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Assessment of recently developed density functionals for the calculation of enthalpies of formation in challenging cases
20 August 1999
Chemical Physics Letters 309 Ž1999. 450–456 www.elsevier.nlrlocatercplett
Assessment of recently developed density functionals for the calculation of enthalpies of formation in challenging cases Angela D. Rabuck, Gustavo E. Scuseria
)
Department of Chemistry and Center for Nanoscale Science and Technology, Mail Stop 60, Rice UniÕersity, Houston, TX 77005-1892, USA Received 13 April 1999; in final form 11 June 1999
Abstract The errors in computed enthalpies of formation for a test set of molecules are compared for some recently developed density functional theory ŽDFT. functionals ŽPBE, VSXC, and PBE1PBE. with B3LYP and experiment. The test molecules studied here are an atypical set that includes molecules found to have larger enthalpy of formation errors for DFT methods. Overall, the B3LYP and VSXC functionals yield the best results, with VSXC yielding slightly better results even though, unlike B3LYP, VSXC does not contain any Hartree–Fock exchange. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction There have been considerable amounts of research into computational thermodynamics, particularly into the development of fast and accurate methods of calculation w1–10x. One such possibility would be density functional theory ŽDFT. methods. DFT methods are considerably less computationally expensive than the Gaussian-2 ŽG2. theory w11x, coupled cluster ŽCC., or Møller–Plesset ŽMP. perturbation theory. In this Letter, we investigate the accuracy of three recently proposed DFT functionals, namely, VSXC w12x, PBE w13,14x, and PBE1PBE w15,16x for predicting molecular enthalpies of formation and we compare their accuracy to B3LYP w17x, which is widely considered among the most accurate DFT methods in studies to date w1–5x.
) Corresponding author. Fax: q1-713-285-5155; e-mail: [email protected]
DFT methods that contain Hartree–Fock ŽHF. exchange Žhybrid functionals. have produced more accurate thermochemical data than nonhybrid functionals ŽDFT methods that do not contain HF exchange. w1,3,5x. Nonhybrid functionals do have computational advantages, thus a nonhybrid functional that produces thermochemical data with greater accuracy would be beneficial. In this Letter, we compare the results of two nonhybrid functionals ŽVSXC and PBE. and one hybrid functional ŽPBE1PBE. to B3LYP, a hybrid functional found to produce accurate thermochemical data. VSXC is a t Žkinetic energy density. dependent functional developed recently by our group w12x. Other t dependent functionals have been recently developed w18–20x. Although VSXC does not contain HF exchange, its accuracy for predicting thermochemical w12x and molecular properties, such as proton transfer barriers w21x and vibrational frequencies w22x, is remarkable. The other two functionals that we are assessing, PBE and the PBE hybrid functional ŽPBE1PBE. are both
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 7 2 8 - 9
A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
based on the generalized gradient approximation. The PBE hybrid combines PBE exchange and correlation with exact exchange using a one-parameter equation w16x. The accuracy of a number of DFT functionals to calculate enthalpies of formation and atomization energies Ža part of enthalpies of formation calculations. have been previously studied with promising results w1,3–7,10x. Bauschlicher w3x compared the accuracy in calculating the atomization energies of three nonhybrid functionals ŽBLYP, BP, and BP86. and two hybrid functionals ŽB3LYP and B3P86. to the accuracy of MP2. The accuracy of both the 6-31GU and 6-311q GŽ3df, 2p. basis sets were also studied. The use of the smaller basis set resulted in over twice the error in accuracy. Curtiss et al. w1,4x. compared the accuracy of atomization energy calculations and enthalpies of formation for four nonhybrid functionals ŽBLYP, PBW91, BP86, and SVWN. and three hybrid functionals ŽB3LYP, B3PW91, and B3P86. to G2 theory. Ventura et al. w6,7x compared the accuracy in enthalpy of formation calculations of DFT functionals, B3LYP and B3PW91, to CCSDŽT. and found that the DFT functionals give equivalent or better results than CCSDŽT. in most cases, when utilizing isodesmic reactions. Mole et al. w10x calculated enthalpy of formations for three nonhybrid DFT functionals ŽLSDA, BLYP, and BPW91. and three hybrid functionals ŽB3LYP, B3P86, B3PW91.. In all of the above studies, B3LYP yielded the best DFT results, with an average absolute deviation over the full G2 test set Ž3.1 kcalrmol. approximately twice that of the G2 theory Ž1.6 kcalrmol. w1x. Of the nonhybrid functionals studied, BLYP yielded the best results with an average absolute deviation of 7.1 kcalrmol over the full G2 test set. This is over twice the average absolute deviation found for the B3LYP functional Ž3.1 kcalrmol. w1x. For this reason, we chose the B3LYP functional for our comparisons. From the previous studies mentioned above, we would expect nonhybrid functionals Žin our case VSXC and PBE. to perform worse than hybrid functionals such as B3LYP. It has been previously shown that B3LYP and other DFT methods yield small enthalpy errors over the extended G2 test set Ž3.1 kcalrmol for B3LYP. w1,2,5x. We investigate the accuracy of the four DFT functionals, mentioned previously, for molecules not
451
included in typical density functional training or test sets, a process which will result in larger enthalpy errors. The molecules studied in this Letter are part of a set of molecules found by previous studies to be challenging cases for DFT methods in the sense that large deviations from experimental values Ž) 10 kcalrmol in many cases. are obtained in their enthalpies of formation 1.
