Report

Quantum Mechanics: Density Functional Theory and Practical Application to Alloys Stewart Clark Condensed Matter Section Department of Physics University of Durham Outline • Aim: To simulate real materials and experimental measurements • Method: Density functional theory and high performance computing • Results: Brief summary of capabilities and performing calculations Introduction to Computer Simulation: Edinburgh, May 2010 What would we like to achieve? • Computers get cheaper and more powerful every year. • Experiments tend to get more expensive each year. • IF computer simulation offers acceptable accuracy then at some point it should become cheaper than experiment. • This has already occurred in many branches of science and engineering. • Possible to achieve this for properties of alloys? Introduction to Computer Simulation: Edinburgh, May 2010 Property Prediction •Property calculation of alloys provided link with experimental measurements: - For analysis For scientific/technological interest •To enable interpretation of experimental results •To predict properties over and above that of experimental measurements Introduction to Computer Simulation: Edinburgh, May 2010 Aim of ab initio calculations Atomic Numbers Solve quantum mechanics for the material Predict physical and chemical properties of systems Introduction to Computer Simulation: Edinburgh, May 2010 From first principles The equipment Application Scientific problemsolving “Base Theory” (DFT) Implementation (the algorithms and program) Setup model, run the code “Analysis Theory” Introduction to Computer Simulation: Edinburgh, May 2010 Research output Properties of materials • Whole periodic table. • Periodic units containing thousands of atoms (on large enough computers). • Structural optimisation (where are the atoms?). • Finite temperature (atomic motion). • Lots of others…if experiments can measure it, we try to calculate it – and then go further… • Toolbox for material properties Introduction to Computer Simulation: Edinburgh, May 2010 The starting point ˆ H E As you can see, quantum mechanics is “simply” an eigenvalue problem Introduction to Computer Simulation: Edinburgh, May 2010 Set up the problem Let’s start defining various quantities Assume that the nuclei (Mass Mi) are at: R1, R2, …, RN Assume that the electrons (mass me) are at: r1, r2, …, rm Now let’s put some details in the SE Hˆ R1 , R 2 ,..., R N , r1 , r2 ,..., rm E R1 , R 2 ,..., R N , r1 , r2 ,..., rm Introduction to Computer Simulation: Edinburgh, May 2010 Summary of problem to solve Hˆ N ,e N ,e R ,r E I i N ,e N ,e R ,r I i Where Hˆ N ,e TˆN Tˆe VˆN N VˆN e Vˆe e Introduction to Computer Simulation: Edinburgh, May 2010 The full problem 2 h 2 2 i I 2 M m I I e i 2 2 e ZIe 1 1 1 4 2 ri R I 2 0 i j ri r j i,I I J R1 , R 2 , ..., R N , r1 , r2 ,..., rm E R1 , R 2 , ..., R N , r1 , r2 , ..., rm 2 Z I Z J e R I R J •Why is this a hard problem? •Equation is not separable: genuine many-body problem •Interactions are all strong – perturbation won’t work •Must be Accurate --- Computation Introduction to Computer Simulation: Edinburgh, May 2010 Model Systems In this kind of first-principles calculation – Are 3D-periodic – Are small: from one atom to a few thousand atoms Bulk alloy Supercells Periodic boundaries Bloch functions Introduction to Computer Simulation: Edinburgh, May 2010 Slab for surfaces First simplification • The electron mass is much smaller than the nuclear mass • Electrons remain in a stationary state of the Hamiltonian wrt nuclear motion Hˆ N ,e N ,e R ,r r R I i N ,e i N ,e I • Nuclear problem is separable (and, as we know, the nucleus is merely a point charge!) Introduction to Computer Simulation: Edinburgh, May 2010 Electrons are difficult! • The mathematical difficulty of solving the Schrodinger equation increases rapidly with N • It is an exponentially difficult problem • The number of computations scales as eN • With modern supercomputers we can solve this directly for a very small number of electrons (maybe 4 or 5 electrons) • Materials contain of the order of 1026 electrons Introduction to Computer Simulation: Edinburgh, May 2010 Density functional theory • Let’s write the Hamiltonian operator in the following way: Hˆ Tˆ Vˆ Uˆ – T is the kinetic energy terms – V is the potential terms external to the electrons – U is the electron-electron term we’ve just classified it into different • so ‘physical’ terms Introduction to Computer Simulation: Edinburgh, May 2010 The electron density • The electronic charge density is given by n ( r) ... r1 , r2 ,..., rn r1 , r2 ,..., rn dr 2 dr 3 ...dr n * • so integrate over n-1 of the dimensions gives the probability, n(r), of finding an electron at r • This is (clearly!) a unique functional of the external potential, V • That is, fix V, solve SE (somehow) for and then get n(r). Introduction to Computer Simulation: Edinburgh, May 2010 DFT • Let’s consider the reverse