On shortranged pairpotentials for longrange electrostatics†‡
Received
10th July 2019
, Accepted 23rd October 2019
First published on 24th October 2019
In computer simulations, longrange electrostatic interactions are surprisingly well approximated by truncated, shortranged pair potentials. Examples are reaction field methods; the Wolf method; and a number of schemes based on cancellation of electric multipole moments inside a cutoff region. These methods are based on the assumption that the polarization of the neglected surroundings can be inferred from a local charge distribution. Multipole moments themselves are only approximations to the true charge distribution, approximations which many times are needed to simplify calculations in complex systems. In this work we investigate a new, generalized pairpotential based on the idea of moment cancellation that covers interactions between electrostatic moments of any type. We find that moment cancellation in itself is insufficient to generate accurate results and a more restricted formalism is needed, in our case to cancel the virtual charges of the imposed moments. Thus, it is unfeasible to cancel higherorder moments with explicit higherorder moments such as dipoles and instead image charges are needed. The proposed pairpotential is general and straight forwardly implementable for any electrostatic moment – monopole, dipole, quadrupole, etc. – with a computational complexity scaling with the number of particles in the system.
Introduction
Accounting for longranged electrostatic interactions in computer simulations is an exquisite task^{1–4} and although formally exact theories exist for repetitive structures subjected to periodic boundary conditions (PBC),^{5,6} these are computationally expensive and may impose artificial symmetry from periodicity. Finiteranged or truncated pairpotentials are fast alternatives to such latticesum models and can be more relevant (and often only valid) for isotropic systems.^{7,8} Research of truncated pairpotentials covers many different approaches and we here highlight the concept of electric multipolar moment cancellation. That is, the effect of long ranged electrostatic interactions is approximated by cancelling one, two, or more electric multipolar moments in the cutoff sphere,^{1,3} mimicking polarization of the surroundings. The newly developed qpotential^{9} for ion–ion interactions is a generalization of this concept, and allows for cancellation of an arbitrary number of moments while being free from empirical damping parameters. In this work we expand the qpotential model to include any type of higherorder interactions, i.e. ion–dipole, dipole–dipole, etc. The methodology applied on dipole–dipole interactions is validated via fluid phase simulations and two distinct strategies for cancelling moments are critically investigated and compared with results from Ewald summation. Our discussion entails connections to existing methodologies and, hence, the next section is a brief overview of the field.
Overview
Developing and deriving approximate pairpotentials for long range electrostatic interactions is an active and longstudied research topic. Instead of rigorously obeying the Poisson equation for the system of interest, an approximate solution is used. For example, for ion–ion interactions, the Coulomb potential is simply scaled with a shortranged function (q), see eqn (1). 
 (1) 
Here k_{c} is the Coulomb constant, e is the elementary charge, z_{i} and z_{j} are the valence of particles i and j respectively, r is the interionic distance, and q = r/R_{c}, where R_{c} is a spherical cutoff after which = 0. The choice of the shortranged function is however delicate. Many variants have been developed and Table 1 gives a nonexhaustive list of such (q). In this table we have selected three main groups of pairpotentials: reaction field methods; damping based methods; and damping free methods. The reaction field methods are based on a cavity occupied by explicit particles which in turn is surrounded by a dielectric continuum. The particles induce electrostatic moments in the implicit medium which in turn induce a field which interacts with the particles. The damping based methods are mainly evolved from the Ewald summation method, neglecting reciprocal space. To avoid a discontinuity at the cutoff distance, the potential and/or its derivative is shifted to zero. The damping free methods are somewhat differently derived: some are empirically fitted; some are derived as approximations to the Poisson equation; and some implicitly incorporate the damping parameter into a polynomial form. Like the damping based methods they usually also shift the potential and its higher order derivatives at the cutoff.
Table 1 Shortranged functions, (q), for various electrostatic schemes for ion–ion interactions where q = r/R_{c} with r being the distance between the charges and R_{c} the cutoff. η = αR_{c}, where α is the commonly used dampingparameter. For the top three reaction field type methods ε_{RF} is the relative permittivity of the surrounding medium, and for the bottom scheme P is the number of cancelled moments. Note that (q) = 0 for q > 1, i.e. for r > R_{c}
Shortrange function, (q) 
Ref. 
Reaction field methods 

