Variation of Ion Polarizability from Vacuum to Hydration: Insights from

Aug 4, 2010 - ... charge equilibration force fields for modeling dynamical charges in classical molecular dynamics simulations. Brad A. Bauer , Sandee...
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Variation of Ion Polarizability from Vacuum to Hydration: Insights from Hirshfeld Partitioning Brad A. Bauer,† Timothy R. Lucas,† Alisa Krishtal,‡ Christian Van Alsenoy,‡ and Sandeep Patel*,† Department of Chemistry and Biochemistry, UniVersity of Delaware, Newark, Delaware 19716, and Department of Chemistry, UniVersity of Antwerp, UniVersiteitsplein 1, B-2610 Antwerp, Belgium ReceiVed: April 24, 2010; ReVised Manuscript ReceiVed: June 11, 2010

The results of iterative Hirshfeld partitioning on the polarizability of monovalent anions (F-, Cl-, and Br-) and Na+ in water clusters ranging from n ) 0 to n ) 25 are presented. In each case, the ions reach a limiting intrinsic polarizability in the fully hydrated state. For F-, Cl-, and Br- using B3LYP/aug-cc-pVDZ, the intrinsic polarizabilities in the condensed-phase limit are 47.2 ( 0.7%, 47.2 ( 0.3%, and 54.2 ( 0.4% of their gasphase value at the corresponding level of theory. The extent of this scaling depends on the basis set (we also consider B3LYP/aug-cc-pVTZ), but intrinsic polarizabilities are generally within 35-55% of the gas-phase value. The sodium cation is the least polarizable in the condensed-phase limit. The average intrinsic polarizability of water in these clusters decreases with the size of the cluster, which is consistent with earlier Hirshfeld analysis of intrinsic polarizabilities of pure water (Krishtal, A.; Senet, P.; Yang, M.; van Alsenoy, C. J. Chem. Phys. 2006, 125, 034312). Further analysis demonstrates that water molecules near ions in sufficiently large clusters (n ) 25) have intrinsic polarizabilities similar to those of water molecules fully coordinated in a pure aqueous cluster. The observed binodal distribution of the water intrinsic polarizability within the cluster is attributed to polarizability differences between interior and exterior water molecules. This observation is in qualitative agreement with arguments based on Pauli’s exclusion principle that suggest a reduced polarizability for condensed-phase water relative to the vacuum value. I. Introduction Electronic polarizability is an important consideration when modeling the electrostatic interactions between atomic or molecular species. Development of polarizable force fields for molecular simulations thus continues today.2,3 Commonly, polarizability is incorporated into classical models using charge equilibration (CHEQ),4-8 Drude oscillator,9-12 and induced dipole formalisms;13-15 mixed quantum mechanical/molecular mechanical (QM/MM) approaches are also viable.16 In the classical models, polarizability is often treated as an empirical parameter used in the development of the force field. Recently, the importance of incorporating polarizability into interfacial simulations of ions in water has been emphasized.17,18 Numerous studies suggest the energetic interfacial preference of larger halides in aqueous salt solutions rather than remaining fully hydrated.17,19,20 These results agree with data from surfacesensitive experimental techniques (VSFG, SHG, and XPS), although a mechanism for the process driving ions to the interface and surface remains a subject of much debate. While multiple explanations for this surface enhancement of ions have been presented,17,21 solvent polarizability, ion polarizability, and ion size are believed to have much influence over this effect. Recent work of Levin and co-workers22,23 provides further support for the importance of ion polarizability in reconciling the results of VSFG,24-30 SHG,31-34 and XPS35 experiments. While force field developers acknowledge the necessity for polarizabilty scaling (relative to the gas phase, that is, Rbulk ) [x (%)]Rgas) for simulations of condensed-phase environments, * To whom correspondence should be addressed. E-mail: [email protected]. † University of Delaware. ‡ University of Antwerp.

there exists a broad distribution of the quantitative specification of this scaling. This is due to the variability in the formalism employed (Drude oscillator versus CHEQ versus point dipole versus multipole, etc.), the parametrization criteria integrated in the fitting protocol, and the protocol to determine firstprinciples-based values of condensed-phase polarizabilities (Schropp et al.36 versus Jungwirth et al.37). When developing their fully polarizable ion-water force field, Lamoureux and Roux12 scaled ion polarizabilities (calculated using high-level ab initio calculations38) by the same factor as the polarizability of their water model. Several theoretical studies have examined the condensed-phase polarizability for ions in the alkali-metal halide series.39,40 Coker39 proposed ion polarizabilities in solution on the basis of an empirical estimation of the compression effect of the solvent (overlap of electronic density and implicated effects of Pauli exclusion). The resulting values of 1.20, 3.65, 4.96, and 7.30 Å3 for the series F- to I- range from 81% to 93% of the gas-phase values. Frediani et al.40 studied the variation of halide polarizability across the water-air interface using a quantum-mechanical continuum solvent model. This study predicted a reduced fluoride polarizability upon entering the condensed phase, qualitatively consistent with the results in ref 39 and arguments based on Pauli’s exclusion principle. However, Frediani et al. observe increased condensed-phase halide polarizability (relative to the gas-phase value) for Cl(when only electrostatic interactions are considered), Br-, and I-. Such an observation may be influenced by the neglect of dispersion interactions or orientational polarization effects at the interface in that study. Jungwirth and Tobias37 investigated the differences in chloride polarizability (relative to the gasphase value) for clusters of an ion hydrated by three and six

