Non-Ewald methods for evaluating the electrostatic interactions of charge systems: similarity and difference

Abstract

In molecular simulations, it is essential to properly calculate the electrostatic interactions of particles in the physical system of interest. Here we consider a method called the non-Ewald method, which does not rely on the standard Ewald method with periodic boundary conditions, but instead relies on the cutoff-based techniques. We focus on the physicochemical and mathematical conceptual aspects of the method in order to gain a deeper understanding of the simulation methodology. In particular, we take into account the reaction field (RF) method, the isotropic periodic sum (IPS) method, and the zero-multipole summation method (ZMM). These cutoff-based methods are based on different physical ideas and are completely distinguishable in their underlying concepts. The RF and IPS methods are “additive” methods that incorporate information outside the cutoff region, via dielectric medium and isotropic boundary condition, respectively. In contrast, the ZMM is a “subtraction” method that tries to remove the artificial effects, generated near the boundary, from the cutoff sphere. Nonetheless, we find physical and/or mathematical similarities between these methods. In particular, the modified RF method can be derived by the principle of neutralization utilized in the ZMM, and we also found a direct relationship between IPS and ZMM.

Appendices

Appendix A: Continuity of energy function

Consider an atomic energy function E_i defined by a cutoff sum (see Eq. 4) of a continuous pairwise function V_ij = q_iq_jV such that

$$ E_{i}(x)=\sum\limits_{j\in\mathcal{R}_{i}}V_{ij}(r_{ij}). $$

(70)

For energetic studies and for keeping energy conservation, e.g., in NEV simulations, the continuity of the energy function is required. We will demonstrate that this requirement can be met by subtracting the boundary term (which is the last term of Eq. 72; see below for detail) or adopting the charge neutrality condition. Specifically, we state that they are equivalent, i.e.,

$$ \begin{array}{@{}rcl@{}} \text{Subtracting the boundary term} =\\ \text{Adopting the charge neutrality condition.} \end{array} $$

(71)

In the following, we assume V_ij(r_c)≠ 0, because V_ij(r_c) = 0 makes E_i continuous with respect to x so that the requirement is already satisfied. Obviously, the continuity of E_i indicates the continuity of the total energy function $x\mapsto E(x)=\frac {1}{2}{\sum }_{j\in \mathcal {N}_{i}}E_{i}(x)$.

To access the problem, just using a simple equivalence, we observe

$$ \begin{array}{@{}rcl@{}} E_{i}(x) & =&E_{i}(x)-\sum\limits_{j\in\mathcal{R}_{i}}V_{ij}(r_{\text{c}} )+\sum\limits_{j\in\mathcal{R}_{i}}V_{ij}(r_{\text{c}})\\ & =&\sum\limits_{j\in\mathcal{R}_{i}}\left[ V_{ij}(r_{ij})-V_{ij}(r_{\text{c} })\right] -V_{ii}(r_{\text{c}})+\sum\limits_{j\in\mathcal{R}^{i}}V_{ij}(r_{\text{c} })\\ & =&\sum\limits_{j\in\mathcal{R}_{i}}q_{i}q_{j}\left[ V(r_{ij})-V(r_{\text{c} })\right] -V(r_{\text{c}}){q_{i}^{2}}\\ &&+V(r_{\text{c}})q_{i}\sum\limits_{j\in \mathcal{R}^{i}}q_{j}, \end{array} $$

(72)

where the relation (4) has been used in the second line. The continuity of E_i and so a continuous time development t↦E_i(x(t)) seem to be guaranteed by the last line of Eq. 72, but they do not hold. This is because the quantity ${\sum }_{j\in \mathcal {R}^{i}}q_{j}$ in the final term, often called the “boundary energy” term, is a fluctuating quantity, since $\mathcal {R}^{i}$ depends on the configuration x (viz., $\mathcal {R}^{i}=\mathcal {R}^{i}(x)$ by definition (3)), so that the sum over it, ${\sum }_{j\in \mathcal {R}^{i}(x)}q_{j}$, is not ensured to be constant in time.

To proceed ahead, one may consider using a uniform/homogeneous approximation (Wu and Brooks 2005) in the boundary energy term such that

$$ \sum\limits_{j\in\mathcal{R}^{i}}q_{j}\simeq\frac{\mathrm{V}_{0}}{\mathrm{V}} \sum\limits_{j\in\mathcal{N}}q_{j} $$

(73)

for any i, where V is the volume of the basic cell and V₀ is the volume of the cutoff sphere. Another possible approximation may be

$$ \sum\limits_{j\in\mathcal{R}^{i}}q_{j}\simeq0 $$

(74)

for any i, which implies a subtraction of the boundary term in the original energy function, viz.,

$$ E_{i}(x)\rightarrow E_{i}(x)-V(r_{\text{c}})q_{i}\sum\limits_{j\in\mathcal{R}^{i} }q_{j}. $$

(75)

The two approximations (73) and (74) are equivalent (Ojeda-May and Pu 2014a) for a large system with V₀ ≪V or for a charge neutral system with

$$ \sum\limits_{j\in\mathcal{N}}q_{j}=0. $$

(76)

