Density Functional Theory Machinery

Transcription

1 Solid State Theory Physics 545 Density Functional Theory- Density Functional Theory Machinery

2 Calculating the Wave Function DFT (and other methods) iterate to self-consistency Guess the wave functions Construct tpotential ti Solve Kohn-Sham Equations No New wave functions Stop Yes Same as old wave function? For a given set of nuclear For a given set of nuclear positions

3 Calculating the Wave Function DFT (and other methods) iterate to self-consistency HΨ nk = ε where H and Ψ, nk nk = K + Vne( r) + Vee( r) + Vxc 2 Guess the wave functions Construct 2 K =, tpotential ti Solve Kohn-Sham Equations 2m No 2 Za Vne( r) = e, r R New wave functions Stop Yes 2 ρ( r ) Vee( r) = e Same as old d rwave r function? δexc Vxc( r) =, δρ ρ ) = f nk Ψ ( r ) nk a nk a 3 2 r,, ) For a given set of nuclear positions

4 Calculating the Wave Function DFT (and other methods) iterate to self-consistency Guess the wave functions Construct tpotential ti Solve Kohn-Sham Equations No New wave functions Stop Yes Same as old wave function? For a given set of nuclear For a given set of nuclear positions

5 Calculating the Wave Function DFT (and other methods) iterate to self-consistency Guess the wave functions Construct tpotential ti Solve Kohn-Sham Equations No New wave functions Stop Yes Same as old wave function? For a given set of nuclear For a given set of nuclear positions

6 Calculating the Wave Function DFT (and other methods) iterate to self-consistency Guess the wave functions Construct tpotential ti Solve Kohn-Sham Equations No New wave functions Stop Yes Same as old wave function? For a given set of nuclear For a given set of nuclear positions

7 Calculating the Wave Function DFT (and other methods) iterate to self-consistency Guess the wave functions Construct tpotential ti Solve Kohn-Sham Equations No New wave functions Stop Yes Same as old wave function? For a given set of nuclear For a given set of nuclear positions

8 Calculating the Wave Function DFT (and other methods) iterate to self-consistency Guess the wave functions Construct tpotential ti Solve Kohn-Sham Equations No New wave functions Stop Yes Same as old wave function? For a given set of nuclear For a given set of nuclear positions

9 Basic Machinery: The theory > computational schemes? Procedure: The iterative self- consistent field the linear-combination-of-atomic-orbitals t i bit l (LCAO) Which kind of basis sets! Benefit from the vast experience of wave function based techniques! How to determine components of Kohn-Sham (KS)? The Coulomb energy The Exchange-correlation energy

10 Solving: 1. Basis set ψ ) c ) n μ μ nφμ unknown Expand in terms of a finite set fk of known wave-functions φ (r ) ˆ hψ ) = ε ψ ) ˆ c hφ ) = ε c φ ) Def n n * 3 h νμ φν ) hφμ ) d r n μ φ μ μ n μ n μ n μ μ ˆ * 3 and S φ ) φ ) d r νμ ν μ μ h c = ε S c νμ μn n νμ μn μ HC =ɛ SC n n n ~ - ~-

11 Introduction of a Basis:The LCAO Ansatz ( ) 2 N 1 ϕ 2 j r 2 dr ( ) + 2+ VXC r1 2 r j 12 A 1A M Z r A ϕ i = A spin orbital f KS ˆK ( * 1 ) ( 1 1 ) ( 2 ) ( 2 ) 2 ( x χ 1 ) i x = j χ j x χi x dx χ j x r The exchange operator LCAO expansion of KS molecular orbitals. As Roothaan ( 1951 ) 12 ε i ϕi Kohn-Sham Integro-differential Equations L c μ i μ = 1 ϕ i= L the set { η μ } complete f KS L L ) cvi ηv ) = εi cvi ηv ) Non- linear Optimization i f KS v= 1 v= 1 η μ linear Optimization η μ Multiply from the left with, Integrate over space

12 L Introduction of a Basis:The LCAO Ansatz L c η r r η r dr ε c η r η r dr 1 i L ( ) KS f ( ) ( ) = ( ) ( ) μ vi μ 1 1 v 1 1 i vi 1 v 1 1 v= 1 v= 1 KS F μν Kohn-Sham matrix S μν Overlap matrix LxL dimensional, real and symmetric i.e. M μ v = M * v ( M ) μ μv = M μv hermitian c11 c12 c1 L ε1 0 0 c21 c22 c 0 ε 2L C= 2 0 ε = cl 1 cl2 c LL 0 0 ε L KS F C = S C ε Analogous to numerical Hartree-Fock methods.

