iMechanica - electronic structure
https://www.imechanica.org/taxonomy/term/1269
enPostdoc positions in computational science at UIUC
https://www.imechanica.org/node/25778
<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/73">job</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/3762">atomistic modeling</a></div><div class="field-item odd"><a href="/taxonomy/term/10902">machine learning</a></div><div class="field-item even"><a href="/taxonomy/term/10730">2D materials</a></div><div class="field-item odd"><a href="/taxonomy/term/2840">Topology optimization</a></div><div class="field-item even"><a href="/taxonomy/term/1269">electronic structure</a></div><div class="field-item odd"><a href="/taxonomy/term/8544">Multi-Physics</a></div><div class="field-item even"><a href="/taxonomy/term/8060">materials design</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>Two postdoc positions in computational science are available at UIUC in the research group of Prof. Harley T. Johnson in the Department of Mechanical Science and Engineering and the Materials Research Lab. </p>
<p><strong>The first position, available immediately, is in the area of machine learning based topology optimization for multiphysics problems.</strong> The project will focus on design of surfaces for thermal- optical and thermo-mechanical applications. The position will require the ability to work closely with national labs partners, and will be for a duration of two years, with the possibility of renewal.</p>
<p><strong>The second position, with a start date in late spring or early summer 2022, is in the area of modeling the mechanics of interfaces in 2D materials.</strong> The work will be based on atomistic modeling using empirical and electronic structure methods, and will involve close collaboration with other modelers and experimentalists at Illinois and elsewhere. The position will be for a duration of two years, with the possibility of renewal.</p>
<p>Postdocs in the group participate in a highly collaborative research environment across UIUC and beyond, and develop professionally through mentorship, networking, and leadership opportunities. More information about recent scholarly work from the group can be found here: <a href="https://mechse.illinois.edu/people/profile/htj">https://mechse.illinois.edu/people/profile/htj</a> .</p>
<p>For questions about the positions, or to submit an application for either position (consisting of a cover letter, a CV, and the names of three references), interested candidates may contact Prof. Johnson by email at <a href="mailto:htj@illinois.edu">htj@illinois.edu</a>. Review of applications will begin immediately and continue until the positions are filled.</p>
</div></div></div>Fri, 11 Feb 2022 17:03:00 +0000Harley T. Johnson25778 at https://www.imechanica.orghttps://www.imechanica.org/node/25778#commentshttps://www.imechanica.org/crss/node/25778Augmented Lagrangian formulation of Orbital-Free Density Functional Theory
https://www.imechanica.org/node/17011
<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/1269">electronic structure</a></div><div class="field-item odd"><a href="/taxonomy/term/3371">DFT</a></div><div class="field-item even"><a href="/taxonomy/term/10009">real-space</a></div><div class="field-item odd"><a href="/taxonomy/term/10010">Augmented Lagrangian</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><span><strong>Abstract</strong></span></p>
<p>We present an Augmented Lagrangian formulation and its real-space implementation for non-periodic Orbital-Free Density Functional Theory (OF-DFT) calculations. In particular, we rewrite the constrained minimization problem of OF-DFT as a sequence of minimization problems without any constraint, thereby making it amenable to powerful unconstrained optimization algorithms. Further, we develop a parallel implementation of this approach for the Thomas–Fermi–von Weizsacker (TFW) kinetic energy functional in the framework of higher-order finite-differences and the conjugate gradient method. With this implementation, we establish that the Augmented Lagrangian approach is highly competitive compared to the penalty and Lagrange multiplier methods. Additionally, we show that higher-order finite-differences represent a computationally efficient discretization for performing OF-DFT simulations. Overall, we demonstrate that the proposed formulation and implementation are both efficient and robust by studying selected examples, including systems consisting of thousands of atoms. We validate the accuracy of the computed energies and forces by comparing them with those obtained by existing plane-wave methods.</p>
<p>The published article can be found here: <a href="http://www.sciencedirect.com/science/article/pii/S0021999114004860">http://www.sciencedirect.com/science/article/pii/S0021999114004860</a></p>
<p>A preprint is available on arXiv: <a href="http://arxiv.org/abs/1405.6456">http://arxiv.org/abs/1405.6456</a></p>
</div></div></div>Mon, 11 Aug 2014 18:12:25 +0000Phanish Suryanarayana17011 at https://www.imechanica.orghttps://www.imechanica.org/node/17011#commentshttps://www.imechanica.org/crss/node/17011Higher-order adaptive finite-element methods for Kohn-Sham density functional theory
https://www.imechanica.org/node/15098
<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/1251">finite elements</a></div><div class="field-item odd"><a href="/taxonomy/term/1269">electronic structure</a></div><div class="field-item even"><a href="/taxonomy/term/4465">density functional theory</a></div><div class="field-item odd"><a href="/taxonomy/term/6602">ab-initio calculations</a></div><div class="field-item even"><a href="/taxonomy/term/8985">real space</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>Dear Colleagues,</p>
<p>
I wish to share with you our recent article on "Higher-order adaptive finite-element method for Kohn-Sham density functional theory", which will soon appear in the Journal of Computational Physics. Below is the abstract and attached is a preprint of the article.