According to either approximation, we get

$$ E_{i}(x)\simeq\sum\limits_{j\in\mathcal{R}_{i}}q_{i}q_{j}\left[ V(r_{ij} )-V(r_{\text{c}})\right] +\text{\text{constant}}\mathrm{,} $$

(77)

where the constant is $-V(r_{\text {c}}){q_{i}^{2}}+V(r_{\text {c}} )\frac {\mathrm {V}_{0}}{\mathrm {V}}q_{i}{\sum }_{j\in \mathcal {N}}q_{j}$ or $-V(r_{\text {c}}){q_{i}^{2}}$, and hence we reach the continuity of E_i. However, considering the neutralization principle (“Neutralization principle”), we see that Eq. 74 does not often hold and large deviations from it are often observed, as Wolf et al. (1992) deeply investigate this issue in ionic systems under condition (76). Therefore, the accuracies in the above approximations would become problematic. Namely, these are approximations that transfer the discontinuity from the original cutoff sum ${\sum }_{j\in \mathcal {R}_{i}}V_{ij}(r_{ij})$ into the boundary energy term and ignore the latter as a constant. In other words, approximation (74), or (73) for a large/neutral system, allows large intrinsic errors against the originally defined energy E_i(x), in exchange for the gain of the continuity of E_i.

Nevertheless, investigation (Ojeda-May and Pu 2014a) of the Madelung energy of an ionic crystal (in the IPS method) suggests that the accuracy of E_i(x) becomes better if we take the approximation (74), which should have a large error against the originally defined energy E_i(x). A possible explanation for this contradiction is that an error that is intrinsically involved in E_i(x) can be extracted by removing the boundary energy. In fact, this explanation can be justified by the charge neutralization principle (Wolf et al. 1999), as follows.

The neutralization principle in the lowest order, discussed in “Neutralization principle” (which is recommended to be read to continue reading below), suggests the replacement of $\mathcal {N}_{i}$ into the neutralized subset ${\mathscr{M}}_{i}^{(l)}$ with l = 0 for calculating the interactions, indicating that E_i(x) should be replaced by

$$ E_{i}^{\prime}(x):=\sum\limits_{j\in\mathcal{M}_{i}^{(0)}}V_{ij}(r_{ij}). $$

(78)

We see that

$$ \begin{array}{@{}rcl@{}} E_{i}^{\prime}(x) & =&{\sum}_{j\in\mathcal{R}_{i}}V_{ij}(r_{ij})\!-{\sum}_{j\in\mathcal{J}_{i}^{(0)}}V_{ij}(r_{ij})\text{ \ (}\because \text{condition~(30a) and Eq.~(31))}\\ & \simeq& E_{i}(x)-{\sum}_{j\in\mathcal{J}_{i}^{(0)}}V_{ij}(r_{\text{c} })\text{\ (}\because\text{condition~(30c) for }l=0\text{)}\\ & =&E_{i}(x)-V(r_{\text{c}})q_{i}{\sum}_{j\in\mathcal{J}_{i}^{(0)}} q_{j}\\ & =&E_{i}(x)-V(r_{\text{c}})q_{i}\left( {\sum}_{j\in\mathcal{R}^{i}}q_{j} -{\sum}_{j\in\mathcal{M}_{i}^{(0)}\cup\{i\}}q_{j}\right) \text{\ \ (} \because\text{Eq.~(31))}\\ & =&E_{i}(x)-V(r_{\text{c}})q_{i}\left( {\sum}_{j\in\mathcal{R}^{i}} q_{j}\right) .\text{\ \ (}\because \text{condition~(30b))} \end{array} $$

(79)

Equation 79 indicates the subtraction approximation (75), so that the statement (71) is proven.

On the total energy, we have

$$ \begin{array}{@{}rcl@{}} E(x) &=&\frac{1}{2}{\sum}_{i\in\mathcal{N}}E_{i}(x)\\ & =&\frac{1}{2}{\sum}_{i\in\mathcal{N}}{\sum}_{j\in\mathcal{R}_{i}}q_{i} q_{j}\left[ V(r_{ij})-V(r_{\text{c}})\right]\\ &-&\frac{V(r_{\text{c}})}{2} {\sum}_{i\in\mathcal{N}}{q_{i}^{2}}+\frac{V(r_{\text{c}})}{2}{\sum}_{i\in \mathcal{N}}q_{i}{\sum}_{j\in\mathcal{R}^{i}}q_{j}. \end{array} $$

(80)

Here, the final term of the second line is still not necessarily zero even for a neutral system, but it can be treated as zero in the approximation (75), or in adopting the neutrality condition, to get

$$ \begin{array}{@{}rcl@{}} \frac{1}{2}\sum\limits_{i\in\mathcal{N}}E_{i}^{\prime}(x)&=&\frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{R}_{i}}q_{i}q_{j}\left[ V(r_{ij} )-V(r_{\text{c}})\right]\\& -&\frac{V(r_{\text{c}})}{2}\sum\limits_{i\in\mathcal{N} }{q_{i}^{2}}. \end{array} $$

(81)

Finally note that adopting the charge neutrality condition, or the approximation with subtracting the boundary term, makes the energy function continuous, but it does not ensure the force function’s continuity. The force function’s continuity is ensured by, e.g., the ZM scheme with l ≥ 1 (Fukuda 2013).

