The Projector Augmented Wave method

Transcription

1 The Projector Augmented Wave method Advantages of PAW. The theory. Approximations. Convergence. 1

2 The PAW method is... What is PAW? A technique for doing DFT calculations efficiently and accurately. An all-electron method with easy-to-control approximations. An elegant theory. A method that works with smooth pseudo wave-functions that can be expanded in a few plane waves (or expressed on coarse grids). Ultra-soft pseudopotentials done right! 2

3 Literature The PAW method was invented by Peter Blöchl in 1994: Projector augmented-wave method, P. E. Blöchl, Phys. Rev. B 50, (1994) Projector augmented wave method: ab initio molecular dynamics with full wave functions, P. E. Blöchl, C. J. Först and J. Schimpl, Bull. Mater. Sci, 26, 33 (2003) Real-space grid implementation of the projector augmented wave method, J. J. Mortensen, L. B. Hansen, and K. W. Jacobsen, Phys. Rev. B, (2005) 3

4 Advantages of PAW No need to deal with inert core electrons. Valence pseudo wave functions are smooth and without nodes inside the augmentation spheres. Access to full all-electron wave functions and density. Useful for orbital-dependent XC-functionals. 4

5 Platinum atom s 2s 3s 4s 5s 6s 2p 3p 4p 5p 3d 4d 5d r (Bohr) 5

6 Augmentation Spheres One cutoff radius for each type of atom. Spheres should not overlap: 6

7 Electron density n = ñ plane waves/coarse grid + a na logarithmic radial grids a ña logarithmic radial grids 7

8 The PAW transformation ψ n ( r) = ˆτ ψ n ( r) ˆτ = 1 + a ( φ a i φ a i ) p a i, i where p a i ( r) = 0 and φ a i ( r) = φa i ( r) for r > ra c, and p a i φ a j = δ ij. p a i = p a nlm( r R a ) = p a nl(r)y lm ( r R a ) ˆτ φ a i = φ a i 8

9 Completeness relations For r R a < rc a we must have: φ a i p a i = 1 i From this follows that inside the augmentation spheres, ψ n and ψ n can be expanded in partial waves and pseudo partial waves respectively: ψ n = i φ a i p a i ψ n. ψ n = i φ a i p a i ψ n, In the interstitial region, we have ψ n = ψ n. 9

10 Electron density (again) n( r) = a n a c ( r R a ) + n f n ψ n ( r) 2 = a n a c + n f n ψn + ai p a i ψ n (φ a i φ a i ) 2 = n a c + f n ψ n 2 a n + f n p a i ψ n (φ a i φ a i ) ψ n p a j (φ a j φ a j ) n aij +2Re f n p a i ψ n φ a i ψ n p a j (φ a j φ a j ) n a i j }{{} ψ n 10

11 Electron density (continued) n = n f n ψ n 2 + aij D a ij(φ a i φ a j φ a i φ a j ) + a n a c, where we have defined atomic density matrices as: D a ij = n p a i ψ n f n ψ n p a j 11

12 Electron density (continued) With these definitions: n a = ij D a ijφ a i φ a j + n a c, ñ a = ij D a ij φ a i φ a j + ñ a c, ñ = n f n ψ n 2 + a ñ a c, we get a very simple expression for the all-electron density: n = ñ + a (n a ñ a ) 12

13 Compensation charges Let Z a ( r) be the nuclear charge for atom a. The Coulomb energy is: ( n( r) + ) ( E C = d rd r a Za ( r R a ) n( r ) + ) a Za ( r R a ) r r = (n + a Z a ) 2 = (ñ + a [n a ñ a + Z a ]) 2 We add and subtract compensation charges localized inside the augmentation spheres: E C = (ñ + a Z a + a [n a ñ a + Z a Z a ]) 2 13

14 Compensation charges (continued) The compensation charges are constructed like this: Z a ( r) = lm Q a lm g a lm( r), where g a lm ( r) = 0 for r > ra c : g a lm( r) = C l r l exp( α a r 2 )Y lm (ˆr), The Q a lm s are chosen such that na ñ a + Z a Z a has no multipole moments: d rr l Y lm (ˆr)(n a ñ a + Z a Z a ) = 0 14

