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Calculating the acidity of silanols and related oxyacids in aqueous solution

JOHN A. TOSSELL* and NITA SAHAIDepartment of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742 USA

(Received January 4, 2000; accepted in revised form April 19, 2000)

Abstract—Ab initio molecular orbital theory was used to calculate deprotonation energies and enthalpies(�Ed, �Hd) of oxyacid monomers and oligomers. Results were interpreted with reference to current phenom-enological models for estimating metal-oxide surface acidities. The ultimate goal is to predict surface aciditiesusing the ab initio method.We evaluated contributions to �Ed and �Hd from the electrostatic potential at the proton, electronic

relaxation, geometric relaxation, solvation, and polymerization for the neutral-charge gas-phase moleculesH2O, CH3OH, HCOOH, SiH3OH, Si(OH)4, Si2O7H6, H3PO4, P2O7H4, H2SO3, H2SO4, HOCl, HClO4,Ge(OH)4, As(OH)3, and AsO(OH)3. �Ed, gas calculated at the modest 6-31G* HF of theory level correlateswell with experimental pKa in solution, because hydration enthalpies for the acid anions (�Hhyd, A�) areclosely proportional to �Ed, gas. That is, anion interaction energies with water in aqueous solution and withH� in the gas phase are closely correlated.Correction for differential hydration between an acid and its conjugate base permits generalization of the

�Ed, gas – pKa correlation to deprotonation reactions involving charged acids. Thus, stable protonated, neutral,and deprotonated species Si(OH)3(OH2)

1�, Si(OH)40, Si(OH)3O

1�, and Si(OH)2O22� have been characterized,

and solution pKa’s for Si(OH)3(OH2)1� and Si(OH)3O

1� were estimated, assuming that the charged species(Si(OH)3(OH2)

1�, Si(OH)3O�1) fit into the same �Ed, gas – pKa correlation as do the neutral acids. The

correlation yields a negative pKa (� �5) for Si(OH)3(OH2)�1.

Calculated �Ed, gas also correlates well with the degree of O under-bonding evaluated using Brown’sbond-length based approach. �Ed, gas increases along the series HClO4 – Si(OH)4 mainly because ofincreasingly negative potential at the site of the proton, not because of differing electronic or geometricrelaxation energies. Thus, pKa can be correlated with underbondings or local electrostatic energies for themonomers, partially explaining the success of phenomenological models in correlating surface pKa of oxideswith bond-strengths.Accurate evaluation of �Hd, gas requires calculations with larger basis sets, inclusion of electron correlation

effects, and corrections for vibrational, rotational, and translational contributions. Density functional and2nd-order Moller-Plesset results for deprotonation enthalpies match well against higher-level G2(MP2)calculations.Direct calculation of solution pKa without resorting to correlations is presently impossible by ab initio

methods because of inaccurate methods to account for solvation. Inclusion of explicit water molecules aroundthe monomer immersed in a self-consistent reaction field (SCRF) provides the most accurate absolutehydration enthalpy (�Hhyd) values, but IPCM values for the bare acid (HA) and anion (A�) give reasonablevalues of �Hhyd, A� – �Hhyd, HA values with much smaller computational expense.Polymers of silicate are used as model systems that begin to approach solid silica, known to be much more

acidic than its monomer, Si(OH)4. Polymerization of silicate or phosphate reduces their gas-phase �Ed, gasrelative to the monomers; differences in the electrostatic potential at H�, electronic relaxation and geometricrelaxation energies all contribute to the effect. Internal H-bonding in the dimers results in unusually small�Ed,gas which is partially counteracted by a reduced �Hhyd. Accurate representation of hydration foroligomers persists as a fundamental problem in determining their solution pKa, because of the prohibitive costinvolved in directly modeling interactions between many water molecules and the species of interest.Fortunately, though, the local contribution to the difference in hydration energy between the neutral polymericacid and its anion seems to stabilize for a small number of explicit water molecules. Copyright © 2000Elsevier Science Ltd

1. INTRODUCTION

The surface charging properties and surface reactivities ofminerals are often interpreted using a surface complexationmodel, in which reactive species on the mineral surfaces arehypothesized and their properties determined by fitting to ex-perimental data (Parks, 1967; Schindler and Stumm, 1987).Several approaches have been developed to obtain the proper-

ties of these surface complexes using theory: (i) either a phe-nomenological model, whose parameters are ultimately fitted toexperiment (Yoon et al., 1979; Bleam, 1993; Sverjensky, 1994;Sverjensky and Sahai, 1996; Hiemstra et al., 1989, Hiemstra etal., 1996); or (ii) nonempirical first-principles theory, employ-ing approximate quantum methods (Goniakowski, et al., 1993)or pair potential approaches (Wasserman, et al., 1999). Otherquantum mechanical approaches have evaluated deprotonationenergies for gas phase systems, and have then determinedaqueous pKa’s using linear free-energy correlations (Rustad etal., 1996; Rustad et al., 1998; Rustad et al., 1999).

*Author to whom correspondence should be addressed ([email protected]).

Pergamon

Geochimica et Cosmochimica Acta, Vol. 64, No. 24, pp. 4097–4113, 2000Copyright © 2000 Elsevier Science LtdPrinted in the USA. All rights reserved

0016-7037/00 $20.00 � .00

4097

In the phenomenological theories, much of the variation inacidity is correlated with changes in the ratio of the metal-oxygen bond strength to the metal-oxygen or metal-protonbond length. The bond strength is defined either using theoriginal Pauling (1940) formulation as ion charge (Z) dividedby coordination number (CN), or using the bond-strength vs.bond-length relationships developed by Brown and Altermatt(1985). Bleam (1993) has shown that for many species thecorrelation with experiment is just as good if the Brown bond-strength alone is used without invoking the metal-oxygen bondlength. There are invariably some species which are anoma-lous, for which additional parameters must be introduced toobtain agreement with experiment. The most important of theseare quartz and amorphous silica, whose surface acidities seemto be seriously underestimated by a one parameter approach.Based on their model for the change in solvation energy of theprotonation reaction, Sverjensky (1994) and Sverjensky andSahai (1996) have introduced an additional dependence ofsurface pKa on the dielectric constant of the solid, which ismuch smaller for quartz than for most all other minerals.Hiemstra et al. (1996) focused instead upon presumed differ-ences in local solvation of the surface ionic groups betweenquartz and other minerals.Rustad et al. (1998) have approached this problem at an

atomistic level, but have employed molecular mechanics withpair potential force-fields and have relied upon linear free-energy relationships (LFER) to relate gas-phase deprotonationenergies to solution pKa’s. For example, they found that usingtheir force-field approach, a protonated silicic acid molecule,Si(OH)3(OH2)

1�, surrounded by 8 water molecules was intrin-sically unstable. However, we will show below that quantummechanical calculations predict Si(OH)3(OH2)

1� plus 4 or 8water molecules to be a well-defined and locally stable species,i.e., it corresponds to a local minimum on the energy surface.They also found that using their calculated energies for a bareSi(OH)3(OH2)

1� ion, they obtained much too positive a solu-tion pKa using the LFER. This was a result of their failure torecognize the difference in hydration energy effects for thedeprotonation of Si(OH)4 versus Si(OH)3(OH2)

1�, as we willshow later in this paper. In their study on the acidity oftrivalent, hexaquo ions (Rustad, et al., 1999), they obtainedgood results using a LFER approach because all the reactionsstudied had the same hydration energy effects.Wasserman et al. (1999) have developed a periodic 2D

approach to the surface acidity problem, again based upon apair potential approach. Giordano et al. (1998) have recentlyaddressed the problem of dissociation of water on a MgOsurface using a first-principles density functional approach. Thefound that dissociation of water was favored, in agreement withexperiment, if sufficient water molecules were available on thesurface.Nortier et al. (1997) have studied surface acidity using a

cluster model and the Hartree-Fock method, although at a fairlylow basis set level, without geometry optimization and usingsmaller clusters then considered here, e.g., only the monomerSi(OH)4 as their model for the silica surface. Nevertheless,their work established that the calculated �Ed, gas correlatedstrongly with the potential at the site of the proton, which inturn correlated strongly with the electrostatic potential from

model charge-distributions. We will show results confirmingthese conclusions later in this paper.Experimental values for acid dissociation constants in water

of common oxyacids are available from a number of sources(Pitzer, 1937; Jolly, 1991; Corbridge, 1990). For silicic acidsdissociation constants have been obtained by Seward (1974),Busey and Mesmer (1977), Sjoberg, et al. (1981) and Sjoberg,et al. (1985). pH values of zero surface charge and inferredsurface acid dissociation constants have been tabulated byParks (1967) and Schindler and Stumm (1987).There have also been several theoretical studies of the acidity

