Calculations of Hydration Entropies of Hydrophobic, Polar, and Ionic

Nov 3, 1999 - A Langevin dipoles solvation model that can determine the entropies of .... George Pontikis , James Borden , Václav Martínek and Jan F...
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10282

J. Phys. Chem. B 1999, 103, 10282-10288

Calculations of Hydration Entropies of Hydrophobic, Polar, and Ionic Solutes in the Framework of the Langevin Dipoles Solvation Model Jan Floria´ n* and Arieh Warshel* Department of Chemistry, UniVersity of Southern California, Los Angeles, California 90089-1062 ReceiVed: June 21, 1999; In Final Form: September 24, 1999

A Langevin dipoles solvation model that can determine the entropies of transfer of molecules from the gas phase to aqueous solution is developed and examined. This computational approach involves the calculation of the hydrophobic part of the hydration entropy using the potential-dependent surface area of the solute. This contribution is augmented by an immobilization entropy term that accounts for the ordering of the solvent dipoles near charged solutes of arbitrary shape. The entropic contributions to the hydration free energies of 55 neutral and 70 ionic solutes at 298 K are calculated using the proposed algorithm and the results are compared to the available experimental data. In addition, it is shown that the hydration entropy contributes significantly to the total activation and reaction entropies of proton transfer and nucleophilic substitution reactions.

1. Introduction The need to model reliably chemical processes in aqueous solution challenges theoretical chemists to utilize all available experimental data for testing their computational models and improving their predictive power. Among the quantities that can be obtained experimentally, the enthalpy and entropy contributions to the activation and reaction free energies are particularly valuable because they can be usually rationalized more easily than the corresponding free-energy differences1 (although electrostatic free-energy contributions are probably an exception to this rule2). The decomposition of the free energy differences in solution into their enthalpy and entropy components requires the evaluation of hydration enthalpies and entropies, i.e., the enthalpy and entropy changes upon the transfer of the reactants, transition state structures, or reaction products from the gas phase to the aqueous solution. However, the evaluation of hydration entropies using all-atom solvent models is significantly more demanding than calculations of the corresponding hydration free energies.3 Thus, modeling studies have exploited faster convergence of free energy calculations by avoiding the harder question about the decomposition of these energies.4,5 In the realm of hydration enthalpies or free energies, the dielectric continuum and dipolar models have been established as computationally efficient alternatives to all-atom simulations.2,6-14 However, the evaluation of hydration entropies still presents a challenge. An important advance in this direction has been made by Rashin and Bukatin15,16 who examined the hydration entropies of spherical solutes in their neutral and charged forms and obtained a promising agreement with relevant experimental information. This study demonstrated that hydration entropies can be calculated by considering available configurations of a single water molecule that is placed near the surface of the solute and is surrounded by the dielectric continuum (mean-field approximation). Although this approach is not practical for systems lacking a spherical symmetry, it provided a useful insight into the charge dependence of hydration entropies and enabled to demonstrate that the hydration entropies of neutral solutes are proportional to their

accessible surface areas. Subsequently, such a proportionality was utilized in calculations of hydration entropies of 50 neutral organic molecules,17 neutral and protonated imidazoles,18 and protonated water clusters.19 However, this approach is not applicable in its present form17 to ion pairs, anions, and small cations.16 In this paper, we evaluate the hydration entropies of neutral and charged solutes of arbitrary shape in the framework of the Langevin dipoles (LD) solvation model.6,12 In this simplified solvation model, the solvent is approximated by polarizable dipoles fixed on a cubic grid. The most recent version of our LD model was parametrized for ab initio calculations of hydration free energies of neutral and ionic molecular solutes12 and applied in theoretical studies of the chemical reactivity,20-23 binding,24 and conformational flexibility25 in aqueous solution. Because lattice and other simplified solvent models have been shown capable of explaining the temperature26 and distance27 dependence of the hydrophobic effect, we decided to extend the applicability of the LD model toward quantitative predictions of hydration entropies. This is done here by introducing a simple term that characterizes the entropy loss upon ordering dipolar solvent molecules in the vicinity of charged solutes and parametrizing it using experimental hydration entropies of atomic ions. To describe properly both hydration entropies and free energies we readjusted several parameters of the LD model and verified the consistency of the hydration free energies obtained with the original and the refined parameter sets. Using the new algorithm and parametrization implemented in the program ChemSol 2.028 we calculate hydration entropies, enthalpies, and free energies of a large set of structurally diverse solutes. These results are compared with predictions obtained previously using all-atom simulations,29 solute accessible surfaces,17 and available experimental data.30-32 2. Computational Methods 2.1. Hydrophobic Entropy. The experimental hydration entropies of neutral solutes have been shown to be proportional to the solute surface area,6,17,33-35 although for polar molecules

10.1021/jp992041r CCC: $18.00 © 1999 American Chemical Society Published on Web 11/03/1999

Modified Langevin Dipoles Solvation Model

J. Phys. Chem. B, Vol. 103, No. 46, 1999 10283

an additional dependence on the solute field has been shown to improve the agreement.36 The linear dependence on the surface area has been attributed to the decrease in the number of accessible configurations for the water molecules interacting with the solute surface1 or alternatively to the redistribution of configuration energies.15,16 However, the hydration entropy includes not only the solvent but also the solute contributions. The latter contributions originate from the changes in the rotational and vibrational entropies of the solute upon going from the gas phase to water (∆∆Svib+rot). (Note that the translational entropy of the solute does not contribute to the hydration entropy due to the use of 1 M standard state for both gas phase and solution.) The proportionality of the total hydration entropy to the surface area of the solute indicates that solute contributions should be reflected by a constant term (otherwise we would not have such a linear proportionality). This observation may reflect a compensation of the changes in solute and solvent contributions or a surface area dependence of ∆∆Svib+rot. At any rate, following the above reasoning, we approximate the hydrophobic entropy at 298 K (∆Sphob, cal/ mol/K) by the formula

