Avgda. Diagonal 643, Barcelona, Spain 2 Departament de Bioquímica i Biología Molecular, Facultat de Química, Universitat de.

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1 Dispersion and Repulsion Contributions to the Solvation Free Energy: Comparison of Quantum Mechanical and Classical Approaches in the Polarizable Continuum Model CARLES CURUTCHET, 1* MODESTO OROZCO, 2,3 F. JAVIER LUQUE, 1 BENEDETTA MENNUCCI, 4 JACOPO TOMASI 4 1 Departament de Fisicoquímica, Facultat de Farmàcia, Universitat de Barcelona, Avgda. Diagonal 643, Barcelona, Spain 2 Departament de Bioquímica i Biología Molecular, Facultat de Química, Universitat de Barcelona, c/. Martí i Franqués 1, Barcelona, Spain 3 Unitat de Modelització Molecular i Bioinformàtica, Institut de Recerca Biomèdica. Parc Científic de Barcelona, c/. Josep Samitier 1, Barcelona, Spain 4 Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Via Risorgimento 35, Pisa, Italy Received 25 January 2006; Accepted 17 March Published online 17 August 2006 in Wiley InterScience ( Abstract: We report a systematic comparison of the dispersion and repulsion contributions to the free energy of solvation determined using quantum mechanical self-consistent reaction field (QM-SCRF) and classical methods. In particular, QM-SCRF computations have been performed using the dispersion and repulsion expressions developed in the framework of the integral equation formalism of the polarizable continuum model, whereas classical methods involve both empirical pairwise potential and surface-dependent approaches. Calculations have been performed for a series of aliphatic and aromatic compounds containing prototypical functional groups in four solvents: water, octanol, chloroform, and carbon tetrachloride. The analysis is focused on the dependence of the dispersion and repulsion components on the level of theory used in QM- SCRF computations, the contribution of those terms in different solvents, and the magnitude of the coupling between electrostatic and dispersion repulsion components. Finally, comparison is made between the dispersion repulsion contributions obtained from QM-SCRF calculations and the results determined from classical approaches. q 2006 Wiley Periodicals, Inc. J Comput Chem 27: , 2006 Key words: solvation; polarizable continuum model; dispersion; repulsion Introduction The passage from the study of compounds isolated in the gas phase to molecular systems in solution is a challenging goal in theoretical chemistry, which has stimulated the development of a variety of computational approaches that combine quantum mechanical (QM) or classical descriptions of the solute with continuum or discrete representations of the solvent molecules. 1 4 The use of effective Hamiltonian methods, which assume a continuum distribution of the solvent, constitutes one of the most powerful theoretical tools to explore the influence of solvation on chemical processes. 5 9 Generally, these meth- Correspondence to: F. J. Luque; fjluque@ub.edu; M. Orozco; modesto@mmb.pcb.ub.es; B. Mennucci; bene@dcci. unipi.it or J. Tomasi; tomasi@dcci.unipi.it *Present address: Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Via Risorgimento 35, Pisa, Italy Contract/grant sponsor: Spanish Ministerio de Ciencia y Tecnología; Contract/grant numbers: BIO CQT This article contains supplementary material available via the Internet at q 2006 Wiley Periodicals, Inc.

2 1770 Curutchet et al. Vol. 27, No. 15 ods decompose the free energy of solvation, DG sol, into separate, more manageable terms, which typically include electrostatic (DG ele ) and nonelectrostatic (DG n ele ) contributions. The former accounts for the work required to assemble the electric charge distribution of the solute in solution, while the latter is typically used to account for van der Waals interactions between solute and solvent, as well as for cavitation, i.e. the work required to create the cavity that accommodates the solute. This partitioning scheme has proven to be extremely valuable to develop different QM self-consistent reaction field (SCRF) continuum models able to predict DG sol with remarkable chemical accuracy Though partitioning of DG sol into components is useful from a practical point of view, the adoption of a partitioning scheme makes it necessary to introduce some assumptions in the specific definition of those terms. In particular, self-consistency between solute and solvent is generally limited to the electrostatic component, thus neglecting the mutual coupling with the solute solvent interactions associated to dispersion and repulsion. In this context, most of the research effort spent in the development of QM-SCRF models in the past decades has dealt with the calculation of the electrostatic interaction, paying attention to issues such as the placement of the boundary between solute and solvent, the treatment of the solute charge distribution, and the formalism used to evaluate the solvent reaction field Likewise, dispersion and repulsion components are determined using classical expressions, which mainly rely on the use of (i) pair potentials expressed as truncated expansions in powers of 1/r that relate suitable chemical fragments of solute and solvent molecules, 25,26 or alternatively (ii) empirical expressions related to the solvent-exposed surface of atoms ,14 16 There have been several attempts to evaluate electrostatic, dispersion, and repulsion contributions to DG sol simultaneously, so that all these forces contribute self-consistently to the solute charge redistribution that takes place upon solvation. The basic formalism in this approach was outlined by Linder, who made use of the solvent fluctuating reaction potential and its effect on the solute polarizability, and has been later implemented into different QM-SCRF continuum formalisms, thus providing a theoretical framework to examine the influence of the mutual coupling between electrostatic, dispersion, and repulsion components on the properties of the solute in liquid environments. In this study, we report a systematic analysis of the dispersion and repulsion contributions to DG sol determined self-consistently with the electrostatic component, using the formalism developed by Amovilli and Mennucci 35,37 within the framework of the polarizable continuum model (PCM). 