Appendix B: Derivation of the reaction field method

We give a simple and systematic derivation of the RF energy formula, Eq. 11, using Eq. 7. For this purpose, we reinterpret the “cavity” using the notion of the charge neutralization, which turns out to be an idea that is really in harmony with this.

A. Cavity

Here, the “cavity” of molecule a, $\mathcal {C}_{a}$, is formally treated as a set of atoms, but not as a set of molecules (for the latter, a resembled notation C_a is used in “Reaction field principle”), and the notation in Eqs. 3 and 4 is followed (“Neutralization principle” should be read before reading below).

In conventional simulations, cavity is often considered as atom-wise to be the all atoms contained within the sphere of radius r_c centered at atom i that is contained in a molecule a (see Fig. 1a).

The new interpretation of the cavity is given molecular-wise and also in a manner that enables atom-based interaction treatment, as follows. Cavity $\mathcal {C}_{a}$ is a subset of atoms contained within the sphere of radius r_c centered at an atom i in a molecule a, so that it can be denoted as ${\mathscr{M}}^{i}$, viz.,

• (a) ${\mathscr{M}}^{i}=\mathcal {C}_{a}\subset \mathcal {R}^{i}=\left \{ j\in \mathcal {N}\text { }|\text { }r_{ij}<r_{\text {c}}\right \}$ for i ∈Mol_a.This subset ${\mathscr{M}}^{i}$ needs not necessarily contain all the atoms of $\mathcal {R}^{i}$ but is assumed to satisfy the following conditions:

• (b) The total charge of these atoms is zero, ${\sum }_{j\in {\mathscr{M}}^{i}} q_{j}=0$, and

• (c) Atoms in $\mathcal {R}^{i}$ but excluded from ${\mathscr{M}}^{i}$ are placed near the boundary of $\mathcal {R}^{i}$.

Note that the conditions (a), (b), and (c) imply that conditions (30a), (30b), and (30c), respectively, hold for l = 0 when we view ${\mathscr{M}}^{i}$ as

$$ \mathcal{M}^{i}=\mathcal{M}_{i}^{(0)}\cup\{i\}. $$

(82)

However, for ${\mathscr{M}}^{i}$ to be defined as a subset associated to each atom i and for the viewpoint of Eq. 82 to be appropriate, it is necessary that ${\mathscr{M}}^{i^{\prime } }=\mathcal {C}_{a}$ is always valid, even when any other atom $i^{\prime }(\neq i)\in$Mol_a is chosen. We will see this is justified in the following subsection.

Before doing so, we specify the definition of interactions. The interaction on individual atom i in a molecule a is considered on the basis of this cavity $\mathcal {C}_{a}$ (rather than the basis of $\mathcal {R}^{i}$ as before), coming from inside $\mathcal {C}_{a}$ (by its whole atoms) and from outside $\mathcal {C}_{a}$ (by the electric field generated from a dielectric continuum). That is, we consider the interactions acting on atom i ∈Mol_a as (i) an energy E_Int coming from inside the cavity $\mathcal {C}_{a}$ and (ii) an energy E_Out coming from outside $\mathcal {C}_{a}$.

B. On (i)

By the definition, the interaction of type (i) acting on an atom $i^{\prime }(\neq i)$ in a molecule a is only from inside $\mathcal {C}_{a}$, wherein $\mathcal {C}_{a}$ does not necessarily coincide with $\mathcal {R}^{i^{\prime } }$ that is the total atoms in the cutoff sphere centered at $i^{\prime }$ (Fig. 1b). Specifically, we assume that a molecule a is sufficiently small such that

($\mathrm {a}^{\prime }$) $\mathcal {C}_{a}$ is a subset of $\mathcal {R}^{i^{\prime }}$

($\mathrm {c}^{\prime }$) Atoms in $\mathcal {R}^{i^{\prime }}$ but excluded from $\mathcal {C}_{a}$ are placed near the boundary of $\mathcal {R}^{i^{\prime }}$.

Note also that it holds from above (a) and (b) that

($\mathrm {b}^{\prime }$) ${\sum }_{j\in \mathcal {C}_{a}}q_{j}=0$.

These conditions (a’), (b), and (c’) also imply that conditions (30a), (30b), and (30c), respectively, hold for l = 0, when $\mathcal {C}_{a}$ is treated as ${\mathscr{M}}^{i^{\prime }}$. That is, we reach a consistent characterization of $\mathcal {C}_{a}$ such that

$$ \forall i\in\text{Mol}_{a},\text{ }\mathcal{M}^{i}=\mathcal{C}_{a} $$

(83)

for any molecule a. Note that it is should be $\mathcal {C}_{a}\subset \cap _{i\in \text {Mol}_{a}}\mathcal {R}^{i}$, but not necessarily $\mathcal {C} _{a}=\cap _{i\in \text {Mol}_{a}}\mathcal {R}^{i}$; Fig. 1b illustrates the case that the equality does not hold. Hence, ${\mathscr{M}}^{i}$ is defined with respect to every atom i, and conditions (30a)–(30c) with l = 0 hold for any i, where e.g., condition (30b) is denoted by

$$ \sum\limits_{j\in\mathcal{M}^{i}}q_{j}=0. $$

(84)

We see that the concept of the neutralized subset is compatible with the concept of the cavity of molecule and will also see below that it fits well with the RF principle.

