- Email: [email protected]
Comment on “Assessment of exchange correlation functionals” [A.J. Cohen, N.C. Handy, Chem. Phys. Lett. 316 (2000) 160–166]
21 July 2000
Chemical Physics Letters 325 Ž2000. 317–321 www.elsevier.nlrlocatercplett
Comment on ‘‘Assessment of exchange correlation functionals’’ wA.J. Cohen, N.C. Handy, Chem. Phys. Lett. 316 ž2000/ 160–166x Reinhart Ahlrichs b
a,)
, Filipp Furche a , Stefan Grimme
b
a Institut fur ¨ Physikalische Chemie, UniÕersitat ¨ Karlsruhe, Kaiserstrasse 12, 76128 Karlsruhe, Germany Organisch-Chemisches Institut, Westfalische Wilhelms-UniÕersitat, Germany ¨ ¨ Corrensstrasse 40, 48149 Munster, ¨
Received 10 May 2000; in final form 30 May 2000
Abstract Additional results of the recently proposed B97GGA-1 density functional wA.J. Cohen, N.C. Handy, Chem. Phys. Lett. 316 Ž2000. 160x for molecular dissociation energies, reaction energies and structure parameters are presented. Compared to the BP86 functional, improved energetics and bond lengths are found for electron-rich main-group compounds, but the new GGA seems to fail for weakly bound molecules. These shortcomings are analyzed in connection with the fitting procedure employed by Cohen and Handy. We discuss whether B97GGA-1 is recommendable for general applications in chemistry. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction There have been occasional requests to implement some of the recently developed density functionals into TURBOMOLE, which could be expected to yield better accuracy than, e.g., BP86 w1,2x or B3LYP w3x. These are the recommended functionals of GGA Žgeneralized gradient approximation. and hybrid type. New functionals are currently proposed in rapid succession. It therefore appeared prudent to wait until matters had settled to some extent. Cohen and Handy ŽCH. w4x have recently presented an assessment of functionals which included ) Corresponding author. E-mail: [email protected]
some very encouraging results. Especially promising appeared B97GGA-1: a functional of GGA type Žno exchange mixing, no second derivatives of the density. derived from Becke’s B97 hybrid w5x, by removing Hartree–Fock exchange and re-optimizing parameters. According to Tables 2 and 3 of CH, B97GGA-1 is only slightly surpassed in accuracy by some hybrid functionals. CH employed an extended training set of molecules to optimize parameters which specify the new functional. Since the training set did not include, e.g., transition metal compounds, we decided to carry out some further test calculations. We do not intend to cover the whole of chemistry, but present a biased sample of compounds we consider as interesting and difficult.
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 6 5 4 - 0
R. Ahlrichs et al.r Chemical Physics Letters 325 (2000) 317–321
318
2. Results In Tables 1 and 2 we collect some results for BP86 and B97GGA-1. TZVPP basis sets from the TURBOMOLE basis set library have been used, which include Ž2p1d. for H and Ž2d1f. for main group atoms. For second row atoms a Ž3d1f. set of polarization functions has been employed. All structure parameters have been optimized within the given molecular symmetry group. This procedure corresponds to real life conditions Žwhere structures are
typically not accurately known. and allows to determine the error in computed structure constants. The atomization energies depend very little on the geometry Žexperiment or MP2 or optimized., the effect is typically less than 1 kcalrmol. For the quadrature a multi-grid scheme w6x ‘‘grid m5’’ w7x has been employed. Open shell cases have been treated in the unrestricted scheme to approximate the states 3 Sy g for O 2 , Al 2 , Si 2 , and 2A 2 for NO 2 . It is expected by the authors that the accuracy of the present results is sufficient for the conclusions drawn below.
Table 1 Deviations d s Ecomp y Eref of computed atomization Žreaction. energies Ecomp from reference values Eref , in kcalrmol Molecules
Li 2 N2 O2 F2 Na 2 Al 2 Si 2 P2 P4 2P2 Cl 2 Cu 2 CH 4 H 2O ŽH 2 O. 2 2H 2 O B 2 H 6 2BH 3 H 3 BCO BH 3 q CO Si 2 H 6 CO CO 2 N2 O NO 2 O3 F2 O O 2 F2 NF3 ClNO ClF3 C 2 F4 SiF4 NiŽCO.4 NiŽCO. 3 q CO
™
™ ™ ™
™
a b c d e f g
Eref
a
24.4 228.6 120.5 38.5 17.0 36.2 74.7 117.2 55.6 b 58.0 47.2 c 419.2 232.2 5.0 d 44.e 25.f 529.3 259.3 389.1 270.5 227.5 147.1 93.3 150.2 g 205.2 191.2 125.6 583.9 572.8 25.c
d ŽBP86.