of various T-O-H functional groups in the gas-phase, whichhave lead to an understanding of differences between deproto-nation energies for C-O-H and Si-O-H groups (Sauer andAhlrichs, 1990; Curtiss et al., 1991) and for monomeric andoligomeric silicate Si-O-H groups (Sauer and Hill, 1994). Cal-culated energies were compared with experimental values ob-tained directly from gas-phase experiments or with valuesestimated from spectroscopic results such as OH stretchingfrequencies. Experimental values for �Hd, gas are available formany weak acids, but only for a few that are relevant to thispaper. For H2O, CH3OH, HCOOH and H2SO4 the experimen-tal �Hd, gas values are 390.8, 381.4, 345.2 and 309.6 kcal/mol(Saur and Ahlrichs, 1990; Blades et al., 1995), with uncertain-ties of a few tenths of a kcal/mol. �Ed, gas have also beencalculated at very high levels for the hypohalous acids (Gluk-hovtsev et al., 1996) and for perchloric, sulfuric and relatedacids (Otto et al., 1997). A very useful scheme for decomposingdeprotonation energies into components was developed earlierby Siggel and Thomas (1986) and has been used by severalresearchers.Information on �Hhyd, HA for neutral acids and their anions

is potentially available from gas-phase measurements, but theexperiments are difficult and results have been obtained foronly a few anions and with very few waters of hydration. Forexample, Blades et al. (1995), Blades et al. (1996) have ob-tained hydration free energies (�Ghyd) for a number of phos-phate anions and varying small numbers of waters. Calculationsby Wu and Houk (1993) with 6-31�G* Hartree-Fock geome-tries and 4th order Moller-Plesset energies reproduce the ex-perimental results of Blades et al. (1995), Blades et al. (1996)reasonably well. The fragmentary nature of the data preventsreliable extrapolation of these hydration free energies for smallnumbers of waters to bulk hydration free energies, althoughsuch a procedure has been carried out for the proton where�Ghyd data is more complete (Tissandier, 1998). However,even with fragmentary data some interesting points emerge.Blades et al. (1995) have noted that their experimental hydra-tion free energies for addition of the 1st molecule of water tovarious oxyanions correlate very well with the experimentalproton affinities of these oxyanions (their Fig. 5). We will showa similar correlation in this paper.There have recently been some calculations of solution acid-

ities which combine high-level quantum mechanical calcula-tion of �Ed, gas with calculated hydration enthalpies of acid andconjugate base, obtained within a polarizable continuum ap-proximation (Amekraz et al. 1996; Wiberg et al., 1996). Suchmethods have already been applied to the As oxyacids (Tossell,1997). However, obtaining energies accurate enough to deter-mine reliable absolute pKa, aq remains very difficult, as dis-

4098 J. A. Tossell and N. Sahai

cussed by Lim et al. (1991). Even for heavily parameterizedmethods (e.g., Cramer and Truhlar, 1992) the �Ehyd values forlarge polyatomic ions show substantial discrepancies from ex-periment.Our goal in this paper is to combine high-level quantum

mechanical calculations of deprotonation energies in the gas-phase with a medium-level description of hydration enthalpiesto obtain acid dissociation enthalpies in solution (i.e., �Ed, gas� �Hhyd � �Hd, aq). We apply this approach both to a rangeof simple organic and inorganic acids and to a number ofsilicate oligomers. The oligomers provide a starting point fordetermining the pKa of the silica surface. We examine corre-lations between:

1. Experimental solution pKa’s and �Ed, gas calculated at sev-eral different quantum mechanical levels;

2. (2) �Ed, gas and differences in �Hhyd between neutral acidand anion;

3. Experimental solution pKa and approximate calculated�Ed, aq; and

4. �Ed, gas and M-O-H bond properties such as O underbond-ings and charge distributions.

We also examine the relationship between �Hd, gas and �Hd, aqfor species of different charge. To do this, we have developedan approach for dealing with the different magnitude of stabi-lization that hydration produces for species of different charge,such as the reactants and products in an acid-dissociationreaction. By accounting for this “differential hydration” be-tween an acid and its conjugate base, reactants and productswith different charges can be shown to fall onto the same plotof gas-phase reaction energy vs. pKa. For example, solutionpKa1 and pKa2 for H2SO3 and H3PO4 are now correctly pre-dicted by the same correlation line. We also examine correla-tions of silicate structure with gas-phase Si-O-H deprotonationenergy and begin what we expect to be a long process ofassessing differences in hydration enthalpy as a function ofsilicate structure. Finally, for water molecules interacting withsilicate surface models we calculate NMR spectral properties todetermine how the detailed interaction of silicate and water canbe probed experimentally.

2. COMPUTATIONAL METHODS

We use the techniques of Hartree-Fock theory, many-bodyperturbation theory and density functional theory (Hehre et al.,1986; Foresman and Frisch, 1996; Jensen, 1999). Recent stud-ies have established that calculation of accurate reaction ener-gies for general reactions requires the inclusion of electroncorrelation through configuration interaction, many-body per-turbation theory or density functional theory. We normallyemploy 2nd order Moller-Plesset many-body perturbation the-ory (MP2, Pople et al., 1976) which is efficient and almostalways yields a better description than the Hartree-Fockmethod, failing only for some pathological open-shell cases.For the density functional calculations we used the BLYPfunctional (Becke, 1992, Lee et al., 1988) and a 6-311�G(2d,p)basis (Hehre et al., 1986). The lowest computational levelemployed was 6-31G* Hartree-Fock, for the larger oligomericspecies. Calculations of hydration enthalpies for supermol-ecules, such as Si(OH)3O

�. . .4H2O, were done at the HF level

with MP2 corrections, ultimately using 6-311�G(2d,p) basis.For the monomeric species we performed computations at the6-311�G(2d,p) MP2//6-311�G(2d,p) HF level and at the6-311�G(2d,p) BLYP//6-311�G(2d,p) BLYP level. We per-formed calculations at the very high composite G2(MP2) level(Curtis et al., 1993) for H2O, CH3OH, HCOOH and SiH3OHand their anions to obtain benchmark values for comparisonwith the computationally less intensive 6-311�G(2d,p) MP2//6-311�G(2d,p) HF results. We also compare our results forHClO, H2SO4 and HClO4 with other results in the literature(Glukhovtsev et al., 1996; Otto et al., 1997). All NMR calcu-lations were performed at the 6-31G* SCF level using theGIAO method (Wolinski et al., 1990). We employed the pro-grams GAMESS (Schmidt et al., 1993) and GAUSSIAN94(Frisch et al., 1994).The acid dissociation reaction for neutral HA in solution is

defined by the reaction:

HA(aq)3 A�(aq) � H�(aq) (1)

with associated deprotonation energy and enthalpy, �Ed, aq and�Hd, aq respectively.

�Ed, gas reactions is calculated corresponding to the reaction:

HA(g)3 A�(g) � H�(g) (2)

Reaction (1) can be written as a sum of reaction (2) and the tworeactions:

H�(g)3 H�(aq) (3)

and

HA(aq) � A�(g)3 HA(g) � A�(aq) (4)

For Eqn. 4,

�H4 � H(HAg) � H(Aaq�) � H(HAaq) � H(Ag

�) (5)

� �HA�, hyd � �HHA, hyd (6)

Therefore, the solution-phase deprotonation energy and en-thalpy can be written as:

�Ed, aq � �E1 � �E2 � �E3 � �E4 (7)

� �Ed, gas � �EH�, hyd � (�EA�, hyd � �EHA, hyd) (8)

and

�Hd, aq � �H1 � �H2 � �H3 � �H4 (9)

� �Hd, gas � �HH�, hyd � (�HA�, hyd � �HHA, hyd) (10)

That is, the enthalpy change for dissociation of HA in aqueoussolution can be decomposed into a gas-phase enthalpy change(�Hd, gas), a hydration enthalpy for the proton (�HH�, hyd), andthe difference between the hydration enthalpies of the anionand the neutral acid (�H4 � �HA�, hyd � �HHA, hyd). �HH�,

hyd is common to all the acid dissociations, so trends in theenthalpy of dissociation in solution depend on �Hd, gas and�H4 � �HA�, hyd � �HHA, hyd.To evaluate hydration enthalpies we use a multipart ap-

proach. For H� we use the newest “experimental” value of thehydration energy (�275.0 kcal/mol), taken from Tissandier etal. (1998), although the best “theoretical” value is probably the

