Correlation between Hydrogen Bonding Association Constants inSolution with Quantum Chemistry Indexes: The Case of SuccessiveAssociation between Reduced Species of Quinones and MethanolAnnia Galano,*,† Martín Gomez,‡ Felipe J. Gonzalez,§ and Ignacio Gonzalez*,†
†Departamento de Química de la Universidad Autonoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, Iztapalapa,C.P. 09340, Mexico D.F., Mexico‡Departamento de Sistemas Biologicos, Universidad Autonoma Metropolitana-Xochimilco, Calzada del Hueso 1100, Col. VillaQuietud, Coyoacan, C.P. 04960, Mexico D.F., Mexico§Departamento de Química, Centro de Investigacion y de Estudios Avanzados del IPN, Av. IPN 2508, Col. San Pedro Zacatenco,C.P. 07360, Mexico D.F., Mexico
*S Supporting Information
ABSTRACT: The functional M05-2X together with the SMD solvent modelhave been used to calculate hydrogen bonding association constants indimethylsulfoxide (DMSO) solution. Data of equilibrium constants inDMSO for the case of electrochemically generated dianions interacting withmethanol have been considered to test the reliability of the chemistrytheoretical approach. From this approach, it was found that the successiveassociation constants involved in the formation of the complexes depend ona linear combination of three quantum chemistry indexes which are theionization energy, the electron affinity, and the charge on the oxygen atomreceiving the methanol molecule. Under this perspective, the stoichiometryof all the dianion−methanol complexes was explained on the basis of therelative strength of the hydrogen bonding interaction compared to that of themethanol−DMSO and methanol dimer complexes. This linear combinationseems to be valid regardless of the nature of the dianion structure and thenumber of methanol molecules in the complex, which is a relevant finding to generalize the applicability of both the functionalM05-2X and the SMD solvent model, to calculate association constants for any other neutral or anionic molecules interacting byhydrogen bonding with proton donors.
■ INTRODUCTION
Even though the nature of hydrogen bonding (HB) is still thesubject of many discussions,1 there are no doubts on theimportance of this kind of interactions in biochemical,chemical, and physical processes.2 In the particular case ofquinones, HB has been described to play important rolesregarding their structures in biological systems and theirfunctions as the active site of quinoenzymes.3−5 It has beenestablished that HB and protonation are fundamental processesthat affect the potentials and mechanisms involved in thereduction of quinones.6 This is particularly important sincequinone-based redox couples have a key role as electron andproton carriers in biological systems.7−11 In addition, quinone−hydroquinone pairs have been the focus of attention over manydecades since they are considered prototypical examples oforganic redox systems.12−17
Quinones are widespread in nature, where they play a largevariety of functions.18 They also have pharmaceuticalapplications, due to their anticancer, antibacterial, and fungicideactivities.18,19 The biological activity of quinones have beenassociated to their redox and acid−base properties,10,18,20−26
which have motivated a great number of electrochemicalstudies in different solvents.27,28 In nonaqueous media, quinonederivatives without acidic moieties (Q) undergo two successivereduction processes yielding two stable anionic intermediates.The first one involves the formation of the semiquinone radical(Q•−), and the second one yields the quinone dianion (Q2−).Due to the fact that the semiquinone radicals are paramagnetichigh energy species, they can disproportionate into thecorresponding quinone and dianion.14,29 In this framework,stabilization of neutral quinones and the reduced species hasbeen achieved by employing hydrogen bonding (HB) donors,which results in the regulation of the redox potential of bothquinones−semiquinone and semiquinone−dianion redox cou-ples.30,31 The presence of HB agents can stabilize thesemiquinone radical and the dianion,18 shifting the tworeduction potentials toward more positive values.32−34 Thisbehavior is in agreement with the fact that the influence of HB
Received: September 13, 2012Revised: October 11, 2012
in biological functions of quinones is well recognized.35,36