13 Introduction of a Basis:The LCAO Ansatz N 2 f expand KS and ρ ) = ϕ j 2 ) i f KS M KS 1 2 ZA ρ2 ) F μν = ημ 1) + dr2+ VXC1) ηv 1) dr1 2 A r 1A r12 1 M 2 Z = A ημ 1) ηv 1) dr1 η 1) μ ηv 1) dr1 2 The electonic kinetic energy A r1 A + ημ 1) VXC 1) ηv 1 ) dr1 The exchange-correlation The electron-nuclear interaction ρ ) + η η 2 μ 1 ) v 1 ) drdr 1 2 r12 M 1 2 Z A hμv = ημ( r1) ηv1) dr1 2 A r1 A Coulomb contribution

14 Introduction of a Basis:The LCAO Ansatz The charge density LCAO scheme ρ 2 ) η μ 1) v 1) drdr 1 2 r η 12 N 2 N L L ρ ) = ϕ ) = c c η ) η ) i μi vi μ v i i μ v Coulomb contribution Density Matrix Basis functions The expansion coefficients P μv N = i c c μi vi Four-center two-electron integrals L L 1 J = P η r η r η r η ( ) ( ) ( ) ) drdr η η η η μ v λσ μ 1 v 1 λ 2 σ λ σ r12 XC V η 1) VXC 1) η μ v = μ v 1) dr1 Exact! The exchange-correlation approximation L L 1 K = P η ( x ) η ( x) η ( x ) η ( x ) dxdx μν λσ μ 1 λ 1 ν 2 σ λ σ r 12 HF exchange integral Due to spin dependence

15 Basis Sets The orbitals Wave function based approaches i High quality Wave-functions Large Sets Orbitals in KS The charge density LCAO scheme Basis functions η μ χ { } N ρ ) = ϕ i ) Cartesian Gaussian-type-orbitals (GTO) Generally used by Hartree-Fock, Configuration-Interaction (CI) η 2 = N x y z exp[ α r ] GTO l m n A normalization factor, ημ η μ = 1 μ v η η 0 μ v Compact (large) diffuse (small) L > 1 (2l + 1) i 2 Less severe Basis Set Requirements To classify the GTO L = l+ m+ n s-functions 0 p-functions 1 L = d-functions 2

16 Slater-type-orbitals (STO) STO n Basis Sets η = Nr 1 exp[ ζ r] Y lm ( Θ, φ ) η The principal quantum number The orbital exponent Contracted Gaussian function (GTO) A CGF GTO = τ d a τ η a a The contraction coefficients Spherical harmonics Correct cusp behavior at r 0 CGF Gaussian, Turbomole GTO DGauss, DeMon Amsterdam Density Functional (ADF) STO Numerical Basis functions DMol Plane waves PW η = exp[ ikr] p = k The projector augmented wave method

17 Basis set: Atomic orbitals s p d f Strictly localised (zero beyond cut-off radius)

18 Basis Sets minimal sets STO-3G only yqualitative results C 1s,2s, and 2-p shell the double-zeta basis set - split-valance 3-21G or 6-31G ( Pople et al ) augmented by polarization functions i.e.functions of higher angular momentum CGF type G(d,p) (Hehre, Ditchfield, and Pople, 1972) - SVP (split-valence polarization) (Ahlrichs et al, 1992) cc -pvqz (correlation-consistent polarized valance quadruple zeta) cc -pv5z (the 5 stands for quintuple) (Dunning, 1989) Combination of the B3LYP technique with (cc-vtz and better) (Raymond and Wheeler, 1999) Optimization of Gaussian Basis Sets ; within HF or correlated wave function based schemes - explicit optimization the LDA approach, DGauss (Godbout et. al., 1992) pnl html

19 Basis Sets Heavier than Krypton (a relativistic) effective core potential ((R)EPC) or pseudopotential (Frenking et. al. 1996, Cundari et. al., 1996) An ECP appropriate for a specific exchange-correlation functional should be generated from atomic calculations employing that functional Sample compounds; transition-metal carbonyls ECP comparison to all-electron calculations (Russo, Martin, and Hay, 1995 van Wuellen,1996)