2. Computational details In this work, the enthalpies of formation are calculated in the same manner as Curtiss et al. w1,2x. The theoretical enthalpies of formation are calculated by first subtracting calculated atomization energies, Ý D 0 , from the known experimental enthalpies of formation of the isolated atoms to obtain D f H Ž0 K.. This is given by D f H o ŽA x B y H z , 0 K. s xD f H o Ž A, 0 K . q yD f H o Ž B, 0 K . q zD f H o Ž H, 0 K . y Ý D 0 .
Ž 1.
The JANAF w24x values are used for the atomic D f H o Ž0 K. where only the values for Al and Si atoms have large uncertainties Ž1.0 and 2.0 kcalrmol, respectively.. The other atomic enthalpies have uncertainties of less than "0.2 kcalrmol. Corrections are added to D f H o Ž0 K. to calculate the theoretical D f H o Ž298 K.. This is given by D f H o Ž A x B y H z , 298 K . s D f H o ŽA x B y H z , 0 K. q H o Ž A x B y H z , 298 K . y H o Ž A x B y H z , 0 K . y x H o Ž A, 298 K . y H o Ž A, 0 K . o
st
o
y y H Ž B, 298 K . y H Ž B, 0 K .
st
y z H o Ž H, 298 K . y H o Ž H, 0 K .
st
.
Ž 2.
The atomic corrections for the standard state of the elements, w H o ŽX, 298 K. y H o ŽX, 0 K.xst , are also taken from the JANAF tables. The molecular correc-
1
These molecules have been taken from a set of 600 species compiled for testing of electronic structure methods w23x.
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A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
tion, w H o ŽA x B y H z , 298 K. y H o ŽA x B y H z , 0 K.x, is the sum of the vibrational energy harmonic approximation w25x using the scaled calculated frequencies, the PV term, and the classical approximations for the rotational Ž RT and 32 RT for linear and nonlinear molecules, respectively. and translational Ž 23 RT. terms. As in previous studies, the resulting D f H o Ž298 K. values will be considered theoretical numbers, even though they contain experimental data. A number of other methods have been utilized to calculate enthalpies of formation. These methods include the use of isodesmic reactions w6–8,26–28x, homonuclear diatomic reactions w6x, and an atom equivalence method w9,10x. Isodesmic reactions, like for example, H 2 O q ClO ™ ClOHq OH q D r H o
Ž 3.
involve using experimental enthalpies of formation for three of the four molecules to obtain the enthalpy of formation for the fourth molecule. This procedure will cancel out a number of errors, but the resulting enthalpy will depend on the isodesmic reaction chosen w29x. Homonuclear diatomic reactions like N2 O4 ™ N2 q 2O 2
Ž 4.
can be used to calculate the enthalpies without including any experimental data. In the above case, the D f H o Ž298 K . of N 2 O4 would be equal to D r H o Ž298 K.. We investigated the possibility of using this procedure for our calculations, but our data indicated that this method would not increase the accuracy over our test set of molecules. Another method proposed to calculate enthalpies is the use of an atom equivalence method for converting DFT energies into enthalpies of formation w9,10x. In this method, the atom equivalence for each molecule has to be derived explicitly from fitting experimental enthalpies before one can calculate the enthalpies of formation of interest. Cioslowski et al. w9x. also included factors for molecular spin and spin multiplicity in their derivation of atom equivalences to increase accuracy.
3. Results and discussion All calculations were done using a developmental version of the Gaussian suite of programs w30x. The
geometries of each test molecule were optimized for each functional using the 6-311 q GŽ3df, 2p. basis set. The zero-point energies and frequencies were calculated using B3LYPr6-311 q GŽ3df, 2p. and scaled by 0.989 w31x. The deviations of the B3LYP, VSXC, PBE1PBE, and PBE enthalpies of formation from experiment are listed in Table 1 for our test set of molecules. A few of the test molecules are included in the original and extended G2 test sets of molecules. The second column of Table 1 lists which molecules are included in the original G2 set ŽA., extended G2 set ŽB., and four additional molecules that were included in the VSXC training set ŽC.. Atomization energies for the molecules in the original G2 set were included in the B3LYP training set. The training set for VSXC included Žamong others. the atomization energies for the molecules in the original G2 set and the four additional molecules noted in Table 1. Atomization