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# Long-ranged contributions to solvation free energies from theory and short-ranged models

Contributed by John D. Weeks, January 13, 2016 (sent for review November 2, 2015; reviewed by Gerhard Hummer, Roland R. Netz, and Frank H. Stillinger)

## Significance

Many important biological and industrial processes, ranging from protein folding and ligand binding to self-assembly of materials at interfaces, take place in solution and are mediated by driving forces rooted in solvation. However, conceptual and computational difficulties arising from long-ranged Coulomb interactions still present a challenge to current approaches. Here we present a framework to determine very accurately the long-ranged contributions to solvation free energies in charged and polar systems from models with only short-ranged, local interactions. We examine a variety of ubiquitous solvation processes, including hydrophobic and ionic hydration, as well as colloidal overcharging. The theory additionally suggests ways to improve density functional theories of solvation by providing insights into commonly used approximations.

## Abstract

Long-standing problems associated with long-ranged electrostatic interactions have plagued theory and simulation alike. Traditional lattice sum (Ewald-like) treatments of Coulomb interactions add significant overhead to computer simulations and can produce artifacts from spurious interactions between simulation cell images. These subtle issues become particularly apparent when estimating thermodynamic quantities, such as free energies of solvation in charged and polar systems, to which long-ranged Coulomb interactions typically make a large contribution. In this paper, we develop a framework for determining very accurate solvation free energies of systems with long-ranged interactions from models that interact with purely short-ranged potentials. Our approach is generally applicable and can be combined with existing computational and theoretical techniques for estimating solvation thermodynamics. We demonstrate the utility of our approach by examining the hydration thermodynamics of hydrophobic and ionic solutes and the solvation of a large, highly charged colloid that exhibits overcharging, a complex nonlinear electrostatic phenomenon whereby counterions from the solvent effectively overscreen and locally invert the integrated charge of the solvated object.

Solvation thermodynamics underlies a vast array of important processes, ranging from protein folding (1, 2) and ligand binding (3) to self-assembly at interfaces (4). Thus, understanding solvation, and driving forces rooted in solvation, has been a focus of chemistry and physics for over a century (5, 6).

Quantitatively successful theories of self-solvation and solvophobic solvation in simple fluids have been developed (7⇓⇓⇓⇓⇓⇓⇓⇓–16). However, a generally useful analytic approach for solvation in complex charged and polar environments is lacking, and solvation is typically studied with computer simulations. Contributions from the long-ranged components of Coulomb interactions in periodic images of the simulation cell are typically evaluated using computationally intense Ewald and related lattice summation techniques (17). These methods generate distorted, system size-dependent interaction potentials (18) and do not scale well in massively parallel simulations (19), adding considerable computational overhead. Moreover, artifacts can arise from spurious interactions between the periodic images of solutes, as observed for proteins in water (20).

The local molecular field (LMF) theory of nonuniform fluids is a promising avenue for substantially improving free energy calculations by removing many of the computational and conceptual burdens associated with long-ranged interactions (14, 21). LMF theory prescribes a way to accurately determine the structure of a full system with long-ranged intermolecular interactions in a general single particle field by studying a simpler mimic system wherein particles interact with short-ranged intermolecular interactions only. An effective field in the mimic system accounts for the averaged effects of the long-ranged “far-field” interactions in the full system.

This approach is especially powerful for studying solvation in charged and polar solvents, where in the simplest case the effective field can represent the interactions between a fixed solute and the solvent. In this paper, we show that when the effective field and induced density around the solute are accurately determined by LMF theory it is very easy to integrate over the solvent structure and accurately compute the far-field contributions to the solvation free energy as well, using quantities determined solely in the short-ranged mimic system, where simulations scale linearly with system size.

