A Continuum Solvent Model of the Multipolar Dispersion Solvation

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A Continuum Solvent Model of the Multipolar Dispersion Solvation Energy Timothy Thomas Duignan, Drew Francis Parsons, and Barry W. Ninham J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp403595x • Publication Date (Web): 09 Jul 2013 Downloaded from http://pubs.acs.org on July 15, 2013

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The Journal of Physical Chemistry

A Continuum Solvent Model of the Multipolar Dispersion Solvation Energy Timothy T. Duignan, Drew F. Parsons,∗ and Barry W. Ninham Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia E-mail: [email protected]

∗ To

whom correspondence should be addressed

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Abstract The dispersion energy is an important contribution to the total solvation energies of ions and neutral molecules. Here, we present a new continuum model calculation of these energies, based on macroscopic quantum electrodynamics. The model uses the frequency dependent multipole polarizabilities of molecules in order to accurately calculate the dispersion interaction of a solute particle with surrounding water molecules. It includes the dipole, quadrupole and octupole moment contributions. The water is modeled via a bulk dielectric susceptibility with a spherical cavity occupied by the solute. The model invokes damping functions to account for solute-solvent wave function overlap. The assumptions made are very similar to those used in the Born model. This provides consistency and additivity of electrostatic and dispersion (quantum mechanical) interactions. The energy increases in magnitude with cation size, but decreases slightly with size for the highly polarizable anions. The higher order multipole moments are essential, making up more than 50% of the dispersion solvation energy of the fluoride ion. This method provides an accurate and simple way of calculating the notoriously problematic dispersion contribution to the solvation energy. The result establishes the importance of using accurate calculations of the dispersion energy for the modeling of solvation.

Keywords: Electrolyte Solutions, Ions, Noble Gas Atoms, Polarizability, Ion Size, Van der Waals Interaction

I. Introduction Dispersion Interactions The dispersion interaction acts between all molecules and arises from the rapid fluctuation of electrons. This is a fascinating and important phenomena as it plays a significant role in the properties of many materials. The short range of the interaction means that this is particularly true for liquids where the molecules are in close proximity, but not chemically bound to each other. 2 ACS Paragon Plus Environment

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The accurate calculation of these energies is therefore an important step in the explanation and prediction of the properties of solutions. A particularly important example of this is the solvation energy of molecules and ions. These are the free energy changes upon moving a solute particle from vacuum into solution. A large literature 1 2 is focused on calculating these energies for molecules of all range of sizes and types, as this is a vital step in understanding chemistry in the condensed phase. The dispersion interaction of the solute particle with the surrounding solvent makes up a significant proportion of the solvation energy. 3 4 It has hardly been considered as a serious player in the classical literature and is ignored or subsumed in ‘effective potentials.’ Classical theories do not account for Hofmeister effects. Quantum interactions must be included in any explanation of ion specificity. 5

Explicit Solvent Approach In explicit solvent simulations, where the behavior of every molecule in a liquid is followed, it is particularly difficult to incorporate this interaction. In classical Molecular Dynamic (MD) simulations the dispersion interaction is included by an empirically parametrized Lennard-Jones potential. The values for these parameters reported in the literature vary over a significant range. 1 They depend on the method of parametrization and the experimental properties they are parametrized against. 6 This fact, combined with the unphysical nature of the mathematical form of the potential, means that the real contribution of dispersion forces is hidden. It also means that for mixed electrolytes predictability is lost. Accurate inclusion of the dispersion interaction in explicit solvent ab initio calculations requires sophisticated correlated quantum mechanical methods with large basis sets. The computational demands of these methods mean the dispersion energy is often included through a separate approximate calculation which is added to a simple DFT level calculation. 7 Methods using this approach are still in development. 8 There are many complications involved in calculating these energies separately: higher order multipole contributions, many body contributions, damping, and 3 ACS Paragon Plus Environment


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repulsion contributions from wave function overlap. These complications are difficult to incorporate, but are particularly important in condensed phases, where the large separation limit can not be used.

Continuum Solvent Approach Because of the computational demands and the complexity of using explicit solvent water models a large amount of research has focused on calculating solvation energies using continuum solvent models. These calculations often include a significant number of fitted parameters. Nonetheless they have been surprisingly successful at reproducing experimental solvation energies. The classic illustration of this is the Born model, 9 which reproduces the solvation energy of the alkali halide ions with surprising accuracy using two temperature dependent fitted parameters. One parameter is added to the cation’s crystal radius to give the value for the radius used in the Born equation. A second, smaller parameter is added to the anion’s crystal radius. 10 11 Generalizations of the Born model, that include a distributed molecular charge, can be applied to a wider class of molecules, such as neutral inorganic atoms, large and possibly polar organic molecules and polyatomic ions. Normally these models require a number of parameters fitted to reproduce a given set of solvation energies, the models are then tested against a larger set of solvation energies in order to determine their accuracy. 12,13 Some of these models 2,13 use accurate quantum mechanical treatments of the solute particle in calculating the electrostatic contribution to the solvation energy. The wave function of the solute is calculated self consistently and the electric field created by the polarization of the dielectric medium is included. For spherical ions this approach is identical to the original Born model calculation. Many of these continuum models account for the dispersion interaction by coarse approximation, incorporating the energy by assuming it can be modeled with a fitted surface tension parameter. This introduces further unknown parameters and ought not to be necessary. The calculation of the dispersion energy should be amenable to calculation at the same level of approximation as the electrostatic energy. 4 ACS Paragon Plus Environment

