Adaptively Compressed Polarizability Operator
|
|
|
- Russell Lucas
- 5 years ago
- Views:
Transcription
1 Adaptively Compressed Polarizability Operator For Accelerating Large Scale ab initio Phonon Calculations Ze Xu 1 Lin Lin 2 Lexing Ying 3 1,2 UC Berkeley 3 ICME, Stanford University BASCD, December 3rd 2016 Ze Xu (UC Berkeley) ACP BASCD / 40
2 Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD / 40
3 Phonon Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD / 40
4 Phonon What is Phonon? In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter... it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles. wikipedia Ze Xu (UC Berkeley) ACP BASCD / 40
5 Phonon Lattice Wave Figure 1: Lattice wave example. Figure from wikipedia. Ze Xu (UC Berkeley) ACP BASCD / 40
6 Phonon Phonon Dispersion Relations Figure 2: Phonon dispersion and densities of states of gallium arsenide (GaAs). Figure from S. Baroni et al., Ze Xu (UC Berkeley) ACP BASCD / 40
7 Phonon Why Care? Ze Xu (UC Berkeley) ACP BASCD / 40
8 Phonon Why Care? Thermal conductivity; specific heat; Ze Xu (UC Berkeley) ACP BASCD / 40
9 Phonon Why Care? Thermal conductivity; specific heat; Elastic neutron scattering; Ze Xu (UC Berkeley) ACP BASCD / 40
10 Phonon Why Care? Thermal conductivity; specific heat; Elastic neutron scattering; Electrical conductivity; Ze Xu (UC Berkeley) ACP BASCD / 40
11 Phonon Why Care? Thermal conductivity; specific heat; Elastic neutron scattering; Electrical conductivity; Electron-phonon interaction related topics; Ze Xu (UC Berkeley) ACP BASCD / 40
12 Phonon Why Care? Thermal conductivity; specific heat; Elastic neutron scattering; Electrical conductivity; Electron-phonon interaction related topics; Ze Xu (UC Berkeley) ACP BASCD / 40
13 Phonon Elastic neutron scattering Thermal conductivity; specific heat; Elastic neutron scattering; Electrical conductivity; Electron-phonon interaction related topics; Figure 3: In elastic neutron-scattering, the neutron bounces off the bombarded nucleus without exciting or destabilizing it. Figure from The Schlumberger Oilfield Glossary. Ze Xu (UC Berkeley) ACP BASCD / 40
14 Phonon Electron-phonon interaction Thermal conductivity; specific heat; Elastic neutron scattering; Electrical conductivity; Electron-phonon interaction related topics; major contributor to electrical resistance in most inorganic metals and semiconductors above zero (very low) temperature; Overheating problem; Ze Xu (UC Berkeley) ACP BASCD / 40
15 Phonon Dynamical Matrix D I,J = 1 2 E tot ({R I }), MI M J R I R J M I mass of the I-th atom, I = 1,..., N A. R I R d position of the I-th atom, I = 1,..., N A. D R dn A dn A Ze Xu (UC Berkeley) ACP BASCD / 40
16 DFT Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD / 40
17 DFT Density Functional Theory Ab initio calculation DFT. most widely used; exact description of ground state properties: electron density, energy, and atomic forces. Ze Xu (UC Berkeley) ACP BASCD / 40
18 DFT Kohn-Sham Density Functional Theory E KS ({ψ i }; {R I }) = 1 N e ψ i (r) 2 dr + V ion (r; {R I })ρ(r) dr 2 i=1 + 1 v c (r, r )ρ(r)ρ(r ) dr dr + E xc [ρ] 2 (1) + E II ({R I }). Ze Xu (UC Berkeley) ACP BASCD / 40
19 DFT Derivative F I = VI (r R I )ρ(r) dr E II({R I }). R I R I 2 E tot ({R I }) = R I R J VI (r R I ) δρ(r) dr + δ I,J R I δr J + 2 E II ({R I }) R I R J. ρ(r) 2 V I R 2 (r R I ) dr I Ze Xu (UC Berkeley) ACP BASCD / 40
20 DFT Derivative F I = VI (r R I )ρ(r) dr E II({R I }). (2) R I R I 2 E tot ({R I }) = R I R J VI (r R I ) δρ(r) dr+δ I,J R I δr J + 2 E II ({R I }) R I R J. ρ(r) 2 V I R 2 (r R I ) dr I (3) Ze Xu (UC Berkeley) ACP BASCD / 40
21 DFT Polarizability Operator δρ(r) = δr J Fréchet derivative χ(r, r ) := operator. δρ(r) V J δv ion (r (r R J ) dr. ) R J δρ(r) δv ion (r ) is the reducible polarizability Ze Xu (UC Berkeley) ACP BASCD / 40
22 DFT KSDFT Cont. Kohn-Sham equations (W. Kohn and L. Sham 1965) H[ρ]ψ i = ( 12 ) + V[ρ] ψ i = ε i ψ i, (4) ε i energy level; ψ i orbitals; N e ψ i (r)ψ j (r) dr = δ ij, ρ(r) = ψ i (r) 2. (5) index i = 1,..., N e called occupied states; i = N e + 1,... unoccupied states; ε g = ε Ne+1 ε Ne band gap; i=1 V[ρ](r) = V ion (r; {R I }) + v c (r, r )ρ(r ) dr + V xc [ρ](r) V ion = I V I, g J = V J R J. Ze Xu (UC Berkeley) ACP BASCD / 40
