J. Phys. Chem. 1994,98, 4301-4303
4301
Acidities of Alcohols and Carboxylic Acids: Effect of Electron Correlation De Ji and T. Darrab Thomas' Department of Chemistry, Oregon State University, Corvallis, Oregon 97331 -4003 Received: February 1, 1994'
The question of the acidity of carboxylic acids relative to alcohols is investigated at a higher level of theory than has been done previously. Formic acid and methanol are used as examples. It is shown through the Hellmann-Feynman theorem that the potential at the acidic proton can be evaluated from the derivative of total energy with respect to proton charge. Using this procedure, the potential energy of the acidic proton in these molecules has been evaluated at the MP2/6-3 ll++G(2d,p) level. Combining these potential energies with acidities calculated at the same level gives the contribution of relaxation in the anion to the acidity. In agreement with earlier, lower level, calculations, these calculations show that the principal factor contributing to the higher acidity of formic acid is the initial-state potential energy of the acidic proton. Relaxation in the anion, including stabilization due to the resonance, contributes little to the acidity difference.
Introduction It has been a long-held view that the greater acid strength of carboxylic acids relative to alcohols arises from resonance delocalization of charge in the carboxylateanion,which stabilizes this anion relative to the alkoxideanion.' Thisview was challenged by Siggel and Thomas? who used both experimental and theoretical results to show that the major factor responsible for the greater acidity of the carboxylic acids is the charge distribution in the neutral molecule. This produces a potential at the acidic hydrogen that is more positive in the carboxylic acids than it is in alcohols, thus favoring proton removal from the acids. Since this idea is not in keeping with the traditional view of acidity, there have been a number of papers discussing it,3-9some of them agreeing with Siggel and Thomas and some presenting arguments in support of the traditional picture. It is apparent that this question is not completely settled in the eyes of the community of physical organic chemists, and it does not appear in many of the recent textbooks.lJ0 It is useful, therefore, to consider some additional theoretical results that lend support to the original case made by Siggel and Thomas. We first recapitulate their argument. The acidity depends on A,??for the reaction
RH
-+
R-
+ H+
and, for convenience, we use D (for deprotonation energy") to represent this energy. Using both classical and quantum mechanical arguments, Siggel and Thomas showed that
D=-V-R
Calculation of the Factors That Influence Activity
Here Vis the potential energy of a unit positive charge at the site of the acidic proton in the neutral molecule; it is negative. V/e is the potential from which the proton must be removed. R represents a relaxation energy arising from the rearrangement of electrons and nuclei accompanying removal of the acidic proton. The energy of the final ionic state and, hence, the deprotonation energy, is lowered by this relaxation, which includes the effect of resonance delocalization in the anion as well as that of any other electronic or geometric rearrangement. The difference in deprotonation energy, AD, between two acids, for instance, formic acid and methanol, is
AD = - A V - AR
In the traditional view of acidity,l the difference between these two acids arises from differential stabilization of the anions by electron rearrangement, with resonance stabilizing the formate ion more than the methoxide ion. Thus AV would be expected to be approximately0 and AD to be approximatelyequal to -AR. Experimentally, Siggel and Thomas determined AV and AR from measurements of gas-phase acidities and core-ionization energies. Theoretically, they evaluated these from ab initio electronic structure calculations of D and V, using eq 1 to obtain R . By both methods they found that AR is small and that the acidity of carboxylic acids relative to alcohols is determined almost entirely by the potential at the proton, which, in turn, isdetermined by the charge distribution in the neutral molecule. We are concerned here with their theoretical calculations. While these are persuasive, they are all based on calculations at the SCF level with small basis sets. They fall short of HartreeFock accuracy and do not include effects of electron correlation. It has been shown by Siggel, Saethre, and Thomasl2 that such calculations give acidities that deviate systematically from the experimental values. In particular, they consistentlyoverestimate relative acidities. In order to obtain accurate acidities, it is necessary to have a basis set that gives SCF results close to the Hartree-Fock value and to include the effects of electron correlation. In view of this problem, one must ask whether or not such high-level calculations would also support the view put forth by Siggel and Thomas. We report here on an investigation of this question. We find that the higher level calculations are indeed in agreement with those obtained at a lower level.