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Indeed, a logical, consistent approach to the problem of solvation is to improve the continuum solvent model to the greatest extent possible before resorting to explicit water models. Models based on water molecules must often ignore many body effects that are subsumed in continuum dielectric descriptions. By improving continuum solvent models we can maximize their accuracy and range of applicability. This should then open the way to go further towards a consistent theory. To achieve this it is necessary to improve the physical modeling, in order to reduce the number of fitted parameters and improve experimental agreement. As already mentioned the role of dispersion interactions is one facet of these solvation models which is normally treated on a particularly inaccurate, indeed arbitrary, level. The development of simple and accurate models of the dispersion energy which can be easily and generally applied is therefore an important task. These interactions are often taken to be proportional to the ‘surface area’ of the solute so that they can be modeled as surface tensions. But this approximation is too crude. These interactions are complex and depend sensitively on the properties of the solute and solvent. In this paper we will address some of this complexity by introducing multipole polarizability and a cavity in the solvent.

Continuum Dispersion Models There are examples in the literature of calculations of the dispersion energy using a continuum solvent model. The Born model requires the molecular charge and dielectric constant. Similarly, these continuum models 14 require accurate knowledge of quantum mechanical properties of the solute ( ) as well as the accurate dielectric susceptibility function of water at imaginary frequencies. ε (iξ ) This function captures the polarization response of the dielectric material to high frequency electric fields. There are several approximations to this function in the literature. 15,16 But for our purposes the differences between them are immaterial. Ref. 14 gives a review of the approaches to the modelling of dispersion energies with a continuum solvent model. Here we discuss two of the most useful methods, reminiscent of and similar



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to our model presented here. The closest model in the literature to the one presented here is that of Amovilli and Mennucci. 4,17 This treats the solute particle on a quantum mechanical level, it finds a self consistent solution for the wave function of a solute particle in a cavity surrounded by a polarizable continuum. This approach is significantly more sophisticated than almost all of the dispersion energy calculations used in the literature. However some problems with this approach are in the neglect of solute-solvent wave function overlap. It does not rest on firm theoretical grounds, and requires quantum mechanical calculations including the presence of the solvent, which are more difficult then straightforward calculations in vacuum. Amovilli’s model is mainly applied to large neutral solutes. An alternative model also similar to the one presented here, was developed and applied to ionic solutes by Mahanty, Boström, and Ninham. 3,18,19 This model is relatively simple and easy to apply. Its accuracy can be improved significantly by removing unphysical aspects of the calculation such as the assumption of a gaussian dipole distribution, and the neglect of the cavity created by the solute. There are also additional separate approaches, which may have potential. 20

Goal The aim of this paper is to find an improved approach to calculate dispersion solvation energies. Building on the background literature outlined above. The approach is to be accurate, and it will draw on and balance the advantages of both the approaches mentioned above. It does so by making the following improvements: One is to include the effect of the overlap of the wave functions of the solute and solvent molecules. This is achieved by use of Tang-Toennies overlap damping functions. 21 The second improvement is to insert the solute into a cavity inside the dielectric medium. This puts the dispersion energy on the same level of theory as the Born model. It allows the use of a single consistent cavity size in the calculation of both contributions. This calculation is placed on a firm theoretical foundations by using Macroscopic Quantum Electrodynamic methods. 22 6 ACS Paragon Plus Environment

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A third development is the inclusion of multipole moments up to the octupole term. This gives a better account of the finite size of the solute particles. Finally, we have endeavored to facilitate ease of use and generalizations of the theory by writing equations in terms of simple fundamental properties that are either already available in the literature, or that can be determined with standard quantum chemistry calculations of the solute particle in vacuum. This last improvement removes the need to perform quantum chemistry calculations in the solvated environment which can be difficult.

II. Theory Dispersion Contribution Scheel and Buhmann have presented an approach to calculating dispersion forces between individual quantum mechanically treated atoms and macroscopic bodies modeled with a dielectric response. 22,23 Some of their results are consistent with earlier work. 18 However they provide a very convenient framework for calculating energies in more complex situations. Here we apply their method to calculate the dispersion interaction of a molecule with bulk solvating water. The solvent is modeled as a continuous macroscopic object with a frequency dependent polarization response. It is assumed that the continuum of water contains a spherical cavity occupied by a solute particle. The solute is modeled with a multipole expansion at the center of the cavity, where each multipole moment has a calculated polarizability. This is the same level of approximation used by the Born model. The result is that the two contributions, electrostatic and dispersion, are consistent with each other.

