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1

MST Continuum Study of the Hydration Free Energies of Monovalent Ionic Species

Carles Curutchet,1 Axel Bidon-Chanal,1 Ignaci Soteras,1 Modesto Orozco,2*

and F. Javier Luque1*

1 Departament de Fisicoquímica. Facultat de Farmàcia. Universitat de Barcelona. Avgda. Diagonal 643. Barcelona 08028. Spain. 2 Departament de Bioquímica i Biología Molecular, Facultat de Química, Universitat de Barcelona, c/. Martí i Franqués 1, 08028, Barcelona, and Unitat de Modelització Molecular i Bioinformàtica, Institut de Recerca Biomèdica. Parc Científic de Barcelona, c/. Josep Samitier 1, 08028 Barcelona, Spain. Send correspondence to FJL (e-mail: [email protected]) or MO (e-mail: [email protected]) Abstract

In this study we revisit the protocol previously proposed within the framework of the

Miertus-Scrocco-Tomasi (MST) continuum model to define the cavity between solute

and solvent for predicting hydration free energies of univalent ions [Orozco, M.; Luque,

F. J. Chem. Phys. 1994, 182, 237]. The protocol relies on the use of a reduced cavity

(around 10-15% smaller than the cavity used for neutral compounds) around the atom(s)

bearing the formal charge. The suitability of this approach is examined here for a series

of 47 univalent ions for which accurate experimental hydration free energies are

available. Attention is also paid to the effect of the charge renormalization protocol used

to correct uncertainties arising from the electron density located outside the solute

cavity. The method presented here provides, with a minimum number of fitted

parameters, reasonable estimates of the hydration free energy of ions (average relative

error of 4.7%) and is able to reproduce solvation in water of both small and large ions.


2

Introduction

The theoretical study of chemical systems in condensed phases has experienced

an extraordinary evolution in the last decades, leading to a number of computational

approaches that combine quantum mechanical (QM) or classical descriptions of the

solute with continuum or discrete representations of the solvent molecules.1 One of

these approaches is the self-consistent reaction field (SCRF) continuum model, where

the solute resides in a cavity carved into a continuum polarizable medium that simulates

the solvent.2 In spite of the rather crude description of the solvent, QM SCRF methods

have found a widespread acceptance due to several reasons. First, the availability of

well-defined formalisms developed to account for the mutual coupling between the

solute’s charge distribution and the solvent’s reaction field (see ref. 3 for a recent

review). Second, the ease of gaining insight into the solvent-induced changes in the

properties of the solute upon transfer from the gas phase to solution.2,4 Third, its low

computational cost, which is only slightly larger than the corresponding QM calculation

in the gas phase.

Though the preceding features have made QM SCRF methods very popular for

the study of molecules in solution, their chemical accuracy has mainly been restricted

by the uncertainty in the location of the boundary between solute and solvent.2 In the

last decade a large research effort has been spent to parametrize the solute cavity in

different SCRF methods, such as the multipolar expansion method developed at Nancy,5

the generalized Born SMx solvation models from Cramer and Truhlar,6 the Polarizable

Continuum model from Tomasi and coworkers7 and the related Miertus-Scrocco-

Tomasi (MST) version developed at Barcelona,8 or the conductor-like screening model

from Klamt and coworkers.9 As a result, the most refined QM SCRF methods predict


3

the free energy of solvation of neutral molecules in a variety of solvents with an average

error clearly below 1 kcal/mol when solute cavities adjusted to the specific electrostatic

model are used.10

For ionic solutes, the uncertainty in the hydration free energies determined from

QM SCRF methods is larger (around 4-5 kcal/mol). The lower predictive accuracy for

ions arises not only for the intrinsic problems in representing the strong solute-solvent

interactions existing between water and ions, but also to the poor quality of most

experimental data. For neutral solutes the solvation free energy can be directly obtained

by measuring partition coefficients of solutes between gas phase and dilute aqueous

solutions in equilibrium. In this way the uncertainty of the measured partition

coefficient increases with increasing solvation free energy, and eventually the

concentration of solute molecules in the gas phase falls below the experimental

detection limits when the solvation free energy (in absolute terms) exceeds ~12

kcal/mol.11 This makes it necessary to determine the solvation free energy of ionic

species by indirect approaches based on the use of suitable thermodynamic cycles. For

example, Pearson12 determined the experimental solvation free energies for a number of

univalent ions using the relationships given by Eqs.1 and 2 for cations and anions,

respectively.

)AH(RTpK.)A(G)H(G)A(B)AH(G asolsolgsol+++ −++= 3032∆∆∆ (1)

)BH(RTpK.)BH(G)H(G)B(B)B(G asolsolgsol 3032++−−= +−− ∆∆∆ (2)

where Bg is the gas phase basicity and solG∆ denotes the free energy of solvation.

Note that according to Eqs. 1 and 2 the quality of the “experimental” solvation

free energies of cations, )AH(Gsol+∆ , and anions, )B(Gsol

−∆ , depend on the accuracy


4

of different thermodynamic quantities, such as the gas phase basicity, the solvation free

energy of the conjugate neutral base, )A(G sol∆ , or acid, )BH(G sol∆ , the pKa constant,

and the solvation free energy of the proton, )H(G sol+∆ . It is also worth noting that

correction terms (not explicitly indicated in Eqs. 1 and 2) arising from the standard state

definition of the chemical species in the gas phase and in solution might be necessary.

Thus, solvation free energies can be defined for the hypothetical process of transferring

a species from a fixed position in the gas phase to a fixed position in the solution phase,

which is equivalent to using the same standard states (typically 1 M concentration) for

gas and solution phases.13 However, experimental data in the gas phase is generally

referred to an ideal gas at 1 atm. of pressure, so that a correction term of RTln(RT) must

be considered.

In his original work Pearson used an estimated )H(G sol+∆ value of -259.5

kcal/mol, which was also adopted by Florian and Warshel in the parametrization of their

Langevin dipole solvation model.14 However, the values of )H(Gsol+∆ used by other

authors vary considerably (from -252 to -263 kcal/mol)15. This range of error is similar

to that determined from theoretical calculations16 and introduces a noticeable range of

uncertainty in the “experimental” hydration free energy of ions. More recently,

Tissandier et al.,17 using a cluster-pair approximation and thermodynamic properties of

ion-water clusters, derived a value of )H(G sol+∆ =-264 kcal/mol. A very similar value

has recently been derived by Tuttle et al and by Coe and coworkers.18 Because this new

)H(G sol+∆ value is considerably more negative than the previous ones, it has a

significant effect on the absolute solvation free energies of ions and the corresponding

acidity/basicity constants.


5

Pliego and Riveros have recently derived new experimental values for the

hydration free energies of 47 univalent ions using as reference the solvation free energy

of the proton derived by Tissandier et al.19 In light of the new reported values, the aim

of this work is to revisit the original parametrization of the MST model20 to predict the

corrected experimental hydration free energies for ions. In turn, this makes it necessary

to solve additional fundamental problems for a continuum description of solvation.

First, how should solute cavities be defined for ionic compounds, which are generally

characterized by a large anisotropy in the charge distribution? Second, how can errors in

the solvation free energy arising from the solute charge density lying outside the cavity

be corrected?

In our previous MST parametrization for ions,20 the van der Waals radii used to

built up the solute/solvent interface were scaled by a factor of 1.10-1.15, which is

around 10% smaller than that recommended for the hydration of neutral solutes.8

Moreover, the inaccuracies in the apparent surface charges due to the solute electron

density lying outside the cavity was treated by means of a simple uniform correction

procedure. However, owing to the small size of the ions examined in that study, the

strategy of reducing the whole cavity might not be appropriate for ionic compounds

having larger, more complex chemical structures, specially when there is high

anisotropy in the charge distribution along the molecule. Then, we explore here the

suitability of our original strategy, but limiting the reduction of the cavity to the atom(s)

that bear the formal charge in the ion. Furthermore, to reduce artefacts arising from the

solute charge density penetration, different charge renormalization procedures are tested

to determine their consistency in predicting the hydration free energies, particularly for

anions having diffuse charge distributions


6

Methods

The MST continuum method. In the MST model the free energy of solvation is

determined as the sum of three contributions: electrostatic, cavitation, and van der

Waals. Following the formalism of the Polarizable Continuum model (PCM) originally

developed by Miertus, Scrocco and Tomasi,7 the electrostatic term (∆Gele) is determined

by using a set of apparent charges spread over the cavity surface (Eq. 3), which account

for the reaction field created in the solvent by the presence of the solute’s charge

distribution. The apparent charges are obtained by solving the Laplace equation with

suitable boundary conditions (Eq. 4).

∑=

=M

1j j

jR

r - r

q V (3)

where VR is the perturbation operator used to couple the solute’s charge distribution and

the solvent reaction field, M is the total number of surface elements, j, into which the

solute/solvent boundary is divided, and qj is the apparent charge at the j surface element,

which is centered at rj.

jT

jj )n

V( S

ε π 4

1-ε - q

∂∂

= (4)

where VT is the total electrostatic potential, which includes both solute and solvent

contributions, n is the unit vector normal to the surface element j, Sj is the area of the

surface element j, and ε is the solvent dielectric constant.


7

The cavitation free energy is determined following Pierotti’s scaled particle

theory21 adapted to molecular shaped cavities by using the procedure proposed by

Claverie (Eq. 5).22 Thus, the cavitation free energy of atom i, ∆GC-P,i, is determined

weighting the contribution of the isolated atom, ∆GP,i, by the ratio between the solvent-

exposed surface of such an atom, Si, and the total surface of the molecule, ST (Eq. 5).

Finally, the van der Waals term (∆GvW) is computed using a linear relationship to the

solvent-exposed surface of each atom (Eq. 6).

iP

N

i T

icav G

S

S∆G ,

1

∆∑=

= (5)

where N is the number of atoms

i

N

iivdw S∆G ∑

==

1

ξ (6)

where ξi is the atomic surface tension, which is determined by fitting to the

experimental free energy of solvation.