15 Compensation charges (continued) Using ρ = ñ + a Z a, ρ a = ñ a + Z a and ρ a = n a + Z a, we get: E C = (ñ + a Z a + a [n a ñ a + Z a Z a ]) 2 = ( ρ + a [ρ a ρ a ]) 2 = ρ ρ a (ρ a ρ a ) + ab (ρ a ρ a )(ρ b ρ b ) Since ρ a ρ a has no multipole moments, we get: E C = ρ a ρ a (ρ a ρ a ) + a (ρ a ρ a ) 2 = ρ 2 + a (ρ a ) 2 a ( ρ a ) 2 (1) 15

16 Finally we have E C = ẼC + a (Ea C Ẽa C ), where ẼC has contributions from all of space: ( ñ( r) + Z Ẽ C = d rd r a a ( r R ) ( a ) ñ( r ) + Z a a ( r R ) a ) r r, and EC a Ẽa C is a correction from each augmentation sphere: ( ) EC a = d rd r (na ( r) + Z a ( r)) n a ( r ) + Z a ( r ) r r, ( ñ a ( r) + Z ) ( ) a ( r) ñ a ( r ) + Z a ( r ) ẼC a = d rd r r r 16

17 Frozen core states. Approximations Truncated multipole expansion of compensation charges. Finite number of projectors, partial waves and pseudo partial waves: Hydrogen: 2 s-type, 1 p-type. Oxygen: 2 s-type, 2 p-type, 1 d-type. Copper: 2 s-type, 2 p-type, 2 d-type. Overlapping augmentation spheres: 17

18 Kinetic energy E kin = Ẽkin + a (E a kin Ẽa kin), where Ẽ kin = 1 2 f n d r ψ n 2 ψn n core E a kin = 1 2 D a ij d rφ a i 2 φ a j 1 2 d rφ a c 2 φ a c ij Ẽ a kin = 1 2 c D a ij d r φ a i 2 φa j ij 18

19 Exchange-correlation energy E xc = Ẽxc + a (E a xc Ẽa xc), where Ẽ xc = d rñɛ xc [ñ] Exc a = d rn a ɛ xc [n a ] Ẽxc a = d rñ a ɛ xc [ñ a ] 19

20 Hamiltonian E = Ẽ + E a (D a δe ij), a δ ψ = f n Ĥ ψ n n Ĥ = ṽ + p a i Hij p a a j, a ij where ṽ = δẽ/δñ = ṽ H + ṽ xc and H a ij = Ea D a ij + lm Q a lm D a ij d rṽ H g a lm The PAW method is a generalized Kleinman-Bylander nonlocal pseudopotential that adapts to the current environment! 20

21 Orthogonality Keep the wave functions orthogonal: δ nm = ψ n ψ m = ψ n Ô ψ m, where and Ô = 1 + a O a ij = p a i Oij p a a j ij d r(φ a i φ a j φ a i φ a j ) 21

22 PBE atomization energy of a nitrogen molecule from ASE import Atom, ListOfAtoms from gridpaw import Calculator a = 8. 0 # size of unit cell h = # grid spacing N = ListOfAtoms ([ Atom ( N, (0, 0, 0), magmom =3)], cell =(a, a, a), periodic =1) calc = Calculator ( nbands =4, xc= PBE, h=h) N. SetCalculator ( calc ) e1 = N. GetPotentialEnergy() d = 1. 1 # bond length N2 = ListOfAtoms ([ Atom ( N, [0, 0, 0]), Atom ( N, [0, 0, 1.1]) ], cell =(a, a, a), periodic =1) calc = Calculator ( nbands =5, xc= PBE, h=h) N2. SetCalculator ( calc ) e2 = N2. GetPotentialEnergy() print 2 * e1 - e2, ev 22

23 Convergence spd E (ev) l max E (ev) Dacapo Blaha et al. Experiment E (ev)