Particularly, it has also been established that studies on HBprocesses involving Q•− and Q2− are relevant to theunderstanding of electron and proton transfers in energy-transducing membranes for respiration and photosynthe-sis.37−39 Therefore, it is not surprising that the quinone HBprocesses is an active field of research.33,34,40−48
It has been recently found that the HB processes between thequinone system and alcohols depend on the concentration ofthe HB agents.34 In the same work, it was proposed that themechanism consists in consecutive association steps, with theirstrength depending on the quinone basicity. The equilibriumconstants were estimated for a set of seven different quinones,as well as the maximum number of alcohol molecules formingthe association complexes. The latter was found to be differentdepending on the particular quinone involved in the process.For example, for the interaction between Q2− and methanol, itvaries from 3 when the process involves tetrachloro-1,4-benzoquinone to 6 for 1,4-benzoquinone; tetramethyl-1,4-benzoquinone; and 2-phenyl-1,4-benzoquinone. For 9,10-anthraquinone and 5,12-naphthacenequinone, the maximummethanol−dianion ratio (CH3OH−Q2−) was found to be 4:1;and for 1,4-naphthoquinone, it was 5:1. All these experimentswere carried out in dimethylsulfoxide (DMSO) solution.Despite of all the experimental investigations conducted so
far on the HB processes involving quinones, and theimportance of such processes, to our best knowledge, thereare only a few theoretical studies dealing with their physicalchemistry insights.49,50 Therefore, it is the main goal of thepresent work to perform a systematic study on a series ofquinones and their HB interactions with methanol. Equilibriumconstants are estimated and compared with the experimentaldata available. Chemical descriptors have been investigated andrelated with the strength of the interactions. Geometricalparameters are reported for the formed complexes. Anexplanation to the variation on the maximum (CH3OH)−Q2− ratio, depending on the quinone, is provided. Theagreement with the available experimental data supports thepresented results.In addition, accurate computation of free energy changes,
and thus of equilibrium constants, for chemical processes insolution has been proven to be particularly challenging.Accordingly, finding computational strategies able of producingreliable values of such equilibrium constants would be animportant milestone in the search for computational methodsreliable enough to quantitatively reproduce experimental resultsin solution. Therefore another important goal of the presentwork is testing the efficiency of the used methodology to thatpurpose, in particular for predicting the equilibrium constantsassociated with the formation of complexes between reducedquinones and HB donors.
■ COMPUTATIONAL DETAILSGeometry optimizations and frequency calculations have beencarried out using the M05-2X functional51 and the 6-311+G(d)basis set. They have been carried out in solution, using theSMD continuum model52 and DMSO as solvent. The M05-2Xfunctional has been chosen for the task at hand because it hasbeen proven to be among the functionals with the bestperformance for calculating binding energies in hydrogen-bonded complexes and in weak-interaction complexes ingeneral.51 The SMD solvent model has been chosen since itsperformance for describing solvation in energies of both neutral
and ionic species, in aqueous and also in nonaqueous solvents,is better than that achieved with other solvent models. Inaddition, the solvent relevant to the present study (DMSO)was included in the parametrization of SMD.52 Geometrieswere fully optimized without imposing any restriction. Localminima were confirmed by the absence of imaginaryfrequencies. All the electronic calculations were performedwith the Gaussian09 package of programs.53 Thermodynamiccorrections at 298.15 K were included in the calculation ofrelative energies. To characterize the HB interactions betweenquinones and methanol, Bader topological analysis54 of thewave functions were performed. The electronic spectra havebeen computed using the time-dependent density functionaltheory (TD-DFT), based on vertical excitations involving the 6lowest lying excited states.
