 77 molecules. This can be compared to B3LYP 49 molecules, HCTH/ molecules and B molecules, again showing improved energetics and a narrower error distribution. In Table IV, we show that these low errors are not an artifact of the fit itself, but are distinct from the functional form and the variables used. Here, we compare the performance of the functionals to the large set, but with the functionals fitted to our formerly used 147 set. The 147 set is a smaller fit set which was used to create the HCTH/147 functional and consists of the G2-1 set of molecules with two extra hydrogen-bonded dimers. This functional, compared to the HCTH/407 functional see Table IV behaved particularly well for charged species, especially anions. By moving TABLE III. Same energetic errors as Table I for the new functionals, compared with B97-1, B3LYP, and HCTH/407. Class Evaluation of Error in -HCTH HCTH/407 B3LYP -HCTH hyb. B97-1 All systems 366 kcal/mol Neutral systems 215 kcal/mol Anionic systems 58 kcal/mol Cationic systems 88 kcal/mol T-metal complexes 5 kcal/mol Ionization potentials 80 ev Electron affinities 58 ev Sum of gradient errors 3441 a.u H 2 OH 2 CO H 2 COH 2 % H 2 OSO 3 H 2 SO 4 % H-bonded dimers 9 Diss. energies % H-bonded dimers 9 Shift of H-bonds %

9 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 Exchange-correlation functionals 9567 FIG. 8. The distribution curve for the error frequency kcal/mol of the -HCTH hybrid functional to the 407 set, shown as a fit to a Gaussian curve. TABLE IV. Errors for the -HCTH and HCTH functionals fit to 147 and 407 systems, evaluated for the 407 set. Class HCTH -HCTH -HCTH hyb. B97-1 Fit To systems Neutrals Anions Cations All molecules TABLE V. The largest energetic errors of the molecules for the B98, VSXC, -HCTH, -HCTH hybrid, and B97-1 functionals in kcal/mol. A positive sign means that the energy is overestimated. Error kcal/mol Molecule B98 VSXC -HCTH -HCTH hyb. B97-1 LiH CN C 2 H ClO C 6 H SiCl FeC 5 H NCCN SiH BH SF FClO Number of systems with errors 15 kcal/mol to the larger set, we were likely to further balance the functional, especially since the error for the neutral molecules hydrocarbons was quite high. Nevertheless, the performance of the HCTH/147 functional to the 407 set led to the conclusion that, generally, the HCTH form is particularly well suited for developing a good GGA functional. On the other hand, we were able to show that some final refinement steps on the functional can be done by extending the training or fitting set. Both steps are of utmost importance, although certain forms allow more flexibility than others. The same holds true for the -HCTH functionals presented in this paper. The error of the -HCTH functional is lowered by 0.5 kcal/mol, putting more emphasis on the atomization energies of neutral molecules. The error of the -HCTH hybrid functionals is lowered by 0.8 kcal/mol by moving from the 147 to the 407 set, again lowering the errors of the neutral molecules at the cost of the errors of the charged species. All this leads to the conclusion that the new functional form itself seems to be a good one; and a carefully chosen fit set may improve it even further. In Table V, some molecules with the largest errors are displayed for the meta-gga functionals. Overall, the -HCTH hybrid has 15 molecules and atoms with errors larger than 15 kcal/mol, also displayed in Table V. This can be compared to B3LYP and VSXC, which each have over 30 of such species in the set. For VSXC, if we take away the atoms, there are only 18 molecules; however, even this comparison is unfair, since all other methods incorporated total energies in their fit, while VSXC did not. It is interesting that for almost all of these large errors, the energies are underestimated, while only for SiCl 4 all methods overestimate its atomization energy by a large amount. Due to inherent problems with DFT in evaluating both CN and NCCN claimed by the authors of the G3 set, those systems have been removed from the G3 set of molecules and the evaluations using the G3 set. Nonetheless, these two molecules were included in our error evaluation. Most of these problematic molecules have quite ionic bonds and some are highly charged like ClO 4 ). Missing from this list are certain multiconfigurational molecules, such as N 2 O 4, which were shown in