23 DFT Sternheimer Equations Q projection onto unoccupied states; Q(ε i H)Qζ ij = Q(ψ i g j ). i = 1,..., N e, j = 1,..., dn a (6) ζ ij defined as the solution to the equation. This is the core part of density-functional perturbation theory (DFPT) (S. Baroni et al. 2001). Ze Xu (UC Berkeley) ACP BASCD / 40
24 DFT How to solve that? Q(ε i H)Qζ ij = Q(ψ i g j ). i = 1,..., N e, j = 1,..., dn a SVD? Frozen phonon approach? (Or finite difference in math) Compression? Ze Xu (UC Berkeley) ACP BASCD / 40
25 DFT How to solve that? cont. Q(ε i H)Qζ ij = Q(ψ i g j ). i = 1,..., N e, j = 1,..., dn a SVD? Frozen phonon approach? (Or finite difference in math) Compression! ACP formulation reduces the computational complexity of phonon calculations from O(N 4 e ) to O(N 3 e ) for the first time (X.et al. 2016). Ze Xu (UC Berkeley) ACP BASCD / 40
26 DFT The Dyson Equation Irreducible polarizability operator χ 0 = δρ δv χ = χ 0 + χ 0 v hxc χ or U = χ 0 G + χ 0 v hxc U, U = χg. (7) So compression of χ 0 can only be done adaptively. Ze Xu (UC Berkeley) ACP BASCD / 40
27 ACP Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD / 40
28 ACP Adaptively Compressed Polarizability Operator Q(ε i H)Qζ ij = Q(ψ i g j ). i = 1,..., N e, j = 1,..., dn a Chebyshev interpolation disentangle the energy dependence on the left Interpolative separable density fitting method compress the rhs vectors. Adaptively compressed cost stays the same across iterations. Ze Xu (UC Berkeley) ACP BASCD / 40
29 ACP Chebyshev Interpolation Pick {ε c } I [ε 1, ε Ne ]. Lagrange interpolation N c ζ = ζ c c=1 c c ε ε c. ε c ε c Ze Xu (UC Berkeley) ACP BASCD / 40
30 ACP Interpolative Decomposition M ij = ψ i g j, or M ij (r) = ψ i (r)g j (r). N µ N µ M ij (r) ξ µ (r)m ij (r µ ) ξ µ (r)ψ i (r µ )g j (r µ ). (8) µ=1 µ=1 (H. Cheng, Z. Gimbutas, P. G. Martinsson, and V. Rokhlin, 2005) Ze Xu (UC Berkeley) ACP BASCD / 40
31 ACP Interpolative Separable Density Fitting Two-step procedure: subsampled random Fourier Transform and QR decomposition. (J. Lu and L. Ying, 2015) Ze Xu (UC Berkeley) ACP BASCD / 40
32 ACP Reconstruction To avoid O(Ne 4 ) complexity, construct N e N c W µ = 2 ψ i ε i ε c ζ cµ ψ i (r µ ). (9) ε c ε c i=1 c=1 c c N µ χ 0 g j W µ g j (r µ ), (10) µ=1 or formally, we have χ 0 χ 0 := W Π T. Ze Xu (UC Berkeley) ACP BASCD / 40
33 ACP Iterative scheme Introduce the following change of variable U = Ũ B, B = v 1 hxcg, (11) Iterative scheme: Ũ = χ 0 [Ũ]v hxcũ + B. (12) (a)construct χ k 0[Ũ k ] = W k (Π k ) T ( ) 1 (b)ũ k+1 = I W k (Π k ) T v hxc B (13) = B + W k ( I (Π k ) T v hxc W k) 1 (Π k ) T v hxc B. Ze Xu (UC Berkeley) ACP BASCD / 40
34 Numerical Examples Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD / 40
35 Numerical Examples One-dimensional Model ϱd DFPT ACP FD phonon frequency ϱd DFPT ACP FD phonon frequency Figure 4: Phonon spectrum for the 1D systems computed using ACP, DFPT, and FD, for both (a) insulating and (b) semiconducting systems. Ze Xu (UC Berkeley) ACP BASCD / 40
36 Numerical Examples One-dimensional Model cont DFPT ACP FD DFPT ACP FD Time (s) 10 3 Time (s) System size: # of Atom System size: # of Atom Figure 5: Computational time of 1D examples. Comparison among DFPT, ACP, and FD for (a) insulating, and (b) semiconducting systems, respectively. Ze Xu (UC Berkeley) ACP BASCD / 40
37 Numerical Examples Two-dimensional Model occupied unoccupied εi index(i) Figure 6: The electron density ρ of the 2D system with defects (a), and the occupied and unoccupied eigenvalues (b). Ze Xu (UC Berkeley) ACP BASCD / 40
38 Numerical Examples Two-dimensional Model cont DFPT ACP ϱd phonon frequency Figure 7: Phonon spectrum for the 2D system with defects. N A = 69. Ze Xu (UC Berkeley) ACP BASCD / 40
39 What s More? Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD / 40
40 What s More? Split Representation of ACP N e χ 0 (r, r ) = 2 i=1 j=n e+1 N e = 2 + N t i=1 j=n e+1 N e i=1 j=n t+1 := χ (1) 0 + χ (2) 0. ψ i (r)ψ j (r)ψ i (r )ψ j (r ) ε i ε j ψ i (r)ψ j (r)ψ i (r )ψ j (r ) ε i ε j ψ i (r)ψ j (r)ψ i (r )ψ j (r ) ε i ε j Ze Xu (UC Berkeley) ACP BASCD / 40
