J. Phys. Chem. 1996, 100, 14665-14671
14665
Density Functional Theory (DFT) Study of Enthalpy of Formation. 1. Consistency of DFT Energies and Atom Equivalents for Converting DFT Energies into Enthalpies of Formation Susan J. Mole, Xuefeng Zhou, and Ruifeng Liu* Department of Chemistry, East Tennessee State UniVersity, Johnson City, Tennessee 37614-0695 ReceiVed: March 15, 1996; In Final Form: June 11, 1996X
A simple atom equivalent method for converting density functional theory (DFT) energies to enthalpies of formation is described and its performance, in conjunction with six DFT methods, was examined. For 23 stable hydrocarbons with well-established experimental data, the root mean square deviations between the calculated and experimental enthalpies of formation range from 1 to 6 kcal/mol. The smallest deviation was obtained with the B3LYP energies, and the largest deviation was obtained with the LSDA energies. The B3LYP atom equivalents of carbon and hydrogen derived from stable hydrocarbons were used without adjustment to calculate the enthalpies of formation of some free radicals and carbocations. The mean deviation between the calculated and experimental results is about 2 kcal/mol, which is of the same order as experimental uncertainties for these highly reactive species.
Introduction Enthalpies of formation are important physical parameters. They are used to assess the stability of a molecule, to estimate the amount of energy released in a reaction, and to calculate other thermodynamic functions. Traditionally, enthalpies of formation are determined by heat of combustion measurements. Such experiments are time consuming and difficult to achieve high accuracy. For unstable species, indirect methods have to be used and the results are often less accurate. Even for some stable molecules, results of different measurements may differ by more than the combined error limits. In view of the important roles enthalpy of formation plays and the fact that the number of experimental measurements has been declining, it is desirable to develop theoretical methods for accurate predictions. There are several computational methods for predicting enthalpies of formation. However, all of them suffer from certain limitations. Highly parametrized empirical (i.e., MM2, MM3, etc.)1 and semiempirical (i.e., MNDO, AM1, MINDO, etc.)2 methods are inexpensive, but they may be less accurate and are not applicable to many compounds due to the lack of reliable parameters. Straightforward ab initio calculation is promising because it is completely unbiased, but to obtain accurate results large basis sets and a sophisticated treatment of electron correlation are required. The G2 theory3 indicates that even at the prohibitively expensive QCISD(T)/6-311+G(3df,2p) level, an empirical correction (HLC) is still required to achieve an accuracy of ∼2 kcal/mol. A more practical approach is the use of isodesmic reactions.4,5 These formal equations usually lead to accurate results by maintaining a constant number of different kinds of bonds on both sides of the equation. Greater precision may be achieved by selecting molecules which have the same number of different kinds of bonds on both sides of the equation. In order to use this approach, however, three out of four of the components of the equation must have experimental values, but these are not always available. Wiberg developed an impressive method6 for accurate prediction of enthalpies of formation based on ab initio HartreeFock (HF) energies. He pointed out that the use of homoiX
Abstract published in AdVance ACS Abstracts, August 1, 1996.
S0022-3654(96)00801-5 CCC: $12.00
sodesmic5 reactions may be considered as a group equivalent scheme and that the zero-point energies as well as the change in energy from 0 to 298 K may be reasonably approximated using group equivalents. In this method, the heat of formation of a molecule is approximated by