Molecular-shaped GEPOL cavities23 are used to define the solute/solvent

interface for the calculation of electrostatic and non-electrostatic (cavitation, van der

Waals) terms. In the last parametrization of the MST model, a dual-cavity strategy was

used,24 so that nonelectrostatic contributions are determined by using a van der Waals

surface mainly built up from Pauling’s radii (in Å; polar hydrogen: 0.9, apolar

hydrogen: 1.2, C and N: 1.5, O: 1.4, F: 1.35, S: 1.75, and Cl: 1.80), whereas the

electrostatic term is determined by using a solvent-exposed surface created by scaling

the atomic radii by a solvent-dependent factor, λ, which adopts values of 1.25, 1.50,

1.60 and 1.80 for the solvation of neutral compounds in water, octanol, chloroform and


8

carbon tetrachloride. Based on the comparison of the ∆Gele values determined from

MST and from Monte Carlo-Free Energy Perturbation (MC-FEP) calculations for seven

small ionic compounds,20 it was found necessary to further reduce the solute cavity by

around 10% to compute the hydration free energy of univalent ions. This reduction was

justified by the fact that due to strong electrostatic interactions solvent molecules are in

general closest to the atoms of charged molecules than in those of the neutral ones.

Charge normalization. Owing to the solute charge density that lies outside the cavity,

and to the numerical errors due to tessellation of the cavity surface (Eq. 4), the sum of

the apparent surface charges, ∑=j

jqQσ , in the MST (PCM) model do not satisfy the

relationship given by Eq. 7,7a,25 where QM denotes the total charge of the solute, and the

error in the charges is given by σσ∆ QQtheor −= , where Mtheor QQε

ε 1−−= .

MQQ

εε

σ1−−= (7)

The MST model8 takes into account the charge normalization by using the

expressions given in the original formalism7a (denoted in the following as the standard

correction (SC) approach), as noted in Eqs. 8a-b, where +σQ and −

σQ denote the sum of

positive and negative apparent charges, respectively.

[ ])2/(1' ++= σσ∆ Qqq jj for 0>jq (8a)

[ ])2/(1' −+= σσ∆ Qqq jj for 0<jq (8b)


9

More elaborate treatments of the charge compensation have been developed by

the Pisa group.26 The most accurate procedure (denoted density-corrected (DC)

approach hereafter) relies on two main features. First, the introduction of separate

factors to correct the apparent charges induced by nuclei ( Zf ) and electrons ( ef ),

which makes possible to correct the numerical errors on the nuclear charges, which are

only affected by the discretization of the cavity (Eq. 9). Second, the explicit calculation

of the charge density outside the cavity (Qout), which is determined from the flux of the

solute electric field through the cavity surface (Eq. 10). Using this new approach the

solvent response induced by the electron charge density inside the cavity (Qin) is then

only affected by the numerical errors due to tessellation of the cavity (Eq. 11).

ZM

j

Zj

Z Qqfε

ε 1−−=∑ (9)

))(4

1()( dsnsEQQQQ sS M

eMin

eMout ⋅⋅+−=−−= ∫π

(10)

where )(sEM is the electric field created at each tessera of the cavity.

eMin

j

ej

e Qqf ,1

εε −−=∑ (11)

For the sake of completeness, let us note that an alternative procedure used by

other groups to alleviate the effects due to the charge density that lies outside the cavity,

where the continuous charge distribution obtained from quantum mechanical or density

functional calculations is replaced by a set of atomic point charges, often supplemented

by point dipoles, which are then used to evaluate the solvent reaction field by means of

finite-difference Poisson-Boltzmann or boundary element techniques.27 This approach,


10

nevertheless, demands that the set of point charges accurately reproduces the charge

distribution of the solute, which then raises the question about the best procedure to

derive the atomic charges.28

Computational details. The set of compounds compiled by Pliego and Riveros19 has

been used to examine the suitability of our previous parametrization of the MST model

to predict the hydration free energy of univalent ions. In order to be consistent with our

previous work,20 all calculations have been performed at the HF/6-31+G(d) level.

Following the standard MST protocol,8,20 the gas phase geometry of the charged

molecules was fully optimized and subsequently used for calculations in solution. The

MST hydration free energies were determined by using a locally modified version of

Monstergauss.29 For the sake of comparison, computations were also performed by

using the Integral Equation Formalism30 of the Polarizable Continuum Model (IEF-

PCM), which has been claimed to be less sensitive to the inaccuracies arising in the

calculation of the electrostatic term. IEF-PCM calculations were performed by using the

code implemented in Gaussian03.31

Results and discussion

About the cavity definition for ionic species. Though the choice of the solute/solvent

boundary is a crucial parameter in the development of any QM SCRF method, the

definition of the solute cavity is even more delicate for ionic compounds, whose

electrostatic field strongly perturbs the solvent, thus making the solvent molecules in the

first hydration shells to exhibit properties clearly different from those of the bulk

solvent. In turn, hydration also changes the electron density distribution of ions,32

leading to differential trends in the electron rearrangement that takes place for cations


11

and anions. Therefore, it can be questioned whether the hydration of ions can be

described in an effective way by means of a suitable refinement of the solute cavity in

continuum models, and whether or not such a refinement is well suited to reproduce the

solvent-induced change in different molecular properties of the solute.33

An attempt often considered to describe the solvation of ions consists of the

addition of an arbitrary number of solvent molecules to the solute, while the rest of the

solvent is treated as a continuum dielectric.34 This approach, however, raises some

problems, such as the increase in the computational task, the need to obtain appropriate

averages of the thermally accessible configurations or the treatment of the librational

motions originated from the weak interactions between solute and solvent molecules.

Moreover, by adding few solvent molecules, the conceptual problem of the solvation of

the solute alone is transformed into the solvation of the solute-solvent cluster. Finally,

the analysis of solute-solvent clusters for medium and large size molecules becomes

even more problematic and computationally more expensive. Because of these

considerations, we prefer here to limit the QM discrete description exclusively to the

solute. This choice, nevertheless, raises the delicate question about the suitability of

transferring the cavity definition optimized for the solvation of neutral solutes to ionic

compounds.

In the original MST parametrization,20 the optimum cavity for the hydration of

ions was defined by scaling the atomic radii by a factor of 1.10-1.15, which implies a

reduction with regard to the scaling factor of 1.25, i. e. the value optimized for the

hydration of neutral solutes.8 A similar correction was also adopted in the UAHF

parametrization of the PCM model,7c where the atomic radii are reduced depending on

the formal charge carried by atoms, though different correction factors are used for

cations and anions. Others authors have also proposed to exploit alternative


12

relationships between the atomic charge and the effective atomic radius,35 or to use

modified set of atomic radii.36 An alternative approach for cavity size comes from the

use of isodensity contours of the electron density, which can be adjusted during the

SCRF calculation as the charge distribution of the solute is polarized by the solvent

reaction field.37 Here the electron density contour that leads to better agreement with the

experimental solvation free energy for ions is larger than that required for neutral

compounds.38

For our purposes here, we have decided to retain the original idea of reducing

the scaling factor (λ) that multiplies the atomic radii in order to build up the solute

cavity used to calculate the ∆Gele component. However, instead of reducing the whole

cavity of the solute, which might be appropriate for small size ions, the scaling down is

here limited to the heavy atom (and the hydrogen atoms attached to it) that bears the

formal charge. Thus, for alcohol cations and anions the reduction in the cavity only

affected the -OH2+ and -O- groups. In the case of ionized carboxylic acids, such a

reduction only affected the oxygen atoms in the –COO- group. Besides retaining the

simplicity of the original approach, this strategy allows us to exploit the same set of

scaling factors that modulate the cavity definition for the calculation of the electrostatic

component of the solvation free energy (see Methods), while introducing a minimum

number of parameters to describe the hydration of ionic species.

Refinement of the MST model for ions. The adjustment of the scaling factor λ used to

determine ∆Gele was performed with the aim to minimize the deviation of the calculated

hydration free energy, ∆Ghyd, from the experimental value for the series of monovalent

ions. Following our previous study,20 MST-HF/6-31+G(d) calculations were performed

by reducing progressively the scaling factor from λ=1.25 to λ=1.00. The ∆Gele values


13

were calculated by using both SC (Eqs. 8a,b) and DC (Eqs. 10 and 11) charge

renormalization schemes. The cavity used to compute the ∆Gele term was built up by

using the set of atomic radii typically used in MST calculations (see Methods). Finally,

the non-electrostatic (cavitation and van der Waals) component, which was estimated by

using the standard atomic surface tensions determined for neutral molecules, was added

to the ∆Gele term to calculate ∆Ghyd. This computational strategy benefits from the fact

that the ∆Gele term is the dominant contribution to the hydration of ionic solutes, and

from the partial cancellation between cavitation and van der Waals components.

Figure 1 shows the profiles of the root mean square deviation (rmsd) and relative

error (in percentage) between experimental and calculated ∆Ghyd values determined for

different scaling factors. The smallest deviation is found for a scaling factor λ ranging

between 1.10 and 1.15, which implies a reduction in the size of the cavity around 90%

relative to that used for neutral molecules. It is also worth noting that such a range of

values is in agreement with the results found previously by comparing the ∆Gele values

determined from MST and MC-FEP calculations for a subset of small ions.20 The

results in Figure 1 also point out that both the rmsd and the relative error are further

reduced by near 1 kcal/mol when the DC charge renormalization scheme is considered.

This finding can be attributed to the larger efficiency of this method to account for the

anisotropies in the charge distribution of the ions.

Table 1 reports the electrostatic component of the hydration free energy

determined from MST calculations by using a scaling factor of 1.15 and the two charge

compensation schemes. For the sake of comparison, it also reports the values

determined by using the IEF-PCM formalism and the same cavity definition for the

univalent ions. Comparison of the MST and IEF results is shown in Figure 2. The

regression equations indicate that there is close agreement between MST and IEF


14

results. Thus, the MST-SC ∆Gele values (Eq. 12) are around 3% larger than the IEF

ones, and such a difference is reduced to only 2% when the MST-DC results are

considered (Eq. 13). The most notable difference between the two charge normalization

schemes, nevertheless, is reflected in the correlation coefficient of the regression

equations, which is significantly larger when the MST-DC results are considered. This

finding points out not only the superiority of the DC charge normalization scheme, but

also the lower sensitivity of the IEF formalism to the errors arising from the penetration

of the charge density outside the solute cavity. Moreover, these results suggest that the

MST-DC optimized cavities can be considered to be transferable to the IEF-PCM

framework.

SC)(MSTG).(.(IEF)∆G eleele −±= ∆00909700 90.0=r 2191.F = (12)

)()005.0(979.0)( DCMSTGIEFG eleele −±= ∆∆ 97.0=r 8748.F = (13)

where r is the Pearson’s correlation coefficient and F stands for the Snedeckor’s

distribution parameter.

