CSIRO PUBLISHING Rapid Communication

www.publish.csiro.au/journals/ajc Aust. J. Chem. 2004, 57, 1205–1210

An Assessment of Theoretical Protocols for Calculation of thepK a Values of the Prototype Imidazolium Cation

Alison M. MagillA and Brian F. YatesA,B

A School of Chemistry, University of Tasmania, Hobart TAS 7001, Australia.B Corresponding author. Email: Brian.Yates@utas.edu.au

The highly accurate complete basis set method CBS-QB3 has been used in conjunction with the conductor-likepolarized continuum (CPCM) method to predict the aqueous pKa values for the three different hydrogen atomsin the imidazolium cation. Excellent agreement was obtained with the available experimental values. The pKa forthe deprotonation of imidazole was also calculated and found to be quite different from the experimental estimate.The protocol for the pKa calculation was carefully analyzed and some recommendations made about the choice oflevels of theory.

Manuscript received: 27 June 2004.Final version: 2 September 2004.

The pKa of a compound is an important property in both thelife sciences and chemistry. The theoretical prediction ofpKa values continues to arouse a considerable amount ofinterest[1–11] and there has been a large number of papers pub-lished in this area even over the last two years.[12–25] Thesepapers have explored different systems, different solvents,and different aspects of the computational methods used toevaluate pKa. Some of the papers have provided contrastingrecommendations about the procedures to be used, but therecent work of Liptak and Shields[26–29] has established awell defined protocol employing state-of-the-art molecularorbital theory, which has been shown to be highly accurateacross a range of different chemical systems. The advantageof this approach is that it is essentially an ab initio one withlittle recourse to experimental data.

Recently we published a systematic evaluation of a seriesof twelve substituted nucleophilic carbenes in which weinvestigated the pKa as a function of substituent effect inboth aqueous and non-aqueous solvents.[30] We employedthe theoretical method of Liptak and Shields and obtainedexcellent agreement with the available experimental data.That study was focussed on the proton detachment from di-N-substituted imidazolium cations. We did not consider theprototype unsubstituted imidazolium cation 1 in that study,but this is a very interesting system as it affords us the oppor-tunity to investigate several different proton detachments.Shown below are the reactions corresponding to detachmentof each of the five protons in 1 (Scheme 1).

It is also possible to consider subsequent proton detach-ment from the neutral imidazole 2 as shown in reaction4 (Scheme 2), which is reminiscent of proton loss in thebiologically important histamine and histidine systems.

This deprotonation of imidazole has been studied exten-sively, both experimentally and theoretically, in order to

Reaction 1

�H�

1 2

NH

H

H

H�N

H

Reaction 2

N

H

H

HN

H

N

NH

H

H

H

1 3

�H�

N

NH

H

H

H

H

�

Reaction 3N

N

H

H

H

H

1 4

�H�

N

NH

H

H

H

H

�

loss of N1 or N3–H

loss of C4 or C5–H

loss of C2–H

Scheme 1.

Reaction 4N

N

H

HH

H

2 5

�H�

N

NH

H

H�

Scheme 2.

understand the nature of zinc–imidazole complexes in vari-ous enzymes.[31–33] Such enzymes are known to be importantin hydration reactions and in some degradation processes.[34]A process analogous to Reaction 4 forms the basis of the‘proton relay’or ‘proton shuttle’mechanism.[35] Despite pre-vious high-level theoretical studies in this area,[2] to the bestof our knowledge the results presented here represent the

© CSIRO 2004 10.1071/CH04159 0004-9425/04/121205

1206 A. M. Magill and B. F. Yates

first highly accurate complete basis set calculations of thepKa values of all four of these reactions.

Reactions 1–4 are particularly interesting to study in lightof two very recent papers. Amyes et al.[36] have provided thefirst experimental determination of the pKa of imidazol-2-ylcarbene 3, while Duarte and coworkers[37] have performed asystematic calculation of the three pKa values for the step-wise deprotonation of histamine. There have also been recentsuggestions[38–43] that imidazole-based systems may bind totransition metals by the C4/C5 positions as well as by C2 orN. These so-called ‘abnormal’ or ‘alternate’ binding modesmay be important in both organometallic catalysis and bio-logical systems, and the pKa values calculated here may helpin understanding the intrinsic driving force for binding at eachposition.

In this paper we present our pKa results for Reactions 1–4and discuss some of the experimental and computationalaspects.

Computational Methods

For the reaction HA+ →A + H+ the pKa may be determined from thefollowing thermodynamic cycle (Scheme 3):

HAgas� Hgas

�Agas �

HAaq� Haq

�Aaq �

�Gs(HA�)º

�Gaqº

�Gs(H�)º�Gs(A)º

º�Ggas

Scheme 3.