39 To this end, calculations have been performed for a series of neutral compounds containing prototypical polar groups in four solvents: water, octanol, chloroform, and carbon tetrachloride. The analysis is focused on the dependence of the dispersion and repulsion components on the level of theory used in computations, the contributions of those terms in different solvents, and the magnitude of the coupling between electrostatic and dispersion repulsion components. Finally, a comparison is made between the dispersion repulsion contributions obtained from QM-SCRF calculations, with the results determined from computational schemes where self-consistency is not considered. Methods Since its first formulation in 1981, the PCM 39 has been largely revised and extended. Here, the integral equation formalism (IEF), the most general version of PCM, including explicitly repulsion and dispersion interactions, will be considered. In the extended IEF-PCM model, the solute, embedded in a cavity placed in the dielectric medium, is described by an effective Hamiltonian, which includes a solvent-specific operator ( ^V R ), collecting not only the electrostatic forces between solute and solvent, but also the repulsion and dispersion analogs [eq. (1)]. ½ ^H o þ ^V R Š ¼ E (1) where ^H o is the Hamiltonian of the solute in vacuo, and C denotes the wavefunction of the solute in solution. In order to obtain solvation quantities, one has to resort to a direct minimization of the free energy functional of the whole solute solvent system, which leads to an effective SCF equation formally equivalent to that of the problem in vacuo. By introducing the standard finite-basis approximation in which molecular orbitals (MO) are expressed in terms of an atomic orbital (AO) basis set, each operator becomes a matrix expressed in the same basis set, and the SCF equation to solve is given by eq. (2), where the effective Fock (or Kohn Sham) matrix is written as a sum of an in vacuo (F 0 ) term plus a collective solvent term (F sol ) arising from electrostatic (el), repulsion (rep) and dispersion (dis) interactions between solute and solvent [eq. (3)]: ðf 0 þ F sol ÞT ¼ STe (2) F sol ¼ F el þ F rep þ F dis (3) where T and e collect the MO coefficients on the AO basis and the orbital energies, respectively, and S is the overlap matrix. Since the form adopted by the electrostatic matrix F el within the PCM formalism has been explained in detail elsewhere (see for instance ref. 43), we limit ourselves to recall that a set of apparent charges is introduced to represent the electrostatic response of the solvent. These charges are placed on each of the finite elements (called tesserae), into which the solute molecular cavity surface is partitioned, and they are univocally determined by the form and shape of such a surface, by the solvent permittivity, and by the solute electrostatic potential on the same surface. These charges induce a reaction potential on the solute that, within the SCF framework, is expressed in terms of two matrices, namely, F el ¼ h el þ X el ðpþ (4) where we have explicitly indicated that X el (P) depends on the solute electron density matrix P ¼ ntt þ.

3 Dispersion and Repulsion Contributions to the Solvation Free Energy 1771 Self-Consistent QM Treatment of Dispersion and Repulsion in the PCM Model The definition of the repulsion and dispersion terms in the PCM model relies on the expressions developed by Amovilli and Mennucci, 35,37 which can be considered a generalization of the theory of intermolecular forces to the specific case of a solute embedded in a cavity surrounded by a continuum solvent. For the sake of brevity, we only report here the final QM expressions obtained in the already introduced finite-basis approximation, namely, G rep ¼ trph rep (5) G dis ¼ trp h dis þ 1 2 X disðpþ where ðh rep Þ ru ¼ bs S in c ru (7) (6) By introducing the same partition of the cavity surface used in the electrostatic term to derive the apparent charges, the repulsion and dispersion matrices [eqs. (7) (9)] can be rewritten in terms of summation over contributions associated to single tesserae (of known area a k, and whose position is given by the vector ~s k ), defined in terms of the local value of the electronic component of the solute electric field normal to the cavity surface (~E ~n) and the related potential V [eqs. (13) and (14)]. S in ru ¼ 1 X 4 k ~E ru ð~s k Þ~nð~s k Þa k (13) ½rsjtuŠ ¼ 1 X V rs ð~s k Þ ~E tu ð~s k Þ~nð~s k Þ 2 k þv tu ð~s k Þ ~E rs ð~s k ~nð~s k Þ ak : ð14þ ðh dis Þ ru ¼ X ½rsjtuŠðS 1 Þ 2 st (8) st ðx dis Þ ru ðpþ ¼ X ½rsjtuŠP st : (9) 2 In eqs. (7) (9) and are the numerical parameters which can be determined from eqs. (10) and (11), where S is the density S of the solvent relative to the density of water at 298 K, n val is the number of valence electrons in the solvent molecule, whose molecular weight is given by M S, and n s and I S are the refractive index and first ionization potential of the solvent, o M is the solute average transition energy, and finally k and k denote suitable scaling parameters (see later and refs. 35 and 37). st ¼ k S n S val M S (10) Classical Descriptions of the Dispersion and Repulsion Components When the mutual coupling between electrostatic and dispersion repulsion contributions is omitted, two main strategies have been adopted to determine the latter terms in PCM calculations. The first approach relies on the use of empirical classical pair potentials. 25,26 For a solute M composed of m fragments surrounded by solvent molecules, each formed by a given number N S of fragments of type s, the average dispersion repulsion energy can be expressed as a discrete sum over the tesserae (k) of the cavity surface [eq. (15)]. X X X G dis rep ¼ s N s a k ~A ms ð~r ms Þ~nð~s k Þ (15) k s where S is the density of the solvent, and the auxiliary functions ~A ms ð~r ms Þ are defined such that m n 2 s ¼ k 1 : (11) 4n s n s þ om I S The average transition energy of the solute can be approximated as follows: o M ¼ 1 v virt X virt e ja e j 1 v occ X occ e i> a e i (12) where e k are Fock eigenvalues, and occ and virt, respectively, denote the number of occupied and virtual orbitals falling in the energy interval ( a, þa) where a has been taken to be 1.1 hartree. 37 Test calculations performed by varying the value of a have shown that the dispersion values are little affected (generally by 5%). ~r~a ms ð~r ms Þ¼U ms ð~r ms Þg ms ð~r ms Þ (16) where g ms is a correlation function between chemical fragments m and s, which within the so-called uniform approximation is approximated as a step function, with values 1 outside the cavity and 0 inside, and U ms is the interaction energy between sites m and s, which can be expressed from empirical pair potentials. An alternative approach to the computation of repulsion and dispersion components, implemented in the framework of the MST continuum model, 14,15 is based on the use of a linear relationship with the solvent-exposed surface of atoms in the solute [eq. (17)]. G dis rep ¼ X i i A i (17) where A i denotes the contribution of atom i to the surface of the cavity, and i is the surface tension of atom i.