∆Hf ) 627.5(EHF - nCH3ECH3 - nCH2ECH2 - ...) Here, the constant is the conversion factor between atomic unit and kcal/mol, EHF is the Hartree-Fock molecular energy, nCH3 is the number of methyl groups, and ECH3 is the group equivalent for a methyl group. Using this equation and HF/631G* energies, the enthalpies of formation of 42 compounds were fitted with a root mean square (rms) error of 1.23 kcal/ mol. A similar scheme, which uses more parameters, was proposed by Schleyer7 and applied to a larger group of compounds. Using a similar method, Allinger et al.8 achieved better results by explicit inclusion of terms to account for effects of populating higher energy conformations, low-lying vibrational states, translational and rotational states, etc. These methods, while successful for classical molecules (molecules that can be represented adequately by a single Lewis structure with localized two electron bonds), cannot be applied to the intermediate sections of potential surfaces, where partial bonds are present, or to nonclassical molecules, because of the lack of reference compounds to provide the necessary empirical corrections. For studying reactions, it is ideal to have an analogous procedure but the empirical corrections should depend only on the atoms present, not on the bonds and groups they form. A procedure which satisfies this requirement was proposed by Dewar,9 although his purpose was to estimate the effective errors in ab initio and semiempirical energies. In his proposal, each element is assigned a single-atom equivalent. The virtue of this approach is its generality: the heat of formation of any species, transition structures, reactive intermediates, nonclassical structures, etc., can, in principle, be estimated in this manner. Dewar’s calculations with HF energies showed that this approach is less accurate, with an average error of ∼6 kcal/mol. As an important error in HF energies is that due to neglecting electron correlation, the performance of this method should be improved when electron correlation is taken into account explicitly in quantum mechanical calculations. However, ab © 1996 American Chemical Society
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initio calculations at a theoretical level that takes into account electron correlation explicitly are much more expensive, and therefore there has been no report of studies using Dewar’s scheme with post-HF energies. Recently the density functional theory (DFT)10,11 has been gaining popularity. When formulated similarly to HF theory, the computational cost of DFT scales similarly as that of HF theory, yet it recovers electron correlation efficiently and produces surprisingly good results on molecular structures, energies, and vibrational frequencies.12 Thus, DFT is promising for developing a general method for accurate prediction of enthalpies of formation of a large variety of compounds. We have recently investigated the consistency of DFT energies for this purpose and found that the combination of Dewar’s scheme with the energies of Becke’s three-parameter hybrid HF/DFT method reproduces the enthalpies of formation of many hydrocarbons satisfactorily. Theory Equations relating enthalpies of formation and quantum mechanical or molecular mechanics energies can be derived in many ways. For simplicity, consider the enthalpies of formation of some hydrocarbon compounds CnHm (the same procedure also applies to other compounds). The enthalpies of formation are given by
∆fHi ) Ei - ECn(graphite) - (m/2)EH2(g) + Xi
(1)
where Ei is the energy of molecule i, ECn(graphite) is the energy of graphite Cn, EH2(g) is the energy of H2(gas), and Xi is the error due to the theoretical method. In practice, it is trivial to calculate EH2(gas), but it is not easy to calculate ECn(graphite). However, both of ECn and EH2 are extensive properties. Therefore
∆fHi ) Ei - n(ECn(graphite)/n) - m(EH2(g)/2) + Xi
(2)
For a theoretical method to be useful in chemistry, it must reproduce the difference between the energies of molecules i and j correctly. This means it must reproduce the difference in enthalpies of formation of molecules i and j correctly. If this is true for all molecules, the error Xi must be an additive sum of errors, xC and xH, of the individual atoms. That is