10.1021/jp103691w  2010 American Chemical Society Published on Web 08/04/2010

Ion Polarizability Variation from Vacuum to Hydration water molecules using Car-Parrinello molecular dynamics (CPMD). They concluded that ion polarizability is significantly reduced in the interface and bulk (representing an approximate reduction to 70% of the gas-phase value), with the extent of this interaction is dependent on the structure of the cluster. Furthermore, this reduction suggested a higher chloride polarizability than was used in their studies of NaCl solutions (RCl) 3.25 Å) at the liquid-vapor interface.41 Morita and Kato42 also considered solvent effects on chloride polarizability. Using ab initio molecular orbital calculations and a dielectric continuum model, they showed that the polarizability of Cl- is reduced by as much as 37% in water and 18% in liquid argon. Such widespread values of ion polarizabilities reflected in parametrized models, ab initio and DFT studies, and empirical estimates36 suggest there is still room for debate regarding the “correct” polarizability of ions in solution. Limited ab initio approaches exist to directly calculate the polarizability of a single ion (or more generally a single molecular entity) within a supramolecular cluster; no experimental techniques can probe this property, leaving only the properties (i.e., polarizability or dipole moment) of an isolated molecule to compare experiment and theory. Earlier attempts to compute single-species polarizability in a “condensed” environment include the work of Morita and co-workers42-44 (using a polarizable continuum solvent model), Salanne et al.45 (dipole polarizabilities of ions and molecular species using Kohn-Sham density functional theory and maximally localized Wannier functions (MLWFs)), Tobias and Jungwirth,37 Heaton et al.46 (using plane-wave DFT calculations), and Hirshfeld partitioning schemes. Postprocessing, electronic density partitioning schemes such as the Hirshfeld method47-49 decompose the polarizability of a many-body system into additive atomic contributions. The Hirshfeld approach has been applied to study the variation of polarizability as a function of the cluster size for small molecules such as water and methanol.1,50 The results of these studies demonstrate that the molecular intrinsic polarizability of these species decreases in large clusters. Given the interest in ion polarizability for the purposes of simulating salt solutions at the water-air interface (and biomolecular systems in general), studies designed to examine the variation in ion polarizability upon hydration are important. The aim of this study is to apply the Hirshfeld partitioning scheme to examine the effects of hydration on ion polarizabilty. This study complements several earlier studies (Coker,39,51 Tobias/Jungwirth,37 Dang et al.19) by applying an alternate methodology for assessing the effects of polarization of single species within the context of a condensed-phase environment. In section II we give a general overview of the iterative Hirshfeld approach from which the polarizabilities of ions and water molecules in a cluster are calculated. We also discuss our computational protocol. We present the results of iterative Hirshfeld calculations for ions in water clusters in section III.A. Primarily, we consider F-, Cl-, and Br- using the B3LYP/aug-cc-pVDZ level. However, we also consider the inclusion of additional polarizable functions within the basis set (aug-cc-pVTZ). Na+-water clusters are also considered. In addition to studying the polarizability variation of the ion with increased cluster size, we consider the effects of the ion on the polarizability of neighboring water molecules. For this analysis, we observe that the polarizability of the water is generally more sensitive to its position within a cluster than the identity of the neighboring ion. The results herein provide further reference for reduced polarizability in a condensed-phase environment and offer

J. Phys. Chem. A, Vol. 114, No. 34, 2010 8985 further insight into the appropriate range of polarizabilities for parametrizing polarizable ion models. II. Theoretical and Computational Methods A. Hirshfeld Partitioning of Polarizabilities. The coupled perturbed Hartree-Fock (CPHF) method52,53 defines each element of the polarizability tensor as

Rab ) -tr[H(a)D(b)]

(1)

where H(a) and D(b) are the dipole moment and first-order density matrices in the a and b Cartesian directions, respectively. Equivalently, this relationship can be expressed using the perturbed quantum mechanical molecular electronic density:

Rab ) -

∫ a δF(b)(r) dr

(2)

where δF(b)(r) represents the perturbed molecular electronic density in the presence of an electric field applied in the b direction. This polarizability can be decomposed into atomic polarizability contributions using the Hirshfeld weight function:

wA(b) )

FA(r)

∑ FB(r)

(3)

B

Here, FA(r) represents the electron density of free atom A, which is placed in the molecular position occupied by atom A. The summation of all free atom electron densities yields the superposition of atomic densities at the location of each atomic site of the molecule; this is defined as the promolecule. From the difference in the density of the molecule and promolecule, the effects of bonding interactions between atoms can be determined. The polarizability of a cluster can be partitioned into a sum of atomic contributions as

Rab )

∑ (RAab + aAqA(b))

(4)

A

A is the ab element of the intrinsic polarizability of where Rab atom A and aAqA(b) is the ab element of charge delocalization; here a is a Cartesian coordinate with respect to the origin of the atom (a - aA). Similarly, the polarizability tensor of molecule M can be obtained by summing over the individual atoms which comprise the molecule

RM ab )

∑ (RAab + aAqA(b))

(5)

A(M)

which can be broken down into intrinsic polarizability and charge delocalization terms:

RM,intrinsic ) ab

∑ RAab

A(M)

(6)

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RQ-trans ) ab

∑ aAqA(b)

Bauer et al.

(7)

A(M)

Defining the molecular polarizability in this form is analogous to utilizing the linear response kernel (χ) of density functional perturbation theory such that it allows for the decomposition of the electron density perturbation (δF(b)(r)) into individual responses to the perpendicular components of an externally applied field (E):

δF(r) ) -

∫ dr′ χ(r, r′)(xEx + yEy + zEz)

) δF(r)(x) + δF(r)(y) + δF(r)(z)

F(r) )

(8)

In the present work we are only concerned with diagonal terms of the polarizability tensor and, therefore, will only be considering cases in which a ) b. Taking the trace of the molecular polarizability tensors allows us to define the isotropic polarizability of a molecule as

Riso )

Rxx + Ryy + Rzz 3

(9)

The traditional Hirshfeld promolecule is representative of the neutral, isolated atoms and does not allow for relaxation of the electron density upon the formation of a molecule nor is it tractable for ions and charged molecules. These deficiencies are mitigated by implementation of an iterative approach to calculating the Hirshfeld promolecule, called the iterative Hirshfeld or Hirshfeld-I method, developed by Bultinck et al.49 Beginning from neutral atomic populations (NA0 ) ZA) and densities (FA0 (r)), one constructs the promolecule per the traditional approach of using the Hirshfeld weighting function (eq 3) and computes the classical Hirshfeld electron populations (NA1 ). It should be noted that the Hirshfeld population does not usually normalize to the isolated atomic population (NA1 * NA0 ). From the Hirshfeld population, isolated atomic densities (F1A(r)) are computed which normalize to the population:

NA1 )

∫ FA1 (r)

(10)

These new isolated atomic densities are used to generate an updated form of the promolecule from which another set of Hirshfeld populations (NA2 ) can be computed. This process is repeated until the populations of two successive iterations reach self-consistency:

∆Ai ) abs(NAi - NAi-1) ) 0

We note, before moving on, that the molecular polarizability expression applied and partitioned in this work is a fully ab initio quantity based on perturbation theory and therefore involves no point polarization approximations. Moreover, the partitioning method itself uses no parameters and is based on a physical quantityselectronic density (no data are fit at any stage of the partitioning process). At the heart of the Hirshfeld partitioning is the decomposition of the total molecular electronic density into atomic contributions:

(11)