(2)
Abstract published in Advance ACS Absmcrs, March IS, 1994.
0022-3654/94/2098-4301$04.50/0
The deprotonation energy, D, can be calculated theoretically as the difference between the energy of the anion and the energy of the neutral molecule. These energies are the output of standard electronic structure calculations. In principle,the potential energy of the proton, V, can be calculated from the electronic wave functions for the neutral species and from the positions of the nuclei. Once D and Vare known, then R is calculated from eq 1. In such programs as Gaussian 86,13 used in these calculations, Vis readily obtained as a standard output of an SCF calculation. This is, however, not the case if one wishes to include electron correlation. The Moller-Plesset (MP) method gives only the effect of electron correlation on the total energy of the molecule and does not give directly any matrix elements, such as the expectationvalue of the potential at the proton. There thus arises 0 1994 American Chemical Society
4302 The Journal of Physical Chemistry, Vol. 98, No. 16, 199'4
a problem of how to include the effect of electron correlation on the potential. We can solve this problem by recognizing that the potential energy of the proton is equal to the derivative of the total energy with respect to proton charge. To see this, we note that the energy, E, of a particle of charge Z e in an electric field V/e is
E=T+VZ
(3)
Ji and Thomas
TABLE 1: Energies Calculated at the 6-311++G(2d,p)// 6-311+G(2d,p) Level with a Variable Charge on the Acidic Proton' formic acid methanol E($)
RHF
MP2
RHF
MP2
E(0.9 e) E(l.O e) E ( l . l e)
-188.74183 -188.83147 -188.93068 -0.94425 (-25.69) -0.942772 (-25.65)
-189.31095 -189.39935 -189.49734 -0.93195 (-25.36)
-1 14.98690 -115.08317 -115.18911 -1.01105 (-27.51) -1 BO964 (-27.47)
-1 15.37387 -115.46837 -115.57247 -0.99300 (-27.02)
dEldZ, where Tis the kinetic energy. Differentiating this with respect to Z gives V = dE/dZ
(4)
In the Appendix, we show that the Hellmann-Feynman theoremI4 can be used to derive this expression for a proton in a molecule. Thus a calculation of the total energy as a function of the charge on the acidic proton can be used to obtain a value of V.
VHF
All energies are in hartrces except those in parentheses, which are in electronvolts.
TABLE 2 De rotonation Energies (eV) Calculated at the 6 3 1 1++G( 2d,pp//6311++G(2d,p) Level' formic acid methanol AD RHF MP2 MP4
Application to Formic Acid and Methanol Siggel et al.l*JS have reported deprotonation energies for formic acid, methane, and methanol calculated at MP4/631 l++G(2d,p)//RHF/6-31 l++G(Zd,p). The results are in excellent agreement with experiment. Using the optimized geometries that they obtained and the same basis set, we have calculated total energiesfor formic acid and methanol as a function of acidic proton charge at both the RHF and MP2 level. The procedure is to add a small charge ( f O . l e) very close to the proton. The total energy is changed by the response of the system to the added charge and also by the Coulomb interaction of the added charge with the proton. This interaction energy is easily calculated, and its effect can be removed. The added charge is so close to the proton that it can be considered as being at the same point as far as the rest of the interactions are concerned. The problem is to choose the distance small enough that this condition is met but far enough that the effect of the interaction between the proton and the added charge is not so large that it dominates the total energy and there is loss of accuracy in the numerical values of the total energy. After some trial calculations, wechose thisdistancetobe0.01 bohr, whichleads toa modification of the total energy of 10 au. Sample calculations in all of the plus and minus Cartesian directions showed that the results are independent of direction to within 0.005 eV. Once the energies have been calculated, the derivatives and, hence, the potential energies are obtained from numerical differentiation.