Given thatpKa = − log Ka

and

�G◦aq = −2.303RT log Ka

it may be seen that

pKa = �G◦aq

/2.303RT

where

�G◦aq = G◦(Aaq) + G◦(H+

aq) − G◦(HA+aq)

= G◦(Agas) + �G◦s (A) + G◦(H+

gas)

+ �G◦s (H

+) − G◦(HA+gas) − �G◦

s (HA+)

Gas-phase free energies (G◦) were calculated with the highly accu-rate CBS-QB3 level of theory.[44] This state-of-the-art method performsan initial geometry optimization and frequency calculation using theB3LYP/CBSB7 level, followed by single point calculations at theCCSD(T)/6-31+G(d†), MP4SDQ/CBSB4 and MP2/CBSB3 levels.Thefinal single point calculation includes a complete basis set extrapolation.The CBS-QB3 method includes a zero-point vibrational energy (ZPVE)correction from the B3LYP/CBSB7 frequency calculation scaled by0.9899. G◦(H+

gas) was taken as −18.4 kJ mol−1. This number is derivedfrom the value of G◦(H+

gas) obtained in the Sackur–Tetrode equation

(−26.3 kJ mol−1), but takes into account a change of reference state(from atm to mol L−1).[29]

Free energies of solvation (�G◦s ) in water were calculated with the

conductor-like polarized continuum model (CPCM).[45] Single-pointcalculations were carried out at the CPCM/HF/6-31G(d), CPCM/HF/6-31+G(d) and CPCM/HF/6-311+G(d,p) levels for geometries optimized

at CPCM/HF/6-31+G(d), and at the CPCM/B3LYP/6-311+G(d,p) levelfor geometries optimized at CPCM/B3LYP/6-31+G(d,p). Althoughsomewhat complicated, we have found advantages in the past[30] inaveraging the solvation energies from these four different calculations inorder to get a better estimate of the true solvation energy. In all solvationcalculations the variablesTSNUM andTSARE, representing the numberof tesserae on each sphere and the area of each tesserae (in Å2), were setat 240 and 0.3, respectively.There is much debate[2,10,29,46–52] surround-ing the value for the free energy of solvation of the proton (�G◦

s (H+aq)).

In our previous work we chose a value of −1095.6 kJ mol−1, whichminimized the error in the pKa calculations compared with referencecompounds. (See ref. [30] for a careful discussion and evaluation of thisvalue.) In this paper we have retained this value for consistency.

All calculated free energies in this paper are at 298.15 K. Allcalculations were carried out with the Gaussian 03 programme.[53]

Results and Discussion

pKa

Calculated free energies of reaction and the associated pKa

values are shown in Table 1 for Reactions 1–4. The calculatedvalues were obtained by averaging pKa values determinedwith the four different solvation free energies as described inthe methods section. A comparison of the results for Reac-tions 1–3 shows that the most acidic hydrogens are thoseattached to the nitrogens (Reaction 1) as might have beenexpected. The hydrogens at the 2, 4, and 5 positions on theimidazolium cation are considerably less acidic. There is abig change in pKa on going from position 1 (or 3) to position2, but a much smaller change on going from position 2 toposition 4 (or 5).

Literature values for the pKa corresponding to Reaction1 are between 6.95 and 7.22,[54] although values towards thelower end of this range are more common. Our calculationsare in excellent agreement with the value quoted in the CRCHandbook of Chemistry and Physics.[55]

The pKa corresponding to Reaction 2 has veryrecently been measured from NMR deuterium exchangeexperiments.[36] This leads to an experimental value of 23.8which is somewhat smaller than our calculated value. In factthe difference between experiment and theory for this valueis larger than the errors which we observed in our earlierstudy.[30] Amyes and coworkers[36] also used the experimen-tal pKa values to determine that 2 is more stable than 3 inwater at 298 K by 95.4 kJ mol−1. This represents the driv-ing force for the 1,2-H shift of the singlet carbene to formthe neutral imidazole. Using our calculated free energies ofthe reaction in Table 1 we estimate a value of 102.0 kJ mol−1

(i.e., �GReaction 1−�GReaction 2) for this same quantity. Onceagain the agreement with experiment is quite reasonable andin fact most of the error is associated with Reaction 2.