Xi ) nxC + mxH
(3)
Substituting eq 3 into eq 2, we get
∆fHi ) Ei - n(ECn(graphite)/n - xC) - m(EH2(g)/2 - xH) ) Ei - nC - mH (4) where C ) ECn/n - xC and H ) EH2/2 - xH are the atom equivalents of C and H to be determined from least-squares fitting of reliable experimental enthalpies of formation instead of from straightforward quantum mechanical calculations. Once the atom equivalents are determined, they can be used to convert quantum mechanical energies into enthalpies of formation. The deviation between the calculated and reliable experimental results is an indication of the consistency of quantum mechanical energies. Computational Details We have studied the performance of the atom equivalent method (eq 4) for calculating enthalpies of formation of hydrocarbons with DFT energies. The DFT energies were
obtained with the following combinations of exchange and correlation functionals: (1) Slater’s exchange functional13 with Vosko-Wilk-Nusair’s correlation functional14 (local spin density approximation, abbreviated LSDA), (2) Becke’s exchange functional15 with Lee-Yang-Parr’s correlation functional16 (BLYP), (3) Becke’s exchange functional with PerdewWang’s correlation functional17 (BPW91), (4) Becke’s threeparameter hybrid DFT/HF method18 with Lee-Yang-Parr’s correlation functional (B3LYP), (5) Becke’s three-parameter hybrid DFT/HF method with Perdew’s correlation functional19 (B3P86), and (6) Becke’s three-parameter hybrid DFT/HF method with Perdew-Wang correlation functional (B3PW91). All DFT calculations were carried out using Gaussian94 program package20 with the 6-311G(d,p) and 6-31G* basis set.21,22 For closed shell molecules, the restricted Hartree-Fock analog of DFT was used. For open shell species (radicals and cations), the unrestricted Hartree-Fock analog of DFT was used. All the molecular structures were fully optimized with each method. DFT energies of the optimized structures were used in the least-squares fitting of experimental enthalpies of formation23 to determine the atom equivalents, or used in the calculation of enthalpies of formation with atom equivalents determined from the least squares fitting. The 6-311G(d,p) instead of the 6-31G* basis set was used in most of the calculations to minimize basis set effects on examination of the consistency of DFT energies. To study basis set dependence, we repeated B3LYP calculations using the smaller 6-31G* basis set. Results and Discussions Twenty-three unstrained small and stable hydrocarbon compounds, whose experimental enthalpies of formation have been well established, were used in the least-squares procedure to determine the atom equivalents for carbon and hydrogen. These molecules contain most of the bond types found in a hydrocarbon compound. Results of the calculations together with the experimental enthalpies of formation, DFT energies obtained with the 6-311G(d,p) basis set, deviations, mean absolute deviations, and rms deviations between the calculated and experimental results are presented in Table 1. The experimental enthalpies of formation were taken from the recent assessment of Pedley23 and were carefully checked for accuracy. As all the molecules are small and stable molecules, the values in Pedley’s collection do not differ significantly from those of much earlier assessments. For example, most of the experimental results included in Table 1 are within 0.1 kcal/mol from the corresponding values used in Schleyer’s study.7 It is shown in Table 1 that with the six DFT methods, the mean absolute deviations between the experimental and calculated enthalpies of formation range from 0.83 to 5.36 kcal/mol. The rms deviations range from 1.03 to 6.54 kcal/mol. Clearly, the B3LYP results are superior to the BLYP results and it is quite surprising that the BPW91 results are superior to the B3PW91 results. The smallest deviation between calculated and experimental results is given by B3LYP. BPW91 and B3PW91 give slightly larger deviations. The largest deviation between the calculated and experimental results was obtained with the LSDA method. Plots of experimental data against the calculated enthalpies of formation are presented in Figure 1. The slope of the correlation line is 1.000. Correlation coefficients of results obtained by B3LYP, BPW91, B3PW91, B3P86, BLYP, and LSDA functionals are 0.9993, 0.9989, 0.9986, 0.9972, 0.9963, and 0.9813, respectively, indicating B3LYP gives the most consistent energies followed by BPW91 and B3PW91 functionals.