energies of all the unmarked molecules were not included in any functional training sets. Table 1 lists the average absolute deviations and root mean squares for the four DFT methods. B3LYP has an average absolute deviation of 3.1 kcalrmol over the extended G2 test set w1x, but this deviation increases to an average absolute deviation of 10.6 kcalrmol for our test set. This increase in deviations illustrates the reason why we chose this test set. VSXC was previously shown to have an average absolute deviation of 2.7 kcalrmol over the extended G2 test set w12x. This increases to 8.8 kcalrmol for our molecule test set. The deviation errors for both DFT methods more than doubles for our group of molecules that are not included in standard testing and training sets. Overall VSXC yields slightly better results than B3LYP. The average absolute deviations for PBE and PBE1PBE over our test set are 38.2 and 11.5 kcalrmol, respectively. Thus, the nonhybrid PBE functional is much worse at predicting accurate enthalpies of formation, but the hybrid PBE functional ŽPBE1PBE. is comparable to B3LYP. PBE has large deviations from experimental enthalpies of formation for most of the molecules studied here and, with the exceptions of SiF4 and AlŽC 2 H 5 . 3 , overbinds Žpositive deviation. nearly all molecules in this test set. PBE does not contain any HF exchange and, as mentioned earlier, DFT meth-
A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
453
Table 1 Deviation ŽExperimenty Theory. of enthalpies calculated by DFT methods from experimental results Žkcalrmol. B3LYP
VSXC
PBE1PBE
PBE
Ref.a
y39.1
y37.2
y18.0
y21.2
y5.9
w9x
y157.4 y369.5 y228.5 y194.1
5.2 y5.0 y3.2 y8.1
13.0 24.6 21.8 y15.2
18.9 33.5 39.5 38.6
52.5 93.3 102.9 98.3
w32x w32x w32x w32x
2.9 25.0 y5.1 4.6
6.6 0.8 y15.1 y16.9
10.2 5.2 y5.4 y21.4
11.4 5.3 y0.5 y21.5
54.0 36.5 46.7 27.7
w32x w32x w32x w33x
2.2 y32.1
10.4 6.4
18.4 6.5
16.2 9.9
93.4 54.2
w32x w32x
y94.1 y33.1 28.0 y22.4 y29.3 y80.3 y139.7
0.7 1.2 y0.1 6.3 y7.9 y7.1 y21.7
6.0 9.7 11.8 18.7 5.1 6.3 y10.3
4.9 6.7 7.1 17.6 18.4 11.5 y5.8
29.3 27.6 24.4 58.8 64.2 48.0 13.9
w32x w32x w32x w32x w32x w34x w9x
y175.7 24.0 y182.4 y291.7 y181.3 y84.8 y70.9 y94.6 y50.8
y21.0 y20.7 y4.7 y18.2 y13.6 y11.7 y7.7 y10.3 y9.0
y17.6 y5.7 6.1 5.6 y5.1 y2.6 y3.3 y2.4 y2.8
y11.6 13.4 0.2 y2.1 y3.3 3.9 y6.2 y3.5 0.1
23.8 52.3 34.6 34.8 30.4 34.3 21.1 31.6 25.2
w32x w32x w32x w32x w32x w32x w32x w32x w32x
y299.8 34.3 14.1 y381.1 y89.6
y12.3 y0.6 y9.0 y18.3 y19.6
y4.2 y1.8 y5.6 y2.3 1.2
y10.5 y4.9 5.0 y14.2 2.1
13.6 4.9 20.1 10.7 19.6
w35x w32x w32x w32x w32x
A B
139.9 y386.0
y3.8 y19.7
5.3 0.3
y1.2 y21.0
6.5 y4.4
w32x w32x
– – – –
yy yy yy yy
10.6 13.3 10.4 37.2
8.8 11.1 24.3 21.4
11.5 15.4 39.5 21.5
38.2 46.6 102.9 5.9
Molecule
Exp.
AlŽC 2 H 5 . 3 C 2 F4 C 4 F8 C 6 F6 C 6 F5 Cl ClNO 2 ClO 2 FClO 3 FOOF
B
C
N2 O4 HNO 3 CO 2 COS CS 2 OCCCO p-OC 6 H 4 O ŽCOCl. 2 C 2 H 5 POCl 2 H 2 SO4 S8 SF4 SF6 SO 2 F2 SO 2 Cl 2 SO 2 SO 3 SOCl 2 F3 PO P2 P4 PF5 PCl 5 Si 2 SiF4
A B B
C
A C
A C
Totals: < x< sx Max. dev. Žq. Max. dev. Žy.
A s molecule is in the original G2 test set; B s molecule is in the extended G2 test set; C s molecule was included in the VSXC training set. a References for the experimental values.
454
A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
ods that include HF exchange are typically more accurate for thermochemical predictions w1,3–5,10x. Because of this, one would expect VSXC to perform poorly in comparison to B3LYP. On the contrary, VSXC is slightly better than B3LYP in the prediction of the enthalpies of formation for a number of these molecules, in particular SiF4 and PCl 5 . B3LYP does not accurately predict the enthalpies for molecules containing centrally bonded S or P atoms. For the nine sulfur compounds listed in section six of Table 1, B3LYP has an average absolute deviation of 13.0 kcalrmol, while VSXC and PBE1PBE have average absolute deviations of 6.1 and 4.9 kcalrmol, respectively. VSXC and PBE1PBE more accurately predict the enthalpies for the five phosphorus compounds listed than B3LYP does. The average absolute deviations for these five
Fig. 2. Histogram of deviations for Ža. PBE1PBE and Žb. PBE for our test set of molecules. Each vertical bar represents deviations in a 1 kcalrmol range.
Fig. 1. Histogram of deviations for Ža. B3LYP and Žb. VSXC for our test set of molecules. Each vertical bar represents deviations in a 1 kcalrmol range.