Reaction 4 corresponds to loss of the proton from imida-zole to form the imidazolate anion. The experimental pKa for

Table 1. Free energies of reaction and pK a values

Reaction �G◦aq [kJ mol−1] pKa (theory) pKa (experiment)

1 40.1 7.02 6.99 (from ref. [54])2 142.1 24.90 23.8 (from ref. [36])3 188.2 32.974 114.1 19.98 14.2 (from ref. [55])

Calculation of the pKa Values of the Prototype Imidazolium Cation 1207

this reaction is not well defined. Although several textbooksquote values between 14.2 and 14.5,[56–58] the original deter-mination appears to be that by Walba and Isensee.[59] Theydid not actually measure the pKa for this reaction but ratherbased their estimate on an assumed difference in hydrolysisconstants between imidazole and 2-phenylimidazole.This ledthem to ‘obtain the approximate value of 14.2 for the pKa’.[59]A later paper by Taft and coworkers[60] discussed the relativeacidity of pyrrole and imidazole and their results also allowone to estimate a pKa of 14 for this reaction. On this basis itwould seem that our calculated value of 19.98 in this workis considerably greater than experiment. This is surprisinggiven the otherwise excellent agreement obtained betweentheory and experiment with the pKa protocol followed hereand it is rather likely that the experimental value is in error.A topic of future work will be to calculate the acidity of2-phenylimidazole at this same level of theory and comparethe theoretical difference in acidity with that assumed byWalba and Isensee.

Duarte and coworkers[37] have recently calculated the step-wise deprotonation of histamine (Scheme 4).The third depro-tonation step is directly analogous to Reaction 4. Althoughthey did not use the high levels of theory that we haveemployed, these authors found pKa values of 15–20 for thethird step. These values are in good agreement with our resultfor Reaction 4 in Table 1.

Proton Affinities

Gas-phase proton affinities of structures 2–5 calculatedat the CBS-QB3 level of theory are presented in Table 2. Asexpected, the proton affinity of the anion 5 is largest. The pro-ton affinity of 2 has been previously studied[61] at the CBS-Qlevel of theory which leads to a value of 936.5 kJ mol−1,a little higher than our result. Our result in Table 2 is inremarkably good agreement with the experimental gas-phasevalue[62] of 934.7 kJ mol−1. The gas-phase proton affinityof 3 has been previously calculated to be 1097 kJ mol−1

at the MP2/6-31G(d,p) level,[63] 1077 kJ mol−1 at the MP2level with an extended basis set,[64] and 1081 kJ mol−1 atB3LYP/6-31G(d).[65] The CBS-QB3 method should be moreaccurate than any of these methods[44] and thus we mightexpect the value in Table 2 to be closer to the true value. Thetheoretical proton affinity of 5 has been studied byYazal andPang[33] as a function of basis set and level of theory. The bestvalues they obtained were in the range 1448–1464 kJ mol−1,which nicely brackets our number in Table 2.

Estimation of Gas-Phase Free Energies

The calculation of absolute pKa values is a challengingtask since a difference of only 5.7 kJ mol−1 in the aqueousfree energy of reaction results in an error of one pKa unit.

�H� �H�

N

NHH

CH2CH2NH2

H N

NH

H

CH2CH2NH2

�

�H�

N

NHH

H

CH2CH2NH3�

N�

NHH

CH2CH2NH3

HH

�

Scheme 4.

Yet the CBS-QB3 method is relatively expensive and wefound in our earlier study[30] that considerable effort (e.g.,two months computing time) was required in order to obtainresults for medium-sized systems. The gas-phase free energycalculations are the most costly part of the theoretical pro-cess described here and so in this section we briefly examinealternative ways of estimating these gas-phase values.

In Table 3 are shown the results of our pKa calculationsin which the free energies of solvation have been kept con-stant but the gas-phase free energies of reaction have beencalculated at different levels of theory. Some of these levelsof theory represent the individual components of the CBS-QB3 calculation. For completeness we have also added thepopular B3LYP/6-31G(d) method, the CBS-4M method,[66]and several variants of the G1, G2, and G3[67–73] levels oftheory. B3LYP/6-31G(d) is a widely applicable inexpensivelevel of theory, while the latter methods may be consideredas (more expensive in some cases) alternatives of CBS-QB3.Liptak and Shields[28] have previously compared the CBS-QB3, CBS-4, G2, G2MP2, and G3 levels of theory for a seriesof carboxylic acids and found that the CBS-QB3 methodgave the best agreement with experiment. Hence, in thisanalysis we have simply compared the various methods withCBS-QB3.

