Accurate calculation of brønsted acidities using low-level ab initio methods

Journal of Molecular Structure (Theo&em), 181 (1988) 305-314 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

ACCURATE CALCULATION OF BR0NSTED LOW-LEVEL AB INITIO METHODS

MICHELE

305

ACIDITIES USING

R.F. SIGGEL*

Department of Chemistry, Institute of Mathematical and Physical Sciences, University of Tromsq N-9001 Tromso, Norway and Department of Chemistry, Oregon State University, Corvallis, OR 97331 (U.S.A.) DE JI and T. DARRAH

THOMAS

Department of Chemistry, Oregon State University, Corvallis, OR 97331 (U.S.A.) LEIF J. SETHRE Department of Chemistry, Institute of Mathematical and Physical Sciences, University of Tromsa, N-9001 Tromsu (Norway) (Received

13 August 1987; in final form 12 January

1988)

ABSTRACT Gas-phase acidities calculated at three different levels of ab initio theory are presented. At the MP4/6-311+ +G (2d,p) level the calculated acidities for methane, methanol, and formic acid, corrected for zero-point energy and temperature, are within 3 kcal mol-’ of the experimental values of AHoz9&At the RHF/3-21+ G//3-21 + G level the calculated acidities of 12 compounds deviate systematically from the experimental values. However, there is a good linear correlation between the theoretical and experimental acidities. This correlation can be used together with acidities calculated at this level to provide predictions of acidities that are probably within 3 kcal mol-’ of the correct values. Acidities for the same compounds calculated at the RHF/STO-3G// STO-3G level scatter about the correlation line, and the reliability of predicted acidities based on such calculations cannot be expected to be better than about 12 kcal mol-‘. Acidities calculated at the same level for a series of substituted phenols, however, correlate well with the experimental values.

INTRODUCTION

The Brensted acidity, which is the ability of a molecule to give up a proton, is a quantity of fundamental chemical interest. Considerable effort has been devoted to the measurement of acidities (both in solution and in the gas phase), to interpretation of the relative strengths of acids and to theoretical calculation of acidities. *Present address: Department

0166-1280/88/$03.50

of Chemistry,

University

of California, Berkeley, CA 94720, U.S.A.

0 1988 Elsevier Science Publishers

B.V.

306

The acidity is the energy difference for the reaction RH + R-+H+ From a theoretical point of view, the acidity is most conveniently defined as the difference in energy, AE’ eq, between the energy minimum for R- and that for RH. For comparisons with experimental values, usually reported as AHoT, additional corrections for the zero-point energy (0.2 to 0.4 eV) and temperature (about 0.1 eV) are necessary. Knowledge of gas-phase acidities is an essential ingredient for understanding the factors that affect acidity. Although many acidities are known from measurements, values for unstable or transient species must often be obtained from theory. Moreover, theoretical calculations can provide information about the factors affecting acidity that are not available from the acidity itself. These include charge-distributions [ 1 ] and potentials at the acidic proton [ 21. It is, therefore, important to know how to make such calculations both accurately and economically, goals that are difficult to achieve simultaneously for large molecules. We are concerned here with two questions: what level of ab initio calculation is necessary to achieve accurate results and how can more economical, lower level calculations be corrected to give results that are in agreement with experiment? The second question is of considerable importance because a high level of calculation is required to give accurate results; at present such calculations require so much computer time and memory that they are feasible only for the smallest molecules. What level of calculation is necessary? Siggel et al. [ 31 have used published results as well as new calculations in investigating this question. They have shown that calculations of the acidities for 9 binary hydrides by Gordon et al. [ 41 at the MP4/6-311+ + G (3df,2pd) //6-31G (cl) level give theoretical acidities with almost no systematic or random deviations from the experimental values. Calculations at the MP4/6-31+ + G (c&p)//6-31G (d) level [ 41 are not quite so good [ 31. Using a basis set that is intermediate between the two used by Gordon et al., MP4/6-311+ +G(2d,p)//6-311+ +G(2d,p), Siggel et al. [ 31 have calculated acidities for methane and formic acid that are within 0.09 eV of the experimental values. Calculations at a somewhat lower level than these, such as MP2/6-31+ G (d) //6-31G (d) [ 51 show systematic deviations between experiment and theory [ 31. Such high-level calculations are prohibitively expensive for the acids that are likely to be of chemical interest and, for the most part, can be done only for the simplest molecules. Siggel et al. [ 31 have explored the use of lower level ab initio calculations (3-21 +G//3-21+ G and 6-31+ G//3-21 + G, with and without corrections for electron correlation) for calculating acidities. They found that all of these methods give acidities that differ systematically from the experimental values. However, they also found that there are good linear relationships between calculated and theoretical acidities and showed how these