4 1772 Curutchet et al. Vol. 27, No. 15 Computational Details To examine the magnitude of dispersion and repulsion contributions to DG sol determined from QM-SCRF and classical calculations, a series of 22 neutral polar molecules were selected. Owing to the large expensiveness of dispersion computations, the dependence of QM-SCRF results on the level of theory was examined for a small subset of compounds (ethane, acetaldehyde, fluorobenzene). Calculations performed at both HF and DFT levels, including SVWN, 44,45 BLYP, B3LYP, 49 HCTH, and PBE 53,54 functionals, and using a variety of basis sets ranging from 6-31 þ G(d,p) to aug-cc-pvtz were compared. From the analysis of these results, B3LYP/aug-cc-pVDZ calculations were carried out for the whole series of compounds. For a given solute, all calculations were performed using the same molecular geometry, which corresponds to the gas-phase geometry optimized at the HF/6-31G(d) level. The use of fixed geometries permits to focus our analysis exclusively on the differences obtained for dispersion and repulsion contributions to DG sol, from QM-SCRF and classical methods in various solvents. Since the different methods examined here to compute dispersion and repulsion were developed making use of different cavity definitions, a careful selection of cavities is needed to make results comparable, though in all cases the standard van der Waals radii implemented in the MST model were adopted. a. QM-SCRF approach. Computations were performed (for all solvents) using a cavity built up by scaling the van der Waals radii by a factor of 1.25 (note that this cavity is used to compute both electrostatic, dispersion, and repulsion components in QM-SCRF calculations). Suitable values of k and k [k ¼ 0.063; k ¼ 0.036; see eqs. (10) and (11)] for this cavity were used 35,37 for both HF and DFT calculations. For the relative density S (in g cm 3 ), refractive index n s and ionization potential I S (in hartrees), calculations were performed using the following values: 1.00, 1.333, and 0.45 for water; 0.83, 1.429, and 0.40 for octanol; 1.48, 1.444, and 0.42 for chloroform; and 1.59, 1.459, and 0.42 for carbon tetrachloride. 55 b. Pairwise potential. Following the original work by Floris and Tomasi, 25 dispersion and repulsion are computed using the cavity obtained by adding the van der Waals radii of suitable groups in the solvent molecule to the van der Waals radii of atoms in the solute (note that the methods that rely on this approach use a different cavity for the electrostatic term). In particular, the following groups and radii (Å; in parenthesis) were considered: O (1.5) and H (1.2) for water; C (1.76), H (bonded to C; 1.2), O (1.5), and H (bonded to O; 1.2) for octanol; and C (2.82), H (1.2), and Cl (1.79) for chloroform and carbon tetrachloride. The interaction energy, U ms [eqs. (15) and (16)], was represented by using the dispersion repulsion pairwise potential implemented in the PCM model by Floris and Tomasi, 25 which exploits the empirical parameters derived by Vigné-Maeder and Claverie 56 from experimental vaporization and sublimation data. In addition, the use of other potentials developed in the context of classical discrete simulations, such as those pertaining to the AMBER (parm94), 57 OPLS- AA (all-atom), 58,59 and MM force fields (Table 1), was investigated. In AMBER and OPLS force fields, the empirical parameters used to evaluate dispersion and repulsion were adjusted to reproduce intermolecular interaction energies and certain properties (for instance, densities, and enthalpies of vaporization) of neat organic liquids. Finally, van der Waals parameters in the MM3 force field were determined by using data from crystal structures, heats of sublimation, and other experimental data of several simple compounds. c. Surface-dependent approach. These methods also use different cavities for electrostatic and dispersion repulsion terms. Particularly, in the usual MST procedure, the solute cavity used to compute dispersion and repulsion is defined from the van der Waals surface originated from the interlocking spheres assigned to the atoms in the solute. The atomic surface tensions [eq. (17)] were determined by fitting to the experimental free energy of solvation in water, octanol, chloroform, and carbon tetrachloride. 14,63 65 Results and discussion Dependence of Dispersion and Repulsion Energies on Level of Theory and Basis Set The dependence of dispersion and repulsion contributions to the solvation free energy in four selected solvents (water, octanol, chloroform, and carbon tetrachloride) was examined for a range of basis sets going from 6-31 þ G(d) to aug-cc-pvtz. Table 2 shows the dispersion (G dis ) and repulsion (G rep ) contributions computed for ethane, acetaldehyde, and fluorobenzene, at both HF and B3LYP levels in the four solvents. Calculations were performed with the 6-31 þ G(d), cc-pvdz, aug-cc-pvdz, ccpvtz, and aug-cc-pvtz basis for the three solutes. The results point out the large dependence of dispersion on the quality of the basis set. In particular, inclusion of diffuse functions is mandatory, as noted in the notable differences found in the G dis values obtained, when passing from cc-pvdz to aug-ccpvdz, and from cc-pvtz to aug-cc-pvtz. Indeed, this effect is more pronounced at the DFT level, as stated for instance in the results obtained for fluorobenzene when passing from cc-pvdz to aug-cc-pvdz regardless of the solvent: the G dis term varies by 4.1/ 4.9 kcal/mol at the B3LYP level, when compared with changes of 2.9/ 3.5 kcal/mol at the HF one. However, it seems that the largest part of the dispersion energy is recovered by the aug-cc-pvdz basis set, as further extension to aug-cc-pvtz leads to less important changes (around 1.2 kcal/mol for ethane and acetaldehyde, and 1.5 for fluorobenzene) in G dis. To further explore the influence of the basis set on G dis,we have utilized extrapolation techniques, recently suggested in the literature, to estimate the basis set convergence of correlated electronic structure calculations based on correlation-consistent polarized basis sets. According to these studies, the complete basis set (CBS) limit for the electron correlation stabilization energy, E corr,? can be extrapolated from the power formula X E corr, ¼ E? corr þax 3 X where E corr is the corresponding energy obtained with a given basis, A is a fitting parameter, and X is the highest angular momentum function in the basis (i.e., for