</p>
<p>
P. Motamarri, N.R. Nowak, K. Leiter, J. Knap, V. Gavini, Higher-order adaptive finite-element methods for Kohn-Sham density functional theory, J. Comp. Phys. 253, 308-343 (2013).
</p>
<p>
Abstract:<br />
We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element discretization of Kohn-Sham density-functional theory (DFT). To this end, we develop an a priori mesh-adaption technique to construct a close to optimal finite-element discretization of the problem. We further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite-elements in conjunction with Gauss-Lobatto quadrature, and a Chebyshev acceleration technique for computing the occupied eigenspace. The proposed approach has been observed to provide a staggering 100-200-fold computational advantage over the solution of a generalized eigenvalue problem. Using the proposed solution procedure, we investigate the computational efficiency afforded by higher-order finite-element discretizations of the Kohn-Sham DFT problem. Our studies suggest that staggering computational savings-of the order of 1000-fold-relative to linear finite-elements can be realized, for both all-electron and local pseudopotential calculations, by using higher-order finite-element discretizations. On all the benchmark systems studied, we observe diminishing returns in computational savings beyond the sixth-order for accuracies commensurate with chemical accuracy, suggesting that the hexic spectral-element may be an optimal choice for the finite-element discretization of the Kohn-Sham DFT problem. A comparative study of the computational efficiency of the proposed higher-order finite-element discretizations suggests that the performance of finite-element basis is competing with the plane-wave discretization for non-periodic local pseudopotential calculations, and compares to the Gaussian basis for all-electron calculations to within an order of magnitude. Further, we demonstrate the capability of the proposed approach to compute the electronic structure of a metallic system containing 1688 atoms using modest computational resources, and good scalability of the present implementation up to 192 processors.
</p>
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<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/file" src="/modules/file/icons/application-octet-stream.png" /> <a href="https://www.imechanica.org/files/KSDFT-FE-higherOrder.pdf" type="application/file; length=944660" title="KSDFT-FE-higherOrder.pdf">KSDFT-FE-higherOrder.pdf</a></span></td><td>922.52 KB</td> </tr>
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</div></div></div>Mon, 12 Aug 2013 10:24:08 +0000Vikram Gavini15098 at https://www.imechanica.orghttps://www.imechanica.org/node/15098#commentshttps://www.imechanica.org/crss/node/15098Minisymposium on Electronic-Structure Methods at USNCCM12
https://www.imechanica.org/node/14221
<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/74">conference</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/1251">finite elements</a></div><div class="field-item odd"><a href="/taxonomy/term/1269">electronic structure</a></div><div class="field-item even"><a href="/taxonomy/term/4465">density functional theory</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>
Dear Colleagues:
</p>
<p>
We would like to invite you to submit a contribution to a minisymposium that we are organizing on <em>Emerging Methods for Large-Scale Quantum-Mechanical Materials Calculations</em> at the 12th US National Congress on Computational Mechanics, to be held July 22-25, 2013 in Raleigh, NC. This minisymposium aims to bring together leading researchers in this emerging area to discuss and exchange ideas on new methods developments for density-functional calculations, mathematical analysis, and applications of ab initio methods in electronic-structure calculations.
</p>
<p>
Information on the minisymposium is available at:<a href="http://12.usnccm.org/MS8_1" target="_blank"> http://12.usnccm.org/MS8_1</a>. The deadline for abstract submission is <strong>March 15, 2013</strong>: please see <a href="http://12.usnccm.org/abstract-submission" target="_blank">http://12.usnccm.org/abstract-submission</a>.
</p>
<p>
We look forward to your participation in the mini-symposium.