∆Sphob ) ∆∆Svib+rot - 0.0463Ω

(1)

in which ∆∆Svib+rot ) ∆∆Srot + ∆∆Svib reflects the constant part of the solute entropy contributions, while the possible dependence of these contributions on the surface area is included in the second term of eq 1. The actual value of ∆∆Svib+rot is 0 and -4.7 cal mol-1 K-1 for atomic and molecular solutes, respectively, and Ω denotes the potential-dependent surface area of the solute.12,37 This area is evaluated as

{

Ω)

∑j f(Vj)

j ∈ surface grid points

1, for |Vj| e Vmin |Vj| - Vmin , for Vmax > |Vj| > Vmin f(Vj) ) 1 Vmax - Vmin 0, for |Vj| g Vmax

}

(2a)

(2b)

where the surface grid points are defined as grid points that lie less than 1.5 Å from the solute boundary. The limiting values of the electrostatic potential, Vmin and Vmax, amount to 0.002 e/Å and 0.038 e/Å, respectively. The use of a surface area that depends on the surface polarity provides a smooth vanishing of the hydrophobic entropy upon going from nonpolar solutes to ionic solutes with high potential near their surface. 2.2. Hydration Entropies of Ions. An empirical solvation model can in principle accommodate any formula for entropy loss upon the transfer of an ion to aqueous solution, if this formula provides a reasonable agreement with the experimental data. Indeed, various empirical expressions that relate the ionic radii of spherical ions to their hydration entropy have been proposed.31,38-41 For arbitrarily shaped solutes, the simple choice is that T∆Shydr (cal/mol/K), at T ) 298 K, is proportional to the hydration free energy (∆Ghydr, kcal/mol). This choice, as implied by the dielectric continuum model,31 can be expressed as

298∆Shydr ) γ∆Ghydr, γ ) 0.0176

(3)

However, this interesting approach was shown to significantly underestimate hydration entropies for atomic ions.31 Although the poor performance of this model could be improved by considering γ as an adjustable parameter, we did not pursue this idea because we believe that the explicit character of the

Figure 1. Schematic representation of the restricted configurational volume of a grid-centered solvent dipole in the presence of an external electrostatic field ξ. In the pure liquid, the rotation of the solvent dipole is assumed to be free and the accessible volume corresponds to the outer shell of a sphere of a radius µ0. In the presence of the external field, the end point of the solvent dipole becomes confined, on the average, to the outer shell determined by the solid angle between the direction of the field and the maximum deviation of the solvent dipole from this direction (shaded area).

LD model could be harnessed to obtain a microscopic description of the hydration entropy. Such a microscopic approach might become especially valuable in heterogeneous solvent environments, such as enzyme active sites, where the continuum dielectric description of the solvent becomes problematic. For this reason, a simple microscopic formulation of the hydration entropy of ions in the framework of the LD solvation model is presented below. For charged solutes, the hydration entropy is dominated by the solvent immobilization in the presence of the solute electrostatic field, ξ. This entropy contribution can be estimated using the concept of accessible configurational volumes introduced by Frank and Evans1,42 and expressed as

∆Simmob )

∑j R ln(Vjsl/Vjpl),

j ∈ solvent dipoles

(4)

where R ) 1.988 cal/mol/K and Vpl and Vsl are the accessible configurational volumes of the solvent dipoles in the pure (bulk) liquid and in the presence of the solute, respectively. The accessible configurational volumes can be expressed as the products of the volumes corresponding to the translational, rotational, and vibrational degrees of freedom of the solvent molecules. The use of the LD solvation model, in which the positions of the solvent dipoles are fixed, forces us to neglect translational contributions to the hydration entropy. Although this contribution is not negligible it is certainly small compared to the changes in rotational degrees of freedom of the solvent molecules. Similarly, the vibrational entropy of water is supposed to be unaffected by the presence of the dissolved ions. Using the above assumptions and the fact that the direction of a solvent dipole in large electrostatic field is bound to lie between 0 and 2R (Figure 1) one obtains the relationship

Vsl )

2R V π pl

(5)

The angle R, which corresponds to the average angle between the direction of the solvent dipole and the external field, is

10284 J. Phys. Chem. B, Vol. 103, No. 46, 1999

Floria´n and Warshel TABLE 1: Atomic Radii Used in the Present LD Model atom type

rLD [Å]

atom type

rLD [Å]

C(sp3) C(sp2,sp) C(sp) O(sp3) O(sp2, inorga) N(sp3,sp2)

2.65 3.00 3.25 2.32 2.65 2.65

S P F Cl Br H

3.2 3.2 2.46 3.16 3.44 b

a sp3 and sp2 oxygens bonded to N, S, and P atoms. b The van der Waals radius of hydrogen is assumed to be linearly dependent on the rLD of the closest heavy atom X: rLD(H) ) kHrLD(X). The value of the constant kH is 0.88 and 0.78 for atom X from the first and second row of the periodic table, respectively.

Figure 2. Immobilization entropy of a single Langevin dipole in the prsence of an external electrostatic field.

determined by the Langevin function43,44 L(x):

cos R ) L(x) ) coth (x) - 1/x

(6)

where (Table 1S, Supporting Information)

x)

µ0 ξ 3kT

(7)

In eq 7, µ0, ξ, k, and T denote, respectively, the magnitude of the solvent dipole, the total electrostatic field at the position of this dipole, the Boltzmann constant, and thermodynamic temperature. Using eqs 4-6, the immobilization entropy of the jth Langevin dipole becomes

∆Sjimmob ) R ln

(

π 2 arccos(L(x))

)

(8)

Equation 8 does not include explicitely the dipole-dipole interaction term. Thus, this equation loses its validity for solvent dipoles positioned in the regions where the magnitude of the solute electrostatic field is comparable to the field from other solvent dipoles. Consequently, we introduced an empirical correction function

∆Sjcor ) -5R arctan

() ( ) x6 x exp 27 32

(9)

which decreases the magnitude of ∆Simmob for small fields. For +1 or -1 charged solutes, this correction affects the solvent dipoles beyond the first hydration shell, whereas for +2 and -2 charged solutes this term affects dipoles outside the first two hydration shells. Finally, a multiplicative constant Ci ) 0.878 that incorporates entropy contributions that cannot be described by the dipolar solvent model was introduced and adjusted by comparing the calculated and experimental hydration entropies of atomic ions. The resulting empirically corrected immobilization entropy is expressed as