307

relationships can be used to predict reasonably accurate acidities where no experimental data are available. Our goals here have been to explore further the quality of calculations based on the 6-311+ + G (2&p) basis set and also to see if calculations based on the STO-3G basis set might be used in predicting acidities. For the first of these goals, we have calculated the acidity of methanol, which provides a different structural type from the two compounds previously considered and is intermediate in acidity between them. Regarding the second goal, we recognize that the STO-3G basis set, which lacks diffuse functions, will give absolute values of the acidities that are too high. However, it represents the simplest basis set that one might expect to use in an ab initio calculation and, thus, provides an extreme test of the linear regression method for predicting experimental acidities from theoretical calculations. HIGH-LEVEL CALCULATIONS FOR THE ACIDITIES OF METHANOL, METHANE AND FORMIC ACID

The calculation for methanol was done using Gaussian 82 [6] and a VAX 8600 computer at the University of Tromso. Geometry optimizations for methanol and its anion were carried out at the RHF/6-311+ + G (24~) level. Using TABLE 1 Energies calculated at the 6-311+ +G(2d,p)//RHF/6-311+

RHF MP2 MP3

MPd(DQ) MP4(SDQ) MP4(SDTQ) dE”,,(RHF)

~E".,WTQ) Corrections”

+G(2d,p)

level

Methanol

Methanoate Methane anion

Methyl anion

Formic acid Formate anion

-115.08317 - 115.46837 - 115.48281 - 115.48490 - 115.48820 - 115.50112

-114.44708 - 114.84925 - 114.85397 - 114.85742 - 114.86306 - 114.88032

-39.51968 -39.71053 -39.72420 -39.72509 -39.72678 -39.73420

-188.83147 - 189.39935 - 189.39625 - 189.39824 - 189.40656 - 189.43206

17.31 (399.2) 16.89 (389.6) -0.32

( - 7.43)

-40.21041 -40.38826 -40.40717 -40.40875 -40.40978 -40.41453

18.79 (433.4) 18.51 (426.9) -0.36

(-8.41)

-188.26233 - 188.84473 - 188.83229 - 188.83519 - 188.84548 - 188.87501

15.49 (357.1) 15.16 (349.6) -0.28

(-6.45)

AH”,,,(SD’W)

16.57 (382.1)

18.15 (418.5)

14.88 (343.1)

AH”mdEx~t.)

16.44 (379.2) 0.09 (2)

18.07 (416.6) 0.04 (1)

14.97 (345.2) 0.09 (2)

Uncertainty

*Corrections for zero-point energy and to convert from AH”, to AHozss. Total energies in hartree,

AE”,, in eV and kcal mol-’ in parentheses.

308 TABLE 2 Structural parameters from geometry optimization at the RHF/6-311+ CD

CENT

Atom

Nl

Z-Matrix (angstroms and degrees)” Length

Methanol 1 1 2 3 2 4 3 5 4 6 5 7 6

+ G (2d,p) level

N2

Alpha

N3

Beta

J

C x 0 H H H H

1 1 1 1 1 3

1.000000 1.399546 1.080914 1.086497 1.086497 0.940803

(c) (v) (v) (vl) (vl) (v)

2 2 2 2 1

3.711727 110.966239 109.963631 109.963631 109.975830

(v) (v) (~2) (~2) (v)

3 3 3 2

0.000 120.000 - 120.000 0.000

(c) (c) (c) (c)

0 0 0 0

Methanoate ion 1 1 C 2 2 0 3 3 H 4 4 H 5 5 H

1 1 1 1

1.324761 1.121943 1.121943 1.121943

(v) (vl) (vl) (vl)

2 2 2

115.114843 (~2) 115.114843 (~2) 115.114843 (~2)

3 3

120.000 (c) -120.000 (c)

0 0

“For a description of the Z-matrix format, see ref. 10, pages 101-104. bThe letter c is used to denote a constant, non-optimized parameter. The letter v denotes a variable, or optimized parameter. A number following the letter v is used to indicate that all parameters with the same number have been constrained to have the same variable value.

