ARTICLE pubs.acs.org/JPCA
Calculations of pKa of Superacids in 1,2-Dichloroethane Aleksander Trummal,*,† Alar Rummel,† Endel Lippmaa,† Ivar Koppel,‡ and Ilmar A. Koppel*,‡ † ‡
National Institute of Chemical Physics and Biophysics, 10 R€avala Blvd., Tallinn 10143, Estonia Institute of Chemistry, University of Tartu, 14A Ravila Street, Tartu 50411, Estonia
bS Supporting Information ABSTRACT: Acidity calculations for some CH and NH superacids in 1,2-dichloroethane (DCE) were carried out using SMD and COSMO-RS continuum solvation models. After comparing the results of calculations with respective experimental pKa values it was found that the performance of SMD/M05-2X/6-31G* method is characterized by the mean unsigned error (MUE) of 0.5 pKa units and the slope of regression line of 0.915. The similar SMD/B3LYP/6-31G* approach was slightly less successful. The strong correlation over entire data set is confirmed by R2 values of 0.990 and 0.984 for M05-2X and B3LYP functionals, respectively. The COSMO-RS method, while providing the value of the linear regression line slope similar to the corresponding values from SMD approach, characterized by rather loose correlation (R2 = 0.823, MUE = 1.7 pKa units) between calculated and experimental pKa values in DCE solution.
’ INTRODUCTION The acids of extreme strength exceeding the threshold defined by the strength of sulfuric acid in particular medium are commonly referred to as superacids.13 A number of processes of great technological relevance1 including fuel cell4 and petrochemistry5 applications are based on the superacidic properties of these acids and the respective acidbase equilibria. The gas-phase acidity (GA) values, both experimental and theoretical,6 are widely used to characterize the acidic strength of superacids. The recent report on equilibrium acidities of many extremely strong acids measured in 1,2-dicloroethane (DCE)3 provides a remarkable extension to the existing data by establishing the self-consistent acidity scale for superacids in the constantcomposition liquid medium. The negligible basicity and inertness of DCE together with its ability to dissolve polar and ionic compounds provide perfect combination for measuring acidity of very strong acids in solution. At this point, it would be highly desirable to complement the measurements by theoretical approach. There are two important considerations worth mentioning. First, the experimental acidity scale in DCE provides additional reference values for testing performance of computational solvation models. Second, a possible good match between calculated and measured pKa values would act as an indirect confirmation of the quality of the experimental acidity scale. Several reports dedicated to the calculation of pKa values and free energies of solvation in organic solvents717 using variants of the dielectric continuum theory18 have been published recently. The following is a brief overview of some results based on popular PCM and COSMO-RS continuum solvation models. Fu et al.14 applied several protocols for the calculation of pKa values in dimethyl sulfoxide (DMSO) and acetonitrile (MeCN). Initially, PCM-based cluster-continuum approach was used to calculate pKa values of organic acids in DMSO with a precision of r 2011 American Chemical Society
1.71.8 pKa units. Later, an improved method based on pure IEF-PCM results combined with B3LYP/6-311þþG(2df,2p) gas-phase acidities predicted experimental pKa values with the errors having standard deviation of 1.4 pKa units. For MeCN, DPCM solvation method was selected to complement B3LYP/ 6-311þþG(2df,2p) gas-phase acidities. The standard deviation of the resulting pKa errors was 1.0 pKa units. The combination of scaled B3LYP/6-311þG** gas-phase acidities with the IEF-PCM method has provided reliable pKa values for diverse set of substituted phenols in both DMSO and MeCN.16 The good predictive power of this approach is characterized with a mean unsigned error (MUE) of 0.6 pKa units for DMSO and 0.7 pKa units for MeCN. The corresponding correlations between the calculated and experimental pKa values resulted in regression line slopes very close to unity. Considering rather good predictive power of the IEF-PCM method19 on calculating nonaqueous pKa's it is reasonable to expect even performance improvement in the new SMD solvation method20 introduced in Gaussian0921 as a recommended procedure for calculating solvation free energies. On the basis of the same bulk electrostatics formalism, the new SMD differs from IEFPCM in using Coulomb solute radii for cavity construction and in more consistent handling of nonbulk electrostatic (CDS) terms. Eckert et al.11 applied COSMO-RS procedure22 combining dielectric continuum theory with a statistical thermodynamics treatment to calculate pKa values for the different classes of organic acids in MeCN. The method predicts pKa of substituted phenols in MeCN with the MUE of 0.8 pKa units. This quite good precision is achieved after applying the results of the correlation between calculated ΔGdiss/RTln(10) and the respective experimental pKa Received: March 15, 2011 Revised: April 26, 2011 Published: May 25, 2011 6641
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values that was established for the large set of diverse compounds. The raw ab initio pKa values deviate more substantially from the corresponding experimental values with MUE and rms error of 3.2 and 3.3 pKa units, respectively. The rms error of ab initio pKa values for the subset of CH acids is much lower, being 0.91.1 pKa units. It appears that the best performance of COSMO-RS could be achieved for the cases where acids form anions with highly delocalized charge that is relevant for several strong CH and NH acids. It is important to note that theoretical nonaqueous pKa values are published mainly for dimethyl sulfoxide, acetonitrile, and tetrahydrofuran solutions, thus leaving the vast majority of widely used organic solvents23 untouched. The obvious reason behind that is the lack of accurate experimental data required for establishing the meaningful comparisons between predicted and measured pKa values. Therefore, the self-consistent experimental acidity scale in DCE defines a new challenge for dielectric continuum theory to make respective pKa predictions. The purpose of this work is to test the performance of COSMO-RS22 and SMD20 solvation methods on calculation of pKa values of some strong CH and NH acids in 1,2-dichloroethane solution.