Now, the energy E_Int, coming from inside $\mathcal {C}_{a}$, is thus represented as

$$ \begin{array}{@{}rcl@{}} E_{\text{Int}}(x) & \equiv&\frac{1}{2}\sum\limits_{a=1}^{M}\sum\limits_{i\in\text{Mol}_{a} }\sum\limits_{j\in\mathcal{C}_{a}\backslash\{i\}}\frac{q_{i}q_{j}}{r_{ij}}\\ & =&\frac{1}{2}\sum\limits_{a=1}^{M}\sum\limits_{i\in\text{Mol}_{a}}\sum\limits_{j\in \mathcal{M}_{i}^{(0)}}\frac{q_{i}q_{j}}{r_{ij}}\\ & =&\frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}_{i}^{(0)}} \frac{q_{i}q_{j}}{r_{ij}}. \end{array} $$

(85)

C. On (ii)

The energy E_Out, which is coming from outside $\mathcal {C}_{a}$, is described by an electric field (reaction field) $\mathcal {E}_{a}$ defined by Eq. 6, which is now represented, using the constant $\gamma \equiv \frac {2(\epsilon _{\text {RF}}-1)}{2\epsilon _{\text {RF}}+1}\frac {1}{r_{\text {c}}^{3}}$, as

$$ \begin{array}{@{}rcl@{}} \mathcal{E}_{a}&=&\gamma\sum\limits_{b\in\mathrm{C}_{a}}\mu_{b}=\gamma{\sum}_{b\in\mathrm{C}_{a}}\sum\limits_{j\in\text{Mol}_{b}}q_{j}\mathbf{r}_{j}\\&=&\gamma \sum\limits_{j\in\mathcal{C}_{a}}q_{j}\mathbf{r}_{j}=\gamma\sum\limits_{j\in\mathcal{M}^{i} }q_{j}\mathbf{r}_{j}, \end{array} $$

(86)

where any i ∈Mol_a can be used in the most right equation (see Eq. (83)). Thus,

$$ \begin{array}{@{}rcl@{}} E_{\text{Out}}(x) &\equiv&-\frac{1}{2}\sum\limits_{a=1}^{M}\left( \mu_{a}\right. \left\vert \text{ }\mathcal{E}_{a}\right) \\ & =&-\frac{1}{2}\sum\limits_{a=1}^{M}\left( \sum\limits_{i\in\text{Mol}_{a}}q_{i} \mathbf{r}_{i}\right. \left\vert \text{ }\gamma\sum\limits_{j\in\mathcal{C}_{a} }q_{j}\mathbf{r}_{j}\right) \\ & =&-\frac{\gamma}{2}\sum\limits_{a=1}^{M}\sum\limits_{i\in\text{Mol}_{a}}\sum\limits_{j\in\mathcal{M}^{i}}q_{i}q_{j}\left( \mathbf{r}_{i}\right. \left\vert \text{ }\mathbf{r}_{j}\right) \\ & =&-\frac{\gamma}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}^{i}} q_{i}q_{j}\left( \mathbf{r}_{i}\right. \left\vert \text{ }\mathbf{r} _{j}\right) \\ & =&\frac{\gamma}{4}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}_{i}^{(0)} }q_{i}q_{j}r_{ij}^{2}, \end{array} $$

(87)

where the last line is obtained as follows:

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}_{i}^{(0)}}q_{i}q_{j} r_{ij}^{2} \end{array} $$

(88a)

$$ \begin{array}{@{}rcl@{}} &=&\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}^{i}}q_{i}q_{j}r_{ij}^{2} \end{array} $$

(88b)

$$ \begin{array}{@{}rcl@{}} &=&2\sum\limits_{i\in\mathcal{N}}q_{i}\left\Vert \mathbf{r}_{i}\right\Vert^{2} \sum\limits_{j\in\mathcal{M}^{i}}q_{j}-2\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in \mathcal{M}^{i}}q_{i}q_{j}\left( \mathbf{r}_{i}\right. \left\vert \text{}\mathbf{r}_{j}\right) \end{array} $$

(88c)

$$ \begin{array}{@{}rcl@{}} &=&-2\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}^{i}}q_{i}q_{j}\left( \mathbf{r}_{i}\right. \left\vert \text{ }\mathbf{r}_{j}\right) . \end{array} $$

(88d)

To get Eq. 88c, we have used Eq. 31 and the consistency condition (Fukuda 2013) for l = 0; and to get Eq. 88d, we have used Eq. 84.