Considering the MST-DC results shown in Table 1, the rmsd between calculated

and experimental hydration free energies amounts to 5.8 kcal/mol (relative error of

5.9%). For the subset of 25 anions the rmsd is 4.7 kcal/mol, but it amounts to 6.9

kcal/mol for the 22 cations. Based on this finding, one might be tempted to use different

scaling factors to build up the solute cavity for anions and cations. However, the larger

rmsd found for cations mainly arises from deviations in protonated R-O-R’ (methanol,

ethanol, dimethyl ether, diethylether) and R-CO-R’ (acetone, methyl phenyl ketone,

methyl ethyl ester) groups, which have relative errors ≥ 10%. When these compounds

are excluded, the rmsd for the subset of cations (3.6 kcal/mol) is similar to that found


15

for anions. It is not clear whether the notable deviation found for these compounds

might stem from short-range interactions with hydrating water molecules not properly

accounted for in the continuum model. Keeping in mind the limited availability of

accurate ∆Ghyd values for ionic compounds and the assumptions intrinsic to the

continuum method, it seems to us not justified, at least on the basis of present results, to

use different scaling factors for anions and cations, a procedure already used in the

context of solvation calculations of ion hydration carried out by using the Born

equation.39 Likewise, attempts to reduce these errors by distinct reshaping of solute

cavities seems not justified and might even yield to non-physical over-trained models.

An alternative strategy to reduce the difference between computed and

experimental data involves the adjustment of the atomic surface tensions, ξi (see Eq. 6),

which in the MST model are typically determined by fitting the experimental free

energies of solvation. Since steric terms are used in continuum models to account

implicitly for effects other than pure bulk electrostatic effects, the fitted parameters for

neutral and charged molecules might not be equal. Accordingly, one can expect that

restricting the refinement of the surface tensions only to the atom(s) bearing the formal

charge would suffice to improve the agreement between calculated and experimental

∆Ghyd values. From a practical point of view, caution is required owing to the limited

availability of data for certain atom types, which can lead to overtraining of the method.

Thus, the only statistically significant improvement was obtained when the surface

tension of the OH+ group was adjusted to -0.51 kcal mol-1 Å-2, which is near 4-fold

more negative than the corresponding value for neutral molecules (-0.14 kcal mol-1 Å-

2).8d With the only correction of this surface tension, the rmsd between calculated and

experimental ∆Ghyd values is 4.6 kcal/mol.


16

The calculated ∆Ghyd and its electrostatic and non-electrostatic components

determined for the univalent anions are given in Table 2. With the exception of the

protonated R-O-R’ and R-CO-R’ groups (see above), the non-electrostatic term is

generally positive and only amounts, on average, to 3.9% of the hydration free energy.

Overall, there is satisfactory agreement between experimental and calculated ∆Ghyd

values, as noted in a relative error of 4.7%, and in the comparison of the regression

equation given in Eq. 14 (see Figure 3), specially keeping in mind the variety of

functional groups and sizes of the ionic compounds.

)DCMST(G).(.(exp)G hydhyd −∆±=∆ 00800191 940.r = 0351.F = (14)

Prediction of pKa values. Since the acidity/basicity of molecules is crucial to

understand chemical reactivity and intermolecular interactions, an intense research

effort has been devoted to check the reliability of theoretical methods, particularly QM

SCRF continuum methods, to predict accurately the pKa. Thus, Chen et al. were able to

predict the pKa values for several molecules by using a model where the reaction

potential was solved via finite-difference solutions to the Poisson-Boltzmann

equation.25c For a small series of imidazole derivatives, Topol et al. predicted pKa

values with an average absolute deviation of 0.8 units from experimental data by

combining results obtained from G2 or density functional calculations with SCRF

estimates of the hydration free energies.25d Schüürmann et al. estimated the acidities of

carboxylic acids by using the PCM-UAHF model at the SCF and MP2 levels, obtaining

good correlations for relative pKa values but not for the absolute ones, arguing that the

major sources of error stem from gas-phase calculations.40 Recently Shields and co-

workers predicted accurate absolute pKa’s for carboxylic acids41 and phenols42 using the


17

C-PCM model along with state-of-the-art gas phase ab initio calculations. Chen and

MacKerell compared pKa differences in substituted pyridines by using semiempirical

and ab initio gas-phase free energies along with IPCM, SCIPCM and SM2 solvation

calculations, and relevant differences between the pKa values obtained at different

levels of theory were found.43 By using the SSC(V)PE model, Chipman checked

different isodensity thresholds to build up cavities in order to predict the pKa’s for

alcohols, acids and amines.38b This study showed that no single isodensity contour can

be applied to neutral compounds, anions and cations, and that good correlations can be

obtained only for compounds sharing the same acidic functional group. Other studies

also have pointed out the difficulty to predict absolute pKa values, which demand highly

accurate estimates of both the gas phase free energy difference between acid and base

partners and of the corresponding solvation free energies.44

To assess the performance of the MST parametrization for ionic compounds, we

have determined the pKa values for a series of carboxylic acids and substituted pyridines

by using the thermodynamic cycle shown in Figure 4 and Eq. 15. These compounds

were chosen because of the availability of experimental data for the gas phase

basicities.45,46 For consistency with the MST parametrization, the experimental

)H(G sol+∆ value determined by Tissandier17 was used. The hydration free energies of

the neutral species were calculated by using the HF/6-31G(d) version of the MST

model.8d Finally, to further compare the effect of the charge normalization scheme both

MST-SC and MST-DC calculations were performed for ionic compounds.

RTGKpK aqaa 303.2/log ∆=−= (15)


18

Table 3 shows the results obtained for the two series of compounds, which

encompasses a range of pKa values larger than 5 units. Comparison of the results

obtained by using the two charge renormalization schemes supports the better

performance of the DC procedure. Thus, whereas both MST-SC and MST-DC methods

gives rise to similar relative pKa variations within the series of compounds (the rmsd

between calculated and experimental pKa differences varies between 0.4 and 1.2 pKa

units), the absolute pKa values determined from MST-DC computations agree better (in

absolute terms) with the experimental values, as noted in a rmsd (in pKa units) of 1.0

(MST-DC) and 2.4 (MST-SC) for carboxylic acids and of 1.9 (MST-DC) and 3.2

(MST-SC) for substituted pyridines.

Taking together the two series of compounds, the rmsd of the pKa values

predicted from MST-DC calculations amount to 1.6 pKa units. This uncertainty can be

mainly ascribed to the error in the hydration free energy of the ionic species, since the

difference between calculated and experimental hydration free energy for neutral

compounds (experimental ∆Ghyd available for acetic acid: -6.8 kcal/mol, pyridine: -4.7

kcal/mol, and 4-methylpyridine: -4.9 kcal/mol) is less than 0.5 kcal/mol. Chen and

McKerell reported rmsd values ranging from 1.22 to 4.14 pKa units depending on the

method used to compute the solvation free energy (see above). For carboxylic acids

alone, standard deviations of 0.4 pKa units have been reported by using Complete Basis

Set models in conjunction with C-PCM calculations.41 However, accurate estimates of

pKa values (i. e., with an error of half pKa unit) are typically obtained by means of an

empirical fitting of the experimental pKa values.44b-d In the absence of those empirical

adjustaments, the pKa values determined from MST-DC calculations compare

satisfactorily with the experimental ones, thus giving support to the computational

strategy used here to refine the MST method for the hydration of ionic compounds.


19

Zwitterion-neutral equilibria. Because of the high anisotropy in the charge

distribution of zwitterions, the equilibrium between neutral and charge-separated forms

is particularly well suited to examine the effect of the charge renormalization scheme

within the framework of the MST model. To this end, we have examined the

equilibrium between neutral and zwitterionic forms of nicotinic and isonicotinic acids

(see Figure 5), a system that was previously studied by Nagy et al.47 In their study the

free energy of tautomerization in water was determined by adding the relative free

energies of hydration to the gas phase free energy difference between neutral and

zwitterionic species. This latter quantity was estimated from calculations carried out at

the MP2/6-311++G(d,p) level and subsequently corrected from zero-point energy,

thermal and entropy corrections determined at the MP2/6-31G(d). At this level, the

neutral forms are predicted to be favoured by 33-35 kcal/mol in the gas phase. To

determine the relative stability of tautomers in water, we adopted the relative gas phase

stabilities determined by Nagy et al., the hydration free energies of the neutral species

were determined from MST/HF6-31G(d) calculations,8d and those of the zwitterionic

compounds were obtained from both MST-SC and MST-DC calculations.

Table 4 reports the hydration free energies of the neutral and zwitterionic species

as well as the predicted tautomerization free energy (∆Gt) in water. As was found for

the calculated pKa values, the results obtained from MST-DC computations agree better

with the experimental data than those determined from MST-SC calculations. Thus,

whereas both neutral and zwitterionic forms of nicotinic acid are predicted to have

similar stability in water from MST-SC calculations, the neutral form is predicted to be

destabilized by around 2.8 kcal/mol according to MST-DC results, which agrees with

the observed preference of the zwitterion in aqueous solution by around 2.0 kcal/mol.


20

With regard to the isonicotinic acid, the MST-SC method predicts again the neutral

form to be preferred in water, whereas the opposite trend is found from MST-DC

calculations, which predict the zwitterion to be more stable by 1.7 kcal/mol, in

agreement also with the experimental data.

Solvent reaction field in SC and DC schemes. The whole of the preceding results

support the better performance of the DC charge renormalization scheme for predicting

the hydration free energy of ionic compounds. This finding agrees with the results

obtained from the analysis of the fractional contributions of specific groups to the

hydration free energy within series of congeneric alkane derivatives,48 since the DC

scheme was found to yield consistently more transferable group contributions than the

SC scheme.

To gain further insight into the origin of the differences noticed between both SC

and DC charge renormalization schemes, we have compared a series of electrostatic

properties determined by using the wavefunction of the fully hydrated solute for a small

subset of charged monosubstituted alkane derivatives (CH3X and CH3(CH2)6X, with X

being NH3+, OH2

+, NH- and O-). This series of compounds is well suited to our purposes

because i) the enlargement of the side chain is expected to play a relatively small

perturbing effect on the hydration of the X unit, and ii) this allows us to concentrate

onto the differences in the solvent reaction field around the X group. To this end, we

have computed the rmsd between the electrostatic potential at selected points in four

layers around the molecule. For our purposes here, such a comparison was performed

by including exclusively those points generated by scaling the atomic radii of atoms in

the group X. Moreover, we have determined the well depth (Emin) of the molecular

interaction potential49 in the vicinity of the group X for the solutes interacting with


21

classical probe particles O- (cations) and H+ (anions). Finally, the results are examined

in light of the solvent’s net apparent charge (Qσ) induced on the surface elements spread

on the portion of the cavity created from the group X. The results are shown in Table 5.