LMF theory presents a general conceptual framework that gives qualitative as well as quantitative insight into many other problems. Its treatment of long- and short-ranged forces makes suggestive connections to other well-established theoretical methods, such as perturbation theory for uniform simple fluids (6, 7), classical density functional theory (DFT) of nonuniform fluids (11), and the successful quasichemical approach for solvation (16). Although our focus in this paper is on the quantitative determination of the solvation free energy, many of these connections will be touched upon in our discussion here and in *Supporting Information*. The treatment of solvation free energies we present here can be readily generalized to determine more complex free energies, including alchemical transformations and potentials of mean force (3), and extended to more general charged and polar mixtures (21) with mobile solutes.

The conceptual development of the LMF approach to solvation thermodynamics is introduced in the next section, with derivations and other technical details given in *Materials and Methods*, *Derivation of the Far-Field Solvation Free Energy* and *Supporting Information*. We first focus on the solvophobic solvation of a repulsive, spherical solute in a Lennard-Jones (LJ) fluid, where most of the ideas can be understood in their simplest form and the basic physics is well understood. We then turn to more challenging and experimentally relevant problems involving the length-scale transition in hydrophobic solvation of an apolar solute in water and its effect on the solvation free energies; the hydration of single ions is discussed in *Supporting Information*. Finally we discuss the solvation and “overcharging” of a large, highly charged colloid in an ionic fluid, a highly nontrivial process involving ion correlations (22) that is completely missed in classic mean field treatments of ionic solutions (23).

## LMF Theory and Truncated Models

For simplicity, we first study solvation in a one-component LJ fluid with pairwise intermolecular interactions. We assume that the intermolecular interactions in the full solvent system are slowly varying at large separations, as is true for both LJ and Coulomb interactions. In the simplest description of solvation using the Grand ensemble, the solvent interacts with an external field *μ*.

In a general perturbation approach (6, 7), the intermolecular interactions of both solute and solvent are usually divided into strong short-ranged reference and slowly varying long-ranged perturbation parts (14):

LMF theory considers a special reference system or “mimic system,” denoted by the subscript R, resulting from a judicious choice of the short- and long-ranged components of

As discussed in detail in ref. 21 and further in *Supporting Information*, extensive previous work has shown that *ϕ* is zero. This equation also holds in other ensembles such as the constant-pressure ensemble where the density exactly approaches

The LMF equation has a functional form suggested by a simple mean field approximation where the averaged effects of the long-ranged intermolecular pair interactions are self-consistently related to the density induced by a single particle mean field. However, this form is derived by an approximate integration over intermolecular forces in the full and mimic systems as described by the exact Yvon–Born–Green (YBG) hierarchy of equations (14, 21) and does not use the traditional mean field ansatz where pair distribution functions are approximated by products of single particle functions. The detailed analysis in ref. 21 and *Supporting Information* shows that quantitatively accurate results from the LMF equation can be expected only when the

LMF theory can be immediately adapted to models of charged and polar systems with pairwise Coulomb interactions (21). Experience has shown that it is advantageous to separate the Coulomb interaction from all charges into short- and long-ranged parts according to*σ* should generally be chosen on the order of typical nearest-neighbor distances. The Coulomb LMF equation analogous to Eq. **4** can be written as (21)**8** we have used the convolution form of

## LMF Thermodynamic Cycle for Solvation

We first consider the solvation of a rigid solute (S) fixed at the origin in a mobile single-component LJ-type solvent (M). Fig. 1 schematically depicts the process of gradually “turning on” the solute–solvent interaction potential

The lower left panel of Fig. 1 shows the core positions of the mobile bulk solvent (M) in a typical configuration. The solvent molecules interact with the full long-ranged pair potential *u*. The lower right panel schematically depicts an equilibrium configuration of the full solute–solvent system, where the solute (S) has been inserted into the fluid.

However, the transformation from a noninteracting point solute on the left into the full solute on the right must generally be carried out in small steps using a series of nonphysical intermediate states as the harshly repulsive solute–solvent core interactions are gradually turned on (3). Moreover, each step of this process requires an accurate treatment of the long-ranged interactions, which for Coulomb interactions adds significant overhead to each time step. We indicate this general difficulty by using a red arrow to connect the two lower panels of Fig. 1.

