Microscopic description of acid–base equilibriumEmanuele Grifonia,b, GiovanniMaria Piccinia,b, and Michele Parrinelloa,b,c,1

aDepartment of Chemistry and Applied Biosciences, Eidgenossische Technische Hochschule (ETH) Zurich, CH-6900 Lugano, Ticino, Switzerland; bInstitute ofComputational Science, Universita della Svizzera Italiana, CH-6900 Lugano, Ticino, Switzerland; and cItalian Institute of Technology, 16163 Genova, Italy

Contributed by Michele Parrinello, January 10, 2019 (sent for review November 29, 2018; reviewed by Christoph Dellago and David E. Manolopoulos)

Acid–base reactions are ubiquitous in nature. Understandingtheir mechanisms is crucial in many fields, from biochemistry toindustrial catalysis. Unfortunately, experiments give only limitedinformation without much insight into the molecular behavior.Atomistic simulations could complement experiments and shedprecious light on microscopic mechanisms. The large free-energybarriers connected to proton dissociation, however, make theuse of enhanced sampling methods mandatory. Here we performan ab initio molecular dynamics (MD) simulation and enhancesampling with the help of metadynamics. This has been madepossible by the introduction of descriptors or collective vari-ables (CVs) that are based on a conceptually different outlook onacid–base equilibria. We test successfully our approach on threedifferent aqueous solutions of acetic acid, ammonia, and bicar-bonate. These are representative of acid, basic, and amphotericbehavior.

acid–base | metadynamics | collective variables | enhanced sampling

Acid–base reactions play a key role in many branches ofchemistry. Inorganic complexation reactions, protein fold-

ing, enzymatic processes, polymerization, catalytic reactions, andmany other transformations in different areas are sensitive tochanges in pH. Understanding the pH role in these reactionsimplies having control over their reactivity and kinetics.

The crucial importance of pH has stimulated the collection ofa large amount of data on acid–base equilibria. These are typi-cally measured in gas and condensed phases, using spectroscopicand potentiometric techniques. However, there are practical lim-itations to the accuracy of these methods especially in condensedphases (1). Furthermore it is very difficult to extract from exper-imental data a microscopic picture of the processes involved. Itis thus not surprising that acid–base equilibrium has been thesubject of intense theoretical activity (1–12).

The acidity of a chemical species in water can be expressed interms of pKa , the negative logarithm of the acid dissociation con-stant. There are two ways of calculating these values, one staticand the other dynamic.

The most standard approach is the static one in which solution-phase free energies, and consequently pKas, are obtained byclosing a Born–Haber cycle composed of gas phase and sol-vation free energies (1, 3–7). While extremely successful inmany cases, the static approach has some limitations. A solva-tion model needs to be chosen and continuum solvent modelshave a limited accuracy. This is particularly true in systemslike zeolites or proteins characterized by irregular cavities inwhich an implicit description of the solvent is challenging. Obvi-ously from such an approach dynamic information cannot begained. Furthermore, there can be competitive reactions thatcannot be taken into account unless explicitly included in themodel.

In principle these limitations could be lifted in a moredynamical approach based on molecular dynamics (MD) sim-ulations in which the solvent molecules are treated explicitly.If one had unlimited computer time, such simulations wouldexplore all possible pathways and assign the relative statisticalweight to the different states. Unfortunately the presence ofkinetic bottlenecks frustrates this possibility by trapping the sys-tem in metastable states, since different protonation states are

separated by large barriers. Furthermore in acid–base reactionschemical bonds are broken and formed. This requires the use ofab initio MD in which the interatomic forces are computed onthe fly from electronic structure theories. This makes the calcu-lation more expensive and reduces further the time scale that canbe explored.

To overcome this difficulty, the use of enhanced samplingmethods (13) that accelerate configurational space explorationbecomes mandatory. A very popular class of enhanced samplingmethods is based on the identification of the degrees of freedomthat are involved in the slow reaction of interest. These degreesof freedom are usually referred to as collective variables (CVs)and are expressed as explicit functions of the atomic coordinatesR. Sampling is then enhanced by adding a bias that is a functionof the chosen CVs (14–16). Furthermore, designing a proper setof good CVs has also a deeper meaning. Successful CVs capturein a condensed way the physics of the problem, identify its slowdegrees of freedom, and lead to a useful modelistic descriptionof the process.