4099Calculating acidity of silanols

value of �267.3 kcal/mol obtained by Tawa et al. (1998). Nineyears after Lim et al. (1991) discussed uncertainties in thequantities determining aqueous pKa’s, the energetics of protonhydration still remain uncertain.For the conjugate acid anions and the neutral acids we

determine the hydration enthalpies using a number of differentprocedures. For the bare acid and its anion we use: (i) thereformulated Born model (RH) approximating the effectiveBorn radius by adding 1.495 Å to the average calculateddistance from the central atom to the oxygen (Rashin andHonig, 1985); (ii) a self-consistent reaction field (SCRF)model, in which the radius is determined from an electrondensity criterion (a surface with electron density 0.001e/au3); or(iii) an IPCM model with the radius determined by the samecriterion (Wiberg et al., 1996; Foresman et al., 1996).For some of the compounds we also generated supermol-

ecules, with four water molecules coordinating the acid or itsanion and then calculated hydration energies for this supermol-ecule using the R&H method, the SCRF method or theSCIPCM method. The supermolecule calculations were done atthe 6 to 311�G(2d,p) MP2/6 to 311�G(2d,p) SCF level. Usingsuch a supermolecule approach, we separate the hydrationenergy into a local term (the bond energy within the supermol-ecule) and a long-distance term (the hydration of the supermol-ecule).For each type of calculation, we focus upon the hydration

energy difference �H4 � �HA�, hyd � �HHA, hyd, since this isthe relevant quantity for determining the enthalpy of aciddissociation in solution. Also, due to anticipated cancellation oferrors, the difference �H4 � �HA�, hyd � �HHA, hyd should beeasier to calculate accurately than the hydration energy ofeither neutral acid or anion alone.It is useful to consider what accuracy we might expect to

attain in evaluating the energetics for the reactions studied.Even an extremely accurate method such as G2(MP2), whichincorporates electron correlation at a very high level, effec-tively extrapolates to the infinite basis set limit, and incorpo-rates some minor empirical corrections, cannot obtain reactionenergies to better average accuracy than �1.5 kcalmol�1 (Cur-tiss et al., 1993). This translates to errors in the equilibriumconstant at room temperature of about a factor of 10. Most ofthe compounds in the set used to test the accuracy of G2(MP2)theory are much smaller than those considered here, andG2(MP2) is a considerably more accurate theory than the6-311�G(2d,p) MP2//6-311�G(2d,p) HF approach which wehave generally employed. Nonetheless, we have found thateven at our lower level of theory, �Hd, gas for this series ofcompounds can be evaluated with an accuracy not greatlyinferior to that of G2(MP2), partly because deprotonation is afairly simple reaction. Invariably, many serious approximationsneed to be made for calculations on such systems to be feasible.We believe we have developed a computationally feasiblemodel and method to describe the relative energetics of suchreactions. It is important that the method be simple enoughcomputationally that we can apply it to a wide range of acids.We shall focus upon the calculation of the energies and

enthalpies for various acid dissociation processes. The valuesof the acid dissociation constant, Ka, are of course determinedby the free energies (�G) for the processes, not the enthalpies.However, using a quantum mechanical approach for the free

acids and anions or even for supermolecules with a smallnumber of explicit waters we have no way to accurately cal-culate the entropy changes for the dissolution process in solu-tion. Pitzer (1937) showed long ago that for a process like Eqn.1 (dissociation of HA in aqueous solution), the entropy changewas about �22 calmol�1 K�1 for a wide range of weak acids.At 298K, this yields a free energy contribution of about �6.5kcalmol�1 corresponding to �4.8 in pKa. That is, the actualpKa’s will be more positive than those obtained from theenthalpy difference alone by �4.8.The alternative to including explicit water molecules is to

use the IPCM and SCIPCM methods for estimating hydrationenergies. The IPCM and SCIPCMmethods do yield free energychanges, but they do not properly account for the decrease inentropy since most of it arises from the local ordering ofsolvent molecules around the ions (Marcus, 1986). For thepresent, therefore, we shall ignore entropic effects completely,anticipating that they will be much the same for the differentacids studied.

3. RESULTS AND DISCUSSION

3.1. Energy vs. Solution pKa Correlations

Calculated gas-phase energeties for the deprotonation of 15different molecules, corresponding to �Ed, gas of Eqn. 2, aregiven in column 2 of Table 1. The molecules are arranged inorder of decreasing �Ed, gas, calculated at the 6-31G* HF level.The order of �Ed, gas is similar, but not identical, to the orderof the experimental solution pKa’s (last column of Table 1).Also given are: �Ed, gas evaluated using the BLYP densityfunctional and the 6-311�G(2d,p) basis; �Hd, gas evaluatedeither at the G2(MP2) level or at the 6-311�G(2d,p) MP2//6-311�G(2d,p) HF level, with vibrational corrections evalu-ated at a scaled 6-31G* SCF level; values of the O underbond-ing (2.0 minus the Brown T-O bond strength, whose calculationis discussed below); the experimental values of �H4 ��HA�, hyd � �HHA, hyd; the calculated IPCM value of �H4;and the experimental solution pKa. �Hd, gas from other high-level calculations are also included in parentheses in column 4.Our 6-311�G(2d,p) MP2//6-311�G(2d,p) HF calculationsgive �Hd, gas values which agree quite well with literaturevalues, obtained mostly by G2 or G2(MP2) methods, and alsoagree well with the few experimental values available. Numer-ous studies have shown that deprotonation enthalpies obtainedat the G2(MP2) level typically agree with experiment to withinbetter than 2 kcalmol�1 (e.g., Glukhovtsev et al., 1996), con-sistent with the results shown in Table 1. Our values typicallylie �2 kcal/mol below the G2(MP2) or experimental values.Thus, we believe that our relative deprotonation enthalpies areat essentially the G2(MP2) level of accuracy. Correlation plotsof several pairs of these quantities are shown in Fig. 1–7.In Fig. (1), we plot the experimental solution pKa’s vs. the

calculated 6-31G* Hartree-Fock gas-phase �Ed, gas for all 15molecules. Clearly the correlation is quite good, with a corre-lation coefficient of 0.890. The experimental pKa values aregenerally known accurately, although those for highly acidmolecules with negative pKa’s and for the silicate and phos-phate dimers may be somewhat less accurate than those forweak monomeric acids like H3PO4. These results indicate that


the overall trend of solution pKa is well reproduced by thegas-phase deprotonation energies. We will use this correlationline (expt. pKa � �55.95 � 0.1720�Ed, gas) to predict pKavalues for other molecules that are not included in Fig. (1) (see

section “Acidities of silicate oligomers”). However, in manycases, the relative ordering of pKa’s for any two acid moleculesare not correctly predicted by the correlation line, especially inthe middle pKa range from �3 to 10.

Table 1. Calculated energetics for deprotonation and hydration, along with calculated underbonding and experimental pKa. Energies and enthalpiesin kcalmol�1.

molecule �Ed,gas6-31G*HF

�Ed,gas6-311

�G(2d,p)BLYP

�Hd,gas6-311�G(2d,p)MP2 @ SCF(G2MP2)exp.a

2 � si Exp.�H4

IPCM�H4

Exp.pKa

H2O 429.3 392.2 388.2 (388.3)390.8

1.19 �99.4 �75.0 15.7

CH3OH 408.4 384.2 379.0(381.8)381.4)

1.01 �64.5 15.2

SiH3OH 376.8 359.2 354.1(355.6)

0.78

HClO 372.7 357.5 354.4(354.8b)

nad �75.7 �68.5 7.5

Ge(OH)4 371.7 0.70 8.6Si(OH)4 368.0 351.7 346.0 0.75 ��63.0 �51.6 9.5As(OH)3 367.0 0.52 �52.8 9.2HCOOH 363.0 345.9 342.2

345.20.46 �69.8 �63.4 3.75

Si2O7H6 347.4 331.3 0.76 7.75H2SO3 346.1 320.7 0.40 �49.3 �52.3 1.8H3PO4 343.2 330.6 325.2 0.53 �51.7 �51.8 2.2AsO(OH)3 335.1 0.46 �49.5 2.2H2SO4 319.1 311.8 307.3 (310.6c)

309.60.35 �51.3 �2.0

P2O7H4 318.1 0.53 1.0HClO4 301.0 309.5 299.9

(299.5c)0.36 �42.2 �7.0

a Experimental values from Blades et al. (1996) and Wiberg et al. (1996).b Gklukhovtsev et al., 1996.c Otto et al., 1997.d na, no bond parameters available for Cl(I) in Brown and Altermatt (1985).

Fig. 1. Correlation plot of experimental solution pKa’s vs. gas-phase6-31G* deprotonation energies, �Ed,gas, (linear regression line is givenon the figure).

Fig. 2. Correlation plot of experimental solution pKa’s vs. gas-phase�Hd,gas, obtained at the 6-311�G(2d,p) MP2//6-311�G(2d,p) HFlevel.