Because it is necessary to have fractional atomic populations, calculation of atomic densities through the use of atomic condensed Fukui functions with a finite difference approach is required. The isolated atomic density of the atom/ion is produced as a weighted average of the next lowest integer and next highest integer occupancies: FANA(r) ) FAlint(NA)(r) + f Alint(NA),+(r)[NA - lint(NA)] ) FAlint(NA)(r) + [FAlint(NA)+1(r) - FAlint(NA)(r)][NA - lint(NA)] ) FAlint(NA)(r)[uint(NA) - NA] + FAuint(NA)(r)[NA - lint(NA)]

(12)

∑ FA(r) ) ∑ wA(r) F(r) A

(13)

A

where wA is the contribution (weight) of atomic electronic density from atom A to the total molecular density. From the molecular (and hence atomic) electronic densities, relations for the intrinsic and charge transfer components of the molecular polarizability follow straightforwardly.54 Thus, within the context of applying partitioning schemes for obtaining properties based on a view of atoms in molecules, the Hirshfeld-I (iterative Hirshfeld) method is a rigorous approach to describing fragment polarizabilities in large, molecular clusters. B. Computational Protocol. Configurations of ion-water clusters were generated from snapshots taken from equilibrated portions of molecular dynamics simulations of a single nonpolarizable ion, Na+, F-, Cl-, Br-, in a periodically replicated cubic cell of 215 TIP4P-QDP water molecules.55 The water model is a rigid geometry fit to the experimental gas-phase structure (HOH angle 104.52° and OH bond 0.957 Å). Ions are parametrized to match ab initio oxygen-ion distances and ion-water binding energies. The parametrized ion-water interaction model features the following binding energy and ion-oxygen distance (with the target values in parentheses): Na+, E ) -23.4 kcal/mol, R ) 2.225 Å (E ) -23.7 kcal/mol, R ) 2.228 Å); F-, E ) -27.4 kcal/mol, R ) 2.669 Å (E ) -25.9 ( 1.5 kcal/mol, R ) 2.436 Å); Cl-, E ) -17.3 kcal/ mol, R ) 3.061 Å (E ) -14.4 ( 1.2 kcal/mol, R ) 3.108 Å); Br-, E ) -13.0 kcal/mol, R ) 3.380 Å (E ) -12.7 ( 0.9 kcal/mol, R ) 3.316 Å). The target values are based on previously reported ab initio values at the MP2/6-311++G** level (for anions)56 and MP2/TZ2P level with basis set superposition error (BSSE) correction (for sodium).57 The ion-oxygen radial distribution functions (RDFs) for the ions parametrized in conjunction with this water model demonstrate good agreement with the experimental, ab initio, and molecular dynamics RDFs in terms of reproducing the position of the first peak maximum for a 1 M NaX solution. The models used here exhibit a F--O peak maximum at approximately 2.85 Å and a Cl--O maximum at 3.2 Å, which can be compared to a range of experimental values (2.6-2.9 Å for F-, 3.1-3.34 Å for Cl-) and values from Monte Carlo, molecular dynamics, and QM/ MM studies of sodium fluoride and sodium chloride (2.6 ( 0.1 and 3.2 ( 0.1, respectively).58,59 The snapshots thus generated were not minimized so as to ensure the ion remained fully hydrated (and we sampled a thermalized ensemble of structures), since it is well-known that certain ions (namely, bromide and chloride) prefer to occupy the interfacial or surface regions of the water-air interface.17,19,20 This has the added advantage that we can analyze the effect of successively adding water molecules to the hydrating sphere, without altering the structure/orientation of molecules closer to the ion. Furthermore, since we are not reporting results of a globally minimized structure, our results are based on a thermodynamic average of multiple snapshots (at least 10) for

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TABLE 1: Intrinsic and Total Gas-Phase Polarizabilities of Water and the Ions Studied in This Worka B3LYP/aug-cc-pVDZ B3LYP/aug-cc-pVTZ B3LYP/cc-pVDZ Hattig and Hess38 Coker39

H2O, intrinsic

H2O, total

7.8586 8.2412 3.8606

9.372 9.742 5.275

Na+

F-

Cl-

Br-

6.797 8.850

23.686 28.781

32.157 41.971

16.65 9.987

36.997 27.736

49.05 36.643

0.24665

a For the monovalent ions in the gas phase, the intrinsic polarizability is equivalent to the total polarizability. All values are expressed in atomic units. The experimental gas-phase polarizability of water is 9.92 au.65

each cluster size. For consistency we generated hydration spheres around ions by successively adding water molecules on the basis of their radial proximity to the ion. We performed ab initio single-point polarizability calculations on these configurations using Gaussian 03.60 Iterative Hirshfeld partitioning was performed using the STOCK program.48 Unless otherwise stated, calculations in this study involving anions were performed at the B3LYP level of theory using the aug-cc-pVDZ basis set, while those involving sodium were performed using B3LYP/cc-pVDZ. Although the B3LYP/aug-cc-pVDZ basis set does not provide as extensive polarizable functions as previous work calculating anion polarizability,38 the basis set does an adequate job representing the gas-phase water polarizability (RDZ ≈ 0.945Rgas, exp). Furthermore, since one of the primary objectives of this work is to approximate the relative reduction of gas-phase ion polarizability in the condensed-phase limit, larger system sizes will be required, which restricts the level of theory permissible under realistic computational costs. We examine the effects of basis set dependence in section III.A.2, although we acknowledge limited system sizes are considered for such analysis. III. Results A. Hydration Effects on Ion Polarizability. We consider variation of ion polarizability upon condensation in an aqueous environment. Of particular interest is the extent of reduction of ion intrinsic polarizability relative to the gas-phase value. Gasphase polarizabilities are shown in Table 1 for the ions studied in this work as well as for isolated water. It is important to note the sensitivity of the gas-phase ion polarizabilities to the basis set. More polarization functions (i.e., pVTZ basis set versus pVDZ) yield a higher gas-phase polarizability (up to a 30% increase). In this sense, we consider the polarizabilities calculated by Hattig and Hess38 to be a limiting value of the ionic polarizability in the gas phase. It is also relevant to distinguish between the total polarizability and intrinsic polarizability. In the case of an ion in vacuum the total and intrinsic polarizabilities are equivalent (null charge transfer). This, however, is not the case for an isolated water molecule. As illustrated in eq 5, the total molecular polarizability is the sum of contributions arising from the intrinsic polarizability (that which is ascribed on an atomic basis) and the charge transfer. In the case of an isolated water molecule, the charge transfer term is described completely by intramolecular charge transfer. Addition of neighboring water molecules introduces an intermolecular charge transfer term that combines with the intramolecular term to yield the total charge transfer contribution. This decomposition provides a powerful tool for developing classical force fields incorporating charge transfer effects. Though in the present work we do not consider intermolecular charge transfer explicitly, we note that the charge equilibration method is one of the few naturally extendable approaches for treating charge transfer. 1. Anion-Water Clusters Using B3LYP/aug-cc-pVDZ. Figure 1 shows the variation of ion polarizability upon hydration by a cluster of n water molecules. We consider the total ion polarizability (Figure 1a) and its decomposition into intrinsic