molecules is 3.0, 7.3, and 12.0 kcalrmol for VSXC, PBE1PBE, and B3LYP, respectively. It is interesting to note that B3LYP underbinds Žnegative deviation. every test molecule containing a centrally bonded S or P atom. VSXC and PBE1PBE also underbind most of the enthalpies for molecules with a centrally bonded S or P atom, but exhibit less difficulties than B3LYP. The inclusion of additional molecules containing centrally bonded S or P atoms in the VSXC training set improved enthalpy of formation predictions for molecules of this type. The distribution of deviations for B3LYP and VSXC are given in Fig. 1. The distribution for VSXC is spread more evenly than for B3LYP. The B3LYP functional has more negative deviations Žunderbinding. than VSXC and also covers a larger
A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
range Žy22 to 38 kcalrmol. than VSXC Žy22 to 25 kcalrmol.. The distribution of deviations for PBE and PBE1PBE are given in Fig. 2. PBE1PBE’s deviations Žy22 to 40 kcalrmol. are more evenly distributed than for B3LYP while covering a similar range of deviations. PBE has considerably more positive deviations Žoverbinding. than the other DFT methods studied here as well as a much larger deviation range Žy6 to 103 kcalrmol.. Adding in corrections for spin–orbit interactions have been found to lower the errors in the calculation of enthalpies of formation on the order of between 0.08 and 0.84 kcalrmol Žvalues for carbon and chlorine atoms, respectively. for each atom w1,5x. In our case, these corrections would not be on a large enough scale to greatly lower the errors and would not affect the trends between methods. 4. Conclusions The accuracy of four DFT methods ŽPBE, VSXC, PBE1PBE, and B3LYP. for calculating enthalpies of formation have been presented in this Letter. Our test set included numerous molecules that have been found problematic for DFT methods. We have found that the deviations in enthalpy of formation errors over this set for the four DFT methods studied are larger than what was found in previous studies for standard test sets. From our results, it appears that DFT functionals need improvement before they can predict accurate enthalpies of formation for classes of molecules not explicitly treated in the functional training sets. There are distinct classes of molecules for which enthalpies need improvement. VSXC Ž8.8 kcalrmol average absolute deviation. without HF exchange is equivalent, or better, than the hybrid functionals, such as B3LYP Ž10.6 kcalrmol.. The placement of additional molecules containing centrally bonded S and P in the fitting set for VSXC improved the accuracy for those types of molecules. Similar procedures could improve the accuracy of numerous other DFT functionals as well. PBE1PBE Ž11.5 kcalrmol average absolute deviation. is comparable to, but overall has larger errors than, B3LYP Ž10.6 kcalrmol.. There are some classes of molecules, such as those mentioned in Section 3, where PBE1PBE predicts better enthalpies than B3LYP. As expected, the PBE func-
455
tional, with an average absolute deviation of 38.2 kcalrmol, does not reproduce experimental enthalpies of formation satisfactorily. Acknowledgements This work was supported by the National Science Foundation ŽGrant No. CHE-9618323., the Welch Foundation ŽGrant No. C-1196., and Gaussian, Inc. A.D.R. is a Marjory Meyer Hasselmann Fellow. We thank Jerzy Cioslowski for providing the test molecules used in this study. References w1x L.A. Curtiss, K. Raghavachari, P.C. Redfern, J.A. Pople, J. Chem. Phys. 106 Ž1997. 1063. w2x L.A. Curtiss, K. Raghavachari, P.W. Deutsch, J.A. Pople, J. Chem. Phys. 95 Ž1991. 2433. w3x C.W. Bauschlicher Jr., Chem. Phys. Lett. 246 Ž1995. 40. w4x K.K. Irikura, D.J. Frurip ŽEds.., Computational Thermochemistry, ACS Symp. Ser. No. 667, Am. Chem. Soc., Washington, DC, 1998. w5x G.A. Petersson, D.K. Malick, W.G. Wilson, J.W. Ochterski, J.A. Montgomery Jr., M.J. Frisch, J. Chem. Phys. 109 Ž1998. 10570. w6x O.N. Ventura, M. Kieninger, R.E. Cachau, J. Phys. Chem. A 103 Ž1999. 147. w7x O.N. Ventura, R.E. Cachau, M. Kieninger, Chem. Phys. Lett. 301 Ž1999. 331. w8x L.A. Curtiss, K. Raghavachari, P.C. Redern, B.B. Stefanov, J. Chem. Phys. 108 Ž1998. 692. w9x J. Cioslowski, G. Liu, P. Piskorz, J. Phys. Chem. A 102 Ž1998. 9890. w10x S.J. Mole, X. Zhou, R. Liu, J. Phys. Chem. 100 Ž1996. 14665. w11x L.A. Curtiss, K. Raghavachari, G.W. Trucks, J.A. Pople, J. Chem. Phys. 94 Ž1991. 7221. w12x T. Van Voorhis, G.E. Scuseria, J. Chem. Phys. 109 Ž1998. 400. w13x J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 Ž1996. 1396. w14x J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 Ž1997. 1396 Žerratum.. w15x M. Ernzerhof, J.P. Perdew, K. Burke, Int. J. Quantum Chem. 64 Ž1997. 285. w16x M. Ernzerhof, G.E. Scuseria, J. Chem. Phys. 110 Ž1999. 5029. w17x A.D. Becke, J. Chem. Phys. 98 Ž1993. 5648. w18x A.D. Becke, J. Chem. Phys. 104 Ž1996. 1040. w19x M. Ernzerhof, G.E. Scuseria, J. Chem. Phys. Žin press.. w20x J.P. Perdew, S. Kurth, A. Zupan, P. Blaha, Phys. Rev. Lett. 82 Ž1999. 2544.