- E ( Z = 0.9 e) v =dd-EZ-- E ( Z = 1.1 e)AZ
(5)
where A 2 = 0.2. To verify the validity of this approximation, for one case we made calculations of E for Z ranging from 0.5 to 1.5 e in steps of 0.1 e. These, together with standard techniques for numerical differentiation,16give accurate values of dE/dZ. It was found that eq 5 gives a value of dE/dZ that is within 0.2% of the more accurately calculated derivative. In order to test this approach,we consider results using Hartree Fock calculations. In the Gaussian programs, the potential energy of the proton can be calculated directly from the wave functions and compared with the value from eq 5 . The results are given in Table 1, where we see the total energies calculated for formic acid and methanol at both the H F and MP2 levels using three different values of the proton charge. Also shown are the values of dE/dZ calculated from eq 5 as well as the values of VHF,which are part of the output from the H F calculations. We note that the values of Vdetermined from numerical differentiation of the HFenergy agree well with thoseobtained directly fromintegration of the charge density. This agreement gives us confidence that
15.49 15.09 15.16 15.25 0.09
17.31 16.85 16.89 16.76 f 0.09
1.82 1.76 1.73 1.51 f 0.12
experiment a The experimentalvalues, which are M O O , are calculatedfrom reported values of W 2 9 (ref ~ 17). See refs 12 and 15 for details of this correction.
TABLE 3: Deprotonation Energy, D, Relaxation Energy, R, and d E / U cakulated at the 6311++C(2d,p)// 6311++G(2d,p) Level. AU Numbers la electronvolts RHF
formic acid methanol D
15.49 V = dEldZ -25.69 Rb 10.20 a
17.31 -27.51 10.20
MP2
formic acid methanol
Aa 1.82 15.09 -1.82 -25.36 0.0 10.54
A = methanol-formic acid. R = -V
16.85 -27.02 10.50'
Aa 1.76 -1.66 -0.10
- D.
the method and our implementation of it are adequate to give us accurate values of V from the Moller-Plesset calculations. The deprotonation energies, D, for formic acid and methanol, calculated at several levels of approximation are given in Table 2, where they are compared with the experimental values. We see that the deprotonation energies calculated at the MP2 and MP4 levels agree with the experimental valuesI7 within the uncertainties. For the deprotonation energy of methanol relative to formic acid, the MP2 and MP4 values differ from the experimental value by slightly more than the uncertainty, but the agreement is close enough to provide confidence that these calculations describe the molecules reasonably well. Table 3 gives the values of D, V, and R for formic acid and methanol. As noted above, R is calculated from D and Vusing eq 1. The last column gives the values of AD, AV, and AZ?. From these results we see that the conclusions reached by Siggel and Thomas on the basis of lower level calculations are confirmed. In particular, we note that, for both the HFand MP2 calculations, the major contributor to the difference in acidity between formic acid and methanol is AV(= -1.7 eV), which is a property of the neutral molecule, and not AZ? (= 0 to -0.1 eV), which contains the influenceof the final state, including resonance delocalization.
Conclusions These results confirm, at a higher level of theory, the conclusion reached by Siggel and Thomas,z namely, that most of the difference between the acidity of alcohols and carboxylic acids arises from the initial-state charge distribution, which produces a potential at the acidic proton that is less attractive in acids than in alcohols. Very little, if any, of the higher acidity of carboxylic acids relative to alcohols results from greater stabilization of the anion.
Acidities of Alcohols and Carboxylic Acids
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4303
AcLaowkdgment. We are pleased to acknowledge J. A. Pople for suggesting the method we have used to evaluate Y at the Moller-Plcsset level. We are grateful to T. X. Carroll for assistance and helpful comments and to M. R. Siggel and L. J. Saethre for help with calculations. This work was supported in part by the National Science Foundation under Grant CHE9014681. Appeadix Let H ( Z ) be a Hermitian operator of the molecular system that dependa on a real parameter Z, and let @(Z)by a normalized eigenvector of H(Z)with eigenvalue E(Z). Then, according to the Hellmann-Feynman thc0rem,l4
vp= E-ZJP - EZ P u~pRop
i rip
For the proton, Zp = 1, we get
Comparing q s 9 and 11 reveals that
-
dk vp dZP
Finally, using q s 6 and 12, we have
References and Notes For a molecule, the Hamiltonianis (under the Born4ppenheimer approximation)
or
(7) Here Teis the electron kinetic energy, V, is the Coulomb energy between electrons, V ,is the Coulomb energy between nuclei, and V, is the Coulomb energy between nuclei and electrons. Now we consider a particular nucleus, p, and let the charge on p vary. We have d h dFe dv, -=-+-+-+-
dZP dZP dZP
dv,
dvm
dZP
dZP
(8)
Since Te and V, are both independent of Z, the first two terms on the right-hand side of q 8 are zero and we have
(9) We can also directly write down the operator for the potential energy of nucleus p.
