Dielectric response with short-ranged electrostatics Stephen J. Coxa,1 aDepartment of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom Edited by John D. Weeks, University of Maryland, College Park, MD, and approved July 14, 2020 (received for review March 28, 2020) The dielectric nature of polar liquids underpins much of their abil- obtained from this study will aid the development of SR inter- ity to act as useful solvents, but its description is complicated molecular potentials, which is often the case with modern by the long-ranged nature of dipolar interactions. This is partic- machine-learning approaches (32–36). It will also help us to ularly pronounced under the periodic boundary conditions com- understand when neglect of LR electrostatics does, and does not, monly used in molecular simulations. In this article, the dielectric have severe consequences on simulated observables. Moreover, properties of a water model whose intermolecular electrostatic it seems likely that strong connections exist between LMFT ideas interactions are entirely short-ranged are investigated. This is and classic density functional theory (see refs. 37 and 38 for dif- done within the framework of local molecular-field theory (LMFT), fering suggestions), and it is hoped that the results that follow which provides a well-controlled mean-field treatment of long- will help the development of such theoretical approaches. The ranged electrostatics. This short-ranged model gives a remarkably results presented here also provide further support to LMFT as a good performance on a number of counts, and its apparent short- theoretical approach for understanding the solvation of charged comings are readily accounted for. These results not only lend species (38, 39). support to LMFT as an approach for understanding solvation behavior, but also are relevant to those developing interaction LMFT and the Dielectric Constant potentials based on local descriptions of liquid structure. LMFT is a statistical mechanical framework based on the Yvon– Born–Green hierarchy of equations that relate molecular corre- dielectric response j liquid water j electrostatic interactions j lations with intermolecular forces (31). The premise of LMFT confined fluids j local molecular field theory is that the intermolecular interactions (or a subset) can be partitioned into SR and LR portions and that there exists a nderstanding the dielectric nature of polar fluids is one mapping to a “mimic” system. This mimic system comprises Uof the principal aims of liquid-state theory and continues intermolecular interactions arising solely from the SR portion CHEMISTRY to be the motivation for both experimental (1–3) and theoreti- and a suitably chosen one-body potential; by construction, the cal (4–13) investigations. As can be immediately deduced from average structure and higher-order correlations of the full system the fact that the free energy of a polar system depends on its are captured. Although LMFT can be applied more generally shape (14), the dipolar interactions that define a polar fluid (40), only its application to electrostatic interactions is consid- are long-ranged (LR). This makes their study both fascinating ered here. Moreover, as detailed derivations have been given and complicated. The importance of understanding the dielec- elsewhere (31), discussion is limited to its most salient features. tric properties of polar liquids cannot be overstated, as they play Let us begin by noting that the Coulomb potential can be a crucial role in phenomena such as solvation, self-assembly, separated exactly into SR and LR contributions, and transport through membranes and nanopores (15–18). Con- sequently, there are broad implications across the biological, 1 erfc(κr) erf(κr) = + ≡ v0(r) + v1(r), [1] chemical, physical, and materials sciences. Of all polar liquids, r r r water stands out owing to its ubiquity across disciplines. It is the −1 focus of this article. where κ defines the length scale over which v0 decays. This In addition to experiments, computer simulation is a widely will be familiar to many as the same procedure taken in Ewald taken approach to investigate the behavior of liquids at the approaches (e.g., ref. 28), where κ is chosen to optimize compu- molecular level. The computational cost associated with the tational efficiency. In contrast, the success of LMFT relies on a microscopic resolution that simulations provide, however, often limits their application to system sizes far below that of samples Significance investigated experimentally. As a result, periodic boundary con- ditions (PBC) are often employed to mitigate spuriously high Much of a liquid’s ability to act as a solvent stems from its degrees of interfacial curvature and surface-to-volume ratios. dielectric properties. The intermolecular forces between polar How to appropriately account for the LR nature of electrostatic molecules can act over very long ranges, which complicates interactions, and the implications this has for dielectric prop- both theoretical descriptions