COSMO BP/TZVP geometry optimizations within RI approximation were carried out first in the conductor limit with Turbomole software package24 for the studied acid and corresponding anion pair. For the resulting solvated structures, COSMO-RS calculations were performed taking DCE as a real solvent and computing the deviations from ideal conductor by evaluating the differences in electrostatic and H-bonding energies according to the default procedure implemented in the COSMOtherm software.25 For any unparametrized solvent like DCE with default LFER constants (c1/RT ln(10) = 1 and c2 = 0) the pKa reading in COSMOtherm corresponds to the difference in free energies for the solvated anion and respective acid. For SMD calculations we use eq 3 together with the following expansion for ΔGs
’ COMPUTATIONAL METHODS The relative experimental pKa values3 anchored to picric acid with arbitrarily assigned value of 0.0 pKa units were used in the present study as a reference. In addition to the selected CH and NH acids, HCl was included in the data set to extend the pKa range for this study. The calculations of pKa are based on the following thermodynamic cycle
RT ln(24.46) reflects the change in the standard conditions from 1 atm to mol/L. The fitting procedure was carried out to minimize MUE of calculated pKa values and estimate c3. It is important to note that the value of c3 affects the intercept rather than the slope of the corresponding linear regression. The structures were optimized both in solution (SMD/B3LYP/ 6-31G* and SMD/M05-2X/6-31G*) and in the gas phase using the same functional and basis set combinations. The ΔGs(A) and ΔGs(AH) values are defined as the differences in SCF energy of the structure in solution and in the gas phase. The internally stored dielectric constant for DCE (ε = 10.125) was used. Both electrostatic and CDS terms were included in the calculation of ΔGs values. When available, experimental GA values2,6,26 were used in this study. For the acids with unknown experimental gas-phase acidity, GA values were calculated27 at G3(MP2) or B3LYP/6-311þG** level of theory. All geometry optimizations, both in the gas phase and in solution, were followed by frequency calculations to confirm the optimized structures to be the true minima on the potential energy surface. All thermal corrections were calculated for the standard state of 1 atm at 298.15K. All SMD and GA calculations were carried out with Gaussian09 software package.21
To calculate pKa the following relations were applied þ ½A s ½Hs ½AHs
ð1Þ
pKa ¼ log Ka
ð2Þ
ΔGs ¼ RT ln Ka
ð3Þ
Ka ¼
pKa ¼
ΔGs RT lnð10Þ
ð4Þ
COSMO-RS calculations are based on the following modification of eq 4 pKa ¼
c1 ðGs ðA Þ Gs ðAHÞÞ þ c2 RT lnð10Þ
ð5Þ
where the slope c1 is expected to be unity and c2 is an intercept. The procedure of obtaining the difference in Gs values is detailed below.