(1) (a) Streitwierer,A.; Heathcock,C. H.; Kolowcr, E.M. Introducrfon ed.; MacMillan: New York, 1992; p 486. (b) Morrison, R. T.;Boyd, R. N. Organic Chemistry, 5th ed.; AUjm and Bacon: Boston, MA, 1987; pp 83-37. (c) McMurry, J. Organic Chemfstry, 3rd ed.; Broolrs/QIe: Pacifi Grove, 1992; p 771. (d) Baker, A. D.; Engel, R. Organic Chemistry; Wsrt Publiahins Co.: St. Paul, MN, 1992; pp 794-795. (2) Si&, M.R.; Thomas, T.D. J. Am. Chem. Soc. 1986,108,4360. (3) Exner, 0.1. Org, Chem. 1988,53,1810. (4) Thoman, T.D.; Carroll, T.X.; S i l , M. R. F. J. Org. Chem. 1988, 53, 1812. (5) Thomas, T.D.;S i e l , M. R.F.; Streitwisrer,A., Jr. 1.Mol. Struct. ITHEOCHEM) 1988.16S. 309. ' (6) Siggel,'M. R.'F.; Stdtwisrsr, A., Jr.; T ~ o ~T.sD., J. Am. Chem. Soc. 1988,110,8022. (7) Taft, R.W.; Koppel, I. A.; Topaom, R. D.; Anvia, F. J. Am. Chem. Soc. 1990,112,2047. (8) Dewar, M.J. S.;Krull, K. J. Chem. Soc., Chem. Commwr. 1990, 333. (9) Pemn,C. L. J. Am. Chem. Soc. 1991,113,2865. (10) An exception, who account( for the acidity of carboxylic ad& in t e m of the explanatim propod in ref8 2 and 6, ir: Solomon, T.W.G. Organic Chemlstry, 5th ed.; John Wiiey and Sow New York, 1992; p 100. (1 1) Previously we have used A (for acidity) to rc-t thh energy. ThQ defiiition can lead to confusion since large A (or 0)c"pon& to a weak acid and vice vena. (12) Si&, M. R. F.; slcthre, L. J.; Thomas,T. D. J. Am. Ch" Soc. 1988,110,91. (13) Gaussian 86, Friach, M. J.; Binkley, J. S.; Schlegel, H. B.; Raghavachari, K.;Mclius, C. F.; Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; ROhlfig, C. M.; Kahn, L. R.; Defreer, D. J.; Scoger, R.; Whiteride, R. A.; Fox, D. J.; Fleuder, R. L.; Poplc, J. A. Gawian, Inc., Pittsburgh, PA, 1984. (14) Lsvine, I. N. Quantum Chemfstry,4th ed.;Prentice Hall: Englswood Cliff6. NJ. 1991: m 445-447. (15) S&l, M k F.; Ji, D.; Thom, T. D.; Saethre, L. J. J. Mol. Srmt. (THEOCHEM) 1988,181, 305. (16) Margmau, H.; Murphy, 0. M. The Mathemrrtlcs ojPhyslcs and Chemistry, 2nd 4.D. :Van Nc&rand Princeton. NJ. 1956 w 467-473. (17) &-, J. E.; McIvcr, R. T., Jr. In Ghr P h e Io; 'Chemistry; Bowers, M. T., Ed.; Academic Resl: New York, 1979; Vol2, p 101. to Organic Chemfsfry,4th