</p>
<p>
Best wishes,
</p>
<p>
N. Sukumar (University of California, Davis)
</p>
<p>
John Pask (Lawrence Livermore National Laboratory)
</p>
<p>
Phanish Suryanarayana (Georgia Institute of Technology)
</p>
</div></div></div>Wed, 20 Feb 2013 03:15:43 +0000N. Sukumar14221 at https://www.imechanica.orghttps://www.imechanica.org/node/14221#commentshttps://www.imechanica.org/crss/node/14221Postdoctoral Research Associate at Shenoy Research Group at University of Pennsylvania
https://www.imechanica.org/node/13930
<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/1269">electronic structure</a></div><div class="field-item odd"><a href="/taxonomy/term/3371">DFT</a></div><div class="field-item even"><a href="/taxonomy/term/7045">Vivek Shenoy</a></div><div class="field-item odd"><a href="/taxonomy/term/7663">UPenn</a></div><div class="field-item even"><a href="/taxonomy/term/8293">VASP</a></div><div class="field-item odd"><a href="/taxonomy/term/8294">Quantum Chemistry</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>
A postdoctoral position with primary focus on first principles modeling is available immediately at <a href="http://www.seas.upenn.edu/directory/profile.php?ID=181">Shenoy Research Group at UPenn</a>. We are looking for a strongly motivated candidate to work on modeling the performance characteristics<br />
of nanomaterials for energy storage. The ideal candidate will have a background<br />
in materials science/computational physics/quantum chemistry <span>with expertise in density functional theory<br />
and electronic structure simulations</span>.<span> </span>This individual will have the opportunity to be directly<br />
involved in complimentary experimental investigations, both at Penn and our<br />
collaborators in industry.<span> </span>
</p>
<p>
Candidates<br />
should send their CV with names of three references to <a href="mailto:vshenoy@seas.upenn.edu">vshenoy@seas.upenn.edu</a></p>
<p>
</p>
</div></div></div>Thu, 20 Dec 2012 20:53:06 +0000Dibakar Datta13930 at https://www.imechanica.orghttps://www.imechanica.org/node/13930#commentshttps://www.imechanica.org/crss/node/13930Journal Club Theme of February 2009: Finite Element Methods in Quantum Mechanics
https://www.imechanica.org/node/4728
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Welcome to the February 2009 issue. In this issue, we will discuss the use of finite elements (FEs) in quantum mechanics, with specific focus on the quantum-mechanical problem that arises in crystalline solids. We will consider the electronic structure theory based on the Kohn-Sham equations of density functional theory (KS-DFT): in real-space, Schrödinger and Poisson equations are solved in a parallelepiped unit cell with Bloch-periodic and periodic boundary conditions, respectively. The planewave pseudopotential approach is the method of choice in such quantum-mechanical simulations, but there has been growing interest in recent years on the use of various real-space mesh approaches, of which finite elements are gaining increasing prominence. Most of us are very well-aware of the use of finite elements in solid and structural mechanics applications, but might be less familiar with its place or potential in quantum-mechanical calculations. To bridge this gap, I provide a brief introduction to the topic and sketch the main ingredients, which is supplemented by links to three review articles for further details. Parts of these journal articles would be very accessible to readers of iMechanica who are familiar with FE and other grid-based methods. In the <a href="node/3837">September 15, 2008 Journal Club issue</a>, the computational challenges in electronic-structure calculations have been outlined by <a href="user/1255">Vikram Gavini</a>; please take a look at the <a href="files/Overview.pdf"> nice overview of DFT</a> that he has written.</p>
<p>
The solution of the equations of DFT provide a means to determine material properties completely from quantum-mechanical first principles (ab initio), without any experimental input or tunable parameters. This facilitates fundamental understanding and robust predictions for a wide range of properties across the gamut of material systems. The quantum-mechanical problem in a crystalline solid consists of determining the electronic charge density (corresponding wavefunction and effective potential) for a system consisting of positively charged nuclei, surrounded by negatively charged electrons. In the all-electron problem, the Coulomb potential Z/r diverges (solution has cusps and oscillates rapidly near nuclear positions), and hence is not readily amenable to accurate numerical calculations for even moderate system sizes. For first principles computations in crystalline solids, the <a href="http://en.wikipedia.org/wiki/Pseudopotential">pseudopotential approach</a> is widely adopted in most quantum molecular dynamics codes. The electrons in the inner shell (core electrons) are tightly bound and do not contribute to any valence bonding. In the pseudopotential approach, the core electrons are frozen in their atomic state, and the divergent Coulomb potential is replaced by a modified potential (pseudopotential) such that the valence states close to the core are less oscillatory, but do not change outside the core region. Only pseudoatomic wavefunctions for the valence (outermost shells) states are solved for in the Schrödinger equation, which makes it numerically tractable via planewaves or with real-space mesh techniques.
</p>
<p>
On using the pseudopotential approach, the KS-DFT equations consist of the solution of single-particle like Schrödinger equations, which are coupled to a Poisson equation. The single-particle Schrödinger equation (atomic units used) for the <em>i</em>th state is:
</p>
<p>
<img src="http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?%20-%5Cfrac%7B1%7D%7B2%7D%20%5Cnabla%5E2%20%5Cpsi_i%20+%20%5Chat%20V_%7Beff%7D%20%5Cpsi_i%20=%20%5Cepsilon_i%20%5Cpsi_i%20%5C,%20," alt="" /></p>
<p>
subject to boundary conditions consistent with Bloch's theorem. In the above equation, <img src="http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?\psi_i" alt="" /> and <img src="http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?\epsilon_i" alt="" /> are the pseudoatomic wavefunction and energy eigenvalue, respectively. The total effective potential now consists of a local ionic part, a non-local ionic part, the electronic (Hartree) potential and the exchange-correlation potential (due to many-body interactions and Pauli's exclusion principle). Typically, to enable predictive capabilities, energy eigenvalues need to be computed within 1 mHa (chemical) accuracy (1 Hartree ~ 27.21 eV). The single-particle pseudowavefunctions are squared sum to form the charge density, which is used in the Poisson equation to solve for the Coulomb potential (local ionic and electronic contributions). Once again <img src="http://physweb.bgu.ac.il/cgi-bin/mimetex.cgi?%5Chat%20V_%7Beff%7D" alt="" /> is formed and the process is repeated until self-consistency is attained (charge density and effective potential do not change). The total energy, forces, etc., can now be computed to enable quantum molecular dynamics simulations.