ŽB97GGA-1.
y4.2 9.0 20.4 13.3 y1.0 0.0 4.7 1.3 2.8 4.3 3.0 9.8 5.6 y0.05 3.6 13. 5.6 5.7 20.2 31.3 34.8 31.7 27.1 51.2 33.5 27.1 31.5 35.0 y9.6 4.
0.9 6.4 14.8 8.7 12.0 y0.4 4.7 y4.2 6.8 0.6 5.1 3.3 0.2 y2.2 y3.6 6. 0.0 y0.1 9.8 12.2 17.8 15.3 14.9 31.5 16.4 11.6 17.4 20.7 y15.1 y3.
Experimental atomization energies from Ref. w9x, unless otherwise stated; ZPE removed. 55.0–56.2, Ref. w9x. Ref. w10x. Ref. w11x. Ref. w12x. Ref. w13x. Ref. w14x.
R. Ahlrichs et al.r Chemical Physics Letters 325 (2000) 317–321 Table 2 Representative structure constants; bond lengths in pm, angles in degrees Molecule
Reference
BP86
B97GGA-1
Parameter
Li 2 N2 O2 F2 Na 2 P2 P4 Cl 2 Cu 2 H 2O
267.0 109.8 120.7 141.7 307.9 189.3 219.5a 198.8 220.0 b 95.9 103.9 291. 121.7 c 157.5 c 159.8 169.8 87.5 183.8 d 114.1 206.4 e 144.0 e 293.f
275.0 110.3 122.0 141.5 309.9 191.2 222.3 202.8 223.7 97.0 104.0 290. 120.2 160.4 165.5 175.4 88.9 182.3 114.9 205.6 143.4 297.5
279.9 109.7 120.5 139.2 326.7 189.8 217.9 199.8 225.9 95.9 103.9 312. 118.3 159.0 162.6 173.1 89.1 182.1 114.3 203.5 142.7 306.
HO HOH OO OO OF ClF1 ClF2,ClF3 F2ClF1 NiC CO FeC CC MnMn
ŽH 2 O. 2 O 2 F2 ClF3
NiŽCO.4 FeCp 2 Mn 2 ŽCO.10 a b c d e f
219–200, Ref. w9x. Ref. w15x. Ref. w14x. Ref. w10x. Ref. w16x. Ref. w17x.
The molecules considered include some cases from the CH training set to verify correctness of the code. We have added some electron-rich compounds ŽNO 2 to ClF3 in Table 1., some electron deficient cases ŽB 2 H 6 ., and very few transition metal compounds, the latter mainly to check structure constants. The new B97GGA-1 functional shows an impressive performance in many respects. The relatively large errors of BP86 in computed atomization energies for O 2 , F2 , CO 2 to C 2 F4 ŽTable 1. are markedly reduced, although they are partly not contained in the training set. The energetics of the two boron compounds and the dissociation energy for the first CO of NiŽCO.4 is satisfying. Computed structure constants, Table 2, also show in general an improvement of B97GGA-1 over BP86. This indicates that the training set employed by CH is sufficiently flexible.