Not unexpectedly, there is close similarity in the results obtained from the two

charge renormalization schemes for CH3NH3+ and CH3OH2

+. However, there are

notable differences when the largest members of the congeneric series, CH3(CH2)6NH3+

and CH3(CH2)6OH2+, are considered. Thus, the electrostatic potential around the group

X is more positive when the DC scheme is used, which reflects the more negative

apparent charge induced on the cavity surface around the NH3+ and OH2

+ groups. In

fact, by enlarging the size of the side chain there is a reduction of 25-32% in Qσ(X)

when the SC scheme is used, which is notably larger than the 8-10% variation found for

the DC results.

For anions the differences between the two charge renormalization schemes are

clear even for the parent anions, CH3NH- and CH3O-, and tend to decrease as the size of

the ionic compound is enlarged. This is particularly true for the former anion, CH3NH-,

where the solvent apparent charge determined in the SC scheme (Qσ(NH-)=+0.91) is

sensibly larger than that obtained in the DC formalism (Qσ(NH-)=+0.78). A detailed

inspection of the results reveals that such a difference stems from the solvent reaction

field created in the portion of the cavity associated to the hydrogen atom. Thus, when

the DC scheme is used, the solvent reaction field is mainly concentrated in the surface

elements pertaining to the nitrogen atom, giving rise to an apparent charge of +0.80,

whereas the apparent charge on the portion of the cavity created by the hydrogen atom

is only -0.02. In contrast, when the SC scheme is considered, both nitrogen and

hydrogen atoms contribute markedly to the solvent reaction field, as noted in apparent

charges of +0.69 and +0.22, respectively. This different distribution of the reaction


22

field in the surroundings of the NH group gives rise to differences in the electrostatic

properties of the solute (see Table 5), leading in turn to different solute-solvent

interaction energy.

In summary, the preceding results clearly point out that the explicit treatment of

anisotropy in the charge distribution leads to subtle, but nonnegligible differences in the

solvent reaction field, which in turn might modulate the calculation of the solvent-

induced change in solute’s properties, as noted in the case of the group contributions to

the solvation free energy.48

Concluding remarks

A satisfactory agreement between predicted and experimental hydration free

energies of univalent ions can be obtained by means of simple, conservative adjustment

in the definition of the cavity used to obtain the electrostatic contribution within the

framework of the MST model. Such a refinement implies a reduction in the cavity of

8% relative to the standard scaling used for neutral species, and is limited to the atom(s)

that bear the formal charge. This allows us to retain the original simplicity of the

approach and to exploit the same set of scaling factors that modulate the cavity

definition for the calculation of the electrostatic term in the MST model.

The results point out the need to account explicitly for the anisotropies in the

charge distribution of ionic species in order to correct the effects due to the solute’s

charge density that lies outside the cavity. The explicit consideration of the anisotropy

in the charge distribution of the ionic compounds leads to a subtle, but nonnegligible

improvement in the calculated electrostatic component of the hydration free energy.

Moreover, it also modulates the magnitude of the solvent reaction field, which in turn

influences the solvent-induced change in the solute’s properties. On the other hand, the


23

IEF values are very close to and exhibit high correlation with the MST-DC results,

which are on average around 2% larger (in absolute value) than the hydration free

energies determined by using the IEF formalism. Therefore, these findings point out that

the optimized cavities should be largely transferible between the two computational

schemes.

The optimized MST-DC method leads to an average relative error of 4.7% in the

hydration free energy for the series of univalent ions containing a variety of functional

groups and molecular sizes. The basic strategy proposed here is still open to further

refinement, at least for the prediction of the hydration free energy, through the

adjustment of the atomic surface tensions used for the charged atoms. However, this

refinement would require the existence of a larger database of experimental values to

avoid overtrained models than can fail when applied to compounds very different to

those used in the parametrization of the model. In any case the robustness of the strategy

is supported by the results obtained for the absolute and relative pKas of carboxylic

acids and substituted pyridines, as well as for the tautomeric equilibrium between

neutral and zwitterionic forms of nicotinic and isonicotinic acids. These examples

suffice to point out that the MST model can be used confidently to measure the

magnitude of the solvent influence of chemically relevant properties and to use this

continuum method as an interpretative tool able to gain insight into the influence of

solvation on chemical processes of ionic species.

Acknowledgments

We are grateful to Prof. J. Tomasi for providing us with his original code of the

PCM model, which was modified by us to carry out the MST calculations. This work

has been supported by the Ministerio de Ciencia y Tecnología (BIO2003-06848 and


24

SAF2002-04282), and the Centre de Supercomputació de Catalunya. C.C. is fellowship

from the Ministerio de Ciencia y Tecnología.


25

Table 1. Electrostatic component of the free energy of hydration (kcal/mol)

determined at the HF/6-31+G(d) level from MST calculations using both SC and DC

charge normalization procedures and from IEF calculations.

Ion MST-SC MST-DC IEF

F- -101.7 -101.7 -98.4 Cl- -75.4 -75.4 -72.5 OCl- -74.6 -76.1 -73.5 OH- -106.4 -104.1 -98.7 HO2

- -92.6 -92.5 -89.4 NH2

- -102.5 -96.8 -90.3 CN- -74.8 -75.0 -71.6 HCC- -81.7 -81.9 -75.8 CH3O

- -90.3 -88.8 -84.7 CH3CH2O

- -89.8 -86.1 -81.4 C6H5O

- -77.0 -70.9 -66.1 HCOCH2

- -76.5 -73.5 -69.4 CH3COCH2

- -77.3 -73.0 -68.9 HCOO- -80.3 -79.8 -76.6 CH3COO- -82.8 -80.7 -76.9 CH2COOEt- -74.0 -70.0 -66.7 CH3CONH- -79.3 -76.4 -72.7 N3

- -72.6 -71.9 -68.9 CH2CN- -70.3 -69.6 -65.8 CH3CHCN- -69.2 -67.1 -64.1 CH2NO2

- -76.5 -73.5 -70.7 HS- -80.3 -76.5 -71.8 CH3S

- -78.4 -76.2 -71.4 C6H5S

- -74.9 -67.4 -63.2 CH3SO3

- -77.8 -76.4 -72.9 H3O

+ -105.2 -106.3 -107.6 CH3OH2

+ -83.8 -84.7 -86.3 CH3CH2OH2

+ -77.3 -79.1 -80.3 (CH3)2OH+ -68.5 -68.9 -69.6 (Et)2OH+ -59.2 -60.1 -61.4 NH4

+ -93.5 -93.7 -94.9 CH3NH3

+ -81.2 -81.3 -82.9 (CH3)2NH2

+ -70.7 -70.6 -72.4 (CH3)3NH+ -60.9 -60.7 -62.3 CH3CH2NH3

+ -76.1 -77.2 -78.9 (Et)3NH+ -52.4 -53.1 -54.1 n-PrNH3

+ -73.1 -75.6 -77.5 n-BuNH3

+ -71.2 -75.2 -76.7 C6H5NH3

+ -69.1 -76.1 -77.8 (CH3COCH3)H

+ -66.5 -67.6 -68.4 (PhCOCH3)H

+ -53.6 -56.8 -58.6


26

(CH3COOEt)H+ -61.2 -63.1 -63.4 HCONH3

+ -78.4 -79.4 -80.4 CH3CONH3

+ -74.2 -74.7 -75.7 C5H5NH+ -59.5 -60.5 -61.8 (CH3)2SH+ -63.7 -63.5 -64.7 (CH3SOCH3)H

+ -63.3 -64.9 -65.7


27

Table 2. Electrostatic and non-electrostatic contributions (kcal/mol) to the free energy

of hydration determined from MST-HF/6-31+G* calculations for the series of univalent

ions used in the parametrization. The electrostatic term was determined by using a

scaling factor of 1.15 and the DC charge renormalization scheme. The experimental

hydration free energies are also given.

Ion ∆Gele ∆Gn-ele ∆Ghyd(MST-DC) Expt. a

F- -101.7 1.7 -99.9 -105.0 Cl- -75.4 0.5 -74.8 -74.6 OCl- -76.1 2.4 -73.7 -80.7 OH- -104.1 0.1 -104.0 -105.0 HO2

- -92.5 2.3 -90.2 -97.3 NH2

- -96.8 0.3 -96.5 -92.2 CN- -75.0 2.9 -72.1 -67.6 HCC- -81.9 -0.8 -82.7 -76.1 CH3O

- -88.8 2.7 -86.0 -95.2 CH3CH2O

- -86.1 3.0 -83.1 -91.1 C6H5O

- -70.9 2.8 -68.1 -71.3 HCOCH2

- -73.5 2.5 -71.0 -75.7 CH3COCH2

- -73.0 2.6 -70.4 -75.6 HCOO- -79.8 4.1 -75.7 -76.2 CH3COO- -80.7 4.2 -76.5 -77.3 CH2COOEt- -70.0 4.0 -66.0 -66.4 CH3CONH- -76.4 2.5 -73.9 -80.1 N3

- -71.9 0.3 -71.6 -70.7 CH2CN- -69.6 3.6 -66.0 -65.7 CH3CHCN- -67.1 4.3 -62.8 -65.6 CH2NO2

- -73.5 4.2 -69.3 -75.9 HS- -76.5 1.8 -74.8 -71.6 CH3S

- -76.2 0.9 -75.3 -73.7 C6H5S

- -67.4 1.1 -66.3 -63.3 CH3SO3

- -76.4 6.1 -70.3 -71.7 H3O

+ -106.3 0.4 -105.9 -110.2 CH3OH2

+ -84.7 -6.9 -91.5 -93.1 CH3CH2OH2

+ -79.1 -6.6 -85.7 -88.4 (CH3)2OH+ -68.9 -3.2 -72.0 -79.8 (Et)2OH+ -60.1 -2.6 -62.6 -71.5 NH4

+ -93.7 0.6 -93.1 -85.2 CH3NH3

+ -81.3 1.2 -80.2 -76.5 (CH3)2NH2

+ -70.6 1.7 -68.9 -68.6 (CH3)3NH+ -60.7 2.3 -58.4 -61.2 CH3CH2NH3

+ -77.2 1.5 -75.8 -73.0 (Et)3NH+ -53.1 3.1 -50.0 -54.7 n-PrNH3

+ -75.2 2.0 -73.2 -70.9 n-BuNH3

+ -75.6 1.7 -73.9 -71.5


28

C6H5NH3+ -76.1 1.4 -74.7 -72.8

(CH3COCH3)H+ -67.6 -5.6 -73.2 -76.8

(PhCOCH3)H+ -56.8 -5.2 -61.9 -65.1

(CH3COOEt)H+ -63.1 -4.0 -67.1 -68.5 HCONH3

+ -79.4 0.6 -78.7 -82.5 CH3CONH3

+ -74.7 0.8 -73.9 -73.8 C5H5NH+ -60.5 1.3 -59.2 -61.1 (CH3)2SH+ -63.5 2.6 -60.9 -64.5 (CH3SOCH3)H

+ -64.9 1.6 -63.3 -68.2 a Data taken from ref.19.


