Multipole Electrostatics Research Articles

Stress tensor and constant pressure simulation for polarizable Gaussian multipole model.

Our previous article has established the theory of molecular dynamics (MD) simulations for systems modeled with the polarizable Gaussian multipole (pGM) electrostatics [Wei et al., J. Chem. Phys. 153(11), 114116 (2020)]. Specifically, we proposed the covalent basis vector framework to define the permanent multipoles and derived closed-form energy and force expressions to facilitate an efficient implementation of pGM electrostatics. In this study, we move forward to derive the pGM internal stress tensor for constant pressure MD simulations with the pGM electrostatics. Three different formulations are presented for the flexible, rigid, and short-range screened systems, respectively. The analytical formulations were implemented in the SANDER program in the Amber package and were first validated with the finite-difference method for two different boxes of pGM water molecules. This is followed by a constant temperature and constant pressure MD simulation for a box of 512 pGM water molecules. Our results show that the simulation system stabilized at a physically reasonable state and maintained the balance with the externally applied pressure. In addition, several fundamental differences were observed between the pGM and classic point charge models in terms of the simulation behaviors, indicating more extensive parameterization is necessary to utilize the pGM electrostatics.

GFN2-xTB-An Accurate and Broadly Parametrized Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions.

An extended semiempirical tight-binding model is presented, which is primarily designed for the fast calculation of structures and noncovalent interaction energies for molecular systems with roughly 1000 atoms. The essential novelty in this so-called GFN2-xTB method is the inclusion of anisotropic second order density fluctuation effects via short-range damped interactions of cumulative atomic multipole moments. Without noticeable increase in the computational demands, this results in a less empirical and overall more physically sound method, which does not require any classical halogen or hydrogen bonding corrections and which relies solely on global and element-specific parameters (available up to radon, Z = 86). Moreover, the atomic partial charge dependent D4 London dispersion model is incorporated self-consistently, which can be naturally obtained in a tight-binding picture from second order density fluctuations. Fully analytical and numerically precise gradients (nuclear forces) are implemented. The accuracy of the method is benchmarked for a wide variety of systems and compared with other semiempirical methods. Along with excellent performance for the "target" properties, we also find lower errors for "off-target" properties such as barrier heights and molecular dipole moments. High computational efficiency along with the improved physics compared to its precursor GFN-xTB makes this method well-suited to explore the conformational space of molecular systems. Significant improvements are furthermore observed for various benchmark sets, which are prototypical for biomolecular systems in aqueous solution.

Efficient minimization of multipole electrostatic potentials in torsion space.

The development of models of macromolecular electrostatics capable of delivering improved fidelity to quantum mechanical calculations is an active field of research in computational chemistry. Most molecular force field development takes place in the context of models with full Cartesian coordinate degrees of freedom. Nevertheless, a number of macromolecular modeling programs use a reduced set of conformational variables limited to rotatable bonds. Efficient algorithms for minimizing the energies of macromolecular systems with torsional degrees of freedom have been developed with the assumption that all atom-atom interaction potentials are isotropic. We describe novel modifications to address the anisotropy of higher order multipole terms while retaining the efficiency of these approaches. In addition, we present a treatment for obtaining derivatives of atom-centered tensors with respect to torsional degrees of freedom. We apply these results to enable minimization of the Amoeba multipole electrostatics potential in a system with torsional degrees of freedom, and validate the correctness of the gradients by comparison to finite difference approximations. In the interest of enabling a complete model of electrostatics with implicit treatment of solvent-mediated effects, we also derive expressions for the derivative of solvent accessible surface area with respect to torsional degrees of freedom.

An optimized intermolecular force field for hydrogen-bonded organic molecular crystals using atomic multipole electrostatics.

We present a re-parameterization of a popular intermolecular force field for describing intermolecular interactions in the organic solid state. Specifically we optimize the performance of the exp-6 force field when used in conjunction with atomic multipole electrostatics. We also parameterize force fields that are optimized for use with multipoles derived from polarized molecular electron densities, to account for induction effects in molecular crystals. Parameterization is performed against a set of 186 experimentally determined, low-temperature crystal structures and 53 measured sublimation enthalpies of hydrogen-bonding organic molecules. The resulting force fields are tested on a validation set of 129 crystal structures and show improved reproduction of the structures and lattice energies of a range of organic molecular crystals compared with the original force field with atomic partial charge electrostatics. Unit-cell dimensions of the validation set are typically reproduced to within 3% with the re-parameterized force fields. Lattice energies, which were all included during parameterization, are systematically underestimated when compared with measured sublimation enthalpies, with mean absolute errors of between 7.4 and 9.0%.