In standard chemical reactions, this is relatively simple sincewell-defined structures can be assigned to reactants and prod-ucts (17–19). This is not the case for acid–base reactions in whicha proton is added to or subtracted from the solute. Once thisprocess has taken place, water ions (H+ or/and OH−) are sol-vated and their structure becomes elusive. In fact, water ionscan rapidly diffuse in the medium via a Grotthuss mechanism(20). They became highly fluxional and the identity of the atomstaking part in their structure changes continuously. The natureof these species is thus difficult to capture in an explicit ana-lytic function of R. However, given the relevance of acid–basereactions, many attempts have been made at defining these enti-ties (8–12). Unfortunately these CVs have an ad hoc natureand, while successful in this or that case, cannot be generallyapplied.

Significance

Acid–base reactions are among the most important chemicalprocesses. Yet we lack a simple way of describing this classof reactions as a function of the atomic coordinates. In fact,once dissolved in water, H+ and its conjugate anion OH−

have a highly fluxional structure difficult to pin down. Herewe solve this issue by taking the point of view of describingacid–base reactions as an equilibrium between the solute andthe whole solvent. This allows identifying generally applica-ble descriptors. As a consequence it is now possible to performquantitative enhanced sampling simulation of acid–base reac-tion in water and in other environments such as the zeolitecavities or at surfaces.

Author contributions: E.G., G.P., and M.P. designed research; E.G. and G.P. performedresearch; E.G., G.P., and M.P. analyzed data; and E.G., G.P., and M.P. wrote the paper.y

Reviewers: C.D., University of Vienna; and D.E.M., University of Oxford.y

The authors declare no conflict of interest.y

Published under the PNAS license.y1 To whom correspondence should be addressed. Email: parrinello@phys.chem.ethz.ch.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1819771116/-/DCSupplemental.y

Published online February 14, 2019.

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To build general and useful CVs we make two conceptualsteps. One is to look at the acid–base process as a reactioninvolving only a few moieties, namely the whole solvent and thereacting residues in the solvated molecule. For example, whenthere is only one type of dissociating residue, we think of theacid–base equilibrium as a reaction of the type

A+H2NON Bq0 +H2N+q1Oq1N , [1]

where N is the number of water molecules, A and B are ageneric acid–base molecule in solution and its conjugate species,respectively, q0 and q1 are integers that can assume values +1and −1 according to the acid–base behavior of the species, andq1 + q0 =0.

This implies that we do not look at the solvent as a set ofmolecules that compete to react with the acid–base species.Rather we consider the solvent in its entirety as one of the twoadducts. Taking this point of view is especially relevant in polarsolvents like water that are characterized by highly structurednetworks. In this case the presence of an excess or a deficiency ofprotons changes locally the network structure and this distortionpropagates along the entire network.

Since the very early days of Wicke and Eigen (21) and Zun-del and Metzger (22), researchers have struggled with how manymolecules should be included in the definition of the perturba-tion (23–25). Given the absence of physical parameters capableof giving a clear and unequivocal answer to this question, the ideaof considering the solvent as a whole circumvents this problem.Thus, the solvent is not just a medium with a passive role, but itis looked at as an ensemble of molecules that contribute collec-tively to the formation of the conjugate acid–base pair. This pointof view is much closer to the original one proposed by Brønstedand Lowry in which the reaction can be seen as a simple exchangeof a hydrogen cation between an acid–base pair.

For the reaction to take place the center of the perturbationhas to move away from the solute. Thus, the second impor-tant step is to monitor the center of the perturbation. Due toGrotthuss-like mechanisms, the perturbation moves along thenetwork. This can lead to different definitions of the defect cen-ter. However, if we tessellate the whole space using Voronoipolyhedra centered on water oxygen atoms, we can assignunequivocally every hydrogen atom to one and only one ofthese polyhedra. The site whose Voronoi polyhedron containsan anomalous number of protons is taken as the center of theperturbation (Fig. 1).