</p>
<p>
The solution of the Poisson equation scales linearly with the number of degrees of freedom N, but the Schrödinger solution varies as the cube of N (see <a href="http://www.cs.sandia.gov/~rmuller/page4/page4.html">this plot</a>). For self-consistency, since repeated Poisson and Schrödinger equations are solved for and in excess of 1000 eigenfunctions may be required for large systems (~100 atoms or more), the solution of the Schrödinger equation is the limiting step in the solution of the equations of DFT. Unlike linear equations that are relatively easy to solve, accurate eigensolutions for thousands of eigenpairs are computationally demanding and algorithmically challenging, since the eigenfunctions also have to meet the orthogonality constraint.
</p>
<p>
<strong>From Planewaves to Real-Space Mesh Techniques</strong>
</p>
<p>
As the preceding discussion indicates, the efficient solution of the Schrödinger equation in DFT computations is of paramount importance. This need is especially pronounced for metallic systems with heavier atoms and under extreme conditions (high-temperature and/or -pressure) whose pseudopotentials are deep and sharply localized. In such instances, planewaves (Fourier bases) are less than ideal since they have the same resolution everywhere, and hence real-space approaches with their ability to have variable resolution in space (adaptive) become more attractive. More importantly, for large-scale electronic-structure calculations, basis-sets that are compactly-supported such as finite elements or wavelets lead to structured sparse system matrices and hence are more amenable to iterative solution schemes and to implementation on massively parallel computational platforms. As a variational method, finite elements are systematically improvable and convergence of the energy eigenvalues is from above (<a href="http://en.wikipedia.org/wiki/Min-max_theorem" target="_blank" title="Min-Max Theorem">min-max theorem</a>). Furthermore, boundary conditions (periodic, Bloch, Dirichlet or a combination of these) are easily incorporated within the weak formulation, which make FEs attractive for modeling crystalline solids, molecules, clusters, or surfaces. These attributes make finite elements particularly promising for density functional calculations, and hence the optimism that in the upcoming years there will emerge wider interest in exploring finite element methods in quantum-mechanical computations. I close with a reading list, and a short summary of each review article.
</p>
<p>
<strong>Reading</strong><br /></p>
<ol><li>T. L. Beck (2000), "Real-Space Mesh Techniques in Density-Functional Theory," <em>Reviews of Modern Physics</em>, Vol. <strong>72</strong>, Number <strong>4</strong>, pp. 1041–1080. [<a href="http://arxiv.org/abs/cond-mat/0006239">arXiv</a>] [<a href="http://prola.aps.org/abstract/RMP/v72/i4/p1041_1">Journal</a>]
<p>
Beck's review discusses finite-difference and finite element formulations in DFT. Emphasis is placed on Poisson and nonlinear Poisson-Boltzmann equations in electrostatics and on solutions of Hartree-Fock and KS-equations (eigenvalue problems) of DFT. First, Beck provides a context for real-space mesh techniques by describing some of the prominent developments so far on electronic structure methods (planewaves, Gaussian, LCAO, etc.). Then, the theory behind KS-DFT and classical electrostatics is described. Second-order FD is briefly discussed, and then a higher-order finite-difference method is used to solve the Poisson equation with a singular charge density. The finite element formulation for the Poisson equation is presented. Multigrid solvers are attractive for real-space methods and the main feature of a multigrid method are discussed. Beck presents energy minimization techniques to solve the Schrödinger eigenproblem, and provides a historical perspective on the developments in finite-difference and finite element methods for self-consistent calculations. Lastly, attention is given to time-dependent DFT calculations.