D. Total energy

Last, combining Eqs. 85 and 87, we have

$$ \begin{array}{@{}rcl@{}} E^{_{\text{MRF}}}(x) & \equiv& E_{\text{Int}}(x)+E_{\text{Out}}(x)\\ & =&\frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}_{i}^{(0)}} \frac{q_{i}q_{j}}{r_{ij}}+\frac{\gamma}{4}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}_{i}^{(0)}}q_{i}q_{j}r_{ij}^{2}\\ & =&\frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}_{i}^{(0)}} q_{i}q_{j}V_{\text{RF}}(r_{ij}), \end{array} $$

(89)

where V_RF is defined by Eq. 10. Thus, a similar procedure to get Eq. 79 leads us to Eq. 11 such that

$$ \begin{array}{@{}rcl@{}} E^{_{\text{MRF}}}(x) & =&\frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in \mathcal{M}_{i}^{(0)}}q_{i}q_{j}V_{\text{RF}}(r_{ij})\\ & =&\frac{1}{2}\sum\limits_{i\in\mathcal{N}}\left[ \sum\limits_{j\in\mathcal{R}_{i}} q_{i}q_{j}V_{\text{RF}}(r_{ij})-V_{\text{RF}}(r_{\text{c}})q_{i}\sum\limits_{j\in\mathcal{R}^{i}}q_{j}\right] \\ & =&\frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{R}_{i}}q_{i} q_{j}\left[ V_{\text{RF}}(r_{ij})-V_{\text{RF}}(r_{\text{c}})\right] -\frac{V_{\text{RF}}(r_{\text{c}})}{2}\sum\limits_{i\in\mathcal{N}}{q_{i}^{2}}.\\ \end{array} $$

(90)

Appendix C: Derivation of the IPS method

We derive Eqs. 2–19, based on the original work (Wu and Brooks 2005) and our additional interpretation.

• (i) First, the number of copies, n(m), of the local region distributed in a shell $D_{m}^{(i)}$ should be defined (see Fig. 3). The number density is set to be irrelevant to m so that n(m)/V_m = n(0)/V₀ = 1/V₀, where V_m is the volume of $D_{m}^{(i)}$, leading to the result

  $$ n(m)=24m^{2}+2. $$

  (91)

  By this setting, the isotropy along the radial direction could be met.

• (ii) To attain the isotropy along the spherical direction, several strategies can be considered. A complete isotropy along the spherical direction cannot be directly attained by an arrangement, such as proposed in (i), of a finite number of spherical regions. Thus, a continuum approximation is defined: each copy of i in $D_{m}^{(i)}$ is extended uniformly along the spherical direction. That is, the charge q_i is supposed to exist not as a point charge, but as a uniform surface density $\rho _{i}^{(m)}$ spread on a sphere $S_{m}^{(i)}\equiv \{\mathbf {r}\in \mathbb {R}^{3}|\left \Vert \mathbf {r}-\mathbf {r}_{i}\right \Vert =R_{m}\}$ (see Fig. 3). The potential energy, generated by this density, acting on the charge q_j is

  $$ {\Psi}_{ij}(m)\equiv{\int}_{S_{m}^{(i)}}q_{j}\widehat{\varepsilon}(s_{ijm} )\rho_{i}^{(m)}d\sigma, $$

  (92)

  where $\widehat {\varepsilon }(r)=1/r$, and s_ijm is the distance between r_j and a surface area dσ on $S_{m}^{(i)}$. After this integration, we get

  $$ {\Psi}_{ij}(m)=\frac{q_{i}q_{j}}{R_{m}}, $$

  (93)

  which we here refer to the spherical interaction (rather than the “random interaction” (Wu and Brooks 2009)). This discussion can be generalized for any function ε_ij(r) and for d-dimensional space $\mathbb {R}^{d}$, and this generalized Ψ_ij(m) depends on r_ij, in principle. Thus, the result, (93), is a special case for the Coulombic function in the three-dimensional space, known as the elementary fact that Coulombic/gravitational potential, which is produced by charge/mass uniformly distributed on the shell, acting on any point charge/mass inside the shell is constant. This simplicity, however, yields a difficulty to construct a method responsible for the IPS principle. That is, although one would like to straightforwardly use these good ideas by applying a definition $\widetilde {\varepsilon }_{ij}(r_{ij};m)=n(m){\Psi }_{ij}(m)$ to Eq. 13a, one finds a divergence in the series $\sum _{m\in \mathbb {N}}\widetilde {\varepsilon }_{ij}(r_{ij};m)=q_{i}q_{j}{\sum }_{m\in \mathbb {N}}(24m^{2}+2)/2mr_{\text {c}}$. Certain remedies are thus needed, as stated below (iii)–(v).

• (iii) The IPS method requires that there should be interactions that are symmetric with respect to the “j–i axis,” which is the axis including both r_i and r_j (i≠j). To meet this requirement, we put (at least two) image regions into each $D_{m}^{(i)}$along the j–i axis. So the image regions arrange, stick close each other, along the j–i axis (see Fig. 3). These image regions yield the energy term

  $$ \sum\limits_{m\in\mathbb{N}}\left[ \varepsilon_{ij}(R_{m}-r_{ij})+\varepsilon_{ij}(R_{m}+r_{ij})\right] , $$

  (94)

  which is called the “axial interaction.” Thus, a candidate of $\widetilde {\varepsilon }_{ij}(r_{ij};m)$ becomes

  $$ (n(m)-2){\Psi}_{ij}(m)+\varepsilon_{ij}(R_{m}-r_{ij})+\varepsilon_{ij} (R_{m}+r_{ij}), $$

  (95)

  which recovers the dependence on r_ij.