Figure 1. (a) Total polarizability and (b) intrinsic polarizability and (c) charge tranfer contributions calculated for F-, Cl-, and Br- as a function of the cluster size using B3LYP/aug-cc-pVDZ. Functional fits for the total and intrinsic polarizability profiles to eq 14 are included as dotted lines. The total and intrinsic polarizabilities in the limit of infinite dilution are reported for each anion in atomic units. For the charge transfer profile, a vertical offset of 1 au is imposed. For clarity, a dotted horizontal line depicting the zero charge transfer is pictured. Each polarizability contribution for an ion in a cluster with a single water, based on a structure optimized at B3LYP/aug-cc-PVDZ, is also included as an “×”.

polarizability (Figure 1b) and charge transfer (Figure 1c) contributions. In the cases of Cl- and Br-, intrinsic polarizability monotonically decreases upon hydration. This is consistent with the behavior of the intrinsic polarizability of water in pure water clusters1 and spatial limitations described by Pauli’s exclusion principle. Fluoride’s intrinsic polarizability, despite ultimately showing a reduction, is enhanced in the presence of a single water. Upon performance of an analogous calculation of fluoride and a single water molecule using a structure optimized at B3LYP/aug-cc-pVDZ, a monotonic reduction similar to the behavior of the other halides is observed. It is reasonable that the structures generated from condensed-phase simulations will not transfer well to near-vacuum environments. As we have previously mentioned, the total polarizability involves contributions arising from both the intrinsic polarizability and charge delocalization. Since each ion is comprised of a single atom, the charge delocalization does not contain an intramolecular contribution; the charge delocalization is strictly due to charge transfer to and from neighboring water molecules. We consider the magnitude of this term as a function of the water cluster size in Figure 1c. As expected, in the case of n ) 0, the charge delocalization contribution is zero since there are no neighboring atoms or molecules. For all anions studied, the charge delocalization shows a sharp increase for the first water added. This quantity is very sensitive to the orientation of the water relative to the ion as suggested by the large standard

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Figure 2. Intrinsic (closed circles) and total (open circles) polarizabilities relative to the gas-phase polarizability for (a) F- (b) Cl-, and (c) Br- as a function of the cluster size using B3LYP/aug-cc-pVDZ. The dotted line represents an empirical fit of eq 15 to the intrinsic polarizability data; the value presented represents the intrinsic polarizability at infinite dilution. As in Figure 1, the total intrinsic ion polarizabilities (relative to the isolated ion polarizability) in optimized clusters with one water are included.

deviations for the first several water molecules added. By the point at which the ion becomes saturated by a first hydration shell, the charge delocalization contribution is essentially zero, suggesting no net charge transfer to or from neighboring water molecules occurs. This observation explains the agreement between the total ion polarizability and the intrinsic ion polarizability in the condensed-phase limit. This convergence is explicitly shown in Figure 2, where the polarizabilities are expressed relative to the gas-phase value (noting Rtotal, gas ) Rintrinsic, gas for monatomic ions). Since the intrinsic polarizability profiles decay exponentially, they were fit to the functional form

Rintrinsic ) A + B exp(-CnH2O)

(14)

for water clusters in the range n ) 1-25. Here, A, B, and C are empirical constants. Such an approach allows for extrapolation of intrinsic polarizability to an infinitely large cluster (we will consider this the condensed-phase limit). In the limit of n f ∞, the equation reduces to Rintrinsic,∞ ) A, suggesting the fitted A value is an appropriate representation of the condensed-phase intrinsic polarizability limit for each ion at a given basis set. We note that the condensed-phase intrinsic polarizabilities for the ions in Figure 1 are not generally applicable since they are not calculated from basis sets capable of capturing the full extent of ion polarizability in the gas phase. It therefore may be more informative to consider the intrinsic polarizability relative to the gas-phase value. This analysis is presented in Figure 2. A similar functional fit was implemented to determine the limit of the intrinsic polarizability scaling:

R ) Ar + Br exp(-CrnH2O) Rgas

(15)

In the limit of n f ∞, the function will reduce to the parameter Ar, which represents the intrinsic polarizability at infinite dilution relative to the gas-phase value. At the B3LYP/aug-cc-pVDZ level, the scaling (percentage of the gas-phase polarizability) is estimated to be 47.2 ( 0.7%, 47.2 ( 0.4%, and 54.2 ( 0.4%

Figure 3. Dependence of the results on the basis set: absolute value of the intrinsic polarizability calculated using B3LYP/aug-cc-pVDZ and B3LYP/aug-cc-pVTZ for each (a) F-, (b) Cl-, and (c) Br-. Panels d, e, and f are the analogous plots considering the intrinsic polarizability relative to the gas-phase value.

for F-, Cl-, and Br-, respectively. These scalings, when applied to current upper limits in gas-phase polarizability (namely, those of Hattig and Hess38), give values of 1.16, 2.59, and 3.94 Å3 for F-, Cl-, and Br-, respectively. These values show considerable reduction compared to the estimates of Coker40 (up to 1 Å3). Similarly, these estimates suggest lower condensed-phase ion polarizabilities than those used by Lamoureux and Roux (1.786, 3.969, and 5.262 Å3).12 2. Anion-Water Cluster Basis Set Dependence. Just as the gas-phase ion polarizability (Table 1) depends on the basis set, one may anticipate the reduction from the the gas-phase value to also depend on the basis set. To explore this, we performed calculations analogous to those in the previous section using B3LYP/aug-cc-pVTZ, which allows us to assess the effect of additional polarizable functions. A comparison of the profiles for absolute intrinsic polarizability and intrinsic polarizability relative to the gas-phase value is shown in Figure 3 for hydration of the anions; the limiting values of intrinsic polarizability for both basis sets determined from a functional fit to eq 15 are included in Table 2. The intrinsic polarizability profiles based on the B3LYP/aug-cc-pVDZ and B3LYP/aug-cc-pVTZ basis sets are qualitatively similar for all anions. In all cases the predicted intrinsic polarizabilities of ions at the condensed-phase limit are higher for the B3LYP/aug-cc-pVDZ data. However, the intrinsic polarizabilities in the condensed-phase limit are remarkably similar for the different basis sets. This convergence may suggest the basis sets implemented in this study are appropriate for condensed-phase analysis, although the gas-phase ion polarizabilities are still underpredicted when compared to the calculations of Hattig and Hess.38 Due to the agreement in the limiting behavior (considering the absolute intrinsic polarizabilities) and the significant differences in the gas-phase polarizabilities between the two sets, the enhanced reduction of condensed-phase intrinsic polarizability is anticipated for the B3LYP/aug-cc-pVTZ set. This reduction would be further enhanced if the absolute intrinsic polarizability limits were made relative to gas-phase values calculated with more extensive basis sets. 3. Sodium-Water Clusters. We now shift our attention to the polarizability (total, as well as intrinsic and charge transfer contributions) of a cation (Na+) upon hydration (Figure 4). As is anticipated for a small, positively charged species, Na+ exhibits a nearly negligible (0.04 Å3) polarizability in the gas phase, as calculated using B3LYP/cc-pVDZ. Previous ab initio

