Quantum Mechanics: Density Functional Theory and Practical Application to Alloys

1. Quantum Mechanics: Density Functional Theory and Practical Application to Alloys Stewart Clark Condensed Matter Section Department of Physics University of Durham

2. 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

3. 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

4. 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

5. Atomic Numbers Aim of ab initio calculations Solve quantum mechanics for the material Predict physical and chemical properties of systems Introduction to Computer Simulation: Edinburgh, May 2010

6. Scientificproblem-solving “BaseTheory”(DFT) Implementation(the algorithmsand program) Setup model,run the code “Analysis Theory” From first principles The equipment Application Researchoutput Introduction to Computer Simulation: Edinburgh, May 2010

7. 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

8. The starting point As you can see, quantum mechanics is “simply” an eigenvalue problem Introduction to Computer Simulation: Edinburgh, May 2010

9. 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 Introduction to Computer Simulation: Edinburgh, May 2010

10. Summary of problem to solve Where Introduction to Computer Simulation: Edinburgh, May 2010

11. The full problem • 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

12. Model Systems • In this kind of first-principles calculation • Are 3D-periodic • Are small: from one atom to a few thousand atoms • Supercells • Periodic boundaries • Bloch functions Bulk alloy Slab for surfaces Introduction to Computer Simulation: Edinburgh, May 2010

13. First simplification • The electron mass is much smaller than the nuclear mass • Electrons remain in a stationary state of the Hamiltonian wrt nuclear motion • Nuclear problem is separable (and, as we know, the nucleus is merely a point charge!) Introduction to Computer Simulation: Edinburgh, May 2010

14. 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

15. Density functional theory • Let’s write the Hamiltonian operator in the following way: • T is the kinetic energy terms • V is the potential terms external to the electrons • U is the electron-electron term • so we’ve just classified it into different ‘physical’ terms Introduction to Computer Simulation: Edinburgh, May 2010

16. The electron density • The electronic charge density is given by • 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 Q and then get n(r). Introduction to Computer Simulation: Edinburgh, May 2010

17. 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

18. Method behind DFT • Assume two potentials V and V’ lead to the same ground state density: • We can do the same again interchanging the dashed and undashed quantities thus: Introduction to Computer Simulation: Edinburgh, May 2010

19. Unique potential • If we add these two final equations we are left with the contradiction • 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

20. Change of emphasis in QM • But by the definition of the lowest energy state we must have • And so the ‘variational principle’ tells us how to solve the problem Introduction to Computer Simulation: Edinburgh, May 2010

21. Don’t bother with the wavefunction! • Express the problem as an energy • 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

22. QM using DFT N-body Schrödinger Equation Density functional theory (Kohn-Sham equations) Both equations Introduction to Computer Simulation: Edinburgh, May 2010

23. Kohn-Sham Equations • Let’s collect all the terms into one to simplify • Where • i labels each particle in the system • ViKSis the potential felt by particlei due to n(r) • n(r) is the charge density Introduction to Computer Simulation: Edinburgh, May 2010

24. 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

25. Variational Method • Schrödinger’s Equation • And the of use the Variational Principle Solve this by Minimising this within DFT Introduction to Computer Simulation: Edinburgh, May 2010

26. 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 interactingN-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

27. Definition of XC Exact XC interaction is unknown Within DFT we can write the exact XC interaction as 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

28. Exchange-Correlation Approximations A simple, but effective approximation to the exchange-correlation interaction is The generalised gradient approximation contains the next term in a derivative expansion of the charge density: Introduction to Computer Simulation: Edinburgh, May 2010

29. 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

30. Zoo of XC approximations B3LYP WDA LDA RPBE SDA WC Meta-GGA sX EXX PBE0 PW91 MP4 OEP CI Semi-Empirical HF CC PBE MP2 Introduction to Computer Simulation: Edinburgh, May 2010

31. 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

32. 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}: Introduction to Computer Simulation: Edinburgh, May 2010

33. Forces • If we take the derivative then: Product rule Product rule Introduction to Computer Simulation: Edinburgh, May 2010

34. Forces II • In DFT we have • 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

35. Phase II Energy Phase I Volume Example: Lattice Parameters • KS equations can be solved to give energy, E • What does that tell us? Common tangent gives transition pressure: P=-dE/dV VII VI Introduction to Computer Simulation: Edinburgh, May 2010

36. Structures without experiment? U(x) start stop x Relative energies of structures: examine phase stability Introduction to Computer Simulation: Edinburgh, May 2010

37. 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

38. Alloys • Alloys are complicated! Phase separated Ordered Random Introduction to Computer Simulation: Edinburgh, May 2010

39. 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

40. 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

41. 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

42. 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

43. VCA Example • NbC1-xNx • C and N are disordered • How does electronic structure vary with x? Introduction to Computer Simulation: Edinburgh, May 2010

44. NbC1-xNx Electronic DoS Introduction to Computer Simulation: Edinburgh, May 2010

45. Electron by electron Introduction to Computer Simulation: Edinburgh, May 2010

46. 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

47. Perturbation Theory • Based on compute how the total energy responds to a perturbation, usually of the DFT external potential v • Expand quantities (E, n, y, v) • Properties given by the derivatives Introduction to Computer Simulation: Edinburgh, May 2010

48. The Perturbations • Perturb the external potential (from the nuclei and any external field): • Nuclear positions  phonons • Cell vectors  elastic constants • Electric fields  dielectric response • Magnetic fields  NMR • But not only the potential, any perturbation to the Hamiltonian: • d/dkatomic charges • d/d(PSP)  alchemical perturbation Introduction to Computer Simulation: Edinburgh, May 2010

49. Example: Phonons • Perturb with respect to nuclear coordinates: • This is equivalent to Introduction to Computer Simulation: Edinburgh, May 2010

50. Atomic Motion Eigenvectors of 2nd order energy give nuclear motion under phonon excitations Introduction to Computer Simulation: Edinburgh, May 2010