our previous paper Ref. 16 to have the largest errors for the GGA functionals tested. Surprisingly, the inclusion of either or Hartree Fock exchange seems to remedy this problem. So far, we have mainly compared the energetics of the functionals for the 407 set. In Table III we provide additional information about the functionals. While -HCTH performs equally well as the GGA HCTH/407 functional for transition TABLE VI. RMS errors of 862 bond lengths and 816 bond angles coming from molecules from the 407 set for the BLYP, HCTH/407, -HCTH, -HCTH hybrid, B97-1, and B3LYP functionals. Error pm, degree Number BLYP HCTH/407 -HCTH -HCTH hybrid B97-1 B3LYP RMS bond lengths Mean bond lengths RMS bond angles Mean bond angles Max1 bond length N 2 O 4 ) Cl 2 ) Cl 2 ) 10.02Cl 2 ) 9.66Cl 2 ) Cl 2 ) Max2 bond length Cl 2 ) 11.97N 2 O 4 ) 11.51N 2 O 4 ) 8.48SI 2 ) 8.19SI 2 ) 7.53SIH 2 )

10 9568 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 A. D. Boese and N. C. Handy metals, the -HCTH hybrid functional is worse, comparable to the other hybrid functionals. The electron affinities and ionization potentials show quite low errors compared to the other functionals, and the -HCTH functional even surpasses the accuracy of the HCTH/407 functional for these species. The gradient errors, which have shown to correlate with the accuracy of geometry for molecules in our last paper Ref. 16; see also Table VI, are even more encouraging; both -functionals give very low gradient errors, from which we would expect quite accurate geometries. Despite the very encouraging results for the 407 set and its geometries, the reaction energies obtained with the new -functionals, listed in Table III, are not significantly more accurate than the hybrid functionals or the HCTH/407 GGA. As mentioned before, the cause might be the error compensation for these reactions, which is still more likely to occur for hybrid DFT functionals. In this case, the -HCTH hybrid overestimates the formation of methane diol by a large amount, similar to the B97-1 functional, which already overestimates this reaction by more than 30%. In general, this coincides with an improved prediction of the formation of sulphuric acid. For the hydrogen-bonded systems, both -HCTH functionals have even larger errors, especially for the H-bond geometry shifts, which are consistently overestimated by 50% compared to the typical 30% for hybrid functionals. The energetics are comparable to other hybrid or meta-gga functionals, surpassing in general GGA functionals other than HCTH/147 and HCTH/407. This might be due to the fact that even with a low weight, the hydrogen bonds are already overestimated for the -HCTH functionals. This contrasts the performance of the HCTH GGA functionals. Thus, increasing the weight of the dimers does not improve the performance of the -HCTH functionals in regards to weak interactions. In our case, we are only fitting to the strong hydrogen bonds in HF 2 and H 2 O 2 which are, in fact, improved, increasing the error for weaker H-bonds. In Table VI, the average geometry errors for most functionals examined are shown. By going from HCTH/407 to -HCTH, the bond lengths are only marginally improved, despite the lower gradient errors of the latter functional. As expected, the hybrid functionals perform better still, with B97-1 showing a marginally better performance than B3LYP. The -HCTH hybrid functional, in contrast, does exceed both other hybrid functionals for the large number of bond lengths which it was tested. For the bond angles, the B97-1 functional renders the lowest error, with the errors of both -HCTH methods surpassing the B3LYP method. We have examined the functionals performance on various other properties and, in general, conclude that -HCTH yields results slightly improved over HCTH/407, and the same holds true for the -HCTH hybrid compared to B97-1. Given the accuracy of HCTH/407 and B97-1, which have been the most accurate functionals we have encountered so far, we find this to be quite remarkable. VII. CONCLUSION By inclusion of the variable into the HCTH functional form, we were able to improve both the HCTH functional and its hybrid form, the B97-1 functional. Thus, its inclusion improves functionals in general, which had not been apparent to us from the previously published functionals. This might indicate