this geometry and basis set, total energies were calculated at the RHF level and with inclusion of electron correlation through MP4 (SDTQ). Results of these calculations are summarized in Table 1 (energies) and Table 2 (optimized geometries ) . In Table 1, we have also included the results of our previous calculations on methane and formic acid [ 31. For comparison with experimental acidities, we have converted the theoretical values of BY,, to dHoZ9s, following standard procedures, as discussed in ref. 3. The corrections are listed in Table 1, where both the theoretical and experimental values of dHoZ9s [ 71 are also given. The theoretical result for methanol is within 0.13 eV (3 kcal mol- ’ ) of the experimental value, or just outside the experimental uncertainty of 2 kcal mol-‘. This accuracy is comparable to, but slightly inferior to, that obtained for methane and formic acid [31. LOWER-LEVEL AB INITIO CALCULATIONS

RHF/3-21-k

G//3-21 + G calculations

Siggel et al. [3] have considered several combinations of lower-level basis sets and treatments of electron correlation to calculate acidities of 12 com-

309

pounds. The compounds, which are listed in Table 3, were chosen to provide several different types of compounds and, within each type, a range of acidities. If theory and experiment are in agreement, a graph of theoretical versus experimental acidity will follow a straight line with unit slope and zero intercept. Disagreement between theory and experiment may be either systematic, with the points following either a different straight line or a curve, or random, with the points scattering about a line. Siggel et al. [3] found that although the calculated acidities deviate systematically from the experimental ones, they follow linear correlations with root-mean-square (RMS) deviations of O.l0.15 eV. A typical result is shown in Fig. 1, where the acidities, dEoeq, calculated at the RHF/3-21+ G//3-21 + G level have been plotted against the experimental values of flH” 298[ 7,8] as squares in the lower part of the figure. These acidities are listed in Table 3. A line (lower, solid) representing the linear regression between the theoretical and experimental values and a reference line (dashed) of unit slope and zero intercept are also shown. The parameters of the linear TABLE 3 Calculated and experimental acidities (eV) Compound

Expt.

1 Monofluoroacetic acid 2 Formic acid 3 Acetic acid 4 Cyclopentadiene 5 2,2-Difluoroethanol 6 Acetone 7 Isopropyl alcohol 8 Ethanol 9 Methanol 10 Ethene 11 Methane 12 Ethane Ally1 alcohol Isopropenyl alcohol

14.64 14.97 15.11 15.44 15.92 15.99 16.22 16.31 16.44 17.61b 18.07 18.2sb

Linear regression parameters r* Intercept Slope Mean deviation” RMS deviationd

RHF/3-21+ 14.58 14.93 15.12 15.85 15.98 16.30 16.83 16.88 16.94 18.23 18.72 18.95 16.65 15.89

G//3-21 + G

RHF/STO-3G//STO-3G 20.30 20.72 20.69 22.07 21.89 23.00 22.81 22.93 22.88 24.28 23.89 22.49 21.04

0.988 -3.008 1.2074

0.872 5.94 eV 1.00

0.362 0.150

5.99 eV 0.45 eV

“Except as noted, from ref. 7. bR.ef. 8. ‘Theory minus experiment. dDeviation between theoretical values and the regression line.

310 25 A

24 -

23 -

p-

1. 2' .= T3 'G 20 (d ‘iTi 19 0 'Z P a 16-

e

AA

A

A

‘A A

A

+

+

12

" 16 -

15 -

13.5

14.5

15.5

Experimental

17.5

16.5

acidity

16.5

(eV)

Fig. 1. Theoretical acidities, AE” eq, plotted versus experimental values of AH” 298’The solid lines show least-squares fits of straight lines to the data. The dashed line has unit slope and zero intercept. (Squares 1 RHF/3-21+ G//3-21 + G. Triangles = RHF/STO-3G//STO-3G. Crosses = RHF/STO-3G for substituted phenols). See Table 3 for names of compounds represented by the squares and triangles. For the phenols the theoretical values are from Press et al. [ 131 and the experimental values from Moylan and Brauman [ 141.

regression line are given in Table 3. It is clear that the data do not fall along the reference line and that the regression line has a slope greater than unity. This result is typical of all calculations that do not use an extensive basis set and include corrections for electron correlation [3]. (Although the calculations have not been corrected for zero-point-energy or for the difference bethese corrections are not the source of the major tween dEoo and LIW’~~~, differences between the calculated and experimental values [ 31.) However, the data do fall close to the linear regression line, with correlation coefficient, r2, of 0.989 and RMS deviation from the line of 0.15 eV. Siggel et al. [3] have shown that such linear regression lines can be used to predict nearly correct acidities of acids for which values might be desired but difficult to obtain. As an example of the predictive value of the linear regression, we consider ally1 alcohol ( CH2 = CHCH,OH), for which a measured acidity is not available and high-level calculations expensive. From a comparison of the experimental acidities of propanol, propylamine and allylamine, Bartmess [ 91 has estimated a value for dHozss of 16.22 2 0.09 eV (374 k 2 kcal mol-’ ) for this molecule.