</p>
</li>
<li>J. E. Pask and P. A. Sterne (2005), "Finite Element Methods in <em>Ab Initio</em> Electronic Structure Calculations," <em>Modelling and Simulation in Materials Science and Engineering</em>, Vol. <strong>13</strong>, pp. R71–R96. [<a href="http://www.iop.org/EJ/abstract/0965-0393/13/3/R01" target="_blank" title="Finite element methods in ab initio electronic structure calculations">Journal</a>]
<p>
A PDF of this paper is uploaded (courtesy of Pask). Pask and Sterne review finite element bases and their use in the self-consistent solution of the Kohn-Sham equations of DFT. The solution of the Schrödinger and Poisson equations is discussed, with particular attention to the imposition of the required Bloch-periodic and periodic boundary conditions, respectively. The use of these solutions in the self-consistent solution of the Kohn-Sham equations and computation of the DFT total energy is then discussed, and applications are given. To impose Bloch-periodic boundary conditions, the wavefunction is rewritten in terms of a periodic function u(<strong>x</strong>), and the variational (weak) form for the Schrödinger equation in terms of u(<strong>x</strong>)is derived. The weak form within the unit cell and expressions for the contributions of the non-local part of the pseudopotential are presented. In general, the trial and test functions can now be complex-valued functions, and the overlap matrix <strong>S</strong> and the local part of the Hamiltonian <strong>H</strong> are Hermitian. Band structure for a Si pseudopotential is presented and the optimal sextic convergence in energy for cubic finite elements is demonstrated. In the context of crystalline calculations, particular attention is given to the handling of the long-range Coulomb interaction via use of neutral charge densities in the Poisson solution. Self-consistent finite element solutions for the band structure of GaAs are shown, and uniform convergence with mesh refinement to the exact self-consistent solution is demonstrated.
</p>
</li>
<li>T. Torsti et al. (2006), "Three Real-Space Discretization Techniques in Electronic Structure Calculations,"<em>Physica Status Solidi. B, Basic Research</em>, Vol. <strong>243</strong>, Number <strong>5</strong>, pp. 1016–1053. [<a href="http://arxiv.org/abs/cond-mat/0601201">arXiv</a>] [<a href="http://www3.interscience.wiley.com/journal/112478903/abstract">Journal</a>]
<p>
In this paper, Torsti and co-workers compare and contrast the performance of finite-differences, finite elements, and wavelets in electronic-structure calculations. The computational problems under consideration to be solved are the single-particle Schrödinger equation for the pseudowavefunction, and the Poisson equation for the Coulomb potential. Basic theory on FD, FE, and wavelets is first presented; hierarchical finite element bases are also touched upon. Solution approaches for linear equation solvers and for generalized eigenproblem solvers are discussed in significant detail. Applications of finite-differences to quantum dots, surface nanostructures and positron calculations are presented, whereas finite element solutions for all-electron calculations for molecules are performed. Finally, the similarities and differences between the different methods of discretization are nicely summarized.
</p>
</li>
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</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/417">Journal Club Forum</a></div></div></div><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-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/838">quantum mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/846">FEM</a></div><div class="field-item even"><a href="/taxonomy/term/1269">electronic structure</a></div><div class="field-item odd"><a href="/taxonomy/term/3371">DFT</a></div></div></div>Sat, 31 Jan 2009 01:45:40 +0000N. Sukumar4728 at https://www.imechanica.orghttps://www.imechanica.org/node/4728#commentshttps://www.imechanica.org/crss/node/4728Journal Club Theme of Sept. 15 2008: Defects in Solids---Where Mechanics Meets Quantum Mechanics
https://www.imechanica.org/node/3837
<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/1269">electronic structure</a></div><div class="field-item odd"><a href="/taxonomy/term/2810">bridging length scales</a></div><div class="field-item even"><a href="/taxonomy/term/2811">defects in materials</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>
Defects in solids have been studied by the mechanics community for over five decades, some of the earliest works on this topic dating back to Eshelby. Yet, they still remain interesting, challenging, and often spring surprises—one example being the observed hardening behavior in surface dominated structures (as discussed in past journal club themes by Wei Cai and Julia Greer). In this journal theme, I wish to concentrate on the underlying physics behind defect behavior and motivate the need to combine quantum mechanical and mechanics descriptions of materials behavior. Through this discussion, I hope to bring forth: (i) The need to bridge mechanics with quantum mechanics; (ii) The challenges in quantum mechanical calculations; (iii) How the mechanics community can have a great impact.