-40.53374 -77.35470 -78.61398 -79.85626 -116.69077 -117.94406 -119.18068 -156.03850 -156.02493 -157.26808 -157.27112 -157.27314 -157.27348 -157.25655 -158.50498 -158.50589 -194.15372 -195.36844 -195.35434 -195.38028 -196.61094 -197.82918 -235.94409
-17.78 54.54 12.55 -20.03 44.19 4.78 -25.02 26.29 34.82 0.02 -1.70 -2.72 -4.04 6.62 -30.02 -32.07 32.10 18.19 25.26 8.13 -18.26 -35.11 -29.47
methane acetylene ethylene ethane propyne propene propane 1,3-butadiene 2-butyne 1-butene (Z)-2-butene (E)-2-butene isobutene cyclobutane n-butane isobutane cyclopentadiene 1,3-pentadiene 1,4-pentadiene cyclopentene cyclopentane n-pentane cyclohexane mean deviation rms deviation Cd Hd
-0.38 -1.75 0.23 0.95 -0.01 0.78 0.74 1.49 1.51 0.51 0.74 0.99 -0.12 0.17 0.08 -1.04 -0.46 1.63 -0.15 -1.00 -1.38 -0.02 -2.24 0.81 1.02
diffc
BLYP
-38.11752 -0.58681
-40.49623 -77.32273 -78.56603 -79.79200 -116.63223 -117.86948 -119.08953 -155.95442 -155.93980 -157.16668 -157.16985 -157.17194 -157.17197 -157.15191 -157.38690 -158.38762 -194.05431 -195.25776 -195.24368 -195.26389 -196.47792 -197.68420 -235.78325
EDFT, aub 1.97 0.62 2.36 2.62 1.80 2.31 1.65 3.38 2.74 1.32 1.63 1.92 0.62 -1.31 0.56 -1.04 -1.87 2.94 1.18 -3.27 -4.31 -0.66 -6.61 2.12 2.52
diffc
B3PW91
-38.11034 -0.59553
-40.51712 -77.31661 -78.57927 -79.82505 -116.63871 -117.89504 -119.13485 -155.97133 -155.95886 -157.20439 -157.20791 -157.20985 -157.21010 -157.19940 -158.44456 -158.44550 -194.07710 -195.28706 -195.27377 -195.30608 -196.53953 -197.75418 -235.85805
EDFT, aub -2.30 -5.15 -2.21 -0.45 -2.51 -0.96 -0.16 -0.81 -0.11 -0.76 -0.23 -0.04 -1.20 2.75 0.05 -1.40 2.13 0.08 -1.18 1.96 2.17 0.13 1.71 1.32 1.79
diffc
BPW91
-38.12381 -0.59201
-40.51989 -77.33899 -78.59493 -79.83357 -116.66721 -117.91665 -119.14899 -156.00677 -155.99346 -157.23166 -157.23547 -157.23742 -157.23735 -157.22486 -157.46429 -158.46504 -194.12251 -195.32852 -195.31415 -195.34339 -196.56924 -197.77951 -235.89253
EDFT, aub -0.40 -3.70 -0.67 0.92 -1.37 0.17 0.58 0.54 0.72 -0.23 0.48 0.68 -0.68 2.14 0.16 -1.42 1.29 1.07 -0.88 0.34 0.09 -0.39 -1.53 0.89 1.18
diffc
B3P86
-38.23207 -0.61470
-40.71398 -77.59660 -78.89799 -80.18234 -117.07910 -118.37435 -119.65284 -156.57269 -156.55965 -157.84436 -157.84796 -157.84977 -157.82025 -157.83960 -159.12326 -159.12488 -194.80120 -196.04902 -196.03574 -196.06886 -197.34116 -198.59359 -236.82103
EDFT, aub
-3.28 -6.30 -3.11 -1.21 -3.45 -1.54 -0.54 -1.20 -0.85 -0.98 -0.40 -0.29 -1.31 2.69 -0.08 -1.20 2.38 0.02 -1.24 2.41 2.95 0.55 3.28 1.79 2.31
diffc
LSDA
-37.89894 -0.59722
-40.30180 -76.88277 -78.15124 -79.40378 -116.00078 -117.26349 -118.50966 -155.12951 -155.11662 -156.36840 -156.37375 -156.37477 -156.37611 -156.37070 -157.61550 -157.61833 -193.03728 -194.24196 -194.22708 -194.27005 -195.50980 -196.72120 -234.62803
EDFT, aub
-9.01 -14.2 -9.74 -5.86 -9.10 -5.67 -3.01 -4.82 -4.37 -3.24 -1.65 -1.94 -2.42 4.85 -0.19 -0.46 6.54 -0.95 -3.22 6.62 8.66 2.45 13.0 5.30 6.51
diffc
a Experimental ∆H °(g) in kcal/mol, taken from the latest assessment of Pedley, ref 23. b DFT/6-311G(d,p) total energy in hartrees. c ∆H °(expt) - ∆H °(calc) in kcal/mol. d Atom equivalents obtained from f f f least-squares fitting.
-38.12760 -0.59461
EDFT, aub
∆Hf(exp)a
name
B3LYP
TABLE 1: Results of DFT/6-311G(d,p) Atom Equivalent Analysis for Hydrocarbons
DFT Study of Enthalpy of Formation J. Phys. Chem., Vol. 100, No. 35, 1996 14667
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Figure 1. Plots of experimental ∆Hf°(gas, 298 K) values for compounds listed in Table 1 against the ∆Hf°(gas, 298 K) values calculated by DFT/6-311G(d,p) atom equivalent analysis using six popular combination of exchange and correlation functionals. Slope 1.000 was taken for the correlation lines.
