Strategies for Solving Kohn- Sham equations

Transcription

1 Strategies for Solving Kohn- Sham equations Peter. E. Blöchl Institute for Theoretical Physics Clausthal University of Technology, Germany 1 1

2 Appetizer: high-k oxides McKee for new transitors source gate oxide gate drain n doped n doped Hu et al. p doped p doped channel silicon waver Metal Oxide Field Effect Transistor (MOSFET) Norga et Ojima,Yoshim 2 2

3 Questions Which functional? LDA,GGA, Meta-GGA, Hybrid,LDA+U,... Which bandstructure method? Gaussian, Slater-orbitals, Pseudopotentials, KKR, LMTO, LAPW, PAW Which relativistic treatment? non-relativistic, scalar relativistic, spin-orbit coupling? Which spin description? non-spin-polarized, spinpolarized, non-colllinear Which kpoint set? Which supercells? How to decouple spurious periodic images? 3 3

4 Starting point: DFT Minimize total energy functional: E[ ψ n,f n ] = f n ψ n ˆp 2 2m ψ n n with E kin + E xc [n(r)] E xc n,m n(r) = f n ψ(r) 2 n electron density d 3 r Solve Kohn-Sham equations: 2 2m 2 + v eff (r) ψ n (r) =ψ n (r) n and v eff (r) = d 3 r e2 [n(r)+z(r)][n(r )+Z(r )] 4π 0 r r E Hartree Λ n,m (ψ n ψ m δ n,m ) constraint Z(r) = Z j δ(r R j ) j charge density of nuclei in electron charges d 3 r e2 [n(r )+Z(r )] 4π 0 r r v Hartree (r) + δe xc δn(r) µ xc (r) 4 4

5 Starting point: DFT Minimize total energy functional: E[ ψ n,f n ] = f n ψ n ˆp 2 2m ψ n n with E kin + E xc [n(r)] E xc n,m n(r) = f n ψ(r) 2 n electron density d 3 r Solve Kohn-Sham equations: 2 2m 2 + v eff (r) ψ n (r) =ψ n (r) n and v eff (r) = d 3 r e2 [n(r)+z(r)][n(r )+Z(r )] 4π 0 r r E Hartree Λ n,m (ψ n ψ m δ n,m ) constraint Z(r) = Z j δ(r R j ) j charge density of nuclei in electron charges d 3 r e2 [n(r )+Z(r )] 4π 0 r r v Hartree (r) + δe xc δn(r) µ xc (r) 5 5

6 Starting point: DFT Minimize total energy functional: E[ ψ n,f n ] = f n ψ n ˆp 2 2m ψ n n with E kin + E xc [n(r)] E xc n,m n(r) = f n ψ(r) 2 n electron density d 3 r Solve Kohn-Sham equations: 2 2m 2 + v eff (r) ψ n (r) =ψ n (r) n and v eff (r) = d 3 r e2 [n(r)+z(r)][n(r )+Z(r )] 4π 0 r r E Hartree Λ n,m (ψ n ψ m δ n,m ) constraint Z(r) = Z j δ(r R j ) j charge density of nuclei in electron charges d 3 r e2 [n(r )+Z(r )] 4π 0 r r v Hartree (r) + δe xc δn(r) µ xc (r) 6 6

7 Starting point: DFT Minimize total energy functional: E[ ψ n,f n ] = f n ψ n ˆp 2 2m ψ n n with E kin + E xc [n(r)] E xc n,m n(r) = f n ψ(r) 2 n electron density d 3 r Solve Kohn-Sham equations: 2 2m 2 + v eff (r) ψ n (r) =ψ n (r) n and v eff (r) = d 3 r e2 [n(r)+z(r)][n(r )+Z(r )] 4π 0 r r E Hartree Λ n,m (ψ n ψ m δ n,m ) constraint Z(r) = Z j δ(r R j ) j charge density of nuclei in electron charges d 3 r e2 [n(r )+Z(r )] 4π 0 r r v Hartree (r) + δe xc δn(r) µ xc (r) 7 7

8 What is the problem? 1. wave function oscillates strongly in the atomic region (Coulomb singularity) 2. wave function needs to be very flexible in the bonding region and the tails (Chemistry) K.Schwarz Talk/WWW From APW to LAPW to (L)APW+lo 3. most electrons (core) are irrelevant 4. relativistic effects 5. tiny but finite nucleus r atomic valence wave functions (s and d) of Pt 8 8

9 Brute force: LCAO Use atom-centered basisset Gaussians (Gaussian, Turbomole, etc) Slater orbitals (ADF) Numeric atomic orbitals (DMOL) LCAO Ansatz: ψ n (r) = α wave function solve eigenvalue problem basis orbital χ α (r) coefficient c α,n [H n O]c n = 0 with H α,β = χ α ˆp 2 2m +ˆv χ β and O α,β = χ α χ β 9 9