The strength of the CBS and Gn methods is that they pro-vide a balanced treatment of basis set and electron correlationeffects. As such, we should not expect MP2 and MP4 to do aswell on their own and these methods have comparatively largedeviations in Table 3. We might expect CCSD(T) to be goodbut because of the small basis set used here it actually hasa fairly large deviation. B3LYP with a large basis set is rea-sonable, but B3LYP/6-31G(d) systematically overestimatesthe pKa values. The best alternatives to CBS-QB3 appear tobe G2, G2MP2, G3MP2B3, and G3MP2. These observationsare consistent with other authors who have concluded that theestimation of pKa values is best carried out with B3LYP orMP2 with large basis sets,[10,11] G2,[2] G2MP2, or G3[26–28]levels of theory. In particularTawa and coworkers[2] evaluatedseveral levels of theory and established a careful methodologyusing G2 for substituted imidazoles.

To help clarify this comparison we have listed the varioustheoretical methods in order of increasing average deviation

Table 2. Gas-phase proton affinities atthe CBS-QB3 level

Structure Proton affinity [kJ mol−1]

2 935.03 1047.14 1125.65 1457.3

1208 A. M. Magill and B. F. Yates

Table 3. pK a values employing gas-phase reaction energies calculated at different levels of theory

Level of theory Reaction 1 Reaction 2 Reaction 3 Reaction 4 Av. dev.A

HF/6-31G(d)B 12.5 30.1 43.7 30.3 7.9B3LYP/6-31G(d)C 10.1 30.7 39.2 28.2 5.8B3LYP/CBSB7C 9.7 27.6 35.9 26.3 3.7B3LYP/CBSB3D 8.1 25.9 33.6 21.1 1.0B3LYP/6-311+G(3df,2p)E 7.9 25.7 33.3 21.1 0.8MP2/CBSB3D 5.6 25.8 32.0 17.5 1.4MP2/GTlargeB 6.0 26.2 32.6 18.2 1.1MP4SDQ/CBSB4D 8.9 26.2 35.5 21.3 1.7CCSD(T)/6-31+G(d†)D 5.3 23.2 31.7 16.6 2.0G1C 6.5 24.3 32.6 18.7 0.7G2MP2C 7.3 24.8 33.0 20.2 0.2G2C 7.2 24.8 33.1 20.1 0.1G3MP2B3C 7.3 24.8 32.8 20.3 0.2G3MP2C 7.4 24.9 33.2 20.3 0.2G3B3C 7.5 25.2 33.3 20.5 0.4G3C 7.6 25.3 33.7 20.6 0.6CBS-4MC 7.2 25.6 34.1 19.5 0.6CBS-QB3C 7.0 24.9 33.0 20.0 0.0

A Average deviation of each level of theory from the CBS-QB3 values.B Includes geometry, ZPVE, thermal and entropy corrections from the G3 level of theory.C Includes geometry optimization and ZPVE, thermal and entropy corrections at this level of theory.D Includes geometry, ZPVE, thermal and entropy corrections from the CBS-QB3 level of theory.E Includes geometry, ZPVE, thermal and entropy corrections from the B3LYP/6-31G(d) level of theory.

Table 4. Accuracy versus time for several levels of theoryThe average timing ratio refers to the average of the ratios of central processing

unit time required relative to the HF/6-31G(d) level of theory

Level of theory Av. dev. of pKa from CBS-QB3 Av. timing ratio

CBS-QB3 0.00 11G2 0.13 39G2MP2 0.18 11G3MP2B3 0.22 6G3MP2 0.23 6G3B3 0.42 24G3 0.55 32CBS-4M 0.63 3G1 0.69 35B3LYP/6-311+G(3df,2p) 0.81 5B3LYP/CBSB3 0.98 5MP2/GTlarge 1.12 9MP2/CBSB3 1.44 6MP4SDQ/CBSB 4 1.74 3CCSD(T)/6-31+G(d†) 2.00 7B3LYP/CBSB7 3.67 3B3LYP/6-31G(d) 5.85 2HF/6-31G(d) 7.93 1

inTable 4 along with the relative time required for the calcula-tions. The timing ratios shown in Table 4 should be taken witha grain of salt. Although all the calculations were run on thesame hardware with the same version of the software, suchratios can be significantly affected by the amount of memoryallocated to a calculation, the IO capabilities of the hardware,and the size of the molecular system. That is, the ratios areexpected to change as the molecule, computer hardware, andjob parameters are changed. Nevertheless some conclusionscan be drawn. Obviously there is a trade off between theaccuracy and the expense of a method. Since G3MP2B3 and