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TABLE 2: Extrapolated Intrinsic Polarizabilities for Ions in an Infinitely Dilute Water Clustera FB3LYP/aug-cc-pVDZ B3LYP/aug-cc-pVTZ Lamoureux and Roux12 Frediani et al.40 Morita and Kato42 a

Cl-

Br-

Rintrinsic,∞

Rintrinsic,∞/Rgas

Rintrinsic,∞

Rintrinsic,∞/Rgas

Rintrinsic,∞

Rintrinsic,∞/Rgas

3.21(5) 3.15(11) 12.05 13.83

0.472(7) 0.356(12) 0.724 0.60

11.18(9) 10.85(25) 26.78 38.30 19.677(485)

0.472(4) 0.382(9) 0.724 0.90 0.634(16)

17.43(12) 17.24(32) 35.51 52.93

0.542(4) 0.411(8) 0.724 0.96

The fractional scaling (Rintrinsic, ∞/Rgas) of the intrinsic polarizability is also presented for each ion and basis set.

Figure 4. (a) Total polarizability, (b) intrinsic polarizability, and (c) charge transfer contributions to the polarizability of Na+ as a function of the cluster size. Data were calculated at the B3LYP/cc-pVDZ level. The polarizability contribution of an optimized cluster with one water is included as an “×”.

estimates of Na+ performed using the self-consistent field method predict a gas-phase polarizability of 0.157 Å3,61 which agrees well with the assumed value of 0.158 Å3.51 Much as the fluoride ion has enhanced intrinsic polarizability (relative to the gas-phase value) when in a cluster with a single water, Na+ shows a similar enhancement of the intrinsic polarizability. This enhancement, however, persists until the sodium is fully coordinated by a first solvation shell. For larger clusters, the sodium intrinsic polarizability approaches zero. We also note that the intrinsic polarizabilities observed for sodium (even when at maximum enhancement) are still below 0.7 atomic units (approximately 0.1 Å3). Such an intrinsic polarizability is an order of magnitude lower than that of gas-phase water and is even less significant when compared to the magnitude of anion intrinsic polarizabilities. This result supports the treatment of sodium as a nonpolarizable entity in molecular dynamics simulations. The charge delocalization contribution for the sodium ion also approaches zero in the condensed-phase limit. Within the uncertainty of the data, the charge delocalization contribution is negligible for most clusters studied (particularly n > 6). B. Effects on Water Polarizability. In this section we will examine the effects ions have on the intrinsic polarizability of surrounding water molecules. In most cases, the presence of several water molecules may convolute the analysis to the extent that polarizability effects cannot easily be attributed to the presence of the ion or neighboring water molecules. Such difficulties do not discourage such an analysis, however, since it is still important to consider the effect cluster size has on water molecules in the presence of an ion. The first analysis we will consider is intrinsic polarizability of the water molecule closest to the ion as a function of the cluster size (Figure 5). The closest molecule was selected as the water featuring the

Figure 5. Intrinsic polarizability of the closest water molecule to (a) F-, (b) Cl-, and (c) Br- as a function of the number of water molecules in the cluster. The profile of the average intrinsic polarizability in each cluster is also included. The intrinsic polarizability of a gas-phase water molecule is denoted as a horizontal dotted line.

shortest oxygen-ion distance within a given cluster; this is generally the molecule in cluster size n ) 1. For comparison, we also include the average intrinsic polarizability in each cluster. Clearly, the data for nH2O ) 1 have effects arising solely from the ion interacting with the water molecule. When the intrinsic polarizability of the water in the n ) 1 clusters is considered, it is observed that the intrinsic polarizability is enhanced over the gas-phase polarizability value (represented as the horizontal dotted line) for each halide studied. The values of the enhancement of intrinsic polarizability are summarized in Table 3. It is clear that this effect is greatest for the smaller, more electronegative ions; enhancements (%) for the series fluoride through bromide decrease as 13.0 ( 0.4%, 7 ( 1%, and 4 ( 1%, respectively. We acknowledge a recent CPMD study by Guardia et al.62 that reported fluoride’s tendency to overpolarize water molecules within the first hydration shell such that the molecular dipole is enhanced on the order of 0.1 D. Furthermore, they observe that the first hydration shell surrounding chloride, bromide, and iodide is also polarized, albeit to a reduced molecular dipole. For the smallest clusters (n ) 1 through n ) 8) the average cluster intrinsic polarizability and the intrinsic polarizability of the molecule closest to the ion are virtually indistinguishable. This result suggests the intrinsic polarizabilities of molecules composing (essentially) the first solvation shell of an ion are similar (on average). The intrinsic polarizability of the closest molecule and the average of all molecules begin to diverge in the larger clusters. This is attributed to the onset of the formation of the second hydration shell, which will impose further spatial restrictions on the interior (first shell) water molecules. The introduction of a second solvation shell introduces additional hydrogen-bonding partners for the interior molecules, which has been previously shown to

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TABLE 3: Analysis of Intrinsic Polarizabilities and Intrinsic Polarizability plus Intramolecular (i + i) Charge Transfer Contributions for Watera -

F ClBr-

Rintrinsic, first/Rintrinsic, gas

Ri+i, first/Rgas

Rintrinsic, avg/Rintrinsic, gas

Ri+i, avg/Rgas

Rintrinsic, clos/Rintrinsic, gas

Ri+i, clos/Rgas

Rintrinsic, clos/Rintrinsic, DDAA

1.130(4) 1.07(1) 1.04(1)