10 Strategies Pseudopotential Augmented waves Chop off singular potential node-less wave functions no core states no information on inner electrons transferability problems Hamann-Bachelet-Schlueter, Kerker, Kleinman-Bylander, Troullier Martins, Ultrasoft,etc start with envelope function replace incorrect shape with atomic partial waves complex basisset retain full information on wave functions and potential LMTO, ASW, LAPW, APW, PAW, (all-electron) 10 10

11 Energy-dependent partial waves ε-dependent radial Schrödinger equation 2 2m 1 r 2 r r ( + 1) r 2 + v eff (r) φ (,r)=0 Fe-d partial wave ε Linear approximation: r[a 0 ] φ(,r)=φ( ν,r)+( ν ) φ( ν,r) +O(( ν ) 2 ) two functions contain most chemical information φ/ 11 11

12 Logarithmic derivative and Wigner rule Logarithmic derivative function describes the scattering properties of an atom D () = rφ (,r) φ (,r) D[ ε] ϕ () =N nodes () π arctan[d ()] 5 4 s 0 ε B ε A ε B εa ε d p ε [ev] 12 12

13 Augmentation (LAPW) Use envelope functions: e.g. plane waves Define atom-centered augmentation spheres Ω R muffin-tin spheres, atomic spheres expand envelope function into spherical harmonics χ G =e i Gr χ 1 G(r) =e i Gr =4π,m i j ( G r )Y,m( G)Y,m (r) keep value and derivative of radial function for each,m Replace radial part of envelope functions with partial waves from the instantaneous spherically averaged potential 13 13

14 Augmented waves Linear methods (O.K. Andersen75) linear augmented plane wave method (LAPW), linear muffin-tin orbital method (LMTO), augmented spherical waves (ASW) Parents (coming back..) Augmented plane wave (APW) method (Slater53), Korringa-Kohn-Rostocker (KKR) method (Korringa47 and Kohn-Rostocker54) Descendent Projector augmented wave (PAW) method (Blöchl94) 14 14

15 Pseudopotentials Important is the logarithmic derivative and its energy derivative value and derivative result from condition of differentiability energy derivative of the log-derivative is related to the norm dd d = 2m 2 r 2 Ω φ2 (,r Ω ) d 3 r φ(, r) 2 Ω Norm Replace partial waves by nodeless (pseudo-) partial waves with the correct norm Invert pseudo partial waves to obtain potential 2 2m 2 + v ps (r) φ(r)! =0 v ps (r) = 1 φ(r) 2 2m 2 φ(r) 15 15

16 Pseudopotentials Invert pseudo partial waves to obtain potential 2 2m 2 + v ps (r) φ(r)! =0 v ps (r) = 1 φ(r) 2 2m 2 φ(r) one potential for each angular momentum Project onto angular momenta before applying potential semi-local potential ˆv ps =ˆv local +,m ˆP,m (ˆv ps, ˆv local ) ˆP,m 16 16

17 Pseudopotentials pseudo-schrödinger equation 2 2m 2 +ˆv ψ(r) =0 with ˆv = ˆv local + ˆP,m (ˆv ps, ˆv local ) ˆP,m,m + d 3 r e2 [ñ(r)+ Z(r)] + δe xc[ñ] 4π 0 r r δn(r) ñ(r) = n valence Z(r) = (N core Z) v eff (ñ(r)],r)] f n ψ n (r) 2 a π 3 e ar2 pseudo charge density d 3 r e2 [ñ at (r)+ Z(r)] 4π 0 r r v eff (ñ at (r)],r)] pseudo core charge + δe xc[ñ at ] δn(r) 17 17

18 Plane waves ψ k,n (r) = G e i( k+ G)r c k,n ( G) with G = g1 i 1 + g 2 i 2 + g 3 i 3 convergence controlled by a single parameter 1 π r = 2 k + G 2 E PW 2EPW Fast Fourier Transform (FFT) map to real space in N ln(n) operations (<< N 2!) u k,n (r) = e igr c k,n ( G) ; c k,n = 1 e igr u k,n (r) N r G kinetic energy and Coulomb energy are diagonal in G-space E kin = V unit-cell 2 2m ( k + G) 2 c k,n ( G) 2 G potential and exchange correlation is diagonal in r-space r 18 18

19 Self-Consistency Cycle 19 19

20 Self-consistency cycle SCF-Cycle 1.Schrödinger equation 2.density from wavefunction 3.potential from density 4.mixing Mixing is critical potential or density mixing charge-sloshing for large systems Mixing is an art! in v eff =(1 p 2 φ n n v eff (r ) v 2m eff ε n φ n =0 e φ n n(r )=Σ * (r ) (r ) n(r ) φ n out v eff (r ) = v ext(r ) + d 3 e r 2 n(r ) 4 πε 0 r r + µ xc (r ) α)vin out eff +αv eff in in out in out v eff =v eff? v eff (r ) 20 20

21 Paradigm shift: Car-Parrinello method eigenvalue problems minimize energy Access to the molecular dynamics Car, Parrinello, PRL 55, 2471(1985) 21 21