∆S˜ immob ) Ci

∑j (∆Simmob + ∆Scor)

j ∈ solvent dipoles (10)

The behavior of the uncorrected (eq 8) and the corrected (eq 10) immobilization entropy as a function of x is presented in Figure 2. Finally, the total hydration entropies are calculated as a sum of the hydrophobic and immobilization entropies

∆Shydr ) ∆Sphob + ∆S˜ immob

(11)

This expression can be applied to both neutral and ionic solutes. Note that it has been suggested that the hydration entropy of large positive ions may be significantly different from that of large negative ions of the same size.16,45 In our model, this entropy does not depend on the sign of the solute charge since we believe that the magnitude of this dependence is still an open question (see discussion in section 3.2.). It should be mentioned here that ∆Sphob could have also been expressed by considering the decrease in the number of configurations available for the solvent dipoles near the nonpolar solute surface relative to the configurations available in the bulk solvent.37 (For a related treatment in the framework of the meanfield approximation see ref 16.) A rigorous implementation of this model would require a complicated treatment of the explicit interactions between each surface molecule and its environment, which involves both the neighboring solvent molecules and the solute surface. In addition, this model might not be more reliable than the use of eq 1. Consequently, the use of an explicit dipolar model for evaluating hydrophobic entropies is not examined in the present paper. Nevertheless, the potential-dependent correction of the solute surface area can be considered as an implicit way of improving the mutual consistency of the hydrophobic and immobilization entropies. The theoretical model described above (eq 11) was used to evaluate hydration entropies of neutral and ionic molecules at 298 K. The dipole configurations were calculated using the iterative Langevin dipoles solvation model6,12 implemented in the program ChemSol 2.0 (LD-CS2).28,46 The calculated hydration entropies, enthalpies, and free energies were defined as the total entropy, enthalpy, and free energy changes associated with the transfer of a solute from a fixed position in the gas to a fixed position in an infinitely diluted aqueous solution. This definition corresponds to the 1 M aqueous solution and 1 M gas phase standard states.47 The solute charge distribution was approximated by a set of potential-derived atom-centered charges. The electrostatic potential of the solute was calculated from the 6-31G* wave function polarized using the PCM model of Tomasi and co-workers48,49 implemented in the Gaussian 94 program.50 The solute boundary was determined using the atomic van der Waals radii that characterize the closest distance between a given atom and the center of a solvent molecule in the first solvation shell (Table 1). For C, H, N, and S atoms, these radii are identical to those used in the ChemSol 1.01.12 versions of the Langevin dipoles model.12 Small adjustments of some parameters of the LD model, including the modification of rLD parameters for O, P, F, and Cl atoms, were necessary to accommodate the newly introducted entropy term. The new features and parameters of the LD solvation model, which are implemented in the program ChemSol 2.028 are presented in detail as Supporting Information (Table 1S).

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J. Phys. Chem. B, Vol. 103, No. 46, 1999 10285

TABLE 2: Calculated and Experimental Contributions to the Hydration Free Energies of Selected Neutral Solutes (kcal/mol) ∆HESb ∆HvdWc -T∆Ssolvd solutea

LD

LD

methane ethane propane butane pentane hexane cyclohexane ethene cyclopentene cyclopentadiene benzene naphthalene methanol ethanol propanol butanol ethanediol THF 1,4-dioxane dimethyl ether diethyl ether phenol formic acid acetic acid H2CO3 acetaldehyde H2O ammonia methylamine ethylamine dimethylamine trimethylamine imidazole pyridine aniline H2S CH3SH C2H5SH thiophenol PH3 CH3PH2 (CH3)2PH (CH3)3P) CH3F CH3Cl HF HCl HNO3 H3PO4 formamide acetamide cytosine CHF2COOH CHCl2COOH

-0.1 0.0 -0.1 -0.1 -0.1 -0.1 0.0 -0.8 -0.5 -1.4 -2.4 -3.3 -7.7 -7.0 -7.1 -7.0 -11.9 -4.9 -6.9 -4.1 -2.7 -7.1 -8.5 -8.4 -9.1 -6.0 -10.7 -6.0 -4.7 -4.7 -3.2 -2.3 -10.0 -5.1 -5.4 -1.4 -2.2 -2.2 -2.2 -0.5 -1.1 -1.4 -1.5 -3.5 -2.3 -6.5 -1.6 -6.7 -14.8 -10.4 -10.4 -19.9 -8.7 -7.5

-3.4 -4.2 -4.9 -5.4 -6.0 -6.7 -5.9 -4.5 -5.6 -5.6 -6.1 -7.5 -3.3 -4.2 -4.8 -5.4 -3.9 -5.0 -5.1 -4.3 -5.9 -6.0 -3.1 -3.7 -3.0 -3.9 -2.0 -2.8 -3.8 -4.4 -4.5 -5.1 -5.0 -5.7 -6.3 -3.8 -4.5 -5.1 -6.8 -4.0 -4.6 -5.1 -5.6 -3.6 -4.3 -2.5 -3.6 -2.4 -3.5 -3.6 -4.1 -5.4 -3.3 -5.3

LD Expe 5.3 6.3 7.0 7.7 8.3 9.2 8.2 6.5 7.8 7.6 8.1 9.3 5.2 6.3 7.1 7.7 6.2 6.7 7.3 6.0 7.8 7.9 5.4 6.1 5.9 5.1 3.8 4.3 5.6 6.3 6.5 7.3 5.7 7.1 8.1 5.1 5.6 6.3 8.4 5.7 6.1 6.7 7.2 4.5 5.1 3.5 4.8 5.1 6.6 4.4 5.1 5.6 6.3 7.6

4.7 5.9 6.4 7.7 7.6 9.4 8.6 4.3 6.1 8.2 5.0 6.9 8.3 9.3 7.1 7.2 5.4 5.8h 5.9 5.3 3.6g 3.6f 5.7f 7.8 7.8f 8.8f 6.6