41 What s More? Silicon with 8 Atoms External Perturbation (avg along z-axis) ACP Response (avg along z-axis) Figure 8: External perturbation and response. Ze Xu (UC Berkeley) ACP BASCD / 40
42 What s More? Silicon with 8 Atoms cont. Error of ACP Response (avg along z-axis) #10-8 Error of FD Response (avg along z-axis) 6 # Figure 9: Error on response. Ze Xu (UC Berkeley) ACP BASCD / 40
43 Conclusion Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD / 40
44 Conclusion Conclusion ACP formulation reduces the computational complexity of phonon calculations from O(N 4 e ) to O(N 3 e ) for the first time. Recently we extend the formulation to cope with real materials. L. Lin, Z. Xu and L. Ying, Adaptively compressed polarizability operator for accelerating large scale ab initio phonon calculations, SIAM Multiscale Model. Simul. accepted, 2016 Ze Xu (UC Berkeley) ACP BASCD / 40
45 Conclusion Conclusion ACP formulation reduces the computational complexity of phonon calculations from O(N 4 e ) to O(N 3 e ) for the first time. Recently we extend the formulation to cope with real materials. L. Lin, Z. Xu and L. Ying, Adaptively compressed polarizability operator for accelerating large scale ab initio phonon calculations, SIAM Multiscale Model. Simul. accepted, 2016 Thanks for your attention. Ze Xu (UC Berkeley) ACP BASCD / 40
Downloaded 05/09/17 to Redistribution subject to SIAM license or copyright; see
MULTISCALE MODEL. SIMUL. Vol. 15, No. 1, pp. 29 55 c 2017 Society for Industrial and Applied Mathematics ADAPTIVELY COMPRESSED POLARIZABILITY OPERATOR FOR ACCELERATING LARGE SCALE AB INITIO PHONON CALCULATIONS
Linear-response excitations. Silvana Botti
from finite to extended systems 1 LSI, CNRS-CEA-École Polytechnique, Palaiseau, France 2 LPMCN, CNRS-Université Lyon 1, France 3 European Theoretical Spectroscopy Facility September 3, 2008 Benasque, TDDFT
Ab initio phonon calculations in mixed systems
Ab initio phonon calculations in mixed systems Andrei Postnikov apostnik@uos.de Outline: Experiment vs. ab initio theory Ways of theory: linear response and frozen phonon approaches Applications: Be x
Introduction to density functional perturbation theory for lattice dynamics
Introduction to density functional perturbation theory for lattice dynamics SISSA and DEMOCRITOS Trieste (Italy) Outline 1 Lattice dynamic of a solid: phonons Description of a solid Equations of motion
Introduction to Engineering Materials ENGR2000. Dr.Coates
Introduction to Engineering Materials ENGR2000 Chapter 18: Electrical Properties Dr.Coates 18.2 Ohm s Law V = IR where R is the resistance of the material, V is the voltage and I is the current. l R A
Compressed representation of Kohn-Sham orbitals via selected columns of the density matrix
Lin Lin Compressed Kohn-Sham Orbitals 1 Compressed representation of Kohn-Sham orbitals via selected columns of the density matrix Lin Lin Department of Mathematics, UC Berkeley; Computational Research
Key concepts in Density Functional Theory (I) Silvana Botti
From the many body problem to the Kohn-Sham scheme European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Temporary Address: Centre
An Approximate DFT Method: The Density-Functional Tight-Binding (DFTB) Method
Fakultät für Mathematik und Naturwissenschaften - Lehrstuhl für Physikalische Chemie I / Theoretische Chemie An Approximate DFT Method: The Density-Functional Tight-Binding (DFTB) Method Jan-Ole Joswig
3: Density Functional Theory
The Nuts and Bolts of First-Principles Simulation 3: Density Functional Theory CASTEP Developers Group with support from the ESF ψ k Network Density functional theory Mike Gillan, University College London
Many-Body Perturbation Theory. Lucia Reining, Fabien Bruneval
, Fabien Bruneval Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Belfast, 27.6.2007 Outline 1 Reminder 2 Perturbation Theory
Preface Introduction to the electron liquid
Table of Preface page xvii 1 Introduction to the electron liquid 1 1.1 A tale of many electrons 1 1.2 Where the electrons roam: physical realizations of the electron liquid 5 1.2.1 Three dimensions 5 1.2.2