TABLE 2: B3LYP/6-311G(d,p) Atom Equivalent Analysis of Strained Hydrocarbons and Benzene Schleyer’s resultsa
present study b
formula
name
EDFT, au
C3H4 C3H6 C4H6 C4H6 C6H6 mean deviation
cyclopropene cyclopropane cyclobutene bicyclobutane benzene
-116.652 87 -117.929 94 -156.015 79 -155.989 78 -232.308 55
c
d
∆Hf(exp)
∆Hf, kcal
diff
∆Hf, kcal
diffd
66.20 12.73 37.45 51.90 19.81
68.00 12.88 38.08 55.40 15.51
1.80 0.15 1.63 3.50 -4.30 2.3
69.81 13.59 38.60 55.69 12.13
3.61 0.86 1.15 3.79 -7.68 3.4
a Results of HF/6-31G* group and bond equivalent analysis of Ibrahim and Schleyer (ref 7), a total of 15 equivalents were used for hydrocarbon. Total energy obtained by B3LYP/6-311G**//B3LYP/6-311G** calculations. c Cox and Pilcher, ref 25. For these compounds, ∆Hf included in ref 23 are either identical to the values in ref 25 or differ by less than 0.1 kcal. d ∆Hf°(expt) - ∆Hf°(calc) in kcal/mol. b
The atom equivalents for B3LYP/6-311G(d,p) energies obtained from the least squares procedure are -38.127 60 and -0.594 61 hartree for carbon and hydrogen, respectively. The B3LYP/6-311G(d,p) energy of H2 is -1.179 57 hartree. The difference between the hydrogen atom equivalent and one-half of H2 energy should be partly due to zero-point vibrational effects. As B3LYP gives the smallest deviations between the calculated and experimental results, we applied the atom equivalents to calculating the enthalpies of formation of some difficult molecules identified by Ibrahim and Schleyer.7 These molecules are cyclopropene, cyclopropane, cyclobutene, bicyclobutane, and benzene. Except for benzene, these molecules are highly strained and therefore require a large basis set with polarization functions to get a good description. Benzene is a nonclassical molecule with strong electron correlation effects. It requires special treatment in the HF based bond and group equivalent approach for accurate description.24 In the present study, we used the atom equivalents obtained from the leastsquares fit of unstrained classical molecules to calculate the enthalpies of formation of the difficult molecules. The calculated results are compared with experimental data and the results
of Ibrahim and Schleyer’s HF/6-31G* group equivalent analysis in Table 2. Mean absolute deviation between our calculated and experimental results is 2.28 kcal/mol. The mean absolute deviation of the HF/6-31G* group equivalent analysis is 3.4 kcal/mol. The improvement on highly strained hydrocarbons is perhaps due mainly to the use of the larger 6-311G(d,p) basis set in our calculations. The improvement in the calculated result of benzene is definitely due to recovering electron correlation by B3LYP. A significant difference between the present study and that of Ibrahim and Schleyer is that we used only two empirical parameters to convert the DFT energies into enthalpies of formation, while Ibrahim and Schleyer used a total of 15 parameters for hydrocarbons. The B3LYP/6-311G(d,p) atom equivalents of carbon and hydrogen obtained from closed shell classical molecules were also used without adjustment to calculate the enthalpies of formation of some hydrocarbon free radicals and carbocations. The results are compared with available experimental data and the results of Ibrahim and Schleyer’s HF/6-31G* group equivalent analysis in Tables 3 and 4. It should be noted that because of the extraordinary reactivity of the radicals and cations,
DFT Study of Enthalpy of Formation
J. Phys. Chem., Vol. 100, No. 35, 1996 14669
TABLE 3: Results of B3LYP/6-311G(d,p) Atom Equivalent Analysis for Free Radicals ref 7a
present study formula
name
EDFT, aub
∆Hf(exp)c
∆Hf
diffe
∆Hf
diffe
CH3 C2H C2H3 C2H5 C3H7 C3H7 C4H9 mean deviation
methyl ethynyl vinyl ethyl n-propyl isopropyl tert-butyl
-39.853 76 -76.629 46 -77.927 03 -79.183 65 -118.507 48 -118.514 16 -157.844 35
35.1 135 70.4 25.9(27.8)d 21.0(22.7)d 18.2(19.1)d 8.7(9.5)d
36.2 138 70.3 28.0 23.6 19.4 11.0
1.1 3 -0.1 2.1(0.2) 2.6(0.9) 1.2(0.3) 2.3(1.5) 1.8(1.0)
33.7 132 68.7 26.3 22.8 19.6 10.7
-1.4 -3 -1.7 0.4(-1.5) 1.8(0.1) 1.4(0.5) 2.0(1.2) 1.7(1.3)