</p>
<p>
<strong>(i) The need to bridge mechanics with quantum mechanics:</strong></p>
<p>Defects play a crucial role in influencing the macroscopic properties of solids—examples include the role of dislocations in plastic deformation, dopants in semiconductor properties, domain walls in ferroelectric properties, and the list goes on. These defects are present in very small concentrations (few parts per million), yet, produce a significant macroscopic effect on the materials behavior through the long-ranged elastic and electrostatic fields they generate. But, the strength and nature of these fields as well as other critical aspects of the defect core are all determined by the electronic structure of the material at the quantum-mechanical length-scale. Hence, there is a wide range of interacting length-scales, from electronic structure to continuum, that need to be resolved to accurately describe defects in materials and their influence on the macroscopic properties of materials. </p>
<p>At this point, I wish to stress the importance of both electronic structure (quantum-mechanical effects) and long-ranged elastic fields by presenting some known results on the energetics of a single vacancy. The vacancy formation energy in aluminum computed from electronic-structure (ab-initio) calculations is about 0.7 eV, of which the contribution of elastic effects (atomic relaxations) is less than 10% of the formation energy, rest is electronic effects (quantum-mechanical effects)! In mechanics, these electronic effects are lumped as the core-energy, which is considered an inconsequential constant, and we deal with only elastic effects. On the other hand, computational materials scientists often work with only core energies as they appear to be the major contribution to the total defect energy. In my opinion, both are equally important and neither can be neglected and I will present some evidence to corroborate this claim. Some recent electronic-structure calculations have been performed to investigate the influence of homogeneous macroscopic strain on the energetics of vacancies (some of which are present in Ho et al. Phys. Chem. Chem. Phys. 2007, 9, 4951), where, in one case atomic relaxations are suppressed and the energetics are solely due to electronic effects and another where atomic relaxations are allowed which contain both electronic effects and elastic interactions with macroscopic fields. In the first case, the vacancy formation energy changed from 0.7 eV at no imposed macroscopic strain to 0.2 eV for 0.15 volumetric strain. This suggests that the defect core energy is very strongly influenced by the macroscopic deformation at the core site, and is not an inconsequential constant! This dependence is quantum-mechanical and there is no obvious way to determine this other than resorting to electronic structure (ab-initio) calculations. On the other hand, in the second case, upon relaxing the atoms and accounting for the elastic effects, the contribution for these elastic effects changed from 10% of the total formation energy at no macroscopic deformation to 50% at 0.15 volumetric strain. These results provide strong evidence that both the core of a defect and the long-ranged elastic fields are equally important in understanding the behavior of defects and these are inherently coupled through the electronic structure of the material. </p>
<p><strong>(ii) The challenges in electronic structure calculations:</strong></p>
<p>The basis of all electronic structure calculations is quantum mechanics which has the mathematical structure of an eigen-value problem. Though the physics behind quantum mechanics has been well-known for almost seven decades, the challenge arises from the computational complexity of the resulting governing equations (Schrodinger’s equation). Unfortunately solutions to the full Schrodinger’s equation are intractable beyond a few electrons (<10) making any meaningful computation of materials properties beyond reach. The direction pursued by the computational physics community in the mid-nineteenth century was beautifully summarized by Paul Dirac: “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of quantum mechanics should be developed, which can lead to an explanation of the main features of the complex atomic systems without too much computation”. These approximate methods are what constitute the electronic structure calculations which are widely used in the present day. The starting point of all electronic structure theories for computing ground-state materials properties is a variational principle, something which is very often seen in mechanics. I have written a brief overview (for readers interested in more details) of various electronic structure calculations and the various approximations involved in arriving at these theories:
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node/3813
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One of the most popular electronic structure theory that is widely used is the density-functional theory (DFT). It has its roots in the seminal work of Kohn, where he rigorously proved that the ground-state properties of a material system are only a function of the electron-density, which has made electronic structure calculations of materials possible. Albeit many theoretical developments in this field and the advent of supercomputing, the computational complexity of these calculations still restricts computational domains to couple of hundred atoms. Thus, historically it was natural to concentrate on periodic properties of materials. DFT has been very successfully in capturing a wide range of bulk properties which include elastic moduli, band-structure, phase transformations, etc. The interest in periodic properties has resulted in the use of a plane-waves as a basis set to compute the variational problem associated with density functional theory. Such a Fourier space formulation has limitations, especially in the context of defects: it requires periodic boundary conditions, thus limiting an investigation to a periodic array of defects. This periodicity restriction in conjunction with the cell-size limitations (200 atoms) arising from the enormous computational cost associated with electronic structure calculations, limits the scope of these studies to very high concentrations of defects that rarely—if ever—are realized in nature. Thus recently, there is an increasing thrust towards using real-space formulations and using finite-element or a wavelet basis, or a finite-difference scheme. The following three articles are good representations of the use of these methods.</p>
<p>1. <a href="http://arxiv.org/abs/cond-mat/9903313">J.E. Pask, B.M. Klein, C.Y. Fong, P.A. Sterne, Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach, Phys. Rev. B. 59 12352 (1999).</a> <br />