G3MP2 are about a factor of two faster than CBS-QB3, ifcomputer resources are at a premium then we would rec-ommend these methods as cheaper alternatives for systemssuch as the ones discussed here. CBS-4M is a surprisinglycost-effective method, however its level of accuracy probablyprecludes it from being used to predict highly accurate pKa

values.[27]

Estimation of Free Energies of Solvation

Although the calculation of the free energy of solvation isa minor cost in the theoretical procedure outlined here, the

Calculation of the pKa Values of the Prototype Imidazolium Cation 1209

Table 5. pK a values employing free energies of solvation calculated at different levels of theory

Level of theory Reaction 1 Reaction 2 Reaction 3 Reaction 4 Av. dev.A

CPCM/HF/6-31G(d)B 7.0 25.6 33.7 18.7 0.7CPCM/HF/6-31+G(d)B 6.8 24.9 32.3 20.1 0.3CPCM/HF/6-311+G(d,p)B 7.0 24.9 32.2 20.0 0.2CPCM/B3LYP/6-311+G(d,p)C 7.3 24.3 33.8 21.1 0.7CPCM/HF/6-31G(d)D 7.0 25.8 33.9 18.9 0.7CPCM/HF/6-31+G(d)D 6.8 25.2 32.6 20.4 0.3CPCM/HF/6-311+G(d,p)D 6.9 25.2 32.5 20.3 0.3CPCM/B3LYP/6-311+G(d,p)D 7.2 24.6 34.0 21.1 0.7CPCM/B3LYP/CBSB7D 7.5 25.1 34.7 19.8 0.6

A Average deviation of each level of theory from the pKa values in Table 1.B Geometry optimized at CPCM/HF/6-31+G(d).C Geometry optimized at CPCM/B3LYP/6-31+G(d,p).D Gas-phase geometry optimized at B3LYP/CBSB7.

process is not entirely unambiguous. Many alternatives existto the CPCM method and even within this method differentlevels of theory may be used. In this final section several ofthese different levels of theory are briefly examined.

In Table 5 are shown the results of our pKa calculationsin which the gas-phase free energies of reaction have beenkept constant at the CBS-QB3 level but the free energies ofsolvation have been calculated at different levels of theory.The first four levels of theory are as described in the methodssection, while the remainder employ gas-phase geometriesrather than those re-optimized with the CPCM method.

In principle one would like to compare the calculated freeenergies of solvation directly with experiment, however, thesevalues are not always determined to the accuracy requiredhere. For example, two papers published within a few monthsof each other in 1997[1,2] report experimental solvation freeenergy differences for Reaction 1, which disagree with eachother by 7 kJ mol−1, or over 1 pKa unit.

The data in Table 5 can be used to assess the value ofre-optimizing the geometry in solution in order to obtain amore accurate free energy of solvation. By comparing theaverage deviations of rows 1 and 5, 2 and 6, 3 and 7, and4 and 8 one can see that re-optimization in the solvent hasvery little (if any) affect on the pKa values for these systems.This is in contrast to the results obtained previously for somephenols[29] where it was found to be necessary to perform ageometry optimization in solution in order to obtain accurateresults.

Overall the results in Table 5 indicate that the bestmethod for the present pKa calculations is CPCM/HF/6-31+G(d,p)//CPCM/HF/6-31+G(d). A slightly less accuratemethod would be CPCM/HF/6-311+G(d,p)//B3LYP/CBSB7(where the geometry is taken from the CBS calculation). Thislatter method has the advantage, however, that there is no needto optimize the geometry in solution, a process that can oftenbe associated with technical difficulties.

Concluding Remarks

The Liptak/Shields protocol for calculating pKa values isclearly a highly accurate state-of-the-art approach, whichinvolves a minimum amount of experimental input. In this

study on the imidazolium cation we obtained excellent agree-ment with experiment for two pKa values while a third wasconsiderably different. This last value requires further workto try and resolve the discrepancy with experiment, however,our values agree with the recent calculations on histamine.The application of this method to substantially larger sys-tems of increased relevance to the biological sciences couldinvolve the use of G3MP2B3 or G3MP2 theories as alterna-tives to CBS-QB3. For the systems studied here it seems thatthere is no need to re-optimize the geometry in solution foraccurate free energy of solvation calculations.

Accessory Materials

A complete listing of Cartesian coordinates, a table of totalenergies, and a table of zero-point vibrational correctionsfor all the structures calculated as part of this work is avail-able from the author, or, until December 2009, the AustralianJournal of Chemistry.

Acknowledgments

This work was supported by the Australian Research Coun-cil and the Australian Partnership for Advanced Comput-ing. A.M.M. is grateful for the provision of an AustralianPostgraduate Award.

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