1.093(4) 1.04(1) 1.02(1)

0.71(1) 0.707(8) 0.717(9)

0.70(1) 0.71(1) 0.71(1)

0.51(5) 0.52(3) 0.53(4)

0.50(5) 0.51(4) 0.52(4)

1.01(9) 1.03(7) 1.05(8)

a The intrinsic polarizability of the first water added to a cluster, the intrinsic polarizability of the water molecule closest to an ion in a cluster (n ) 25), and the average intrinsic polarizability of all water molecules in a cluster (n ) 25) (all relative to the intrinsic gas-phase polarizability of water) are included. Analogous ratios considering the combined contribution of intrinsic polarizability and intramolecular charge transfer are expressed relative to the total gas-phase polarizability of water. The intrinsic polarizability of the water closest to an ion in a cluster (n ) 25) relative to the intrinsic polarizability of a fully coordinated water (Rintrinsic, DDAA ) 3.97 au) in a pure water cluster is included in the last column.

Figure 6. Distribution of intrinsic polarizabilities for (a) F-, (b) Cl-, and (c) Br-. The population of all water molecules is represented as a dashed line, whereas the populations of water in the interior and exterior of the cluster are represented as bars. The distinction of an interior molecule is based on an ion-oxygen distance less than the position of the first minimum in the model’s RDF; an exterior molecule has a distance greater than this position.

Figure 7. Intrinsic polarizability of water molecules as a function of their radial distance from (a) F-, (b) Cl-, and (c) Br-.

reduce water’s intrinsic polarizability.1 Since the exterior water molecules are less constrained spatially and have (on average) more unsatisfied hydrogen bonds, they contribute to a higher average cluster intrinsic polarizability. Figures 6 and 7 further support the claim that there is a significant difference in the intrinsic polarizability between the interior and exterior water molecules. Figure 6 considers the intrinsic polarizability dis-

tributions for all water molecules and distinguishes between those within interior and exterior molecules. For this analysis, only clusters of size n ) 20 were considered. The distinction between interior and exterior was based on the ion-oxygen distance relative to the position of the first minimum in the RDF; water molecules with a distance less than the minimum were considered interior, whereas water molecules at distances greater than the first minimum were considered exterior. The intrinsic polarizability distribution for all water molecules within the cluster shows a bimodal distribution in which a smaller peak (or shoulder) appears for molecules of lower polarizability. Distinguishing between molecules in the first and second solvation shells clearly shows that the molecules in the first solvation shell represent the majority of the low intrinsic polarizability population. This result is consistent for all ion-water clusters studied. It is also interesting to observe the width of the intrinsic polarizability distribution is essentially unaltered by the identity of the ion solvated. This is further suggested by the negligible differences in the average scaled intrinsic polarizability of each cluster and the average scaled intrinsic polarizability for the closest water molecule to the ion (Table 3). On the basis of these results, we suggest the intrinsic polarizability of water in hydrated environments is influenced by the surrounding water molecules more than neighboring ions (in dilute concentrations at least). This observation agrees with first-principle density functional calculations by Krekeler et al.63 that showed, as the number of water molecules in a water-ion cluster was increased, the molecular polarization was determined by water-water interactions rather than water-ion interactions. Similar results have also recently been reported by Zhao et al. using the quantum theory of atoms in molecules (QTAIM).64 They observed that hydrogen bonding with water molecules in the second solvation shell had a greater effect on the magnitude of the polarizability of the first solvation shell waters than did their interaction with the ion. To further assess the influence of an ion on neighboring water molecules, we take the ratio of the intrinsic polarizability of a water molecule closest to an ion (near the condensed-phase limit, n ) 25) to the intrinsic polarizability of a fully hydrated water molecule in a pure water cluster. The latter quantity is taken as 3.97 au, the average value for a water molecule with both donor and acceptor bonds satisfied.1 For all ion clusters, this ratio is unity within the error associated with this quantity (1.01 ( 0.09, 1.03 ( 0.07, and 1.05 ( 0.08 for F-, Cl-, and Br-, respectively). This result suggests there is no discernible difference in intrinsic polarizability for a fully solvated water near an ion versus a fully solvated water molecule in a pure aqueous environment. We could alternatively consider the intrinsic polarizability of the water molecules as a function of their distance from the solvated ion (as is shown in Figure 7). As in the analysis for Figure 6, we consider only clusters of n ) 20 water molecules. These plots demonstrate nearly linear dependence of the intrinsic polarizability on the distance from the ion. This result is somewhat deceptive, however. The limited cluster size does not

Ion Polarizability Variation from Vacuum to Hydration

Figure 8. (a) Profile of the intrinsic polarizability of the closest water to Na+ as a function of the cluster size and profile of the average polarizability of all the waters in a cluster as a function of the cluster size. (b) Distribution of intrinsic water polarizabilities for all water molecules (black dashed lines), interior water molecules (red bars), and exterior water molecules.

allow molecules in the second solvation shell to be completely hydrated, but rather partially hydrated (that is, like molecules at the liquid-vapor interface). Such treatment allows these molecules to adopt higher intrinsic polarizabilities than they otherwise would have if they were completely hydrated. If the system were substantially larger, we would expect molecules within the first several solvation shells (those that were completely hydrated) to reach a condensed-phase limit (as may be approached in the case of the closest molecules to the ion in the current system). It is important to note that the intrinsic polarizability of the outer water molecules remains well below the gas-phase value. This is expected on the basis of the effects the of cluster size on the intrinsic polarizability.1 Finally, we address the effects of sodium on the water polarizability. Figure 8a demonstrates the intrinsic polarizability of the closest water molecule to the ion and the average intrinsic polarizability of all water molecules within the cluster for a given size. Similar to the case of the anions, water intrinsic polarizability is initially enhanced in the vicinity of the sodium. Furthermore, the average over all water molecules and the intrinsic polarizability of the closest molecule to the ion are closely matched until they ultimately diverge at cluster sizes n > 15. Figure 8b displays the distribution of intrinsic polarizabilities for waters surrounding the sodium ion. Again, a bimodal distribution is observed which can be attributed to the differences in intrinsic polarizabilities for interior molecules (which take on lower values) and exterior molecules (which take on higher values). We note that the width of the distribution is reduced from those of distributions involving anions. However, we attribute such effects to the cc-pVDZ basis set, which predicts a significantly lower gas-phase polarizability for water. IV. Discussion and Conclusions We have reported the results of the Hirshfeld partitioning scheme applied to calculating the polarizability profile of monovalent ions in water clusters. Consistent with theoretical arguments, polarizability of anions is reduced in condensedphase environments relative to the gas phase. Although the nature of the change in polarizability reduction as a function of the cluster size demonstrates a dependence on the identity of the ion and on the basis set used, results from this analysis consistently show reductions ranging from 0.35 to 0.55 times