Expressions We first review the most general equations derived in the theory from which the specific expressions for the solvation energies can be derived. Their dispersion energy of a solute particle is



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known as a Casimir-Polder interaction energy. The dipole dispersion energy is: 22 4π kB T UD (rA ) = c2

) ( 1 ∑ 1 − 2 δn0 ξn2α (2)(iξn) • G(1)(rA, rA, iξn) n=0 ∞

(1)

Here the α (2) (iξn ) is the dipole polarizability tensor of rank two, it includes a factor of 1/4πε0 and so has units of m−3 . The summation is evaluated at the Matsubara frequencies: ξn = 2π kB T n/¯h. G(1) (r, r′ , iξn ) is the scattering part of the dyadic Greens function for the Helmholtz equation with the boundary conditions determined by the shape of the dielectric material. 22 The • denotes the Frobenius inner product. The next step is the inclusion of the quadrupole energy, given in Eq. (15) of Ref. 24. We drop the second term in that equation because it is a thermalized ground state atom 25 giving: 4π k T B UQ (rA ) = 2 c

) ( [ ] → − ← − 1 2 (4) (1) ′ ′ ∑ 1 − 2 δn0 ξn α (iξn) • ∇ ⊗ G (r, r , iξn) ⊗ ∇ n=0 ∞

(2)

r,r′ →rA

From Ref. 26 the octupole contribution is: 4π k T B UO (rA ) = 2 c

( ) [→ ] − → − ← − ← − 1 ∑ 1 − 2 δn0 ξn2α (6)(iξn) • ∇ ⊗ ∇ ⊗ G(1)(r, r′, iξn) ⊗ ∇ ′ ⊗ ∇ ′ n=0 ∞

r,r′ →rA

(3)

α (4) (iξn ) is the quadrupole polarizability tensor of rank 4, it has units of units of m−5 . α (6) (iξn ) is the octupole polarizability tensor of rank 6, it has units of units of m−7 . Definitions of these quantities are given in Ref. 26. G(1) (r, r′ , iξn ) is the scattering part of the dyadic Greens function 22 with the boundary conditions of a spherical cavity in a dielectric medium. This is known. 27 The Greens function can be simplified by taking the non-retarded limit, ie. by calculating limξ /c→∞ (ξ /c)2 G(1) (r, r, iξ ). 28 We have compared the dipole dispersion energy for the retarded and non-retarded cases, establishing that the contribution due to retardation is negligible. We are dealing with isotropic solute particles, it is therefore useful to rewrite the polarizability


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tensors in terms of the standard scalar isotropic polarizabilities, which are: ⟨ ⟩ 2 ωn0 n Q0l 0 1 2e2 αl (iω ) = ∑ ω2 + ω2 4πε0 h¯ n=1 n0

(4)

After some algebra, we can relate α (2) (iξn ), α (4) (iξn ) and α (8) (iξn ) to a general isotropic tensor of the same rank multiplied by α1 (iξn ), α2 (iξn ) and α3 (iξn ) respectively. Making these substitutions and simplifying we arrive at the following expressions for the dispersion contribution to the solvation energy. 6kB T GD = − 3 Rcav

( ) 1 ε (iξn ) − 1 ∑ 1 − 2 δn0 α1 (iξn) 2ε (iξn) + 1 n=0

(5)

15kB T GQ = − 5 2Rcav

( ) 1 ε (iξn ) − 1 ∑ 1 − 2 δn0 α2(iξn) 3 ε (iξ ) + 1 n n=0 2

(6)

28kB T GO = − 7 3Rcav

∞

∞

) ( ε (iξn ) − 1 1 ∑ 1 − 2 δn0 α3(iξn) 4 ε (iξ ) + 1 n n=0 3 ∞

(7)

A full retarded expression for the dipole term (Eq. 5) has been given before in an equivalent formulation. (Eq. 49 of Ref. 28) At small separations between solvent and solute molecules the higher order multipole moments will be of significant magnitude. As far as we are aware they have not been derived before. We equate these expressions with free energy changes rather than internal energies. This is based on the same justification for why the Born model gives free energies rather than internal energies. Both calculations account for the energy change of the total system, including the energy cost of polarizing the dielectric material. They depend on temperature through the temperature dependence of the dielectric susceptibility and possibly the cavity radius, this gives an entropic component. These equations are consistent with a pure Lifshitz calculation of the dispersion interaction of a dielectric sphere with a surrounding medium. In order to show this we need the nth order polarizability of a dielectric sphere. This can be determined by calculating the electrostatic multipole 9 ACS Paragon Plus Environment


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moment induced on a dielectric sphere in a dielectric medium. Where the sphere has been placed in an externally applied static electric field equal to the the nth order term in the expansion of the potential in terms of Legendre polynomials. 29 The polarizability of a dielectric sphere of radius RS and dielectric constant εS in vacuum, is:

αnS (iξ ) = R2n+1 S

εS (iξn ) − 1 εS (iξn ) + n+1 n

(8)

Substituting this expression into Eq. 5, Eq. 6, and Eq. 7 above reproduces Eq. (33) of Ref. 30 to first order.