5 Dispersion and Repulsion Contributions to the Solvation Free Energy 1773 Table 1. Empirical Pair Potentials Used to Describe Dispersion Repulsion Interactions in this Work. 6 Claverie U ms ¼ k ms 4: eð 12:35rms msþ 0:214 ms r ms " # MM3 U ms ¼ e ms 1: e 12:0rms 6 r ms 2:25 r ms r ms OPLS 12 6 U ms ¼ 4e ms ms ms r ms r ms r 12 2 AMBER ms U ms ¼ e rms 6 ms r ms r ms p k ms ¼ k m k s, ms ¼ 2 ffiffiffiffiffiffiffiffi r m r s p e ms ¼ ffiffiffiffiffiffiffiffiffi e m e s, rms ¼ r m þ r s p e ms ¼ ffiffiffiffiffiffiffiffiffi p e m e s, ms ¼ ffiffiffiffiffiffiffiffiffiffi m s p e ms ¼ ffiffiffiffiffiffiffiffiffi e m e s, rms ¼ r m þ r s aug-cc-pvdz and aug-cc-pvtz X is two and three, respectively). Since the expression of G dis given in eq. (6) stems from a generalization of the theory of intermolecular forces to the specific case of a solute in a continuum solvent, it is reasonable to assume a similar basis set convergence for G dis [eq. (18)]. G X dis ¼ G1 dis þ AX 3 : (18) After having performed two calculations using basis sets with highest angular momentum functions X and Y, and obtained dispersion contributions G X dis and G Y dis, respectively, it can be easily shown that elimination of the linear parameter A leads to the following expression for the value of G? dis, G 1 dis ¼ GX dis X3 G Y dis Y3 X 3 Y 3 : (19) By using eqs. (18) and (19), and the dispersion values determined from B3LYP calculations with the correlation-consistent polarized basis sets (Table 2), one can estimate the corresponding CBS dispersion contributions for ethane, acetayldehyde, and fluorobenzene in water, octanol, chloroform, and carbon tetrachloride, which are given in Table 3. Comparison of the G? dis values with the results obtained from calculations performed using the aug-cc-pvdz and aug-cc-pvtz basis sets indicates that convergence to CBS limit should increase (in absolute value) the dispersion contribution by 11.6% (r ¼ 0.99) and 3.9% (r ¼ 1.00), respectively. In contrast to the preceding results, the repulsion contribution is much less sensitive to basis set variations. For instance, the results determined for fluorobenzene with cc-pvdz to aug-cc-pvdz basis differ by 0.6 and 0.9 kcal/mol at both HF and B3LYP levels. Furthermore, extension of the basis set from aug-cc-pvdz to aug-ccpvtz has no relevant effect on the G rep values. Therefore, even though B3LYP estimates are slightly more sensitive to basis set effects than do the HF ones, the results given in Table 2 clearly show that repulsion can be estimated with an uncertainty lower than 0.5 kcal/mol, even without including explicitly diffuse functions. On the basis of the trends mentioned earlier, it is not surprising that the net contribution to DG sol arising from dispersion and repulsion (G dis rep ) also exhibits a notable dependence on the basis set, which reflects the balance between the basis set modulation of dispersion and the relative basis set insensitivity of repulsion. Thus, comparison of the B3LYP G dis rep contributions reported in Table 2 with the estimated CBS values (G? dis-ref ; see Table 3) for ethane, acetaldehyde, and fluorobenzene in the four solvents shows that the latter are 17.2% (r ¼ 0.99) and 5.0% (r ¼ 1.00) larger (in absolute value) than the corresponding values determined from calculations with aug-cc-pvdz and aug-ccpvtz basis sets, respectively. We have also examined the dependence of the G dis and G rep contributions on the geometrical parameters. In particular, it seems worth to examine how their contributions can be affected by the geometrical relaxation induced upon solvation of the solute. To this end, additional computations were performed at the B3LYP/ aug-cc-pvdz level, for a series of compounds of different polarity (ethane, acetaldehyde, fluorobenzene, acetic acid, acetamide, and benzamide) in the four solvents, using the gas phase and the solvent-relaxed geometry. The results given in Table 4 indicate that the solvent-induced change in the geometrical parameters gives rise to negligible variations (generally less than 0.05 kcal/mol) in the magnitude of both dispersion and repulsion components. Until here, our attention has been focused on the influence exerted by the basis set on the QM-SCRF values of dispersion and repulsion. Now, we extend our analysis to the differences encountered by using not only HF and B3LYP levels of theory, but also a variety of DFT functionals corresponding to different exchange-correlation functionals. To this end, calculations were also performed for ethane and acetaldehyde, using the aug-cc-pvdz basis at the HF level and with BLYP, B3LYP, HCTH, PBE, and SVWN functionals. Let us note that for a given solute the differences in G rep can only arise from the differences in the solute electron density obtained at the HF and DFT levels of theory. However, changes in G dis can be induced not only by