iMechanica - long-range interactions
https://imechanica.org/taxonomy/term/5035
enNew paper - Discrete-to-Continuum Limits of Long-Range Electrical Interactions in Nanostructures
https://imechanica.org/node/26592
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/636">nanostructures</a></div><div class="field-item odd"><a href="/taxonomy/term/12315">atomic to continuum modeling & coupling</a></div><div class="field-item even"><a href="/taxonomy/term/13766">electrical interaction</a></div><div class="field-item odd"><a href="/taxonomy/term/13767">continuum limit</a></div><div class="field-item even"><a href="/taxonomy/term/13768">dipole-dipole interaction</a></div><div class="field-item odd"><a href="/taxonomy/term/5035">long-range interactions</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>Authors - </strong>Prashant K. Jha, Timothy Breitzman, and Kaushik Dayal </p>
<p><strong>Journal - </strong>Archive for Rational Mechanics and Analysis </p>
<p><strong>Abstract</strong></p>
<p>We consider electrostatic interactions in two classes of nanostructures embedded in a three dimensional space: (1) helical nanotubes, and (2) thin films with uniform bending (i.e., constant mean curvature). Starting from the atomic scale with a discrete distribution of dipoles, we obtain the continuum limit of the electrostatic energy; the continuum energy depends on the geometric parameters that define the nanostructure, such as the pitch and twist of the helical nanotubes and the curvature of the thin film. We find that the limiting energy is local in nature. This can be rationalized by noticing that the decay of the dipole kernel is sufficiently fast when the lattice sums run over one and two dimensions, and is also consistent with prior work on dimension reduction of continuum micromagnetic bodies to the thin film limit. However, an interesting contrast between the discrete-to-continuum approach and the continuum dimension reduction approaches is that the limit energy in the latter depends only on the normal component of the dipole field, whereas in the discrete-to-continuum approach, both tangential and normal components of the dipole field contribute to the limit energy.</p>
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<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://imechanica.org/files/ARMA_discrete-to-continuum_nanostructures_JhaBreitzmanDayal.pdf" type="application/pdf; length=1198054" title="ARMA_discrete-to-continuum_nanostructures_JhaBreitzmanDayal.pdf">PDF of article</a></span></td><td>1.14 MB</td> </tr>
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</div></div></div>Sun, 02 Apr 2023 06:29:43 +0000Prashant K. Jha26592 at https://imechanica.orghttps://imechanica.org/node/26592#commentshttps://imechanica.org/crss/node/26592Linear scaling solution of the all-electron Coulomb problem in solids
https://imechanica.org/node/7981
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/449">electrostatics</a></div><div class="field-item odd"><a href="/taxonomy/term/846">FEM</a></div><div class="field-item even"><a href="/taxonomy/term/3371">DFT</a></div><div class="field-item odd"><a href="/taxonomy/term/5034">all-electron</a></div><div class="field-item even"><a href="/taxonomy/term/5035">long-range interactions</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In this manuscript (available at <a href="http://arxiv.org/abs/1004.1765" title="Linear scaling solution of the all-electron Coulomb problem in solids">http://arxiv.org/abs/1004.1765</a>), we present a systematically improvable, linear scaling formulation for the solution of the all-electron Coulomb problem in crystalline solids. In an infinite crystal, the electrostatic (Coulomb) potential is a sum of nuclear and electronic contributions, and each of these terms diverges and the sum is only conditionally convergent due to the long-range 1/r nature of the Coulomb interaction. In the all-electron quantum-mechanical problem in solids, there are three distinct divergences that must be addressed simultaneously: (1) the 1/r divergence of the electrostatic potential at the nuclei; (2) the divergence of both potential and energy lattice sums due to the long-range Coulomb interaction; and (3) the infinite self energies of the nuclei. </p>
<p>We achieve linear scaling by introducing smooth, strictly local neutralizing densities to render nuclear interactions strictly local, and solving the remaining neutral Poisson problem for the electrons in real space. In so doing, the all-electron problem is decomposed into analytic strictly-local nuclear, and numerical long-range electronic parts; with required numerical solution in the Sobolev space <img src="http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?H^1" alt="" />, so that convergence is assured and approximation is optimal in the energy norm. Expressions for the Coulomb energy per unit cell, analytically excluding the divergent nuclear self-energy, are derived. Rapid variations in the required neutral electronic potential in the vicinity of the nuclei are efficiently treated by an enriched finite element solution, using local radial solutions as enrichments (see <a href="node/3168" title="Classical and enriched formulations . . . ">this paper</a>). We demonstrate the accuracy and convergence of the approach by direct comparison to standard Ewald sums for a lattice of point charges, and demonstrate the accuracy in quantum-mechanical calculations with an application to crystalline diamond.</p>
<p>For some background material on density-functional theory and all-electron calculations, the discussions in the <a href="node/3837" title="September 2008 Journal Club">September 2008</a><a href="node/3837" title="September 2008 Journal Club"></a> and <a href="node/4728" title="Feb 2009 Journal Club">February 2009</a> journal club issues are pertinent. </p>
</div></div></div>Tue, 13 Apr 2010 02:05:24 +0000N. Sukumar7981 at https://imechanica.orghttps://imechanica.org/node/7981#commentshttps://imechanica.org/crss/node/7981Error | iMechanica