29

Table 3. Calculated and experimental pKa values for series of substituted pyridines and

carboxylic acids. MST calculations have been performed by using both SC and DC

charge renormalization methods. Absolute and relative pKa values are given in plain and

in italics, respectively.

Compound ∆Ggasa ∆Gsol(A) ∆Gsol(AH+)

(SC) pKa

b

(SC) ∆Gsol(AH+)

(DC) pKa

b

(DC) pKa

(exp) c

Pyridine derivatives H 213.7 -4.9 -58.2 2.2

0.0 -59.2 2.9

0.0 5.2 0.0

p-CH3 218.0 -5.3 -54.9 2.6 0.4

-56.2 3.6 0.7

6.0 0.8

m-CH3 216.6 -5.2 -55.1 1.8 -0.4

-56.7 3.0 0.1

5.7 0.5

m-F 206.7 -3.5 -61.2 0.3 -1.9

-62.3 1.1 -1.8

3.0 -2.2

p-Cl 210.4 -3.9 -59.4 1.4 -0.9

-60.8 2.4 -0.5

3.8 -1.4

m-Cl 207.5 -4.0 -60.0 -0.3 -2.5

-61.4 0.7 -2.3

2.8 -2.4

o-OCH3 220.9 -4.9 -52.9 3.6 1.4

-54.5 4.8 -1.9

6.6 1.4

m-OCH3 216.7 -4.6 -54.2 1.7 -0.5

-55.9 2.9 0.0

4.8 -0.4

p-CN 202.5 -5.5 -64.4 -1.9 -4.1

-67.8 0.6 -2.3

1.9 -3.4

m-CN 201.7 -6.1 -65.5 -2.2 -4.4

-69.0 0.4 -2.5

1.4 -3.9

m-NO2 200.2 -5.4 -66.4 -2.1 -4.3

-69.8 0.4 -2.5

1.2 -4.0

mse d rmsd

-3.2 -0.2 3.2 0.4

-1.8 0.5 1.9 0.8

Carboxylic acids Formic 336.5 -7.1 -76.2 2.5

0.0 -75.7 2.9

0.0 3.8 0.0

Acetic 339.5 -6.8 -78.6 2.7 0.2

-76.5 4.3 1.4

4.8 1.0

Cyanoacetic 321.8 -9.4 -67.1 0.1 -2.4

-66.1 0.9 -2.0

2.5 -1.3

Chloroacetic 327.0 -8.3 -68.7 1.9 -0.6

-69.0 1.7 -1.2

2.9 -0.9

Pivalic 335.9 -4.7 -75.5 0.8 -1.7

-69.8 5.0 2.1

5.0 1.2


30

mse rmsd

-2.2 -0.2 2.4 1.2

-0.8 -0.3 1.0 0.6

All compounds mse rmsd

-2.9 -0.2 3.0 0.7

-1.5 0.3 1.6 0.7

a Data taken from refs. 45 and 46. b A correction term of 1.9 kcal/mol was added to the gas phase free energy

difference to account for the change in standard states at 298.15 K. c Values taken from ref. 50. d rms: root-mean square deviation; mse: mean signed deviation.


31

Table 4. Calculated and experimental free energies of tautomerization between neutral

and zwitterionic forms of nicotinic (NA) and isonicotinic (INA) acids. MST calculations

were performed by using both SC and DC charge renormalization methods.

Compound ∆Ggasa ∆Gsol(A) ∆Gsol(AH+)

(SC) ∆Gaq (SC)

∆Gsol(AH+) (DC)

∆Gaq (DC)

∆Gaq (exp.) a

NA (A)

(B)

33.3

33.5

-9.3

-9.0

-42.6

-42.6

0.0

0.0

-45.3

-45.3

-2.7

-2.7

-2.0

INA 35.1 -8.6 -42.1 1.6 -45.4 -1.7 -2.5

a Data taken taken from ref. 47

Cur

utch

et e

t al.,

Oct

ober

200

4, r

evis

ed v

ersi

on

32

Tab

le 5

. D

iffe

renc

es in

the

elec

tros

tati

c po

tent

ial c

ompu

ted

in la

yers

aro

und

the

mol

ecul

e, th

e w

ell d

epth

of

the

mol

ecul

ar in

tera

ctio

n po

tent

ial,

and

the

net a

ppar

ent c

harg

e in

duce

d on

the

cavi

ty s

urfa

ce f

or s

erie

s of

con

gene

ric

ioni

c co

mpo

unds

.

Com

poun

d X

rm

sd a in

laye

r E

min

b Q

σ(X

) c

x 1.

4 x

1.6

x 1.

8 x

2.0

MS

T-S

C

MS

T-D

C

MS

T-S

C

MS

T-D

C

CH

3NH

3+ N

H3+

0.0

0.0

0.0

0.0

-123

.8

-123

.8

-0.7

1 -0

.71

CH

3(C

H2)

6NH

3+ N

H3+

2.1

1.8

1.6

1.4

-118

.6

-121

.5

-0.5

3 -0

.65

CH

3OH

2+ O

H2+

0.5

0.4

0.3

0.3

-130

.1

-129

.7

-0.5

7 -0

.59

CH

3(C

H2)

6OH

2+ O

H2+

3.6

3.0

2.5

2.2

-127

.1

-127

.5

-0.3

9 -0

.53

CH

3NH

- N

H-

6.6

5.5

4.4

3.7

-187

.6

-192

.3

+0.

91

+0.

78

CH

3(C

H2)

6NH

- N

H-

1.6

1.3

1.1

0.9

-190

.6

-190

.6

+0.

79

+0.

72

CH

3O-

O-

0.9

0.8

0.6

0.5

-169

.6

-168

.5

+0.

81

+0.

77

CH

3(C

H2)

6O-

O-

0.6

0.5

0.5

0.4

-168

.2

-168

.2

+0.

76

+0.

75

Cur

utch

et e

t al.,

Oct

ober

200

4, r

evis

ed v

ersi

on

33

a rm

sd b

etw

een

the

elec

tros

tati

c po

tent

ials

com

pute

d fr

om th

e w

avef

unct

ions

det

erm

ined

fro

m M

ST

-SC

and

MS

T-D

C c

alcu

lati

ons

for

the

ioni

c

com

poun

ds. L

ayer

s de

fine

d by

sca

ling

the

van

der

Waa

ls r

adii

by

1.4,

1.6

, 1.8

and

2.0

.

b Min

imum

ene

rgy

of th

e m

olec

ular

inte

ract

ion

pote

ntia

l for

the

ioni

c co

mpo

unds

inte

ract

ing

wit

h cl

assi

cal p

robe

par

ticl

es O

- (ca

tion

s) a

nd H

+

(ani

ons)

.

c Net

app

aren

t cha

rge

spre

ad o

n th

e ca

vity

sur

face

of

the

grou

p X

det

erm

ined

fro

m b

oth

SC

and

DC

cha

rge

reno

rmal

izat

ion

sche

me.


34

Legends to Figures

Figure 1. Representation of the change in the root-mean-square deviation (rmsd; solid

line) and the relative error (in percentage; dashed line) between the experimental

hydration free energies and the MST values obtained by varying the scaling factor used

to correct the atomic radii of charged atoms in the calculation of the electrostatic

component of the hydration free energy (see text for details). The electrostatic free

energy of solvation was determined by using both SC (black) and DC (white) charge

renormalization schemes (see text for details). The rmsd values are in kcal/mol.

5,0

6,0

7,0

8,0

9,0

10,0

1,00 1,05 1,10 1,15 1,20 1,25

Scaling Factor

rmsd

/rel

ativ

e er

ror


35

Figure 2. Comparison of the electrostatic component of the free energy of hydration

(kcal/mol) determined from MST calculations by using a scaling factor of 1.15 and both

SC (black) and DC (white) charge renormalization schemes and IEF calculations at the

HF/6-31+G(d) level. The plots displays the ideal regression line y=cx with slope c equal

to unity.

40.0

60.0

80.0

100.0

120.0

40.0 60.0 80.0 100.0 120.0

-∆Gele(MST)

- ∆G

ele(

IEF

)


36

Figure 3. Comparison of the experimental and calculated hydration free energies

(kcal/mol) for the series of univalent ions. Theoretical estimates were obtained from

MST-HF/6-31+G(d) calculations carried out by using a scaling factor of 1.15 and the

DC charge renormalization schemes. The plots displays the ideal regression line y=cx

with slope c equal to unity.

40,0

60,0

80,0

100,0

120,0

40,0 60,0 80,0 100,0 120,0

-∆Ghyd(MST-DC)

- ∆G

hyd

(exp

t)


37

Figure 4. Thermodynamic cycle used for the calculation of the pKa values for the series

of substituted pyridines and carboxylic acids.

∆Ggas AH+ A- + H+ gas phase

∆Gsol(AH+) ∆Gsol(A-) ∆Gsol(H

+)

AH+ A- + H+ water ∆Gaq


38

Figure 5. Representation of the neutral and zwitterionic species of nicotinic (top) and

isonicotinic (bottom) acids.

