DFT in practice. Sergey V. Levchenko. Fritz-Haber-Institut der MPG, Berlin, DE

Transcription

1 DFT in practice Sergey V. Levchenko Fritz-Haber-Institut der MPG, Berlin, DE

2 Outline From fundamental theory to practical solutions General concepts: - basis sets - integrals and grids, electrostatics, molecules and clusters versus periodic systems - scalar relativity - solution of the eigenvalue problem

3 Equations to be solved Hohenberg-Kohn theorems Electron density Ψ,,, ; { } [ ], = Ψ,,, ; { } Ψ,,, ; { } Variational principle: = 0,,?

4 Equations to be solved = Ψ Ψ = Ψ,, Ψ,, Ψ,,, Ψ,, + + Ψ,,, Ψ,, = =, +, with one- and two-particle density matrices:, = Ψ,, Ψ,,, = Ψ,, Ψ,,, Problem: kinetic energy and electron-electron interaction energy as a functional of ( ) --?

5 Equations to be solved Kohn-Sham approach: + = + ( ) + [ ] where functions represent the density: = = 0 1 = 0 (Generalized) Kohn-Sham equations: = + + +, [{ }], =

6 = Solving Kohn-Sham equations + + +, [{ }], =

7 Self-consistent field method Initial guess: e.g. density Update potential n 0 V j+1 Calculate potential V 0 Construct Hamiltonian n j+1 = n j +f(δn) Update density / density matrix (mixer, preconditioner) No Obtain new eigenvalues, eigenvectors, and density n j Converged? Yes

8 Convergence When is the calculation finished? Answer: When property does not change Very loose Convergence threshold Very strict Typical properties to converge: Change of the density D n Change of the sum of eigenvalues Change of the total energy Options differ between codes

9 Self-consistent field method (SCF) Initial guess: e.g. density Update potential n 0 V j+1 Calculate potential V 0 Construct Hamiltonian n j+1 = n j +f(δn) Update density / density matrix (mixer, preconditioner) No Obtain new eigenvalues, eigenvectors, and density n j Converged? Yes

10 Initial guess Importance of the initial guess Electronic structure exhibits multiple minima Qualitatively different behavior depending on guess Common issue for magnetic solids 1 O 2 3 O 2

11 Initial guess Importance of the initial guess Electronic structure exhibits multiple minima Qualitatively different behavior depending on guess Common issue for magnetic solids How do we find the right 1 O 2 initial guess? 3 O 2

12 Initial guess Random initialization Traditional approach (for bandstructure codes)

13 Initial guess Superposition of spheric atomic densities Most common (for bandstructure codes)

14 Initial guess Extended Hückel Model [1] Semi-empirical method: - Linear combination of atomic orbitals - Hamiltonian parameterized: - Density from orbitals: [1] R. Hoffmann, J Chem. Phys (1963), 1397; [2] R. S. Mulliken, J. Chem. Phys. (1946) 497; [3] M. Wolfsberg and L. Helmholtz, J. Chem. Phys (1952), 837

15 Self-consistent field method (SCF) Initial guess: e.g. density Update potential n 0 V j+1 Calculate potential V 0 Construct Hamiltonian n j+1 = n j +f(δn) Update density / density matrix (mixer, preconditioner) No Obtain new eigenvalues, eigenvectors, and density n j Converged? Yes

16 Density Update Calculate new density from Kohn-Sham orbitals Replace old density by new density n j 1 = n j D n Naive Mixing Example: H 2, d=1.5å, projected in 1 dimension

17 Density Update Calculate new density from Kohn-Sham orbitals Replace old density by new density n j 1 = n j D n Naive Mixing Example: H 2, d=1.5å, projected in 1 dimension Typically does not work - Best case: Osciallating, non-converging results - Worst case: Bistable, apparently converged solution

18 Density Update Linear Mixing Calculate new density from Kohn-Sham orbitals Replace old density by new density n j 1 = n j ad n Example: H 2, d=1.5å, projected in 1 dimension

19 Density Update Linear Mixing Calculate new density from Kohn-Sham orbitals Replace old density by new density n j 1 = n j ad n Example: H 2, d=1.5å, projected in 1 dimension

20 Density Update Linear Mixing Calculate new density from Kohn-Sham orbitals Replace old density by new density n j 1 = n j ad n Example: H 2, d=1.5å, projected in 1 dimension

21 Density Update Linear Mixing Calculate new density from Kohn-Sham orbitals Replace old density by new density n j 1 = n j ad n Example: H 2, d=1.5å, projected in 1 dimension system dependent No clear recipe to choose ideal α Guaranteed convergence

22 Density Update Pulay Mixing [1] (Direct Inversion of Iterative Subspace) Define residual: Change between input and output density Predict residual of next step from previous steps Idea: obtain b minimizing norm Pitfall: Only use limited number of previous steps [1] P. Pulay, Chem. Phys. Lett. 73, 393 (1980).