Noncovalent Interactions and Impact of Charge Penetration Effects in Linear Oligoacene Dimers and Single Crystals

Noncovalent interactions determine in large part the thermodynamic aspects of molecular packing in organic crystals. Using a combination of symmetry-adapted perturbation theory (SAPT) and classical multipole electrostatics, we describe the interaction potential energy surfaces for dimers of the oligoacene family, from benzene to hexacene, including up to 5000 configurations for each system. An analysis of these surfaces and a thorough assessment of dimers extracted from the reported crystal structures underline that high-order interactions (i.e., three-body nonadditive interactions) must be considered in order to rationalize the details of the crystal structures. A comparison of the SAPT electrostatic energy with the multipole interaction energy demonstrates the importance of the contribution of charge penetration, which is shown to account for up to 50% of the total interaction energy in dimers extracted from the experimental single crystals; in the case of the most stable cofacial model dimers, this con...

High-Resolution Crystal Structures of Protein Helices Reconciled with Three-Centered Hydrogen Bonds and Multipole Electrostatics

Theoretical and experimental evidence for non-linear hydrogen bonds in protein helices is ubiquitous. In particular, amide three-centered hydrogen bonds are common features of helices in high-resolution crystal structures of proteins. These high-resolution structures (1.0 to 1.5 Å nominal crystallographic resolution) position backbone atoms without significant bias from modeling constraints and identify Φ = -62°, ψ = -43 as the consensus backbone torsional angles of protein helices. These torsional angles preserve the atomic positions of α-β carbons of the classic Pauling α-helix while allowing the amide carbonyls to form bifurcated hydrogen bonds as first suggested by Némethy et al. in 1967. Molecular dynamics simulations of a capped 12-residue oligoalanine in water with AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications), a second-generation force field that includes multipole electrostatics and polarizability, reproduces the experimentally observed high-resolution helical conformation and correctly reorients the amide-bond carbonyls into bifurcated hydrogen bonds. This simple modification of backbone torsional angles reconciles experimental and theoretical views to provide a unified view of amide three-centered hydrogen bonds as crucial components of protein helices. The reason why they have been overlooked by structural biologists depends on the small crankshaft-like changes in orientation of the amide bond that allows maintenance of the overall helical parameters (helix pitch (p) and residues per turn (n)). The Pauling 3.613 α-helix fits the high-resolution experimental data with the minor exception of the amide-carbonyl electron density, but the previously associated backbone torsional angles (Φ, Ψ) needed slight modification to be reconciled with three-atom centered H-bonds and multipole electrostatics. Thus, a new standard helix, the 3.613/10-, Némethy- or N-helix, is proposed. Due to the use of constraints from monopole force fields and assumed secondary structures used in low-resolution refinement of electron density of proteins, such structures in the PDB often show linear hydrogen bonding.

Multipolar force fields and their effects on solvent dynamics around simple solutes.

The performance of multipole (MTP) and point charge (PC) force fields in classical molecular dynamics (MD) simulations of condensed-phase systems for both equilibrium and dynamical quantities is compared. MTP electrostatics provides an improved description of the anisotropic electrostatic potential, which is especially important to describe key, challenging interactions, such as lone pairs, π-interactions, and hydrogen bonds. These chemical environments are probed by focusing on the hydration properties of two molecules: N-methylacetamide and phenyl bromide. Both, equilibrium and dynamical, quantities are affected by the quality of the electrostatic model. The alteration of the first solvation shell in MTP simulations is validated by comparing with lifetimes and correlation times of solute-solvent interactions from experiment. The improved dynamical behavior found in the MTP simulations—observed for molecules parametrized using very different protocols—suggests that a systematic improvement of both equilibrium and dynamical quantities when using MTP electrostatics is possible.

Toward transferable interatomic van der Waals interactions without electrons: the role of multipole electrostatics and many-body dispersion.