2. <a href="http://arxiv.org/abs/cond-mat/9805262">T.A. Arias, Multiresolution analysis of electronic structure: semicardinal and wavelet basis, Rev. Mod. Phys. 71, 267 (1999).</a> <br />
3. <a href="http://www.math.ucsb.edu/~cgarcia/OFDFT/Garcia-Cervera_CommCompPhys_2_p334_2007.pdf">C.J. Garcia-Cevera, An efficient real-space method for orbital-free density functional theory, Comm. Comp. Phys. 2, 334 (2006).<br /></a>
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<br /><strong>(iii) How the mechanics community can have a great impact.</strong></p>
<p>Although the use of real-space formulations seems to provide freedom from periodicity, the computational complexity still restricts calculations to a few hundred atoms. However, an accurate description of defects requires resolution of the electronic structure of the core as well as the long-ranged elastic effects. There have been some multi-scale methods based on embedding schemes that have been proposed which address this problem. One representative article for these methods is the following:</p>
<p>4. <a href="http://arxiv.org/abs/cond-mat/0506006">G. Lu, E. Tadmor, and E.Kaxiras, "From electrons to finite elements: A concurrent multiscale approach for metals" Phys. Rev. B 73, 024108 (2005).<br /></a><br /><br />
The philosophy behind these embedding schemes is to embed a refined electronic structure calculation (inside a small domain) in a coarser atomistic simulation using empirical potentials, which in turn is embedded in a continuum theory. Valuable as these schemes are, they suffer from some notable shortcomings. In some cases, uncontrolled approximations are made such as the assumption of separation of scales, the validity of which can not be asserted. Moreover, these schemes are not seamless and are not solely based on a single electronic structure theory. In particular, they introduce undesirable overlaps between regions of the model governed by heterogeneous and mathematically unrelated theories.</p>
<p>I feel there is tremendous potential for the mechanics community to contribute in the development of multi-scale schemes solely based on electronic structure calculations which are seamless, have controlled approximations, assure the notion of convergence, and provide insights into the behavior of defects. To start the discussion and motivate, let me provide an analogy. The electronic structure of a defect in a material has similar structure to a composite problem with a damage zone. Homogenization techniques and adaptive finite-element basis sets are common solutions to such composite problems! With a little care, I believe it is possible for the mechanics community to make a huge impact in electronic structure calculations of defects. <br />
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</div></div></div>Sun, 14 Sep 2008 21:50:53 +0000Vikram Gavini3837 at https://www.imechanica.orghttps://www.imechanica.org/node/3837#commentshttps://www.imechanica.org/crss/node/3837Postdoctoral Position at UC Davis in Computational Materials Science
https://www.imechanica.org/node/2809
<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/73">job</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/447">Finite Element Method</a></div><div class="field-item odd"><a href="/taxonomy/term/449">electrostatics</a></div><div class="field-item even"><a href="/taxonomy/term/1269">electronic structure</a></div><div class="field-item odd"><a href="/taxonomy/term/1981">Kohn-Sham equations</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>Update:</strong> The position has been filled; thanks to all who responded.
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A post-doctoral position is immediately available at UC Davis. The individual will work on a joint project led by myself and <a href="http://physci.llnl.gov/Research/Metals_Alloys/People/Pask" target="_blank" title="Metals and Alloys Group at LLNL">John Pask at LLNL</a> on the development and application of a new finite-element based approach for large-scale quantum mechanical materials calculations.
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The research is aimed at getting beyond standard planewave based methods for large-scale quantum molecular dynamics, and addressing a range of problems that have been inaccessible so far by such accurate, quantum-mechanical means. The key ingredient of the new finite-element based approach, as currently formulated, is the partition-of-unity basis which allows known physics to be built into the basis set without sacrificing locality or systematic improvability; thus allowing for a substantial reduction in basis size while retaining natural and efficient parallelization.
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The candidate should possess a strong background in finite elements and computational mathematics, with excellent programming skills. Knowledge and research experience in quantum mechanics, electronic-structure calculations, and/or nanomechanics will be advantageous. Interested individuals should send a copy of their curriculum vitae, a list of three references, and a short statement of research interests (upto a page) to me via e-mail. Please send a single PDF file to me at <a href="mailto:nsukumar@ucdavis.edu" target="_blank">nsukumar@ucdavis.edu</a>
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</div></div></div>Tue, 04 Mar 2008 20:16:22 +0000N. Sukumar2809 at https://www.imechanica.orghttps://www.imechanica.org/node/2809#commentshttps://www.imechanica.org/crss/node/2809Journal Club Theme of September 2007: Quantum Effects in Solid Mechanics
https://www.imechanica.org/node/1865
<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/139">Carbon nanotube</a></div><div class="field-item odd"><a href="/taxonomy/term/838">quantum mechanics</a></div><div class="field-item even"><a href="/taxonomy/term/1268">deformation</a></div><div class="field-item odd"><a href="/taxonomy/term/1269">electronic structure</a></div><div class="field-item even"><a href="/taxonomy/term/1270">quantum dot</a></div><div class="field-item odd"><a href="/taxonomy/term/1271">quasicontinuum</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>
Since the early 1990s, when <strong><a href="http://en.wikipedia.org/wiki/Quantum_dot">quantum dots</a></strong> and quantum wires began to attract the attention of physicists, and when <strong><a href="http://en.wikipedia.org/wiki/Carbon_nanotube">carbon nanotubes</a></strong> were discovered, mechanics related issues have begun to emerge as important in understanding properties of nanostructures. These structures were first considered useful mostly for their electronic or optical applications, yet deformation has been seen to play an important role in their functional characteristics. Advances in modeling also have begun to link electronic structure with mechanical properties of materials at larger length scales, particularly when microstructural or crystallographic effects influence bulk behavior.