a Results of HF/6-31G* group and bond equivalent analysis of Ibrahim and Schleyer (ref 7), a total of 15 equivalents were used for hydrocarbon. Total energy obtained by B3LYP/6-311G**//BLYP/6-311G** calculation. c McMillen and Golden, ref 25. d Holmes et al., ref 27. e ∆Hf°(expt) ∆Hf°(calc) in kcal/mol. b
TABLE 4: Results of B3LYP/6-311G(d,p) Atom Equivalent Analysis for Carbocations ref 7a
present study formula
name
CH3 C2H3 C2H5 C3H3 C3H3 C3H5 C3H7 C4H7 C4H9 C5H9 mean deviation
methyl vinyl ethyl cyclopropenyl propargyl allylic isopropyl sec-butyl tert-butyl cyclopentyl
EDFT,
aub
-39.491 36 -77.607 78 -78.883 27 -115.760 63 -115.720 11 -117.001 21 -118.243 66 -157.573 95 -157.593 76 -195.685 78
∆Hf(exp)
∆Hf, kcal
diffg
261c 266.9d 215.6e 256c 281c 225e 191e 183e 166e 191f
264 270.6 216.5 255 280 223 189 181 168 191
3 3.7 0.9 -1 -1 -2 -2 -2 2 0 1.8
∆Hf, kcal
diffg
256 273 219.0 250 285 222 190 183 166 192
-5 6.1 3.4 -6 4 -3 -1 0 0 1 3.0
a Results of HF/6-31G* group and bond equivalent analysis of Ibrahim and Schleyer (ref 7), a total of 15 equivalents were used for hydrocarbon. Total energy obtained by B3LYP/6-311G**//B3LYP/6-311G** calculation. c Bowers, M. T. Gas Phase Ion Chemistry; Academic Press: New York, 1979; Vol. 2. d Wagman, D. D.; Evans, W. H.; Parker, V. P.; Schumm, R. H.; Halow, I.; Bailey, S. M.; Churney, K. L.; Nuttall, R. L. The NBS Tables of Chemical Thermodynamic Properties, J. Chem. Ref. Data, Suppl. 1982, No. 2, 11. e Rosenstock, H. M.; Draxl, K.; Steiner, B. W.; Herron, J. T. J. Phys. Chem. Ref. Data 1977, 6, Suppl. 1. f Allinger, N. L.; Dodziuk, H.; Rogers, D. W.; Naik, S. N. Tetrahedron 1982, 38, 1593. g ∆H °(expt) - ∆H °(calc). f f b
experimental results of these species were obtained indirectly and may be associated with larger uncertainties than those of the closed shell stable molecules. Table 3 compares the calculated and experimental results of the free radicals. Compared to McMillen and Golden’s compilation26 of experimental results, the mean absolute deviation between the experimental and our calculated enthalpies of formation is 1.8 kcal/mol, which is about the same as that obtained in Ibrahim and Schleyer’s HF/6-31G* group equivalent analysis. However, Ibrahim and Schelyer defined two parameters specifically for the radicals, and the enthalpies of formation of the free radicals were used in the least-squares procedure to determine these parameters. In the present study, none of the experimental data of the radicals and cations were used to optimize the atom equivalents. Therefore, our results of the radicals and cations are a priori. Table 3 shows that almost all our calculated results are overestimates compared to the experimental data. This may indicate that the listed experimental data are systematically too low. Indeed, the recent experimental studies27 on ethyl, n-propyl, iso-propyl, and tert-butyl radicals derived enthalpies of formation (numbers in the parentheses in Table 3) systematically higher than those of McMillen and Golden’s compilation. When compared to the recent experimental results, the mean absolute deviation between experimental and our calculated results is 1.0 kcal/mol, which is comparable to that of stable classical molecules. Table 4 compares the calculated enthalpies of formation of carbocations with experimental results. The mean absolute deviation between our calculated results and experimental enthalpies of formation is 1.8 kcal/mol. The mean absolute deviation of Ibrahim and Schleyer’s HF/6-31G* group equivalent analysis is 3.0 kcal/mol. We should note again that we
