The Projector Augmented Wave method

The Projector AugmentedWave method• Advantages of PAW.• The theory.• Approximations.• Convergence.1

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 thatcan be expanded in a few plane waves (or expressed on coarsegrids).• Ultra-soft pseudopotentials done right!2

LiteratureThe PAW method was invented by Peter Blöchl in 1994:• ”Projector augmented-wave method”, P. E. Blöchl, Phys. Rev. B50, 17953 (1994)• ”Projector augmented wave method: ab initio moleculardynamics with full wave functions”, P. E. Blöchl, C. J. Först andJ. Schimpl, Bull. Mater. Sci, 26, 33 (2003)• ”Real-space grid implementation of the projector augmentedwave method”, J. J. Mortensen, L. B. Hansen, and K. W.Jacobsen, Phys. Rev. B, 71 035109 (2005)3

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

Platinum atom43211s2s3s4s5s6s2p3p4p5p3d4d5d0-1-20 1 2 3 4r (Bohr)5

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

Electron densityn= ñ plane waves/coarse grid+ ∑ a na logarithmic radial grids− ∑ a ñalogarithmic radial grids7

The PAW transformationψ n (⃗r) = ˆτ ˜ψ n (⃗r)ˆτ = 1 + ∑ a∑(|φ a i 〉 − | ˜φ a i 〉)〈˜p a i |,iwhere ˜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 i8

Completeness relationsFor |⃗r − ⃗R a | < rc a we must have:∑| ˜φ a i 〉〈˜p a i | = 1iFrom this follows that inside the augmentation spheres, ψ n and ˜ψ ncan be expanded in partial waves and pseudo partial wavesrespectively:| ˜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

Electron density (again)n(⃗r) = ∑ an a c (|⃗r − ⃗R a |) + ∑ nf n |ψ n (⃗r)| 2= ∑ an a c + ∑ nf n∣ ∣∣∣∣˜ψn + ∑ ai〈˜p a i | ˜ψ n 〉(φ a i − ˜φ a i )∣2= ∑ n a c + ∑ f n | ˜ψ n | 2a 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∑⎪⎬ai 〈 ˜ψ n |˜p a j 〉(φ a j − ˜φ a j )n a ij⎪⎭} {{ }˜ψ n10

Electron density (continued)n = ∑ nf n | ˜ψ n | 2 + ∑ aijD a ij(φ a i φ a j − ˜φ a i ˜φ a j ) + ∑ an 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

Electron density (continued)With these definitions:n a = ∑ ijD a ijφ a i φ a j + n a c ,ñ a = ∑ ijD a ij ˜φ a i ˜φ a j + ñ a c ,ñ = ∑ nf n | ˜ψ n | 2 + ∑ añ a c ,we get a very simple expression for the all-electron density:n = ñ + ∑ a(n a − ñ a )12

Compensation chargesLet Z a (⃗r) be the nuclear charge for atom a. The Coulomb energy is:(∫ n(⃗r) + ∑ ) (E C = d⃗rd⃗r ′ a Za (⃗r − ⃗R a ) n(⃗r ′ ) + ∑ )a Za (⃗r ′ − ⃗R a )|⃗r − ⃗r ′ |= (n + ∑ aZ a ) 2= (ñ + ∑ a[n a − ñ a + Z a ]) 2We add and subtract compensation charges localized inside theaugmentation spheres:E C = (ñ + ∑ a˜Z a + ∑ a[n a − ñ a + Z a − ˜Z a ]) 213

Compensation charges (continued)The compensation charges are constructed like this:˜Z a (⃗r) = ∑ lmQ 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 multipolemoments: ∫d⃗rr l Y lm (ˆr)(n a − ñ a + Z a − ˜Z a ) = 014

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 2 + 2˜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 2 + 2 ∑ 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

Finally ...... we have E C = ẼC + ∑ a (Ea C − Ẽa C ), where ẼC has contributionsfrom all of space:(∫ ñ(⃗r) + ∑ ˜ZẼ C = d⃗rd⃗r ′ 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 = d⃗rd⃗r ′ (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 = d⃗rd⃗r ′ |⃗r − ⃗r ′ |16

• Frozen core states.Approximations• Truncated multipole expansion of compensation charges.• Finite number of projectors, partial waves and pseudo partialwaves:– 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

Kinetic energyE kin = Ẽkin + ∑ a(E a kin − Ẽa kin),whereẼ kin = − 1 2∑∫f nd⃗r ˜ψ ∗ n∇ 2 ˜ψn∑∫n∑core∫E a kin = − 1 2D a ijd⃗rφ a i ∇ 2 φ a j − 1 2d⃗rφ a c∇ 2 φ a cijẼ a kin = − 1 2∑∫cD a ijd⃗r ˜φ a i ∇ 2 ˜φa jij18

Exchange-correlation energyE xc = Ẽxc + ∑ a(E a xc − Ẽa xc),where∫Ẽ xc = d⃗rñɛ xc [ñ]∫Exc a = d⃗rn a ɛ xc [n a ]∫Ẽxc a = d⃗rñ a ɛ xc [ñ a ]19

HamiltonianE = Ẽ + ∑ ∆E a (D a δEij),aδ ˜ψ= f n Ĥ ˜ψ nn∗Ĥ = − 1 2 ∇2 + ṽ + ∑ ∑|˜p a i 〉∆Hij〈˜p a a j |,aijwhere ṽ = δẼ/δñ = ṽ H + ṽ xc and∆H a ij = ∆Ea∂D a ij+ ∑ lm∂Q a lm∂D a ij∫d⃗rṽ H˜g a lmThe PAW method is a generalized Kleinman-Bylander nonlocalpseudopotential that adapts to the current environment!20

OrthogonalityKeep the wave functions orthogonal:δ nm = 〈ψ n |ψ m 〉 = 〈 ˜ψ n |Ô| ˜ψ m 〉,whereandÔ = 1 + ∑ a∆O a ij =∫∑|˜p a i 〉∆Oij〈˜p a a j |ijd⃗r(φ a i φ a j − ˜φ a i ˜φ a j )21

PBE atomization energy of anitrogen moleculefrom ASE import Atom , ListOfAtomsfrom gridpaw import Calculatora = 8.0 # size of unit cellh = 0.18 # grid spacingN = 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 lengthN2 = 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

Convergencespd 11 22 221 222 321 231∆E (eV) 10.016 10.141 10.520 10.519 10.519 10.514l max 0 1 2∆E (eV) 10.520 10.574 10.560Dacapo Blaha et al. Experiment∆E (eV) 9.611 10.546 9.90923

The Projector Augmented Wave method

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