Finite Element Implementation of Orbital-Free Density Functional Theory for Electronic Structure Calculations Soumya Swayamjyoti Report/Preprint No. 12–I–02 Institute of Applied Mechanics (CE) · Micromechanics of Materials Group University of Stuttgart, 70550 Stuttgart, Pfaffenwaldring 7, Germany ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄·⋄·⋄·⋄ ⋄⋄ ⋄·⋄ ⋄ ⋄ ⋄ ⋄·⋄ ⋄⋄··⋄·⋄·⋄·⋄⋄·⋄··⋄⋄ ⋄⋄··⋄·⋄·⋄⋄·⋄··⋄⋄ ⋄··⋄⋄··⋄·⋄·⋄⋄··⋄⋄··⋄ ·⋄··⋄⋄··⋄·⋄·⋄·⋄·⋄·⋄·⋄·⋄·⋄··⋄ ·⋄⋄·⋄⋄·⋄⋄·⋄⋄·⋄⋄·⋄⋄·⋄⋄·⋄⋄·⋄ ·⋄··⋄··⋄··⋄··⋄··⋄··⋄··⋄··⋄ ·⋄ ·⋄⋄·⋄·⋄·⋄·⋄⋄·⋄ ·⋄ ·⋄ ·⋄⋄· ⋄··⋄·⋄·⋄·⋄ ·⋄ ·⋄⋄· ⋄··⋄·⋄·⋄·⋄⋄· · ⋄··⋄·⋄··⋄·⋄·⋄··⋄ · ··⋄·⋄···⋄··⋄··⋄·⋄·· ···⋄··⋄··⋄··⋄··⋄·· Contents 1. Acknowledgements. 2. Introduction. 3. Orbital-Free Density Functional Theory. 3.1. The Orbital-Free Density Functional Theory Framework. 3.2. The Variational Problem. 3.3. Finite Element Formulation. 4. Multi-dimensional Optimization Methods for Orbital-Free DFT. 4.1. Steepest Descent Method. 4.2. Conjugate Gradient Method. 4.3. Root-Finding Newton Method. 5. Numerical Implementation. 5.1. Mesh Generation. 5.2. Implementation with DUNE Numerics. 6. Numerical Examples. 6.1. Atoms. 6.2. Molecules. 6.3. Aluminium Clusters. 6.4. Computational Time. 7. Conclusion and Future Directions. A. Required Derivatives. A.1. Derivatives with respect to {ui}. A.2. Derivatives with respect to {φi}. Master Thesis supervised by Dipl.-Phys. V. Schauer & Jun.-Prof. Dr. C. Linder Stuttgart, April 2012 (modified Dec 2013) Acknowledgements 1 1. Acknowledgements First of all, I would like to thank Jun.-Prof. Christian Linder for giving me the opportunity to do my master thesis in his group. I thank him for his encouragement and support during this interesting study. Additionally, I would like to express my gratitute to him for his help and advice during a period of personal difficulties. I want to thank Dipl.-Phys Volker Schauer for his guidance and supervision of this thesis. I am thankful to him for his constant support and help. It was a great learning experience to work with him. I am grateful to him for his valuable suggestions. The discussions we had pertaining to this thesis and about physics in general motivated my interest in computational physics to a great extent. I would also like to thank him for his friendship during my studies in Stuttgart. I seize this opportunity to wish him the best in his future endeavors. Many thanks to Prof. Christian Miehe for his timely advice and institutional support. I would like to thank my PhD supervisors Dr. Peter M. Derlet and Prof. J¨org L¨offler for the confidence they reposed in my profile following my seminar at Paul Scherrer Institut and subsequent discussions. Thanks to our collaborators Prof. Vikram Gavini and his group (in particular, Mr. Phani Sudheer Motamarri) for useful inputs. Thanks to all my colleagues in the Micromechanics of Materials Group and the In- stitute of Applied Mechanics (Chair I) of University of Stuttgart for the cordial work atmosphere. I thank Prof. Christian Linder, Prof. Eugenio Onate and Prof. Pedro Diez for their help during my PhD applications. A strong reference by Prof. Christian Linder as my Master Thesis Supervisor, a reference letter of Outstanding Performance in the first semester in Barcelona by Prof. Eugenio Onate and support of my abilities by Prof. Pedro Diez as the coordinator of my master program helped me find a very good PhD position at Paul Scherrer Institut and ETH Zurich. It is also relevant to point out that I enjoyed Prof. Christian Linder’s interactive lectures on Atomistic Modeling of Materials very much during my Masters. I look upto Prof. Eugenio Onate for the pioneering research he has done in the field of computational mechanics and his active management of CIMNE Barcelona as its director and founder. I acknowledge Prof. Pedro Diez’s, Prof. Sergio Zlotnik’s and Mr. Gautam Ethiraj’s excellent coordination of the Erasmus Mundus Master Program in Computational Mechanics from Barcelona and Stuttgart. Thanks to Prof. Sergio Zlotnik for organizational support for hosting the next EMA General Assembly in Barcelona in June 2013. I gratefully acknowledge the contributions to the formative years of my scientific career of Prof. K. Swaminathan as my Bachelor Thesis Advisor and Prof. Lakshman Nandagiri as a mentor at my home university National Institute of Technology Karnataka, Surathkal. I thank Dr. Brijesh Eshpuniyani as one of the first persons to introduce me to research while interning with him at IIT Delhi and IIT Kanpur during two consecutive summers. Special mention to Mr. Gautam Ethiraj (Stuttgart) and Ms. Lelia Zielonka (Barcelona) for their help with administrative affairs during my studies in the Erasmus Mundus Master of Science in Computational Mechanics. I thank my teachers at University of Stuttgart and UPC Barcelona