■ RESULTS AND DISCUSSIONThe electrochemical reduction of quinones in aprotic mediumgives rise to semiquinones and dianions, whose stabilization canbe favored by hydrogen bonding (HB) interactions with weakproton donors such as alcohols. In this framework, previousvoltammetric studies on the reduction of several quinones inthe presence of methanol allowed establishing successiveassociation steps and their respective equilibrium constants.33,34
From these studies, it was demonstrated that experimentalvalues of the association constants for the semiquinones aresignificantly smaller and less precise than in the case ofdianions.34 Therefore, the HB association processes involvingthe dianions of six different quinones and up to four moleculesof methanol, in DMSO solution, have been considered to testthe performance of the functional used in this work to predictthe equilibrium constants of the association steps. The studiedquinones are 1,4-benzoquinone (BQ), tetrachloro-1,4-benzo-quinone (TCBQ), tetramethyl-1,4-benzoquinone (TMBQ), 2-phenyl-1,4-benzoquinone (2PBQ), 1,4-naphthoquinone (NQ),and 9,10-anthraquinone (AQ) (Figure 1).
Before the test of functional M05-2X for the estimation ofequilibrium hydrogen bonding constants, selected geometricalparameters of the fully optimized geometries of these quinones,as well as those of the corresponding semiquinone radicals anddianions are reported in Table 1. According to these data, as thereduction proceeds (Q→Q•− → Q2−), the CO distancesincrease, while the CC distances become larger and the C−Cdistance shorter. This is in agreement with previousdescriptions of gradual geometrical change from quinonoid tobenzenoid form, as quinones are successively reduced.55
Figure 1. Quinones studied in this work.
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There are no experimental data reported for the geometriesof the reduced species, but there are for some of the studiedneutral quinones (Table 2). By comparing the values in Tables
1 and 2, it can be observed that the calculated geometries are inexcellent agreement with the experimental data, which supportsthe reliability of the geometries obtained with the M05-2Xfunctional. The mean unsigned error was found to be only0.012 Å. The maximum absolute error was 0.05 Å andcorresponds to the C−C distance in NQ. The geometricalparameters presented in this work are also in good agreementwith those values obtained from theoretical calculations atdifferent levels of theory not only for quinones but also for theirfirst reduced species (Q•−).56
Because of the large amount of calculations involved in thepresent work, it was unfeasible to perform exhaustiveconformational analyses for all the quinone−methanol clusters.Alternatively, we used BQ2− as reference molecule and testeddifferent conformations in DMSO as solvent. For the complexwith BQ2−−CH3OH ratio 1:1, where two different orientationsof the methanol molecule were tested (Supplementary Figure
1S). The first one (a) corresponds to a conventional HBbetween the H atom in the OH moiety of methanol and one ofthe O atoms in BQ2−, while the second one (b) corresponds toa nonconventional HB between the same H atom in methanoland the π orbitals in the quinone ring. It was found that theGibbs free energy of structure (a) is lower than that of structure(b) by 6.56 kcal/mol.For the 1:2 BQ2−−CH3OH cluster, five configurations were
tested (Supplementary Figure 2S), three of them (c, d, and e)involving only conventional HB and two of them (f and g)including one nonconventional HB. On the basis of thefindings for the 1:1 cluster, structure (a) was taken as a startingpoint, and a second methanol molecule was added at differentlocations. As it was the case for the clusters with only onemethanol molecule, the clusters involving nonconventional HBhave significantly higher Gibbs free energies than those formedonly by conventional HB. Among the latter ones (c, d, and e),the difference in Gibbs free energies are very small and belowthe currently accepted accuracy of theoretical calculations. Itmeans that any of them could be used to model the 1:2 BQ2−−CH3OH cluster. For the 1:3 and 1:4 BQ2−−methanol clusters,two orientations were tested in each case (SupplementaryFigures 3S and 4S). As for the other cases, one of themcorresponds to conventional HB and the other one not. Thecomplexes formed through the conventional HB were found tobe those with lower Gibbs free energies, by 5.93 and 6.49 kcal/mol, for the 1:3 and 1:4 complexes, respectively.According to the information gathered from the conforma-