Exact exchange (spin current) density-functional methods for magnetic properties, spin-orbit effects, and excited states
Exact exchange (spin current) density-functional methods for magnetic properties, spin-orbit effects, and excited states DFT with orbital-dependent functionals A. Görling Universität Erlangen-Nürnberg
Time-dependent density functional theory
Time-dependent density functional theory E.K.U. Gross Max-Planck Institute for Microstructure Physics OUTLINE LECTURE I Phenomena to be described by TDDFT Some generalities on functional theories LECTURE
Time-dependent density functional theory (TDDFT)
Advanced Workshop on High-Performance & High-Throughput Materials Simulations using Quantum ESPRESSO ICTP, Trieste, Italy, January 16 to 27, 2017 Time-dependent density functional theory (TDDFT) Ralph
Electronic Structure Theory for Periodic Systems: The Concepts. Christian Ratsch
Electronic Structure Theory for Periodic Systems: The Concepts Christian Ratsch Institute for Pure and Applied Mathematics and Department of Mathematics, UCLA Motivation There are 10 20 atoms in 1 mm 3
Density Functional Theory. Martin Lüders Daresbury Laboratory
Density Functional Theory Martin Lüders Daresbury Laboratory Ab initio Calculations Hamiltonian: (without external fields, non-relativistic) impossible to solve exactly!! Electrons Nuclei Electron-Nuclei
Quantum Mechanical Simulations
Quantum Mechanical Simulations Prof. Yan Wang Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332, U.S.A. yan.wang@me.gatech.edu Topics Quantum Monte Carlo Hartree-Fock
The GW Approximation. Manish Jain 1. July 8, Department of Physics Indian Institute of Science Bangalore 1/36
1/36 The GW Approximation Manish Jain 1 Department of Physics Indian Institute of Science Bangalore July 8, 2014 Ground-state properties 2/36 Properties that are intrinsic to a system with all its electrons
Key concepts in Density Functional Theory (II) Silvana Botti
Kohn-Sham scheme, band structure and optical spectra European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Temporary Address:
Time-dependent density functional theory (TDDFT)
04/05/16 Hands-on workshop and Humboldt-Kolleg: Density-Functional Theory and Beyond - Basic Principles and Modern Insights Isfahan University of Technology, Isfahan, Iran, May 2 to 13, 2016 Time-dependent
MD simulation: output
Properties MD simulation: output Trajectory of atoms positions: e. g. diffusion, mass transport velocities: e. g. v-v autocorrelation spectrum Energies temperature displacement fluctuations Mean square
Chemistry, Physics and the Born- Oppenheimer Approximation
Chemistry, Physics and the Born- Oppenheimer Approximation Hendrik J. Monkhorst Quantum Theory Project University of Florida Gainesville, FL 32611-8435 Outline 1. Structure of Matter 2. Shell Models 3.
Phonons In The Elk Code
Phonons In The Elk Code Kay Dewhurst, Sangeeta Sharma, Antonio Sanna, Hardy Gross Max-Planck-Institut für Mikrostrukturphysik, Halle If you are allowed to measure only one property of a material, make
Density Functional Theory - II part
Density Functional Theory - II part antonino.polimeno@unipd.it Overview From theory to practice Implementation Functionals Local functionals Gradient Others From theory to practice From now on, if not
Exchange-Correlation Functional
Exchange-Correlation Functional Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Depts. of Computer Science, Physics & Astronomy, Chemical Engineering & Materials Science, and Biological
Linear response theory and TDDFT
Linear response theory and TDDFT Claudio Attaccalite http://abineel.grenoble.cnrs.fr/ CECAM Yambo School 2013 (Lausanne) Motivations: +- hν Absorption Spectroscopy Many Body Effects!!! Motivations(II):Absorption
Electronic band structure, sx-lda, Hybrid DFT, LDA+U and all that. Keith Refson STFC Rutherford Appleton Laboratory
Electronic band structure, sx-lda, Hybrid DFT, LDA+U and all that Keith Refson STFC Rutherford Appleton Laboratory LDA/GGA DFT is good but... Naive LDA/GGA calculation severely underestimates band-gaps.