23 Density Update Pulay Mixing [1] (Direct Inversion of Iterative Subspace) Construct optimal density minimizing residual norm, form new density Learning from previous steps Ideal case: convergence independent of a Rules of thumb: - a = for insulators - a = 0.05 for metals [1] P. Pulay, Chem. Phys. Lett. 73, 393 (1980).

24 Density Update Pulay Mixing [1] (Direct Inversion of Iterative Subspace) Construct optimal density minimizing residual norm, form new density Learning from previous steps Ideal case: convergence independent of a Rules ofmuch thumb: faster than linear mixing - a = No 0.2 guarantee for insulators for congerence - a = 0.05 for metals [1] P. Pulay, Chem. Phys. Lett. 73, 393 (1980).

25 Density Update Kerker (charge) preconditioning Problem: Charge Sloshing Charge bounces back and forth (Lack of) balance between long-range and short-range charge re-distribution Typical for metals Often encountered in defective systems Localized defects (vacancies, interstitials) Surfaces, Slabs, Thin Films

26 Density Update Charge preconditioning Solution: Make α depend on ( ) Concept: Treat density change in Fourier space: = + Δ = + Δ = 1 Ω Δ Damp long-range oscillations: q 0 : Screening parameter = + q >> q 0 : G a Normal mixing q << q 0 : G aq 2 /q 0 2 Damping

27 Density Update Charge preconditioning = + q 0 : Screening parameter 8 layer Al slab, PBE calculation no a priori known ideal choice for q 0 Reasonable guess from Thomas-Fermi:

28 Density Update A special problem: Metallic systems Metallic properties determined by Fermi surface Fermi-surface: Collection of k-points at which e = e Fermi At e Fermi occupation changes from 1 to 0 Problem: Frequent level switching during SCF

29 Density Update A special problem: Metallic systems Solution: Replace step function by a more smooth function Fermi [1] Gaussian [2] Methfessel-Paxton [3] [1] N. Mermin, Phys. Rev.137, A1441 (1965).[2] C.-L. Fu, K.-H. Ho, Phys. Rev. B 28, 5480 (1983). [3] M. Methfessel, A. Paxton, Phys. Rev. B 40, 3616 (1989).

30 Density Update A special problem: Metallic systems Numerical Implications Level switching is damped Much faster convergence Total energy now depends on σ, no longer variational Back-extrapolation possible Pitfall: Adsorption of molecules Physical implications: States become fractionally occupied Cu (111), 5 layer slab, Gaussian smearing, PBE calculation, 12x12x1 k- points In DFT, well defined as ensemble average Broadening is not electron temperature (except: Fermi)

31 Summary SCF Ground state electron density determined iteratively Initial guess needs initial thought, can change results qualiatively Density update by Linear mixing (slow) Pulay mixing Convergence acceleration by Preconditioner Broadening of states

32 Representing KS states: Basis sets

33 The tool for this workshop: FHI-aims (NAOs)

34 Our choice: Numeric atom-centered basis functions

35 Our choice: Numeric atom-centered basis functions

36 Constructing a basis-set library

37 Iterative selection of NAO basis functions

38 Result: Hierarchical basis set library for all elements

39 Accuracy: (H 2 O) 2 hydrogen bond energy

40 Basis-set superposition error (BSSE) By construction, our NAO basis sets are close to completeness for very small BSSE for atomic dimers

41 Basis-set superposition error (BSSE)

42 Using numeric atom-centered basis functions

43 Numeric atom-centered basis functions: Integration

44 Overlapping atom-centered grids: Partitioning of unity

45 Integration in practice: Large systems - small errors!

46 Hartree potential (electrostatics): Overlapping multipoles

47 Electrostatics: convergence with

48 Periodic systems

49 Structure optimization

50 Global structure search Born-Oppenheimer energy surface can contain several minima Constitution isomery Configuration isomery Conformation isomery Polymorphism System in equilibrium is given by ensemble average over all minima Often dominated by global minimum (but watch out for tautomers) 50

51 Global structure search Methods to find the global minimum: o Basin Hopping o Molecular dynamics: o Simulated annealing o Minima Hopping o Metadynamics o Cluster expansion o Genetic algorithm o Diffusion methods

52 Local structure optimization Once we have a reasonable guess, find closest minimum Different approaches possible: Mapping of the potential energy surface Requires a lot of calculations Typically only for dynamics a CCO d CC 52

53 Local structure optimization Once we have a reasonable guess, find closest minimum Different approaches possible: Mapping of the whole potential energy surface Gradient free method: e.g., simplex Gradient-based methods Calculate gradient (a.k.a. forces ) =

54 Total energy gradient Search for minimum by following the gradient = Ψ Ψ + Ψ + Ψ = + = + First term vanishes Pulay forces Second term survives for atom-centered basis functions (vanish when basis set approaches completeness)

55 Total energy gradient Search for minimum by following the gradient = Ψ Ψ + Ψ + Ψ Additional contributions from atom-centered approximations Multipole expansion Relativistic corrections (Integration grids) All straightforward but lengthy Once we have the Force, how do we find the minimum?