The LMF treatment of solvation in Fig. 1 introduces a thermodynamic cycle involving a short-ranged mimic system that eliminates most of the problems arising from conventional treatments of long-ranged forces. Moreover, it provides a natural and physically suggestive way of partitioning the free energy into short and long-ranged components that are conceptually related to elements of the quasichemical theory of solvation (16, 25). These connections will be discussed in detail in a future publication; here we focus on the important far-field contribution to the solvation free energy, which plays a fundamental role in all partitioning schemes.

The LMF thermodynamic cycle includes the two upper panels of Fig. 1, which describe solvation in the short-ranged mimic system. The upper left panel illustrates a configuration of the strong coupling or mimic solvent (M_{0}); the red color indicates truncated solvent–solvent interactions, _{0} panels.

Solvation in the mimic system involves insertion of a mimic solute (S_{R}), described by the renormalized potential**2** and the last term on the right in the LMF Eq. **4**. This latter term is also slowly varying because of the integration over _{0} + S_{R} panel is very similar to that around the full solute in the M

We exploit this fact in determining the far-field component of the solvation free energy: the free energy change between the lower and upper panels on the left and right sides of Fig. 1, indicated by the vertical green arrows. By a “bottom-up” functional integration over the effective field and induced density as connected by the LMF equation and paying close attention to constant terms, we derive in *Materials and Methods*, *Derivation of the Far-Field Solvation Free Energy* a simple, analytic expression for the far-field component of the solvation free energy**11** is our main result. It can be immediately generalized as in Eq. **7** for charged and polar systems as

The validity of Eq. **11** relies on the mean field form and accuracy of the LMF Eq. **4**, which can generally be justified only for particular slowly varying choices of the *Materials and Methods*, *Derivation of the Far-Field Solvation Free Energy* and *Supporting Information* for further discussion of this important conceptual point.

To complete the cycle we must calculate the solvation free energy in the short-ranged mimic system,

Moreover, Eq. **9** allows us to write *Supporting Information*.

## Length Scale Dependence of Solvophobic Solvation

The crux of the LMF treatment of solvation is that accurate thermodynamic properties follow immediately from a good description of the induced solvent structure. To illustrate this point, we first show that LMF theory can quantitatively capture the drying in an LJ fluid observed at the surface of a harshly repulsive solute with an effective hard sphere diameter

As illustrated in Fig. 2*A*, the WCA solvent strongly wets the solute surface at large *A*, *Inset*. The field

Given this accurate description of structure from LMF theory, we now study the solvation thermodynamics of harshly repulsive spherical solutes of various sizes in LJ and WCA fluids. Detailed simulation results are shown in Fig. 2*B*. The Gibbs free energies of solvation, **11** is then used to obtain the far-field correction to the WCA solvation free energies. The LMF free energies recover the length-scale transition and reproduce the LJ solvation free energies with quantitative accuracy.

The computational utility of LMF theory becomes more apparent in systems with costly long-ranged electrostatic interactions. We first illustrate this by studying hydrophobic solvation of repulsive spheres in water, in complete analogy to solvophobic solvation in LJ fluids. In the short-ranged SC reference system, extended simple point charge (SPC/E) water is modeled in a Gaussian-truncated (GT) fashion, wherein the *D*).

Drying at the surface of a large repulsive sphere associated with breaking local hydrogen bonds should also be qualitatively captured by GT water, as shown in Fig. 3*A*. However, small differences in the nonuniform water density

These differences are most clearly seen in the Gaussian smoothed charge density, **8**. Smoothed charge densities for GT and full water are shown in Fig. 3*B*. Water is polarized at the solute interface even in the full system, evidenced by the positive lobe in *r*. This buildup of polarization is effectively screened, consistent with the subsequent negative peak and *r*. In GT water, however, only local hydrogen bonding constraints are optimized and a much larger positive peak at small *r* is observed, indicating overpolarization at the solute surface. Moreover, this peak is largely unscreened; only a small peak develops at large *r*, and

The picture provided by *C*, further illustrates that long-ranged electrostatics—mainly dipole-dipole interactions in the case of water—significantly influences the polarization of interfacial water. GT water overorients its O–H bonds toward the solute with respect to the full water model, because the drive to form a dangling O–H bond at a nonpolar interface arises from local, short-ranged interactions (9, 27). The dipolar screening in full water reduces this tendency to form dangling bonds.