3.5 4.3

∆Hsolv

∆Gsolv

LD

Expe

LD

-3.5 -4.2 -4.9 -5.5 -6.1 -6.8 -6.0 -5.3 -6.1 -7.0 -8.5 -10.8 -11.0 -11.2 -11.9 -12.4 -16.1 -9.9 -11.9 -8.4 -8.6 -13.2 -11.7 -12.2 -12.1 -10.1 -12.8 -8.9 -8.6 -9.2 -7.8 -7.4 -15.2 -10.9 -11.7 -5.2 -6.6 -7.4 -9.1 -4.5 -5.7 -6.6 -7.1 -7.1 -6.7 -9.1 -5.3 -9.2 -18.4 -14.2 -14.8 -25.5 -11.9 -12.8

-2.7 -4.1 -4.8 -5.6 -5.3 -7.0 -7.3 -3.1

1.8 2.0 2.1 2.2 2.2 2.4 2.3 1.2 1.7 0.6 -0.4 -1.5 -5.8 -4.9 -4.8 -4.7 -9.8 -3.2 -4.7 -2.3 -0.8 -5.3 -6.3 -6.2 -6.2 -5.0 -9.0 -4.6 -3.0 -2.9 -1.3 -0.1 -9.5 -3.8 -3.6 -0.2 -1.1 -1.1 -0.7 1.1 0.4 0.1 0.1 -2.6 -1.6 -5.6 -0.5 -4.1 -11.8 -9.8 -9.7 -19.9 -5.6 -5.2

-7.0 -10.6 -10.1 -10.8 -13.0 -14.0 -16.7 -10.6 -10.9 -7.8h -12.9 -10.6 -12.0 -10.0g -7.9f -10.3f -12.3 -12.1f -12.0f -11.3

-3.7 -4.9

Expe 1.9 1.8 2.0 2.2 2.3 2.6 1.2 1.3 0.6 -0.9 -2.4 -5.1 -5.0 -4.8 -4.7 -9.6 -3.5 -5.1 -1.9 -1.6 -6.6 -6.7 -3.5 -6.4 -4.3 -4.6 -4.5 -4.3 -3.2 -10.3 -4.7 -4.9 -0.7 -1.2 -1.2 -2.6 0.6

-0.2 -0.6

-9.7

a All-trans conformers of alkanes, C (trans) conformers of alkyl s alcohols, all-gauche conformer of ethanediol. b The electrostatic part of the hydration enthalpy, ∆HES ) ∆HQµ + ∆Hrelax + ∆Hbulk (Table 1S). c The van der Waals part of the hydration enthalpy. d Equation 11. e The experimental hydration entropies were evaluated from the hydration free energies and enthalpies reported by Cabani et al.32 Note that the hydration enthalpies of Cabani et al. were transformed to the 1 M gas standard state (∆Hhydr (1 M) ) ∆Hhydr (Cabani) + 0.6 kcal/ mol). f From reference 63. g From reference 64. h From reference 65

3. Results and Discussion 3.1. Neutral Solutes. The calculated hydration entropies (-T∆Shydr), enthalpies (∆Hhydr), and free energies (∆Ghydr) are compared in Table 2 with the corresponding experimental data for a selected set of neutral solutes. The ∆Ghydr values for linear saturated hydrocarbons show only a very small dependence on

their length. This behavior is the consequence of the compensating behavior of the contributions to ∆Ghydr from -T∆Shydr and ∆Hhydr. These contributions grow (in absolute value) approximately by 0.7-0.9 kcal/mol per each -CH2- group. An exception to this trend occurs upon going from butane to pentane, for which experimental ∆Hsolv decreases from -8.6 to -8.3 kcal/mol. Because these ∆Hhydr were reported by different research groups (see ref 32 and references therein) they reflect most probably differences in the experimental techniques used for measuring ∆Hhydr. In view of this inconsistency, we estimate the absolute accuracy of the experimental T∆Shydr values as 1 kcal/mol. The agreement between the calculated and observed hydration entropies and enthalpies falls within this margin for saturated hydrocarbons, chloro- and fluorohydrocarbons, unsaturated alcohols, and acetic acid. A slightly better accuracy and generally similar trends were obtained for these and other classes of neutral compounds by Rashin and coworkers,17 who also parametrized the hydration entropy in terms of the accessible surface areas. Such a similarity is consistent with the use of the same methodology while employing different parametrizations. It should be noted, however, that our approach involves a smaller number of empirical parameters because our expression for the accessible surface area (eq 2) does not depend on the type of solute atoms. Note also that ref 17 evaluated hydration entropies only for neutral solutes, because it was found that models based on the solute surface area are not appropriate for estimating the hydration entropy of negatively charged solutes and also for small positively charged solutes.16 For methylamines, our calculations tend to underestimate the experimental -T∆Shydr by about 1.5 kcal/mol, but this error is small compared to the differences between the calculated and observed hydration enthalpies and free energies. Similar disagreement with the available experimental data has been reported by several research groups that used all-atom,29,51-54 dielectric continuum,10,17,55,56 and dipolar12,57,58 solvation models. Our decomposition of the hydration free energies indicates that the main source of the discrepancy between the theoretical and experimental results for methylamines lies in their enthalpic part. The deficiencies in the calculations of the enthalpic term are most probably associated with the neglect of charge-transfer to the solvent.12,54 In contrast to amines, our model overestimates the magnitude of hydration entropies of unsaturated hydrocarbons and alcohols. This deficiency is not serious considering the fact that, besides van der Waals radii, our model (eq 1-3) does not involve parameters that depend on atom types. Indeed, the calculated substituent effects on the hydration entropies seem to agree well with their experimental counterparts even for challenging molecules such as amines or alcohols. For example, our prediction that the replacement of hydrogen in the O-H bond by the methyl group in the series H2O f CH3OH f CH3OCH3 increases -T∆Shydr by 1.4 and 0.8 kcal/mol is consistent with the experimental data (Table 2). In contrast, a constant -T∆Shydr of 4.6 kcal/mol was calculated for H2O, CH3OH, and CH3OCH3 by the finite-difference free-energy perturbation method.29 3.2. Charged Solutes. The hydration entropies predicted by our model (eq 11) are compared in Table 3 with the available experimental data for atomic ions.1,30,31 Clearly, the observed hydration entropies depend strongly on the magnitude of the charge of the solvated ion. This feature, as well as the average magnitudes of the hydration entropies in the given charge group, served as the main criteria that we used to adjust the constant Ci and the function Scor in eqs 9 and 10. In addition, the function Scor was adjusted to reproduce the observed variation of the