Wave Function Overlap A problem arises with the expressions above for ions where the water molecules are very close to the solute. The multipole expansion does not converge. Instead of decreasing, the higher order moment contributions to the solvation energy increase in magnitude as the order increases. This problem can be attributed to the fact that the multipole expansion used is only valid at large separation, ie. large cavity sizes, where there is negligible overlap of the electronic shells of the interacting atoms. 32 At such small separations the wave-function of the ions overlap with those of the water molecules, which leads to the rapidly diverging multipole expansion. Usually, to account for this effect in the theory of intermolecular interactions, semi-empirical damping functions have to be introduced. These reduce the size of the dispersion interaction. Higher orders are damped more and the damping becomes increasingly large the smaller the separation. 21,31 This effect is put forward as a reason the polarizability of water molecules needs to be reduced in polarizable water simulations. 33 One of the best methods of correcting for this effect in the context of calculating pair wise interactions are the Tang-Toennies damping functions: 21


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(

(br)k 1− ∑ exp−br k! k=0

fm (r) =

m

) (9)

The b parameters are Born-Mayer parameters, which are related to the spatial spread of the wave function. They are used in the Born-Mayer repulsive potential. 34 The r value gives the distance between the centers of the two interacting molecules. The multipole expansion of the dispersion interaction between two atoms including these functions up to the octupole moment becomes 5

∑ f2n(r)C2n/r2n

(10)

n=3

We can apply these functions to the dispersion solvation energy problem with some minor adjustments. We assume that the damping of the dispersion interaction of the solute with the bulk solvent is the same as the damping of the vacuum pairwise dispersion interaction between the solute and the closest water molecule. This assumption can be justified. The damping is due to the wavefunction overlap with the nearby water molecules which restricts the degree of polarization of the solute. This means the solute will have a smaller dispersion interaction with all of the solvent molecules. So we simply multiply each of Eq. 5, Eq. 6, and Eq. 7 by a damping function of the form of Eq. 9 The value of m is determined by the distance behavior of the interaction type. To determine the correct distance behavior we need to be able to compare Eq. 5, Eq. 6, and Eq. 7 with the point-wise interactions which the Tang-Toennies damping functions were developed for. We can do this by using the expression for the multipole polarizability of a spherical cavity in water, this expression is calculated in the same way as Eq. 8. If Rcav is the size of the cavity and ε is the dielectric constant of the medium the expression is:

αncav (iξ ) = R2n+1 cav ε (iξn )

1 − ε (iξn ) 1 + n+1 n ε (iξn )


(11)


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If we substitute this into Eq. 6 we arrive at: ) ∞ ( 1 15kB T GQ = 10 ∑ 1 − 2 δn0 α2(iξn)α2cav (iξn) 2Rcav ε (iξn ) n=0

(12)

Apart from a constant factor, this has the same form as the quadrupole-quadrupole contribution to the pairwise interaction energy. (Eq. (3.50) of Ref. 31) This indicates that Eq. 6 is analogous to a quadrupole-quadrupole pairwise interaction with a 1/r10 dependence. Hence, the damping function of the quadrupole dispersion solvation energy term should take m = 10. By similar reasoning, m = 6 and m = 14 are used for the dipole and octupole terms respectively. The other two parameters in the Tang-Toennies equations are the b values and the r separation parameter, which we will refer to as RS . The b parameters used in the Tang-Toennies functions need to be an average of the two molecules. We use a well tested 35 combining rule for the determination of this parameter for two unlike atoms coming into contact,

b12 = 2b1 b2 /(b1 + b2 )

(13)

Where b1 and b2 are the Born-Mayer parameters for the solute-solute and water-water interactions. The determination of these parameters will be discussed below. The final expressions, including the damping functions, for the dipolar, quadrupolar and octupolar dispersion contribution to the solvation energy are: (

(bRS )k GD = − 1 − ∑ exp−bRS k! k=0 (

6

(bRS )k GQ = − 1 − ∑ exp−bRS k! k=0 (

10

(bRS )k exp−bRS GO = − 1 − ∑ k! k=0 14

) 6kB T R3cav

( ) ε (iξn ) − 1 1 ∑ 1 − 2 δn0 α (iξn) 2ε (iξn) + 1 n=0

(14)

15kB T 2R5cav

) ( 1 ε (iξn ) − 1 ∑ 1 − 2 δn0 α2(iξn) 3 ε (iξ ) + 1 n n=0 2

(15)

28kB T 3R7cav

( ) 1 ε (iξn ) − 1 ∑ 1 − 2 δn0 α3(iξn) 4 ε (iξ ) + 1 n n=0 3

(16)

)

)

∞

∞

∞

Here b = b12 as given in Eq. 13. Once the parameters are known, these expressions allow a very


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simple and quick calculation which gives the dispersion solvation energy.