311

Calculations at the RHF/3-21+ G//3-21 + G level (Table 3) give 16.65 eV, in significant disagreement with the estimate based on experimental data. However, if the theoretical value is used in conjunction with the regression line shown in Fig. 1, the predicted value is 16.28 eV, in good agreement with the experimental estimate of 16.22. From the RMS deviation of the points from the line of Fig. 1 and from the slope of this line, we estimate a statistical uncertainty in such theoretically estimated values of about 0.13 eV (3 kcal mol-‘). Calculations using the 3-21+ G basis set with corrections for electron correlation or the 6-31+ G basis set with and without such corrections give theoretical acidities for ally1 alcohol ranging from 16.15 to 16.77 eV [3]. However, when each of these values is used with an appropriate linear regression line [ 31 to predict the experimental acidity, the range of predicted values is 16.1816.29 eV, all within 0.07 eV of the value suggested by Bartmess. A similar treatment of isopropenyl alcohol ( CH2 = C (OH) CH3) is equally successful [ 31. STO-3G calculations For larger molecules, even as simple a basis set as 3-21+ G can tax the available time and memory of typical computer facilities. It is, therefore, useful to test the predictive value of the linear regression procedure for even lower levels of basis sets. For this purpose, we have calculated acidities at the STO-3G// STO-3G level for eleven of the twelve acids that were considered by Siggel et al. [ 31, omitting only cyclopentadiene for which there were convergence problems in the geometry optimization at the STO-3G level. The calculated acidities are given in Table 3. These results are shown in Fig. 1 as triangles, together with the line representing the linear regression between the theoretical and experimental data. The values of the slope, intercept, and correlation coefficient (r’) as well as the mean deviation (theory minus experiment ) and RMS deviation are given in Table 3. It is immediately apparent that these calculations predict theoretical acidities that are too high by about 6 eV (140 kcal mol-’ ). This is a well known effect of using a basis set that does not include diffuse functions and is, therefore, inadequate to describe anions accurately [ 10-121. The slope of the regression line in Fig. 1 is almost exactly unity, in contrast to the slopes of 1.1-1.2 discussed by Siggel et al. [3] for most other low-level ab initio calculations. We will see below that it is also in contrast to what is found for STO-3G calculations of the acidities of substituted phenols. However, from the scatter of the points about the line and using standard statistical formulas, we find that the uncertainty (standard deviation) in the value of the slope is 0.12; the difference between 1.0 and 1.2 is, therefore, probably not significant. Of most interest from the point of view of using STO-3G calculations to predict reasonably accurate acidities is the RMS deviation of the points from

312

the regression line (0.45 eV) and the correlation coefficient, r2, (0.872). These quantities reflect the likelihood that the theoretically calculated acidity for another compound similar in structure to those considered here will fall on the line. They therefore indicate whether the linear regression parameters can be used in conjunction with the theoretical acidity to predict the true acidity. The RMS deviation of 0.45 eV is significantly worse than the values of 0.15,0.12 and 0.10 found by Siggel et al. [3] for acidities calculated using better basis sets and treatments of electron correlation (RHF/3-21+ G//3-21 + G, RHF/ 6-31+ G//3-21 +G and MP2/6-31 +G//3-21+ G, respectively). We expect, therefore, that the predictive value of the STO-3G calculations will be noticeably worse than that of the better calculations. That this is the case can be seen from consideration of the acidities of ally1 and isopropenyl alcohol. The theoretically calculated values for the acidities of these two substances are given in Table 3. From these and the linear regression parameters, also given in Table 3, we predict that the acidities should be 16.49 eV and 15.05 eV, respectively. These can be compared with the predictions made by Bartmess [ 91, on the basis of systematics of experimental acidities, of 16.22 and 15.65 eV or with the predictions of 16.27 and 15.65 eV, respectively, based on RHF/321+ G//3-21 + G calculations and the appropriate linear regression parameters. It is clear that the STO-3G calculations have only a limited predictive value for acidities of compounds of this sort. Acidities for substituted phenols have been calculated at the RHF/STO-3G level using standard geometries by Pross et al. [ 131. Their theoretical values are plotted against the experimental ones [ 141 in Fig. 1 as crosses, together with a linear regression line. The slope of the regression line is 1.205, in agreement with slopes found for a number of other such correlations when the calculations are done at a low level [ 31, but in contrast to the results discussed above. The value of the correlation coefficient, r2, (0.944) and the RMS deviation of the points from the regression line (0.09 eV) indicate that calculations at this level, when used in conjunction with the regression parameters might be useful in predicting acidities of substituted phenols. This good correlation is probably due to the close structural similarity of the compounds. However, the range of acidities is narrow, only about 1 eV. It is, therefore, questionable whether calculations at this level will be of great value in understanding any but the gross features of the relative acidities of phenols. CONCLUSIONS