20 The Calculation of the Coulomb Term various strategies to compute the classical electrostatic contribution in KS 1 1 J[ ρ] = ρ ) ρ ) dr dr 2 r With regular wave-function based Four-center two-electron integrals L L 1 Jμv = P λσ η 1) η μ v 1) λ 2) σ 2 ) drdr 1 2 r η η λ σ 12 4 L the computational bottle-neck in HF ρ ) J = η ) η ) dr dr μv μ μv κ μ κ 2 1 v r 12 K ω ) J c ) ) dr dr = η η { ω } κ 2 1 v r 12 κ auxiliary set K ρ) ρ) c ω ) = κ κ approximate density K, 2-3 times larger then L 4 10 Eh Baerends, Ellis, and Ros,1973 for STO, extended by Sambe and Felton, 1975 to CGF κ c.a kcal/mol

21 The Calculation of the Coulomb Term determination ti of the coefficients i c κ (constraint ρ dr ) = N ) 2 F= [ ρ) ρ)] dr Minimizing the functions 2 1ηv r12 (Dunlap, Connolly, and Sabin, 1979) applicable with Gaussians only [ ρ ) ρ )][ ρ ) ρ )] dr dr F = r Minimizing the Coulomb self-repulsion of the residual density ρ ) J η ) ) dr dr Turbomole (RI-J resolution of the identity ) μv = η η μ ADF approach DGauss K ρ 2 ) ρ 2 ) ω 2 ) κ V Coul1) = dr2 dr2 = c κ dr2 = V Coul1) r12 r12 κ r12 P J η 1) η 1)V 1) dr1 η 1) η 1)V p ) W μv μ v Coul p p Coul p p grid points 12 Weights in the numerical quadrature

22 The Calculation of the Coulomb Term Dmol, Poisson`s equation 2 V 1) = 4 π ρ) Coul solved numerically on the grid (Delley, 1990 or Becke and Dickson, 1988) Four-center two-electron integrals V Coul L L 1 J = P η ) η ) η ) η ) drdr μv λσ μ 1 v 1 λ 2 σ λ σ r12 4 ON ( ) asymptotic scaling for very large systems : N 3 N ` versus myth ` 4 to 3 ON ( ) (Johnson, 1995) Employing fitted densities; Beneficial or not? ON 2 ( ) (Von Armin and Alhrichs, 1998) Turbomole 1000 basis functions one order of magnitude faster then explicit computing for J μν majority of integrals close to zero, because of the vanishing overlap of the basis functions

23 Techniques to Handle V XC V XC μν ) V ) η ) = η dr μ Grid-based methods for V XC Choose a grid XC Evaluate V XC at each point ρ α, ρ β derivatives at each point (for GGA) Apply a quadrature ν V XC μν P Wpη p grid-based and grid-free methods μ and their ) V ) η ) p XC p ν p

24 Grid-based Techniques Choice of a grid V=V(ρ V(ρ ); ρ has cusps at the nuclei simple cartesian grids are not suitable I = I Solution: atom-centered t grids A I A Constructing F ' s : M I = F() r dr F() r = F () r FA() r = WA() r F() r I = I = A A A F A ) A A WA ) 1as r 0 W () r 0 close to B A dr W A 1 0 A A

25 Grid-based Techniques I A π 2π 2 rad ang = ( ) F ( ) A r, θ, φ r sinθ drdθ dω Wp Wq FA rp, θ q, φq P Q grid points Grid pruning: grids of varying density Problems No invariance to rotation Numerical noise Gradients and frequencies (low modes) suffer Can be solved by using standard orientation T Gradients may not vanish at a minimum d E ~ XC = f dx t Octahedral symmetry of L. grids doubly degenerate modes are not exactly equal P p Q q Lebedev grids t d W dx t + W t d dx f t

26 Grid-free Techniques V XC μν ) V ) η ) = η dr μ XC ν VV XC is a functional of ρ δe GGA V XC E [ ] ( XC ρ α, ρ β = f ρα, ρ β, ρα, ρ β )dr XC δρ Let us use a matrix representation for ρ : R μν () r ˆ ρ) η ) = η dr μ Goal: Obtain matrix elements of V XC ν by operating on matrix R f ( R) V XC