N

O

OH

N

OH

O

N

O

O

H

N

OHO

N

OO

H

-

+(A) (B)

+

-


39

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1

Development of distributed models of atomic polarizability for molecular simulations

Carles Curutchet,1 Ignacio Soteras,1 Axel Bidon-Chanal,1 Cristophe Chipot, 2* Modesto

Orozco,3* and F. Javier Luque1*

1 Departament de Fisicoquímica. Facultat de Farmàcia. Universitat de Barcelona. Avgda. Diagonal 643. Barcelona 08028. Spain. 2 Equipe de Dynamique des Assemblages Membranaires. Unité Mixte de Recherche CNRS/UHP 7565. Institut Nancéien de Chimie Moléculaire. Université Henri Poincaré – Nancy I. BP 239, 54506 Vanœuvre-lès-Nancy Cedex, France. 3 Departament de Bioquímica i Biología Molecular, Facultat de Química, Universitat de Barcelona, c/. Martí i Franqués 1, 08028, Barcelona, and Unitat de Modelització Molecular i Bioinformàtica, Institut de Recerca Biomèdica. Parc Científic de Barcelona, c/. Josep Samitier 1, 08028 Barcelona, Spain. Send correspondence to FJL (e-mail: [email protected]) or MO (e-mail: [email protected])

Abstract

This study is focused on the development of atomic distributed polarizability models for the treatment of induction effects in molecular simulations. Different models of dipolar polarizabilities are derived for a series of aromatic compounds from the OPEP (Optimal Partitioning of Electrical Properties) suite of programs. These models are obtained by fitting the induction energies computed in a grid surrounding the molecule by using a fast perturbational approach which permits to reproduce the induction energies determined at the MP2/Sadlej level of theory. Isotropic atomic dipole polarizability models are considered by imposing suitable restrains in the fitting procedure, and by exploring all-atom and united-atom models. The goodness of the distributed models is assessed by comparing the molecular polarizabilities recovered from the models and those obtained from MP2/Sadlej calculations. The behaviour of the models is further examined in the context of cation-π interactions by computing the polarization interaction energy of the aromatic systems with a sodium cation represented by a non-polarizable point charge, and by comparing them to MP2/Sadlej QM/MM calculations. In general, excellent results are obtained for isotropic all-atom models, with deviations around 1 kcal/mol. Energy profiles that characterize the approach of the sodium cation towards the center of the aromatic ring of benzene are also compared to QM/MM calculations and demonstrate the goodness of the models for different orientations. Finally, the transferability of the atomic polarizabilities within the series of benzene derivatives is also examined.


2

Introduction

The assumption that induction effects can be accounted for in an average sense

by means of a simple, appropriate parameterization is at the origin of the durable success of pairwise, additive potential energy functions for cost-effective statistical simulations of liquids. An important ingredient in the development of such nonpolarizable, effective force fields consists of increasing artificially the polarity of the participating molecules to compensate for missing, through-space intermolecular induction phenomena.1 A popular approach for implicit polarization is based upon the observation that, compared to the experimental gas-phase quantities, molecular dipole moments computed at the Hartree-Fock (HF) level of theory, using a split-valence 6-31G(d) basis set, are systematically exaggerated.2 In a number of instances where explicit nonadditivity effects can be neglected, intermolecular potentials parametrized with net atomic charges derived from the HF/6-31G(d) electrostatic potential provide a reasonable description of the properties of the liquid,3 although explicit, mutual polarization of the constituent molecules and their environment is clearly absent. The advantage of representing the system using pairwise, additive potential energy functions lies in the cost-effectiveness of the statistical simulation, avoiding the computationally demanding estimation of induced moments at the expense of sampling of the phase space. Such an implicit polarization approach is, however, not equivalent to a rigorous, atomic-level description of the molecular response to a nonuniform, external electric field, thus constituting the inherent limitation of the approach. This is particularly relevant in statistical simulations of ionic species or highly polarizing systems interacting with polarizable ones. Furthermore, explicit polarization has proven to be pivotal in those cases where intramolecular induction effects are sizable.4,5 The massive increase of computational resources witnessed in recent years, pushing back the limitations of statistical simulations, has allowed molecular assemblies of appreciable complexity to be tackled over realistic time-scales, using sophisticated potential energy functions. The generalized availability of such resources has inevitably contributed to the recent interest of nonadditive force fields, opening the way to the development of alternative approaches for modelling induction effects in chemical systems. To a large extent, the success of molecular simulations relying on polarizable potential energy functions is closely coupled to the ability of the modeller to design such functions. Arguably, the most popular scheme for constructing models of atomic polarizabilities is the one proposed by Applequist.6 In essence, his method is based on a self-consistent determination of parameters characterizing the atomic polarizability that see each other through screened dipole-dipole interactions. Just like its refined version devised by Thole,7 this heuristic approach implies a substantial component of arbitrariness in the parameterization of the model. In contrast, the partitioning scheme put forward by Stone,8-9 combining an earlier formulation of the susceptibility function of the charge density10 with the distributed multipole analysis (DMA) method,11 is clearly more rigorous. Rational models of distributed polarizabilities are derived from high-quality quantum mechanical (QM) calculations of the response of an isolated molecule to an external perturbation, thereby providing one and two-center charge-charge, charge-dipole, dipole-dipole, etc. parameters. A pivotal aspect of this approach concerns the partitioning of the response charge density into atomic and nonlocal contributions. Closely related, an alternative procedure supplying reasonably transferable, distributed polarizabilities12 relies on a topological partitioning of the


3

molecular space into atomic regions, according to the theory of “atoms in molecules” (AIM) devised by Bader.13 The derived polarizability parameters have been shown to reproduce accurately, in a given region, the induced moments due to a local electrostatic potential and its successive derivatives experienced at another site. Inherently nonlocal,8,9,13 the resulting models include a plethora of terms that rapidly become cumbersome to handle, especially in the context of statistical simulations. For the latter, the number of polarizability components should be reduced significantly to yield compact and tractable sets of parameters, limited to low-order terms, e.g., charge flow and one-center dipole polarizabilities, capable of describing induction phenomena at a minimal cost. As a result, the above methods are not necessarily compatible with such requirements, and the quality of the generated models strongly depends on their level of sophistication. For instance, limiting the description to atomic dipole polarizabilities, accurate reproduction of molecular quantities beyond the order of the dipole is hopeless. Atomic point charges derived from the quantum mechanical electrostatic potential computed on a grid of points around the molecule14-16 are largely utilized for parameterizing the Coulomb part of all-purpose force fields. In the spirit of this approach, a number of schemes targeted at the construction of models of distributed polarizabilities have been devised, based on a least-squares fitting procedure to the induction energy.17-20 Their strength resides in the possibility to generate easily compact and flexible sets of polarizability parameters at any given order. Contrasting with the electrostatic potential, directly available from the wave function, the induction energy is somewhat more difficult to attain. In its most straightforward formulation, referred to as finite perturbation (FP),19 it can be estimated by considering the interaction of the molecule with a nonpolarizable point charge. Whereas potential derived charges imply a single QM calculation to map the entire space around the molecule, discretized in the form of a grid of Np points, evaluating induction energies for the same grid via FP would require Np distinct QM calculations, hence constituting the computational bottleneck of the approach. Clearly, the significant computational effort involved in the determination of induction energies, including usually intramolecular electron correlation and employing large basis sets to warrant an appropriate reproduction of the molecular polarizabilities, limits the applicability of FP to sufficiently small, prototypical systems. In light of these conclusions, two alternatives have been developed concurrently, aimed at a faster estimation of induction energies and relying essentially upon a single QM calculation – a sine qua non condition for the development of nonheuristic models of distributed polarizabilities. The first scheme, based on second-order perturbation theory (PT), allows the generation of large grids of induction energies derived from one QM calculation at the HF level.21,22 In essence, it relies upon an uncoupled form of the HF equations and provides only an approximation of the exact induction energy. Using adequately chosen scaling factors, this perturbative approach has been shown to yield induction energies in good agreement with the FP quantities.22 The second, more elaborate scheme consists of computing induction energies from fully distributed models of polarizabilities obtained within the framework of the AIM theory.12 The key feature of this approach lies in its ability to map grids of induction energies from a single QM calculation at a sophisticated level of approximation, followed by the topological partitioning of the electron density response into atomic regions.23

These two approaches have been compared in the evaluation of induction energy

grids and the derivation of distributed polarizability models at the MP2 level of theory


4

and considering different choices of basis set24. The results pointed out that both strategies supply induction energies of comparable quality in terms of error with respect to the reference FP values for neutral molecules, provided that the PT HF/Saldej induction energy estimates are scaled appropriately to reproduce the MP2 quantities, this approach being more difficult to apply in the case of anions. Nevertheless, the strategy based on second-order perturbation theory constitutes a very appealing alternative due to its limited computational cost an ease of implementation. Moreover, this study showed the importance of sampling the space surrounding the molecule sufficiently far from the nuclei in the computed induction energy grids in order to obtain appropriate distributed models, and the suitability of the Sadlej26 basis set at the MP2 level of theory as an optimal compromise in terms of computational cost and agreement with experimental molecular polarizabilities.

The purpose of this study is to explore the strategy relying on the PT calculation

of induction energies for the derivation of distributed polarizability models able to introduce polarization effects in classical force fields in a systematic and well-defined way. As a test case, distributed models are derived for a series of aromatic compounds in order to explore the behaviour of this strategy in the context of cation-π interactions, where polarization effects play an important role25. The models are obtained by considering isotropic dipole atomic polarizabilities, and by exploring both united-atom and all-atom models, and the quality of the fitted parameters is assessed by comparing the molecular polarizabilities recovered from the models and those obtained from MP2/Sadlej calculations. The behaviour of the models are then examined in the context of cation-π interactions by computing the polarization interaction energy of the aromatic systems with a sodium cation represented by a non-polarizable point charge, and by comparing them to MP2/Sadlej QM/MM calculations. In addition, energy profiles that characterize the approach of the sodium cation towards the centre of the aromatic ring of benzene are also compared to QM/MM calculations to explore the goodness of the models for different orientations. Finally, the transferability of the atomic polarizabilities within the series of benzene derivatives is also examined.

Methods

A series of aromatic systems shown in Figure 1 were chosen in order to test the performance of the atomic distributed polarizability models in the prediction of polarization energies in cation-π complexes. The benzene derivatives were optimized at the MP2/6-31G(d,p) level of theory. Then, induction energy grids were generated by using the approach based on second-order perturbation theory. This approach permits the evaluation of the whole grid of induction energies from a single quantum chemical calculation, which was performed at the HF/Sadlej level of theory. Then, the induction energy estimates were scaled by a distance-dependent function in order to reproduce MP2/Sadlej estimates24. This choice is justified as the Sadlej26 basis set used at the MP2 level of theory offers a good compromise between computational cost and agreement with experimental molecular polarizabilities. The grids were constructed by taking about 1000 points going from 2 to 5 times the van der Waals atomic radii of the nearest atom, in order to sample the space surrounding the molecule sufficiently far from the nuclei. Molecular polarizabilities were also computed at the MP2/Sadlej level in order to assess the goodness of the distributed models.