Reaction field^{10} 

7


11


Damping based methods 
erfc(qη) 
Realspace Ewald^{5} 
[η = 0 → 1] 

erfc(qη) − qerfc(η) 
Wolf^{1} 
[η = 0 → 1 − q] 


12

[η = 0 → 1 + q − q^{2}] 


13

[η = 0 → (1 − q)^{2}] 
14


3




Damping free methods 
(1 − q)^{3} 
15

(1 − q)^{4} 
16

(1 + q)(1 − q)^{3} 
17

(1 + 2q + 2q^{2})(1 − q)^{4} 
4


18


qPotential^{9} 
In addition to Table 1 there are many other schemes: effective potentials,^{19} image methods,^{2,20} latticemethods,^{5,6,21–23} and the fastmultipolemethod^{24,25} among others. However, many times the charge distribution of a system is anisotropic and it is convenient to approximate it by electric point multipole moments instead of point charges. These moments each account for a portion of the charge distribution and do therefore inherently contain the (now implicit) charges. Several of the mentioned ion–ion approaches have also been expanded to, or do cover, multipole moment (i.e. higherorder) interactions.^{6,10,26–28} Each scheme has (dis)advantages and the choice of the summation method is sensitive to the specific system. Often shortrange pairpotentials for electrostatics are derived based on the assumption of a homogeneous and isotropic distribution beyond the cutoff region. Therefore such approaches should be used with care in systems where this might not be true, for example at interfaces.^{29} In the following we expand the theory of the qpotential to higher order interactions and connect the formalism to existing schemes presented in Table 1.
Theory
Multipolar energy in a periodic system
For a cuboidal unit cell with periodic boundaries, the total interaction energy between N particles with electric point moments of type v and w is 
 (2) 
Here the prime indicates that i ≠ j when n = 0, r_{ij} is the distancevector between the centers of moments i and j, ∘ is the Hadamard product, and the size of the cuboid cell is described by sidelengths L = (L_{x},L_{y},L_{z}). The lth order interactiontensor T_{l}(r), using l ∈ _{0}, is introduced as 
 (3) 
where r = r, which gives ion–ion interactions using l = 0, ion–dipole using l = 1, dipole–dipole using l = 2, and so forth. Note that the factor 1/2 in eqn (2) is included only for like type moments v and w.
Derivation of the multipolar qpotential
Consider a local region of an isotropic system – a spherical cutoff sphere, for example – where the contained charge distribution is selfconsistently polarized by its surroundings. We now describe both the local charge distribution and the surroundings as two multipoles. These will be oppositely polarized and, given that the local region is sufficiently large, perfectly cancel each other. This physical relation is the main idea behind the qpotential,^{9} where image charges (or moments) of the local region are used to generate the opposing multipole of the surroundings (see Fig. 1). Perfect cancellation of the local and surrounding multipole moments is achieved by requiring that their sum results in zerotensors, leading to an effectively shortranged potential. Therefore, for a reasonably large cutoff region, the presented approach is valid for both PBC and nonPBC systems. A similar observation has been made in lattice systems^{6,30} where the effective Coulomb interaction rapidly decays as r^{−5}.

 Fig. 1 Illustration of how dipole moment cancellation is used to approximate the exact electrostatic energy using two different schemes. The pairpotential truncation neglects all interactions beyond the cutoff radius which is corrected for using either image charges (q_{0}) or imagelike dipoles (q_{2}). The former is found to be more appropriate.  
By enforcing total moments described by zerotensors,^{1,3,9}i.e. achieving a shortranged potential, eqn (2) is simplified to a single cell n = 0, or for nonPBC systems, a local region. This has previously been done for ion–ion interactions (see especially eqn (9) in ref. 9) and generalizing this procedure, we obtain the modified interaction tensor for any type of electrostatic interactions,