tions tested for the BQ2−−CH3OH clusters, the moreenergetically favored interactions are conventional HBinvolving the H atom in the OH moiety of methanol andone of the O atoms in BQ2−. Therefore, for all of the otherquinones studied in this work, these are the only interactionsthat have been taken into account. Moreover, their startinggeometries have been constructed based on the conformationsthat were found to have the lowest Gibbs free energy for theBQ2−−CH3OH system.In order to calculate the equilibrium constants of the
different successive association steps, the number of equivalentconfigurations, i.e., with similar energies, has been included. Tothat purpose, we have considered the two p orbitals of eachoxygen atom in the quinone that can be involved in the HBwith every methanol molecule. This means that the degeneracynumber (σ) is equal to 4, 3, 2, and 1 for the formation ofclusters 1:1, 1:2, 1:3, and 1:4, respectively, when computedaccording to the successive association steps mechanism:
+ ↔ ···− −Q CH OH [Q CH OH]23 3
2(step 1)
··· + ↔ ···− −[Q CH OH] CH OH [Q (CH OH) ]32
3 3 22
(step 2)
··· + ↔ ···− −[Q (CH OH) ] CH OH [Q (CH OH) ]3 22
3 3 32
(step 3)
··· + ↔ ···− −[Q (CH OH) ] CH OH [Q (CH OH) ]3 32
3 3 42
(step 4)
Each of the successive equilibrium constants (Ki), conditionedhere to DMSO as solvent, has then been calculated with eq 1,while the global equilibrium constants (βn) have been obtained,from the values of Ki, according to eq 2.
σ= −ΔK eiG RT/i (1)
Table 1. Bond Distances (Å) of the Quinones (Q),Semiquinones (Q•−), and Dianions (Q2−); Obtained fromFull Geometry Optimizations at the M05-2X/6-311+G(d)Level of Theory
aC−X represents single bonds between the C atoms in the quinoneand the substituent (X = H for BQ, 2PBQ, NQ and AQ; X = C forTMBQ; and X = Cl for TCBQ).
Table 2. Experimental Bond Distances (Å) of the StudiedQuinones (Q)
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∏β ==
Kni
n
i1 (2)
The calculated values of βn are reported in Table 3, togetherwith the experimental values obtained in DMSO. Theagreement is very good with the largest discrepancycorresponding to the TCBQ2−−(CH3OH)3 and TCBQ2−−(CH3OH)1 complexes. In these cases, the calculated valueswere found to be 4.5 and 2.1 times lower than the experimentalone. For all the other βn values, this factor was lower than 1.5times. The correlation between calculated and experimental βnvalues is shown in Figure 2. The linear correlation has an r2
value of 0.92, a slope close to one (∼1.1), and an interceptclose to zero (∼−0.2). Accordingly, it can be stated that thepresented calculations properly reproduce the experimentalbehavior of the successive association phenomenon. Therelevance of this finding arises from the fact that associationequilibrium constants for different quinones structure werereproduced with the same functional. At our knowledge, this isa first example of electronic structure calculations in which theHB association phenomena has been described for electro-chemically generated species. Although this functional was usedfor a particular set of quinones, this work opens the possibilityto predict single or successive association constants for othersystems.
Because of the fact that the experimental successiveassociation constants have been properly reproduced, we havecalculated also the equilibrium constants related with theformation of the methanol dimer and the methanol−DMSOcomplex to explain why the maximum number of methanolmolecules forming the association complexes is not the samefor different quinones, Their values were found to be 2.37 ×10−3 and 5.63 × 10−2 M−1, respectively. Therefore, complex-ation processes with values of Ki lower than 5.63 × 10−2 M−1
are less favored than the formation of the methanol−DMSOcomplex, and the corresponding complexes are not expected tobe observed. That might explain why the TCBQ2−−(CH3OH)3complex was not detected in the experiments.Other linear correlation involving the experimental global
equilibrium constants has been obtained from the geometries ofoptimized quinone dianion−(CH3OH)i complexes (Figures3−8). Among the 1:1 complexes, the shortest interaction
distance was found for TMBQ and the longest one for TCBQ.Moreover, a linear correlation between the experimental β1values and the HB distance in these complexes was found withr2 = 0.996 (Figure 9, n = 1). Similar correlations were tested forthe complexes with more than one methanol molecules, usingthe average HB distance. As the number of methanol moleculesincreases, the correlation decreases. While for the 1:2complexes r2 is still quite good (r2 = 0.982), for the 1:3 and
Table 3. Global Equilibrium Constants (βn), in M−n, for the Hydrogen Bonded Complexes Formed between the Dianions ofStudied Quinones and n Molecules of Methanol in DMSO66
Figure 2. Calculated (calcd) vs experimental (exptl) global equilibriumconstants (βn) in DMSO.