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w21x S. Sadhukhan, D. Munoz, ˜ C. Adamo, G.E. Scuseria, Chem. Phys. Lett. Žin press.. w22x J. Jaramillo, G.E. Scuseria, Chem. Phys. Lett. Žin press.. w23x These molecules have been taken from a set of 600 species compiled for testing of electronic structure methods: J. Cioslowski, M. Schimeczek, G. Liu, V. Stoyanov Žto be published.. w24x M.W. Chase, Jr., C.A. Davies, J.R. Downey, Jr., D.J. Frurip, R.A. McDonald, A.N. Syverud, J. Phys. Chem. Ref. Data 14, Suppl. No. 1 Ž1985.. w25x G.N. Lewis, M. Randall, Thermodynamics, 2nd ed., McGraw-Hill, New York, 1961 Žrevised by K.S. Pitzer, L. Brewer... w26x Y.G. Lazarou, P. Papagiannakopoulos, Chem. Phys. Lett. 301 Ž1999. 19. w27x E.W. Ignacio, H.B. Schlegel, J. Phys. Chem. 94 Ž1990. 7439. w28x B.M. Messer, M.J. Elrod, Chem. Phys. Lett. 301 Ž1999. 10. w29x J.B. Foresman, A. Frisch, Exploring Chemistry with Electronic Structure Methods, 2nd ed., Gaussian, Inc., Pittsburgh, PA, 1996. w30x M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, V.G. Zakrzewski, J.A. Mont-
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Chemical Physics Letters 309 Ž1999. 450–456 www.elsevier.nlrlocatercplett
Assessment of recently developed density functionals for the calculation of enthalpies of formation in challenging cases Angela D. Rabuck, Gustavo E. Scuseria
)
Department of Chemistry and Center for Nanoscale Science and Technology, Mail Stop 60, Rice UniÕersity, Houston, TX 77005-1892, USA Received 13 April 1999; in final form 11 June 1999
Abstract The errors in computed enthalpies of formation for a test set of molecules are compared for some recently developed density functional theory ŽDFT. functionals ŽPBE, VSXC, and PBE1PBE. with B3LYP and experiment. The test molecules studied here are an atypical set that includes molecules found to have larger enthalpy of formation errors for DFT methods. Overall, the B3LYP and VSXC functionals yield the best results, with VSXC yielding slightly better results even though, unlike B3LYP, VSXC does not contain any Hartree–Fock exchange. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction There have been considerable amounts of research into computational thermodynamics, particularly into the development of fast and accurate methods of calculation w1–10x. One such possibility would be density functional theory ŽDFT. methods. DFT methods are considerably less computationally expensive than the Gaussian-2 ŽG2. theory w11x, coupled cluster ŽCC., or Møller–Plesset ŽMP. perturbation theory. In this Letter, we investigate the accuracy of three recently proposed DFT functionals, namely, VSXC w12x, PBE w13,14x, and PBE1PBE w15,16x for predicting molecular enthalpies of formation and we compare their accuracy to B3LYP w17x, which is widely considered among the most accurate DFT methods in studies to date w1–5x.
) Corresponding author. Fax: q1-713-285-5155; e-mail: [email protected]
DFT methods that contain Hartree–Fock ŽHF. exchange Žhybrid functionals. have produced more accurate thermochemical data than nonhybrid functionals ŽDFT methods that do not contain HF exchange. w1,3,5x. Nonhybrid functionals do have computational advantages, thus a nonhybrid functional that produces thermochemical data with greater accuracy would be beneficial. In this Letter, we compare the results of two nonhybrid functionals ŽVSXC and PBE. and one hybrid functional ŽPBE1PBE. to B3LYP, a hybrid functional found to produce accurate thermochemical data. VSXC is a t Žkinetic energy density. dependent functional developed recently by our group w12x. Other t dependent functionals have been recently developed w18–20x. Although VSXC does not contain HF exchange, its accuracy for predicting thermochemical w12x and molecular properties, such as proton transfer barriers w21x and vibrational frequencies w22x, is remarkable. The other two functionals that we are assessing, PBE and the PBE hybrid functional ŽPBE1PBE. are both
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 7 2 8 - 9
A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
based on the generalized gradient approximation. The PBE hybrid combines PBE exchange and correlation with exact exchange using a one-parameter equation w16x. The accuracy of a number of DFT functionals to calculate enthalpies of formation and atomization energies Ža part of enthalpies of formation calculations. have been previously studied with promising results w1,3–7,10x. Bauschlicher w3x compared the accuracy in calculating the atomization energies of three nonhybrid functionals ŽBLYP, BP, and BP86. and two hybrid functionals ŽB3LYP and B3P86. to the accuracy of MP2. The accuracy of both the 6-31GU and 6-311q GŽ3df, 2p. basis sets were also studied. The use of the smaller basis set resulted in over twice the error in accuracy. Curtiss et al. w1,4x. compared the accuracy of atomization energy calculations and enthalpies of formation for four nonhybrid functionals ŽBLYP, PBW91, BP86, and SVWN. and three hybrid functionals ŽB3LYP, B3PW91, and B3P86. to G2 theory. Ventura et al. w6,7x compared the accuracy in enthalpy of formation calculations of DFT functionals, B3LYP and B3PW91, to CCSDŽT. and found that the DFT functionals give equivalent or better results than CCSDŽT. in most cases, when utilizing isodesmic reactions. Mole et al. w10x calculated enthalpy of formations for three nonhybrid DFT functionals ŽLSDA, BLYP, and BPW91. and three hybrid functionals ŽB3LYP, B3P86, B3PW91.. In all of the above studies, B3LYP yielded the best DFT results, with an average absolute deviation over the full G2 test set Ž3.1 kcalrmol. approximately twice that of the G2 theory Ž1.6 kcalrmol. w1x. Of the nonhybrid functionals studied, BLYP yielded the best results with an average absolute deviation of 7.1 kcalrmol over the full G2 test set. This is over twice the average absolute deviation found for the B3LYP functional Ž3.1 kcalrmol. w1x. For this reason, we chose the B3LYP functional for our comparisons. From the previous studies mentioned above, we would expect nonhybrid functionals Žin our case VSXC and PBE. to perform worse than hybrid functionals such as B3LYP. It has been previously shown that B3LYP and other DFT methods yield small enthalpy errors over the extended G2 test set Ž3.1 kcalrmol for B3LYP. w1,2,5x. We investigate the accuracy of the four DFT functionals, mentioned previously, for molecules not
451
included in typical density functional training or test sets, a process which will result in larger enthalpy errors. The molecules studied in this Letter are part of a set of molecules found by previous studies to be challenging cases for DFT methods in the sense that large deviations from experimental values Ž) 10 kcalrmol in many cases. are obtained in their enthalpies of formation 1.