10286 J. Phys. Chem. B, Vol. 103, No. 46, 1999 TABLE 3: Comparison of the Calculated and Experimental Hydration Entropies and Free Energies (kcal/mol) of Atomic Ions

Li+ Na+ K+ Rb+ Cs+ Ag+ Au+ FClBrIMg2+ Ca2+ Sr2+ Ba2+ V2+ Cr2+ Mn2+ Fe2+ Co2+ Ni2+ Zn2+ Cd2+ Sn2+ Hg2+ Pb2+ Y3+ La3+

rLDa (Å)

-T∆Shydrc rPb d (Å) calcd Franke Marcusf Noyesg

∆Ghydr calcdh expi

2.15 2.58 3.06 3.25 3.58 2.25 1.77 2.46 3.16 3.44 3.80 1.82 2.38 2.70 2.92 1.91 1.89 1.93 1.84 1.57 1.50 1.55 1.98 2.44 1.94 2.55 1.75 2.30

0.60 0.95 1.33 1.48 1.69 1.26 1.37 1.36 1.81 1.95 2.16 0.65 0.99 1.13 1.35 0.88 0.84 0.80 0.76 0.74 0.72 0.74 0.97 1.12 1.10 1.20 0.93 1.15

-121.0 -122.1 -98.2 -98.4 -80.9 -80.6 -75.4 -75.5 -67.2 -67.8 -114.7 -114.5 -145.7 -145.1 -104.0 -107j -77.9 -78.1j -70.4 -69.2 -62.9 -60.3 -454.2 -454.2 -377.8 -379.5 -339.0 -339.7 -314.1 -314.0 -441.8 -441 -445.5 -444.8 -438.3 -436.4 -451.4 -451.8 -483.5 -481 -491.3 -492.8 -486.7 -483.3 -433.4 -429.1 -370.4 -371.4 -437.3 -434.9 -355.4 -356.5 -858.2 -859.0 -777.9 -778.8

6.7 5.9 5.1 4.8 4.3 6.5 7.3 6.1 5.0 4.6 4.2 20.2 17.6 16.6 15.9 19.8 19.9 19.7 20.1 21.1 21.2 21.1 19.4 17.4 19.6 17.0 37.4 36.7

7.6 5.8 3.3 2.6 2.1 7.9 3.7 2.5 1.2 20.9 15.3 14.8 12.4

20.1

8.8 6.6 3.9 3.3 2.9 7.0 8.9 8.4 4.0 2.9 1.2 22.2 16.6 15.9 13.2 20.0 20.6 19.5 24.4 22.7 23.7 21.3 19.0 15.0 16.6 13.5 33.1 31.1

7.6 5.2 2.8 2.0 1.8 5.8 8.3 4.2 3.0 1.5 19.2 15.1 19.6 11.5 11.2 18.6 21.1 24.3 24.5 19.2 16.5 13.6 14.2 11.1 27.3 27.1

a Atomic radii used for LD-CS2 calculations. These radii characterize the distance between the center of the given atom and the center of the closest solvent molecule. b Ionic atomic radii derived from bond distances in crystals.66 c The contribution of ∆Shydr to ∆Ghydr at 298 K (kcal/mol). d Equation 11. e Reference 1. The data reported were estimated as the difference between the experimental partial molar entropy of an ion dissolved in aqueous solution67 and the absolute entropy of the same ion in the gas phase (calculated using the ideal gas approximation). We subtracted 9.3 e.u. from the hydration entropies given by Frank and Evans to convert these results to the standard state used by us (see the text). f Reference 30 g The entropy of charging an ion in aqueous solution.31 This entropy differs from the hydration entropy, -T∆Shydr, by the hydration entropy of an neutral atom with the same size as the ion, -T∆Sneut, i.e., -T∆Shydr ) -T∆SNoyes T∆Sneut. Here we assume that ∆Sneut ) 0. h LD-CS2 solvation model. i Reference 31. j Reference 12.

Figure 3. Relationship between the hydration entropies and free energies of atomic ions.

experimental T∆Shydr with ∆Ghydr of atomic ions (Figure 3).59 Interestingly, the slope γ ) 0.0436 derived from the experimental data is more consistent with the calculated data than with the experimental hydration entropies. This trend indicates that the experimental data are not described well by the simple

Floria´n and Warshel TABLE 4: Comparison of the Calculated and Experimental Hydration Entropies, Enthalpies, and Free Energies (kcal/ mol) of Ionic Moleculesa LD-CS1c

LD-CS2 soluteb cyclopentadieneOHMeOEtOHCOOMeCOOPhOHCO3CO32H3O+ MeOH2+ EtOH2+ CNNH4+ MeNH3+ Me2NH2+ Me3NH+ anilineH+ imidazoleH+ pyridineH+ acetonitrileH+ SHMeSEtSPhSPH4+ MePH3+ Me2PH2+ Me3PH+ CHF2COOCHCl2COOCH2NO2cytosineH+g formamideH+f acetamideH+f H2PO4HPO42PO43-

expt

-T∆Shydr ∆Hhydr ∆Ghydr ∆Ghydr ∆Ghydrc -T∆Shydrd 5.2 6.5 5.8 5.8 5.3 5.3 5.3 5.4 14.7 6.1 5.4 5.5 4.9 5.3 4.9 4.7 4.3 6.1 4.6 4.4 4.6 5.0 5.2 5.5 5.6 4.7 4.5 4.3 4.2 5.3 5.3 5.2 5.1 5.3 4.9 5.8 14.1 29.5

-72 -119 -103 -98 -82 -83 -79 -81 -280 -105 -87 -82 -81 -88 -79 -73 -67 -73 -67 -63 -72 -82 -82 -80 -77 -76 -71 -67 -63 -76 -75 -83 -71 -81 -73 -81 -261 -563