22 Fictitious Lagrangian L = f n ψ n ˆm ψ ψ n M j R 2 j n j fictitious kinetic energy E kin of nuclei Equations of motion Car-Parrinello method ˆm ψ ψ n f n = Ĥ ψ nf n E DF T / ψ n Car, Parrinello, PRL 55, 2471(1985) E DFT {f n, ψ n, R j } Λ n,m (ψ n ψ m δ n,m ) n,m constraints ˆm ψ ψf n m friction ψ m Λ m,n force of constraint M j Rj = Fj M j α R R j Rj E DF T friction Discretize using Verlet algorithm ẍ ẋ x(t + ) 2x(t)+x(t ) 2 x(t + ) x(t ) 2 Constraints using [Ryckaert,Cicotti, Berendsen, J. Comp.Phys. 23, 327 (1977)] 22 22

23 K-points and Brillouin-zone integration 23 23

24 K-points and Band structure Wave functions in a periodic crystal can be written as Bloch states: periodic function modulated by a plane wave periodic ψ k,n (r) = u k,n (r)e i kr ψ k,n (r) = χ α r ( R α + t) α Ω 0 t e i k( R α +t) c α, k,n z L x U S X Z W K y 24 24

25 Monkhorst-Pack Special Points Replace integral by sum over equispaced grid A = n 1 Fermi function d 3 k f( n ( V k)) A n ( k) g V g matrix element Brillouin-zone integral H.J. Monkhorst and J.D.Pack, PRB13, 5188 (1976) 1 N k N k Corresponds to an interpolation by a plane wave expansion j=1 f( n ( k))a n (k j ) k-point sum f( n ( k))a n ( k)= {t} e i kt c n (t) rapid (exponential) convergence poor description of the Fermi surface control grid density with a cutoff t <R x ;(R x 20) 25 25

26 Special points for metals produce instabilities in the self-consistency non-differential total energies Finite temperatures Introduce finite temperatures to smoothen density of states: Mermin functional [Mermin, PR137, A1441 (1965)] F ({ ψ n,f n },T) free energy = E({ ψ n,f n }) + k B T [f n ln(f n )+(1 f n )ln(1 f n )] n internal energy U TS with entropy S produce finite temperature Fermi distribution f n = effectively smears out energies εn by kbt 1 1+e 1 k B T ( n µ) Density of states 26 ε 26

27 Tetrahedron method Jepsen, Andersen, sol. st. comm. 9, 1763 (1971); Lehmann,Taut, phys. stat. sol. 54, 469 (1972) linear interpolation inside tetrahedra analytic integration of interpolated functions. Blöchl correction No misweighting: produces special-point ε result for filled bands corrects for linear over-estimation of partially filled bands Blöchl, Jepsen, Andersen, PRB49, (1994) µ k x k y Writes result as a weighted sum k x from A = tetrahedron method w n ( k i ) A n ( k i ) ki,n 27 27

28 k-point convergence Tetrahedron method + correction (preliminary) 10 0 Al Fe E-E 0 /ev E-E 0 /ev E-E 0 /ev MgO R x E-E 0 /ev R x Si k < 2π R x 28 28

29 Supercells How to deal with non-crystals? Make super-crystals out of them! molecules surfaces interfaces point defects amorphous (disordered) keep 0.6 nm vacuum between hydrogen atoms, - more for lone-pairs, dangling bonds etc, electropositive atoms keep 1 nm between defects in a solid - more for very dispersive bands choose isotropic cells (z.b. face-centered cubic) - maximize distance/volume, i.e. accuracy/cost 29 29

30 Frozen-core approximation 30 30

31 Frozen-core approximation Core electrons do not take part in chemical binding Rounding errors: core energy are 10 7 times larger than chemical accuracy (2 32 =10 9 ) Most electrons are core electrons are savings possible? 31 31

32 Frozen-core approximation Import atomic core wave functions as basisfunctions for the crystal n c ψ fc n = m=1 ψ at mu m,n One-particle expectation values (kinetic energy, density, etc.) are invariant under a unitary transformation of core states  fc = = n c n=1 n c m,o=1 ψ fc n  ψfc n = ψ at m  ψat o n c n c n=1 m,o n c n U o,n U n,m δ m,o U m,nψ at m  ψat o U o,n = n c n=1 ψ at n  ψat n =  at 32 32

33 Frozen-core approximation Frozen-core wave functions are not identical to atomic wave functions! errors in the total energy are second-order in the deviation from the frozen-core wave function quality of the F-C approximation depends on the exact solution of the atomic problem 33 33

34 Spins 34 34

35 Spin orbitals: Spins an electron is represented by two wave functions, one for spin and one for spin spin-density is a three-dimensional vector Obtain non-collinear spin density as 3-d vector field in 3-d s(r) = 2 ψ (r) ψ (r) ψ (r)ψ (r)+ψ (r)ψ (r) iψ (r)ψ (r)+iψ (r)ψ (r) ψ (r)ψ (r) ψ (r)ψ (r) 35 35

36 Spins Non-collinear: use two-component wave functions most general formulation spin-polarized, collinear (local spin density approximation, LSDA) one of the two components are set to zero electrons are in eigenstates of sz spin density points along z-direction non-spin polarized all electrons are paired zero spin density 36 36

37 Done 37 37