319
Problems with B97GGA-1 actually occur within the training set: The dissociation energy ŽDe . of Na 2 is 12 kcalrmol too large, an error of 70% of De . In contrast, the computed binding energy of the water dimer Žwith respect to 2H 2 O. is only 2.8 kcalrmol ŽBP86 and reference value 5.0 kcalrmol.. The water dimer is not contained in the 93 training set, but in the larger 147 set employed for assessment by CH. The bond distances are also not satisfying in these cases: 19 pm too large for Na 2 and 21 pm too large for the OO distance in the water dimer. These features suggest that B97GGA-1 has problems with small electron densities. This is also seen for the MnMn distance in Mn 2 ŽCO.10 which is 13 pm too large. These problems occur to a lesser degree for Li 2 and Cu 2 : against the normal trend there is an increase in bond length and in De Žas compared to BP86.. In all these cases BP86 performs better than B97GGA-1. The largest percentage deviation in the reaction energies given in Table 1 for BP86 occurs for the dissociation of H 3 BCO to BH 3 and CO, the computed value being 13 kcalrmol larger then the reference value. Dative bonding is in general difficult to describe as is confirmed by this case. There is another way to look at the energetics of molecules. Atomization energies are a very convenient way to discuss the accuracy of methods for molecular electronic structure treatments. In chemistry one usually is interested in reactions between molecules and the corresponding reaction energies which are differences of atomization energies. In this sense it is instructive to consider the extreme deviations from experimental Žor accurate. atomization energies presented in Table 3. There are two effects visible: B97GGA-1 reduces the error interval by about 20% and shifts the atomization energies towards smaller values. Both effects improve the mean deviation from the reference values but only the first improves accuracy for reaction energies.
Table 3 Maximum deviations from reference atomization energies, in kcalrmol Functional
Min.
Max.
Interval
BP86 B97GGA-1
y9.5 ŽSiF4 . y15.1 ŽSiF4 .
35.0 ŽC 2 F4 . 20.7 ŽC 2 F4 .
44.5 35.8
320
R. Ahlrichs et al.r Chemical Physics Letters 325 (2000) 317–321
3. Conclusions The above results can be summarized in two ways. The optimistic view would emphasize that B97GGA-1 shows impressive progress over BP86, it just fails for cases where small densities are of importance. The pessimistic view would be that the errors seen for Na 2 , which is not a difficult case, and for the water dimer exclude its use for much of chemistry and that, on the whole, there is only moderate improvement over BP86. Somewhat similar conclusions could be drawn from other recent investigations. Mateev, Staufer, Mayer and Rosch ¨ w10x have tested the PBE functional w18x, which is found to perform similar as BP86, as well as two recent variants, revPBE w19x and RPBE w20x. The authors conclude that the improvement in atomization energies obtained with revPBE and RPBE is accompanied by slightly poorer structure constants, and bond lengths are even larger than for BP86 or PBE. Adamo, Ernzerhof and Scuseria w8x have investigated the PKZB meta-GGA w21x, which includes the Laplacian of the density and the kinetic energy density, in comparison to PBE. It is also found that improved energetics goes together with poorer bond distances. We note in passing that the assessment of BP86, PBE and PKZB presented by CH and that from Ref. w10x and w8x are not in agreement. Various formal properties of the exchange-correlation functional are known which impose constraints on approximate functional forms, and extensive parameter fitting without these constraints has been questioned w22x. In any case it appears important to consider the target function employed to optimize functional parameters. A minimization of mean absolute or mean square deviations from reference values has some drawbacks: errors of 5 kcalrmol in the De of N2 or the binding energy of Na 2 or the water dimer are of different quality. If this is accepted, it is inevitable to introduce appropriate indiÕidual weights. A related strategy is necessary if gradients Žwith respect to nuclear coordinates. are included in the target function, as done by CH, since otherwise there is a strong bias towards stiff molecules. A special Žlarge. weighting for weakly bound molecules has actually been employed in the HCTH re-parametrizations presented by Boese, Doltsinis, Handy and Sprik w23x. The resulting disso-
ciation energy of the water dimer agrees within 0.5 kcalrmol with the reference value w24x. One should finally consider to include simple reaction energies in the target function: many molecules of Table 1 or the training set of CH can be produced from diatomics, e.g. NO 2 from 1r2 N2 and O 2 . These reaction energies have a direct bearing for chemistry and their inclusion can only improve the usefulness of the resulting functional for chemistry.
4. Note on the implementation of BP86 The TURBOMOLE BP86 results deviate from those obtained by the GAUSSIAN and Q-Chem DFT codes. This has been noted repeatedly and has caused confusion. We would like to stress here that our implementation strictly follows the prescription given by Perdew w1x and is in agreement with that in CADPACK.
Acknowledgements We thank R.D. Amos, A.D. Boese, A.J. Cohen and N.C. Handy for very valuable help especially with the B97GGA-1 functional. This work was supported by the Deutsche Forschungsgesellschaft, SFB 195 Ž‘Lokalisierung von Elektronen in makroskopischen und mikroskopischen Systemen’..