Figure 3. Structures of the BQ2−−(CH3OH)n clusters.
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1:4 complexes, r2 is significantly lower (r2 = 0.853 and 0.771,respectively).In order to confirm the interactions leading to the complex
formation, Bader topological analyses were performed, andseveral critical points were identified. The electronic chargedensity ρ(r) and its Laplacian,∇2ρ(r), are reported inSupplementary Table 1S. For all the complexes, at least onebond critical point (BCP) was found between the dianion ofthe quinone and every methanol molecule involved, confirmingthe existence of the intermolecular interaction for each pair offragments. Correlations between the experimental βn values andthe electronic charge density, at the BCP, were also investigated(Figure 10). For the complexes with more than one methanolmolecule, the ρ(r) was taken as the sum of all the individualvalues. Clear trends were found in all the cases. However, as thenumber of methanol molecules increases, the correlationworsens.
Despite the fact that the relationships involving the averageHB distance or involving the sum of the electronic chargedensity at the BCPs weakens as the number of methanolmolecules increases, they seem to be good descriptors of thestrength of the HB interactions leading to the complexformation, at least qualitatively. The decreasing of r2 with thenumber of methanol molecules can be attributed to the wayused to consider more than one interaction at the same time.Averaging the HB distances or summing the ρ(r) values mightnot be as accurate as desired. However, we did not find anotherway of considering all the interactions together.Searching for alternative indexes that provide a better
quantitative description of the complex formation, we havealso calculated several global reactivity indexes. They werecalculated according to the Perdew−Levy approximation, whichis the analogue to Koopmans theorem approximation67 forDFT framework. Perdew et al.68,69demonstrated that, in exactDFT, the ionization energy (IE) is
= −IE E g( )NHOMO (3)
Figure 4. Structures of the TMBQ2−−(CH3OH)n clusters.
Figure 5. Structures of the TCBQ2−−(CH3OH)n clusters.
Figure 6. Structures of the 2PBQ2−−(CH3OH)n clusters.
Figure 7. Structures of the NQ2−−(CH3OH)n clusters.
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where EHOMO(gN) represents the energy of the highestoccupied molecular orbital (HOMO) of the N-electron system,from which an electron is removed.Similarly the electron affinity (EA) was calculated as
= −EA E g( )NLUMO (3)
where ELUMO(gN) represents the energy of the lowestunoccupied molecular orbital (LUMO) of the N-electronsystem.The absolute hardness (η), defined by Parr and Pearson as
the second derivative of the electronic energy of the systemwith respect to the number of electrons at a constant externalpotential,70
η = ∂∂ ν
⎛⎝⎜
⎞⎠⎟
EN
12
r
2
2( ) (4)
was evaluated based on the commonly used finite differenceapproximation, leading to
η = −IE EA2 (5)
The electronegativity (χ) of Pauling and Mulliken71wascalculated according to the Mulliken formula72 as
χ = +IE EA2 (6)
The electrophilicity (ω) has been calculated as proposed byParr et al.73 for the ground-state parabola mode as
ω = +−
IE EAIE EA
( )8( )
2
(7)
The electroaccepting power (ω+) and the electrodonantingpower (ω−) indexes, which have been recently presented byGazquez et al.,74 have been calculated as
ω = +−
+ IE EAIE EA
( 3 )16( )
2
(8)
and
ω = +−
− IE EAIE EA
(3 )16( )
2
(9)
IE, EA, and the reactivity indexes were calculated for thesystem receiving the next methanol molecule (CH3OH = M),i.e., they were calculated for Q2−−Mi−1 for the formation ofeach Q2−−Mi complex. The obtained values for the mentionedindexes are reported in Supplementary Tables 2S and 3S,together with the partial charge on the O atom (qO) in Q2−
involved in the formation of the Q2−−Mi complex. No directrelationship was found for any of these descriptors alone.However, multiple linear regressions were tested, and it wasfound that only a linear combination of IE, EA, and qO correctlydescribes the formation of the complexes, regardless of thenumber of methanol molecules involved, on the contrary ofwhich was previously described for the HB distance and theelectronic charge density at the BCPs. The values of thecoefficients are reported in Table 5.Apparently such combination of IE, EA, and qO accounting
for the structure−reactivity relationships are relevant for thecomplexation processes of the quinone dianions with differentnumbers of methanol molecules. The r2 values were found to
Figure 8. Structures of the AQ2−−(CH3OH)n clusters.