Fig. 1. Two examples of partitioning the space. (Left) We show a convec-tional approach in which the distance from the oxygen atom is used todefine its surroundings. Clearly artificial superpositions can be seen. (Right)The Voronoi tessellation does not suffer from these shortcomings.

Fig. 2. Smooth tessellation of a 2D space with cells centered on the threewater molecule oxygen atoms. The flat blue regions represent the portionof space in which the function assumes a value of 1 and the yellow onesrepresent the borders among cells. This surface has been obtained with avalue of λ= 4.

This point of view gives the method a very general nature,making it applicable to every acid–base system, without the needof fixing beforehand the reacting pairs. Thus, it is possible toexplore all of the relevant protonation states even in systemscomposed of more than one acid–base pair.

This general approach allows defining CVs without having toimpose specific structures or select the identity of the atomsinvolved. We test our method by performing metadynamicssimulations in a weak acid case (acetic acid), in a weak base(ammonia), and in an amphoteric species (bicarbonate) cho-sen as benchmarks because of their comparable strength, butdifferent acid–base behavior.

MethodsAs discussed above we introduce two CVs, one related to the protonationstate and another that locates the charge defects and measures their rela-tive distance. Both of these CVs need a robust definition for assigning thehydrogen atoms to the respective acid–base site. To achieve this result wepartition the whole space into Voronoi polyhedra centered on the acid–base sites i located at Ri . The sites include all of the atoms able to breakand form bonds with an acid proton. The standard Voronoi space partitionis described by a set of index functions wi(r) centered on the different Rissuch that wi(r) = 1 if the ith atom is the closest to r and wi(r) = 0 otherwise.For their use in enhanced sampling methods CVs need to be differentiable.To this effect we introduce a smooth version of the index functions, ws

i (r).These are defined using softmax functions

wsi (r) =

e−λ|Ri−r|∑m

e−λ|Rm−r| , [2]

where i and m run over all the acid–base sites and λ controls the steepnesswith which the curves decay to 0, that is, the selectivity of the function.With an appropriate choice of λ this definition achieves the desired resultas shown in Fig. 2. In such a way, a hydrogen atom in a position Rj is assignedto the polyhedron centered on the site i with the weight wi(Rj). Then, thetotal number of hydrogen atoms assigned to the ith acid–base site is

Wi =∑j∈H

wsi (Rj), [3]

where the summation on j runs over all the hydrogen atoms.One can associate to each acid–base site a reference value W0

i that countsthe number of bonded hydrogen atoms in the neutral state. The differencebetween the instantaneous value of hydrogen atoms and the reference one is

δi = Wi −W0i . [4]

When different from zero, δi will signal whether the ith site has gained orlost a proton. In the case of water oxygen atoms, a hydronium ion has aδi =+1 while a hydroxide ion has δi =−1.

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We then group the acid–base sites in species. For instance, in the caseof the simplest amino acid glycine in aqueous solution the number ofspecies Ns will be equal to 3. All water oxygen atoms belong to one species;then one counts in another species the two carboxylic oxygen atoms, andfinally one considers as the third species the nitrogen atom of the aminogroup.

In the spirit of this work we count the total excess or defects of protonsassociated to each species:

qk =∑i∈k

δi. [5]

This implies that we are not interested in the specific identity of the reactedsite, but whether or not the kth species in its entirety has increased (qk =

+1), has decreased (qk =−1), or has not changed its number of protons. Ifwe consider a solute with only one reactive moiety, then each possible stateof the system can be described by one of the three 2D vectors (0,0), (−1,1),or (1,−1).

In the general case each protonation state can be described by a vector~q = (q0, q1, . . . qNs−1) with dimension equal to the number of inequivalentreactive sites, Ns. A more exhaustive explanation is provided in SI Appendix.

For use in enhanced sampling these vectors need to be expressed as ascalar function f = f(~q) such that, for each physically relevant ~q, f attainsvalues able to distinguish the different overall protonation states. There areinfinitely many ways of constructing a scalar from a vector. Possibly the sim-plest choice is to write f(~q) = ~X ·~q and, to distinguish between differentprotonation states, to choose ~X = (20, 21, 22, . . . 2Ns−1).