References w1x w2x w3x w4x w5x w6x
J.P. Perdew, Phys. Rev. B 33 Ž1986. 8822. A.D. Becke, Phys. Rev. A 36 Ž1988. 3098. A.D. Becke, J. Chem. Phys. 98 Ž1993. 5648. A.J. Cohen, N.C. Handy, Chem. Phys. Lett. 316 Ž2000. 160. A.D. Becke, J. Chem. Phys. 107 Ž1997. 8554. D.J. Tozer, M.E. Mura, R.D. Amos, N.C. Handy, in: F. Bernardi, L. Rivail ŽEds.., ECCC 1 Computational Chemistry: FECS Conference held at the University Henri Poincare, ´ Nancy, France, May 1994, vol. 330, of AIP Conference Proc., AIP Press, New York, 1995, p. 3. w7x K. Eichkorn, F. Weigend, O. Treutler, R. Ahlrichs, Theor. Chem. Acc. 97 Ž1997. 119. w8x C. Adamo, M. Ernzerhof, G.E. Scuseria, J. Chem. Phys. 112 Ž2000. 2643. w9x M. Haser, O. Treutler, J. Chem. Phys. 102 Ž1995. 3703. ¨
R. Ahlrichs et al.r Chemical Physics Letters 325 (2000) 317–321 w10x A. Mateev, M. Staufer, M. Mayer, N. Røsch, Int. J. Quantum Chem. 75 Ž1999. 863. w11x W. Klopper, M. Schutz, ¨ Ber. Bunsenges. Phys. Chem. 99 Ž1995. 469. w12x C.L. Yu, S.H. Bauer, J. Phys. Chem. Ref. Data 27 Ž1998. 807. w13x C.W. Bauschlicher Jr., A. Ricca, Chem. Phys. Lett. 237 Ž1995. 14. w14x K. Andersson, B.O. Roos, Int. J. Quantum Chem. 45 Ž1993. 591. w15x K.P. Huber, G. Herzberg, Constants of Diatomic Molecules, vol. 4, Molecular Spectra and Molecular Structure, Van Nostrand Reinhold, New York, 1979. w16x A. Haaland, J. Lusztyk, D.P. Novak, J. Brunvoll, K.B. Starowieyski, J. Chem. Soc. Chem. Commun., 1974, p. 54.
321
w17x A.F. Wells, Structural Inorganic Chemistry, 5th edn., Clarendon Press, 1994, Oxford. w18x J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 Ž1996. 3865. w19x Y. Zhang, W. Yang, Phys. Rev. Lett. 80 Ž1998. 890. w20x B. Hammer, L.B. Hansen, J.K. Nørskov, Phys. Rev. B 59 Ž1999. 7413. w21x J.P. Perdew, S. Kurth, A. Zupan, P. Blaha, Phys. Rev. Lett. 82 Ž1999. 2544. w22x S. Kurth, J.P. Perdew, P. Blaha, Int. J. Quantum Chem. 75 Ž1999. 889. w23x A.D. Boese, N.L. Doltsinis, N.C. Handy, M. Sprik, J. Chem. Phys. 112 Ž2000. 1670. w24x A.J. Cohen, N.C. Handy, private communication.