-67 -112 -97 -92 -77 -78 -73 -76 -265 -99 -82 -77 -76 -83 -74 -68 -63 -67 -63 -59 -67 -77 -76 -75 -71 -71 -67 -63 -59 -71 -69 -78 -66 -75 -68 -75 -247 -534

-65 -115 -97 -93 -81 -82 -80 -101 -83 -77 -75 -81 -74 -67 -62 -64 -62 -59 -68 -75 -74 -72 -69 -72 -68 -63 -59 -75 -71 -82 -66 -76 -69 -70 -247 -536

-66 -110 -98 -94 -80 -82 -75 -105 -87 -81 -75 -81 -73 -66 -59 -68 -64 -58 -69 -76 -76 -74 -65 -73e -63 -57 -53 -70 -66 -80 -67 -78 -70 -68 -245 -536

10.1 5.9 10.8 8.4 16.1 8.9 4.4 6.6 6.6

5.6

21.7

10.5 18.0 28.6

a The calculated and experimental values refer to T ) 298 K and 1 M concentrations of the solute in both the solution (extrapolated to infinitly diluted aqueous solution) and the gas phase. b Me, Et, and Ph denote methyl, ethyl, and phenyl groups, respectively. c Reference 12; the uncertainty of the experimental ∆Ghydr is approximately 5-7 kcal/ mol. d Reference 30. e Reference 68. f Protonation at the oxygen atom. g N3-protonated cytosine.

linear formula. In fact, the variation of the experimental -T∆Shydr with ∆Ghydr for each group of the singly and doubly charged ions would be described more accurately if a negative constant term was added to eq 3. The need for introducing this term is probably related to the ability of ions to break the structure of water beyond the first or second hydration shell.1,40,60 However, since the pertinent experimental evidence about the contribution of these effects to the hydration entropies of molecular ions is either missing or nonquantitative, we did not attempt to account for the structure-breaking properties of ions in our model. Consequently, the LD model yields -T∆Shydr values that are somewhat overestimated for larger atomic ions, whereas they are underestimated for atomic ions with small radii. The hydration entropies, enthalpies, and free energies calculated for molecular ions are presented in Table 4. The experimental entropies listed in this table were taken from the compilation of Marcus.30 Because it is difficult to estimate the absolute accuracy of these data, they were not used for calibration purposes. Nonetheless, if we disregard the experimental -T∆Shydr values of acetate and PH4+ ions that seem to be unrealistically large, the agreement between the experimental

Modified Langevin Dipoles Solvation Model

J. Phys. Chem. B, Vol. 103, No. 46, 1999 10287

TABLE 5: Changes in the Hydration Entropies Associated with Reactions of Polar or Ionic Solutes reaction type

reactantsa

productsa

(0) + (0) f (-1) + (+1) (0) + (0) f (-1) + (+1) (0) + (0) f (-1) + (+1) (0) + (0) f (-1) + (+1) (0) + (0) f (0) (0) + (-1) f (-1) (0) + (-2) f (-2) (-1) + (-1) f (-2)

phenol + Me3N acetic acid + imidazole H2O+imidazole 2H2O H2O+acetaldehyde H2O + methyl phosphateH2O + methyl phosphate2OH- + methyl phosphate-

phenolate + Me3NH acetate- + imidazole+ OH- + imidazole+ OH- + H2O+ acetal hydrate methylphosphoranemethylphosphorane2methylphosphorane2-

∆(-T∆Shydr)b +

-5.6 -1.9 1.6 5.0 -1.9 -4.0 -4.1 1.5

a The calculated hydration entropies (-T∆Shydr, T ) 298 K) for acetal hydrate, methyl phosphate-, methyl phosphate2-, methylphosphorane-, and methylphosphorane2- amount to 7.0, 5.9, 14.2, 5.7, and 13.9 kcal/mol, respectively. For the remaining compounds see Tables 2 and 4. b The contribution of the hydration entropy to the reaction free energy in aqueous solution, -T∆Shydr(products) + T∆Shydr(reactants) (kcal/mol), for T ) 298 K. Individual reactants and products are assumed to be at infinite separation.

and calculated hydration entropies is quite reasonable. The hydration entropies of large solutes were found to be smaller for charged molecules than for their neutral counterparts. For example, the calculated -T∆Shydr of protonated aniline (6.1 kcal/ mol) is notably smaller than for neutral aniline (8.1 kcal/mol). The magnitude of this difference decreases with decreasing solute size, until an effective solute radius of about 3 Å is reached. For smaller radii, the hydration entropy of singly charged solutes becomes larger than for neutral solutes of comparable size (cf. NH3 anf NH4+ or H2O and OH- in Tables 2 and 4). This behavior is consistent with the calculations based on the mean-sphere approximation (see also below).16,45 Substantial differences in the magnitude of the hydration entropies of large positive and negative ions were implied previously by calculations of the configurational entropy associated with the interaction between a single water molecule and 0, (0.5, and (1 charged spheres (the mean-field approximation).16,45 In contrast, our results suggest that positively and negatively charged solutes of comparable size have similar hydration entropies. Compare, for example, the calculated -T∆Shydr for protonated aniline (6.1 kcal/mol) and phenolate anion (5.3 kcal/mol) or protonated (5.5 kcal/mol) and deprotonated (5.8 kcal/mol) ethanol. The symmetry of the hydration entropy of positive and negative ions is a consequence of the insensitivity of the interaction energy between the charge and an orientable dipole to the sign of this charge. The mean-field approximation results16,45 indicate that such entropic symmetry can be only a rough approximation. However, it should be realized that Rashin’s and Bukatin’s evaluation of hydration entropy involved a single water molecule on a surface of a sphere embedded in a continuum environment. Although this is clearly a step in the right direction, such a model does not account quantitatively for the compensating structural effects of other water molecules (e.g., hydrogen bonding to the water molecules in the next solvation shell). Thus, it is quite possible that the mean-sphere approximation16,45 overestimated the dependence of the orientational entropy on the sign of the charge of the solute. Unfortunately, such a difference cannot be determined experimentaly since we do not have ions of opposite charges and identical size. Thus, a quantitative evaluation of the entropic effect of the charge sign should involve microscopic all-atom simulations. As far as the hydration free energies are concerned, the new LD-CS2 model yields a standard deviation from the experimental data that is identical to that from the previous version of the LD model, denoted here as LD-CS1. In addition, both LD-CS2 and LD-CS1 models perform similarly within each group of solutes, such as amines or alcohols. Thus, the conclusions of our previous work12 (and the discussion therein) about the performance of the LD-CS1 model can be extended to the LD-CS2 model.