Welcome to the Solid State
Max Planck Institut für Mathematik Bonn 19 October 2015 The What 1700s 1900s Since 2005 Electrical forms of matter: conductors & insulators superconductors (& semimetals & semiconductors) topological insulators...
Electrochemistry project, Chemistry Department, November Ab-initio Molecular Dynamics Simulation
Electrochemistry project, Chemistry Department, November 2006 Ab-initio Molecular Dynamics Simulation Outline Introduction Ab-initio concepts Total energy concepts Adsorption energy calculation Project
Multi-Scale Modeling from First Principles
m mm Multi-Scale Modeling from First Principles μm nm m mm μm nm space space Predictive modeling and simulations must address all time and Continuum Equations, densityfunctional space scales Rate Equations
First-principles Molecular Dynamics Simulations
First-principles Molecular Dynamics Simulations François Gygi University of California, Davis fgygi@ucdavis.edu http://eslab.ucdavis.edu http://www.quantum-simulation.org MICCoM Computational School, Jul
Magnetism. Eric Bousquet. University of Liège. Abinit School, Lyon, France 16/05/2014
Magnetism University of Liège eric.bousquet@ulg.ac.be Abinit School, Lyon, France 16/05/2014 Outline Origin of magnetism: Inside an atom Between 2 atoms Interaction with ligands Interaction through ligands
Time reversible Born Oppenheimer molecular dynamics
Time reversible Born Oppenheimer molecular dynamics Jianfeng Lu Mathematics Department Department of Physics Duke University jianfeng@math.duke.edu KI-Net Conference, CSCAMM, University of Maryland, May
Localized Optimization: Exploiting non-orthogonality to efficiently minimize the Kohn-Sham Energy
Localized Optimization: Exploiting non-orthogonality to efficiently minimize the Kohn-Sham Energy Courant Institute of Mathematical Sciences, NYU Lawrence Berkeley National Lab 16 December 2009 Joint work
Neutral Electronic Excitations:
Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra Claudio Attaccalite http://abineel.grenoble.cnrs.fr/ Second Les Houches school in computational physics: ab-initio
Solid State Theory: Band Structure Methods
Solid State Theory: Band Structure Methods Lilia Boeri Wed., 11:00-12:30 HS P3 (PH02112) http://itp.tugraz.at/lv/boeri/ele/ Who am I? Assistant Professor, Institute for Theoretical and Computational Physics,
Semiclassical Electron Transport
Semiclassical Electron Transport Branislav K. Niolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 64: Introduction to Solid State Physics http://www.physics.udel.edu/~bniolic/teaching/phys64/phys64.html
Lecture 2: Background dconcepts
Optoelectronics I Lecture : Background dconcepts M. Soroosh Assistant Professor of Electronics Shahid Chamran University 1 Face Centered Crystal (FCC) Body Centered Crystal (BCC) Bragg s Law William Lawrence
SUPPLEMENTARY INFORMATION
SUPPLEMENTARY INFORMATION Half-Heusler ternary compounds as new multifunctional platforms for topological quantum phenomena H. Lin, L.A. Wray, Y. Xia, S.-Y. Xu, S. Jia, R. J. Cava, A. Bansil, and M. Z.
Algorithms and Computational Aspects of DFT Calculations
Algorithms and Computational Aspects of DFT Calculations Part I Juan Meza and Chao Yang High Performance Computing Research Lawrence Berkeley National Laboratory IMA Tutorial Mathematical and Computational
Key concepts in Density Functional Theory (II)
Kohn-Sham scheme and band structures European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Present Address: LPMCN Université
Etudes ab initio des effets de la température sur le spectre optique des semi-conducteurs
Etudes ab initio des effets de la température sur le spectre optique des semi-conducteurs Woyten Tielesch Laboratoire de Physique de la Matière Condensée et Nanostructures January 27, 2011 Outline Outline
Orbital dependent correlation potentials in ab initio density functional theory
Orbital dependent correlation potentials in ab initio density functional theory noniterative - one step - calculations Ireneusz Grabowski Institute of Physics Nicolaus Copernicus University Toruń, Poland
Atoms? All matters on earth made of atoms (made up of elements or combination of elements).