We estimate polarizabilities of atoms in molecules without electron density, using a Voronoi tesselation approach instead of conventional density partitioning schemes. The resulting atomic dispersion coefficients are calculated, as well as many-body dispersion effects on intermolecular potential energies. We also estimate contributions from multipole electrostatics and compare them to dispersion. We assess the performance of the resulting intermolecular interaction model from dispersion and electrostatics for more than 1300 neutral and charged, small organic molecular dimers. Applications to water clusters, the benzene crystal, the anti-cancer drug ellipticine-intercalated between two Watson-Crick DNA base pairs, as well as six macro-molecular host-guest complexes highlight the potential of this method and help to identify points of future improvement. The mean absolute error made by the combination of static electrostatics with many-body dispersion reduces at larger distances, while it plateaus for two-body dispersion, in conflict with the common assumption that the simple 1/R(6) correction will yield proper dissociative tails. Overall, the method achieves an accuracy well within conventional molecular force fields while exhibiting a simple parametrization protocol.

Spherical tensor multipolar electrostatics and smooth particle mesh Ewald summation: a theoretical study.

The point-charge approximation, typically used by classical molecular mechanics force-fields, can be overcome by a multipolar expansion. For decades multipole moments were only used in the context of the rigid body approximation but recently it has become possible to combine multipolar electrostatics with molecular flexibility. The program DL_MULTI, which is derived from DL_POLY_2, includes efficient multipolar Ewald functionality up to the hexadecapole moment but the code is restricted to rigid bodies. The incorporation of flexibility into DL_MULTI would cause too large an impact on its architecture whereas the package DL_POLY_4 offers a more attractive and sustainable route to handle multipolar electrostatics. This package inherently handles molecular flexibility, which warrants sufficiently transferable atoms or atoms that are "knowledgeable" about their chemical environment (as made possible by quantum chemical topology and machine learning). DL_MULTI uses the spherical multipole formalism, which is mathematically more involved than the Cartesian one but which is more compact. DL_POLY_4 uses the computationally efficient method of smooth particle mesh Ewald (SPME) summation, which has also been parallellized by others. Therefore, combining the strengths of DL_POLY_4 and DL_MULTI poses the challenge of merging SPME with multipolar electrostatics by spherical multipole. In an effort to recast as clearly as possible the principles behind DL_MULTI, its key equations have been reformulated by the more streamlined route involving the algebra of complex numbers, and some of these equations' peculiarities clarified. This article explores theoretically the repercussions of the merging of SPME with spherical multipole electrostatics (as implemented in DL_MULTI). Difficulties in design and implementation of possible future code are discussed.

Leveraging Symmetries of Static Atomic Multipole Electrostatics in Molecular Dynamics Simulations

Multipole (MTP) electrostatics provides the means to describe anisotropic interactions in a rigorous and systematic manner. A number of earlier molecular dynamics (MD) implementations have increasingly relied on the use of molecular symmetry to reduce the (possibly large) number of MTP interactions. Here, we present a CHARMM implementation of MTP electrostatics in terms of spherical harmonics. By relying on a systematic set of reference-axis systems tailored to various chemical environments, we obtain an implementation that is both efficient and scalable for (bio)molecular systems. We apply the method to a series of halogenated compounds to show (i) energy conservation; (ii) improvements in reproducing thermodynamic properties compared to standard point-charge (PC) models; (iii) performance of the code; and (iv) better stabilization of a brominated ligand in a target protein, compared to a PC force field. The implementation provides interesting perspectives toward a dual PC/MTP resolution, à la QM/MM.

Scoring Multipole Electrostatics in Condensed-Phase Atomistic Simulations

Permanent multipoles (MTPs) embody a natural extension to common point-charge (PC) representations in atomistic simulations. In this work, we propose an alternative to the computationally expensive MTP molecular dynamics simulations by running a simple PC simulation and later reevaluate-"score''-all energies using the more detailed MTP force field. The method, which relies on the assumption that the PC and MTP force fields generate closely related phase spaces, is accomplished by enforcing identical sets of monopoles between the two force fields-effectively highlighting the higher MTP terms as a correction to the PC approximation. We first detail our consistent parametrization of the electrostatics and van der Waals interactions for the two force fields. We then validate the method by comparing the accuracy of protein-ligand binding free energies from both PC and MTP-scored representations with experimentally determined binding constants obtained by us. Specifically, we study the binding of several arylsulfonamide ligands to human carbonic anhydrase II. We find that both representations yield an accuracy of 1 kcal/mol with respect to experiment. Finally, we apply the method to rank the energetic contributions of individual atomic MTP coefficients for molecules solvated in water. All in all, MTP scoring is a computationally appealing method that can provide insight into the multipolar electrostatic interactions of condensed-phase systems.

