## Abstract

Atomic interactions in a large class of functional materials, essentially—all dielectrics, polarizable solids and ionic solids, involve long-range Coulomb interactions. Yet atomistic-to-continuum multiscale methods, such as the quasicontinuum method, are currently applicable only in cases where the atomic interactions are short-ranged. This restriction on the nature of atomic-level interactions for multiscale methods excludes numerous materials that are central to various scientific and industrial applications. A number of studies have pointed out unphysical artifacts in molecular simulations upon using the direct cutoff-based truncated sum to evaluate Coulomb interactions in ionic solids. However, recently it is understood that the artifacts of the direct cutoff-based truncated sum can be significantly minimized if a suitable correction term is added. In this work we examine whether or not the Wolf summation method, one of the prominent cutoff-based methods, is suitable to carry out the accumulation of the Coulomb interactions within the context of atomistic-to-continuum multiscale methods. In this regard, we choose the quasicontinuum method from the existing collection of multiscale methods for demonstration. It is found that the Wolf summation method can be applied if the atomistic system has charge ordering; otherwise, the error in the accumulation of the Coulomb interactions could be large.

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## Notes

All problems formulated using the QC method are essentially atomistic in nature though coarse-graining assimilates the formulation with continuum concepts.

The term “fully nonlocal” is introduced in Ref. [26] for the first time to mark the key difference to the existing QC versions employing a combination of nonlocal description for regions in fully atomistic resolution and a local description in coarse-grained regions using FE interpolation functions along with an interface region at the interface of these regions. The fully nonlocal QC method enables seamless and continuous scale transition by gradual coarse-graining.

Even though residual and spurious forces are conservative in the energy-based scheme, they are non-physical and can typically be quantified in defect-free infinite systems.

Conceptually, the potential energy and relevant thermodynamic quantities computed in the coarse-grained region can then be thought of to be based on finite elements containing sampling atoms as opposed to being based on representative atoms.

Interface region is the region sandwiched between between atomistic region (with the smallest mesh elements) and coarse-grained region (with large mesh elements).

Here we adopt the representation of the FE space with vector-valued shape functions and scalar-valued DoFs. We construct the

*d*vector-valued shape functions associated with the same node using the scalar-valued shape functions (associated to corresponding scalar-valued DoFs). A more commonly used representation with scalar-valued shape functions and vector-valued DoFs is an alternative form of Eq. (4).If the atomistic system consists of ions then the computation of the energy and forces result in a computational complexity of \(\mathcal{O}({\mathsf N}^2)\) if the (long-range) Coulomb potential is used directly without any additional treatment.

In mathematics, however, a scalar valued function \(f :\mathbb R^{d} \rightarrow {\mathbb {R}}\) defined on a

*d*-dimensional space is a rapidly decreasing function if \(f({{\mathbf {\mathsf{{r}}}}})\) and all its derivates decay faster in \({\left| {{\mathbf {\mathsf{{r}}}}}\right| }\) than \({\left| {{\mathbf {\mathsf{{r}}}}}\right| }^{-n}, \forall \,\, n \in {\mathbb {Z}}^{+}\). This is an informal definition of a Schwartz function; a rigorous definition of Schwartz function space involves the use of distributions.Instead of considering an isolated system of \({\mathsf N}\) charged atoms, one could also consider a more general case of a system of \({\mathsf N}\) charged atoms within a finite or an infinite ionic crystal. Since the goal of the original work was to calculate the total Madelung energy of a finite portion of a perfect crystal, an infinite ionic crystal was considered to start with; though (in hindsight) the summation expression so formulated is valid for both finite and infinite crystals. Here, for the sake of simplicity we consider a finite ionic crystal.

In this section, we focus our attention exclusively on computing the total energy (as opposed to problems related to computing the kinematic response due to a quasi-static load). Note that in the one-dimensional nanoindentation problem in “One-dimensional nanoindentation problem” force computations are also performed in addition to the energy computations, thereby studying the suitability of the Wolf summation method to model the Coulomb potential not just for energy computations but also for force computations. In actuality, ionic crystals consist of at least two different chemical elements (or species). Even in the ionic crystals with just two different chemical elements, the interaction between two ions of the same element type could be different from that of two different element types. The interaction between two ions of different element types could be modeled using a different interaction potential or by a common interaction potential with different parameters. It is therefore required that the kinematic response of ions in ionic crystals be treated with sophisticated techniques such as “Cascading Cauchy-Born” kinematics [14, 39].

Note that all the above computational experiments are conducted with finite ionic crystals with surfaces.

For coarse-graining Coulomb interactions the cutoff radius \(\,{{\mathsf r}}^{}_c\) and the damping coefficient \(\alpha \) are chosen identically similar to the way they are chosen for conventional (fully) atomistic simulations.

We base the notion of applicability of the Wolf summation method on how relative errors in computing energy (using the QC approach) compare with that of a short-ranged potential.

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## Acknowledgements

VB and PS, DD, and BE would like to thank the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) for the financial support under the research group project FOR-1509 Ferroic Functional Materials, Grant no. DA 1664/2-1, and Grant no. EI 453/2-1, respectively. PS also gratefully acknowledges the support by the Cluster of Excellence “Engineering of Advanced Materials”, Research Area A3.

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Boddu, V., Davydov, D., Eidel, B. *et al.* Cutoff-Based Modeling of Coulomb Interactions for Atomistic-to-Continuum Multiscale Methods.
*Multiscale Sci. Eng.* **1**, 299–317 (2019). https://doi.org/10.1007/s42493-019-00027-z

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DOI: https://doi.org/10.1007/s42493-019-00027-z