did not use any empirical parameters specifically for the carbocations, while the HF/6-31G* group equivalent analysis defined some parameters specifically for the carbocations, and the experimental enthalpies of formation of the carbocations were used in their calculations to determine these parameters. It is shown that with B3LYP/6-311G(d,p) energies, the atom equivalent method reproduces the experimental enthalpies of formation of several classes of hydrocarbon compounds accurately with errors similar to experimental uncertainty. It is of interest to know if the smaller 6-31G* basis set works as well. For this purpose, we carried out B3LYP/6-31G* calculations on the 23 stable classical hydrocarbon compounds and fit the B3LYP/6-31G* energies by the least-squares procedure. Results of these calculations are compared with results of B3LYP/6-311G(d,p) analysis in Table 5. It is shown that the mean absolute and rms deviations between the calculated and experimental results are 1.17 and 1.72 kcal/mol, respectively, with the B3LYP/6-31G* energies. The linear correlation coefficient for the the experimental versus calculated enthalpies of formation is 0.9983. Thus, for large molecules, the smaller 6-31G* basis set may be used in the atom equivalent analysis to estimate enthalpies of formation. Conclusions We have examined the performance of a simple atom equivalent method for accurate prediction of enthalpies of formation from the 6-311G(d,p) energies of six popular density functional methods. The best performance was achieved by the B3LYP functional which gives a rms deviation of about 1 kcal/mol between the calculated and reliable experimental enthalpies of formation of 23 stable classical molecules. The
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TABLE 5: Results of B3LYP/6-31G* and B3LYP/6-311G(d,p) Atom Equivalent Analysis B3LYP/6-311G(d,p) formula
name
∆Hf(expt)
CH4 C2H4 C2H4 C2H6 C3H4 C3H6 C3H8 C4H6 C4H6 C4H8 C4H8 C4H8 C4H8 C4H8 C4H10 C4H10 C5H6 C5H8 C5H8 C5H8 C5H10 C5H12 C6H12 mean deviation rms deviation C H
methane acetylene ethyl ethane propyne propene propane 1,3-butadiene 2-butyne 1-butene (Z)-2-butene (E)-2-butene isobutene cyclobutane n-butane isobutane cyclopentadiene 1,3-pentadiene 1,4-pentadiene cyclopentene cyclopentane n-pentane cyclohexane
-17.78 54.54 12.55 -20.03 44.19 4.78 -25.02 26.29 34.82 0.02 -1.70 -2.72 -4.04 6.62 -30.02 -32.07 32.10 18.19 25.26 8.13 -18.23 -35.11 -29.47
BPW91 and B3PW91 functionals also perform satisfactorily, with rms deviations of less than 2 kcal/mol. The worst performance was given by LSDA, with a rms deviation of 6.5 kcal/mol which is similar to the value obtained by Dewar’s HF/ 6-31G* analysis. The B3LYP/6-311G(d,p) atom equivalents obtained from the least-squares fit of experimental enthalpies of formation of stable classical molecules were applied, without change, to calculate the enthalpies of formation of some free radicals and carbocations. The calculated results for these open shell species are very satisfactory. The mean absolute deviation is about 2 kcal/mol which is on the order of experimental uncertainties for these highly reactive species. We also determined B3LYP/6-31G* atom equivalents for carbon and hydrogen and showed that the rms deviation of B3LYP/6-31G* atom equivalent analysis is less than 2 kcal/mol. Thus, for large molecules one may estimate enthalpies of formation by the smaller 6-31G* basis set with little loss of accuracy. Compared to the bond and group equivalent methods, the atom equivalent method has the advantage that it employs the smallest number of empirical parameters and these parameters depend only on the atoms in the molecule and not on the bonds and groups they form. We are currently examining the effect of including zero-point vibrational energy explicitly in the calculations and determining atom equivalents for nitrogen, oxygen, fluorine, sulfur, and chlorine. Results of these studies will be published in a forthcoming paper. At completion of this study, it came to our attention that a similar atom equivalent analysis of the enthalpies of formation of a large number of hydrocarbons, using HF/6-31G* energies, was carried out by W. C. Herndon (Abstract 457, Joint Southeast-Southwest Regional ACS Meeting, Nov 29-Dec 1, 1995, Memphis, TN). In addition to atom equivalents for carbon and hydrogen, he also defined another parameter (∼0.94) to scale the HF/6-31G* energies. For a group of 65 hydrocarbons, the mean deviation between the calculated and experimental enthalpies of formation is only 1.1 kcal/mol. This is very impressive. However, we believe the Hartree-Fock based method will have trouble to deal with conjugated open shell