Limiting assumptions in molecular modeling: electrostatics

Molecular mechanics attempts to represent intermolecular interactions in terms of classical physics. Initial efforts assumed a point charge located at the atom center and coulombic interactions. It is been recognized over multiple decades that simply representing electrostatics with a charge on each atom failed to reproduce the electrostatic potential surrounding a molecule as estimated by quantum mechanics. Molecular orbitals are not spherically symmetrical, an implicit assumption of monopole electrostatics. This perspective reviews recent evidence that requires use of multipole electrostatics and polarizability in molecular modeling.

Molecular Dynamics of β-Hairpin Models of Epigenetic Recognition Motifs

The conformations and stabilities of the β-hairpin model peptides of Waters (Riemen, A. J.; Waters, M. L. Biochemistry 2009, 48, 1525; Hughes, R. M.; Benshoff, M. L.; Waters, M. L. Chemistry 2007, 13, 5753) have been experimentally characterized as a function of lysine ε-methylation. These models were developed to explore molecular recognition of known epigenetic recognition motifs. This system offered an opportunity to computationally examine the role of cation-π interactions, desolvation of the ε-methylated ammonium groups, and aromatic/aromatic interactions on the observed differences in NMR spectra. AMOEBA, a second-generation force field (Ponder, J. W.; Wu, C.; Ren, P.; Pande, V. S.; Chodera, J. D.; Schnieders, M. J.; Haque, I.; Mobley, D. L.; Lambrecht, D. S.; DiStasio, R. A., Jr.; Head-Gordon, M.; Clark, G. N.; Johnson, M. E.; Head-Gordon, T. J. Phys. Chem. B 2010, 114, 2549), was chosen as it includes both multipole electrostatics and polarizability thought to be essential to accurately characterize such interactions. Independent parametrization of ε-methylated amines was required from which aqueous solvation free energies were estimated and shown to agree with literature values. Molecular dynamics simulations (100 ns) using the derived parameters with model peptides, such as Ac-R-W-V-W-V-N-G-Orn-K(Me)(n)-I-L-Q-NH(2), where n = 0, 1, 2, or 3, were conducted in explicit solvent. Distances between the centers of the indole rings of the two-tryptophan residues, 2 and 4, and the ε-methylated ammonium group on Lys-9 as well as the distance between the N- and C-termini were monitored to estimate the strength and orientation of the cation-π and aromatic/aromatic interactions. In agreement with the experimental data, the stability of the β-hairpin increased significantly with lysine ε-methylation. The ability of MD simulations to reproduce the observed NOEs for the four peptides was further estimated for the monopole-based force fields, AMBER, CHARMM, and OPLSAA. AMOEBA correctly predicted over 80% of the observed NOEs for all 4 peptides, while the three-monopole force fields were 40-50% predictive in only 2 cases and approximately 10% in the other 10 examples. Preliminary analysis suggests that the decreased cost of desolvation of the substituted ammonium group significantly compensated for the reduced cation-π interaction resulting from the increased separation due to steric bulk of the ε-methylated amines.

Polarizable atomic multipole X-ray refinement: weighting schemes for macromolecular diffraction

In the past, weighting between the sum of chemical and data-based targets in macromolecular crystallographic refinement was based on comparing the gradients or Hessian diagonal terms of the two potential functions. Here, limitations of this scheme are demonstrated, especially in the context of a maximum-likelihood target that is inherently weighted by the model and data errors. In fact, the congruence between the maximum-likelihood target and a chemical potential based on polarizable atomic multipole electrostatics evaluated with Ewald summation has opened the door to a transferable static weight. An optimal static weight is derived from first principles and is demonstrated to be transferable across a broad range of data resolutions in the context of a recent implementation of X-ray crystallographic refinement using the polarizable AMOEBA force field and it is shown that the resulting models are balanced with respect to optimizing both R(free) and MolProbity scores. Conversely, the classical automatic weighting scheme is shown to lead to underfitting or overfitting of the data and poor model geometry. The benefits of this approach for low-resolution diffraction data, where the need for prior chemical information is of particular importance, are also highlighted. It is demonstrated that this method is transferable between low- and high-resolution maximum-likelihood-based crystallographic refinement, which proves for the first time that resolution-dependent parameterization of either the weight or the chemical potential is unnecessary.