Figure (2) is a correlation plot of pKa vs. �Hd, gas values,evaluated at the 6-311�G(2d,p) MP2/6-311�G(2d,p) HFlevel. The correlation coefficient is now slightly higher (0.924)than in Fig. (1). However, the number of data points is smallerthan in Fig. (1), because the necessary calculations are muchmore demanding. There are two main contributions to thedifference between the 6-31G* SCF energies (column 2 ofTable 1) and the 6-311�G(2d,p) MP2/6-311�G(2d,p) HF en-thalpies (column 4 of Table 1). First, the deprotonation ener-gies are lower in the higher level calculation due to the largerbasis set, particularly the inclusion of diffuse functions, and dueto the inclusion of correlation effects. Second, and more im-

portantly, the vibrational zero-point energy is reduced duringthe reaction because of the loss of an O-H bond in the disso-ciation process (roughly �6 to �7 kcal/mol). This secondreason also explains why the BLYP energies in column 3 ofTable (1) are consistently � 5 kcalmol�1 larger than the6-311�G(2d,p) MP2 enthalpy values in column 4.As stated previously, �Hd, aq can be decomposed into

�Hd, gas, �HH�, hyd and �H4 � �HA�, hyd � �HHA, hyd.Assuming �HH�, hyd is �275.0 kcalmol�1 (Tissandier et al.,1998), knowing the experimental �Hd, aq for a few of the acids(Pitzer, 1937), and using our most accurate calculated values of�Hd, gas at the 6-311�G(2d,p) MP2 or G2(MP2) levels, we candetermine an “experimental” value for �H4. These “experimen-tal” values of �H4 are collected in column 6 of Table 1. Earlierresults based on hydration enthalpies of salts and semiempiricalestimates of lattice energies gave about �73 kcalmol�1 for thehydration enthalpy difference between H3PO4 and H2PO4

�1

Fig. 3. Correlation plot of experimental difference in hydrationenthalpies of A� and HA, �H4, vs. gas-phase �Hd,gas obtained at thethe 6-311�G(2d,p) MP2//6-311�G(2d,p) HF level.

Fig. 4. Correlation plot of experimental solution pKa’s vs. estimated�Hd,aq in solution (6-311�G(2d,p) MP2//6-311�G(2d,p) HF �Hd,gas,�275.0 kcalmol�1 (experimental hydration enthalpy of H�) plusIPCM estimates of hydration enthalpy of HA and A�).

Fig. 5. Correlation plot of 6-311�G(2d,p) MP2//6-311�G(2d,p) HF�Hd,gas and 6-311�G(2d,p) BLYP �Ed,gas.

Fig. 6. Correlation plot of 6-31G* �Ed,gas vs. (2.0 – Z/CN).


(George, 1970), which is considerably larger in magnitude thanour result of about �52 kcalmol�1. The number obtained byGeorge (1970) has often been quoted as an accurate experi-mental value, when it is in fact only a rough estimate.The experimental �H4 is shown in Figure 3 versus the

highest level calculated �Hd, gas values. The number of datapoints is only five, since we have been able to find experimental�Hd, aq for only five of the acids. Nonetheless, the correlationis very strong, with a correlation coefficient of 0.933. Thestrong correlation obtained in Figure 3 tells us that the �Hd, gasis strongly correlated with the hydration enthalpy difference,�H4. Since �H4 is dominated by the hydration enthalpy of theanion (�Hhyd, A�), this also indicates a strong correlationbetween �Hhyd, A� and �Ed, gas. That is, the more stronglybasic an anion is, the more strongly it interacts both withprotons in the gas-phase and with groups of water molecules insolution. This is consistent with the correlation of hydrationfree energy and proton affinity noted earlier by Blades et al.,1995 (their Fig. 5). This basically means that if an anioninteracts very strongly with H� it will also interact verystrongly with an environment of water molecules. Since�Ed, gas is strongly correlated with �Hd, gas, it correlates wellwith the solution pKa, whose variation depends upon both �Hd,gas and �H4.Note, however, that all the molecules considered in Figure 3

are small, compact monomers, so the hydration of the moleculewill not be hindered by size or shape effects. Conceivably, sucha correlation might be poorer if size effects hindered the hy-dration.The experimental solution pKa is compared to the approxi-

mate calculated solution �Hd, aq in Fig. 4. The calculatedsolution �Hd, aq was obtained as the value of �Hd, gas at6-311�G(2d,p) MP2 level, plus an estimate of �H4 fromIPCM calculations on bare A� and HA (column 7 of Table 1),plus �HH�, hyd from Tissandier et al. (1998). The number ofpoints in the correlation is fairly small (six) but we now obtaina correlation coefficient of 0.988. We also obtain the correctorder of �Hd, aq relative to experimental pKa for almost all

molecules, except CH3OH and H2O, which are still in invertedorder.Somewhat surprisingly, this improvement in the correlation

occurs even though the IPCM values of �H4 are not reallyhighly accurate, as will be discussed later in the section “Othermethods for computing hydration enthalpies.” The IPCM resultfor the bare acids and anions is not our most accurate or mostcomputationally intensive estimate of the A� and HA hydra-tion energies. In fact, the IPCM calculation is a trivial compu-tation compared to the high-level gas-phase energy optimiza-tions. This observation suggests that if we are able to carry outthe gas-phase enthalpy calculation at the G2(MP2) level orclose to it, and obtain even a reasonably accurate estimate ofthe hydration enthalpy difference (�H4) we can accuratelypredict solution pKa’s.As shown in Figure 5, there is also a good correlation

between either the 6-311�G(2d,p) BLYP �Ed, gas or the6-311�G(2d,p) MP2 �Hd, gas and the 6-31G* HF �Ed, gasvalues. Therefore, in principle for even larger molecules wecould use the less computationally demanding BLYP energiesto accurately predict MP2 level enthalpies. We could then addto these gas-phase enthalpies the experimental value of�HH�, hyd and IPCM values of �H4, and correlate that sumwith solution pKa’s.

3.2. Deprotonation Energies and Bonding Character

It is also useful and desirable to interpret the deprotonationenergies and solution pKa’s in structural terms. A number ofdifferent quantities, such as T electronegativity, T-O bondstrength (where T is the central atom), etc. might be correlatedwith the deprotonation energies. We first focus on the Paulingbond strength (Z/CN) of the T atom (Pauling, 1961), where Zis the formal charge and CN is the coordination number. Theunderbonding of the O in a T-O-H linkage is defined as itsvalence of 2.0 minus the sum of the bond strengths it receives,or 2.0 � Z/CN using the Pauling definition. We can plot the�Ed, gas calculated at 6-31G* HF level for 20 neutral moleculesvs. 2.0 � Z/CN. This results in a linear correlation with acorrelation coefficient of �0.699 (Fig. 6). The Brown bond-strength is defined as (Brown and Altermatt, 1985):

si � exp [(r0 � r)/b]

where the r0 and b parameters are1.624 Å for Si-O and 0.37Å�1 from their Table 1, and r is obtained from the geometry ofthe acid anion at the 6-31G* HF level. For example, forSi(OH)3O

�1, with a calculated Si-O distance of 1.542 Å, theSi-O bond strength (si) is 1.248, and the underbonding (2 � si)is 0.752. Using the Brown approach the correlation of �Ed, gaswith underbonding is now substantially improved as shown inFigure 7, with a correlation coefficient of 0.805. Since ourcalculated underbondings are simply related to calculated bonddistances in the anions, this correlation establishes a closerelationship between the structure of the monomeric anion andits protonation energy in the gas-phase. Of course, as noted byBleam (1993), the underbonding (2 � si) also correlates di-rectly with the experimental solution pKa’s.

Fig. 7. Correlation plot of 6-31G* �E vs. underbonding of O, 2� si,where si is obtained from the bond-length based approach of Brownand Altermatt, 1985.


3.3. Acidities of Silicate Oligomers

We now consider in more detail the relative acidities ofdifferent oligomeric silicate species. We have calculated gas-phase deprotonation energies at the 6-31G* HF level for thesilicate monomer, dimer, 3-ring trimer, 4-ring tetramer, double4-ring octamer and for Si7O19H10, a fully hydroxylated versionof an incompletely condensed silsesquioxane (Si7O12H10)which has been proposed as a model for the cristobalite surface(Feher et al., 1989) The calculated �Ed, gas are collected inTable 2. The double 4-ring, Si8O20H8, had been studied previ-ously by Sauer and Hill (1994). They found that Si8O20H8 hada smaller deprotonation energy than Si(OH)4, which was con-sistent with the experimental value estimated from vibrationalspectroscopy and also with the increased acidity of silica sur-faces compared to the silicate monomer.We find that the deprotonation energies are indeed smaller

for the oligomeric species than for the monomer, consistentwith a lower pKa for solid silica than for Si(OH)4. However,both Si2O7H6 and Si7O19H10 have unusually small deprotona-tion energies compared to the other oligomers. This is a resultboth of the greater geometrical flexibility of the anions in thesetwo cases and of the presence of internal H-bonding inSi2O7H5

�. If we temporarily ignore these two molecules, wefind a fairly systematic decrease in deprotonation energy as the

degree of polymerization of the oligomer increases, with avalue of 368 kcalmol�1 for the Q0 Si(OH)4 species, around 360kcal for the rigid Q2 3-ring and 4-ring species and around 353kcalmol�1 for the Q3 double 4-ring species. This decrease indeprotonation energy is associated with a decrease in the Si-O�

bond distance in the anion, from�1.54 to 1.53 to 1.51 Å for theQ0, Q2 and Q3 species, giving larger values of si and conse-quently smaller magnitudes of underbonding 2-si. The decreasein Si-O� distance is also associated with the presence of moreSi-Obridging bonds, which are typically longer than Si-OHbonds. It is worthwhile noting the difference in behavior of themonomeric acids, such as the HClO4–Si(OH)4 series, com-pared to the behavior of the silicate oligomers in Figure 7. Theoligomers fall below the correlation line defined by the mono-mers. There is a suggestion that they may define a separate linewith lower deprotonation energies, although, admittedly, thereare few points and considerable scatter.The optimized 6-31G* HF geometries of Si2O7H6, Si2O7H5