that the short-range expansion of the exchange-correlation hole used in the development of functionals does not necessarily yield the best functional form. The new functional form obtained is similar to the HCTH form, but has an extra term that simulates some nonlocality. Although it has been previously claimed, we cannot confirm other authors observations that it in fact simulates Hartree Fock exchange, 31,39 since the exact exchange mixing coefficient when including Hartree Fock exchange is not affected by the inclusion of the extra variable. From the performance of the -HCTH and its hybrid compared to their corresponding GGA functionals, the overall energy error of our test set has been lowered by about 10% and 20%, respectively. This is remarkable since the new density functionals are approaching a mean absolute error of less than 4 kcal/mol for the large 407 set. These errors of the -HCTH can be compared to the best current hybrid functionals, while the -HCTH hybrid functional shows the lowest error of all functionals tested to date. Although the functionals show larger errors for hydrogen bonds, it seems to be difficult to improve upon this property within this functional form. The accuracies of the predicted geometries of the new meta-gga (-HCTH functional still do not surpass other commonly used hybrid functionals, yet the -HCTH hybrid is clearly an improvement. Overall, we were able to present new higher quality functionals, lowering the error of even the best functionals. ACKNOWLEDGMENT One of the authors A.D.B. is grateful for support by the EPSRC and the Gottlieb Daimler-und Karl-Benz-Stiftung. 1 S. J. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, J. P. Perdew and Y. Wang, Phys. Rev. B 45, A. D. Becke, Phys. Rev. A 38, J. P. Perdew, Phys. Rev. B 33, J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, A. D. Becke, J. Chem. Phys. 98, L. A. Curtiss, K. Raghavachari, G. W. Trucks, and J. A. Pople, J. Chem. Phys. 94, L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 106, L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 109, L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 112, A. D. Becke, J. Chem. Phys. 107, F. A. Hamprecht, A. J. Cohen, D. J. Tozer, and N. C. Handy, J. Chem. Phys. 109, see also supplementary material to Ref. 16, EPAPS Document No. E-JCPSA A. D. Boese, N. L. Doltsinis, N. C. Handy, and M. Sprik, J. Chem. Phys. 112, A. D. Boese and N. C. Handy J. Chem. Phys. 114, C. Tuma, A. D. Boese, and N. C. Handy, Phys. Chem. Chem. Phys. 1, ; note that the HCTH/120 functional is denoted HCTH-38 in this paper. 18 N. C. Handy and A. J. Cohen, Mol. Phys. 99, A. D. Becke, Int. J. Quantum Chem. 23,

11 J. Chem. Phys., Vol. 116, No. 22, 8 June 2002 Exchange-correlation functionals R. Neumann and N. C. Handy, Chem. Phys. Lett. 266, D. J. Tozer and N. C. Handy, J. Chem. Phys. 109, M. Grüning, O. V. Gritsenko, S. J. A. van Gisbergen, and E. J. Baerends, J. Chem. Phys. 114, G. Menconi, P. J. Wilson, and D. J. Tozer, J. Chem. Phys. 114, R. Neumann and N. C. Handy, Chem. Phys. Lett. 252, A. D. Becke and M. R. Roussel, Phys. Rev. A 39, R. Neumann and N. C. Handy, Chem. Phys. Lett. 246, A. D. Becke, J. Chem. Phys. 104, A. D. Becke, J. Chem. Phys. 109, A. J. Cohen and N. C. Handy, Chem. Phys. Lett. 316, R. Ahlrichs, F. Furche, and S. Grimme, Chem. Phys. Lett. 325, T. Van Voorhis and G. E. Scuseria, J. Chem. Phys. 109, J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys. Rev. Lett. 82, E. Proynow, H. Chermette, and D. R. Salahub, J. Chem. Phys. 113, E. I. Proynov, A. Vela, E. Ruiz, and D. R. Salahub, Int. J. Quantum Chem., Quantum Chem. Symp. 29, E. I. Proynov, S. Sirois, and D. R. Salahub, Int. J. Quantum Chem. 64, J. S. Francisco and I. H. Williams, J. Am. Chem. Soc. 115, M. Hofmann and P. von R. Schleyer, J. Am. Chem. Soc. 116, A. D. Rabuck and G. E. Scuseria, Theor. Chem. Acc. 104, A. D. Becke, J. Chem. Phys. 112, J. P. Perdew and K. Burke, Int. J. Quantum Chem. 57, E. H. Lieb and S. Oxford, Int. J. Quantum Chem. 19, G. K-L. Chan and N. C. Handy, Phys. Rev. A 59, S. Kurth, J. P. Perdew, and K. Burke, Int. J. Quantum Chem. 75, C. F. von Weizsäcker, Z. Phys. 96, J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, P. P. Rushton, S. J. Clark, and D. J. Tozer, Phys. Rev. B 63, P. J. Wilson, T. J. Bradley, and D. J. Tozer, J. Chem. Phys. 115,