5

Figure 1. Aromatic structures considered in this study.

Isotropic dipole atomic components were fitted following different criteria in order to derive the distributed models, either by adopting an all-atom (AA) or united-atom (UA) approach. At the UA models, atomic polarizabilities were fitted for all atoms except for hydrogens. In order to test the transferability of the models, three different schemes were used for either AA or UA schemes. First, fitting to the induction energy grids was carried out without any restrain (no restrains - NR). Second, symmetric atoms were forced to share the same polarizability components (symmetry - S). Finally, all carbons and hydrogens pertaining to aromatic CH groups were forced to have the same components at each different system (atom type – AT). The atomic polaribilities obtained by using these three procedures were subsequently compared and used to discuss the transferability of these values between different aromatic systems.

The final behaviour of the models was analyzed by comparing the fitted

molecular polarizabilities with the pure QM estimates. In addition, polarization interaction energies between a sodium cation and the aromatic systems were computed both from the distributed models and from QM/MM calculations, where the sodium cation was simulated as a nonpolarizable point charge. The sodium cation was added at 2.47 Å above the aromatic ring for all systems (see ref. 25).The polarization energies derived from the isotropic distributed models were computed following the procedure proposed first by Applequist27, and subsequently refined by Thole7. On the other hand, QM/MM estimates were determined by two different approaches. First, by computing the full variational polarization interaction energy of the point charge with the aromatic system, and second, by using the second-order perturbational approach used to build up the induction energy grids, i.e., by computing perturbationally the HF/Sadlej polarization energy and then scaling it to reproduce MP2/Sadlej values. This final calculation also permits to analyze the performance of the fast second-order perturbational approach in comparison with the more costly variational calculation.

11 12 13 14 15

6 7 8 9 10

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6

Finally, polarization energy profiles determined for the approach of the sodium cation to the aromatic ring of benzene were calculated using the isotropic all-atom symmetry (AA-S) model and from QM/MM MP2/Sadlej calculations, with the aim to investigate the behaviour of the model at different orientations of the cation.

Results and discussion

Table 1 shows the polarization energies obtained variationally for the cation-π complexes at the MP2/Sadlej level and the values obtained using the perturbational approach mentioned before. The polarization energies determined perturbatively are able to reproduce the variational results with a deviation of only 9% (r2=0.97) and a much lower computational cost, thus giving support to its use as a general strategy for the derivation of atomic polarizabilities adjusted to a given molecule.

Table 1. Variational and perturbational QM/MM MP2/Sadlej polarization interaction energies (kcal/mol) between the aromatic systems and the Na+ cation. Compound Variational Perturbational

1 -21.4 -23.8 2 -25.2 -28.3 3 -25.1 -27.6 4 -28.7 -30.4 5 -21.2 -23.7 6 -22.4 -24.9 7 -24.3 -26.5 8 -22.3 -25.2 9 -22.2 -25.3 10 -21.0 -23.5 11 -21.0 -23.7 12 -22.4 -23.4 13 -20.5 -21.3 14 -18.9 -20.9 15 -19.8 -22.3

msea 2.3 mue 2.3 rms 2.5

a mse: mean signed deviation (kcal/mol); mue: mean unsigned deviation (kcal/mol); rms: root-mean square deviation (kcal/mol)

The comparison of molecular polarizabilities obtained from the different models with the values determined variationally at the MP2/Sadlej level are shown in Table 2. A very close agreement is found when an all-atom description is used, as noted in mean deviations less than 1% when atom-type restraints are imposed in the fitting of atomic polarizabilities. When a united-atom description is used, the agreement between molecular polarizabilities recovered from the distributed models and computed variationally is slightly worse, even though the mean deviation is only 4%. At this point, it must be pointed out that the molecular polarizability was not used as an adjustable parameter in the fitting procedure to derive the distributed models, and so the agreement obtained gives strong support to the method.


7

Table 2. Trace of the molecular polarizability (a.u.) obtained from the quantum mechanical calculation (MP2/Sadlej) and from different isotropic distributed models (All-atom/United-atom) obtained by imposing different kind of restrains: no restrains (NR), symmetry (S), atom type (AT). Compound All-atom United-atom QM NR S AT NR S AT

1 208.3 214.2 214.2 214.2 230.1 230.6 230.6 2 360.5 345.4 346.3 345.7 355.7 356.3 362.4 3 308.4 301.8 301.8 301.3 309.6 309.6 314.8 4 399.2 366.2 366.5 364.1 375.4 375.5 383.8 5 208.1 212.8 212.9 212.9 219.6 219.9 222.9 6 225.2 230.1 230.1 230.1 237.6 238.1 241.3 7 247.7 253.7 253.7 253.5 261.5 261.8 265.0 8 251.9 256.5 256.6 256.7 263.0 263.1 266.2 9 259.2 257.5 257.6 257.6 265.3 265.6 268.3 10 207.8 211.2 211.3 211.3 213.8 213.9 213.9 11 209.9 214.1 214.6 214.6 214.6 214.9 214.9 12 165.1 175.4 175.6 175.3 191.3 191.3 191.2 13 145.9 154.5 154.7 155.9 170.4 170.4 170.2 14 147.7 156.8 156.8 158.2 171.1 171.1 171.2 15 192.2 196.0 196.0 196.6 213.4 213.4 213.8

msea -0.6 -0.8 -0.7 -10.3 -10.6 -12.9 mue 8.1 8.1 8.5 14.2 14.3 14.9 rms 11.4 11.3 12.0 17.0 17.1 17.3


Another parameter considered to asses the goodness of the fittings is the

deviation found between the induction energies computed quantum-mechanically and the values recovered from the distributed models along the grid points (see Table 3). In general, a good agreement is found for all the models, with small deviations ranging from 0.12·10-3 to 0.27·10-3 (in a.u.).

Table 3. Root-mean square deviation (10-3 a.u.) between the quantum mechanically and the recovered induction energies computed at the grids from different isotropic distributed models (All-atom/United-atom) obtained by imposing different kind of restrains: no restrains (NR), symmetry (S), atom type (AT). Compound All-atom United-atom NR S AT NR S AT

1 0.11 0.11 0.11 0.36 0.36 0.36 2 0.13 0.14 0.14 0.21 0.22 0.30 3 0.12 0.12 0.13 0.20 0.20 0.27 4 0.17 0.17 0.26 0.24 0.25 0.40 5 0.11 0.11 0.11 0.24 0.24 0.28 6 0.12 0.12 0.12 0.26 0.27 0.31 7 0.13 0.13 0.13 0.26 0.26 0.30 8 0.11 0.11 0.12 0.21 0.21 0.24 9 0.13 0.13 0.13 0.22 0.22 0.25 10 0.10 0.10 0.10 0.17 0.17 0.17 11 0.11 0.11 0.11 0.12 0.12 0.12 12 0.11 0.11 0.11 0.25 0.25 0.25 13 0.10 0.10 0.11 0.23 0.23 0.23 14 0.11 0.11 0.13 0.25 0.25 0.25 15 0.11 0.11 0.12 0.36 0.36 0.36

Mean 0.12 0.12 0.13 0.24 0.24 0.27


8

In order to further investigate the behaviour of the models, the polarization energy of the aromatic systems interacting with the Na+ cation (placed at its optimum position to establish a cation-π interaction) was computed from the distributed models and compared to the variational MP2/Sadlej QM/MM calculations. Table 4 shows the results computed by neglecting the coupling between atomic induced dipoles, i. e., by considering only the electric field generated by the point charge of the sodium cation. On the other hand, Table 5 shows the results determined by computing the electric field generated by both the sodium cation and by the atomic induced dipoles following an iterative procedure, which usually converges in one or two cycles. The polarization energies of the different models with the Na+ cation show a good agreement, especially when an all-atom description is used. In this case, the differences between the polarization energies obtained at the QM and classical levels are lower than 2%, while this difference increases to 15% when united-atom models are considered. The results also indicate that the mutual coupling between the induced dipoles for the different cation-π complexes have a relalitively small effect in the computed induction energies. These results suggest, then, that the mutual coupling could be neglected, thus avoiding the self-consistent procedure that must be carried out to reach convergence between the induced dipoles, with the corresponding saving in computational cost specially for simulations of complex systems. Table 4. Polarization interaction energy (kcal/mol) between the Na+ cation and the aromatic systems determined from the variational quantum mechanical calculation and from different isotropic distributed models (All-atom/United-atom) obtained by imposing different kind of restrains (no restrains, symmetry, atom type). Energies computed without coupling between atomic induced dipoles. Compound All-atom United-atom QM NR S AT NR S AT

1 -21.4 -22.2 -22.3 -22.3 -29.1 -29.2 -29.2 2 -25.2 -23.2 -22.9 -23.6 -20.9 -20.7 -24.3 3 -25.1 -21.7 -21.7 -23.7 -21.8 -21.8 -24.0 4 -28.7 -24.1 -24.0 -23.0 -19.1 -19.1 -22.0 5 -21.2 -21.4 -21.5 -21.5 -24.2 -24.2 -25.6 6 -22.4 -22.5 -22.6 -22.6 -25.6 -25.7 -27.0 7 -24.3 -23.8 -23.9 -24.0 -26.5 -26.5 -27.6 8 -22.3 -22.4 -22.4 -22.4 -25.0 -25.0 -25.9 9 -22.2 -23.4 -23.4 -23.4 -23.5 -23.5 -24.1 10 -21.0 -20.6 -20.7 -20.7 -21.7 -21.7 -21.7 11 -21.0 -20.4 -20.5 -20.5 -20.4 -20.5 -20.5 12 -22.4 -21.6 -21.7 -21.5 -27.9 -27.9 -27.9 13 -20.5 -19.0 -19.1 -19.5 -25.0 -25.0 -25.0 14 -18.9 -19.3 -19.3 -19.8 -25.3 -25.3 -25.4 15 -19.8 -20.5 -20.5 -20.8 -27.4 -27.4 -27.4

msea 0.7 0.7 0.5 -1.8 -1.8 -2.7 mue 1.2 1.2 1.1 4.2 4.2 4.0 rms 1.8 1.8 1.8 5.1 5.1 4.8



9

Table 5. Polarization interaction energy (kcal/mol) between the Na+ cation and the aromatic systems determined from the variational quantum mechanical calculation and from different isotropic distributed models (All-atom/United-atom) obtanied by imposing different kind of restrains (no restrains, symmetry, atom type). Energies computed coupling atomic induced dipoles. Compound All-atom United-atom QM NR S AT NR S AT