�

and Si2O7H51�. . .H2O are shown in Figure 8. Movement of the

Si-O� group in the anion to facilitate the internal H-bonding isevident in the figure. The O..H..O distance of 2.66 Å is withinthe range of H-bonded distances for oxyacids (Wells, 1975).The decomposition of the deprotonation energy shown in Table3 and discussed below also establishes that geometric relax-ation is considerably larger for Si2O7H6 than for the othersilicates (and slightly larger for Si7O19H10).Using the linear regression equation relating the experimen-

tal pKa and the gas-phase 6-31G* �Ed, gas shown in Fig. 1, ourpKa estimate for Si(OH)4 is 7.3, reasonably close to the exper-imental value of 9.5 (Sjoberg et al., 1981). If we first assumethat differences in hydration enthalpies between neutral acidsand their anions (i.e., the value of �H4) are about the same forthe oligomers as for Si(OH)4 (although they are probably notexactly the same, as we discuss later), we would expect thedeprotonation energies and pKa’s of the oligomers to follow thesame trend as that shown in Figure 1 for the monomers. Thus,using the same linear regression we can estimate pKa’s for theoligomeric species.Estimated pKa values for oligomers are shown in Table 2.

All the oligomers show somewhat smaller pKa values than themonomer. The flexible dimer and Si7O19H10 species showmore dramatic lowerings of pKa. Of course, we can anticipatethat for the Si2O7H5

1� ion, which shows internal H-bonding, theenthalpy of hydration might be reduced, because the internalH-bond must be partially broken to form the H-bond to thewater molecule. This will result in a reduced stability in aque-ous solution. We will return to this point later in “Trends inlocal hydration enthalpies. . ..”

3.4. Acidities of Polyprotic Acids

We can also use our 6-31G* values of �Ed, gas corrected byapproximate hydration enthalpies (such as IPCM values of�H4), to estimate solution pKa’s for species with differentcharges, or equivalently different order pKa’s, such as pKa2 forthe polyprotic acids. For example, we have characterized thespecies Si(OH)3(OH2)

1� and SiO2(OH)22� as bare molecules at

the 6-311�G(2d,p) MP2//6-311�G(2d,p) HF level, and assupermolecules with four waters at the 6-31G* HF level. Wefind them both to be stable molecules, corresponding to local

Table 2. Calculated 6-31G* HF deprotonation energies, underbond-ings at O, calculated pKa and experimental values of pKa, for positiveand negative ions. �Ed,gas values have been corrected by calculatedIPCM hydration energies as described in text. Energies in kcalmol�1.Qn refers to degree of polymerization of Si-O bonds, where n is thenumber of Si-O bonds (per tetrahedron) that are polymerized or bridg-ing.

molecule �Ed,gas6-31G* HF

2.0 � si calc. pKa ��55.95 �0.1720�Ed,gas

exp. pKa

neutral silicatesSi(OH)4 Q0 368.0 0.75 7.3 9.5a

Si2O7H6 Q1 347.4 0.76 3.8 7.75a

Si3O9H6 Q2 360.0 0.71 6.0Si4O12H8 Q2 361.9 0.70 6.3Si7O19H10 Q2 347.5 0.72 3.8Si8O20H8 Q3 353.2 0.65 4.8 amorphous

silica 6.71b

neutral models for charged silicatesSi(OH)3ONa 392.2 11.5Si(OH)3(OH2)Cl 324.0 �0.2

charged silicates (energies corrected for hydration)Si(OH)3(OH2)

� 298.2 �4.7Si2O7H7

� 308.4 �2.9Si3O9H7

� 307.7 �3.0Si(OH)3O

� 395.0 12.0 12.7aluminate

NaAl(OH)4 401.0 13.0 �14.5other neutrals

H2SO3 346.1 3.6 1.8a

H3PO4 343.2 3.1 2.2a

other anions (energies corrected for hydration)H2SO3

� 389.9 11.1 8.3H2PO4

� 372.4 8.1 7.1

a compounds from Table (1) that were used to obtain the correlationcalc.pKa � �55.95 � 0.1720�Ed,gas.b Schindler and Stumm (1987).


minima on the energy surface. The optimized geometry forSi(OH)3(OH2)

1�. . .4H2O is shown in Figure 9. We have alsoestablished that the Si(OH)3(OH2)

1� is at least a local energy

minimum, even in the presence of eight water molecules,contrary to the Molecular Dynamics result of Rustad et al.(1998).

Fig. 8. (a) 6-31G* HF optimized geometry of Si2O7H6(b) 6-31G* HF optimized geometry of Si2O7H5�1(c) 6-31G* HF

optimized geometry of Si2O7H5�1. . .H2O

Table 3. Calculated components of 6-31G* gas-phase deprotonation energy, along with potential determined charges, and sums of electrostatic H�

� O and H� � T interactions. Energies in kcalmol�1.

moleculepotentialat proton

electronicrelaxationenergy

geometricrelaxationenergy R(O™H) R(T™H)

electrostaticenergy fromformal chargeand bondstrength

potential derivedcharges (PDC)

electrostaticenergy

from PDC

NBO chargeselectrostaticenergy fromNBO chargesOnbr T H Onbr T H

HClO 589.9 �217.1 �0.1 0.951 2.126 542 �0.49 0.03 0.49 �70 �0.71 0.20 0.51 �110HClO4 535.6 �216.6 �18.0 0.960 2.102 416 �0.48 1.05 0.49 0 �0.78 2.56 0.54 �73H2SO4 554.1 �214.4 �20.6 0.954 2.103 459 �0.56 1.20 0.48 2 �0.97 2.92 0.54 �67H3PO4 576.3 �214.8 �18.5 0.955 2.126 500 �0.62 1.22 0.47 12 �1.07 2.80 0.54 �35Si(OH)4 604.7 �218.1 �18.6 0.947 2.225 552 �0.79 1.33 0.45 35 �1.16 2.59 0.51 �10Si2O7H6 603.5 �219.7 �36.4 0.948 2.217 551 �0.78 1.24 0.46 41 �1.16 2.64 0.51 �5P2O7H4 571.6 �216.5 �37.0 0.951 2.125 503 �0.64 1.10 0.48 25 �1.06 2.86 0.54 �41Si3O9H6 600.8 �221.6 �19.2 0.944 2.273 557 �0.80 1.51 0.44 27 �1.16 2.66 0.51 �10Si4O12H8 605.4 �222.2 �21.3 0.944 2.270 557 �0.81 0.62 0.50 45 �1.16 2.69 0.51 �7Si7O19H10 595.3 �222.7 �25.1 0.943 2.288 559 �0.86 0.93 0.51 86 �1.17 2.68 0.52 �11Si8O20H8 597.6 �223.3 �22.1 0.947 2.232 552 �0.80 1.11 0.50 58 �1.16 2.70 0.52 �3


The pKa values for these species can be estimated based onthe 6-31G* HF energies for the bare molecules. Consider thereactions:

H2A�(aq)3 HA(aq) � H�(aq) (11)

HA�(aq)3 A2�(aq)� H�(aq) (12)

The energetics of these two equations differ from that of Eqn.1 partly because of different hydration enthalpy contributionsfrom the acid and conjugate base, due to their different charges.For Eqn. 1 the relevant hydration energy is given by Eqn. 4,which is dominated by the �Hhyd of the mononegative product.This results in a substantial stabilization of the product side ofthe equation, and thus reduces the overall reaction enthalpy.For reaction (12), the reactant side is stabilized by the hydra-tion energy of the monopositive reactant, so that the overallreaction enthalpy is increased compared to its gas-phase value.For reaction (13), the product side is stabilized by the hydrationenergy of the dinegative anion. This stabilization is only par-tially compensated by the hydration energy of the mononega-tive reactant, so that the reaction enthalpy is decreased com-pared to the gas-phase value. Thus, for Eqn. 1, (12) and (13) wehave, respectively:

�Ed, aq, 1 � �Ed, gas, 1 � �HH�, hyd � �HA�, hyd � �HHA, hyd(13)

�Ed, aq, 11 � �Ed, gas, 11 � �HH�, hyd � �HHA, hyd � �HH2A�, hyd

(14)

�Ed, aq, 12 � �Ed, gas, 12 � �Hhyd, H� � �HA2�, hyd � �HHA�, hyd

(15)

If we assume that all the species containing the acid moiety Aare of similar size, then we can estimate the hydration enthalpycontribution from only the magnitude of charge on the reactantsand products, i.e., we assume:

�HH2A�, hyd � �HHA�, hyd � �HA�, hyd

�HA2�, hyd � 4 � �HA�, hyd

Also, we assume that

�HHA, hyd � 0

Then, we can re-write Eqns. 13–15 as

�Hd, aq, 1 � �Hd, gas, 1 � �HH�, hyd � �HA�, hyd (16)