question: for a given n(r), does this come from a unique V? • Can two different external potentials, V and V’, give rise to the same electronic density? Introduction to Computer Simulation: Edinburgh, May 2010 Method behind DFT • Assume two potentials V and V’ lead to the same ground state density: E |H | ' | H | ' ' | H ' V V ' | ' ' | H ' | ' ' | V V ' | ' E ' ' | V V ' | ' • We can do the same again interchanging the dashed and undashed quantities thus: E ' E | V ' V | Introduction to Computer Simulation: Edinburgh, May 2010 Unique potential • If we add these two final equations we are left with the contradiction E ' E E E ' • so our initial assumption must be incorrect • That is, there cannot be two different external potentials that lead to the same density • We have a one-to-one correspondence between density, n(r), and external potential, V(r). Introduction to Computer Simulation: Edinburgh, May 2010 Change of emphasis in QM • But by the definition of the lowest energy state we must have ' | H | ' | H | ' | F | ' ' | V | ' | F | | V | n ' ( r)V ( r) dr F n ' ( r) n ( r)V ( r) dr F n ( r) E n ' ( r) E n ( r) • And so the ‘variational principle’ tells us how to solve the problem Introduction to Computer Simulation: Edinburgh, May 2010 Don’t bother with the wavefunction! • Express the problem as an energy E n Ts n E H n V n E xc n • And solve variationally with respect to the density Degrees of freedom in the density, n, versus energy E[n] Introduction to Computer Simulation: Edinburgh, May 2010 QM using DFT N-body Schrödinger Equation 2 2 h 1 e 2 2 m 2 4 0 e i j n 1 ri r j exact r e 2 4 0 ZI ri R I i,I 1 e 2 2 4 0 I J r , r ,... E r , r ,... 1 2 1 2 R I R J ZIZJ ... r , r , r ,... r , r , r ,...dr * 1 2 3 1 2 3 2 dr 3 ... Density functional theory (Kohn-Sham equations) 2 h 1 e 2 i 2 4 0 2 m e n ( r' ) r r' dr ' e 2 4 0 n DFT I Z I n (r ' ) r' R I dr ' 1 e 2 2 4 0 I J r i* r i r i Both equations n exact r n DFT r Introduction to Computer Simulation: Edinburgh, May 2010 ZIZJ RI RJ xc n r i r i i r Kohn-Sham Equations • Let’s collect all the terms into one to simplify 1 2 KS i V i n r i r i i r 2 • Where – i labels each particle in the system KS is the potential felt by particle i due to n(r) – V i – n(r) is the charge density Introduction to Computer Simulation: Edinburgh, May 2010 Kohn-Sham Equations • The Kohn-Sham (KS) equations are formally exact • The KS particle density is equal to the exact particle density • We have reduced the 1 N-particle problem to N (coupled) 1-particle problems • We can solve 1-particle problems! Introduction to Computer Simulation: Edinburgh, May 2010 Variational Method Schrödinger’s Equation And the of use the Variational Principle Solve this Hˆ E by Minimising this within DFT E Hˆ Introduction to Computer Simulation: Edinburgh, May 2010 DFT: The XC approximation • Basically comes from our attempt to map 1 N-body QM problem onto N 1-body QM problems • Attempt to extract single-electron properties from interacting N-electron system • These are quasi-particles “DFT cannot do…” : This statement is dangerous and usually ends incorrectly (in many publications!) Should read: “DFT using the ??? XC-functional can be used to calculate ???, but that particular functional introduces and error of ??? because of ??? Introduction to Computer Simulation: Edinburgh, May 2010 Definition of XC Exact XC interaction is unknown Within DFT we can write the exact XC interaction as E xc [ n ] 1 n(r ) 2 n xc ( r , r ' ) | r r '| drdr ' This would be excellent if only we knew what nxc was! This relation defines the XC energy. It is simply the Coulomb interaction between an electron an r and the value of its XC hole nxc(r,r’) at r’. Introduction to Computer Simulation: Edinburgh, May 2010 Exchange-Correlation Approximations A simple, but effective approximation to the exchange-correlation interaction is E LDA xc [ n ( r )] n ( r ) hom xc [ n ( r )] dr The generalised gradient approximation contains the next term in a derivative expansion of the charge density: GGA E xc [ n ( r )] n ( r ) xc GGA [ n ( r ), n ( r )] dr Introduction to Computer Simulation: Edinburgh, May 2010 Hierarchy of XC appoximations • LDA depends only on one variable (the density). • GGA’s require knowledge of 2 variables (the density and its gradient). • In principle one can continue with this expansion. • If quickly convergent, it would characterise a class of many-body systems with increasing accuracy by functions of 1,2,6,…variables. • How fruitful is this? As yet, unknown, but it will always be semi-local. Introduction to Computer Simulation: Edinburgh, May 2010 Zoo of XC approximations LDA B3LYP RPBE WC SDA Meta-GGA PW91 Semi-Empirical sX EXX PBE0 PBE WDA HF OEP MP2 Introduction to Computer Simulation: Edinburgh, May 2010 CI MP4 CC Structure Determination – – – – Minimum energy corresponds to zero force Much more efficient than just using energy alone Equilibrium bond