differences in the solute electron density, but also by differences in the estimated average transition energy of the solute, o M [eqs. (11) and (12)]. Table 5 shows G dis and G rep contributions computed at the different levels of theory along with their estimates of o M.Themain differences in G dis are found between the results obtained from SVWN and HF computations, the former being and kcal/mol more negative than the HF values for ethane and acetaldehyde, respectively. In all cases, the DFT G dis values are more negative than the HF ones, though BLYP, HCTH, and PBE methods yield G dis values that are around 0.4 kcal/mol lower in magnitude than those obtained from SVWN calculations. As it should be expected, the hybrid B3LYP functional falls somewhat between pure DFT functionals and HF, still giving rise to G dis values that are kcal/mol more negative than the HF ones. Inspection of the average transition energies computed at the different levels of theory (Table 5) indicates that these discrepancies

6 1774 Curutchet et al. Vol. 27, No. 15 Table 2. Dispersion (G dis ) and Repulsion (G rep ) Components of the Solvation Free Energy (kcal/mol) Determined at HF and B3LYP Levels with Various Basis Sets for Ethane, Acetaldehyde, and Fluorobenzene in Four Solvents. G dis G rep G dis rep Basis set HF B3LYP HF B3LYP HF B3LYP C 2 H 6 Water 6-31þG(d) cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz Octanol cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz Chloroform cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz Carbon tetrachloride cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz C 2 H 4 O Water 6-31þG(d) cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz Octanol cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz Chloroform cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz Carbon tetrachloride cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz C 6 H 5 F Water 6-31þG(d) cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz Octanol cc-pvdz aug-cc-pvdz (Continued)

7 Dispersion and Repulsion Contributions to the Solvation Free Energy 1775 Table 2. (Continued) G dis G rep G dis-rep Basis set HF B3LYP HF B3LYP HF B3LYP cc-pvtz aug-cc-pvtz Chloroform cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz Carbon tetrachloride cc-pvdz aug-cc-pvdz cc-pvtz aug-cc-pvtz largely reflect the differences in the values of o M estimated with DFT methods with respect to the HF value. In contrast, very similar G rep contributions are obtained from DFT calculations, they being in turn slightly larger (around kcal/mol) than the HF values. Finally, the results in Table 5 also point out that the influence exerted by the solvent on G dis and G rep is properly captured at all levels of theory. The preceding findings suggest that the magnitude of the dispersion repulsion component is largely modulated by the dispersion component, which in turn depends on the set of orbital energies determined for occupied and virtual orbitals. At this point, it is worth noting that the orbitals derived from DFT calculations not only exhibit symmetry and shape properties comparable to those derived at the HF level, but even there is a linear relationship between the orbital energies determined at the two levels of theory. 69,70 Therefore, it is reasonable to assume that the differences observed from HF and DFT results might be corrected by means of suitable adjustment of the k and k scaling parameters [eqs. (10) and (11)]. On the other hand, several studies 71,72 have shown that hybrid DFT methods are able to estimate ionization potential and electron affinities with good accuracy, leading to absolute deviations from experiments (below 0.2 ev), smaller than those found for methods based on local density approximation and gradient-corrected functionals. It seems, therefore, that computations at the B3LYP level might be a satisfactory compromise between chemical accuracy and computational cost. Nevertheless, the development of more elaborate hybrid functionals 73,74 can be valuable to obtain more reliable estimates of the dispersion and repulsion components. Overall, the preceding results indicate that the repulsion contribution to DG sol is little affected by both the flexibility of the basis sets and the level of theory used in computations. In contrast, the quality of the basis set has a notable influence on the dispersion component, which makes it necessary to use very flexible basis sets. These findings, therefore, represent a serious limitation to the application of QM-SCRF calculations of dispersion and repulsion for large solutes in solution, making it con- Table 4. Dispersion (G dis ) and Repulsion (G rep ) Components of the Solvation Free Energy (kcal/mol) of Selected Compounds in Four Solvents Determined Using the Gas Phase and the Solvent-Relaxed Geometry Obtained from B3LYP/aug-cc-pVDZ Calculations. Table 3. Dispersion (G dis ) and Dispersion þ Repulsion (G dis-rep ) Components of the Solvation Free Energy (kcal/mol) of Ethane, Acetaldehyde, and Fluorobenzene in Four Solvents Determined from Extrapolation to Complete Basis Set Limit from Two-Point Fit Using the Values Obtained from aug-cc-pvdz and aug-cc-pvtz Calculations. Solvent G dis G dis-rep C 2 H 6 C 2 H 4 O C 6 H 5 F C 2 H 6 C 2 H 4 O C 6 H 5 F Water Octanol Chloroform Carbon tetrachloride Compound G dis G rep Gas phase Solution Gas phase Solution Water C 2 H C 2 H 4 O C 6 H 5 F CH 3 COOH CH 3 CONH C 6 H 5 NH Octanol C 2 H 4 O Chloroform C 2 H 4 O Carbon tetrachloride C 2 H 4 O