2. Computational details In this work, the enthalpies of formation are calculated in the same manner as Curtiss et al. w1,2x. The theoretical enthalpies of formation are calculated by first subtracting calculated atomization energies, Ý D 0 , from the known experimental enthalpies of formation of the isolated atoms to obtain D f H Ž0 K.. This is given by D f H o ŽA x B y H z , 0 K. s xD f H o Ž A, 0 K . q yD f H o Ž B, 0 K . q zD f H o Ž H, 0 K . y Ý D 0 .
Ž 1.
The JANAF w24x values are used for the atomic D f H o Ž0 K. where only the values for Al and Si atoms have large uncertainties Ž1.0 and 2.0 kcalrmol, respectively.. The other atomic enthalpies have uncertainties of less than "0.2 kcalrmol. Corrections are added to D f H o Ž0 K. to calculate the theoretical D f H o Ž298 K.. This is given by D f H o Ž A x B y H z , 298 K . s D f H o ŽA x B y H z , 0 K. q H o Ž A x B y H z , 298 K . y H o Ž A x B y H z , 0 K . y x H o Ž A, 298 K . y H o Ž A, 0 K . o
st
o
y y H Ž B, 298 K . y H Ž B, 0 K .
st
y z H o Ž H, 298 K . y H o Ž H, 0 K .
st
.
Ž 2.
The atomic corrections for the standard state of the elements, w H o ŽX, 298 K. y H o ŽX, 0 K.xst , are also taken from the JANAF tables. The molecular correc-
1
These molecules have been taken from a set of 600 species compiled for testing of electronic structure methods w23x.
452
A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
tion, w H o ŽA x B y H z , 298 K. y H o ŽA x B y H z , 0 K.x, is the sum of the vibrational energy harmonic approximation w25x using the scaled calculated frequencies, the PV term, and the classical approximations for the rotational Ž RT and 32 RT for linear and nonlinear molecules, respectively. and translational Ž 23 RT. terms. As in previous studies, the resulting D f H o Ž298 K. values will be considered theoretical numbers, even though they contain experimental data. A number of other methods have been utilized to calculate enthalpies of formation. These methods include the use of isodesmic reactions w6–8,26–28x, homonuclear diatomic reactions w6x, and an atom equivalence method w9,10x. Isodesmic reactions, like for example, H 2 O q ClO ™ ClOHq OH q D r H o
Ž 3.
involve using experimental enthalpies of formation for three of the four molecules to obtain the enthalpy of formation for the fourth molecule. This procedure will cancel out a number of errors, but the resulting enthalpy will depend on the isodesmic reaction chosen w29x. Homonuclear diatomic reactions like N2 O4 ™ N2 q 2O 2
Ž 4.
can be used to calculate the enthalpies without including any experimental data. In the above case, the D f H o Ž298 K . of N 2 O4 would be equal to D r H o Ž298 K.. We investigated the possibility of using this procedure for our calculations, but our data indicated that this method would not increase the accuracy over our test set of molecules. Another method proposed to calculate enthalpies is the use of an atom equivalence method for converting DFT energies into enthalpies of formation w9,10x. In this method, the atom equivalence for each molecule has to be derived explicitly from fitting experimental enthalpies before one can calculate the enthalpies of formation of interest. Cioslowski et al. w9x. also included factors for molecular spin and spin multiplicity in their derivation of atom equivalences to increase accuracy.