We can now integrate over structure and obtain the LMF contribution to the solvation free energy as in Eq. **12**. Owing to the similar local structure of the full and GT water systems, accurate estimates of *Supporting Information* for more details). The electrostatic LMF correction determined in this manner brings the hydration free energies into quantitative agreement with those of the full system, as shown in Fig. 3*D*.

## Solvation and Overcharging in Colloidal Systems

We now turn to a case where long-ranged electrostatic interactions play a major role in determining both the structure and thermodynamics of the system, solvation of a charged colloid in an ionic fluid. Large, highly charged colloidal particles can seemingly invert their charge when immersed in solutions of multivalent counterions (23). Multivalent counterions adsorb to the surface of the colloid at densities high enough that the net charge contained inside the first solvation shell is opposite to that of the colloid. This phenomenon, known as overcharging, is highly nontrivial (22) and cannot be captured by classic mean field theories of ionic solutions (23).

Here we consider the solvation of a large (*ε* as modeled in ref. 33. The charge is uniformly smeared over the colloid surface, and the colloid and counterions are modeled as charged WCA species (7). The SC system is constructed with a smoothing length of *a* is the nearest-neighbor distance defined setting

As a measure of the solvation structure, we monitor the integrated charge,*A*. This quantity describes how the solute charge is screened by the solvent charges. We find overcharging of the colloid by the ionic solvent, with a maximum inverted charge of

Overcharging in the SC system is substantially less than that in the full system, indicating that long-ranged electrostatics plays an important role in determining the colloid solvation structure. Additionally, the SC system does not capture the local neutrality of the system, because its *A*.

These large corrections to the SC system suggest that long-ranged interactions will make a significant contribution to the colloidal solvation free energy

We also obtained the free energy of charging the colloid to intermediate charges (*Q*) between 0 and *Q* to a good approximation, as shown in Fig. 4*B*, *Inset*. The charge dependence is dominated by the LMF contribution, which itself is quadratic in *Q*.

The quadratic dependence of *Q* can be rationalized by noting that the long-ranged electrostatic behavior is very insensitive to many atomic-scale details because of Gaussian smoothing. Accurate approximations to the potential *Supporting Information*, this approximation yields**14** captures the behavior of **15** is equivalent to the self-interaction term of ref. 36 and highlights the quadratic dependence of *Q* and its inverse dependence on *σ*. Although it is very accurate in many cases involving small ionic solutes (36), we find it to be only qualitatively accurate for this highly charged system.

The use of simple theoretical constructs in Eq. **14** that capture only a few long wavelength properties illustrates how LMF theory can be used in combination with existing frameworks to obtain both qualitative insight and accurate predictions.

## Conclusions and Outlook

In this paper we have shown that LMF theory quantitatively describes both the structure and thermodynamics of solvation in general molecular systems using only short-ranged models. Moreover, this enables the study of nonneutral systems, as illustrated in *Supporting Information*, by the estimation of an ion hydration free energy. We expect our approach to be of significant importance to the study of large biomolecular and materials systems, where poor scaling of lattice summation techniques (19) limits the length and time scales accessible to simulations [the most efficient particle-mesh algorithms for lattice summations scale like *N* is the number of particles in the system (17)]. Models with only short-ranged interactions scale linearly with system size, so the use of LMF theory in conjunction with highly optimized biomolecular simulation packages should allow researchers to study length and time scales previously inaccessible without any loss in accuracy. Moreover, LMF theory can be used with commodity hardware such as graphics processing units without the need for additional algorithm development (37), offering another source of increased efficiency.

The general theoretical framework presented here is not limited to simple solvation and can be used to determine generic free energies as a function of an arbitrary order parameter by recasting more complex processes in the language of solvation (3). Thus, we expect this theory of solvation to find wide use and significantly affect free energy computations across molecular science.