56 Geometry update Steepest descent Follow negative gradient to find minimum Step-length α variable Guaranteed but slow convergence Oscillates near minimum Not suitable for saddle points Improved versions exists Line minimization (optimal step length) Conjugated gradient [1] [1] M. Hestenes, E. Stiefel (1952). "Methods of Conjugate Gradients for Solving Linear Systems"

57 Geometry update (Quasi)Newton Methods Approximate PES by quadratic function Δ + Δ + Δ Δ... Hessian Find minimum: Newton: calculate exact H Expensive! Cheaper method needed [1] J. Nocedal and S. J. Wright, Numerical optimization (Springer, 2006)

58 Geometry update (Quasi)Newton Methods Approximate PES by quadratic function Δ + Δ + Δ Δ... Hessian Find minimum: Newton: calculate exact H Quasi-Newton: approximate H Update as search progresses [1] [1] J. Nocedal and S. J. Wright, Numerical optimization (Springer, 2006)

59 Geometry update (Quasi)Newton Methods Guess initial Hessian Simple choice: Scaled unit matrix Chemically motivated choice (e.g.: Lindh [1]) streching bending torsion k parameterized [1] Update as search progresses [2] [1] R. Lindh et al., Chem. Phys. Lett. 241, 423 (1995). [2] J. Nocedal and S. J. Wright, Numerical optimization (Springer, 2006)

60 Geometry update (Quasi)Newton Methods Initial Hessian is critical for performance May even influence result (combination with convergence thresholds) Diagonal Hessian Model Hessian (Fischer) Courtesy of Elisabeth Wruss Lower in energy

61 Challenges of Quasi-Newton Methods Soft degrees of freedom can cause large ΔR Step control needed: Line search method: If new point is worse than old, interpolate Trust radius method Enforce upper limit for ΔR Evaluate quality of quadratic model Adjust ΔR max based on q

62 Conclusions (Global optimization: PES feature-rich, methods to find global minima exist) Local geometry optimization: Follow gradient Hellman-Feynman from moving potentials Pulay from moving basis functions + additional terms Quasi-Newton method de-facto standard Requires approximation and update of Hessian Step control by line search or trust radius method

63 Vibrations in the harmonic approximation

64 Why calculating vibrations? Vibrations give important information about the system: Classification of stationary points (minimum / saddle point) If saddle-point: Provides search direction Thermodynamic data Zero-point energy Partition sum Finite-temperature effects Connection to experiment: Infrared intensities: derivative of dipole moment Raman intensities: derivative of polarizabilty

65 How good is the harmonic approximatino? Morse-potential: Harmonic oscillator: Harmonic osc. Morse:

66 How good is the harmonic approximation? Morse-potential: Harmonic oscillator: Harmonic osc. Harmonic osc. Reasonable model for small displacements (~10% of bond length) Maximum displacement: Morse: About 0.1Å at room temperature

67 Vibrations Expand Energy in Taylor series: Hessian H

68 Calculating Vibrations Solve Newton s equations of motion: Exponential ansatz: Leads to generalized eigenvalue problem: Interpretation of results: Single negative ω: Transition state ~ /

69 From vibrations to thermodynamics Partition function Free energy for finite temperature Hessian from geometry optimization is not sufficent Analytic second derivative using perturbation theory [1] Numerical differentiation Computationally very expensive Contains parameter (displacement), needs to be checked carefully Often single displacement is not sensible to sample all vibrations [1] S. Baroni et al., Rev. Mod. Phys. 73, 515 (2001).

70 Conclusions Often calculated in harmonic approximation Yield information about stability of geometry Required for temperature effects Anharmonic effects via molecular dynamics

71 Relativity

72 Simple approximation to scalar relativity

73 Simple approximation to scalar relativity

74 Fixing ZORA

75 Atomic ZORA and scaled ZORA in practice

76 Computational scaling: two subproblems

77 CPU time versus system size

78 CPU time versus system size

79 Parallel eigenvalue solvers - the problem

80 Parallel eigenvalue solvers - the problem

81 A massively-parallel dense eigensolver: ELPA

82 The key improvement: two-step trigonalization

83 The key improvement: Two-step trigonalization

84 ELPA: Performance

85 ELPA: Performance (Cray XC30, 2013)

86 Generalized Kohn-Sham DFT: Exact exchange

87 Resolution of identity (RI)

88 Hybrid functionals -- scaling with system size Periodic GaAs, HSE06 hybrid functional

89 Hybrid functionals -- CPU scaling Periodic GaAs, HSE06 hybrid functional

90 Summary

91 Our approach to all-electron simulations: FHI-aims Igor Ying Zhang (FHI) Karsten Reuter (TU Munich) Volker Blum (Duke Univ.) Patrick Rinke (Aalto Univ., Finland) Ville Havu (Aalto Univ., Finland)