 (4) 
Here
P ∈
index the number of cancelled moments,
is the
qanalogue of the binomial coefficient, and (
a;
q)
_{P} is the
qPochhammer symbol.
^{9} Since the multiplicative factor to the original interactiontensor is a
qanalogue, we index the modified interactiontensor with a superscript
q and from here on include the derived potential in the
qpotential notation. Like for the Coulomb
qpotential, we deduce the number of cancelled higherorder moments
P − 1 to equal the number of derivatives of
T^{q}_{l}(
r) with respect to
r to be zero at the cutoff. From now on, when we mention the derivative of
T^{q}_{l}(
r), it is with respect to
r.
Moment cancellation schemes
We now detail two moment cancellation schemes – see Fig. 1 – differing in how they cancel moments. The first approach, which has previously been validated for ion–ion interactions,^{9} uses image charges (l = 0) while the second approach uses imagelike dipoles (l = 2) to cancel higherorder moments. We later revisit the meaning of imagelike dipoles.
The image charge approach is derived using the ion–ion interactiontensor from the Coulomb qpotential, where higherorder interactions are described by applying the gradient(s) to the same, that is ∇^{k}T^{q}_{0}(r) where k ∈ _{0}. This approach thus cancels the (implicit) charges of the moments and their higherorder moments by using image charges. Note that there is an arbitrariness in image charge positioning while still achieving moment cancellation, yet the qpotential is based on a physically inspired scheme.^{9,20,31} For each use of the gradientoperator on T^{q}_{0}(r), one loses the highest order derivative of the interactiontensor to be zero at the cutoff. This is expressed in eqn (5), for any l, where the gradientoperator is applied k times, leaving p ≤ P − 1 − k higherorder derivatives zero at the cutoff.

 (5) 
Therefore, if for example we require the dipole–dipole interactionenergy (
l +
k = 2) using ∇
^{2}T^{q}_{0} to be zero at the cutoff distance (
i.e. no higherorder cancellation, or
p = 0), then we must use 0 + 2 ≤
P − 1 →
P ≥ 3. This is equivalent to cancelling the implicit charge (
P ≥ 1), dipole (
P ≥ 2) and quadrupole (
P ≥ 3) moment.
The second approach simply makes use of the dipole–dipole interactiontensor T^{q}_{2} directly, and thus 0 + 0 ≤ P − 1 → P ≥ 1 is enough for the interactionenergy to be zero at the cutoff distance. Here the explicit dipole moment is cancelled but not the implicit charges, nor necessarily the quadrupole moment.
We define q_{l}(P) to be the qpotential, where the cancellation is based either on image charges (l = 0) or on imagelike dipoles (l = 2), which cancel P − 1 higherorder moments where P is counted from what type of base moment is used in the cancellation procedure. When not using the index (P), like in q_{l}, we refer to the general group of pairpotentials using l but for any P.
The presented qpotential approaches can be viewed as a generalization of global electroneutrality. Since longrange electrostatic interactions solely come in the form of ion–ion, ion–dipole, dipole–dipole, and ion–quadrupole interactions, we recognize that to accurately compensate for the neglected contributions outside of the cutoff region we generally need to cancel at least up to the quadrupolemoment. Higher order interactions are shortranged and thus given a large enough cutoff region, these interactions will be explicitly accounted for. However, the cutoff region will also need to be large enough as to capture the local anisotropy of the system^{8} with an upper limit of R_{c} ≤ min(L)/4. Thus, the longrange interactions determine P and the shortrange ones together with the lengthscale of the local anisotropy determine R_{c}.
Now we address the use of imagelike dipoles. For a charge at position r relative to the origin, its image charge in a conductive sphere with radius R_{c} centered in the origin has a charge scaled by −R_{c}/r as compared to the charge at r. Since potentials due to charges are isotropic, the charge and image charge have the same angular (in)dependence with regard to the origin. For dipoles however this is not true. Given a dipole moment μ not perpendicular to r, there is an angular alteration to its imagedipole μ′; see eqn (6) where the hats indicate normalized vectors.

μ′ ∝ μ − 3(·μ)  (6) 
In this work we have used the exact image moment positions, yet fixed the angular part of the image moments, in the dipole example that is
μ′ ∝
μ. Thus these dipoles are imagelike. We later discuss how this may affect the results.
Summarizing the above, the final interaction tensor for dipole–dipole interactions using the qpotential is given by