8 1776 Curutchet et al. Vol. 27, No. 15 Table 5. Dispersion (G dis ) and Repulsion (G rep ) Components of the Solvation Free Energy (kcal/mol) and Average Transition Energies (hartrees) Determined at HF and DFT Levels (Using Different Exchange- Correlation Functionals) with the aug-cc-pvdz Basis Set for Ethane and Acetaldehyde in Four Solvents. Water Octanol Chloroform Carbon tetrachloride Level of theory o M G dis G rep G dis G rep G dis G rep G dis G rep C 2 H 6 HF B3LYP BLYP HCTH PBE SVWN C 2 H 4 O HF B3LYP BLYP HCTH PBE SVWN venient, if not necessary, to develop strategies to alleviate the expensiveness of these computations. At this point, let us remark that the computation of G dis rep for fluorobenzene at the B3LYP/ aug-cc-pvtz level required around 13 days CPU time, using an Intel Xeon 2800 processor. A potential approach to reduce the cost of these computations consists of the partition of the basis set into two parts, one needed to construct the density matrix, and the other, orthogonalized to the first, describing a complementary space, which enables a substantial reduction in the number of integrals that have to be computed to determine G dis. 37 Alternatively, the adoption of basis set extrapolation schemes (mentioned earlier) might be a promising approach to reduce the computational burden of these computations. However, all these approaches are not expected to reduce enough the cost of pure QM-SCRF calculations of dispersion repulsion, reinforcing the interest in using less-expensive empirically based approaches (see later). Dispersion and Repulsion Contributions in Different Solvents Inspection of the results given in Tables 2 5 shows that dispersion and repulsion contributions to DG sol are also affected by the nature of the solvent. To examine in more detail the dependence of G dis and G rep on the nature of the solvent, these contributions were determined from QM-SCRF calculations performed at the B3LYP/aug-cc-pVDZ level for a variety of neutral polar solutes corresponding to aliphatic and aromatic compounds, containing prototypical organic functional groups in water, octanol, chloroform, and carbon tetrachloride. Table 6 reports the addition of dispersion and repulsion (G dis rep ) components determined for the different solvents (separate G dis and G rep are available as Supporting Information). The comparison of the G dis, G rep, and G dis rep contributions determined in the four solvents is shown in Figure 1. Inspection of Figure 1 shows a clear distinction between the values of G dis and G rep obtained for the series of methyl- and benzene-substituted compounds, as expected from the different size and electronic structure of the methyl and benzene moieties. Thus, G dis values in water range from 8.3 to 13.2 and from 19.2 to 23.8 kcal/mol for aliphatic and aromatic compounds, Table 6. Sum of Dispersion and Repulsion Components of the Solvation Free Energy (kcal/mol) Determined from B3LYP/aug-cc-pVDZ Calculations for the Series of Selected Compounds in Four Solvents. Compound Water Octanol Chloroform Carbon tetrachloride CH 3 CH CH 3 CHO CH 3 CN CH 3 CONH CH 3 COOH CH 3 F CH 3 NH CH 3 NO CH 3 OCH CH 3 OH CH 3 SH PhCH PhCHO PhCN PhCONH PhCOOH PhF PhNH PhNO PhOCH PhOH PhSH

9 Dispersion and Repulsion Contributions to the Solvation Free Energy 1777 carbon tetrachloride and chloroform, but it is less stabilizing in octanol and water by around 9 and 17%, respectively. With regard to the repulsion component, solvation in chloroform and carbon tetrachloride gives rise to similar G rep values, which are in turn lower than those obtained in octanol and water by around 6 and 33%. As a result, the net contribution due to dispersion and repulsion (G dis rep ) becomes less stabilizing, as the permittivity of the solvent increases. Thus, compared to the G dis rep values obtained in carbon tetrachloride, the corresponding values determined in water, octanol, and chloroform are lower (in absolute value) by around 29, 12, and 3%, respectively. Let us note that these trends are also reflected in the extrapolated G? dis and G? dis rep values reported in Table 3. For instance, passing from water to carbon tetrachloride makes G? dis to enlarge its magnitude by 2.0, 2.6, and 4.4 kcal/mol for ethane, acetaldehyde, and fluorobenzene (i.e., a variation close to 20%), whereas such a difference amounts to 2.8, 3.7, and 5.5 kcal/mol for the net dispersion þ repulsion contribution (around 35% variation). Overall, these findings point out the notable dependence of the dispersion repulsion contribution on the nature of the solvent. Clearly, such a dependence is not as large as that found for the electrostatic term. However, present results point out that the solvent-dependence of the dispersion and repulsion terms is far from being negligible. Coupling Between Dispersion Repulsion and Electrostatic Components Figure 1. Representation of repulsion, dispersion, and dispersion þ repulsion components of the solvation free energy (kcal/mol) for a series of selected compounds in water (~), octanol (n), and chloroform (l), relative to the values obtained in carbon tetrachloride. respectively, and the corresponding G rep values vary from 2.1 to 4.2 and from 5.6 to 7.5 kcal/mol, respectively. Nevertheless, all the compounds included in the two congeneric series fit well the regression lines shown in Figure 1, thus indicating that they are similarly affected by the change in the solvent. Comparison of the G dis values obtained in the different solvents indicate that the dispersion contribution is very similar in Owing to the expensiveness of the QM-SCRF computation of G dis, it is customary to determine the magnitude of dispersion (and repulsion) contributions to DG sol from empirical approaches, which neglect the mutual coupling of these terms with the electrostatic