ΔGs ¼ GAðAHÞ þ ΔGs ðA Þ ΔGs ðAHÞ þ c3
ð6Þ
The c3 parameter introduced in eq 5 combines the unknown absolute values of ΔGs(Hþ) and pKa of picric acid in DCE c3 ¼ RT lnð24:46Þ þ ΔGs ðHþ Þ pKa ðpicricÞRT lnð10Þ ð7Þ
’ RESULTS AND DISCUSSION The results of COSMO-RS calculations are presented in Table 1. The results in Table 1 are based on the following linear regression established for the calculated ΔGs and respective experimental pKa values pKa ¼ 0:571ΔGs 161:461,
R 2 ¼ 0:628,
c1 ¼ 0:779
A closer look at individual pKa errors reveals that CH(CN)3 deviates strongly from the overall correlation. Notably, the GA value of 294.8 kcal/mol calculated for this compound in the present work at G3(MP2) level is consistent with the experimental estimate of 293.0 kcal/mol based on fractional additivities of CN substitutions, the calculations at MP2/6-311þþG** level 6642
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Table 1. Experimental and Calculated pKa Values and Differences in Gibbs Free Energies in Solution (in kcal/mol, 1 kcal/mol = 4.184 kJ/mol) expt3
ΔGs
calcd
error
calcdb
error
HCl
0.4
275.7
4.1
3.7
3.9
3.5
4-NO2C6H4SO2NHSO2C6H4-4-Cl CH(CN)3
2.4 6.5
277.4 260.2
3.1 12.9
0.7 6.4
2.8
0.4 3.0
9.0
272.5
5.9
3.1
6.0
Tf2NH
11.9
265.3
10.0
1.9
10.8
1.1
(C4F9SO2)2NH
12.2
264.3
10.6
1.6
11.5
0.7
(C2F5SO2)2NH
12.3
264.1
10.7
1.6
11.6
0.7
CF2(CF2SO2)2NH
13.1
261.0
12.4
0.7
13.6
0.5
(FSO2)3CH
13.6
256.0
15.3
1.7
16.9
3.3
Tf3CH mean error
16.4
260.3
12.8
3.6 0.0
14.1
2.3 0.0
C6F5CHTf2a
a
MUE
2.5
1.7
rms error
3.0
2.1
Tf denotes CF3SO2 group. b CH(CN)3 excluded from correlation.
Table 2. Experimental and Calculated pKa Values M05-2X/6-31G* 3
a
expt
calcd
error
calcd
b
error
B3LYP/6-31G* c
error
calcd
error
calcdb
error
calcdc
error 0.1
calcd
HCl
0.4
0.5
0.1
0.5
0.1
0.4
0.0
0.4
0.0
0.4
0.0
0.3
4-NO2C6H4SO2NHSO2C6H4-4-Cl
2.4
1.2
1.2
1.0
1.4
0.6
1.8
0.4
2.0
0.1
2.3
0.6
1.8
CH(CN)3 C6F5CHTf2a
6.5 9.0
6.0 7.8
0.5 1.2
6.0 8.0
0.5 1.0
6.1 7.8
0.4 1.2
5.9 7.4
0.6 1.6
5.9 7.9
0.6 1.1
5.9 7.7
0.6 1.3 0.4
Tf2NH
11.9
11.8
0.1
11.8
0.1
11.7
0.2
11.8
0.1
11.8
0.1
12.3
(C4F9SO2)2NH
12.2
12.4
0.2
12.2
0.0
11.8
0.4
12.2
0.0
12.0
0.2
12.0
0.2
(C2F5SO2)2NH
12.3
12.9
0.6
12.8
0.5
12.3
0.0
12.8
0.5
12.8
0.5
12.8
0.5
CF2(CF2SO2)2NH
13.1
13.1
0.0
13.2
0.1
13.0
0.1
13.2
0.1
13.2
0.1
13.2
0.1
(FSO2)3CH
13.6
14.2
0.6
14.4
0.8
14.4
0.8
14.3
0.7
14.5
0.9
14.7
1.1
Tf3CH
16.4
17.1
0.7
17.8
1.4
16.8
0.4
17.1
0.7
17.4
1.0
17.7
1.3
mean error MUE
0.1 0.5
0.0 0.6
0.3 0.5
0.2 0.6
0.2 0.7
0.1 0.7
rms error
0.7
0.8
0.8
0.9
0.9
0.9
Tf denotes CF3SO2 group. b Using gas-phase geometry in solution. c Using Gibbs free energy values from frequency calculations.