Chapter 1 Atoms? All matters on earth made of atoms (made up of elements or combination of elements). Atomic Structure Atom is the smallest particle of an element that can exist in a stable or independent
Tutorial on DFPT and TD-DFPT: calculations of phonons and absorption spectra
Tutorial on DFPT and TD-DFPT: calculations of phonons and absorption spectra Iurii Timrov SISSA Scuola Internazionale Superiore di Studi Avanzati, Trieste Italy itimrov@sissa.it Computer modelling of materials
Chapter 4: Bonding in Solids and Electronic Properties. Free electron theory
Chapter 4: Bonding in Solids and Electronic Properties Free electron theory Consider free electrons in a metal an electron gas. regards a metal as a box in which electrons are free to move. assumes nuclei
Introduction to DFTB. Marcus Elstner. July 28, 2006
Introduction to DFTB Marcus Elstner July 28, 2006 I. Non-selfconsistent solution of the KS equations DFT can treat up to 100 atoms in routine applications, sometimes even more and about several ps in MD
Adaptive Finite Element Approximations for Density Functional Models
HPPC Methods and Algorithms for Materials Defects, IMS Singapore, February 12, 2015 Adaptive Finite Element Approximations for Density Functional Models Aihui Zhou LSEC, Institute of Computational Mathematics
The frequency-dependent Sternheimer equation in TDDFT
The frequency-dependent Sternheimer equation in TDDFT A new look into an old equation Miguel A. L. Marques 1 Centre for Computational Physics, University of Coimbra, Portugal 2 LPMCN, Université Claude
c 2017 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 39, No. 6, pp. B1178 B1198 c 17 Society for Industrial and Applied Mathematics COMPUTING LOCALIZED REPRESENTATIONS OF THE KOHN SHAM SUBSPACE VIA RANDOMIZATION AND REFINEMENT ANIL
Fast Algorithms for the Computation of Oscillatory Integrals
Fast Algorithms for the Computation of Oscillatory Integrals Emmanuel Candès California Institute of Technology EPSRC Symposium Capstone Conference Warwick Mathematics Institute, July 2009 Collaborators
Module 6 1. Density functional theory
Module 6 1. Density functional theory Updated May 12, 2016 B A DDFT C K A bird s-eye view of density-functional theory Authors: Klaus Capelle G http://arxiv.org/abs/cond-mat/0211443 R https://trac.cc.jyu.fi/projects/toolbox/wiki/dft
Semiconductor Physics and Devices Chapter 3.
Introduction to the Quantum Theory of Solids We applied quantum mechanics and Schrödinger s equation to determine the behavior of electrons in a potential. Important findings Semiconductor Physics and
The electronic structure of solids. Charge transport in solids
The electronic structure of solids We need a picture of the electronic structure of solid that we can use to explain experimental observations and make predictions Why is diamond an insulator? Why is sodium
Electronic Structure Methodology 1
Electronic Structure Methodology 1 Chris J. Pickard Lecture Two Working with Density Functional Theory In the last lecture we learnt how to write the total energy as a functional of the density n(r): E
TDDFT in Chemistry and Biochemistry III
TDDFT in Chemistry and Biochemistry III Dmitrij Rappoport Department of Chemistry and Chemical Biology Harvard University TDDFT Winter School Benasque, January 2010 Dmitrij Rappoport (Harvard U.) TDDFT
Key concepts in Density Functional Theory
From the many body problem to the Kohn-Sham scheme ILM (LPMCN) CNRS, Université Lyon 1 - France European Theoretical Spectroscopy Facility (ETSF) December 12, 2012 Lyon Outline 1 The many-body problem
Unit III Free Electron Theory Engineering Physics
. Introduction The electron theory of metals aims to explain the structure and properties of solids through their electronic structure. The electron theory is applicable to all solids i.e., both metals
GW Many-Body Theory for Electronic Structure. Rex Godby
GW Many-Body Theory for Electronic Structure Rex Godby Outline Lecture 1 (Monday) Introduction to MBPT The GW approximation (non-sc and SC) Implementation of GW Spectral properties Lecture 2 (Tuesday)
DENSITY FUNCTIONAL THEORY FOR NON-THEORISTS JOHN P. PERDEW DEPARTMENTS OF PHYSICS AND CHEMISTRY TEMPLE UNIVERSITY
DENSITY FUNCTIONAL THEORY FOR NON-THEORISTS JOHN P. PERDEW DEPARTMENTS OF PHYSICS AND CHEMISTRY TEMPLE UNIVERSITY A TUTORIAL FOR PHYSICAL SCIENTISTS WHO MAY OR MAY NOT HATE EQUATIONS AND PROOFS REFERENCES
OVERVIEW OF QUANTUM CHEMISTRY METHODS
OVERVIEW OF QUANTUM CHEMISTRY METHODS Outline I Generalities Correlation, basis sets Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods III Density
Electrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
Performance ofhybrid density functional methods,screened exchange and EXX-OEP methodsin the PAW approach p.1/26
Performance of hybrid density functional methods, screened exchange and EXX-OEP methods in the PAW approach Georg Kresse, J Paier, R Hirschl, M Marsmann Institut für Materialphysik and Centre for Computational
Lecture 4: Band theory
Lecture 4: Band theory Very short introduction to modern computational solid state chemistry Band theory of solids Molecules vs. solids Band structures Analysis of chemical bonding in Reciprocal space
Pseudopotentials for hybrid density functionals and SCAN
Pseudopotentials for hybrid density functionals and SCAN Jing Yang, Liang Z. Tan, Julian Gebhardt, and Andrew M. Rappe Department of Chemistry University of Pennsylvania Why do we need pseudopotentials?