Figure 9. Relationship between the experimental global equilibriumconstants (βn) in DMSO and average HB distance in the complexes.
Figure 10. Relationship between the experimental global equilibriumconstants (βn) and the sum of the electronic charge density at theBCPs.
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be 0.967, 0.98, 0.96, and 0.977 for the complexes with 1, 2, 3,and 4 methanol molecules, respectively (Figure 11). The results
obtained with this combination, which can be defined as adescriptor directly related to the formation of HB complexsolution, are quite promising. Therefore, this descriptor mightbe useful for other HB complexation processes, albeit thisremains to be tested since such generalization escapes thepurposes of the present work. Moreover, the weight of thedifferent coefficients in the multiple linear regressions allows fordirectly quantifying the relative importance IE, EA, and qO onthe complexation process, which can be highly useful to thedetailed understanding of such process for different systems.
■ CONCLUSIONSThe functional M05-2X together with the SMD solvent modelhave shown to be adequate for describing binding energies andsolvation both in neutral and ionic species interacting byhydrogen bonds. The reliability of these calculations was testedin the case of the hydrogen bonding interaction of methanoland a set of quinone dianions. For all the dianion−methanolsystems, the number of methanol molecules involved in theobserved complexes has been explained based on the relativestrength of these complexes compared to that of themethanol−DMSO and methanol dimer complexes, althoughthe last one was found to be the weakest. That is, the electronicstoichiometry of the dianion−methanol complexes is mainlydetermined by the dianion basicity and the strength of thehydrogen bonding interactions between DMSO and methanol.From this approach, it was found that the successive
association constants involved in the formation of thecomplexes depend on a linear combination of three quantumchemistry indexes which are the ionization energy, the electronaffinity, and the charge on the O atom receiving the methanol
molecule. This combination seems to be valid regardless of thenumber of methanol molecules in the complex. Because of thefact that the parameters before mentioned could be calculatedwith the M05-2X and SMD approach for any molecule, aninteresting perspective of this work could be the use of thecoefficients of the linear combination to predict associationconstants for other HB systems.
■ ASSOCIATED CONTENT*S Supporting InformationElectronic charge density and its Laplacian at the BCP criticalpoints. Ionization energies, electron affinities, and Mullikencharge in the O involved in the HB interaction. Absolutehardness, electronegativity, electrophilicity, electrodonantingpower, and electroaccepting power of the H acceptors.Optimized geometries of the complexes. This material isavailable free of charge via the Internet at http://pubs.acs.org.
■ ACKNOWLEDGMENTSA.G. acknowledges the Laboratorio de Visualizacio n yComputo Paralelo at Universidad Autonoma Metropolitana-Iztapalapa for the access to its computer facilities. M.G.acknowledges CONACYT (Project #CB2006-1-62198) for thefinancial support.
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Table 5. Coefficients of the Multiple Linear Regressiona
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The Journal of Physical Chemistry A Article
dx.doi.org/10.1021/jp309085g | J. Phys. Chem. A XXXX, XXX, XXX−XXXH