Multipole electrostatics in hydration free energy calculations

Hydration free energy (HFE) is generally used for evaluating molecular solubility, which is an important property for pharmaceutical and chemical engineering processes. Accurately predicting HFE is also recognized as one fundamental capability of molecular mechanics force field. Here, we present a systematic investigation on HFE calculations with AMOEBA polarizable force field at various parameterization and simulation conditions. The HFEs of seven small organic molecules have been obtained alchemically using the Bennett Acceptance Ratio method. We have compared two approaches to derive the atomic multipoles from quantum mechanical calculations: one directly from the new distributed multipole analysis and the other involving fitting to the electrostatic potential around the molecules. Wave functions solved at the MP2 level with four basis sets (6-311G*, 6-311++G(2d,2p), cc-pVTZ, and aug-cc-pVTZ) are used to derive the atomic multipoles. HFEs from all four basis sets show a reasonable agreement with experimental data (root mean square error 0.63 kcal/mol for aug-cc-pVTZ). We conclude that aug-cc-pVTZ gives the best performance when used with AMOEBA, and 6-311++G(2d,2p) is comparable but more efficient for larger systems. The results suggest that the inclusion of diffuse basis functions is important for capturing intermolecular interactions. The effect of long-range correction to van der Waals interaction on the hydration free energies is about 0.1 kcal/mol when the cutoff is 12Å, and increases linearly with the number of atoms in the solute/ligand. In addition, we also discussed the results from a hybrid approach that combines polarizable solute with fixed-charge water in the HFE calculation.

Polarizable Atomic Multipole X-Ray Refinement: Hydration Geometry and Application to Macromolecules

We recently developed a polarizable atomic multipole refinement method assisted by the AMOEBA force field for macromolecular crystallography. Compared to standard refinement procedures, the method uses a more rigorous treatment of x-ray scattering and electrostatics that can significantly improve the resultant information contained in an atomic model. We applied this method to high-resolution lysozyme and trypsin data sets, and validated its utility for precisely describing biomolecular electron density, as indicated by a 0.4–0.6% decrease in the R- and Rfree-values, and a corresponding decrease in the relative energy of 0.4–0.8 Kcal/mol/residue. The re-refinements illustrate the ability of force-field electrostatics to orient water networks and catalytically relevant hydrogens, which can be used to make predictions regarding active site function, activity, and protein-ligand interaction energies. Re-refinement of a DNA crystal structure generates the zigzag spine pattern of hydrogen bonding in the minor groove without manual intervention. The polarizable atomic multipole electrostatics model implemented in the AMOEBA force field is applicable and informative for crystal structures solved at any resolution.

X-Ray Crystallography Refinement as Ewald Intended: From Drug Design to Ribosome Crystals

Precise protein-ligand complexes determined via X-ray crystallography experiments are critical for rational drug design. Here we demonstrate both improved statistical precision and greater chemical information content for a range of systems beginning with the trypsin-benzamidine complex shown below and scaling up to ribosome refinements. Keys to our method include use of the AMOEBA polarizable atomic multipole force field combined with a space group particle mesh Ewald (SG-PME) electrostatics algorithm that exploits space group symmetry to increase speed and reduce memory requirements. At high resolution (better than 1.0 A), we achieve lower R-free statistics by using the AMOEBA electron distribution within the X-ray scattering model. At all resolutions, rigorous treatment of polarizable atomic multipole electrostatics via SG-PME improves molecular conformation and orients the water hydrogen bonding network.View Large Image | View Hi-Res Image | Download PowerPoint Slide

Molecular Dynamics Simulations of Bromine Clathrate Hydrates

A polarizable force field that explicitly includes contributions from exchange repulsion, dispersion, charge penetration, and multipole electrostatics was developed to describe the interaction between bromine and water. This force field was combined with a polarizable force field for water and used in molecular dynamics simulations to calculate the relative energetics of three bromine clathrate hydrates. The simulations predict the tetragonal structure (Allen, K. W.; Jeffrey, G. A. J. Chem. Phys. 1963, 38, 2304) to be the most stable, with the CS-I and CS-II cubic structures being less stable. Although the CS-II species is not the most stable energetically, we argue that it could be formed under conditions of low bromine concentration, in agreement with recent measurements (Goldschleger, I. U.; Kerenskaya, G.; Janda, K. C.; Apkarian, V. A. J. Phys. Chem. A 2008, 112, 787) that provide evidence for three different bromine hydrate crystal types.