Chemical Physics Letters 325 Ž2000. 317–321 www.elsevier.nlrlocatercplett
Comment on ‘‘Assessment of exchange correlation functionals’’ wA.J. Cohen, N.C. Handy, Chem. Phys. Lett. 316 ž2000/ 160–166x Reinhart Ahlrichs b
a,)
, Filipp Furche a , Stefan Grimme
b
a Institut fur ¨ Physikalische Chemie, UniÕersitat ¨ Karlsruhe, Kaiserstrasse 12, 76128 Karlsruhe, Germany Organisch-Chemisches Institut, Westfalische Wilhelms-UniÕersitat, Germany ¨ ¨ Corrensstrasse 40, 48149 Munster, ¨
Received 10 May 2000; in final form 30 May 2000
Abstract Additional results of the recently proposed B97GGA-1 density functional wA.J. Cohen, N.C. Handy, Chem. Phys. Lett. 316 Ž2000. 160x for molecular dissociation energies, reaction energies and structure parameters are presented. Compared to the BP86 functional, improved energetics and bond lengths are found for electron-rich main-group compounds, but the new GGA seems to fail for weakly bound molecules. These shortcomings are analyzed in connection with the fitting procedure employed by Cohen and Handy. We discuss whether B97GGA-1 is recommendable for general applications in chemistry. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction There have been occasional requests to implement some of the recently developed density functionals into TURBOMOLE, which could be expected to yield better accuracy than, e.g., BP86 w1,2x or B3LYP w3x. These are the recommended functionals of GGA Žgeneralized gradient approximation. and hybrid type. New functionals are currently proposed in rapid succession. It therefore appeared prudent to wait until matters had settled to some extent. Cohen and Handy ŽCH. w4x have recently presented an assessment of functionals which included ) Corresponding author. E-mail: [email protected]
some very encouraging results. Especially promising appeared B97GGA-1: a functional of GGA type Žno exchange mixing, no second derivatives of the density. derived from Becke’s B97 hybrid w5x, by removing Hartree–Fock exchange and re-optimizing parameters. According to Tables 2 and 3 of CH, B97GGA-1 is only slightly surpassed in accuracy by some hybrid functionals. CH employed an extended training set of molecules to optimize parameters which specify the new functional. Since the training set did not include, e.g., transition metal compounds, we decided to carry out some further test calculations. We do not intend to cover the whole of chemistry, but present a biased sample of compounds we consider as interesting and difficult.
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 6 5 4 - 0
R. Ahlrichs et al.r Chemical Physics Letters 325 (2000) 317–321
318
2. Results In Tables 1 and 2 we collect some results for BP86 and B97GGA-1. TZVPP basis sets from the TURBOMOLE basis set library have been used, which include Ž2p1d. for H and Ž2d1f. for main group atoms. For second row atoms a Ž3d1f. set of polarization functions has been employed. All structure parameters have been optimized within the given molecular symmetry group. This procedure corresponds to real life conditions Žwhere structures are
typically not accurately known. and allows to determine the error in computed structure constants. The atomization energies depend very little on the geometry Žexperiment or MP2 or optimized., the effect is typically less than 1 kcalrmol. For the quadrature a multi-grid scheme w6x ‘‘grid m5’’ w7x has been employed. Open shell cases have been treated in the unrestricted scheme to approximate the states 3 Sy g for O 2 , Al 2 , Si 2 , and 2A 2 for NO 2 . It is expected by the authors that the accuracy of the present results is sufficient for the conclusions drawn below.
Table 1 Deviations d s Ecomp y Eref of computed atomization Žreaction. energies Ecomp from reference values Eref , in kcalrmol Molecules
Li 2 N2 O2 F2 Na 2 Al 2 Si 2 P2 P4 2P2 Cl 2 Cu 2 CH 4 H 2O ŽH 2 O. 2 2H 2 O B 2 H 6 2BH 3 H 3 BCO BH 3 q CO Si 2 H 6 CO CO 2 N2 O NO 2 O3 F2 O O 2 F2 NF3 ClNO ClF3 C 2 F4 SiF4 NiŽCO.4 NiŽCO. 3 q CO
™
™ ™ ™
™
a b c d e f g
Eref
a
24.4 228.6 120.5 38.5 17.0 36.2 74.7 117.2 55.6 b 58.0 47.2 c 419.2 232.2 5.0 d 44.e 25.f 529.3 259.3 389.1 270.5 227.5 147.1 93.3 150.2 g 205.2 191.2 125.6 583.9 572.8 25.c
d ŽBP86.
ŽB97GGA-1.
y4.2 9.0 20.4 13.3 y1.0 0.0 4.7 1.3 2.8 4.3 3.0 9.8 5.6 y0.05 3.6 13. 5.6 5.7 20.2 31.3 34.8 31.7 27.1 51.2 33.5 27.1 31.5 35.0 y9.6 4.
0.9 6.4 14.8 8.7 12.0 y0.4 4.7 y4.2 6.8 0.6 5.1 3.3 0.2 y2.2 y3.6 6. 0.0 y0.1 9.8 12.2 17.8 15.3 14.9 31.5 16.4 11.6 17.4 20.7 y15.1 y3.
Experimental atomization energies from Ref. w9x, unless otherwise stated; ZPE removed. 55.0–56.2, Ref. w9x. Ref. w10x. Ref. w11x. Ref. w12x. Ref. w13x. Ref. w14x.