3.3. Contributions of Hydration Entropies to Reaction Free Energies. The variation in the entropy of hydration affects both the reaction rate constants and equilibria. Although the detailed investigation of these phenomena exceeds the scope of the present paper, we felt that useful insights would be obtained by identifying the reaction types in which the changes in hydration entropy are expected to have the largest contributions (Table 5). First, we examine proton transfer reactions between neutral solutes of various polarity and size. In these reactions, the change in the hydration entropy represents a dominant part of the total reaction entropy because the contribution from gas-phase entropy changes is very small. Our calculations indicate that the hydration entropy stabilizes the charged products if the proton transfer occurs between two solutes that contain hydrophobic surfaces. On the other hand, the change in the entropy of the solvent decreases the solution acidity of small polar solutes such as HF or H2O. In particular, the -T∆Shydr term contributes as much as 5 kcal/mol to the free energy cost of the formation of the hydroxide and hydronium ions from neutral water molecules. The remaining reactions considered in Table 5 represent nucleophilic additions. In this case the molecularity of the reactants and products is not the same, and the total entropy changes will be dominated by the translational entropy. However, the results presented in Table 5 for the attack of a water molecule on the phosphorus atom in methyl phosphate monoand dianion (leading to the formation of the pentacovalent phosphorane intermediate21) indicate that hydration entropy changes are not always negligible. Consequently, these changes should be taken into account when one draws mechanistic conclusions from experimentally observed entropy contributions to activation or reaction free energies in aqueous solution.61 4. Concluding Remarks The capability of the LD-CS2 solvation model to explicitly provide hydration entropies, enthalpies, and free energies of both neutral and charged solutes can be used to predict the entropic component of chemical processes in aqueous solution. For such a theoretical prediction of reaction or activation entropies, the thermodynamic cycle presented in Figure 4 should be considered. Note that this cycle includes in its hydration entropy all of the contributions to ∆S(Aaq f Baq) that cannot be determined by standard quantum chemical programs. In particular, the hydration entropy includes the change in the entropy of the solvent and the contribution from the change in the rotational and vibrational degrees of freedom of the solute upon hydration. The latter is approximately accounted for by a constant term plus a possible linear dependence on the surface area (eq 1). However, it is not clear whether this approximation is valid for large flexible solutes, very polar solutes, unstable reaction

10288 J. Phys. Chem. B, Vol. 103, No. 46, 1999

Figure 4. Thermodynamic cycle for the evaluation of the entropic part of activation barriers and reaction free energies in 1 M aqueous solution (∆Saq(1 M)). The symbols A, B, Sgas, Str(1 M), Srot, Svib, and ∆Shydr denote solute A, solute B, total gas-phase entropy, translational entropy for a solute with 1 M concentration, and rotational, vibrational, and hydration entropy, respectively. Note that standard quantum mechanical programs typically report the translational entropies for ideal gas at 298 K, i.e., for the concentration 1/24.5 M. The transfer of the solute from 1/24.5 M to 1 M gas is associated with the change in the solute translational entropy by -4.5 cal/K/mol.

intermediates, and transition states. Here, a useful methodology for the calculation of the change in the vibrational frequencies of the solute immersed in aqueous solution in the framework of the polarized continuum model has recently become available.62 In addition, work on the implementation of the harmonic vibrational analysis in solution using the LD solvation model is in progress in our laboratory. Acknowledgment. This work was supported by the NIH Grant GM24492. Supporting Information Available: Differences in the implementation and parametrization of the LD-CS1 and LDCS2 solvation models (Table 1S). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Frank, H. S.; Evans, M. W. J. Chem. Phys. 1945, 13, 507. (2) Warshel, A.; Russell, S. T. Q. ReV. Biol. 1984, 17, 283. (3) Levy, R. M.; Gallicchio, E. Annu. ReV. Phys. Chem. 1999, 49, 531. (4) Warshel, A. Computer Modeling of Chemical Reactions in Enzymes and Solutions; John Wiley & Sons: New York, 1991. (5) Kollman, P. Chem. ReV. 1993, 93, 2395. (6) Warshel, A. J. Phys. Chem. 1979, 83, 1640. (7) Rashin, A. A. J. Phys. Chem. 1990, 94, 1725. (8) Cramer, C. J.; Truhlar, D. G. J. Am. Chem. Soc. 1991, 113, 8305. (9) Tomasi, J.; Persico, M. Chem. ReV. 1994, 94, 2027. (10) Tannor, D. J.; Marten, B.; Murphy, R.; Friesner, R. A.; Nicholls, A.; Honig, B. J. Am. Chem. Soc. 1994, 116, 11875. (11) Tawa, G. J.; Martin, R. L.; Pratt, L. R.; Russo, T. V. J. Phys. Chem. 1996, 100, 1515. (12) Floria´n, J.; Warshel, A. J. Phys. Chem. B 1997, 101, 5583. (13) Smith, B. J.; Hall, N. E. J. Comput. Chem. 1998, 19, 1482. (14) Cramer, C. J.; Truhlar, D. G. Chem. ReV. 1999, 99, 2161. (15) Rashin, A. A.; Bukatin, M. A. J. Phys. Chem. 1991, 95, 2942. (16) Rashin, A. A.; Bukatin, M. A. J. Phys. Chem. 1994, 98, 386. (17) Rashin, A. A.; Young, L.; Topol, I. A. Biophys. Chem. 1994, 51, 359. (18) Topol, I. A.; Tawa, G. J.; Burt, S. K.; Rashin, A. A. J. Phys. Chem. A 1997, 101, 10075. (19) Tawa, G. J.; Topol, I. A.; Burt, S. K.; Caldwell, R. A.; Rashin, A. A. J. Chem. Phys. 1998, 109, 4852. (20) Floria´n, J.; Warshel, A. J. Am. Chem. Soc. 1997, 119, 5473. (21) Floria´n, J.; Warshel, A. J. Phys. Chem. B 1998, 102, 719. (22) Floria´n, J.; Åqvist, J.; Warshel, A. J. Am. Chem. Soc. 1998, 120, 11524. (23) Hall, N. E.; Smith, B. J. J. Phys. Chem. A 1998, 102, 3985. (24) Floria´n, J.; Sponer, J.; Warshel, A. J. Phys. Chem. B 1999, 103, 884.