From Data or Dogma? The Myth of the Ideal Helix

In the course of building detailed surface representations for peptidomimetics, we were motivated to analyze the detailed surfaces of helices in proteins. Unexpectedly, we observed few ideal helical forms in high-resolution protein structures. Instead of a bimodal distribution matching the well-known alpha and 3(10) helical forms, we observed a smooth, single-peaked population, characterized by intermediate helices with shared hydrogen bonds. Bifurcated, or three-center hydrogen bonds, have been well-documented in small molecules and peptides, but they're rarely highlighted in the context of folded proteins. The data suggests shared hydrogen bonds are a major component of helices in proteins.Contrary to the Pauling-Corey-Branson models, we did not restrict our analysis to single hydrogen bonds--shared three-center hydrogen bonds were included. High resolution (<2.0 A), electron density data is sharp enough that a helical structure should be unambiguous and accurately modeled. These helices are centered on an intermediate helical form. At poorer resolutions (2.0-5.0 A), electron density is ambiguous; refinement fitting methods are employed to model ideal structures into the data. Here there is an enriched population of ideal structures. A structurally representative subset of proteins reveals the same trends as the whole PDB. The data support the observation that ideal helical parameters do not accurately describe the distribution of real helices in proteins.Hydrogen bonds are a polar (and polarizable) moiety and an accurate model must account for this. We present data comparing molecular dynamics simulations using popular monopole force fields (OPLS-AA and CHARMM) with a next generation force field (AMOEBA) implementing polarizability and multipole electrostatics. AMOEBA simulations are shown to quantitatively reproduce the experimentally observed trends in helical populations. These results emphasize the importance of using appropriate force field potential models when simulating hydrogen bonded structures in proteins.

Yukawa multipole electrostatics and nontrivial coupling between electrostatic anddispersion interactions in electrolytes

An exact treatment of screened electrostatics in electrolyte solutions is presented. Inelectrolytes the anisotropy of the exponentially decaying electrostatic potentialfrom a molecule extends to the far field region. The full directional dependenceof the electrostatic potential from a charged or uncharged molecule remains inthe longest range tail (i.e. from all multipole moments). In particular, the rangeof the potential from an ion and that from an electroneutral polar particle isgenerally exactly the same. This is in contrast to the case in vacuum or pure polarliquids, where the potential from a single charge is longer ranged than that from adipole, which is, itself, longer ranged than the one from a quadrupole etc. Theorientational dependence of the exponentially screened electrostatic interaction betweentwo molecules in electrolytes is therefore rather complex even at long distances.These facts are formalized in Yukawa multipole expansions of the electrostaticpotential and the pair interaction free energy based on the Yukawa function familyexp(−κr)/rm, wherer is the distance,κ is a decayparameter and m is a positive integer. The expansion is formally exact for electrolytes with molecular solventand in the primitive model, provided the non-Coulombic interactions between the particlesare sufficiently short ranged. The results can also be applied in the Poisson–Boltzmannapproximation. Differences and similarities to the ordinary multipole expansion ofelectrostatics are pointed out.On the other hand, when the non-Coulombic interactions between theconstituent particles of the electrolyte solution contain a dispersion1/r6 potential, the electrostatic potential from a molecule decays like a power law for longdistances rather than as a Yukawa function. This is due to nontrivial coupling between theelectrostatic and dispersion interactions. There remains an exponentially decayingcomponent in the electrostatic potential, but it becomes oscillatory in the presence of thedispersion interactions. For weak dispersion forces and low electrolyte concentrations, thewavelength is, however, long compared to the decay length of the exponentialdecay. In other cases the qualitative behaviour may be substantially differentfrom the conventional picture. The Green function for the electrostatic potential(the ‘screened Coulomb potential’) in simple electrolytes ultimately decays likeconst/r6 forlarge r, where the constant prefactor depends on the ratio between the strengthof the dispersion forces and the square of the average ionic charge,(q++|q−|)/2.