The work of Gordon et al. on the acidity of hydrides shows that calculations at the MP4/6-311+ + G (3df2pd) //6-31G (d) level are capable of giving theoretical acidities that are within the experimental uncertainty of the measured values. For a somewhat lower level basis set, such as 6-311+ +G(2d,p)//6311+ + G (2d,p) the acidities calculated at the MP4 level for methane, meth-

313

anol and formic acid differ from the measured values by amounts that are almost equal to, but slightly greater than, the experimental uncertainties. For a slightly lower level calculation, MP4/6-31+ + G (c&p) //6-31G (d), considered by Gordon et al., the root-mean-square deviations from experimental values are 0.14 eV, compared with typical experimental uncertainties of 0.09 eV. These basis sets, therefore, represent the minimum that is needed to obtain accurate calculations. Because the expense of such calculations limits their application to any but the simplest molecules, it is worthwhile considering the possibility that lower level ab initio calculations can provide useful information on acidities. The results presented here show that acidities calculated at the RHF/3-21+ G//321 + G level can be used to predict unknown acidities with satisfactory accuracy. Similar analysis presented elsewhere [ 31 on results based on the 6-31+ G basis set and/or different treatments of electron correlation show that these can also be used for similar predictions with reasonable accuracy. Except for the substituted phenols, there is considerable random deviation of acidities calculated at the RHF/STO-3G level from a linear relationship between experiment and theory. Such calculations will therefore not be very useful for predicting absolute acidities or understanding relative acidities. ACKNOWLEDGMENTS

This work was supported in part by the National Science Foundation, the Norwegian Council for Scientific and Industrial Research and the Norwegian Marshall Fund. We are indebted to John Bartmess for providing us with his estimates of the acidities of ally1 and isopropenyl alcohol.

REFERENCES 1 2 3 4 5 6

7 8 9 10

T.D. Thomas, M.R.F. Siggeland A. Streitwieser, Jr., J. Mol. Struct. (Theochem), 165 (1988) 309. M.R. Siggel and T.D. Thomas, J. Am. Chem. Sot., 108 (1986) 4360. M.R.F. Siggel, T.D. Thomas and L.J. Saethre, J. Am. Chem. Sot., 110 (1988) 91. MS. Gordon, L.P. Davis, L.W. Burggraf and R. Damrauer, J. Am. Chem. Sot., 108 (1986) 7889. J. Gao, D.S. Garner and W.L. Jorgensen, J. Am. Chem. Sot., 108 (1986) 4784. J.S. Binkley, M.J. Frisch, D.J. DeFrees, K. Rahgavachari, R.A. Whiteside, H.B. Schlegel, E.M. Fluder and J.A. Pople, Department of Chemistry, Carnegie-Mellon University, Pittsburgh, PA. J.E. Bartmess and R.T. McIver, Jr., in M.T. Bowers (Ed.), Gas Phase Ion Chemistry, Vol. 2, Academic Press, New York, 1979, p. 101. C.H. DePuy, V.M. Bierbaum and R. Damrauer, J. Am. Chem. Sot., 106 (1984) 4051. J.E. Bartmess, private communication, 1986.. W.J. Hehre, L. Radom, P. v. R. Schleyer and J.A. Pople, Ab Initio Molecular Orbital Theory, Wiley, New York, 1986, p. 312.

314 11 12 13 14

See ref. 8 of J. Chandrasekhar, J.G. Andrade and P. v. R. Schleyer, J. Am. Chem. Sot., 103 (1981) 5609. R. Ahlrichs, Chem. Phys. Lett., 34 (1975) 570. A. Pross, L. Radom and R.W. Taft, J. Org. Chem., 45 (1980) 818. C.R. Moylan and J.I. Brauman, Ann. Rev. Phys. Chem., 34 (1983) 187.