Density Functional Theory
Chemistry 380.37 Fall 2015 Dr. Jean M. Standard October 28, 2015 Density Functional Theory What is a Functional? A functional is a general mathematical quantity that represents a rule to convert a function
ELEMENTARY BAND THEORY
ELEMENTARY BAND THEORY PHYSICIST Solid state band Valence band, VB Conduction band, CB Fermi energy, E F Bloch orbital, delocalized n-doping p-doping Band gap, E g Direct band gap Indirect band gap Phonon
We have arrived to the question: how do molecular bonds determine the band gap? We have discussed that the silicon atom has four outer electrons.
ET3034Tux - 2.2.2 - Band Gap 2 - Electrons in Molecular Bonds We have arrived to the question: how do molecular bonds determine the band gap? We have discussed that the silicon atom has four outer electrons.
Exchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride. Dimer. Philip Straughn
Exchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride Dimer Philip Straughn Abstract Charge transfer between Na and Cl ions is an important problem in physical chemistry. However,
Spin gaps and spin flip energies in density functional theory
Spin gaps and spin flip energies in density functional theory Klaus Capelle Universidade Federal do ABC Carsten Ullrich University of Missouri-Columbia G. Vignale University of Missouri-Columbia Charge
Development of density functional theory for plasmon assisted superconductors. Ryotaro Arita Univ. Tokyo/PRESTO
Development of density functional theory for plasmon assisted superconductors Ryotaro Arita Univ. Tokyo/PRESTO In collaboration with Ryosuke Akashi (Univ. Tokyo) Poster 28 (arxiv:1303.5052, 1305.0390)
Vibrational Spectroscopy Methods
Vibrational Spectroscopy Methods Keith Refson STFC Rutherford Appleton Laboratory September 10, 2012 Phonons and Spectroscopy: Workshop: Frankfurt 2012 1 / 39 Motivations from experimental spectroscopy:
Band calculations: Theory and Applications
Band calculations: Theory and Applications Lecture 2: Different approximations for the exchange-correlation correlation functional in DFT Local density approximation () Generalized gradient approximation
The electronic structure of materials 1
Quantum mechanics 2 - Lecture 9 December 18, 2013 1 An overview 2 Literature Contents 1 An overview 2 Literature Electronic ground state Ground state cohesive energy equilibrium crystal structure phase
Computational Methods. Chem 561
Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational
Microscopic Ohm s Law
Microscopic Ohm s Law Outline Semiconductor Review Electron Scattering and Effective Mass Microscopic Derivation of Ohm s Law 1 TRUE / FALSE 1. Judging from the filled bands, material A is an insulator.
Modified Becke-Johnson (mbj) exchange potential
Modified Becke-Johnson (mbj) exchange potential Hideyuki Jippo Fujitsu Laboratories LTD. 2015.12.21-22 OpenMX developer s meeting @ Kobe Overview: mbj potential The semilocal exchange potential adding
Reactivity and Organocatalysis. (Adalgisa Sinicropi and Massimo Olivucci)
Reactivity and Organocatalysis (Adalgisa Sinicropi and Massimo Olivucci) The Aldol Reaction - O R 1 O R 1 + - O O OH * * H R 2 R 1 R 2 The (1957) Zimmerman-Traxler Transition State Counterion (e.g. Li
Time-Dependent Density-Functional Theory
Summer School on First Principles Calculations for Condensed Matter and Nanoscience August 21 September 3, 2005 Santa Barbara, California Time-Dependent Density-Functional Theory X. Gonze, Université Catholique
LEAST-SQUARES FINITE ELEMENT MODELS
LEAST-SQUARES FINITE ELEMENT MODELS General idea of the least-squares formulation applied to an abstract boundary-value problem Works of our group Application to Poisson s equation Application to flows
Optimized Effective Potential method for non-collinear Spin-DFT: view to spin-dynamics
Optimized Effective Potential method for non-collinear Spin-DFT: view to spin-dynamics Sangeeta Sharma 1,2, J. K. Dewhurst 3, C. Ambrosch-Draxl 4, S. Pittalis 2, S. Kurth 2, N. Helbig 2, S. Shallcross
From Last Time Important new Quantum Mechanical Concepts. Atoms and Molecules. Today. Symmetry. Simple molecules.