## Materials and Methods

### Simulation Details.

All simulations were performed with the DL_POLY 2.18 software package appropriately modified to incorporate the various interaction potentials used throughout this work (38). In the full systems, electrostatic interactions were evaluated using Ewald summation (17). Unless otherwise noted, LJ interactions and the real space part of the electrostatic interactions were truncated at 9 Å. Simulations were performed in the isobaric-isothermal (constant NPT) ensemble to ensure that the bulk density far from the solute remains constant across all systems.

### Hard Sphere Solvation in an LJ Fluid and Water.

Hard spheres were modeled as the WCA repulsive portion of an integrated, “9-3” LJ potential following previous work (27). To compute free energies, simulations were performed with repulsive spheres of varying radii, in increments of

### Colloid Solvation Free Energy Calculations.

Solvation free energies of purely repulsive spheres in the electrolyte solution were computed in analogy to those in water and the LJ fluid. To compute the charging free energy, we vary the charge on the colloid in increments of *Q* by varying the number of coions between simulations performed in the NPT ensemble with *Q* is equivalent to pulling them from this reservoir of constant neutrality. The charging free energy was then evaluated using BAR (39). All colloid simulations were performed with a constant dielectric constant of

### Derivation of the Far-Field Solvation Free Energy.

We derive an exact expression for the solvation free energy difference in the Grand ensemble *λ*,

The solute–solvent field *λ* dependence to be specified later, such that when *λ*.

The grand partition function at a particular value of *λ* can then be written as

where

where

In the following it is useful to rewrite the solute–solvent energy and chemical potential terms in Eq. **17** using the microscopic configurational density

as

By differentiating the grand free energy *λ*, we immediately obtain a well-known exact result

Here *λ*. We have assumed that **16** and note from Eqs. **17**–**20** that

We now integrate over *λ* to obtain the free energy difference between the mimic system at *λ* dependence of *λ*, **21** over this special path yields

Eq. **23** gives the free energy difference associated with the rightmost green arrow in Fig. 1. The free energy of the remaining green arrow is obtained when

Because the solvation free energy is **23** to write

with

Eq. **24** is an exact formula for the solvation free energy difference corresponding to the green vertical arrows in Fig. 1. However, it may seem too complicated for practical use because it requires exact values of *μ* and of the partially coupled nonuniform pair distribution function *λ* even though the nonuniform singlet density does not change along the chosen path.

However, we show here that when LMF theory provides an accurate self-consistent description of the induced single particle density **24** involving only

**22** the exact result

Eq. **24** can then be exactly rewritten (41) as

The free energies in Eq. **27** are all functionals of the common density

Using Eq. **26** we can functionally differentiate Eq. **27** to obtain a formally exact relation between

Here we have moved the chemical potentials terms inside the curly braces and chosen constants so the term inside the braces vanishes for a uniform bulk system where

The LMF Eq. **4**, derived independently by an approximate integration of the first member of the YBG hierarchy of equations relating intermolecular forces to induced structure (14, 21), gives a separate and often very accurate relation between **28** as

where constants have again been chosen such that the term in the curly braces vanishes in the uniform bulk. Using the LMF Eq. **4** the term in braces can be exactly rewritten as

Assuming the accuracy of the LMF approximation for **29** from Eq. **28**. The potential terms cancel and we can then formally perform functional integrals over **30** we obtain

Thus, when using a properly chosen mimic system, the complicated expression on the left side of Eq. **31** involving the *λ*-dependent nonuniform pair distribution functions and exact values of *μ* and

The terms on the left side of Eq. **31** are the last two terms in the exact expression for the solvation free energy in Eq. **24**. Thus, it can be accurately approximated as

which is the basic LMF solvation formula Eq. **11**.