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<img src="http://www.mechse.uiuc.edu/johnson/indexo6.jpg" alt="Electron density around an edge dislocation core in GaN" title="Electron density around an edge dislocation core in GaN" hspace="10" vspace="10" width="188" height="193" align="left" />The question of how<em> solid mechanics, through the effects of deformation, connects to the quantum mechanical behavior of electrons in a solid</em> arises from some of these considerations. The usual way to approach the problem is to note that a crystalline material can be characterized by an electronic energy band structure, from which many electrical, optical (and even thermal or mechanical) properties can be derived. When the crystal is strained or otherwise mechanically perturbed, for example, through the presence of defects, the bandstructure changes. If one can write down a constitutive relation for this coupling, a complete model may be possible. There are several complications to consider, however. For example, in confined geometries, electrons are not always best described by bulk bandstructure, because their properties may be dominated by quantization effects due to the boundaries. Also, the connection between bandstructure and deformation is a two-way coupled problem, in general, so that deformation may affect bandstructure, and changes in electronic structure may also induce strain. With these complications in mind, one must confront many subtle issues in building models for quantum effects in solid mechanics. For example, what level of accuracy is necessary to understand deformation at this scale? Is continuum mechanics sufficient, or should one adopt an atomistic approach, and if so, is an electronic structure method necessary? Can quantum mechanical effects be understood using a homogenization technique such as an effective mass model, or should one consider every electron in the solid? These issues impact a wide range of interesting problems in applied physics.
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Three simple questions are posed here to better frame the discussion for this edition of the journal club. The issue of quantum effects in solid mechanics is important for each of these questions, and central to the journal articles selected for discussion. Each of the three representative articles has had significant scholarly impact in the short time since it was published. Readers may have differing opinions about the depth of the physics or mechanics in each of the three papers. Hopefully this will be a thread for discussion in the journal club. Here are the questions, and the papers chosen to represent them:
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<em>1. How does strain affect optical properties at the nanoscale?</em>
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<a href="http://prola.aps.org/pdf/PRB/v52/i16/p11969_1">M. Grundmann, O. Stier, and D. Bimberg, "InAs/GaAs<strong> </strong>pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure," Phys. Rev. B<strong> 52</strong>, 11969 - 11981 (1995).</a>
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In this work the authors compute the full strain tensor field arising from lattice mismatch between an idealized quantum dot and the underlying substrate. They then compute the effect of the strain distribution on the optical transition energies and find very good agreement between their model and the results of photoluminescence and optical absorption data from experiments. Many other authors have subsequently taken up issues addressed in this paper, including a number of mechanicians interested in accurately calculating strain distributions for such small scale structures.
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<em>2. How does strain affect electrical conduction at the nanoscale?</em>
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<a href="http://prola.aps.org/pdf/PRB/v60/i19/p13824_1">A. Rochefort, P. Avouris, F. Lesage, and D. H. Salahub, "Electrical and mechanical properties of distorted carbon nanotubes," Phys. Rev. B <strong>60</strong>, 13824 - 13830 (1999).</a>
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As in the first paper, the authors here consider the coupling between electronic structure and mechanical behavior in a simple, idealized nanometer scale structure. Here they model electrical conduction in deformed single-wall carbon nanotubes, using a molecular mechanics approach to study deformation. There have been numerous studies on this topic in the time since this paper appeared, including many analyses (atomistic, continuum, and multi-scale) of deformation in carbon nanotubes. But, like the first paper, this study is significant because it treats the problem of <em>coupling</em> between quantum mechanics and deformation.
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<em>3. How does electronic structure influence elasticity?</em>
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<a href="http://prola.aps.org/pdf/PRB/v59/i1/p235_1">E. B. Tadmor, G. S. Smith, N. Bernstein, and E. Kaxiras, "Mixed finite element and atomistic formulation for complex crystals," Phys. Rev. B <strong>59</strong>, 235 - 245 (1999).</a>
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This work is different in spirit than the first two papers. Here the authors present details of a quasicontinuum method applicable for complex crystalline materials (such as silicon) where one is interested in connecting the underlying electronic structure with the macroscopic mechanical behavior. Their approach extends <a href="http://www.qcmethod.com/">the method first presented by Tadmor, Ortiz, and Phillips</a> to include an atomistic formulation based on tight-binding, which is one of the simplest atomic scale approaches that explicitly accounts for electrons. This work opened the door to many studies that have followed, including recent work linking density functional theory atomistics to continuum models via the quasicontinuum methodology.
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Note: A student trained in elasticity and continuum mechanics needs to be familiar with basic solid state physics and quantum mechanics to really get into this area. Some quantum mechanics and solid state texts are fairly accessible to solid mechanicians, but for a reader with a mechanics of materials perspective, the book by <strong><a href="http://www.amazon.com/Crystals-Defects-Microstructures-Modeling-Across/dp/0521793572">Phillips (Crystals, Defects and Microstructures, Cambridge, 2001)</a></strong> is an excellent place to start.
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</div></div></div>Fri, 31 Aug 2007 20:26:57 +0000Harley T. Johnson1865 at https://www.imechanica.orghttps://www.imechanica.org/node/1865#commentshttps://www.imechanica.org/crss/node/1865