(294.2 kcal/mol), and the scaled value of 295.4 kcal/mol.2 It seems that in case of CH(CN)3, the default BP/TZVP method could be insufficient for Turbomole COSMO and subsequent COSMO-RS calculations to make reasonable pKa prediction. The exclusion of this outlier from the correlation leads to the alternative set of pKa values based on the following linear regression pKa ¼ 0:662ΔGs 186:301,
R 2 ¼ 0:823,
c1 ¼ 0:903
It is evident that even after excluding the data for deviating CH(CN)3 from overall correlation there are still difficulties with the remaining acids. Although the slope of the regression line (0.903) can be regarded as reasonable, the individual pKa errors for several acids are still too high, translating into MUE of 1.7 pKa units and low R2 (0.823). On the basis of the previous experience with COSMO-RS one could expect the large deviation for HCl due to its anion with strongly localized charge. However, the difficulties in COSMO-RS handling of several solvated CH acids seem rather surprising because it is exactly the situation where COSMO-RS is expected to demonstrate its best performance.
The results of pKa calculations using SMD method are given in Table 2. The MUE for calculated pKa are 0.5 and 0.6 pKa units for M05-2X and B3LYP functionals, respectively. The largest individual errors are 1.2 pKa units for both 4-NO2C6H4SO2NHSO2C6H4-4-Cl and C6F5CHTf2 at M05-2X level and 2.0 and 1.6 pKa units, respectively, for the same compounds at B3LYP level. For all other compounds, the differences between results obtained by two alternative methods are small and it is evident that the slightly better performance of M05-2X over B3LYP is based on the better description of 4-NO2C6H4SO2NHSO2C6H4-4-Cl and C6F5CHTf2 solvation by the former method. The plots of linear regression of calculated pKa vs experimental values are presented in Figure 1. The slopes of the regression lines are 0.915 and 0.883 while the values of R2 are 0.990 and 0.984 for M05-2X and B3LYP, respectively. The better results achieved by the former method derive from its superior performance on 4-NO2C6H4SO2NHSO2C6H4-4-Cl and C6F5CHTf2. 6643
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Figure 1. Correlation between the experimental and calculated pKa values in DCE.
The standard approach implemented in this study to calculate ΔGs values follows the best practice of (a) using the difference in SCF energies in solution (including Gibbs free energy contributions from nonelectrostatic terms) and gas phase to calculate the ΔGs and (b) reoptimizing the geometry in solution instead of using gas-phase geometry in solution phase. Some alternative procedures for calculating ΔGs were also evaluated in the present study. Using gas-phase geometries for computing SCF energies of solvated structures increases MUE by up to 0.1 pKa units for both M05-2X and B3LYP methods over against the standard procedure. For individual compounds, the largest effect was observed for Tf3CH at M05-2X level. The similar small increases of mean unsigned and/or rms errors were observed for the pKa calculations where ΔGs values were computed as the differences in Gibbs free energies based on thermal contributions from the nuclear motions in gas phase and solution. For M05-2X, the effect of solution phase optimization is more visible than for B3LYP. Comparison of thermal motion based Gibbs free energies to SCF energies for ΔGs calculations reveals more prominent effect in case of B3LYP. These results indicate that the standard approach provides somewhat better performance as compared to alternative (and less justified) ways of calculating ΔGs values. This outcome is consistent with the recent comments by Ho et al.28 on the correct use of continuum solvation models. On the basis of results of the regression analysis, it is possible to establish two relations between the value of the free energy of solvation of proton and the value of pKa of picric acid in DCE at 298.15 K ΔGs ðHþ Þ ¼ 1:364pKa ðpicricÞ 270:8
ð8Þ
ΔGs ðHþ Þ ¼ 1:364pKa ðpicricÞ 270:5
ð9Þ
The former expression (M05-2X) has some preference over the latter one (B3LYP) based on somewhat better performance demonstrated by M05-2X method.
’ CONCLUSIONS The predictive power of the SMD method for calculation of pKa values of CH and NH superacids in DCE is respectable. Both tested functionals (M05-2X and B3LYP) demonstrated solid performance with M05-2X results having slight advantage over B3LYP. The mean unsigned error of 0.5 pKa units (M05-2X)
compares well with the corresponding values of 0.6 and 0.7 obtained by using IEF-PCM method on substituted phenols in dimethyl sulfoxide and acetonitrile, respectively. The correlation between calculated and experimental pKa values over the entire data set is high (R2 = 0.990) and the value of the slope of the regression line (0.915) is quite reasonable. These results also indirectly validate the experimental pKa values used in the present study. In contrast, COSMO-RS method, while providing the value of the linear regression line slope similar to the corresponding values from SMD approach, is characterized by a very loose correlation between calculated and experimental pKa values in DCE solution. The significant deviation was observed for CH(CN)3 and to a smaller extent also for a couple of other CH acids as well as for HCl. For HCl, the observed deviation was not totally unexpected. For the remaining CH and NH acids, however, considering goodto-satisfactory precision of COSMO-RS pKa predictions in acetonitrile, especially for the set of CH acids as a reference, the present results look rather discouraging. The observed difficulties could arise from the established procedure of direct calculation of the Gibbs free energies for the solvated structures that is specific to current COSMO-RS implementation. The combination of high quality GA values with the free energies of solvation for pKa calculations proved to be more reliable alternative.
