Projector augmented wave Implementation

Transcription

1 Projector augmented wave Implementation Peter. E. Blöchl Institute for Theoretical Physics Clausthal University of Technology, Germany 1

2 = Projector augmented wave all-electron method no shape approximations Year method Blöchl cumulated citations ab-initio molecular dynamics consistent theoretical framework 2

3 PAW: Historical Background All-electron methods Scattered waves (APW,KKR) Energy independent basissets (LMTO,LAPW) Pseudopotential methods empirical pseudopotentials first-principles pseudopotentials ab-initio molecular dynamics (Car-Parrinello method) energy- and potential-independent wave functions (Projector augmented wave method) Used in main packages: GPAW,CP-PAW, VASP, ABINIT, NWCHEM, PWPAW, ESPRESSO,

4 Why going beyond LAPW? Car-Parrinello method requires a unique total energy functional of the basis functions E[{c n }, { R j }] with ψ n = α χ α c α,n Linear methods use potential-dependent basis functions Freezing partial waves in conventional augmented wave methods is inaccurate PAW overcomes the problem by exploiting additive augmentation. 4

5 PAW augmentation ψ = all-electron pseudo α φ α p α ψ ψ + ψ 1 1-center, all-el. 1-center, pseudo ψ 1 α φ α p α ψ = + - = - + 5

6 PAW-transformation theory PAW is about a transformation between two representations of the wave functions ( watch the tilde!!! ) Total energy in terms of true or auxiliary wave functions Kohn-Sham equations: Expectation values: all-electron ψ true = ˆT pseudo ψ auxiliary E = E[ ψ n ]=E[ ˆT ψ n ]=E [ ψ n ] Ĥ n ψ n =0 ˆT Ĥ ˆT ˆ H ˆT ˆT n ψ n =0 ˆÕ  = ψn  ψ n = ψ n ˆT  ˆT ψ n ˆÃ 6

7 PAW transformation theory Exploit atomic behavior... use sum of local corrections all-electron ψ true = ˆT pseudo ψ auxiliary ˆT = ˆ1+ R Ŝ R ψ = ˆT ψ = ψ + R Ŝ R ψ Ŝ R local Ŝ R ψ = ψ ψ pairs of all-electron partial waves and pseudo partial waves φ α = ˆT φ α ŜR φ α = φ α φ α Use projector operator ˆ1 = α obtain expression for Ŝ R = ŜRˆ1 = α φ α p α with p α φ β = δ α,β Bi-orthogonality Ŝ R φ α p α = α φ α φ α p α 7

8 PAW transformation theory ˆT = ˆ1+ R Ŝ R Ŝ R = α φ α φ α p α all-electron ψ true = ˆT ψ = ψ + α pseudo ψ auxiliary φ α φ α p α ψ ψ = all-electron pseudo α φ α p α ψ ψ + ψ 1 1-center, all-el. 1-center, pseudo ψ 1 α φ α p α ψ 8

9 Partial waves all-electron partial waves φ α integrate Schrödinger equation outward have the correct nodal structure 2 Fe pseudo partial waves smooth inside identical to ae partial waves outside n-ncore nodes φ α usually constructed by adjusting an dependent potential r [a 0 ] s d p 9

10 Projector functions projector functions have the angular momentum character of their partial waves are localized within the augmentation region have (normally) an increasing number of nodes 10

11 PAW augmentation + = ψ = all-electron pseudo α φ α p α ψ ψ + ψ 1 1-center, all-el. 1-center, pseudo ψ 1 α φ α p α ψ =

12 Wave functions: Expectation values I ψ = all-electron pseudo ψ + α φ α p α ψ ψ 1 1-center, all-el. 1-center, pseudo ψ 1 α φ α p α ψ Expectation values: ψ Â ψ all-electron = ψ Â ψ pseudo + ψ 1 Â ψ1 1-center, all-el. ψ 1 Â ψ 1 1-center, pseudo ψ Â ψ = + ψ Â ψ+ ψ Â ψ1 ψ Â ψ 1 +ψ 1 Â ψ+ψ 1 Â ψ1 ψ 1 Â ψ 1 ψ 1 Â ψ ψ 1 Â ψ1 ψ 1 Â ψ 1 + ψ 1 Â ψ 1 + ψ 1 Â ψ 1 = ψ Â ψ + ψ 1 Â ψ1 ψ 1 Â ψ 1 + ψ ψ 1 Â ψ 1 ψ 1 + ψ 1 ψ 1 Â ψ ψ 1 0insideΩ R 0outsideΩ R 0outsideΩ R 0insideΩ R 12

13 Expectation values II  = n valence f n ψ n  ψ n + n core φ c n  φ c n pseudo + D α,β φ α  φ β + φ c n  φc n α,β n core 1-center, all-electron D α,β φ α  φ β φ c n  φ c n α,β n core 1-center, pseudo 1-center density matrix D α,β = p β ψ n f n ψ n p α n valence core states are included electron density: n(r) = ñ v (r)+ñ c (r) + n 1 v(r)+n c (r) ñ 1 v(r)+ñ c (r) ñ(r) n 1 (r) ñ 1 (r) 13

14 Total energy Also, the total energy is written as sum of an extended plane wave part two one-center expansions for each site Ẽ = Ψ n 2 2 2m Ψ Compensation density: n + d 3 r v(r)ñ(r) n e + 1 d 3 r d 3 r e2 [ñ(r)+ Z(r)][ñ(r )+ Z(r )] 2 4π 0 r r + E xc [ñ(r)] E 1 R = i,j R ẼR 1 = i,j R N c,r D i,j φ j 2 2 φ i + φ c 2m n 2 2 φ c e 2m n e n R d 3 r d 3 r e2 [n 1 (r)+z(r)][n 1 (r )+Z(r )] 4π 0 r r D i,j φ j 2 2 2m φ i + d 3 r v(r)ñ 1 (r) e d 3 r E [{ ψ n,r j }]=Ẽ + R + E xc [n 1 (r)] ER 1 Ẽ1 R smart zero d 3 r e2 [ñ 1 (r)+ Z(r)][ñ 1 (r )+ Z(r )] 4π 0 r r + E xc [ñ 1 (r)] contains pseudized core (non-linear core correction) 14