This leads to the following definition for the CV that is used to describethe protonation state of the system,

sp =

Ns−1∑k=0

2k · qk, [6]

where k are the indexes used to label the respective reactive site groups. InSI Appendix an example is worked out in detail. Of course the CV is madecontinuous by the use of Wi in the calculation of the δi needed to evaluateqk in Eq. 5.

The second CV is a summation of distances between all acid–base sitesmultiplied by their partial charge δi ,

sd =∑

i,m>i

−rim · δi · δm, [7]

where the indexes i and m run over all the acid–base sites belonging to dif-ferent k groups, and rim is the distance between the two atoms. In this way,just the acid–base pair that has exchanged a proton gives a contributiondifferent from zero. Eq. 7 is valid only when one single-conjugate acid–basepair is present. However, due to the action of bias, it may occur occasion-ally that several acid–base pairs are formed. To avoid sampling these veryunlikely events we apply a restraint on the number of pairs. Further detailsare provided in SI Appendix.

ResultsWe have applied our method to three aqueous solutions of aceticacid, ammonia, and bicarbonate as representations of a weakacid, a weak base, and an amphoteric compound, respectively.The setups of all three simulations are identical except for theidentity of the solvated molecules. This ensures that the outcomereflects the different chemistry of these three systems and thatthere is no bias due to the initial condition.

Each simulation of the systems was performed with Born–Oppenheimer MD simulations combined with well-temperedmetadynamics (14, 26) using the CP2K package (27) patchedwith PLUMED 2 (28) and strongly constrained and appro-priately normed functional (29) for the xc energy, Exc . SeeSI Appendix for details.

In Fig. 3 we plot the free-energy surfaces (FESs) as a functionof sp and sd . These FESs vividly reproduce the expected behav-ior. They all have a minimum at sp =0 that corresponds to thestate in which no charges are present in the solvent. In the aceticacid FES (Fig. 3A) a second minimum close to sp =−1 reflectsits acid behavior. By contrast, the ammonia FES (Fig. 3B) showsa second minimum close to sp =1. The shapes of ammonia andacetic acid FESs are approximately related by a mirror symmetry

0

2

4

6

8

− 1 − 0.5 0 0 .5 1

0

2

4

6

8

0

2

4

6

8

s d

sp

0

20

40

60

80

100

s d8

16

24

32

40

s d

8

16

24

32

40A

B

C

Fig. 3. (A–C) Free-energy surfaces along sp and sd of acetic acid (A), ammo-nia (B), and bicarbonate (C) in aqueous solution. Color bars indicate the freeenergy expressed in kJ·mol−1 units. The CV sd is expressed in angstroms.

reflecting their contrasting behavior. Similarly the bicarbonatesymmetric FES (Fig. 3C) mirrors its amphoteric character.

As the conjugate pair is formed sd starts to assume positivevalues corresponding to the separation and diffusion of the con-jugate pair. Compared with the undissociated state in which onlysd =0 is allowed, states where a conjugate pair is present show anelongated shape of the basins along this variable. This is causedby the diffusive behavior of the hydronium and hydroxide ions insolution that makes accessible a continuum range of distances.Moreover, along this CV we can observe a barrier around 1.5corresponding to the breaking of the covalent bond between thehydrogen atom and the acid–base site.

ConclusionsThe general applicability of this method to systems with differentnatures is an important step made in their understanding anddescription. The scheme can be extended to include quantumnuclear effects with the use of path integral molecular dynamics

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(30). This would be of quantitative significance since for instancepKa values are affected by deuteration. Moreover, the absenceof assumptions or impositions about reactive candidates or reac-tion paths allows extending this method to systems of increasingcomplexity which cannot be addressed with traditional methods.Examples of questions that can now be answered are tautomeric

equilibria in biochemical processes and acid behavior in zeolitesand on the surface of oxides exposed to water.

ACKNOWLEDGMENTS. Calculations were carried out on the ETH Eulercluster and on the Monch cluster at the Swiss National Supercomput-ing Center. This research was supported by the European Union GrantERC-2014-AdG-670227/VARMET.

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