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Introduction to DFT and the Plane-Wave Pseudopotential Method

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Introduction to DFT and the Plane-Wave Pseudopotential Method Introduction to DFT and the plane-wave pseudopotential method Keith Refson STFC Rutherford Appleton Laboratory Chilton, Didcot, OXON OX11 0QX 23 Apr 2014 Parallel Materials Modelling Packages @ EPCC 1 / 55 Introduction Synopsis Motivation Some ab initio codes Quantum-mechanical approaches Density Functional Theory Electronic Structure of Condensed Phases Total-energy calculations Introduction Basis sets Plane-waves and Pseudopotentials How to solve the equations Parallel Materials Modelling Packages @ EPCC 2 / 55 Synopsis Introduction A guided tour inside the “black box” of ab-initio simulation. Synopsis • Motivation • The rise of quantum-mechanical simulations. Some ab initio codes Wavefunction-based theory • Density-functional theory (DFT) Quantum-mechanical • approaches Quantum theory in periodic boundaries • Plane-wave and other basis sets Density Functional • Theory SCF solvers • Molecular Dynamics Electronic Structure of Condensed Phases Recommended Reading and Further Study Total-energy calculations • Basis sets Jorge Kohanoff Electronic Structure Calculations for Solids and Molecules, Plane-waves and Theory and Computational Methods, Cambridge, ISBN-13: 9780521815918 Pseudopotentials • Dominik Marx, J¨urg Hutter Ab Initio Molecular Dynamics: Basic Theory and How to solve the Advanced Methods Cambridge University Press, ISBN: 0521898633 equations • Richard M. Martin Electronic Structure: Basic Theory and Practical Methods: Basic Theory and Practical Density Functional Approaches Vol 1 Cambridge University Press, ISBN: 0521782856 • C. Pisani (ed) Quantum Mechanical Ab-Initio Calculation of the properties of Crystalline Materials, Springer, Lecture Notes in Chemistry vol.67 ISSN 0342-4901. Parallel Materials Modelling Packages @ EPCC 3 / 55 Motivation Introduction Synopsis The underlying physical laws necessary for the mathematical theory Motivation of a large part of physics and the whole of chemistry are thus Some ab initio codes completely known, and the difficulty is only that the application of Quantum-mechanical approaches these laws leads to equations much too complicated to be soluble. Density Functional Theory P.A.M. Dirac, Proceedings of the Royal Society A123, 714 (1929) Electronic Structure of Condensed Phases Total-energy calculations Basis sets Plane-waves and Nobody understands quantum mechanics. Pseudopotentials How to solve the R. P. Feynman equations Parallel Materials Modelling Packages @ EPCC 4 / 55 Some ab initio codes Introduction Synopsis Numerical Motivation Planewave Gaussian Some ab initio codes CRYSTAL FHI-Aims Quantum-mechanical CASTEP PEtot approaches CP2K SIESTA Density Functional VASP PARATEC Dmol Theory AIMPRO PWscf Da Capo Electronic Structure of ADF-band Condensed Phases Abinit CPMD OpenMX Total-energy calculations Qbox fhi98md Basis sets DFT GPAW Plane-waves and PWPAW SFHIngX PARSEC Pseudopotentials (r), n(r) How to solve the DOD-pw NWchem equations Octopus JDFTx LAPW LMTO O(N) WIEN2k LMTART ONETEP Fleur LMTO Conquest exciting FPLO BigDFT Elk http://www.psi-k.org/codes.shtml Parallel Materials Modelling Packages @ EPCC 5 / 55 Introduction Quantum-mechanical approaches Quantum-mechanics of electrons and nuclei The Schr¨odinger equation Approximations 1. The Hartree approximation The Hartree-Fock approximation Practical Aspects Density Functional Quantum-mechanical approaches Theory Electronic Structure of Condensed Phases Total-energy calculations Basis sets Plane-waves and Pseudopotentials How to solve the equations Parallel Materials Modelling Packages @ EPCC 6 / 55 Quantum-mechanics of electrons and nuclei Introduction • Quantum mechanics proper requires full wavefunction of both electronic and Quantum-mechanical nuclear co-ordinates. approaches Quantum-mechanics of • First approximation is the Born-Oppenheimer approximation. Assume that electrons and nuclei The Schr¨odinger electronic relaxation is much faster than ionic motion (me << mnuc). Then equation wavefunction is separable Approximations 1. The Hartree approximation R R R r r r The Hartree-Fock Ψ=Θ( 1, 2, ..., N )Φ( 1, 2, ..., n approximation { } { } Practical Aspects Ri are nuclear co-ordinates and ri are electron co-ordinates. Density Functional • Therefore can treat electronic system as solution of Schr¨odinger equation in Theory fixed external potential of the nuclei, Vext Ri . Electronic Structure of { } Condensed Phases • Ground-state energy of electronic system acts as potential function for nuclei. • Can then apply our tool-box of simulation methods to nuclear system. Total-energy calculations • Basis sets B-O is usually a very good approximation, only fails for coupled Plane-waves and electron/nuclear behaviour for example