Electronic Properties For the calculation of the dispersion energies various properties are also needed. The dielectric function of water at imaginary frequencies (ε (iξn )) has been determined by Dagastine. 15 Accurate polarizabilities are essential in calculating these interactions. This means high level correlated quantum calculations are necessary for quantitative results. 19 We use the polarizabilities of the halides ions and the noble gas atoms up to the octupole moment calculated at the TDMP2 level of theory by Hättig and Hess. 36,37 The alkali ion polarizabilities for lithium up to rubidium were provided by Jim Mitroy based on a semi-empirical calculation. 38 The polarizabilities of all of these atoms and ions were also calculated using the Amsterdam Density Functional (ADF) software package (v2010.01) 39–41 using the response program 42,43 with the Model LB94 functional 44 and ZORA/QZ4P 45 or ET/ET-QZ3P-3DIFFUSE 46 basis sets, which gave reasonable agreement with the calculations of Hättig and Hess for the dipole and quadrupole polarizabilities. For instance the dispersion solvation energy of argon calculated using these ADF polarizabilities is within 1.5% of the dispersion solvation energy calculated from the polarizabilities given in Ref. 36, where augcc-pV5Z basis sets with additional diffuse functions are used. However, there was a significant difference in the octupole polarizability. The ADF calculation underestimates the polarizability by over 50%, compared to the more sophisticated approaches. The imaginary frequency polarizabilities of cesium, copper(I) and silver(I) were calculated using this software. These properties were used to calculate the dispersion solvation energy for these ions. So far as we are aware these properties have not been published. The inaccuracy of the octupole polarizability does lead to significant uncertainty in the octupole contribution to the solvation energy for these ions. These polarizabilities are for molecules in the gas phase. There is significant evidence 47–52 that anions in condensed phase have polarizabilities which are significantly reduced from the gas phase values. This effect is much weaker for cations. We therefore follow Lamoureux and Roux 33,53 in reducing the polarizability of all of the anions by a factor of 0.724 from their gas phase values and 13 ACS Paragon Plus Environment


we leave the cation and noble gas polarizabilities the same as in gas phase. The single factor of 0.724 was chosen for simplicity although there is evidence this effect may be stronger for smaller anions and larger cations. 51

Size Parameters Born-Mayer Parameters A definition of the radii of the noble gas atoms with coordination number 6 is given in Ref. 54. These radii were chosen to be consistent with ionic radii of coordination number 6, that are given in Ref. 55 and Ref. 56. We will refer to these two sets of radii as RI . The b parameters are known for the noble gases. 34 As shown in Figure 1 these values are approximately linearly related to the

5

-1

Born Mayer b value (Å )

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He Ne

4.5

4 Ar

3.5

Kr

Xe

3

1

1.2

1.4 1.6 Radius (RI) (Å)

1.8

2

Figure 1: The monotonic relationship between the Born-Mayer b parameters 34 and the inherent radius for noble gas molecules is shown. RI values for the noble gas atoms. We therefore use a simple interpolation of this relationship to determine the b parameters for the ions. Similarly the b parameter for water can be determined by using the value of 1.38Å for RI . This water molecule radius is determined with the same method 14 ACS Paragon Plus Environment

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used to determine the noble gas radii. 54

RS Parameters We now need the RS parameter, which gives the distance between the center of the two molecules. A reasonable choice for this parameter can be determined from solute-oxygen radial distribution functions. These functions give the probability distribution of finding a water molecule (oxygen atom) as a function of distance from the solute center. These functions normally have a strong initial peak at some small separation which corresponds with the first water layer around the solute. The distance to this peak therefore serves as a good choice for the solute-water separation parameter, (RS ). Ref. 57–59 compile a large number of experimental measurements of the RS parameter, determined from diffraction and X-ray absorption fine structure (EXAFS) experiments, for many of the important ions. We use the average of these values to get a good estimate of the RS parameter. The values determined in this way are presented in Table 1. Additional justification for this choice of ion-water molecule distance, can be gained from comparison with ab initio quantum mechanical calculations. We performed structure optimizations of ion-water clusters at the MP2 level, with six water molecules placed with D2h symmetry around a central ion. The TURBOMOLE package (v6.4) 62,63 was used with the jobex program. 64–66 The aug-cc-pVQZ 67–70 basis sets were used for the anions up to bromide and water, aug-cc-pVQZPP were used for iodide, 71,72 copper and silver. 73 The def2-QZVPP 74,75 basis sets were used for the alkali cations. This allowed a value for the ion-oxygen distance to be determined theoretically. These values are given in Table 2 along with the experimentally determined values for RS , showing they are largely consistent with each other. Usefully, this method allows a determination of the size of the copper (I) ion in water, which can not be determined experimentally due to its poor solubility. The RS parameters for the noble gas atoms are taken from averaging values from MD calculations and EXAFS measurements of the radial distribution functions. 60,61 The exception is helium, which we determine from the linear extrapolation of RI and RS for the other noble gas atoms.