J. Phys. Chem. A, Vol. 114, No. 34, 2010 8991 the gas-phase anion polarizability. While the choice of basis sets was necessarily restricted in our investigation to sample reasonably sized clusters, our results lend some insight into force field development. If we are to consider calculations by Hattig and Hess38 as true upper bounds for the gas-phase polarizability of anions and consider the range of polarizability reductions calculated here to be transferable to higher levels of theory and basis sets, then we can estimate absolute values for condensedphase ion polarizability. Upon such an exercise, we find 0.88-1.16, 1.51-2.59, and 2.99-3.94 Å3 to be appropriate polarizability ranges for F-, Cl-, and Br-, respectively, for the purposes of condensed-phase simulation. These ranges are comparable (albeit reduced by as much as 1.5 Å3) to the polarizable ion force fields of Lamoureux and Roux12 and Jungwirth and Tobias.41 Parametrizing polarizable force fields using such target polarizabilities is likely to reduce the amount of surface enhancement observed in interfacial simulations as suggested by Figure 4 of ref 17. With respect to the water polarizabilities calculated in this work, we reported average condensed-phase limit intrinsic polarizabilities as low as 51 ( 5% of the gas-phase value. Water molecules taking on such low intrinsic polarizabilities relative to the gas-phase value represented those in the interior of the cluster, where spatial constraints are maximized. Furthermore, within the interior of the cluster, the potential for hydrogen bond formation is greater than near the interface, which is prone to dangling hydrogen bonds. Krishtal et al.1 have previously reported the influence of hydrogen bonding on low intrinsic polarizabilities of water. We note, however, that this anticipated intrinsic polarizability scaling for a condensed environment (∼50% of the gas-phase value) is significantly lower than values predicted by Schropp and Tavan (68%)36 and Morita (91-93%),43 as well as the scaling utilized in empirical water models such as TIP4P-FQ (76%)6 and SWM4-DP (72.4%).12 Although the average intrinsic polarizability scaling for all waters in a cluster (∼70%) agrees well within the range of these values, this comparison is not direct since “interfacial” waters are included in the average and are likely to increase the intrinsic polarizability calculated for the cluster. We have primarily investigated the behavior of the intrinsic polarizability of water relative to its intrinsic polarizability in vacuum, neglecting the contributions from charge delocalization. In the Hirshfeld partitioning formalism used in this study, charge delocalization can be further decomposed into inter- and intramolecular contributions1 as

∑ aAq(b)a ) ∑ (aA - aM)q(b)a + aMQ(b)M

A(M)

A(M)

) CTintra + CTinter

(16)

Here the intramolecular charge transfer term is derived from the charge delocalization at each atomic position (aA) relative to the geometrical center of the molecule (aM), and the intermolecular delocalization includes the net induced charge on a molecule from an external field applied in the b Cartesian direction. Within the context of the parametrization of polarizable force fields that restrict charge transfer to within a molecule, treatment of intramolecular charge transfer contributions would also be of interest to this study. Considering the total contribution of the intrinsic polarizability and intramolecular charge transfer (Table 3), we note that the general conclusions drawn from intrinsic water polarizability analysis are still appropriate. In the case of the combined contribution, we took polarizability relative to the total gas-phase polarizability (which considers both intrinsic and intramolecular charge delocalization contributions). It is worthwhile to mention that for monatomic ions the

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charge transfer contribution is solely attributed to intermolecular interactions. Therefore, inclusion of charge transfer contributions would not be meaningful for monatomic ions for applications to polarizable force fields (specifically those in which charge is conserved within each molecular entity). However, in the gas phase and in the condensed-phase limit such contributions approach zero. This results in equivalent total and intrinsic polarizabilities for monatomic ions in the gas phase and in the condensed-phase limit. The goal of this study was to further the understanding of the nature of ion polarizability upon hydration. Since our initial aim is toward the parametrization of water-ion force fields, our study presents important clues for appropriate ion polarizabilities in aqueous environments. It may be insightful to study the nature of ion polarizability upon condensation in nonaqueous environments. Certainly, one could expect the response of ion polarizability in a hydrophobic or inert environment to differ considerably from that in water.42 Moreover, although we sought a single value for a condensed-phase ion polarizability, a transferable ion model that self-consistently modulates its polarizability in response to the local environment would be optimal for studying liquid-vapor or water-lipid interfaces. Certainly this work demonstrates that polarizability is a dynamic property that is influenced by the local environment. Such considerations may be important when developing the next generation of polarizable force fields. Acknowledgment. A.K. acknowledges the Fund for Scientific ResearchsFlanders (FWO) for a postdoctoral position. References and Notes (1) Krishtal, A.; Senet, P.; Yang, M.; van Alsenoy, C. J. Chem. Phys. 2006, 125, 034312. (2) Illingworth, C. J.; Domene, C. Proc. R. Soc. London, A 2009, 465, 1701–1716. (3) Lopes, P. E. M.; Roux, B.; MacKerell, A. D., Jr. Theor. Chem. Acc. 2009, 124, 11–28. (4) Rick, S. W.; Berne, B. J. J. Am. Chem. Soc. 1996, 118, 672. (5) Rick, S. W.; Stuart, S. J.; Bader, J. S.; Berne, B. J. J. Mol. Liq. 1995, 65/66, 31. (6) Rick, S. W.; Stuart, S. J.; Berne, B. J. J. Chem. Phys. 1994, 101, 6141. (7) Patel, S.; Brooks, C. L., III J. Comput. Chem. 2004, 25, 1. (8) Patel, S.; Brooks, C. L., III J. Comput. Chem. 2004, 25, 1504. (9) Lamoureux, G.; MacKerell, A. D., Jr.; Roux, B. J. Chem. Phys. 2003, 119, 5185. (10) Lamoureux, G.; Roux, B. J. Chem. Phys. 2003, 119, 3025. (11) Anisimov, V. M.; Lamoureux, G.; Vorobyov, I. V.; Huang, N.; Roux, B.; Alexander, D.; MacKerell, A. D., Jr. J. Chem. Theory Comput. 2005, 1, 153. (12) Lamoureux, G.; Roux, B. J. Phys. Chem. B 2006, 110, 3308. (13) Ren, P.; Ponder, J. W. J. Comput. Chem. 2002, 23, 1497. (14) Ren, P.; Ponder, J. W. J. Phys. Chem. B 2003, 107, 5933. (15) Caldwell, J. W.; Kollman, P. A. J. Phys. Chem. 1995, 99, 6208. (16) Lin, H.; Truhlar, D. G. Theor. Chem. Acc. 2007, 117, 185–199. (17) Vrbka, L.; Mucha, M.; Minofar, B.; Jungwirth, P.; Brown, E. C.; Tobias, D. J. Curr. Opin. Colloid Interface Sci. 2004, 9, 67. (18) Warren, G. L.; Patel, S. J. Phys. Chem. C 2008, 112, 7455. (19) Dang, L. X.; Rice, J. E.; Caldwell, J.; Kollman, P. A. J. Am. Chem. Soc. 1991, 113, 2481–2486. (20) Perera, L.; Berkowitz, M. L. J. Chem. Phys. 1991, 95, 1954. (21) Eggimann, B. L.; Siepmann, J. I. J. Phys. Chem. C 2008, 112, 210. (22) Levin, Y. Phys. ReV. Lett. 2009, 102, 147803. (23) Levin, Y.; dos Santos, A. P.; Diehl, A. Phys. ReV. Lett. 2009, 103, 257802. (24) Chen, Z.; Ward, R.; Tian, Y.; Baldelli, S.; Opdahl, A.; Shen, Y. R.; Somorjai, G. A. J. Am. Chem. Soc. 2000, 122, 10615–10620. (25) Liu, W.-T.; Zhang, L.; Shen, Y. R. J. Chem. Phys. 2006, 125, 144711. (26) Liu, D.; Ma, G.; Levering, L. M.; Allen, H. C. J. Phys. Chem. B 2004, 108, 2252–2260.