T^{q}(r) = T_{2}(r)a(q) + Ir^{−3}b(q)  (7) 
where
I is the identity matrix,

 (8) 
for
l = 0 and
k = 2 (
i.e. using image charges), whereas
a(
q) = (
q^{3};
q)
_{P} and
b(
q) = 0 for
l = 2 and
k = 0 (
i.e. using imagelike dipoles).
In the original study^{9} ion–ion interactions (i.e. the q_{0}potential) were investigated, and indeed the potential was found to be isotropic enough for watersystems. From a theoretical pointofview the q_{0} approach is more isotropic than that of q_{2}, due to the isotropy of the image charges and anisotropy of the imagelike dipoles. However, this does not exclude the q_{2}potential from being isotropic itself. Similarly it does not prove that the q_{0} approach is isotropic enough for being an efficient pairpotential using dipole–dipole interactions. That is why we in the next section investigated the presented potentials using numerical simulations.
Metropolis Monte Carlo simulations
Metropolis Monte Carlo simulations of dipolar Stockmayer fluids^{32} in the canonical ensemble were performed using the Faunus software,^{33} with N = 3000, density ρ* = 0.924, temperature T* = 1.333, and squared dipole moment μ*^{2} = 3.470, where stars indicate reduced units. The equilibration used N × 10^{5} steps each consisting of a combined translational and rotational Monte Carlo attempt. Production runs were ten times longer. The Ewald summation method was used as a reference, and comparisons will always be with regard to this scheme if not stated otherwise. The used cutoff was R_{c}* = 4 for the pairpotentials while Ewald summations used a realspace cutoff of half the boxlength, a reciprocalspace spherical integer cutoff of 9, a dampingparameter equal to π/R_{c}, and a conducting surrounding dielectric medium. The q_{0}potential was tested from P = 3 and higher values, and the q_{2}potential from P = 1. These values were chosen since their respective interaction tensors are zero at the cutoff distance for such P, with no higherorder derivatives cancelled. Thus P = 3 for q_{0} and P = 1 for q_{2} are directly comparable in this regard.
Results
Radial distribution functions
To validate the developed pairpotentials, we first compare radial distribution functions, g(r), with reference Ewald summation results. Fig. 2 shows the logarithm of the ratio between q_{l}potential and Ewald radial distribution functions. It is clear that q_{0} is closer to the reference than q_{2}, and that the largest discrepancy is between the smallest Pvalues for both q_{0} and q_{2}. However, while for q_{0} going from P = 3 to P = 4 gives more accurate results, for q_{2} going from P = 1 to P = 2 gives (slightly) less accurate results. Yet, no q_{2}potential rivals the results from any q_{0}potential.

 Fig. 2 The logarithm of the ratio between pairpotential and reference Ewald (black lines) radial distribution functions using q_{0} (top) and q_{2} (bottom). The insets show the (overlapping) radial distribution functions.  
Dipole–dipole correlations
Fig. 3 shows dipole–dipole correlation differences to Ewald results, and it is clear that q_{2} is incapable of capturing the system properties, even being negative at small distances. We also see that q_{0}(P = 4), i.e. up until the first derivative of the interactiontensor is zero at the cutoff, is closest to the reference akin to q_{0}(P = 5) (see ESI‡), and that increasing P beyond these values gives worse agreement.