15 1 2 d 3 r Ewald-like trick compensation density depends on the density d 3 r e2 [ñ(r)+ Z(r)][ñ(r )+ Z(r )] 4π 0 r r Z(r) sum of Gaussians with angular momenta localized in augmentation regions, thus poor plane wave convergence introduce smooth, extended compensation density [ñ + Z] =[ñ + S]+[ Z S] d 3 rr Y,m (r) [ñ + Z] [n + Z] =0 Obtain a smooth Hartree energy + an external potential + and a term calculated analytical with Gaussians S(r) 15

16 Ewald-like trick 1 2 d 3 r d 3 r e2 [ñ(r)+ Z(r)][ñ(r )+ Z(r )] 4π 0 r r [ñ + Z][ñ + Z] = (ñ + S)+( Z S) (ñ + S)+( Z S) = [ñ + S][ñ + S] + 2[ñ + S][ Z S]+[ Z S][ Z S] = [ñ + S][ñ + S]+ 2[ñ][ Z S]+2[ S][ Z] 2[ S][ S] + [ Z][ Z] 2[ S][ Z] + [ S][ S] = [ñ + S][ñ + S] convergent in G-space + 2[ñ][ Z S] inherits plane wave cutoff + [ Z][ Z] [ S][ S] analytical with Gaussians 16

17 Approximations Frozen core approximation control by adding semicore states to valence states for relaxed core see [Marsmann, Kresse, JCP125, (2006)] truncated plane wave expansion 30 Ry normal (50Ry accurate) truncated partial-wave expansion 1-2 per angular momentum All terms are in their natural numerical representation extended, smooth terms are expressed by plane waves local terms are expressed on radial grids and multiplied with spherical harmonics. Full charge density is calculated no transferability problems! 17

18 Accuracy VASP: [Paier,Hirschl, Marsman, Kresse JCP122, (1005)] GPAW: [Carsten Rostgaard] PBE atomization energies relative to Gaussian: 18

19 Potentials and Forces potentials and forces are obtained strictly as analytic derivatives of the total energy no ambiguity no further approximations effective Schrödinger equation has the form of a separable pseudopotential, that adjusts to the instantaneous electronic structure ˆp 2 2m +ṽ(ˆr) n + α,β but implies p α (dh α,β n do α,β ) p β ψ n =0 ˆp 2 2m + v(ˆr) n ψ n =0 19

20 What is needed: all-electron partial waves pseudo partial waves projector functions core density Nice to have: Atomic input (Setup) p α one-center kinetic energy one-center overlap φ α φα dt α,β = φ α ˆp 2 Usually kept in a separate database. Can also be constructed on the fly. n c (r) = f n φ n (r) 2 n core 2m φ β φ α ˆp 2 do α,β = φ α φ β φ α φ β 2m φ β 20

21 all-electron atomic calculation pseudize potential pseudize partial waves One-center terms Construct Setups (I) dt α,β = φ α ˆp 2 v(r) ṽ(r) φ α (r) φ α (r) pre-projector functions from closure relation p α = 2 p α = β 2m φ β φ α ˆp 2 2m 2 +ṽ + α,β 2 2m 2 +ṽ α dh α,β = φ α ˆp 2 p α (dh α,β do α,β ) p β φ α =0 φ α final projectors from biorthogonality condition p φ 1 p β α,β 2m +ˆv φ β φ α ˆp 2 2m + ˆṽ φ β 2m φ β and do α,β = φ α φ β φ α φ β p α φ β = δ α,β 21

22 Construct Setups II choose Gaussian decay for compensation density unscreening: obtain ṽ(r) = v(r)+ v(r) from d 3 r e2 [ñ(r )+ Z(r )] 4π 0 r r Z(r) + µ xc ([ñ], r) Ẽ = Ψ n 2 2 2m Ψ n + d 3 r v(r)ñ(r) n e + 1 d 3 r d 3 r e2 [ñ(r)+ Z(r)][ñ(r )+ Z(r )] 2 4π 0 r r E 1 R = D i,j φ j 2 2m e 2 φ i + N c,r φ c n 2 2m e 2 φ c n + E xc [ñ(r)] 22

23 Comparison Pseudopotential approximation: linearize 1-center energies about atomic density impose norm-conservation PAW acts like a pseudopotential on the fly LAPW (linear augmented plane waves): projector functions have the form of a δ-function and its derivative at the surface of a muffin tin sphere but most of the PAW numerics does not work 23

24 PAW Implementations GPAW (wiki.fysik.dtu.dk/gpaw) CP-PAW (orion.pt.tu-clausthal.de/paw/) PWPAW (pwpaw.wfu) VASP (cms.mpi.univie.ac.at/vasp) Abinit ( Socorro (dft.sandia.gov/socorro Quantum Espresso ( NWChem ( 24

25 References Projector augmented wave method, Blöchl, Phys. Rev. B Electronic structure methods:augmented Waves, Pseudopotentials and the Projector Augmented Wave method, Blöchl, Kästner, Först, chapter in Handbook of Materials Modelling, Vol 1, Sidney Yip (Ed.) Springer 2005 and arxiv:cond-mat/

26 Take-home message: PAW is an all-electron method PAW is not a pseudopotential method The pseudopotential method is an approximation of PAW 26