Floria´n and Warshel (25) Floria´n, J.; Strajbl, M.; Warshel, A. J. Am. Chem. Soc. 1998, 120, 7959. (26) Silverstein, K. A. T.; Haymet, A. D. J.; Dill, K. A. J. Am. Chem. Soc. 1998, 120, 3166. (27) Kolomeisky, A. B.; Widom, B. J. Chem. Soc., Faraday Trans. 1999, 112, 81. (28) Floria´n, J.; Warshel, A. ChemSol, Version 2.0; University of Southern California: Los Angeles, 1999. The program can be downloaded from the anonymous ftp server ftp.usc.edu, directory /pub/warshel/cs2, or from the web page http://apriori.usc.edu/florian. (29) Kubo, M. M.; Gallicchio, E.; Levy, R. M. J. Phys. Chem. 1997, 101, 10527. (30) Marcus, Y. Ion Properties; Marcel Dekker: New York, 1997. (31) Noyes, R. M. J. Am. Chem. Soc. 1962, 84, 513. (32) Cabani, S.; Gianni, P.; Mollica, V.; Lepori, L. J. Solution Chem. 1981, 10, 563. (33) Privalov, P. L.; Makhatadze, G. I. J. Mol. Biol. 1993, 232, 660. (34) Giesen, D. J.; Cramer, C. J.; Truhlar, D. G. J. Am. Chem. Soc. 1994, 98, 4141. (35) Nemethy, G.; Scheraga, H. A. J. Chem. Phys. 1962, 36, 3401. (36) Lee, F. S.; Chu, Z. T.; Warshel, A. J. Comput. Chem. 1993, 14, 161. (37) Luzhkov, V.; Warshel, A. J. Comput. Chem. 1992, 13, 199. (38) Powell, R. E.; Latimer, W. N. J. Chem. Phys. 1951, 19, 1139. (39) Laidler, K. Can. J. Chem. 1956, 34, 1107. (40) Marcus, Y. J. Chem. Soc., Faraday Trans. 1 1986, 82, 233. (41) Marcus, Y. Biophysical Chemistry 1994, 51, 111. (42) Frank, H. S. J. Chem. Phys. 1945, 13, 478. (43) Debye, P. J. W. Polar Molecules; Chemical Catalog Co.: New York, 1929. (44) Langevin, P. Annales de Chimie Physique 1905, 8, 70. (45) Rashin, A. A.; Bukatin, M. A. Biophys. Chem. 1994, 51, 167. (46) To distinguish among different parametrizations of the Langevin dipoles solvation model we denote the results obtained by the programs ChemSol1.x and ChemSol2.x as LD-CS1 and LD-CS2, respectively. (47) Ben-Naim, A. J. Phys. Chem. 1978, 82, 792. (48) Miertus, S.; Scrocco, E.; Tomasi, J. Chem. Phys. 1981, 55, 117. (49) Miertus, S.; Tomasi, J. Chem. Phys. 1982, 65, 239. (50) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Cioslowski, J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; D. J. Fox; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; HeadGordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94, Revision D.2; Gaussian, Inc.: Pittsburgh, PA, 1995. (51) Ding, Y.; Bernardo, D.; Krogh-Jespersen, K.; Levy, R. J. Phys. Chem. 1995, 99, 11575. (52) Meng, E. C.; Caldwell, J. W.; Kollman, P. A. J. Phys. Chem. 1996, 100, 2367. (53) Morgantini, P. Y.; Kollman, P. A. J. Am. Chem. Soc. 1995, 117, 6057. (54) Spector, T. I.; Kollman, P. A. J. Phys. Chem. B 1998, 102, 4004. (55) Orozco, M.; Jorgensen, W. L.; Luque, F. J. J. Comput. Chem. 1993, 14, 1498. (56) Marten, B.; Kim, K.; Cortis, C.; Friesner, R. A.; Murphy, R. B.; Ringnalda, M. N.; Sitkoff, D.; Honig, B. J. Phys. Chem. 1996, 100, 11775. (57) Malcolm, N. O. J.; McDouall, J. J. W. J. Mol. Struct. (THEOCHEM) 1996, 366, 1. (58) Lipinski, J.; Bartkowiak, W. J. Phys. Chem. A 1997, 101, 2159. (59) A related proportionality between T∆Shydr and ∆Hhydr (which is similar to ∆Ghydr for ions) has been shown previously in Figure 3 of ref 7. (60) Marcus, Y. J. Solution Chem. 1994, 23, 831. (61) Åqvist, J.; Kolmodin, K.; Floria´n, J.; Warshel, A. Chem. Biol. 1999, 9, R71. (62) Barone, V.; Cossi, M.; Tomasi, J. J. Comput. Chem. 1998, 19, 404. (63) Jones, F.; Arnett, E. Prog. Phys. Org. Chem. 1974, 11, 263. (64) Ben-Naim, A.; Marcus, Y. J. Chem. Phys. 1984, 81, 2016. (65) Wagman, D.; Evans, W.; Parker, V.; Schumm, R.; Halow, I.; Bailey, S.; Churney, K.; Nuttall, R. J. Phys. Chem. Ref. Data: NBS Tables of Chemical Thermodynamic Properties; American Institute of Physics: New York, 1982. (66) Pauling, L. The Nature of the Chemical Bond; Cornell University Press: Ithaca, N. Y., 1960. (67) Latimer, W. M.; Pitzer, K. S.; Smith, W. V. J. Am. Chem. Soc. 1938, 60, 1829. (68) Pearson, R. G. J. Am. Chem. Soc. 1986, 108, 6109.