superconductivity, quantum crystals Pseudopotentials such as He and cases of strong quantum motion such as H in KDP. How to solve the equations Parallel Materials Modelling Packages @ EPCC 7 / 55 The Schr¨odinger equation Introduction • Ignoring electron spin for the moment and using atomic units (¯h = me = e =1) Quantum-mechanical approaches 1 Quantum-mechanics of 2 ˆ R r ˆ r r r electrons and nuclei + Vext( I , i )+ Ve-e( i ) Ψ( i )= EΨ( i ) The Schr¨odinger − 2 ∇ { } { } { } { } { } equation Approximations 1. The where 1 2is the kinetic-energy operator, Hartree approximation − 2 ∇ The Hartree-Fock Zi approximation Vˆext= is the Coulomb potential of the nuclei, − R r Practical Aspects Xi XI | I − i| Density Functional ˆ 1 1 Theory Ve-e= is the electron-electron Coulomb interaction 2 rj ri Electronic Structure of Xi Xj6=i | − | Condensed Phases and Ψ( r )=Ψ(r1,... rn) is a 3N-dimensional wavefunction. { i} Total-energy calculations • This is a 3N-dimensional eigenvalue problem. Basis sets • E-e term renders even numerical solutions impossible for more than a handful of Plane-waves and electrons. Pseudopotentials • Pauli Exclusion principle Ψ( ri ) is antisymmetric under interchange of any 2 How to solve the r r { } r r equations electrons. Ψ(... i, j ,...)= Ψ(... j , i,...) − 2 • Total electron density is n(r)= ... dr2 ...drn Ψ( r ) | { i} | R R Parallel Materials Modelling Packages @ EPCC 8 / 55 Approximations 1. The Hartree approximation Introduction • Substituting Ψ(r1,... rn)= φ(r1) ...φ(rn) into the Schr¨odinger equation Quantum-mechanical approaches yields Quantum-mechanics of electrons and nuclei 1 2 The Schr¨odinger + Vˆext( RI , r)+ VˆH(r) φn(r)= Enφn(r) equation − 2 ∇ { } Approximations 1. The Hartree approximation r′ ′ n( ) The Hartree-Fock where the Hartree potential: VˆH(r)= dr is Coulomb interaction approximation Z r′ r Practical Aspects r| − | r 2 of an electron with average electron density n( )= i φi( ) . Sum is over Density Functional all occupied states. | | Theory P • φ(rn) is called an orbital. Electronic Structure of Condensed Phases • Now a 3-dimensional wave equation (or eigenvalue problem) for φ(r ). • n Total-energy calculations This is an effective 1-particle wave equation with an additional term, the Basis sets Hartree potential • r r Plane-waves and But solution φi( ) depends on electron-density n( ) which in turn depends on Pseudopotentials φi(r). Requires self-consistent solution. How to solve the • This is a very poor approximation because Ψ( ri ) does not have necessary equations antisymmetry and violates the Pauli principle.{ } Parallel Materials Modelling Packages @ EPCC 9 / 55 The Hartree-Fock approximation Introduction • Approximate wavefunction by a slater determinant which guarantees Quantum-mechanical antisymmetry under electron exchange approaches Quantum-mechanics of electrons and nuclei φ1(r1, σ1) φ1(r2, σ2) ... φ1(rn, σn) The Schr¨odinger r r r equation φ2( 1, σ1) φ2( 2, σ2) ... φ2( n, σn) r r 1 Approximations 1. The Ψ( 1,... n)= . Hartree approximation . √n! . The Hartree-Fock approximation r r r φn( 1, σ1) φn( 2, σ1) ... φn( n, σn) Practical Aspects Density Functional • Substitution into the Schr¨odinger equation yields Theory Electronic Structure of Condensed Phases 1 2 + Vˆext( RI , r)+ VˆH(r) φn(r) (1) Total-energy calculations − 2 ∇ { } ∗ r′ r′ Basis sets ′ φm( )φn( ) dr φm(r)= Enφn(r) (2) Plane-waves and − Z r′ r Pseudopotentials Xm | − | How to solve the equations • Also an effective 1-particle wave equation. The extra term is called the exchange potential and creates repulsion between electrons of like spin. • Involves orbitals with co-ordinates at 2 different positions. Therefore expensive to solve. Parallel Materials Modelling Packages @ EPCC 10 / 55 Practical Aspects Introduction • Practical solution of Hartree-Fock developed by John Pople, C. Roothan and Quantum-mechanical others. (Nobel Prize 1998). approaches Quantum-mechanics of • Key is to solve 1-particle effective Hamiltonian in a self-consistent loop. electrons and nuclei SCF The Schr¨odinger Sometimes known as methods (Self Consistent Field). equation • Hartree-Fock yields reasonable values for total energies of atoms, molecules. Approximations 1. The • Hartree approximation Basis of all quantum chemistry until 1990s. The Hartree-Fock • Error in Hartree-Fock energy dubbed correlation energy. approximation • Failures: Excitation energies too large. Practical Aspects • Completely fails to reproduce metallic state. (Predicts logarithmic singularity in Density Functional Theory DOS at ǫF.) Electronic Structure of • Various more, accurate (and expensive) methods such as MP2, MP4, Condensed Phases Coupled-Cluster, full CI are based on HF methods, and give approximations to Total-energy calculations the correlation energy. Basis sets Plane-waves and Pseudopotentials How to solve the equations
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