Eq. **32** requires the accuracy of the LMF Eq. **4** and exploits its simple mean field form in the “bottom-up” functional integration that gives the associated free energy change. This strongly contrasts with the usual treatment of mean field theory in classic DFT (11). In this “top-down” approach the free energy itself including constant terms is expressed as a general functional of the nonuniform density. It then is written as a sum of reference and perturbation parts, with the reference part often chosen for convenience as a hard sphere system for which accurate analytic functionals have been developed. This initial choice fixes the form of the remaining potential

However, this DFT perspective provides no physical insight into what reference system should be used or when and why the mean field product approximation would be expected to be accurate. Nevertheless, as noted by Archer and Evans (42) (AE), when the crude mean field approximation is used and the associated single particle potential is calculated by functional derivatives as in Eq. **26**, one obtains an equation very similar to the LMF Eq. **4**. Indeed, AE assert that the LMF equation “follows directly from the standard mean-field DFT treatment of attractive forces.”

However, the detailed derivation of the LMF equation from the YBG hierarchy in ref. 21 and *Supporting Information* makes no use of the crude mean field approximation or other concepts from DFT. It shows that the resulting mean field form and quantitive accuracy requires a well chosen

As emphasized in ref. 21, the LMF equation (and in particular its use here to obtain solvation free energies) “is not a blind assertion of mean-field behavior but rather a controlled and accurate approximation, provided that we choose our mimic system carefully.” See *Supporting Information* for further discussion of this important point and a detailed derivation of the LMF equation.

## Derivation of the LMF Equation and Relation to DFT

For simplicity we consider solvation in a simple one-component LJ-like system, although these ideas can be readily extended to Coulomb systems, using methods described in ref. 21. LMF theory and most perturbation approaches based on classic DFT consider a restricted class of density-matched reference systems, denoted here by a subscript

As would be expected, the utility and accuracy of any simple perturbation approach requires a well-chosen reference

LMF theory uses the exact YBG hierarchy of equations relating liquid structure and forces to derive a formally exact equation satisfied by **S2** the analogous YBG equation for a density matched reference system (14, 21). Using Eq. **S1** we find exactly**S4** formally determines

As argued in detail in ref. 21, considerable simplifications arise when we can choose an optimal separation where the short-ranged

These common strong forces should generate very similar features in the conditional densities in both the full and mimic systems at short length scales where **S6** is small. Similarly the gradient of the slowly varying **S7** is small at short distances, precisely the region where the conditional and singlet densities differ most from each other. At larger distances, assuming there are no intrinsic long-ranged correlations as seen near critical points, and so on, the conditional and singlet densities approach one another and the integrand is again small. Thus, in many cases the term **S7** is also very small.

To derive the LMF equation we assume both terms can be neglected. The remaining term **S5** can then be integrated exactly. Imposing the boundary condition that the density reduces to

Fortunately, experience has shown that LMF theory gives very accurate results in many cases both for nonuniform LJ fluids with walls or fixed solutes, and even better results have been found for charged and polar systems by properly separating the Coulomb potential. When the accuracy of the LMF equation can be established, *Materials and Methods*, *Derivation of the Far-Field Solvation Free Energy* in the main text shows that simple and very accurate approximations for the solvation free energy can then be found by a “bottom-up” functional integration over the LMF potential and induced density.

Archer and Evans (42) (AE) have recently argued that DFT offers an alternate pathway to the LMF equation, asserting that it “follows directly from the standard mean-field DFT treatment of attractive forces.” Here we simplify their argument and assess the validity of this statement.

The standard DFT perturbation treatment focuses directly on the intrinsic free energy functional **25** of the main text and that of an appropriately chosen density matched reference system **S9** as in Eq. **26** of the main text.

In practice accurate analytic approximations for **27** of the main text, this can be exactly written as**S10** by replacing the nonuniform pair correlation function by that of a uniform system at some intermediate or averaged density (44, 45), it is not clear how this should be chosen in very nonuniform systems and the results depend strongly on the choice made.

By default, most researchers have used the crude but much simpler mean field approximation discussed by AE. For any density-matched reference system **S10** can be safely ignored by approximating**S9** then can be written as**26** of the main text we then have**S13** reduces to the mean field van der Waals approximation for the bulk chemical potentials (14):**S13** and taking **S8**, as noted by AE.