3. Results and discussion All calculations were done using a developmental version of the Gaussian suite of programs w30x. The
geometries of each test molecule were optimized for each functional using the 6-311 q GŽ3df, 2p. basis set. The zero-point energies and frequencies were calculated using B3LYPr6-311 q GŽ3df, 2p. and scaled by 0.989 w31x. The deviations of the B3LYP, VSXC, PBE1PBE, and PBE enthalpies of formation from experiment are listed in Table 1 for our test set of molecules. A few of the test molecules are included in the original and extended G2 test sets of molecules. The second column of Table 1 lists which molecules are included in the original G2 set ŽA., extended G2 set ŽB., and four additional molecules that were included in the VSXC training set ŽC.. Atomization energies for the molecules in the original G2 set were included in the B3LYP training set. The training set for VSXC included Žamong others. the atomization energies for the molecules in the original G2 set and the four additional molecules noted in Table 1. Atomization energies of all the unmarked molecules were not included in any functional training sets. Table 1 lists the average absolute deviations and root mean squares for the four DFT methods. B3LYP has an average absolute deviation of 3.1 kcalrmol over the extended G2 test set w1x, but this deviation increases to an average absolute deviation of 10.6 kcalrmol for our test set. This increase in deviations illustrates the reason why we chose this test set. VSXC was previously shown to have an average absolute deviation of 2.7 kcalrmol over the extended G2 test set w12x. This increases to 8.8 kcalrmol for our molecule test set. The deviation errors for both DFT methods more than doubles for our group of molecules that are not included in standard testing and training sets. Overall VSXC yields slightly better results than B3LYP. The average absolute deviations for PBE and PBE1PBE over our test set are 38.2 and 11.5 kcalrmol, respectively. Thus, the nonhybrid PBE functional is much worse at predicting accurate enthalpies of formation, but the hybrid PBE functional ŽPBE1PBE. is comparable to B3LYP. PBE has large deviations from experimental enthalpies of formation for most of the molecules studied here and, with the exceptions of SiF4 and AlŽC 2 H 5 . 3 , overbinds Žpositive deviation. nearly all molecules in this test set. PBE does not contain any HF exchange and, as mentioned earlier, DFT meth-
A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
453
Table 1 Deviation ŽExperimenty Theory. of enthalpies calculated by DFT methods from experimental results Žkcalrmol. B3LYP
VSXC
PBE1PBE
PBE
Ref.a
y39.1
y37.2
y18.0
y21.2
y5.9
w9x
y157.4 y369.5 y228.5 y194.1
5.2 y5.0 y3.2 y8.1
13.0 24.6 21.8 y15.2
18.9 33.5 39.5 38.6
52.5 93.3 102.9 98.3
w32x w32x w32x w32x
2.9 25.0 y5.1 4.6
6.6 0.8 y15.1 y16.9
10.2 5.2 y5.4 y21.4
11.4 5.3 y0.5 y21.5
54.0 36.5 46.7 27.7
w32x w32x w32x w33x
2.2 y32.1
10.4 6.4
18.4 6.5
16.2 9.9
93.4 54.2
w32x w32x
y94.1 y33.1 28.0 y22.4 y29.3 y80.3 y139.7
0.7 1.2 y0.1 6.3 y7.9 y7.1 y21.7
6.0 9.7 11.8 18.7 5.1 6.3 y10.3
4.9 6.7 7.1 17.6 18.4 11.5 y5.8
29.3 27.6 24.4 58.8 64.2 48.0 13.9
w32x w32x w32x w32x w32x w34x w9x
y175.7 24.0 y182.4 y291.7 y181.3 y84.8 y70.9 y94.6 y50.8
y21.0 y20.7 y4.7 y18.2 y13.6 y11.7 y7.7 y10.3 y9.0
y17.6 y5.7 6.1 5.6 y5.1 y2.6 y3.3 y2.4 y2.8
y11.6 13.4 0.2 y2.1 y3.3 3.9 y6.2 y3.5 0.1
23.8 52.3 34.6 34.8 30.4 34.3 21.1 31.6 25.2
w32x w32x w32x w32x w32x w32x w32x w32x w32x
y299.8 34.3 14.1 y381.1 y89.6
y12.3 y0.6 y9.0 y18.3 y19.6
y4.2 y1.8 y5.6 y2.3 1.2
y10.5 y4.9 5.0 y14.2 2.1
13.6 4.9 20.1 10.7 19.6
w35x w32x w32x w32x w32x
A B
139.9 y386.0
y3.8 y19.7
5.3 0.3
y1.2 y21.0
6.5 y4.4
w32x w32x
– – – –
yy yy yy yy
10.6 13.3 10.4 37.2
8.8 11.1 24.3 21.4
11.5 15.4 39.5 21.5
38.2 46.6 102.9 5.9
Molecule
Exp.
AlŽC 2 H 5 . 3 C 2 F4 C 4 F8 C 6 F6 C 6 F5 Cl ClNO 2 ClO 2 FClO 3 FOOF
B
C
N2 O4 HNO 3 CO 2 COS CS 2 OCCCO p-OC 6 H 4 O ŽCOCl. 2 C 2 H 5 POCl 2 H 2 SO4 S8 SF4 SF6 SO 2 F2 SO 2 Cl 2 SO 2 SO 3 SOCl 2 F3 PO P2 P4 PF5 PCl 5 Si 2 SiF4
A B B
C
A C
A C
Totals: < x< sx Max. dev. Žq. Max. dev. Žy.
A s molecule is in the original G2 test set; B s molecule is in the extended G2 test set; C s molecule was included in the VSXC training set. a References for the experimental values.
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A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
ods that include HF exchange are typically more accurate for thermochemical predictions w1,3–5,10x. Because of this, one would expect VSXC to perform poorly in comparison to B3LYP. On the contrary, VSXC is slightly better than B3LYP in the prediction of the enthalpies of formation for a number of these molecules, in particular SiF4 and PCl 5 . B3LYP does not accurately predict the enthalpies for molecules containing centrally bonded S or P atoms. For the nine sulfur compounds listed in section six of Table 1, B3LYP has an average absolute deviation of 13.0 kcalrmol, while VSXC and PBE1PBE have average absolute deviations of 6.1 and 4.9 kcalrmol, respectively. VSXC and PBE1PBE more accurately predict the enthalpies for the five phosphorus compounds listed than B3LYP does. The average absolute deviations for these five
Fig. 2. Histogram of deviations for Ža. PBE1PBE and Žb. PBE for our test set of molecules. Each vertical bar represents deviations in a 1 kcalrmol range.