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Table 1: The solute crystal size of coordination number 6 (RI ), solute-oxygen distance (RS ), Born-Mayer parameter (b) , and cavity size (Rcav ). RI (Å) 0.76a 1.02a 1.38a 1.52a 1.67a 0.77a 1.15a 1.33a 1.81a 1.96a 2.20a 1.08b 1.21b 1.64b 1.78b 1.38b

Atom/Ion Li+ Na+ K+ Rb+ Cs+ Cu+ Ag+ F− Cl− Br− I− He Ne Ar Kr H2 O

RS (Å) 2.06 ± 0.02c 2.35 ± 0.02c 2.79 ± 0.03c 2.92c 3.11 ± 0.04c 2.25h 2.41 ± 0.01c 2.68 ± 0.06c 3.20 ± 0.01c 3.35 ± 0.03c 3.64 ± 0.01c 3.08i 3.21 ± 0.01d 3.55 ± 0.01d 3.73 ± 0.03d −

−1

b(Å ) Rcav (Å)g 5.49e 1.22 e 5.00 1.51 e 4.33 1.95 e 4.06 2.08 3.77e 2.27 e 5.47 1.41 e 4.76 1.57 4.42e 1.84 e 3.47 2.36 e 3.18 2.51 e 2.71 2.80 4.89f 2.24 f 4.64 2.37 f 3.84 2.71 f 3.52 2.89 e 4.33 −

a Ref.

55 b Ref. 54 c Average of values from Ref. 57–59 d Ref. 60,61 e Interpolation of b versus. R for noble gas particles I f Ref. 34 g R − 0.84Å h ab initio calculation S i Extrapolated by linear relationship between R and I RS

Table 2: A comparison between theoretical and experimental estimates of the ion-oxygen distance in water Atom/Ion Li+ Na+ K+ Rb+ Cs+ Ag+ F− Cl− Br− I−

RS (Å)(Experiment) 2.06 ± 0.02 2.35 ± 0.02 2.79 ± 0.03 2.92 3.11 ± 0.04 2.41 ± 0.01 2.68 ± 0.06 3.20 ± 0.01 3.35 ± 0.03 3.64 ± 0.01

RS (Å)(Theory) 2.10 2.35 2.73 2.93 3.12 2.46 2.75 3.23 3.32 3.54


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These values for the ion size in water are largely consistent with the very intuitive expression:

RS = RI + RWat

(17)

originally observed by Marcus, 76 who took RWat = 1.38Å. The only exceptions to this are silver (I) and copper (I), which have RS values roughly 0.1Å smaller than this rule predicts. Cavity Size Parameter The cavity size parameter is the last that needs to be determined. The solvation energies are quite sensitive to the value of this parameter through the 1/Rncav terms. The cavity size defines the spherical region where the relative dielectric function has the value of unity as it is for a vacuum. Outside this spherical cavity the bulk water dielectric function is used. This change occurs at the position where the polarization of the water becomes significant. This will occur in the first hydration layer around the solute. In principle the dielectric function will increase continuously from 1 to the bulk water value in the region between RI and RS from the solute center. For simplicity however, we model it as a discontinuous boundary, somewhere within this region. So we define Rcav as the distance from the center of the solute particle at which the dielectric changes discontinuously from vacuum to that of the solvent. We assume that it can be written as Rcav = RS − Radj

(18)

Where Radj is fixed for all solutes and 0 < Radj < 1.38Å. To recapitulate, the principle behind the calculations presented here was to use a model that calculates the dispersion contribution to the solvation energy at the same level at which the Born model calculates the electrostatic energy. Indeed, as this discussion does not depend on whether this is the frequency or static dielectric function it is reasonable to conclude that the cavity size for the electrostatic (Born) contribution should be the same as for the dispersion contribution. An expression for the electrostatic cavity size parameter has been determined from a more 17 ACS Paragon Plus Environment


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sophisticated derivation of the Born equation using an MSA model. The expression is 77

Rcav = RI + 0.54Å

(19)

We can combine Eq. 17, Eq. 18 and Eq. 19 to arrive at

Radj = 0.84Å

(20)

Using this value for the cavity size to calculate the electrostatic (Born) energy of solvation for the lithium ion gives a value within 5% of the experimental solvation energy. For this ion the electrostatic component should dominate as its small size and polarizability mean that its cavity formation and dispersion solvation energies will be minimal. This indicates that this choice of cavity size parameter is appropriate. Because of the consistency of the dispersion and electrostatic contribution we therefore use this value for the cavity size in determining the dispersion contribution to the solvation energy for all of the molecules studied in this paper. Figure 2 provides a visualization of the choices for the size parameters.

Figure 2: Diagram of an ion in a cavity visualizing size parameters.


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Israelachvili 79 has shown that the surface tension of a wide range of weakly polar liquids can be estimated using a Lifshitz calculation. The separation between the two dielectric surfaces is taken to be l = σ /2.5. Where σ is the atomic spacing of the liquid. This expression is derived by comparison with a molecular level calculation assuming pairwise additivity. From Figure 3 it is

Figure 3: Diagram of two interacting water interfaces. clear that Radj can be written in terms of l as

Radj =

σ − l 3σ = 2 10

(21)

For water a reasonable choice for σ is 2 × 1.38Å = 2.76Å which gives Radj = 0.83Å. Very similar to the estimate based on MSA model.