�Hd, aq, 11 � �Hd, gas, 11 � �HH�, hyd � �HA�, hyd (17)

�Hd, aq, 12 � �Hd, gas, 12 � �HH�, hyd � 3 �HA�, hyd (18)

Thus, the magnitude of the overall stabilization due to hydra-tion in Eqn. 20 is 1 unit, while in Eqn. 21 the stabilization is�1unit, and in Eqn. 22 it is 3 units where each unit is �HA�, hyd.Using Eqn. 20 as a “reference” level, we can determine themagnitude of stabilization due to all other effects except hy-dration, by adding 2�Hhyd, HA to Eqn. 21 and adding�2�Hhyd, HA to Eqn. 22. This brings all the three equations tothe same “level” of stabilization due to hydration. Any differ-ences in energy are now purely due to non-solvation effects.Thus, Eqns. 20–22 now become:

�Hd, aq, 2 � �Ed, gas, 1 � �HH�, hyd � �HA�, hyd (19)



For example, the hydration energy of the silicate monomer ofcharge magnitude 1 is �56.1 kcalmol�1 according to theRashin and Honig (1985) model. The correction for the mo-nopositive ions Si(OH)3(OH2)

�, Si2O7H7� and Si3O9H7

�, willthen be 2 � �56.1 kcalmol�1. Adding this correction factor tothe �Ed,gas values of 195.0, 205.2 and 204.5 kcalmol

�1, re-spectively, yields corrected energy values (�Ed, aq,i) of 298.2,308.4 and 307.7 kcalmol�1, for the three silicate cations. Usingother estimates of the hydration enthalpy gives fairly similarresults. Using the least squares equation shown in Figure 1, thisgives estimated solution pKa values of �4.7, �2.9 and �3.0for these three cationic species. If we had used forSi(OH)3(OH2)

� the Molecular Dynamics deprotonation energyof 175 kcalmol�1 from Rustad et al. (1998) with our hydrationcorrection, we would have obtained a corrected energy of�278kcalmol�1 and an estimated pKa of about �8.1. Thus, thehydration correction brings even the MD value to at least arough approximation of our ab initio value of �4.7.For the dimer and cyclic trimer cations, note that the most

stable cation has the proton bonded to a bridging O, not aterminal OH group. This is the main reason for the differencein deprotonation energy and pKa between these oligomericcationic species and the monomer.In the same way, we can obtain a corrected deprotonation

energy for Si(OH)3O1� by adding twice the mononegative

hydration energy of �51.6 kcal/mol to the gas-phase deproto-nation energy of 498.2 kcal/mol to get �Eaq � 395.0 kcal/mol.This value projects a pKa of 12.0 for this species, or equiva-lently, a value of 12.0 for pKa2 of Si(OH)4. Using IPCM valuesfor the hydration enthalpy of the monoanion, we can apply thesame procedure to determine the pKa2 of H2SO3 and H3PO4.We obtain values of 11.1 and 8.1 compared with experimentalvalues of 8.3 and 7.1 (Table 2). Although our projected valuesdiffer significantly from experiment, our approach does give

Fig. 9. 6-31G* HF optimized geometry of Si(OH)3(OH2)�1 . . .

4H2O.


pKa2– pKa1 differences of 7.5 and 5.0 for the two acids, whichmatch well against the experimental differences of 6.5 and 4.9,supporting the validity of our scheme for referencing the dep-rotonation energies to a common level.We should note that there is a simpler way to approach the

problem of calculating higher or lower order pKa’s. We couldsimply add ions to the charged silicate species to give a neutralcompound, e.g., Si(OH)3ONa to model pKa2 for the monomerand Si(OH)3(OH2)Cl to model deprotonation of the cation.Gas-phase deprotonation energies evaluated at the 6-31G* HFlevel for these species and the correlation line from Figure 1give pKa values of 11.5 and �0.2, reasonably close to thevalues of 12.0 and �4.7 evaluated from our referencingscheme. However, this approach suffers from the presence ofspecific interactions which might not be present for silica. Forexample, the species with formula Si(OH)3(OH2)Cl is actuallySi(OH)4 complexed with HCl, which probably gives it too higha deprotonation energy, and too positive a pKa.We can now estimate pH values which would give zero

charge (pHPZC) for the silicate monomer and the silicate oli-gomers. For the monomer we would average 7.3 for Si(OH)4and �4.7 for Si(OH)3(OH2)

1� to give 1.3, while for the cyclictrimer we would average 6.0 for Si3O9H6 and �3.0 forSi3O9H7

1� to give 1.5.As an exercise, we can even stretch our analysis to include

tetrahedral aluminate groups, Al-O-H. If we choose a simpleNaAl(OH)4 model for the aluminate group and simply calculateits gas-phase deprotonation energy at the 6-31G* HF level weget 401.0 kcal/mol, which projects a pKa of 13.0. Early studies(e.g., Parks, 1967, as discussed in Blum and Lasaga, 1991)strongly supported the idea that tetrahedral Al-O-H linkageswere less strongly acidic than tetrahedral Si-O-H linkages, by�5 pKa units, which is close to the difference we obtaincomparing Si(OH)4 and NaAl(OH)4. Such a comparison iscertainly a crude one, but it seems to capture a good deal of thebasic physics of the problem.

3.5. Decomposition of the Deprotonation Energy

To obtain a better understanding of the deprotonation energywe can decompose it into a sum of terms as done by Siggel andThomas (1986). We can write:

�E � �V � �Re � �RN (22)

using a symbolism like that in Eqn. 9 of Siggel and Thomas(1986), in which the terms on the right are:

�V: the potential at the site of the proton in the neutralmolecule;

��Re: the energy lowering due to relaxation of the electrondensity in going from the neutral to the anion; and

��RN: the energy lowering due to relaxation of the nuclearpositions between the neutral and the anion. The effect ofnuclear or geometric relaxation was actually ignored in Sig-gel and Thomas (1986) so they did not have the��RN term.

Sauer and Ahlrichs (1990) used the same approach to show thatthe difference in deprotonation energy between CH3OH andSiH3OH was determined mainly by the difference in �V, withonly minor contributions from electronic and geometric relax-ation. Calculated energy components for HClO, the HClO4 –

Si(OH)4 series and for some silicate and phosphate oligomersare given in Table 3.The first point to note is that �V, or equivalently, the energy

for removal of a proton from the frozen nuclear and electroniccharge distribution of the neutral, correlates well with theoverall �Ed, gas (given in Table 1). For the set of compoundsshown in Table 3, the correlation coefficient is 0.80. This strongcorrelation occurs because the electronic relaxation and nuclearrelaxation energies are considerably smaller in magnitude thanthe potential at the proton and are fairly constant from onecompound to another (with the exception of the nuclear relax-ation energy in some of the oligomers).Although the oxyanions ClO4� – Si(OH)3O

� differ by�20% in calculated total electric dipole polarizabilities (JAT,unpublished results), their electronic relaxation energies (whichare related to the polarization energies for the anion) differ byonly a few percent. Likewise, there is very little contribution ofdifferential nuclear relaxation energies to the deprotonationenergy differences, except for the silicate and phosphatedimers.The cyclic silicate trimer shows almost the same nuclear

relaxation energy as the silicate monomer. Examining the sil-icates in more detail, we see that the oligomers generally havesmaller potentials at the proton, more negative electronic re-laxation energies, and slightly more negative geometric relax-ation energies. The difference in electronic relaxation energyindicates that the oligomeric silicates are somewhat more po-larizable by the proton than is the monomeric silicate anion.We can also try to analyze the differences in potential at the

proton using a model based on the charge density. Although thecharge on an atom within a molecule cannot be uniquelydefined, there are several approaches that seem to give mean-ingful relative effective charges. One popular approach is tocalculate the potential at an array of points outside the molecule(i.e., near the surface defined by the atomic van der Waalsradii), and to perform a least squares fit of effective atomiccharges to best reproduce these potentials. We have chosen theversion of this approach developed by Spackman (1996) andimplemented in the program GAMESS. The potential derivedcharges (PDC) obtained using this method, are given in Table3. Alternatively, we can define charges using a natural bondorbital (NBO) analysis, pioneered by Weinhold (Reed et al.,1988) and implemented in GAUSSIAN. The NBO charges aredefined for all levels of quantum mechanical treatment (e.g., forcorrelated methods) and are stable toward basis set expansion,in contrast to Mulliken charges which tend to change erraticallywith basis set (Jensen, 1999, pp. 217–234). NBO charges forare also given in Table 3. Note that the PDC and NBO chargesare quite different, in both magnitude and trend, and that thePDC charges seem to change rather erratically with polymer-ization.Several phenomenological schemes focus on an electrostatic

attraction of the proton and its nearest neighbour oxygen,coupled with the electrostatic repulsion of the proton and itssecond nearest neighbor metal cation (Parks, 1967; Hiemstra etal., 1989; Sverjensky and Sahai, 1996). Sverjensky and Sahai(1996) calculated this term using the formal charges of �1 and�2 for the proton and its nearest neighbor O, and approximatedthe residual charge on T as the strength of the T-O bond, givenby Z/CN. That is, in our terminology, they use the formula:


�V � 2/R(O � H) � Z/[CN * R(T� H)] (23)

Using the bond distances calculated for the neutral molecules,we have calculated electrostatic energies using this chargemodel which we list as “electrostatic energies from formalcharge and bond strength” in column 6 of Table 3. We see thattheir absolute values are actually fairly close to the calculatedvalues of the potential at the proton from our 6-31G* HFcalculations and their trend in the HClO4 – Si(OH)4 series is thesame as that of the potentials at the proton, although the changeis somewhat exaggerated.A correlation plot of the potentials at the proton from the

6-31G* HF calculations vs. this approximation for the electro-static energy is shown in Figure 10. The strong correlationindicates that this simple model must incorporate much of thebasic physics that determines the deprotonation energy. How-ever, detailed inspection of Table 3 indicates that these modelelectrostatic energies (at least in this simple local approxima-tion) do not correctly reproduce the trends with oligomerizationfor either the Si or P species.We have also calculated the sum of the electrostatic energies

for the proton-O and proton-T interactions, using both thepotential derived charges (PDC) and the natural bond orbital(NBO) charges from Table 3 with our calculated bond dis-tances. We list these energies in Table 3. We find that the trendin potential at the proton along the HClO4 – Si(OH)4 series isalso reproduced in a qualitative way by both the PDC and NBOelectrostatic energies, although it is considerably underesti-mated. However, the difference between silicate monomer andoligomers is not reproduced at all, and HClO appears to beanomalous for both PDC and NBO charges.A correlation plot of the potential at the proton vs. the PDC

and NBO electrostatic energies is shown in Figure 10. The PDCcharges give a poor correlation while the NBO charges yield afair correlation, which is not as good as the formal charge plusbond strength model. These failures cause us to doubt that anyof the electrostatic approaches described above represents a

viable physical model for predicting changes in acidity witholigomerization, at least for these gas-phase molecules.

3.6. Implications for Phenomenological Models

The results in the preceding paragraphs may be summarizedas:

1. Deprotonation energies are well correlated with the under-bondings of the oxygens in the anions, and with the elec-trostatic potentials at the sites of the protons.

2. Silicate and phosphate oligomers are more acidic (smallerdeprotonation energies) than the corresponding monomers,with the acidity generally increasing with degree of poly-merization.

3. For the dimer and cyclic silica trimer, the most stable formof the cationic species has the proton bonded to a bridgingoxygen, not to a terminal OH group, which is mainly re-sponsible for the large increase in their acidities relative tothe monomer.

4. The anomalously low acidity of the flexible silica dimer maybe related to the fact that the terminal oxygen atom in thedeprotonated form of the acid tends to bend back in towardsthe molecule forming internal H-bonds. The calculatedO..H..O distance is 2.66 Å, which is in the range of H-bonding distances observed in acid salts or hydrates (Wells,1975). It is not clear to what extent this effect would persistif the flexible dimer were explicitly solvated by a sufficientnumber of water molecules. Accurate calculation of ener-gies for explicitly solvated large oligomers is computation-ally very expensive.

5. Contributions of the geometric relaxation energies and elec-tronic relaxation energies to the trends in deprotonationenergy are relatively small - most of the change in depro-tonation energy arises from changes in the potential at theproton.

It should be remembered that our calculations were done ongas-phase molecules and have been extended to obtain approx-imate solution pKa’s. The implications of our results for phe-nomenological models that predict surface pKa’s must, there-fore, be taken as conditional. Equally important, it should berealized that our calculations were restricted to anionic oxyac-ids such as [Si(OH)3O]

1�, and did not include any cationicmetal acids such as [Al(OH)2 (H2O)4]

1�. So the interpretationof phenomenological surface complexation models, whichwere developed mainly for metal-oxides, in terms of our resultsis, perhaps, an overextension of our results. With this caveat inmind, we may then continue as follows.Point (1) provides validation for several of the models that

have been used in the past to relate the deprotonation energy tounderbondings or electrostatic potentials. Point (2) suggeststhat the anomolously high acidity of silica compared to theSi(OH)4 monomer may be an intrinsic property of the silicateoligomers (as indicated also by the results of Sauer and Hill,1994). If this polymerization effect is properly accounted for,there may be no need to invoke any other parameters related toion stabilization or hydration such as the dielectric constant(Sverkjensky, 1994; Sverjensky and Sahai, 1996) or the localhydration of the ions (Hiemstra et al., 1996). Point (3) appearsto support one feature of the revised MUSIC model of Hiemstra

Fig. 10. Correlation plot of 6-31G* calculated potential at proton vs.formal charge plus bond strength, PDC and NBO electrostatic energies.Open squares are formal charge plus bond strenth, filled diamonds arePDC and filled squares are NBO electrostatic energies.


et al. (1996), because that model assumes that the acidic protonon the positively-charged surface site is bonded to a bridgingrather than a terminal oxygen.However, to obtain a critical test of Hiemstra et al.’s (1996)

model of surface hydration we would need to include sufficientwaters of hydration around the larger oligomers in our calcu-lations, to determine if the local hydration effects assumed byHiemstra et al. are actually present. Also, to critically test theSverjensky and Sahai (1996) model, we would need to calcu-late the pKa’s for monomers and oligomers of metal-oxidesother than silica, such as Al2O3 or TiO2.

3.7. Other Methods for Computing Hydration Enthalpies

The relative hydration enthalpies for A� and HA in Table 1were obtained using the very simple IPCM method. Othermuch more complicated and demanding approaches are cer-tainly possible and even desirable. An excellent example is thecalculation of the proton hydration enthalpy and free energy byTawa et al. (1998). In their approach the proton was hydratedwith a number of explicit water molecules and this “supermol-ecule” was then immersed in a polarizable continuum. Resultsconverged well with respect to the size of the supermoleculewhen it contained only four water molecules and was treated atthe BLYP or MP2 level with a 6-311�G(d,p) basis. Suchcalculations are possible but are computationally very demand-ing for species such as Si(OH)4. Part of the problem is theflexibility of the supermolecule due to the relatively weakinteraction of water with the solute (particularly for the neutralacids), coupled with several different possible H-bondingschemes involving the waters, leading to slow, tedious conver-gence of the geometry optimizations.We have calculated optimized energies at the

6-311�G(2d,p) MP2//6 to 311�G(2d,p) HF level for super-molecules consisting of either H2O, HClO, H3PO4, Si(OH)4 orone of their corresponding conjugate acids coordinated to fourwater molecules. However, we have not been able to carry outa force constant analysis for the neutral solute supermolecules.The calculated supermolecule interaction energy values aregiven in Table 4 as the local terms, along with the long-rangecontributions to the hydration enthalpy difference calculatedusing either the Rashin and Honig prescription, the SCRFmethod or the SCIPCM method (we encountered severe con-vergence problems using the IPCM method).The first point to note in Table 4 is that the local contribu-

tions and the long-range contributions to the hydration enthalpydifference are roughly equal in magnitude, so that both must beconsidered to get reasonable hydration enthalpies. The secondpoint is that while the R&H and SCRF values are quite similar,

the SCIPCM values are often much smaller in magnitude. Thisdifference arises from a much larger stabilization of the neutralmolecule in the SCIPCM calculations. It appears that theSCIPCM results are less reliable since they regularly lead tounderestimates of the difference of acid and anion hydrationenthalpy differences, although one could possibly avoid thisproblem by adjusting the electron density criterion for definingthe isodensity surface. A third point is that even though the twomore reliable methods (the semiempirical R&H and the SCRF)show some differences, the calculated total differential hydra-tion enthalpies match quite well against the experimental val-ues. Still, we should remember that some potentially importantterms have been ignored, e.g., the change in zero point energyduring the formation of the supermolecule due to our failure toobtain the vibrational spectra. Also, the hydration enthalpy forthe proton is uncertain by �8 kcalmol�1. Thus, it appears thatit is not yet possible to calculate absolute deprotonation enthal-pies in solution with chemical accuracy. The situation is notmuch better for the relative values, since the calculated hydra-tion enthalpy difference seems to be larger in magnitude forH3PO4 than for Si(OH)4, which is opposite to the experimentalvalues of �H4.