Although this and related derivations of an LMF-like equation may have the “advantage of concision” (46), we consider them quite misleading from a fundamental point of view. They seem to link the quantitative success of the LMF equation to the accuracy of the crude mean field approximation in **S11** or **S12**. However, the magnitude and nature of possible errors that this direct and uncontrolled imposition of a mean field form could induce both on the free energy itself and on correlation functions and fields given by functional derivatives of the approximate form are by no means obvious.

Closely related problems with the use of the crude mean field approximation are apparent even in the direct derivation of the LMF equation from the YBG hierarchy. If Eq. **S11** is used in the exact Eqs. **S5**–**S7**, the terms involving conditional densities vanish identically for any density matched reference system

The LJ fluid represents one of the few cases where the mimic system can be well approximated by a hard sphere reference system and almost all quantitatively successful applications of the mean field approximation in DFT have been for the LJ fluid. However, it seems highly implausible even for the LJ fluid with a near-optimal WCA choice of reference system that the crude mean field approximation in Eq. **S12** could accurately reproduce the exact free energy in Eq. **S10** over a wide range of densities.

This is immediately clear for a uniform fluid where **S9** reduces to the usual Helmholtz free energy *F* and Eq. **S10** gives the exact uniform fluid perturbation expression for **S10** by its value at

This approximation can be motivated by mean field ideas applied to structure and forces, where the net effects of the long-ranged attractive forces from **S10**, quantitative errors (14) ranging from 10 to 15 percent for **S14**.

LMF theory shows how and why the LMF equation and natural generalizations to Coulomb systems can nevertheless be used to give quantitatively accurate results for the structure induced by the solute and for the excess solvation thermodynamics both for the nonuniform LJ fluid and for a wide variety of charged and polar systems as well. However, its success in this particular context should not be taken as evidence for the more general validity of **S11** or **S12**.

## Solvation in the Mimic System

To determine the solvation free energy in the mimic system, **9** of the main text. Thus, we write

The slowly varying nature of **S16** provides a simple and accurate estimate of

Similar statements hold in the context of electrostatic interactions, enabling the estimation of the corresponding free energy due to turning on

## Numerically Practical Form of the Long-Ranged Contribution to the Solvation Free Energy

There exists a cancellation of terms between the solvation free energy of the mimic system and Eq. **11** of the main text, clearly observed when using Eq. **S16**. Although the separation of free energy contributions in this manner is favorable for physical interpretation, such cancellations can lead to numerical inaccuracies in some contexts. To avoid such inaccuracies, we can combine Eq. **S16** with Eq. **11** of the main text to obtain a useful expression for the total long-ranged component of the solvation free energy, **S18** removes the cancelling terms and thus is numerically advantageous when the Gaussian approximation is accurate.

For completeness, we note that the electrostatic analog of Eq. **S18** can be readily obtained by substituting charge densities for densities, as is done in ref. 21 for the LMF equation. This leads to

## Linear Response Theory for the Charge Density

In this section, we briefly review the linear response formalism of Hu and Weeks (26) for determining the charge density of the mimic system directly from a system with some effective slowly varying electrostatic potential *N* is the number of charges in the system and

We now determine

The charge density around a repulsive sphere with radius *A* for the full, GT, and mimic (LMF) systems. The charge density in the mimic system was determined using Eq. **S22** with *A*. The formalism of Hu and Weeks (26) corrects the charge density of the GT system with quantitative accuracy, bringing it into good agreement with that of the full system. The LMF potential *B*.

## Debye–Hückel Approximation for the LMF Free Energy Contribution to Colloid Solvation

The electrostatic LMF contribution to the solvation free energy is

The charge density of solvent ions can be written using Debye–Hückel theory as (35)

Using the approximation in Eq. **S31** for the charge density, the LMF free energy can be written as**S39** provides a qualitative description of

## Ion Hydration

We now consider ion hydration as an example where the system has a net charge and long-ranged electrostatics play a significant role in solvation thermodynamics. Traditional approaches to ion solvation, such as the successful formalism developed by Hummer et al. (36), involve the simulation of a single ion in a dielectric solvent. The electrostatics are treated by Ewald summation in these systems and therefore require the presence of a neutralizing background charge density. In addition, significant finite size effects due to the periodicity of the Ewald sum are present (20), although successful finite-size corrections to the solvation free energy have been developed (21, 36, 47).