lengths, angles, etc. Minimum enthalpy corresponds to zero force and stress – Can therefore minimise enthalpy w.r.t. supercell shape due to internal stress and external pressure – Pressure-driven phase transitions Introduction to Computer Simulation: Edinburgh, May 2010 Nuclear Positions? • Up until now we assume we know nuclear positions, {Ri} • What if we don’t? • Guess them or take hints from experiment • Get zero of force wrt {Ri}: F R E (R) E H Introduction to Computer Simulation: Edinburgh, May 2010 Forces • If we take the derivative then: E R H R R H H R H R H E | | R R R E R H R | H R H R E Product rule Introduction to Computer Simulation: Edinburgh, May 2010 Product rule Const 0 Forces II • In DFT we have Hˆ ( r, R ) h 2 2m r V e e n r V n e n r, R V n n n R V xc n r • But only Vn-e and Vn-n depends on R and derivative are taken analytically • We get forces for free! • Optimise under F to obtain {Ri} • Add in nuclear KE to obtain finite temperature Introduction to Computer Simulation: Edinburgh, May 2010 Example: Lattice Parameters Energy •KS equations can be solved to give energy, E •What does that tell us? Common tangent gives transition pressure: Phase II P=-dE/dV Phase I VII VI Introduction to Computer Simulation: Edinburgh, May 2010 Volume Structures without experiment? U(x) start stop x Relative energies of structures: examine phase stability Introduction to Computer Simulation: Edinburgh, May 2010 Summary so far • Can get electronic density and energy • Can use forces (and stresses) to optimise structure from an “intelligent” initial guess • Minima of energy gives structural phase information Introduction to Computer Simulation: Edinburgh, May 2010 Alloys • Alloys are complicated! Phase separated Ordered Introduction to Computer Simulation: Edinburgh, May 2010 Random Ordered • Ordered alloys are “easy” (usually!) • The have a repeated unit cell Can perform calculation on this unit cell: •Electronic structure •Band Structure •Density of States •Etc… Introduction to Computer Simulation: Edinburgh, May 2010 Disordered • There are two main approaches: • The Supercell approach – Make a large unit cell with species randomly distributed as required – Characterises microscopic quantities • The Virtual Crystal Approximation (VCA) – Make each atom behave as if it were an average of various species AxB1-x – Encapsulates only average quantities Introduction to Computer Simulation: Edinburgh, May 2010 Supercell Approach • Need large unit cell • Computationally expensive • A lot of atoms • Require check on statistics (how many possible random configurations?) Introduction to Computer Simulation: Edinburgh, May 2010 Virtual Crystal Approximation • What is an “average” atom? • Put x of one atom and 1-x of the other atom at every site Introduction to Computer Simulation: Edinburgh, May 2010 VCA Example • NbC1-xNx • C and N are disordered • How does electronic structure vary with x? Introduction to Computer Simulation: Edinburgh, May 2010 NbC1-xNx Electronic DoS Introduction to Computer Simulation: Edinburgh, May 2010 Electron by electron Introduction to Computer Simulation: Edinburgh, May 2010 Relation to experiment • We solve the problem and get energy, E and density n(r) – experiments don’t measure these! • An experiment: “Different” Radiation or Particle (k-q) Radiation or Particle (k) Material (q) Introduction to Computer Simulation: Edinburgh, May 2010 Perturbation Theory Based on compute how the total energy responds to a perturbation, usually of the DFT external potential v Expand quantities (E, n, , v) E E (0) E (1 ) E 2 (2) ... • Properties given by the derivatives 1 E n E (n) n! n 1 E 2 and in particular E (2) Introduction to Computer Simulation: Edinburgh, May 2010 2 2 The Perturbations Perturb the external potential (from the nuclei and any external field): • Nuclear positions • Cell vectors • Electric fields • Magnetic fields phonons elastic constants dielectric response NMR But not only the potential, any perturbation to the Hamiltonian: • d/dk • d/d(PSP) atomic charges alchemical perturbation Introduction to Computer Simulation: Edinburgh, May 2010 Example: Phonons • Perturb with respect to nuclear coordinates: E E0 E lk u lk l ,k , 1 2! E l ,k , lk l ' k ' u lk u l ' k ' K l ',k ', • This is equivalent to f ( x ) f x 0 2 df dx x x0 1 d f 2! dx x K 2 2 Introduction to Computer Simulation: Edinburgh, May 2010 x0 Atomic Motion Eigenvectors of 2nd order energy give nuclear motion under phonon excitations Introduction to Computer Simulation: Edinburgh, May 2010 Summary • First principles electronic structure calculations: Extremely powerful technique in condensed matter • Applicable to many different sciences Condensed Matter Physics Chemistry Biology Material Science Surfaces Geology • Simulate experimental measurements • Computationally possible, but still requires good computer resources Introduction to Computer Simulation: Edinburgh, May 2010