component. Accordingly, self-consistency in the solute charge redistribution that takes place upon solvation is modulated only by the electrostatic interaction between solute and solvent. In order to explore the reliability of such an assumption, the electrostatic contribution (G ele ) to DG sol was determined from QM-SCRF calculations at the B3LYP/aug-ccpVDZ level performed with and without coupling, with dispersion and repulsion forces. Figure 2 shows the representation of the G ele values obtained for the whole series of neutral compounds in water, octanol, chloroform, and carbon tetrachloride (data given in Supporting Information). The results indicate that the explicit inclusion of dispersion repulsion forces has a minor effect in the magnitude of G ele, which on average are reduced (in absolute value) by around 2 4%. The influence of the coupling between electrostatic and dispersion repulsion is even lower for the dipole moment, which remains nearly unaffected, as noted in differences less than 0.5% (see Supporting Information). Overall, these results support the separate calculation of electrostatic and dispersion repulsion contributions to DG sol, as generally adopted in QM-SCRF continuum models. Comparison Between QM-SCRF and Classical Treatments of Dispersion and Repulsion Two main strategies have been adopted in the framework of the PCM model to compute dispersion repulsion, avoiding more

10 1778 Curutchet et al. Vol. 27, No. 15 purposes here, we have checked the results obtained, using parameters taken from MM3, AMBER, and OPLS force fields. The results in Table 7 indicate that there is no quantitative agreement between the G dis rep values determined from these empirical parameters and from QM-SCRF computations, as noted in rmsds of *7.7 kcal/mol (Table 7). This finding, however, is not surprising, since the van der Waals parameters defined in classical pairwise force fields were developed to maintain a subtle balance between electrostatic and van der Waals forces, using a discrete description of both solute and solvent molecules. Therefore, their transferability to the context of QM-SCRF methods, where the solvent molecules are replaced by a continuum, is not guaranteed. Indeed, the use of a solventaccessible surface like that originally adopted for Claverie formalism (see refs. 25 and 26) may not be necessarily the best Figure 2. Representation of the electrostatic component to the solvation free energy (kcal/mol) for the series of compounds in water, octanol, chloroform, and carbon tetrachloride determined from QM- SCRF B3LYP/aug-cc-pVDZ calculations with and without coupling with the dispersion and repulsion contributions. costly QM-SCRF procedures. First, the use of empirical pair potentials 25,26 based on the parameters derived by Vigné-Maeder and Claverie. 56 Second, the use of a linear relationship with the solvent-exposed surface of atoms in the solute, as implemented in the MST model, 14,63 65 where the atomic surface-tensions were derived by fitting to experimental solvation free energies. Table 7 reports the results of the statistical analysis for the comparison of the dispersion repulsion term determined using the empirical pair potentials taken from Claverie (G dis rep data available as Supporting Information) and the QM-SCRF values extrapolated to CBS limit from B3LYP/aug-cc-pVDZ calculations. Inspection of the results reported in Table 7 shows that the empirical parameters developed by Claverie provide G dis rep values that deviate notably from the QM-SCRF ones, as noted in a root-mean square deviation (rmsd) of near 8 kcal/mol and a scaling factor (c) close to 0.6. However, a more delicate feature is the fact that there is not a systematic variation between the G dis rep values determined from Claverie and QM-SCRF formalisms in the four solvents, as noted in a correlation coefficient (r) of 0.6. While there is good agreement between QM-SCRF and Claverie G dis rep values in water (rmsd of 1.9 kcal/mol and a slope of 0.87), the ability to reflect consistently the QM-SCRF G dis rep values determined for the rest of solvents is less satisfactory, as noted in the variation of the slope, which varies from 0.87 in water to 0.47 in carbon tetrachloride (Table 7). It seems, therefore, that the parameters derived by Claverie, though useful for water, do not have a universal validity for the treatment of dispersion and repulsion in different solvents. The simplicity and inexpensiveness of the pair-potential approach makes it convenient to explore the transferability of van der Waals parameters implemented in current empirical force fields developed for classical discrete simulations. For our Table 7. Statistical Analysis of the Comparison of the Dispersion Repulsion Component (G dis-rep, kcal/mol) Determined by Using Empirical Pairwise Potential and Solvent-Exposed Surface Formalisms with QM-SCRF B3LYP/aug-cc-pVDZ Values Extrapolated to Basis Set Limit for the Series of Compounds in Four Solvents. Solvent mse a rmsd a c b r b Claverie All Water Octanol Chloroform Carbon tetrachloride MM3 All Water Octanol Chloroform Carbon tetrachloride AMBER All Water Octanol Chloroform Carbon tetrachloride OPLS All Water Octanol Chloroform Carbon tetrachloride MST solvent-exposed surface All Water Octanol Chloroform Carbon tetrachloride a Mean-signed error and root-mean square deviation (kcal/mol) determined with regard to the QM-SCRF dispersion repulsion values. b Slope and regression coefficient of the linear regression equation y ¼ cx, where x and y stand, respectively, for the QM-SCRF and empirical pair-potential or surface-dependent estimates of the dispersion repulsion component.