∆Hf(calc) -17.40 56.29 12.32 -20.98 44.20 4.00 -25.76 24.80 33.31 -0.53 -2.44 -3.71 -3.92 6.70 -30.45 -31.03 32.56 16.56 25.41 9.13 -16.88 -35.09 -27.13 -38.12760 -0.59461
diff -0.38 -1.75 0.23 0.95 -0.01 0.78 0.74 1.49 1.51 0.51 0.74 0.99 -0.12 -0.08 0.43 -1.04 -0.46 1.63 -0.14 -1.00 -1.38 -0.02 -2.34 0.81 1.02
B3LYP/6-31G* ∆Hf(calc) -15.47 60.74 14.63 -19.64 46.78 5.40 -24.94 25.78 34.03 0.30 -2.02 -3.36 -3.60 5.24 -30.22 -30.70 30.87 16.54 26.31 7.19 -18.91 -35.45 -30.20 -38.11705 -0.59417
diff -2.31 -6.20 -2.08 -0.39 -2.60 -0.62 -0.08 0.52 0.79 -0.32 0.32 0.64 -0.44 1.38 0.20 -1.37 1.23 1.65 -1.05 0.94 0.65 0.34 0.73 1.17 1.72
species due to neglecting electron correlation and significant spin contamination of the unrestricted Hartree-Fock wave function. Acknowledgment. This study was partially supported by a grant from the Research and Development Committee of East Tennessee State University. References and Notes (1) Allinger, N. L.; Yuh, Y. H.; Lii, J.-H. J. Am. Chem. Soc. 1989, 111, 8551 and references therein. (2) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J. Am. Chem. Soc. 1985,107, 3902. (3) Curtis, L. A.; Raghavachari, K.; Truck, G. W.; Pople, J. A. J. Chem. Phys. 1991, 94, 7221. (4) Hehre, W. J.; Ditchfield, R.; Radom, L.; Pople, J. A. J. Am. Chem. Soc. 1970, 92, 4796. (5) George, P.; Trachtman, M.; Bock, C. W.; Brett, A. M. Tetrahedron 1976, 32, 317. (6) K. B. Wiberg, J. Comput. Chem. 1984, 5, 197; J. Org. Chem. 1985, 50, 5285. (7) Ibrahim, M. R.; Schleyer, P. v. R. J. Comput. Chem. 1985, 6, 157. (8) Allinger, N. L.; Schmitz, L. R.; Motoc, I.; Bender, C.; Jabanowski, J. K. J. Am. Chem. Soc. 1992, 114, 2880 and references therein. (9) Dewar, M. J. S.; Storch, D. M. J. Am. Chem. Soc. 1985, 107, 3898. Dewar, M. J. S. J. Phys. Chem. 1985, 89, 2145. (10) Perdew, J. Phys. ReV. 1986, B33, 8822. Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: Oxford, U.K., 1989. (11) Kohn, W.; Sham, L. J. Phys. ReV. 1965, A140, 1133. (12) Andzelm, J.; Wimmer, E. J. Chem. Phys. 1992, 96, 1280. Gill, P. M. W.; Johnson, B. G.; Pople, J. A.; Frisch, M. J. Chem. Phys. Lett. 1992, 197, 449. Johnson, B. G.; Gill, P. M. W.; Pople, J. A. J. Chem. Phys. 1993, 98, 5612. Wiberg, K. B.; Cheeseman, J. R.; Ochterski, J. W.; Frisch, M. J. J. Am. Chem. Soc. 1995, 117, 6535. (13) Slater, J. C. Quantum Theory of Molecules and Solids, Vol. 4: The Self-Consistent Field for Molecules and Solids; McGraw-Hill: New York, 1974. (14) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (15) Becke, A. D. Phys. ReV. 1988, A38, 3098. (16) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. 1988, B37, 785. Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H. Chem. Phys. Lett. 1989, 157, 200. (17) Perdew, J. P.; Wang, Y. Phys. ReV. 1992, B45, 13244. (18) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (19) Perdew, J. P. Phys. ReV. 1986, B33, 8822.
DFT Study of Enthalpy of Formation (20) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Cioslowski, J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; HeadGordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94, Revision B.1; Gaussian, Inc.: Pittsburgh, PA, 1995. (21) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J. Chem. Phys. 1980, 72, 650. Frisch, M. J.; Pople, J. A.; Binkley, J. A. J. Chem. Phys. 1984, 80, 3265. (22) Hehre, W. J.; Ditchfield, R. D.; Pople, J. A. J. Chem. Phys. 1972, 56, 2257.
J. Phys. Chem., Vol. 100, No. 35, 1996 14671 (23) Pedley, J. B. Thermochemical Data and Structures of Organic Compounds; Thermodynamics Research Center: College Station, TX, 1994, Vol. 1. (24) Schulman, J. M.; Peck, R. C.; Disch, R. L. J. Am. Chem. Soc. 1989, 111, 5675. Peck, R. C.; Schulman, J. M.; Disch, R. L. J. Am. Chem. Soc. 1990, 112, 6637. (25) Cox, J. D.; Pilcher, G. Thermochemistry of Organic and Organometallic Compounds; Academic Press: London, 1976. (26) McMillen, D. F; Golden, D. M. Annu. ReV. Phys. Chem. 1982, 33, 493. (27) Holmes, J. L.; Lossing, F. P.; Maccoll, A. J. Am. Chem. Soc. 1988, 110, 7339.
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