27 Grid-free Techniques Example: n ˆ ρ ) ˆ ρ) ˆ ρ) ˆ ρ) n times n R R R R n n η μ ) ρ ) η ν ) dr = ( R ) μν in a matrix form Question: Will this work for a more general case V XC = f(ρ) )? i.e. V XC = μν [ f ( R) ] μν n times R11 R21 f 1 ) 0 R R f R 2 ) LL R 11 R12 r1 0 orthogonolize diagonalize apply f ( x) R21 R22 0 r2 RLL rl F 11 F12 F11 F12 undiagonalize unorthogonolize F21 F22 F21 F22 f ) F L LL F LL

28 Grid-free Techniques = η μ Real V XC : Terms proportional to ) [ f ) g ) ] η ) dr μ fˆ gˆ ν = μ fˆ Iˆ gˆ μ ˆ ρ π ˆ γ π γ ρ ν = γ γ Resolution of the identity μ ν μ ˆ ρ π γ π γ ˆ ρ ν ν ρ 4 3 and ) f ) π ) dr π ) g ) η ) dr { π } is an auxillary basis set η γ λ λ ν ; λ s ρ 4 3 ρ

29 Grid-free Techniques How do we get a matrix representation for ρ? R R μν μν L L η ) ) ) ρ ) = P η ) η ) μ ρ ην = dr L L ) η ) η ) η ) Use Resolution of the identity = P η dr λ σ λσ μ λ K ρ ) ~ ρ ) c ω ) ρ ) ρ ) ω ) R μν L L K = κ κω κ σ ρ ~ ν K λ σ λσ λ σ Four-center integral! [ ] dr ) = ω κ ω κ ) η ) ω ) dr ω ) η ) η ) P λσ ημ ν κ = dr λ σ κ κ κ λ σ

30 Footnote Why Resolution of the identity? ρ () r = ρ() r () r or κ [ ] dr () r ω κ ω κ ρ = κ ω κ ω κ ρ I = κ ω κ ω κ

31 Advantages of Grid-free Techniques Smooth reproducible error due to finite {π }, k {ω k } Error independent on the choice of coordinate system However, very large {π }, {ω k k } are needed

32 Towards Linear Scaling KS Theory Computational Bottlenecks Evaluation of J Diagonalization of KS Matrix Numerical Quadrature of fe XC and dv XC Modified Fast Multipole Method: Partitioning physical space around point charges Use multipole expansions within far-field field region Partitioning of J 1 r S () r + L() r = ) 1 f ) f r + r S(r) L(r) ( ) r

33 Applying DFT A computational problem: near the nucleus, the electron wave function oscillates very rapidly

34 Applying DFT A computational problem: rapid oscillation Tabulated functions Need LOTS of data For Fourier Transforms, need LOTS of frequencies

35 Applying DFT A computational problem: rapid oscillation Tabulated functions Lots of data Several ways to deal with this Most popular: Pseudopotentials

36 Applying DFT A computational problem: rapid oscillation Tabulated functions Lots of data Several ways to deal with this Most popular: Pseudopotentials

37 Projector Augmented Wave Method Most recent and advanced electronic structure calculation method Developed by Peter Blöchl Fast and accurate Does not sacrifice information in the core region Easy comparison to experimental results

38 Solving: 2. Boundary conditions o Isolated object (atom, molecule, cluster): open boundary conditions (defined at infinity) 3D Periodic object (crystal): Periodic Boundary Conditions Mixed: 1D periodic (chains) 2D periodic (slabs)

39 k-point sampling Electronic quantum states in a periodic solid labelled by: Band index k-vector: vector: vector in reciprocal space within the first Brillouin zone (Wigner-Seitz cell in reciprocal space) Other symmetries (spin, point (spin, point-group representation ) occ Approximated by 2 3 ρ ) = ψ n) d k sums over selected n k B. Z k points

40 Some materials properties p C Si Na Cu Au Exp. LAPW Other PW PW DZP a (Å) B (GPa) E c (ev) a (Å) B (GPa) E c (ev) a (Å) B (GPa) E c (ev) a (Å) B (GPa) E c (ev) a (Å) B (GPa) E c (ev)