III. Results and Discussion Dispersion contributions to the solvation energy have been calculated using the methods outlined here. The results for the noble gas atoms and some monoatomic and monovalent ions are presented in Table 3. They are also plotted as a function of size in Figure 4 (ions) and Figure 5 (noble gas atoms) 19 ACS Paragon Plus Environment


Table 3: The contributions to the dispersion solvation energies of some simple ions and noble gas atoms. Atom/Ion Li+ Na+ K+ Rb+ Cs+ Cu+ Ag+ F− Cl− Br− I− He Ne Ar Kr

GDisp −7.1 −21.5 −46.2 −61.6 −74.9 −112.5 −119.8 −100.5 −90.3 −93.3 −85.4 −6.5 −12.0 −28.2 −33.4

GDipole −6.5 (91%) −17.9 (83%) −35.1 (76%) −44.7 (73%) −52.4 (70%) −75.4 (67%) −90.9 (76%) −47.3 (47%) −46.8 (52%) −49.0 (53%) −47.1 (55%) −5.6 (86%) −9.8 (82%) −21.1 (75%) −24.5 (74%)

GQuadrupole −0.6 (8%) −3.0 (14%) −8.9 (19%) −13.2 (21%) −19.3 (26%) −21.1 (19%) −22.1 (18%) −31.0 (31%) −27.0 (30%) −28.0 (30%) −25.1 (29%) −0.8 (12%) −1.8 (15%) −5.5 (19%) −6.8 (20%)

GOctupole −0.0 (1%) −0.6 (3%) −2.1 (5%) −3.7 (6%) −3.2 (4%) −16.0 (14%) −6.8 (6%) −22.2 (22%) −16.5 (18%) −16.3 (17%) −13.2 (15%) −0.1 (2%) −0.4 (3%) −1.6 (6%) −2.1 (6%)

The percentages of the total dispersion energy are in brackets. The free energies are given in units of kJmol-1

0 Li

-1

Dispersion Energy (kJ mol )

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+ +

Na

-50

-100

-150 0

K

Anion Dispersion Cation Dispersion Anion Dipole Cation Dipole Anion Quadrupole Cation Quadrupole Anion Octupole 0.5

+

+

Rb

+

Cs

-

Cl F

-

-

I

-

Br

1 1.5 Radius (RI)(Å)

2

2.5

Figure 4: The dispersion contribution to the solvation energy of the alkali halide ions as a function of their inherent radius.


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0 -1

Dispersion Energy (kJ mol )

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He

-10

Ne

-20 Noble Dispersion Noble Dipole Noble Quadrupole Noble Octupole

-30

-40

1

1.2

Ar

1.4 1.6 Radius (RI) (Å)

Kr

1.8

Figure 5: The dispersion contribution to the solvation energy of the noble gas atoms as a function of their inherent radius. These values cannot be directly compared with experiment as there are important contributions from the cavity formation and electrostatic energies which contribute to the solvation energies. These will need to be taken into account before comparison with experimental solvation energies can be made. This work has been completed and submitted for publication. 11

Magnitude These dispersion energies are of significant size when compared with the experimental solvation energies. 80,81 For example iodide’s dispersion energy is 34% of its solvation energy. The conclusion is inescapable. The dispersion energy makes an important contribution. This is true a fortiori for the anions. The particular importance of the dispersion interaction for the anions has been noted before. 82,83 This helps explain why anions play a more important role in many ion-specific effects. 82,84 Their large size and polarizability means the electrostatic interaction is comparatively weak and so the dispersion interaction may play an increasingly important role. There are two reasons for the importance of the dispersion energy in the solvation of anions. 21 ACS Paragon Plus Environment


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The first is that their excess electron makes them highly polarizable, resulting in a larger dispersion interaction. Although the electron does make the ions larger, this effect is not significant enough to compensate for the increased polarizability. Secondly, due to the charge on an ionic solute, the water dipoles are pulled significantly closer to the ions than they are to the neutral noble gas solutes. This is clear from the RS parameters. The value of RS for argon is larger than for any of the ions including iodide. The short range nature of the dispersion interaction means that this pulling in of the water molecules significantly increases the magnitude of the dispersion energy. This reduction in the RS parameter occurs for both cations and anions. But the lower polarizability of cations, due to the missing electron, mitigates this effect for the smaller alkali ions resulting in a dispersion energy similar to the noble gas atom’s dispersion energy, and hence relatively small compared with the electrostatic contribution. This is not true however, for copper (I) and silver (I), which are comparatively small, so this pulling in of water molecules is particularly pronounced. For instance, silver (I) has the same dipole polarizability as rubidium, but it has an RS value only slightly larger than that of sodium. This results in a relatively large dispersion energy for these ions as well.