11 Dispersion and Repulsion Contributions to the Solvation Free Energy 1779 choice, when van der Waals parameters adopted from empirical force fields are used. Clearly, more work seems necessary to investigate in detail these points. In spite of the preceding discussion, it is worth noting that there is a good correlation between the pair potential-derived G dis rep values computed using MM3, AMBER, and OPLS force fields, and the QM-SCRF ones determined for all the compounds in the four solvents (correlation coefficients ranging from 0.92 and 0.94; see Table 7). This finding suggests that a proper scaling of the G dis rep values would suffice to implement an inexpensive procedure for the evaluation of dispersion and repulsion in continuum solvation calculations. In fact, the statistical comparison points out that a consensus scaling factor close to 0.56 could be used to correct the pair potential-derived G dis rep values. However, two exceptions can be noted from inspection of the data in Table 7. First, the different behavior observed for the results in water, as determined from MM3 or AMBER (and OPLS) computations, which mainly reflects the different magnitude of the van der Waals parameters used in those force fields for the water molecule. Thus, AMBER and OPLS only assign van der Waals parameters to the oxygen atom, whose hardness (e ¼ kcal/mol; r ¼ Å (in AMBER) and ¼ Å (in OPLS)) is approximately threefold smaller than that used for the oxygen atom of water in MM3 (O: e ¼ kcal/mol; r ¼ 1.82 Å; H:e ¼ kcal/mol; r ¼ 1.60 Å). Second, the deviation observed in AMBER and OPLS for the results determined in octanol, where besides the intrinsic difference relative to the magnitude of the van der Waals parameters, the large anisotropy of the solvent molecules might also create additional problems to capture dispersion effects in the framework of the pair-potential approach. Finally, Table 7 also reports the statistical analysis for the G dis rep values determined by using the surface-dependent relationship optimized in the framework of the MST model. The results indicate that the MST-derived G dis rep values exhibit not only a good correlation with the QM-SCRF results, but that there is also satisfactory quantitative agreement, as noted in a rmsd of 3.1 kcal/mol and a slope of This finding can be attributed, at least partly, to the fact that the surface-dependent G dis rep values were obtained using atomic surface tensions optimized in the framework of MST computations performed at the B3LYP level within the IEF formalism. 24 For our purposes here, present results also support the use of a surface-dependent approach as an inexpensive strategy to evaluate dispersion and repulsion in continuum calculations. Conclusions The QM-SCRF calculation of dispersion and repulsion represents a valuable extension of the capability of current continuum models toward a more accurate description of the influence exerted by the solvent on the solute properties. Regarding the particular expressions developed in the framework of the PCM model, 35,37 the Pauli repulsion term is proportional to the fraction of solute electrons outside the cavity and to the average number of solvent valence electrons per unit volume, whereas the dispersion term depends on the solvent refractive index and ionization potential, and on solute response functions approximated in terms of the ground state electron density and of an average transition energy. The results presented here for solvation free energies show that the level of theory and, particularly, the quality of the basis set have a notable influence on the dispersion component. In contrast, the repulsion component is little affected by both the flexibility of the basis sets and the level of theory used in computations. These results, therefore, represent a serious limitation to the application of QM-SCRF calculations of dispersion and repulsion for large solutes in solution, making it convenient, if not necessary, to develop strategies to alleviate the expensiveness of these computations, but are able to reflect the differential trends of solvation on the magnitude of dispersion and repulsion. At this point, present results also support the separate calculation of electrostatic and dispersion repulsion contributions to DG sol, which is an assumption generally adopted in current QM-SCRF continuum models. On the basis of these grounds, present results suggest that van der Waals parameters taken from empirical force fields developed for classical discrete simulations can be transferred to the context of continuum solvation methods. Efforts, however, are still necessary to further refine the details of the pairpotential approach adopted here in conjunction with those parameters, such as the use of optimized scaling factors or the definition of suitable cavities adjusted to improve the quantitative prediction of dispersion and repulsion. Alternatively, our results also support the use of surface-dependent relationships together with optimized atomic surface tensions. Clearly, a detailed comparison of the G dis rep values estimated from these approaches with the QM-SCRF results is essential not only to extend continuum calculations to large molecular systems in solution, where QM-SCRF computations would not be feasible, but also to improve our understanding of the role played by dispersion and repulsion in modulating the solvent-dependent change in the solute properties. Acknowledgments We are grateful to the Centre de Supercomputació de Catalunya for computational facilities. References 1. Warshel, A. Computer Modeling of Chemical Reactions in Enzymes and Solutions; Wiley: New York, Cramer, C. J.; Truhlar, D. G. Structure and Reactivity in Aqueous Solution, Vol. 568; American Chemical Society: Washington, van Gunsteren, W. F.; Weiner, P. K.; Wilkinson, A. J., Eds. Computer Simulation in Biomolecular Systems: Theoretical and Experimental Applications, Vol. 3; Kluwer: Dordrecht, Orozco, M.; Luque, F. J. Chem Rev 2000, 100, Tomasi, J.; Persico, M. Chem Rev 1994, 94, Rivail, J. L.; Rinaldi, D. In Computational Chemistry: Reviews of Current Trends; Leszczynski, J., Ed.; World Scientific: Singapore, 1995; p.139.