R. Ahlrichs et al.r Chemical Physics Letters 325 (2000) 317–321 Table 2 Representative structure constants; bond lengths in pm, angles in degrees Molecule
Reference
BP86
B97GGA-1
Parameter
Li 2 N2 O2 F2 Na 2 P2 P4 Cl 2 Cu 2 H 2O
267.0 109.8 120.7 141.7 307.9 189.3 219.5a 198.8 220.0 b 95.9 103.9 291. 121.7 c 157.5 c 159.8 169.8 87.5 183.8 d 114.1 206.4 e 144.0 e 293.f
275.0 110.3 122.0 141.5 309.9 191.2 222.3 202.8 223.7 97.0 104.0 290. 120.2 160.4 165.5 175.4 88.9 182.3 114.9 205.6 143.4 297.5
279.9 109.7 120.5 139.2 326.7 189.8 217.9 199.8 225.9 95.9 103.9 312. 118.3 159.0 162.6 173.1 89.1 182.1 114.3 203.5 142.7 306.
HO HOH OO OO OF ClF1 ClF2,ClF3 F2ClF1 NiC CO FeC CC MnMn
ŽH 2 O. 2 O 2 F2 ClF3
NiŽCO.4 FeCp 2 Mn 2 ŽCO.10 a b c d e f
219–200, Ref. w9x. Ref. w15x. Ref. w14x. Ref. w10x. Ref. w16x. Ref. w17x.
The molecules considered include some cases from the CH training set to verify correctness of the code. We have added some electron-rich compounds ŽNO 2 to ClF3 in Table 1., some electron deficient cases ŽB 2 H 6 ., and very few transition metal compounds, the latter mainly to check structure constants. The new B97GGA-1 functional shows an impressive performance in many respects. The relatively large errors of BP86 in computed atomization energies for O 2 , F2 , CO 2 to C 2 F4 ŽTable 1. are markedly reduced, although they are partly not contained in the training set. The energetics of the two boron compounds and the dissociation energy for the first CO of NiŽCO.4 is satisfying. Computed structure constants, Table 2, also show in general an improvement of B97GGA-1 over BP86. This indicates that the training set employed by CH is sufficiently flexible.
319
Problems with B97GGA-1 actually occur within the training set: The dissociation energy ŽDe . of Na 2 is 12 kcalrmol too large, an error of 70% of De . In contrast, the computed binding energy of the water dimer Žwith respect to 2H 2 O. is only 2.8 kcalrmol ŽBP86 and reference value 5.0 kcalrmol.. The water dimer is not contained in the 93 training set, but in the larger 147 set employed for assessment by CH. The bond distances are also not satisfying in these cases: 19 pm too large for Na 2 and 21 pm too large for the OO distance in the water dimer. These features suggest that B97GGA-1 has problems with small electron densities. This is also seen for the MnMn distance in Mn 2 ŽCO.10 which is 13 pm too large. These problems occur to a lesser degree for Li 2 and Cu 2 : against the normal trend there is an increase in bond length and in De Žas compared to BP86.. In all these cases BP86 performs better than B97GGA-1. The largest percentage deviation in the reaction energies given in Table 1 for BP86 occurs for the dissociation of H 3 BCO to BH 3 and CO, the computed value being 13 kcalrmol larger then the reference value. Dative bonding is in general difficult to describe as is confirmed by this case. There is another way to look at the energetics of molecules. Atomization energies are a very convenient way to discuss the accuracy of methods for molecular electronic structure treatments. In chemistry one usually is interested in reactions between molecules and the corresponding reaction energies which are differences of atomization energies. In this sense it is instructive to consider the extreme deviations from experimental Žor accurate. atomization energies presented in Table 3. There are two effects visible: B97GGA-1 reduces the error interval by about 20% and shifts the atomization energies towards smaller values. Both effects improve the mean deviation from the reference values but only the first improves accuracy for reaction energies.
Table 3 Maximum deviations from reference atomization energies, in kcalrmol Functional
Min.
Max.
Interval
BP86 B97GGA-1
y9.5 ŽSiF4 . y15.1 ŽSiF4 .
35.0 ŽC 2 F4 . 20.7 ŽC 2 F4 .