for helping me build up my knowledge base in computational me- Acknowledgements 2 chanics. Thanks to my colleague and my best friend in Europe Mr. Anuj Sharma for a very generous friendship in Stuttgart. I thank Mr. Rodolfo Fleury (Swansea) and Mr. Fabien Poulhaon (Nantes) for the good times in Barcelona. I wish Anuj, Rodolfo and Fabien (who are also my classmates in the Erasmus Mundus Master of Science in Computational Mechanics) the very best in their doctoral studies at TU Darmstadt, Oxford University and Ecole Centrale Nantes respectively. I thank my family in India for their unconditional love and support. I gratefully acknowledge the financial support of the European Commission. My stud- ies in Europe were funded by the prestigious Erasmus Mundus Scholarship (A grant of 42k EUR by the European Commission for two years of master studies). I thank the selection consortium comprising of CIMNE (International Center for Numerical Methods in Engineering, Barcelona), University of Stuttgart, Ecole Centrale Nantes and Swansea University. Introduction 3 2. Introduction At the fundamental level, ground state properties of materials depend only on elec- tron density and hence are obtainable from first principles. However, a complete ab-initio calculation by solving the Schrodinger equation is computationally ambitious for a wide range of length scales involved in material modeling and thus separation of scales become necessary. At the microscopic scale, one can aim at ab-initio electronic structure calcula- tions in the framework of approximate theories developed over the last decades, the most prominent being the density functional theory which was pioneered by [24] and [25]. In the Kohn-Sham Density Functional Theory (KSDFT) the kinetic energy is computed by solv- ing an eigenvalue problem involving the wave-functions or orbitals of a material system. The computational complexity associated with this calculation is high and thus limits the potential system size to a few hundred atoms. This has inspired interest in orbital-free ki- netic energy functionals. This form of density functional theory where the kinetic energy is modeled by orbital-free kinetic energy functionals is referred to as Orbital-Free Density Functional Theory (OFDFT). This theory has its roots in the Thomas-Fermi model which finds mention in the early papers of [39], [27], [26]. In the context of material science, the electronic structure of a material determines the behavior of the material at the macroscopic level, which is the length scale of interest for engineering applications. Processes that occur at a certain length scale govern the behaviour of the material at larger length scales. Hence, it becomes necessary to success- fully bridge scales that can couple quantum mechanical, molecular mechanical or classical atomistic and continuum mechanics simulations in a unified theory. A feature of these multiscale simulations is to use accurate and computationally complex techniques to treat regions of small length scale and “coarse-grain” the rest of the system, that is, use less accurate and computationally faster methods for the rest of the system. In the atomistic framework, each atom is tracked during a deformation process and this information is used for predicting the macroscopic properties. While such resolution gives the exact description of a material, the computational demand associated with large system sizes makes most problems intractable in a completely atomistic framework. This is where the quasi-continuum (QC) methods come into play. Extensively discussed in [38], [42] and [16], QC methods aims at the right balance between fully atomistic and fully continuum techniques. They involve an adaptive use of both methodologies, that is, retaining the atomistic resolution in spatial regions of the problem only where required and use of continuum appoximations elsewhere to reduce the computational expense. Initially, the QC method was limited to atomistic interactions at the highest resolution. But it turns out that, a complete material description depends on the electronic structure, in particular, to predict material phenomenon such as bond breaking at the crack tip during failure in solids. To put the present work in context, we have implemented numerical methods to compute electronic structure of materials. The mathematical frameworks used in the current study can be programmed into a QC formalism. A natural question that arises here is our choice of OFDFT as the quantum mechanical sub-framework in the larger framework of multiscale method development. Arguably, Kohn-Sham-DFT will suit better for it’s accuracy and theoretical robustness, refer [52] and [41] for recent developments in finite-element implementation of KSDFT. However, a direct implementation of Kohn-