• (iv) Introduce a parameter ξ to ensure that the force at the distance of r_c becomes zero, which is required for a stable MD simulation. Although we expect the axial interaction defined by Eq. 95 to yield the results that the force at the distance of r_c is zero, it is not automatically ensured. To solve this problem, recall the strategy that we may duplicately put copies of the local region. Namely, we can put ξ copies along the j–i axis for the two opposite side, so that Eq. 95 will be changed into

  $$ \begin{array}{@{}rcl@{}} \overset{\circ}{\widetilde{\varepsilon}}_{ij}(r_{ij};m)&\equiv&(n(m)-2\xi ){\Psi}_{ij}(m)+\xi\varepsilon_{ij}(R_{m}-r_{ij})\\&& +\xi\varepsilon_{ij} (R_{m}+r_{ij}). \end{array} $$

  (96)

  The specific process to determine the value of ξ ensuring the force continuity is seen in (v), and ξ will be 1 for the three-dimensional Coulombic interaction. Note that ξ will not be an integer for a general interaction (e.g., LJ), so that its interpretation as the number of copies should also be generalized.

• (v) Regardless of the introduction of ξ, the sum of Eq. 96 with respect to $m\in \mathbb {N}$ does not converge due to the presence of the right first term. To remedy this problem, a certain regularization is introduced. Since $\widetilde {\varepsilon } _{ij}(r_{ij};m)$ should be a potential energy (acting on j from all copies of i), the energy zero can be freely chosen. We define

  $$ \widetilde{\varepsilon}_{ij}(r_{ij};m)\equiv\overset{\circ}{\widetilde {\varepsilon}}_{ij}(r_{ij};m)-\overset{\circ}{\widetilde{\varepsilon}} _{ij}(0;m), $$

  (97)

  then the first term of Eq. 96 vanishes,

  $$ \begin{array}{@{}rcl@{}} \widetilde{\varepsilon}_{ij}(r_{ij};m)&=&\xi\lbrack\varepsilon_{ij}(R_{m} -r_{ij})-\varepsilon_{ij}(R_{m})\\ &&\quad+\varepsilon_{ij}(R_{m}+r_{ij})-\varepsilon _{ij}(R_{m})], \end{array} $$

  (98)

  and the sum is convergent:

  $$ \begin{array}{@{}rcl@{}} {\Phi}_{ij}^{\text{IPS}}(r) :&=&\sum\limits_{m\in\mathbb{N}}\widetilde{\varepsilon }_{ij}(r;m)\\ & =&q_{i}q_{j}\xi\sum\limits_{m\in\mathbb{N}}\left[ \frac{1}{R_{m}-r}-\frac{1} {R_{m}}+\frac{1}{R_{m}+r}-\frac{1}{R_{m}}\right] \\ & =&-\frac{q_{i}q_{j}\xi}{2r_{\text{c}}}\left[ \psi\left( 1-\frac {r}{2r_{\text{c}}}\right) + \psi\left( 1+\frac{r}{2r_{\text{c}}}\right) + 2\gamma\right] , \end{array} $$

  (99)

  where ψ is the digamma function and γ is the -Mascheroni constant. Equation 13a is thus

  $$ \begin{array}{@{}rcl@{}} E_{ij}^{\text{IPS}}(r_{ij}) & =&q_{i}q_{j}\widehat{E}^{\text{IPS}} (r_{ij})=\varepsilon_{ij}(r_{ij})+{\Phi}_{ij}^{\text{IPS}}(r_{ij})\\ & =&\frac{q_{i}q_{j}}{r_{ij}}-\frac{q_{i}q_{j}\xi}{2r_{\text{c}}}\left[ \psi\left( 1-\frac{r_{ij}}{2r_{\text{c}}}\right) +\psi\left( 1+\frac {r_{ij}}{2r_{\text{c}}}\right) +2\gamma\right] . \\ \end{array} $$

  (100)

  Note that this result implies that all the spherical interactions (93) do not contribute to the IPS potential and that the only axial interactions (94) define the IPS potential. Using the properties of the digamma function, we observe that $-D\widehat {E}^{\text {IPS}}(r_{\text {c}})=0$ is attained only if

  $$ \xi=1. $$

  (101)

  Thus, we have Eq. 18 along with Eq. 2.

• (vi) Continuity of the energy function is required in general, as detailed in Appendix A. The condition of the energy continuity in the IPS method, $\widehat {E}^{\text {IPS}}(r_{\text {c}})=0$, is attained only if $\xi =1/(1-2\ln 2)$ (due to $\psi \left (3/2\right ) =\psi \left (1/2\right ) +2=-\gamma -2\ln 2+2$), which is not equal to the condition of the force continuity, Eq. 101. Thus, if we use Eq. 101, an alternative treatment for the energy continuity is required.

