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 Introduction Total-energy calculations Basis sets Plane-waves and Pseudopotentials How to solve the equations Parallel Materials Modelling Packages @ EPCC 2 / 55
Synopsis A guided tour inside the “black box” of ab-initio simulation. Introduction Synopsis The rise of quantum-mechanical simulations. • Motivation Wavefunction-based theory • Some ab initio codes Density-functional theory (DFT) • Quantum-mechanical Quantum theory in periodic boundaries • approaches Plane-wave and other basis sets • Density Functional SCF solvers • Theory Molecular Dynamics • Electronic Structure of Condensed Phases Recommended Reading and Further Study Total-energy calculations Jorge Kohanoff Electronic Structure Calculations for Solids and Molecules, • Basis sets Theory and Computational Methods , Cambridge, ISBN-13: 9780521815918 Plane-waves and Dominik Marx, J¨ urg Hutter Ab Initio Molecular Dynamics: Basic Theory and • Pseudopotentials Advanced Methods Cambridge University Press, ISBN: 0521898633 How to solve the Richard M. Martin Electronic Structure: Basic Theory and Practical Methods: • equations 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 The underlying physical laws necessary for the mathematical theory Synopsis of a large part of physics and the whole of chemistry are thus Motivation Some ab initio codes completely known, and the difficulty is only that the application of Quantum-mechanical these laws leads to equations much too complicated to be soluble. approaches Density Functional P.A.M. Dirac, Proceedings of the Royal Society A123 , 714 (1929) Theory Electronic Structure of Condensed Phases Total-energy calculations Basis sets Nobody understands quantum mechanics. Plane-waves and Pseudopotentials R. P. Feynman How to solve the equations Parallel Materials Modelling Packages @ EPCC 4 / 55
Some ab initio codes Introduction Numerical Gaussian Planewave Synopsis Motivation FHI-Aims CRYSTAL Some ab initio codes PEtot CASTEP Quantum-mechanical SIESTA CP2K approaches VASP PARATEC Dmol Density Functional AIMPRO Theory Da Capo PWscf ADF-band Electronic Structure of CPMD Abinit Condensed Phases OpenMX Total-energy calculations fhi98md Qbox DFT GPAW Basis sets PWPAW SFHIngX PARSEC (r), n(r) Plane-waves and Pseudopotentials NWchem DOD-pw How to solve the JDFTx Octopus equations LAPW O(N) LMTO 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 Quantum-mechanical approaches Density Functional 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 First approximation is the Born-Oppenheimer approximation . Assume that • Quantum-mechanics of electrons and nuclei electronic relaxation is much faster than ionic motion ( m e << m nuc ). Then The Schr¨ odinger wavefunction is separable equation Approximations 1. The Hartree approximation Ψ = Θ( { R 1 , R 2 , ..., R N } )Φ( { r 1 , r 2 , ..., r n } The Hartree-Fock approximation Practical Aspects R i are nuclear co-ordinates and r i 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, V ext { R i } . Electronic Structure of Ground-state energy of electronic system acts as potential function for nuclei. • Condensed Phases Can then apply our tool-box of simulation methods to nuclear system. • Total-energy calculations B-O is usually a very good approximation, only fails for coupled • Basis sets electron/nuclear behaviour for example superconductivity, quantum crystals Plane-waves and such as He and cases of strong quantum motion such as H in KDP. Pseudopotentials 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 = m e = e = 1 ) • Quantum-mechanical approaches � − 1 � 2 ∇ 2 + ˆ V ext ( { R I } , { r i } ) + ˆ Quantum-mechanics of V e-e ( { r i } ) Ψ( { r i } ) = E Ψ( { r i } ) electrons and nuclei The Schr¨ odinger equation where − 1 2 ∇ 2 is the kinetic-energy operator, Approximations 1. The Hartree approximation Z i The Hartree-Fock ˆ � � V ext = − | R I − r i | is the Coulomb potential of the nuclei, approximation Practical Aspects i I V e-e = 1 1 Density Functional ˆ � � | r j − r i | is the electron-electron Coulomb interaction Theory 2 i j � = i Electronic Structure of and Ψ( { r i } ) = Ψ( r 1 , . . . r n ) is a 3N-dimensional wavefunction. Condensed Phases This is a 3N-dimensional eigenvalue problem. • Total-energy calculations E-e term renders even numerical solutions impossible for more than a handful of • Basis sets electrons. Plane-waves and Pseudopotentials Pauli Exclusion principle Ψ( { r i } ) is antisymmetric under interchange of any 2 • How to solve the electrons. Ψ( . . . r i , r j , . . . ) = − Ψ( . . . r j , r i , . . . ) equations d r 2 . . . d r n | Ψ( { r i } ) | 2 � � Total electron density is n ( r ) = . . . • Parallel Materials Modelling Packages @ EPCC 8 / 55
Approximations 1. The Hartree approximation Introduction Substituting Ψ( r 1 , . . . r n ) = φ ( r 1 ) . . . φ ( r n ) into the Schr¨ odinger equation • Quantum-mechanical yields approaches Quantum-mechanics of � − 1 � electrons and nuclei 2 ∇ 2 + ˆ V ext ( { R I } , r ) + ˆ V H ( r ) φ n ( r ) = E n φ n ( r ) The Schr¨ odinger equation Approximations 1. The n ( r ′ ) � Hartree approximation where the Hartree potential : ˆ d r ′ V H ( r ) = | r ′ − r | is Coulomb interaction The Hartree-Fock approximation i | φ i ( r ) | 2 . Sum is over Practical Aspects of an electron with average electron density n ( r ) = � Density Functional all occupied states. Theory φ ( r n ) is called an orbital . • Electronic Structure of Now a 3-dimensional wave equation (or eigenvalue problem) for φ ( r n ) . • Condensed Phases This is an effective 1-particle wave equation with an additional term, the • Total-energy calculations Hartree potential Basis sets But solution φ i ( r ) depends on electron-density n ( r ) which in turn depends on • Plane-waves and φ i ( r ) . Requires self-consistent solution. Pseudopotentials This is a very poor approximation because Ψ( { r i } ) does not have necessary How to solve the • 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 φ 1 ( r 1 , σ 1 ) φ 1 ( r 2 , σ 2 ) . . . φ 1 ( r n , σ n ) � � electrons and nuclei � � The Schr¨ odinger φ 2 ( r 1 , σ 1 ) φ 2 ( r 2 , σ 2 ) . . . φ 2 ( r n , σ n ) � � 1 equation � � Ψ( r 1 , . . . r n ) = √ . . . Approximations 1. The � � ... . . . � � n ! Hartree approximation . . . � � The Hartree-Fock � � φ n ( r 1 , σ 1 ) φ n ( r 2 , σ 1 ) . . . φ n ( r n , σ n ) approximation � � Practical Aspects Density Functional Substitution into the Schr¨ odinger equation yields • Theory Electronic Structure of � − 1 � 2 ∇ 2 + ˆ V ext ( { R I } , r ) + ˆ Condensed Phases V H ( r ) φ n ( r ) (1) Total-energy calculations φ ∗ m ( r ′ ) φ n ( r ′ ) � Basis sets � d r ′ − φ m ( r ) = E n φ n ( r ) (2) | r ′ − r | Plane-waves and m Pseudopotentials How to solve the Also an effective 1-particle wave equation . The extra term is called the equations • 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
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