Fig. 1. Histogram of deviations for Ža. B3LYP and Žb. VSXC for our test set of molecules. Each vertical bar represents deviations in a 1 kcalrmol range.
molecules is 3.0, 7.3, and 12.0 kcalrmol for VSXC, PBE1PBE, and B3LYP, respectively. It is interesting to note that B3LYP underbinds Žnegative deviation. every test molecule containing a centrally bonded S or P atom. VSXC and PBE1PBE also underbind most of the enthalpies for molecules with a centrally bonded S or P atom, but exhibit less difficulties than B3LYP. The inclusion of additional molecules containing centrally bonded S or P atoms in the VSXC training set improved enthalpy of formation predictions for molecules of this type. The distribution of deviations for B3LYP and VSXC are given in Fig. 1. The distribution for VSXC is spread more evenly than for B3LYP. The B3LYP functional has more negative deviations Žunderbinding. than VSXC and also covers a larger
A.D. Rabuck, G.E. Scuseriar Chemical Physics Letters 309 (1999) 450–456
range Žy22 to 38 kcalrmol. than VSXC Žy22 to 25 kcalrmol.. The distribution of deviations for PBE and PBE1PBE are given in Fig. 2. PBE1PBE’s deviations Žy22 to 40 kcalrmol. are more evenly distributed than for B3LYP while covering a similar range of deviations. PBE has considerably more positive deviations Žoverbinding. than the other DFT methods studied here as well as a much larger deviation range Žy6 to 103 kcalrmol.. Adding in corrections for spin–orbit interactions have been found to lower the errors in the calculation of enthalpies of formation on the order of between 0.08 and 0.84 kcalrmol Žvalues for carbon and chlorine atoms, respectively. for each atom w1,5x. In our case, these corrections would not be on a large enough scale to greatly lower the errors and would not affect the trends between methods. 4. Conclusions The accuracy of four DFT methods ŽPBE, VSXC, PBE1PBE, and B3LYP. for calculating enthalpies of formation have been presented in this Letter. Our test set included numerous molecules that have been found problematic for DFT methods. We have found that the deviations in enthalpy of formation errors over this set for the four DFT methods studied are larger than what was found in previous studies for standard test sets. From our results, it appears that DFT functionals need improvement before they can predict accurate enthalpies of formation for classes of molecules not explicitly treated in the functional training sets. There are distinct classes of molecules for which enthalpies need improvement. VSXC Ž8.8 kcalrmol average absolute deviation. without HF exchange is equivalent, or better, than the hybrid functionals, such as B3LYP Ž10.6 kcalrmol.. The placement of additional molecules containing centrally bonded S and P in the fitting set for VSXC improved the accuracy for those types of molecules. Similar procedures could improve the accuracy of numerous other DFT functionals as well. PBE1PBE Ž11.5 kcalrmol average absolute deviation. is comparable to, but overall has larger errors than, B3LYP Ž10.6 kcalrmol.. There are some classes of molecules, such as those mentioned in Section 3, where PBE1PBE predicts better enthalpies than B3LYP. As expected, the PBE func-
455
tional, with an average absolute deviation of 38.2 kcalrmol, does not reproduce experimental enthalpies of formation satisfactorily. Acknowledgements This work was supported by the National Science Foundation ŽGrant No. CHE-9618323., the Welch Foundation ŽGrant No. C-1196., and Gaussian, Inc. A.D.R. is a Marjory Meyer Hasselmann Fellow. We thank Jerzy Cioslowski for providing the test molecules used in this study. References w1x L.A. Curtiss, K. Raghavachari, P.C. Redfern, J.A. Pople, J. Chem. Phys. 106 Ž1997. 1063. w2x L.A. Curtiss, K. Raghavachari, P.W. Deutsch, J.A. Pople, J. Chem. Phys. 95 Ž1991. 2433. w3x C.W. Bauschlicher Jr., Chem. Phys. Lett. 246 Ž1995. 40. w4x K.K. Irikura, D.J. Frurip ŽEds.., Computational Thermochemistry, ACS Symp. Ser. No. 667, Am. Chem. Soc., Washington, DC, 1998. w5x G.A. Petersson, D.K. Malick, W.G. Wilson, J.W. Ochterski, J.A. Montgomery Jr., M.J. Frisch, J. Chem. Phys. 109 Ž1998. 10570. w6x O.N. Ventura, M. Kieninger, R.E. Cachau, J. Phys. Chem. A 103 Ž1999. 147. w7x O.N. Ventura, R.E. Cachau, M. Kieninger, Chem. Phys. Lett. 301 Ž1999. 331. w8x L.A. Curtiss, K. Raghavachari, P.C. Redern, B.B. Stefanov, J. Chem. Phys. 108 Ž1998. 692. w9x J. Cioslowski, G. Liu, P. Piskorz, J. Phys. Chem. A 102 Ž1998. 9890. w10x S.J. Mole, X. Zhou, R. Liu, J. Phys. Chem. 100 Ž1996. 14665. w11x L.A. Curtiss, K. Raghavachari, G.W. Trucks, J.A. Pople, J. Chem. Phys. 94 Ž1991. 7221. w12x T. Van Voorhis, G.E. Scuseria, J. Chem. Phys. 109 Ž1998. 400. w13x J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 Ž1996. 1396. w14x J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 Ž1997. 1396 Žerratum.. w15x M. Ernzerhof, J.P. Perdew, K. Burke, Int. J. Quantum Chem. 64 Ž1997. 285. w16x M. Ernzerhof, G.E. Scuseria, J. Chem. Phys. 110 Ž1999. 5029. w17x A.D. Becke, J. Chem. Phys. 98 Ž1993. 5648. w18x A.D. Becke, J. Chem. Phys. 104 Ž1996. 1040. w19x M. Ernzerhof, G.E. Scuseria, J. Chem. Phys. Žin press.. w20x J.P. Perdew, S. Kurth, A. Zupan, P. Blaha, Phys. Rev. Lett. 82 Ž1999. 2544.
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