44.5 35.8
320
R. Ahlrichs et al.r Chemical Physics Letters 325 (2000) 317–321
3. Conclusions The above results can be summarized in two ways. The optimistic view would emphasize that B97GGA-1 shows impressive progress over BP86, it just fails for cases where small densities are of importance. The pessimistic view would be that the errors seen for Na 2 , which is not a difficult case, and for the water dimer exclude its use for much of chemistry and that, on the whole, there is only moderate improvement over BP86. Somewhat similar conclusions could be drawn from other recent investigations. Mateev, Staufer, Mayer and Rosch ¨ w10x have tested the PBE functional w18x, which is found to perform similar as BP86, as well as two recent variants, revPBE w19x and RPBE w20x. The authors conclude that the improvement in atomization energies obtained with revPBE and RPBE is accompanied by slightly poorer structure constants, and bond lengths are even larger than for BP86 or PBE. Adamo, Ernzerhof and Scuseria w8x have investigated the PKZB meta-GGA w21x, which includes the Laplacian of the density and the kinetic energy density, in comparison to PBE. It is also found that improved energetics goes together with poorer bond distances. We note in passing that the assessment of BP86, PBE and PKZB presented by CH and that from Ref. w10x and w8x are not in agreement. Various formal properties of the exchange-correlation functional are known which impose constraints on approximate functional forms, and extensive parameter fitting without these constraints has been questioned w22x. In any case it appears important to consider the target function employed to optimize functional parameters. A minimization of mean absolute or mean square deviations from reference values has some drawbacks: errors of 5 kcalrmol in the De of N2 or the binding energy of Na 2 or the water dimer are of different quality. If this is accepted, it is inevitable to introduce appropriate indiÕidual weights. A related strategy is necessary if gradients Žwith respect to nuclear coordinates. are included in the target function, as done by CH, since otherwise there is a strong bias towards stiff molecules. A special Žlarge. weighting for weakly bound molecules has actually been employed in the HCTH re-parametrizations presented by Boese, Doltsinis, Handy and Sprik w23x. The resulting disso-
ciation energy of the water dimer agrees within 0.5 kcalrmol with the reference value w24x. One should finally consider to include simple reaction energies in the target function: many molecules of Table 1 or the training set of CH can be produced from diatomics, e.g. NO 2 from 1r2 N2 and O 2 . These reaction energies have a direct bearing for chemistry and their inclusion can only improve the usefulness of the resulting functional for chemistry.
4. Note on the implementation of BP86 The TURBOMOLE BP86 results deviate from those obtained by the GAUSSIAN and Q-Chem DFT codes. This has been noted repeatedly and has caused confusion. We would like to stress here that our implementation strictly follows the prescription given by Perdew w1x and is in agreement with that in CADPACK.
Acknowledgements We thank R.D. Amos, A.D. Boese, A.J. Cohen and N.C. Handy for very valuable help especially with the B97GGA-1 functional. This work was supported by the Deutsche Forschungsgesellschaft, SFB 195 Ž‘Lokalisierung von Elektronen in makroskopischen und mikroskopischen Systemen’..
References w1x w2x w3x w4x w5x w6x
J.P. Perdew, Phys. Rev. B 33 Ž1986. 8822. A.D. Becke, Phys. Rev. A 36 Ž1988. 3098. A.D. Becke, J. Chem. Phys. 98 Ž1993. 5648. A.J. Cohen, N.C. Handy, Chem. Phys. Lett. 316 Ž2000. 160. A.D. Becke, J. Chem. Phys. 107 Ž1997. 8554. D.J. Tozer, M.E. Mura, R.D. Amos, N.C. Handy, in: F. Bernardi, L. Rivail ŽEds.., ECCC 1 Computational Chemistry: FECS Conference held at the University Henri Poincare, ´ Nancy, France, May 1994, vol. 330, of AIP Conference Proc., AIP Press, New York, 1995, p. 3. w7x K. Eichkorn, F. Weigend, O. Treutler, R. Ahlrichs, Theor. Chem. Acc. 97 Ž1997. 119. w8x C. Adamo, M. Ernzerhof, G.E. Scuseria, J. Chem. Phys. 112 Ž2000. 2643. w9x M. Haser, O. Treutler, J. Chem. Phys. 102 Ž1995. 3703. ¨
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