Today From Last Time Important new Quantum Mechanical Concepts Indistinguishability: Symmetries of the wavefunction: Symmetric and Antisymmetric Pauli exclusion principle: only one fermion per state Spin
Pseudopotentials: design, testing, typical errors
Pseudopotentials: design, testing, typical errors Kevin F. Garrity Part 1 National Institute of Standards and Technology (NIST) Uncertainty Quantification in Materials Modeling 2015 Parameter free calculations.
Recent developments of Discontinuous Galerkin Density functional theory
Lin Lin Discontinuous Galerkin DFT 1 Recent developments of Discontinuous Galerkin Density functional theory Lin Lin Department of Mathematics, UC Berkeley; Computational Research Division, LBNL Numerical
Introduction to Density Functional Theory
Introduction to Density Functional Theory S. Sharma Institut für Physik Karl-Franzens-Universität Graz, Austria 19th October 2005 Synopsis Motivation 1 Motivation : where can one use DFT 2 : 1 Elementary
Review of Optical Properties of Materials
Review of Optical Properties of Materials Review of optics Absorption in semiconductors: qualitative discussion Derivation of Optical Absorption Coefficient in Direct Semiconductors Photons When dealing
Optical Properties of Semiconductors. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India
Optical Properties of Semiconductors 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/semi2013 Light Matter Interaction Response to external electric
Electronic Structure of Clusters and. Nanocrystals
Electronic Structure of Clusters and Nanocrystals James R. Chelikowsky and Yousef Saad Department of Chemical Engineering and Materials Science Department of Computer Science Institute for the Theory of
THE UNIVERSITY OF CHICAGO FIRST PRINCIPLES SIMULATIONS OF VIBRATIONAL SPECTRA OF AQUEOUS SYSTEMS A DISSERTATION SUBMITTED TO
THE UNIVERSITY OF CHICAGO FIRST PRINCIPLES SIMULATIONS OF VIBRATIONAL SPECTRA OF AQUEOUS SYSTEMS A DISSERTATION SUBMITTED TO THE FACULTY OF THE INSTITUTE FOR MOLECULAR ENGINEERING IN CANDIDACY FOR THE
PHY492: Nuclear & Particle Physics. Lecture 4 Nature of the nuclear force. Reminder: Investigate
PHY49: Nuclear & Particle Physics Lecture 4 Nature of the nuclear force Reminder: Investigate www.nndc.bnl.gov Topics to be covered size and shape mass and binding energy charge distribution angular momentum
The Oxford Solid State Basics
The Oxford Solid State Basics Steven H. Simon University of Oxford OXFORD UNIVERSITY PRESS Contents 1 About Condensed Matter Physics 1 1.1 What Is Condensed Matter Physics 1 1.2 Why Do We Study Condensed
Introduction to Density Functional Theory
1 Introduction to Density Functional Theory 21 February 2011; V172 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT 2 3 4 Ab initio Computational
Time Dependent Density Functional Theory. Francesco Sottile
An Introduction Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Belfast, 29 Jun 2007 Outline 1 Introduction: why TD-DFT? 2 (Just)
Physics 541: Condensed Matter Physics
Physics 541: Condensed Matter Physics In-class Midterm Exam Wednesday, October 26, 2011 / 14:00 15:20 / CCIS 4-285 Student s Name: Instructions There are 23 questions. You should attempt all of them. Mark
Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals. 5 nm
Metals Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals 5 nm Course Info Next Week (Sept. 5 and 7) no classes First H/W is due Sept. 1 The Previous Lecture Origin frequency dependence
Time Dependent Density Functional Theory. Francesco Sottile
Applications, limitations and... new frontiers Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Vienna, 19 January 2007 /55 Outline
Nonadiabatic Molecular Dynamics with Kohn-Sham DFT: Modeling Nanoscale Materials. Oleg Prezhdo
Nonadiabatic Molecular Dynamics with Kohn-Sham DFT: Modeling Nanoscale Materials Oleg Prezhdo CSCAMM Mar 11, 2010 Outline Nonadiabatic MD with Kohn-Sham DFT Advantages & Validity Quantum Backreaction &
PH575 Spring Lecture #28 Nanoscience: the case study of graphene and carbon nanotubes.
PH575 Spring 2014 Lecture #28 Nanoscience: the case study of graphene and carbon nanotubes. Nanoscience scale 1-100 nm "Artificial atoms" Small size => discrete states Large surface to volume ratio Bottom-up
Density Functional Theory
Density Functional Theory Iain Bethune EPCC ibethune@epcc.ed.ac.uk Overview Background Classical Atomistic Simulation Essential Quantum Mechanics DFT: Approximations and Theory DFT: Implementation using
Solid State Theory: Band Structure Methods
Solid State Theory: Band Structure Methods Lilia Boeri Wed., 11:15-12:45 HS P3 (PH02112) http://itp.tugraz.at/lv/boeri/ele/ Plan of the Lecture: DFT1+2: Hohenberg-Kohn Theorem and Kohn and Sham equations.