  For this, going back to Eq. 13b (see also Eq. (17a)), we have

  $$ \begin{array}{@{}rcl@{}} E_{i}^{\text{IPS}}(x) & =&\sum\limits_{j\in\mathcal{R}_{i}}E_{ij}^{\text{IPS} }(r_{ij})+{\Phi}_{ii}^{\text{IPS}}(r_{ii})\\ & =&\sum\limits_{j\in\mathcal{R}_{i}}q_{i}q_{j}\widehat{E}^{\text{IPS}}(r_{ij}) \end{array} $$

  (102)

  using ${\Phi }_{ii}^{\text {IPS}}(0)=0$ for the Coulomb function (owing to $\psi \left (1\right ) =-\gamma$). Thus, applying (72) in the scheme described in Appendix A, the atomic energy is

  $$ \begin{array}{@{}rcl@{}} E_{i}^{\text{IPS}}(x)&=&\sum\limits_{j\in\mathcal{R}_{i}}q_{i}q_{j}\left[ \widehat {E}^{\text{IPS}}(r_{ij})-\widehat{E}^{\text{IPS}}(r_{\text{c}})\right] \\&-&\widehat{E}^{\text{IPS}}(r_{\text{c}}){q_{i}^{2}}+\widehat{E}^{\text{IPS} }(r_{\text{c}})q_{i}\sum\limits_{j\in\mathcal{R}^{i}}q_{j}, \end{array} $$

  (103)

  and the last term may be approximated as

  $$ \widehat{E}^{\text{IPS}}(r_{\text{c}})q_{i}\sum\limits_{j\in\mathcal{R}^{i}} q_{j}\simeq\widehat{E}^{\text{IPS}}(r_{\text{c}})\frac{\mathrm{V}_{0} }{\mathrm{V}}q_{i}\sum\limits_{j\in\mathcal{N}}q_{j} $$

  (104)

  via Eq. 73 or approximated just as zero via Eq. 74. The total energy of the IPS method is described by Eq. 19, i.e.,

  $$ \begin{array}{@{}rcl@{}} E^{\text{IPS}}(x)&=&\frac{1}{2}\sum\limits_{i\in\mathcal{N}}E_{i}^{\text{IPS} }(x)\\ & =&\frac{1}{2}{\sum}_{i\in\mathcal{N}}{\sum}_{j\in\mathcal{R}_{i}}q_{i} q_{j}\left[ \widehat{E}^{\text{IPS}}(r_{ij})-\widehat{E}^{\text{IPS} }(r_{\text{c}})\right]\\ &&-\frac{\widehat{E}^{\text{IPS}}(r_{\text{c}})}{2} {\sum}_{i\in\mathcal{N}}{q_{i}^{2}}\\ &&+\frac{\widehat{E}^{\text{IPS}}(r_{\text{c}} )}{2}{\sum}_{i\in\mathcal{N}}q_{i}{\sum}_{j\in\mathcal{R}^{i}}q_{j}, \end{array} $$

  (105)

  where the last term may be approximated similarly above. In particular, subtracting the boundary term, which is equivalent to adopting the charge neutrality condition (see Eq. 71), yields

  $$ \begin{array}{@{}rcl@{}} E^{\text{MIPS}}(x)&:=&\frac{1}{2}{\sum}_{i\in\mathcal{N}}{\sum}_{j\in\mathcal{R}_{i}}q_{i}q_{j}\!\left[ \widehat{E}^{\text{IPS}}(r_{ij}) - \widehat {E}^{\text{IPS}}(r_{\text{c}})\right]\\ && -\frac{\widehat{E}^{\text{IPS} }(r_{\text{c}})}{2}{\sum}_{i\in\mathcal{N}}{q_{i}^{2}} \end{array} $$

  (106)

which we refer to as a modified IPS method. Finally, we note an alternative approach that has recently been proposed (Wu and Brooks 2019). By studying the reference point of $\widetilde {\varepsilon }_{ij}$ so far, we were able to make the sum converge by building it as (97). The resulting ${\Phi }_{ij}^{\text {IPS}}(r)$ was Eq. 99. However, Wu and Brooks (2019) argue that the origin of ${\Phi }_{ij}^{\text {IPS}}(r)$ itself can also be shifted. As a result, the boundary term can also be adjusted, and they try to do this in their “homogeneity condition.” They showed numerically the effectiveness of this method for a functional form of $\varepsilon _{ij}(r)\propto ar^{-12}-br^{-n_{\text {c}}}$ with various integers n_c. It would be desirable to study it in depth for the Coulomb type function.

Appendix D: Derivation of the ZMM

Here, we first briefly derive the energy formula of ZMM (Fukuda 2013), Eq. 34, starting from Eq. 32b . We consider an approximation of $\widehat {\varepsilon }(r)=1/r$ near r = r_c such that

$$ \widehat{\varepsilon}(r)=\widehat{\varepsilon}_{l}(r)+o\left( \left\vert r-r_{\text{c}}\right\vert^{l}\right) \text{ \ }(r\rightarrow r_{\text{c}}), $$

(107)

where $\widehat {\varepsilon }_{l}(r)$ is a polynomial of the form

$$ \widehat{\varepsilon}_{l}(r)\equiv\sum\limits_{m=0}^{l}a_{m}^{(l)}r^{2m}. $$

(108)

As shown in literature (Fukuda 2013, Appendix A), Eq. 107 is equivalent to

$$ D^{m}\widehat{\varepsilon}(r_{\text{c}})=D^{m}\widehat{\varepsilon} _{l}(r_{\text{c}})\text{ for\ }m=0,\ldots,l, $$