Size Dependence An interesting observation clear from Figure 4 is that the magnitude of the dispersion contribution to the solvation energy actually decreases with ion size for the halides. This means that the fluoride ion actually has a slightly larger dispersion solvation energy than iodide. This is a surprising and counterintuitive result. However, it can be understood if we consider the increase in polarizability compared to their isoelectronic noble gas atoms. Fluoride has 6 times the dipole polarizability of neon, whereas bromide has less than three times the polarizability of Krypton. So as the atoms become larger the relative effect of one less proton becomes less pronounced. The dispersion interaction is therefore less enhanced. This is also consistent with certain calculations of the Lennard-Jones dispersion potentials, fitted by comparison to experiment, which assign to fluoride the largest dispersion interaction. 85


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There is significant disjunction between anions, cations and noble gas solutes in the relationship between their size and their dispersion energy contribution. This is shown in Figure 4 and Figure 5. These results clearly indicate that the relationship between dispersion energy and size is far from trivial and depends sensitively on the electronic properties of the solute. It can not be assumed that solutes of the same size will have the same dispersion energy. This fact means there will be be significant error in models which assume the dispersion energy of all solutes scales with their size. 86,87

Multipoles Figure 5 shows that the dipole contribution gives a reasonable estimate of the dispersion energy for the smaller noble gas atoms. But as these atoms get larger the higher orders, particularly quadrupole, start to result in a significant correction. This is also true for the alkali cations as shown in Figure 4. Table 3 shows that copper(I) and silver(I) have surprisingly significant higher order contributions given their size. This is due to the pulling in of the water molecules. For the anions, Figure 4 shows that the higher order multipole moments are very important. This is particularly true for small anions such as fluoride, where the higher order multipole moment contributions comprise over half of the total dispersion energy. This is understandable as these terms account for the variation of the electric field over the region that the molecule occupies. In other words they account for the finite size of the ion. The electron cloud around an anion is quite disperse and so this finite size contribution is significant.

Bulk Surface Tension Puzzle These results may help explain an additional, separate puzzle: that the surface tension increments of electrolytes decrease with the size of the halide ions. 88 This indicates that large anions are less repelled from the surface than small anions. In addition to this there is evidence that large anions such as iodide are adsorbed to the air-water surface. 89 This has caused problems with simple models which attempt to explain ion specific surface tension increments purely on the basis of the



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dispersion interaction. In these models to reproduce surface tension increments, which decreases with larger anion size, the fitted polarizability of anions must decrease with size as well. 82 This is unphysical. 84 Intuitively, if dispersion interactions are the origin of the ion-specificity of surface tensions, it would be expected that the larger ions, because of their larger polarizability would have stronger dispersion interactions with the air-water interface. As this interaction is repulsive, these larger ions would be more strongly repelled. In this way surface tension increment would be expected to increase with halide ion size. The simplest explanation, which is that dispersion forces can be neglected, is clearly incorrect. 82,85 Our current model provides a satisfactory explanation. The dispersion interaction of an ion with an air-water interface is equivalent to the change in the ion’s dispersion solvation energy as it approaches the surface. This results in a dispersion repulsion from the air-water interface due to the loss of the dispersion solvation energy. 90 This model entails that the dispersion solvation energy of fluoride is larger than for iodide. This means that fluoride will be more strongly repelled from the air-water interface than iodide, which is consistent with the experimental surface tension measurements. This counter intuitive result is due to the fact that it is not polarizability alone which determines the dispersion interaction energy, but rather polarizability and size. This does not necessarily mean that the dispersion energy can completely explain electrolyte surface tensions. The ion specificity observed can be plausibly attributed to other effects such as the release of the cavity formation energy, 91,92 or a static dipole moment induced on the ion at the surface. 93 One of these effects may explain the observation of iodide adsorbed to the air water interface.


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IV. Conclusion In summary this model provides a method for calculating the dispersion contribution to the solvation energy. It is based on the continuum model and uses the same level of approximation as the Born model, it is simple to apply using properties available in the literature. Counter intuitively, the dispersion solvation energy increases with cation size, but decreases with anion size. The higher order multipole moment contributions are essential, making up over 50% of fluoride’s dispersion solvation energy. In further work we will combine this model with a calculation of the electrostatic and cavity contributions to the solvation energy in order to compare with experiment. This model would be advanced further by more accurate calculation of the frequency dependent higher order polarizabilities of additional molecules. Generalizations of the equations could be made to anisotropic molecules and to cases where the multipole moment is not at the center of the cavity. These improvements would allow the application of the model to a much wider class of molecules. In addition the determination of the Green’s function for more general situations would be very useful. For example an ellipsoidal cavity, a cavity near a surface, or two cavities near each other. This would provide a calculation of the contribution of the dispersion energy to a wider class of experimental properties such as osmotic coefficients, surface forces, and surface tensions.

Acknowledgement We would like to thank Stefan Scheel, Stefan Buhmann for very useful discussions, and Stan van Gisbergen for helpful information. This work was supported by the NCI National Facility at the ANU.

Notes and References (1) Hunenberger, P.; Reif, M. Single-Ion Solvation: Experimental and Theoretical Approaches to Elusive Thermodynamic Quantities; The Royal Society of Chemistry, 2011.



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Page 32 of 32

A Continuum Solvent Model of the Multipolar Dispersion Solvation

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