and molecular simulations. Here erties, has a long history (19–28). While certainly not the only the dielectric properties of a model system with only short- method available to deal with electrostatic interactions under ranged intermolecular interactions are investigated, and this PBC, Ewald summation techniques are now widely considered short-ranged model’s behavior is rationalized on a theoret- the de facto standard (29). On the one hand, it is hard to ical basis. This work will likely facilitate the development argue against the success that Ewald approaches have enjoyed of both efficient short-ranged interaction potentials and our as a computational tool. On the other hand, they are not espe- understanding of fluids under confinement. cially intuitive and risk masking simple physical interpretations of liquid-state behavior. Author contributions: S.J.C. designed research, performed research, analyzed data, and In this article, the dielectric properties of liquid water wrote the paper.y whose intermolecular electrostatic interactions are entirely The author declares no competing interest.y short-ranged (SR) are investigated. Specifically, the framework This article is a PNAS Direct Submission.y provided by local molecular-field theory (LMFT) (30, 31) is Published under the PNAS license.y exploited to recast LR electrostatic interactions in a mean- 1Email: [email protected] field, yet well-controlled, fashion. Aside from demonstrating This article contains supporting information online at https://www.pnas.org/lookup/suppl/ how LMFT’s performance can be understood within the exist- doi:10.1073/pnas.2005847117/-/DCSupplemental.y ing statistical mechanical framework for polar liquids, the insight www.pnas.org/cgi/doi/10.1073/pnas.2005847117 PNAS Latest Articles j 1 of 7 Downloaded by guest on September 29, 2021 choice of κ such that the mimic system accurately captures the Dielectric Response of Bulk Liquid Water one-body density and correlations of the full system. In what fol- An infinite system is not a realizable object, even in a computer lows κ−1 = 4:5 A,˚ which has been previously demonstrated to be simulation that employs PBC. Thus while can be “determined” a reasonable choice (31). Instead of computing LR electrostatic within the SCA framework from arguments based on an infinite interactions explicitly, the effects of v1 are accounted for by a sample, it still needs to be established how measurable quantities static restructuring potential, like the polarization response, or fluctuations at zero field, are affected. In this section, such properties are investigated for bulk Z 0 0 0 liquid water under PBC. This is probably the closest realizable VR(r)= V(r) + dr nR(r )v1(jr − r j), [2] system to the infinite geometry considered above. Nevertheless, it is important to bear in mind that there is now an implicit where nR is the average charge density in the mimic system, “boundary at infinity” (19, 20, 22, 46). V is an external electrostatic potential that would be applied Fig. 1 A and B shows how the polarization P responds to either to the full system, and the integral is understood to be taken a uniform electric field E or electric displacement field D applied over all space. As VR is to be chosen such that nR = n, where along z, respectively (e.g., refs. 48–50). Results are shown for n is the average charge density of the full system, Eq. 2 defines the cases that LR electrostatics are both calculated explicitly a self-consistent relationship between VR and nR. In a “pure” (“Ewald”) and neglected entirely (“SCA”). In fact, using SCA LMFT approach, Eq. 2 can be solved either by brute force or by instead of a full LMFT treatment can perhaps be justified here: exploiting linear-response theory (41). As the focus of this arti- VR = V on account of the fact that uniform fields induce charge cle is on understanding dielectric properties within the LMFT density only at physical boundaries, which, at least explicitly, are framework, here a more pragmatic approach is instead taken: absent in the current geometry. (−@z V = E or D accordingly.) At The self-consistent cycle is “short-circuited” by using n obtained constant E a degree of nonlinear response is observed at larger from a simulation of the full system as the initial, and only, guess fields, while the response to constant D is linear to an excellent (42). While Eq. 2 has a simple mean-field form, it is important approximation. In either case, the Ewald and SCA approaches to stress that it is not derived from a mean-field ansatz. It repre- are virtually indistinguishable over the range of field strengths sents a controlled approximation provided that the mimic system studied. This observation is corroborated by the probability dis- is chosen carefully. It should also be noted that Eq. 2 has been tributions at zero field, pE and pD , of the z component of the derived with nonuniform systems in mind and that, for uniform total dipole moment of the simulation cell M = ΩP, shown in systems, more sophisticated LMFT approaches exist (42–44).