(109)

which yields the coefficients $\{a_{m}^{(l)}\}_{m=0,1,\ldots ,l}$ to be uniquely determined (see below for some specific cases). Use of this approximation exerts an effect, owing to the condition defined by Eq. 30c, in the evaluation of the excess energy (see ibid., Appendix B):

$$ \frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{J}_{i}^{(l)}}q_{i} q_{j}\widehat{\varepsilon}(r_{ij})\simeq\frac{1}{2}\sum\limits_{i\in\mathcal{N}} \sum\limits_{j\in\mathcal{J}_{i}^{(l)}}q_{i}q_{j}\widehat{\varepsilon}_{l}(r_{ij}). $$

(110)

The polynomial (108) fits the algebraic condition, Eq. 30b, so that the RHS of Eq. 110 can be simplified, after some algebra with the help of the consistency condition (see ibid., Appendix C), such that

$$ \frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{J}_{i}^{(l)}}q_{i} q_{j}\widehat{\varepsilon}_{l}(r_{ij})=\frac{1}{2}\sum\limits_{i\in\mathcal{N}} \sum\limits_{j\in\mathcal{R}^{i}}q_{i}q_{j}\widehat{\varepsilon}_{l}(r_{ij}). $$

(111)

Substitution of Eqs. 110 and 111 into Eq. 32b yields

$$ E(x)\simeq\frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{R}_{i}} q_{i}q_{j}\left[ \widehat{\varepsilon}(r_{ij})-\widehat{\varepsilon} _{l}(r_{ij})\right] -\frac{a_{0}^{(l)}}{2}\sum\limits_{i\in\mathcal{N}}{q_{i}^{2}}, $$

(112)

which is Eq. 34.

As described in “Neutralization principle,” we can generalize $\widehat {\varepsilon }(r)$ into the damped form V (r) (Eq. 36), and the result is given in Eq. 35. This energy formula can also be represented as

$$ \begin{array}{@{}rcl@{}} E_{\text{ZM}}^{(l)}(x)&=&\frac{1}{2}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in \mathcal{R}_{i}}q_{i}q_{j}\left[ u^{(l)}(r_{ij})-u^{(l)}(r_{\text{c} })\right] \\&-&\frac{1}{2}\left[ u^{(l)}(r_{\text{c}})+\frac{2\alpha}{\sqrt{\pi }}\right] \sum\limits_{i\in\mathcal{N}}{q_{i}^{2}}, \end{array} $$

(113)

where

$$ u^{(l)}(r)\equiv\left\{ \begin{array} [c]{cc} V(r)-\sum\limits_{m=1}^{l}\widetilde{a}_{m}^{(l)}r^{2m} & \text{for }l\geq1\\ V(r) & \text{for }l=0 \end{array} \right\} , $$

(114)

because V (r_c) = V_l(r_c), which corresponds to Eq. 109 with m = 0 for α ≥ 0, yielding

$$ \widetilde{a}_{0}^{(l)}=u^{(l)}(r_{\text{c}}) $$

(115)

and

$$ u^{(l)}(r)-u^{(l)}(r_{\text{c}})=V(r)-V_{l}(r) $$

(116)

for any l ≥ 0.

We finally describe the polynomials for l = 3 and 4, which are used in “Relationship between the IPSp method and the ZMM” and “Other relationships” (2), respectively. When l = 3, the pair potential function of the ZMM (Fukuda et al. 2014), u^(l), is represented as

$$ \begin{array}{@{}rcl@{}} u^{(3)}(r) &=&V(r)\\ & +&\left( \frac{15}{16}\frac{d_{1}}{r_{\text{c}}}+\frac{7}{16}d_{2}+\frac {1}{16}d_{3}r_{\text{c}}\right) r^{2}\\ & -&\left( \frac{5}{16}\frac{d_{1}}{r_{\text{c}}^{3}}+\frac{5}{16}\frac {d_{2}}{r_{\text{c}}^{2}}+\frac{1}{16}\frac{d_{3}}{r_{\text{c}}}\right) r^{4}\\ & +&\left( \frac{1}{16}\frac{d_{1}}{r_{\text{c}}^{5}}+\frac{1}{16}\frac {d_{2}}{r_{\text{c}}^{4}}+\frac{1}{48}\frac{d_{3}}{r_{\text{c}}^{3}}\right) r^{6}, \end{array} $$

(117)

where d_n ≡ (−)ⁿDⁿV (r_c) is (−)ⁿ times the n th derivatives of V at r_c. Moreover, if α = 0 then $d_{n}=\frac {n!}{r_{\text {c}}^{n+1}}$ so that Eq. 117 yields

$$ u^{(3)}|_{\alpha=0}(r)=\frac{1}{r}+\frac{35}{16}\frac{r^{2}}{r_{\text{c}}^{3} }-\frac{21}{16}\frac{r^{4}}{r_{\text{c}}^{5}}+\frac{5}{16}\frac{r^{6} }{r_{\text{c}}^{7}}. $$

(118)

Similarly, for l = 4 and α = 0, we have

$$ u^{(4)}|_{\alpha=0}(r)=\frac{1}{r}+\frac{105}{32}\frac{r^{2}}{r_{\text{c}} ^{3}}-\frac{189}{64}\frac{r^{4}}{r_{\text{c}}^{5}}+\frac{45}{32}\frac{r^{6} }{r_{\text{c}}^{7}}-\frac{35}{128}\frac{r^{8}}{r_{\text{c}}^{9}}. $$

(119)