3.8. Trends in Local Hydration Enthalpies fromSupermolecule Calculations

Although the absolute hydration enthalpies from our super-molecule calculations leave much to be desired, it may beuseful to consider some trends seen in these calculated ener-gies, for different numbers of water molecules, and silicateanions of different size. All the calculations described in thissection were done at the HF level using the 6-31G* basis.Results are given in Table 5.We first studied the stabilization of the monomer anion

Si(OH)3O1�, relative to the neutral monomer Si(OH)4, pro-

duced by adding various numbers of water molecules to eachspecies. The calculated geometry for Si(OH)3O

1� with fourwaters is shown in Figure 11a. The differential hydration en-ergies are 16.5, 14.2 and 21.0 kcal/mol, for 1, 2 and 4 watermolecules respectively. This indicates that although the totalinteraction energy with water of either the neutral or the anionincreases with the number of waters, the differential hydrationof the anion compared to the neutral does not change verymuch. It also indicates that most of the local differential hy-dration occurs with the addition of the first water, as expected.The interaction of the first water with the Si-O� group of theanion is considerably more favorable than the Si-O-H. . .waterinteraction in the neutral.On the other hand, the stabilization of Si2O7H5

1�, compared

Table 4. Calculated local and long-range contributions to the difference of A� and HA hydration enthalpies. Enthalpies in kcalmol�1.

HA

Local(supermolecule

calc.)

Long-RangeTotal Differential

Hydration Enthalpies

Exp. �H4R&H SCRF SCIPCM R&H SCRF SCIPCM

H2O �46.2 �36.9 �38.5 �31.7 �83.1 �84.7 �77.9 �99.4HClO �46.7 �32.1 �40.2 �13.2 �78.8 �86.9 �59.9 �75.7H3PO4 �26.7 �32.7 �37.3 �12.9 �59.4 �64.0 �39.6 �51.8Si(OH)4 �31.5 �30.1 �34.3 �14.6 �61.6 �65.8 �46.1 ��63.0


to Si2O7H6, produced by adding a single water molecule toeach is only 3.4 kcalmol�1, compared to 16.5 kcalmol�1 forthe monomer. This reflects the fact that the Si-O� group inSi2O7H5

1� is already internally H-bonded. However, when wa-ter is present this internal H-bond is partially broken. Theaddition of other water molecules produces a larger differentialstabilization for the Si2O7H5

1� anion, so that by the time 7water molecules have been added (Fig. 11b), the energy dif-ference has been reduced by 14 kcalmol�1 from the gas-phasevalue.For Si3O9H6 the stabilization of anion compared to neutral

produced by a single water molecule is actually larger than that

produced by three waters. It is clear that the local contributionto the hydration energy difference between neutral acid andanion converges for a relatively small number of waters. How-ever, there seems to be considerable scatter in the results, whichmay be due to our simple approach to geometry optimization,where the global minimum has not necessarily been reached.

3.9. Calculation of NMR Shieldings of 29Si and 1H andElectric Field Gradients at O for Silicates of VariousCharges and Degrees of Hydration

To better understand the nature of the hydration of the silicaoligomers and the surface of silica, it would be desirable tohave a spectral probe for the local geometry of the water aboutthe silanol group. 1H and 29Si-NMR are potentially usefulprobes. Fleischer et al. (1993) established that the 1H-NMRshielding was correlated with the deprotonation energy for arestricted set of molecules, relevant as models for surfacehydroxides. Civalleri et al. (1999) found that 1H-NMR shield-ings obtained for model silicates interacting with water couldreproduce some of the experimental data for water on silicasurfaces. Moravetski et al. (1996) observed that interaction ofSi(OH)4 or Si(OH)3O

1� with water increased the Si-NMRshielding.We have calculated 29Si and 1H-NMR shieldings for a range

of species as shown in Table 6. We find that protonation of bareSi(OH)4 increases the

29Si shielding: the first deprotonationdecreases 29Si shielding substantially, and the second deproto-nation causes little change. When the silicate species is part ofa supermolecule with four waters this trend in 29Si shielding istempered and even reversed slightly, with the deprotonatedspecies actually being slightly more shielded. Both bare silicateand supermolecule results, thus, indicate that aSi(OH)3(OH2)

1� species might be distinguishable by 29Si-NMR but that the other species would have very similar shifts.For the cyclic trimer, Si3O9H6, the

29Si shielding is larger, asexpected for a more polymerized species, but the change upondeprotonation is about the same.The 1H-NMR shows considerable change from the �OH2

proton of Si(OH)3(OH2)1� to the �OH protons of

Si(OH)2O22�, and this difference persists in the supermolecules.

This indicates that 1H-NMR might be useful for determiningthe charge of the silicate group if the spectrum were wellresolved. The calculations indicate that hydration deshields theprotons of Si(OH)4 by 1 to 2 ppm, in the same direction asobserved experimentally (Caillerie et al., 1997) but of smallermagnitude. Note that the neutral monomer and cyclic trimergive essentially the same 1H-NMR although they differ indeprotonation energy by �8 kcalmol�1, so that small changesin deprotonation energy associated with oligomerization are

Fig. 11. (a) 6-31G* SCF optimized geometry for Si(OH)3O�1..

4H2O(b) 6-31G* SCF optimized geometry of Si2O7H5�1. . .7H2O.

Table 5. Energy difference at 6-31G* HF level of n3 acid and anion,hydrated by differing numbers of water molecules, for various silicates.Energies in kcalmol�1.

molecule/n 0 1 2 3 4 7

Si(OH)4 368.0 353.8 348.1 — 349.0 —Si2O7H6 347.4 344.0 — — — 334.8Si3O9H6 360.0 343.5 — 351.8 — —


probably not discernable from the 1H-NMR. On the other hand,if we consider the series Si(OH)3(OH2)

1� – Si(OH)3O1� we

see a substantial change in 1H shielding, where the species withthe larger deprotonation energy, Si(OH)3 O1�, is moreshielded. A similar trend was found by Fleischer et al. (1993)(their Fig. 5).We find that �OH2 oxygens and �OH oxygens have rather

similar electric field gradients (EFG), 1.3 to 1.5 atomic units,corresponding roughly to quadruple coupling constants of 6.8to 7.8 MHz. These values are distinct from �O� oxygenswhich have much smaller electric field gradients. Hydrationreduces the EFG of the�OH oxygens and increases that for the�O� oxygens, but by only a modest amount. Thus, 17O NMRis potentially a means for assessing the interactions of neutraland ionized silanols with water.In closing, we should caution that some basic questions

concerning oxyanion speciation may still be unresolved. Wehave assumed that the species formed by acid dissociation ofSi(OH)4 is the 4-coordinate Si(OH)3O

�. An alternative is thatOH� adds to Si(OH)4 to form 5-coordinate Si(OH)5

�1. Weldon,et al. (personal communication) have recently presented IRspectroscopic and computational evidence for 5-coordinate an-ionic centers on the surface of silica. Our own computationalstudies on Si(OH)3O

�. . .H2O and Si(OH)5� indicate that their

relative energies are very close and fairly sensitive to the levelof treatment. At the 6-31G* HF level assuming the 5-coordi-nate rather than the 4-coordinate anionic species would changethe deprotonation energy by �5 kcal/mol, projecting a changein pKa of only �0.8 units. Therefore, at this level of accuracywe can not really distinguish between 4-coordinate and 5-co-ordinate conjugate base species.

4. CONCLUSIONS

Our results indicate that it is presently impossible to calcu-late accurate absolute solution pKa’s completely from firstprinciples due to (i) inaccuracies in the available methods forevaluating hydration enthalpies of ions; and (ii) uncertainty inthe proton hydration enthalpy. On the other hand, deprotona-tion energies in the gas phase can be calculated with an accu-racy of a few kcalmol�1, at least for monomeric oxyacids.However, it is possible to estimate solution pKa’s reliably byusing a correlation between experimental pKa values and cal-culated gas-phase or solution enthalpies. This correlation be-comes better as the level of the calculation is improved, but it

is already quite good at the 6-31G* HF level. Approximatecorrection for differential hydration enthalpies makes it possi-ble to generalize this correlation so that several different typesof acid dissociation reactions can be treated.We find that gas-phase deprotonation energies are strongly

correlated with the underbonding at the O, and that calculatedpotentials at the proton are correlated with electrostatic ener-gies obtained using a simple charge model, based on formal orNBO charges. Thus, it is possible to establish a good correla-tion of pKa’s with underbondings or with electrostatic energiesfor the monomeric oxyacids.Polymerization invariably decreases the deprotonation en-

ergy, with changes in the potential at the proton, the electronicrelaxation energy and the geometric relaxation energy all con-tributing. At least part of this variation is correlated withchanges in the underbonding of the O, but none of the electro-static models are able to reproduce the trend. The phosphateand silicate dimers have particularly small deprotonation ener-gies due to internal H-bonding, which is compensated to someextent by reduced hydration enthalpies.Hydration energies for the larger silicate species are difficult

to calculate accurately but it appears generally that the differ-ence in the local contributions to hydration enthalpies betweenthe neutral acid and anion reaches a constant value for arelatively small number of explicit molecules of water. Thissuggests that determining the relative energetics of hydration ofacid and anion entirely from quantum mechanics will soon befeasible.

Acknowledgments—This work was supported by DOE grant DE-FG02to 94ER14467 to JAT and by a NSF Earth-Sciences Post-DoctoralFellowship to NS. We thank an anonymous referee for pointing out anincorrect termination (�H instead of �OH) for the Si7. . . oligomer inthe original manuscript.

Associate editor: G. Sposito

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