’ ASSOCIATED CONTENT
bS
Supporting Information. GA values, M05-2X/6-31G* and B3LYP/6-31G* energies, free energies, SMD/M05-2X/6-31G* and SMD/B3LYP/6-31G* energies, free energies, solvation free energies, and structures of the studied acids and corresponding anions. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: (A.T.) aleksander.trummal@kbfi.ee; (I.A.K) ilmar.
[email protected].
’ ACKNOWLEDGMENT This work was in part supported by Grant 8162 from the Estonian Science Foundation and by target financing Grants SF0690021s09 and SF0180089s08 from the Estonian Ministry of Education and Research. The authors appreciate the suggestion 6644
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The Journal of Physical Chemistry A of one of the referees to use relative experimental acidities instead of arbitrarily fixing the absolute pKa value of picric acid in DCE.
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ARTICLE
Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, € Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; S.; Daniels, A. D.; Farkas, O.; Fox, D. J. Gaussian 09, Revision B.01; Gaussian, Inc.: Wallingford, CT, 2009. (22) (a) Eckert, F.; Klamt, A. AIChE J. 2002, 48, 369. (b) Klamt, A.; Eckert, F. Fluid Phase Equilib. 2000, 172, 43. (23) Reichardt, C.; Welton, T. Solvents and Solvent Effects in Organic Chemistry, 4th ed.; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, 2011. (24) (a) TURBOMOLE V6.2 2010, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH, 19892007, TURBOMOLE GmbH, since 2007; available from http://www.turbomole. com. (b) Ahlrichs, R.; Baer, M.; Haeser, M.; Horn, H.; Koelmel, C. Chem. Phys. Lett. 1989, 162, 165. (c) Treutler, O.; Ahlrichs, R. J. Chem. Phys. 1995, 102, 346. (d) Arnim, M. v.; Ahlrichs, R. J. Comput. Chem. 1998, 19, 1746. (e) Eichkorn, K.; Treutler, O.; Oehm, H.; Haeser, M.; Ahlrichs, R. Chem. Phys. Lett. 1995, 240, 283. (f) Eichkorn, K.; Treutler, O.; Oehm, H.; Haeser, M.; Ahlrichs, R. Chem. Phys. Lett. 1995, 242, 652. (g) Eichkorn, K.; Weigend, F.; Treutler, O.; Ahlrichs, R. Theor. Chem. Acc. 1997, 97, 119. (h) Weigend, F. Phys. Chem. Chem. Phys. 2006, 8, 1057. (i) Sierka, M.; Hogekamp, A.; Ahlrichs, R. J. Chem. Phys. 2003, 118, 9136. (j) Weigend, F. Phys. Chem. Chem. Phys. 2002, 4, 4285. (k) Armbruster, M. K.; Weigend, F.; van W€ullen, C.; Klopper, W. Phys. Chem. Chem. Phys. 2008, 10, 1748. (25) Eckert, F.; Klamt, A. COSMOtherm, Version C2.1, Revision 01.10; COSMOlogic GmbH&CoKG: Leverkusen, Germany, 2009. (26) Bartmess, J. NIST Chemistry WebBook, NIST Standard Reference Database Number 69; National Institute of Standards and Technology: Gaithersburg MD, 2005; http://webbook.nist.gov. (27) GA values calculated in this work (in kcal/mol) and computational methods: CH(CN)3 C6F5CHTf2
294.8 287.7
G3(MP2) B3LYP/6-311þG** B3LYP/6-311þG**
(C4F9SO2)2NH
283.6
(FSO2)3CH
280.0
G3(MP2)
Tf3CH
275.3
B3LYP/6-311þG**
(28) Ho, J.; Klamt, A.; Coote, M. L. J. Phys. Chem. A 2010, 114, 13442.
6645
dx.doi.org/10.1021/jp202434p |J. Phys. Chem. A 2011, 115, 6641–6645