1 -21.4 -22.5 -22.6 -22.6 -29.1 -29.2 -29.2 2 -25.2 -23.2 -22.7 -23.4 -20.9 -20.7 -24.3 3 -25.1 -21.8 -21.8 -23.8 -21.8 -21.8 -24.0 4 -28.7 -24.2 -24.1 -23.4 -19.1 -19.1 -22.0 5 -21.2 -22.0 -22.1 -21.9 -26.9 -26.9 -26.7 6 -22.4 -22.7 -22.8 -22.7 -25.6 -25.7 -27.0 7 -24.3 -23.9 -24.0 -24.1 -26.5 -26.5 -27.6 8 -22.3 -23.5 -23.5 -23.2 -28.3 -28.3 -27.7 9 -22.2 -23.7 -23.7 -23.8 -22.2 -22.1 -22.7 10 -21.0 -21.1 -21.2 -21.2 -23.0 -23.2 -23.2 11 -21.0 -21.0 -21.1 -21.1 -21.6 -21.7 -21.7 12 -22.4 -22.2 -22.2 -22.0 -27.9 -27.9 -27.9 13 -20.5 -19.6 -19.7 -19.9 -25.0 -25.0 -25.0 14 -18.9 -23.7 -23.7 -26.0 -33.0 -33.0 -33.0 15 -19.8 -21.2 -21.3 -21.2 -27.4 -27.4 -27.4

msea 0.0 0.0 -0.3 -2.8 -2.8 -3.5 mue 1.5 1.6 1.5 5.1 5.2 4.7 rms 2.2 2.3 2.6 6.4 6.5 6.1

a mse: mean signed deviation (kcal/mol); mue: mean unsigned deviation (kcal/mol); rms: root-mean square deviation (kcal/mol) Finally, the energy profiles that characterize the approach of the sodium cation to benzene were calculated from the isotropic all-atom symmetry (AA-S) model and from QM/MM calculations. As shown in Figure 2, approximation profiles were computed along three orientations: (i) along the middle of a C-C bond (x orientation), (ii) along the C-H bond (y orientation), and (iii) perpendicular to the center of the aromatic ring (z orientation). In all cases, polarization energies at the “hard particle radius” (defined as the point where dispersion and repulsion cancel each other, see ref. 25) and in points closer to and further away from the molecule were computed. The results given in Figure 3 show the good behaviour encountered for the model along these three orientations. Figure 2. Orientations of cation-aromatic interactions in this study. The z axis corresponds to the cation-π interaction.

x

y z

H HH

HH

H


10

Figure 3. Polarization energy profiles that characterize the approach of Na+ to benzene determined from QM/MM MP2/Sadlej calculations and from the isotropic all-atom symmetry model of distributed polarizabilities. Distances (in Å) are referred in all cases to the hard sphere radius, the point where dispersion and repulsion cancel each other. Hard sphere radii are: 4.58 Å (x axis), 4.97 Å (y axis), and 3.24 Å (z axis). Profiles computed without coupling between atomic induced dipoles.


11

In order to further investigate the transferability of atomic polarizability components between the different systems considered, the atomic polarizabilities obtained for the C and H atoms (all-atom) and the C-H group (united-atom) in the different distributed models are shown in Table 6. Table 6. Isotropic polarizability components (a.u.) for C and H pertaining to aromatic CH groups derived from different isotropic models. Mean values and standard deviations are given for the whole series of benzene derivatives considered in the study.

All-atom αC NR 10.2 6.8 S 10.4 5.9 AT 9.6 2.3 αH NR 3.5 1.1 S 3.4 0.9 AT 3.6 0.4

United-atom αC NR 19.1 8.8 S 19.1 8.4 AT 16.7 8.1 Inspection of the mean values obtained for the series of complexes shows that no relevant differences are found in the atomic polarizabilities obtained either without restraints or by imposing symmetry or atom-type equivalences. The atomic polarizability of C and H oscillates (in a.u.) between 9.6 and 10.4, and between 3.4 and 3.6, and for the C-H group adopts values between 16.7 and 19.1. However, it must be noted that notably large standard deviations are found, which suggests the existence of marked differences in the atomic polarizabilities obtained for these atoms in different molecules. This is clearly reflected in the results shown in Tables 7 (all-atom) and 8 (united-atom), where the mean atomic polarizabilities obtained for each compound are given. For instance, the C polarizability obtained without restraints oscillates between 4.0 and 18.2 (a.u.), while the C-H polarizability oscillates between 12.7 and 31.5. These results seriously question the assumption that distributed models of atomic polarizabilities can be transferable between structurally related molecules. Table 7. Isotropic polarizability components (a.u.) for C and H pertaining to aromatic CH groups derived from different all-atom isotropic models. For the atom-type model (AT) components derived for each aromatic system are given. In cases where diferent values can be found at each aromatic system, i.e. no restrains (NR) and symmetry (S) models, mean, maximum and minimum values, along with standard deviations are given for each compound. Atom-type No restrains Symmetry αC αC αC Mean Max Min Dev Mean Max Min Dev 1 8.2 8.1 8.9 5.8 1.1 8.2 8.2 8.2 0.0 2 10.8 11.5 23.8 -0.6 9.1 12.5 13.6 11.5 1.1 3 11.0 11.6 21.3 4.2 6.5 11.6 21.3 4.2 6.5 4 11.5 16.1 29.9 2.3 9.3 16.5 30.2 7.8 8.8 5 8.6 9.8 13.7 4.4 3.9 9.7 12.1 7.5 2.3 6 8.6 9.9 13.8 3.8 4.7 9.9 12.4 6.7 3.0 7 9.5 9.9 14.0 5.5 3.4 9.9 12.4 8.2 1.7 8 8.6 10.0 15.9 1.2 6.2 10.0 13.5 4.8 4.7 9 8.6 9.2 12.7 4.0 4.2 9.2 11.0 6.8 2.2 10 10.9 10.6 12.3 8.5 1.7 10.9 10.9 10.9 0.0 11 18.5 18.2 19.7 16.4 1.6 18.5 18.5 18.5 0.0 12 8.4 8.4 12.7 4.4 3.6 8.4 9.9 6.8 1.8 13 7.6 6.5 13.2 -0.7 6.8 6.6 12.3 0.9 6.6 14 7.2 4.0 6.9 -1.8 5.0 4.0 6.9 -1.8 5.0 15 6.8 5.3 19.2 -4.9 12.2 5.3 18.5 -4.0 12.0


12

αH αH αH Mean Max Min Dev Mean Max Min Dev 1 3.7 3.8 4.7 3.5 0.5 3.7 3.7 3.7 0.0 2 3.5 3.5 6.1 1.4 1.6 3.1 3.2 2.9 0.2 3 3.4 3.3 5.3 0.9 1.5 3.3 5.3 0.9 1.5 4 3.9 2.6 5.0 0.4 1.4 2.5 3.6 0.4 1.3 5 3.6 3.4 4.5 3.0 0.6 3.4 3.6 3.2 0.2 6 3.8 3.6 4.6 2.9 0.6 3.6 3.7 3.2 0.2 7 3.7 3.7 4.5 3.0 0.6 3.7 4.0 3.2 0.3 8 3.7 3.5 4.9 2.7 0.9 3.5 4.0 2.9 0.5 9 3.6 3.6 4.7 3.0 0.7 3.5 3.7 3.3 0.2 10 3.2 3.3 3.8 2.9 0.4 3.2 3.2 3.2 0.0 11 2.1 2.2 2.8 1.6 0.6 2.1 2.1 2.1 0.0 12 3.7 3.6 4.7 2.8 0.8 3.6 4.2 3.1 0.6 13 3.5 3.7 5.6 2.1 1.7 3.7 5.0 2.3 1.6 14 3.8 4.4 5.6 3.4 1.2 4.4 5.6 3.4 1.2 15 4.0 4.1 5.6 2.2 1.6 4.1 5.4 2.3 1.6 Table 8. Isotropic polarizability components (a.u.) for C pertaining to aromatic CH groups derived from different united-atom isotropic models. For the atom-type model (AT) components derived for each aromatic system are given. In cases where diferent values can be found at each aromatic system, i.e. no restrains (NR) and symmetry (S) models, mean, maximum and minimum values, along with standard deviations are given for each compound. Atom-type No restrains Symmetry αC αC αC Mean Max Min Dev Mean Max Min Dev 1 12.8 12.8 15.9 10.5 2.1 12.8 12.8 12.8 0.0 2 18.9 22.0 33.6 10.1 8.4 22.2 29.0 15.3 7.3 3 18.4 21.4 32.6 10.5 8.9 21.4 32.6 10.5 8.9 4 19.9 24.2 36.5 16.1 8.0 24.1 34.8 16.3 7.7 5 14.9 19.0 31.3 6.0 11.2 19.0 29.7 8.8 10.5 6 15.1 19.7 36.1 3.2 13.4 19.6 31.8 7.6 12.1 7 16.2 20.7 34.5 5.3 13.3 20.7 33.1 8.1 12.5 8 15.0 18.8 29.6 7.6 10.5 18.8 29.3 8.5 10.4 9 15.9 19.1 28.8 9.7 8.4 19.1 27.4 12.0 7.8 10 21.9 21.8 25.7 17.7 3.9 21.9 21.9 21.9 0.0 11 31.5 31.5 31.9 31.1 0.4 31.5 31.5 31.5 0.0 12 13.5 14.0 15.5 12.2 1.5 14.0 15.1 12.8 1.4 13 12.8 13.0 13.8 12.4 0.7 13.0 13.6 12.4 0.7 14 12.9 12.6 13.3 12.1 0.6 12.6 13.3 12.1 0.6 15 12.7 12.4 17.3 6.8 4.5 12.4 17.1 6.8 4.6

Concluding remarks The procedure described in this study constitutes a promising alternative for the derivation of distributed models of atomic polarizabilities. The models obtained are able to reproduce either the polarization energies computed variationally or the global polarizability of the molecule. No relevant differences are found in the magnitude of the atomic polarizabilities due to the use of symmetry or atom type-based restraints compared to the values derived without imposing any restraint. On the other hand, marked differences are found in the values obtained for different molecules, thus difficulting the definition of a single set of atomic values transferable between molecules. These findings suggest that the procedure adopted can be an efficient alternative for the definition of distributed models of atomic polarizabilities.


13

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