LMF theory provides a useful alternative to periodic lattice summation techniques when studying ion solvation. Aside from the efficient simulation of purely short-ranged systems afforded by LMF theory, the conceptual difficulties associated with a nonuniform effective electrostatic potential that depends on the size of the simulation cell can be eliminated. By using the iteration scheme in the following section, the LMF potential will display the asymptotic behavior predicted by classic electrostatics, *r*.

Following Hummer et al. (36), we consider the calculation of the solvation free energy of a charged methane-like particle in SPC water. In this case, methane is modeled in the united atom scheme, such that methane (Me) is represented as a single LJ particle with Me–water interaction parameters of

We separate the solvation free energy **S40** is the solvation free energy of the ion in the SCA system. This can be divided into a free energy of inserting an uncharged cavity into the GT variant of SPC water, *Q* using the Bennett acceptance ratio (39).

The hydration free energy of a methane-like cation in SPC water determined from LMF theory-based free energy calculations is compared with the results of ref. 36 in Table S1. The total solvation free energy

## Stable Iteration of the LMF Equation for Charged Systems

In this section we detail a stable iteration scheme for self-consistently solving the LMF equation when the system has a net charge. We consider here an ion with a point charge

We first separate the electrostatic LMF **8** in the main text into short- and long-ranged components (21) according to**S43** shows that

Classic electrostatics tells us that the asymptotic form of the polarization potential*ϵ* is the known dielectric constant of the solvent (SPC/E water). We can generate this asymptotic form by approximating**S44**, consistent with the idea of water as a linear dielectric medium that screens all but a fraction

Eqs. **S41**–**S43** show that **S43**. To that end we treat *σ*-smoothed) system, and use LMF theory to map this system onto a new mimic system with an additional smoothing length *l*. This mapping separates **S47** we combined the Gaussian convolutions in the definitions of **S46**.

We now choose *l* sufficiently large that we can use Eq. **S45** to accurately approximate the charge density in Eq. **S47**. This gives a simple analytic expression for the final long-ranged component*l* on the order of *σ* suffice for this approximation to hold, and the results presented herein were obtained with

This stable iteration scheme for systems with a net charge requires the self-consistent solution of the intermediate-ranged portion of the renormalized field**S49** at each iteration.

## Acknowledgments

We thank Jocelyn Rodgers, Zhonghan Hu, and Giacomo Fiorin for helpful comments on the manuscript. This work was supported by National Science Foundation Grants CHE0848574 and CHE1300993.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: jdw{at}umd.edu.

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2009.

Author contributions: R.C.R., S.L., and J.D.W. designed research; R.C.R., S.L., and J.D.W. performed research; R.C.R., S.L., and J.D.W. analyzed data; and R.C.R. and J.D.W. wrote the paper.

Reviewers: G.H., Max Planck Institute of Biophysics; R.R.N., Free University of Berlin; and F.H.S., Princeton University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1521570113/-/DCSupplemental.

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- Abstract
- LMF Theory and Truncated Models
- LMF Thermodynamic Cycle for Solvation
- Length Scale Dependence of Solvophobic Solvation
- Solvation and Overcharging in Colloidal Systems
- Conclusions and Outlook
- Materials and Methods
- Derivation of the LMF Equation and Relation to DFT
- Solvation in the Mimic System
- Numerically Practical Form of the Long-Ranged Contribution to the Solvation Free Energy
- Linear Response Theory for the Charge Density
- Debye–Hückel Approximation for the LMF Free Energy Contribution to Colloid Solvation
- Ion Hydration
- Stable Iteration of the LMF Equation for Charged Systems
- Acknowledgments
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