 Fig. 3 Dipole–dipole correlation differences to the reference result using q_{0} (top) and q_{2} (bottom). The insets show the dipole–dipole correlation 〈(0)·(r*)〉. The black lines are Ewald results.  
In the ESI,‡ we have included more results from the simulations (energies, mean squared dipole moments), and presented results for different cutoffs. Those data qualitatively show the same behaviour of q_{0} and q_{2} as shown so far and thus further verify the outcome of the analysis in the next section.
Discussion
The q_{2}potentials cancel moments by using imagelike dipoles, yet the simulation results are far from the Ewald summation reference. By expanding the ion–ion interactiontensor to dipolar systems, i.e. by using the q_{0}potentials which cancel moments by means of image charges, we get more reasonable outcomes. Though both approaches make use of moment cancellation there is a salient difference in how they do so, which is reflected in the simulation results. We conclude that moment cancellation in itself is insufficient whereas neutralizing the (implicit) charges of the moments and cancelling their higherorder moments provide accurate results. This contrasts other works which like the q_{2}potential uses explicit dipolecancellation, but needs to introduce an arbitrary damping parameter to remedy poor results.^{27,28} A single higherorder cancellation is sufficient in many regards to accurately retrieve valid results when using the q_{0}potential. This is convenient from a computational pointofview since molecular dynamics equivalently requires zero force at the cutoff. A modest increase in the cancellationorder improves the results slightly.
There are two main theoretical differences between the q_{0} and q_{2}approaches beyond the ones mentioned in the previous paragraph. First, the q_{0} image particles can be interpreted as having a nonlocalized chargedistribution on the shell of a sphere, whereas the q_{2} image particles are true pointparticles (see Fig. 1). Thus the nature of the q_{0}potential is more isotropic than that of the q_{2}potential, a feature which seemingly affects the results. Second, the q_{0} image particles are exact image particles, whereas the q_{2} image particles are imagelike as discussed earlier. Since we have found that q_{0} provides more accurate results than q_{2}, which might be a consequence of the just mentioned facts, there may be a corresponding version of the q_{2}potentials which utilizing explicit dipole cancellation and generates accurate results using nonlocalized image dipole distributions, or exact image particles. For example, by using a nonlocalized dipole moment at the cutoff sphere (P = 1) instead of imagelike dipoles, we regain the reactionfield method^{10} using ε_{RF} = ∞. By comparing our results to those in a previous study using the described reactionfield method on an identical system as simulated here,^{28} we note that such an approach is more accurate than q_{2} yet less so than q_{0}.
Furthermore, we note that the currently proposed formalism is only one of formally infinitely many using the concept of electrostatic cancellation of moments. This since the position of the image moments can be altered while keeping the feature of momentcancellation intact (see eqn (7) in the derivation of the qpotential^{9}). The positions chosen in this work, and for the Coulomb qpotential, are however physically based and in line with previous works on image moments.^{9,20,31}
We have found that q_{0} using P = 4–5 gives most accurate results for the simulated bulk Stockmayersystem, like for the ionic qpotential for watersystems.^{9} From theory we might however expect P = ∞ to give most accurate results since all moments then are cancelled and no explicit interactions with the surrounding are sustained. Nonetheless, most systems exhibit some form of local fluctuations and a too large value of P will therefore suppress this inherent quality. Higher order interactions than dipole–dipole or ion–quadrupole are shortrange and thus can be captured with a large enough cutoff region. Hence, infinite cancellation is not necessarily needed. Assuming that the fluctuations of the cutoff region will decay while increasing its size, a large P in combination with a large enough R_{c} will still give accurate results. Thus, the order of the fluctuations in the system seems to determine the upper limit of P while the order of longrange interactions determines the lower limit, P = 3.
Finally, the presented formalism based on the T^{q}_{l}(r)tensors is applicable, though not tested, for other electrostatic interactions. For example, the ion–quadrupole interaction makes use of the same interactiontensor as in the dipole–dipole interaction. We therefore surmise our presented results to be generalizable to this case, i.e. q_{0} will render more accurate results than q_{2}. Though there is little need to use summation methods for even higherorder electrostatic interactions, the procedure is valid for similar cases outside the electrostatic framework in which arbitrary higherorder moments need cancellation.
Conclusion
We have expanded a formalism accounting for longranged Coulomb (ion–ion) interactions by means of moment cancellation to include higherorder electrostatic interactions. The expansion was done in two fundamentally different ways: cancellation of the implicit charges inherent to any higherorder electrostatic moment and their higherorder moments by means of image charges; and cancellation of the explicit moments and their higherorder moments by means of images of said moments. The results unambiguously show moment cancellation to be insufficient to generate accurate results, rather we find that cancellation of the (implicit) charges inherent to the used moments gives comparable results to the standard Ewald summation.
The expanded validated formalism is general and straight forwardly implementable for higherorder electrostatic interactions with a computational cost proportional to the number of particles in the system. The method utilizes a cutoff region large enough to represent the sample, and a cancelling parameter directly connected to the relevant higherorder moments in this region, and it therefore avoids any arbitrary dampingparameter.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
We thank Swedish Research Foundation (grant number 201704372) for financial support and LUNARC in Lund for computational resources.
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Footnotes 
† A C++ implementation of the derived potential as well as other truncation based schemes are available at https://doi.org/10.5281/zenodo.3522058. 
‡ Electronic supplementary information (ESI) available: Detailing cutoff effects; derivation of selfenergies and the dielectric constant. See DOI: 10.1039/c9cp03875b 

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