Bauer et al. (27) Richmond, G. L. Chem. ReV. 2002, 102, 2693–2724. (28) Walker, D. S.; Richmond, G. L. J. Am. Chem. Soc. 2007, 129, 9446– 9451. (29) Walker, D. S.; Richmond, G. L. J. Phys. Chem. C 2007, 111, 8321– 8330. (30) Schrodle, S.; Richmond, G. L. Appl. Spectrosc. 2002, 62, 389. (31) Petersen, P. B.; Saykally, R. J. Chem. Phys. Lett. 2004, 397, 51. (32) Petersen, P. B.; Saykally, R. J.; Mucha, M.; Jungwirth, P. J. Phys. Chem. B 2005, 109, 10915. (33) Petersen, P. B.; Saykally, R. J. Annu. ReV. Phys. Chem. 2006, 57, 333. (34) Petersen, P. B.; Saykally, R. J. J. Phys. Chem. B 2006, 110, 14060. (35) Ghosal, S.; Hemminger, J. C.; Bluhm, H.; Mun, B. S.; Hebenstreit, E. L. D.; Ketteler, G.; Ogletree, D. F.; Requejo, F. G.; Salmeron, M. Science 2005, 307, 563. (36) Schropp, B.; Tavan, P. J. Phys. Chem. B 2008, 112, 6233. (37) Jungwirth, P.; Tobias, D. J. Phys. Chem. A 2002, 106, 379–383. (38) Hattig, C.; Hess, B. A. J. Chem. Phys. 1998, 108, 3863. (39) Coker, H. J. Phys. Chem. 1976, 80, 2084. (40) Frediani, L.; Mennucci, B.; Cammi, R. J. Phys. Chem. B 2004, 108, 13796–13806. (41) Jungwirth, P.; Tobias, D. J. Phys. Chem. B 2000, 104, 7702–7706. (42) Morita, A.; Kato, S. J. Chem. Phys. 1999, 110, 11987. (43) Morita, A. J. Comput. Chem. 2002, 23, 1466. (44) Ishida, T.; Morita, A. J. Chem. Phys. 2006, 125, 074112. (45) Salanne, M.; Vuilleumier, R.; Madden, P. A.; Simon, C.; Turq, P.; Guillot, B. J. Phys.: Condens. Matter 2008, 20, 494207. (46) Heaton, R. J.; Madden, P. A.; Clark, S. J.; Jahn, S. J. Chem. Phys. 2006, 125, 144104. (47) Hirshfeld, F. L. Theor. Chim. Acta 1977, 44, 129. (48) Rousseau, B.; Peeters, A.; van Alsenoy, C. Chem. Phys. Lett. 2000, 324, 189. (49) Bultinck, P.; van Alsenoy, C.; Ayers, P.; Carbo-Dorca, R. J. Chem. Phys. 2007, 126, 144111. (50) Krishtal, A.; Senet, P.; van Alsenoy, C. J. Chem. Theory Comput. 2008, 4, 426–434. (51) Coker, H. J. Phys. Chem. 1976, 80, 2078. (52) Dodds, J.; McWeeny, R.; Raynes, W. T.; Riley, J. P. Mol. Phys. 1977, 33, 611–617. (53) Karna, S. P.; Talapatra, G. B.; Prasad, P. N. J. Chem. Phys. 1991, 95, 5873. (54) Krishtal, A.; Senet, P.; van Alsenoy, C. J. Chem. Phys. 2009, 131, 044312. (55) Bauer, B. A.; Warren, G. L.; Patel, S. J. Chem. Thoery Comput. 2009, 5, 359–373. (56) Kim, J.; Lee, H. M.; Suh, S. B.; Majumdat, D.; Kim, K. S. J. Chem. Phys. 2000, 113, 5259. (57) Kim, J.; Lee, S.; Cho, S. J.; B. J.; Mhin, K. S. K. J. Chem. Phys. 1995, 102, 839. (58) Tongraar, A.; Rode, B. M. Phys. Chem. Chem. Phys. 2003, 5, 357. (59) Ignaczak, A.; Gomes, J. A. N. F.; Cordeiro, M. N. D. S. Electrochim. Acta 1999, 45, 659. (60) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision D.02; Gaussian, Inc.: Wallingford, CT, 2004. (61) Mahan, G. Phys. ReV. A 1980, 22, 1780–1785. (62) Guardia, E.; Skarmoutsos, I. J. Chem. Theory Comput. 2009, 5, 1449–1453. (63) Krekeler, C.; Hess, B.; Site, L. D. J. Chem. Phys. 2006, 125, 054305. (64) Zhao, Z.; Rogers, D. M.; Beck, T. L. J. Chem. Phys. 2010, 132, 014502. (65) Murphy, W. F. J. Chem. Phys. 1977, 67, 5877.

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