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VASP: Basics (DFT, PW, PAW, … ) VASP: Basics (DFT, PW, PAW, … ) University of Vienna, Faculty of Physics and Center for Computational Materials Science, Vienna, Austria Outline ● Density functional theory ● Translational invariance and periodic boundary conditions ● Plane wave basis set ● The Projector-Augmented-Wave method ● Electronic minimization The Many-Body Schrödinger equation Hˆ (r1,...,rN )=E (r1,...,rN ) 1 1 ∆i + V (ri)+ (r1,...,rN )=E (r1,...,rN ) 0−2 ri rj 1 i i i=j X X X6 | − | @ A For instance, many-body WF storage demands are prohibitive: N (r1,...,rN ) (#grid points) 5 electrons on a 10×10×10 grid ~ 10 PetaBytes ! A solution: map onto “one-electron” theory: (r ,...,r ) (r), (r),..., (r) 1 N ! { 1 2 N } Hohenberg-Kohn-Sham DFT Map onto “one-electron” theory: N (r ,...,r ) (r), (r),..., (r) (r ,...,r )= (r ) 1 N ! { 1 2 N } 1 N i i i Y Total energy is a functional of the density: E[⇢]=T [ [⇢] ]+E [⇢]+E [⇢] + E [⇢]+U[Z] s { i } H xc Z The density is computed using the one-electron orbitals: N ⇢(r)= (r) 2 | i | i X The one-electron orbitals are the solutions of the Kohn-Sham equation: 1 ∆ + V (r)+V [⇢](r)+V [⇢](r) (r)=✏ (r) −2 Z H xc i i i ⇣ ⌘ BUT: Exc[⇢]=??? Vxc[⇢](r)=??? Exchange-Correlation Exc[⇢]=??? Vxc[⇢](r)=??? • Exchange-Correlation functionals are modeled on the uniform-electron-gas (UEG): The correlation energy (and potential) has been calculated by means of Monte- Carlo methods for a wide range of densities, and has been parametrized to yield a density functional. • LDA: we simply pretend that an inhomogeneous electronic density locally behaves like a homogeneous electron gas. • Many, many, many different functionals available: LDA, GGA, meta-GGA, van-der-Waals functionals, etc etc An N-electron system: N = O(1023) Hohenberg-Kohn-Sham DFT takes us a long way: (r ,...,r ) (r), (r),..., (r) 1 N ! { 1 2 N } (#grid points)N N (#grid points) ⇥ Nice for atoms and molecules, but in a realistic piece of solid state material N= O(1023)! Translational invariance: Periodic Boundary Conditions Translational invariance implies: ikR nk(r + R)= nk(r)e and ikr nk(r)=unk(r)e unk(r + R)=unk(r) All states are labeled by Bloch vector k and the band index n: • The Bloch vector k is usually constrained to lie within the first Brillouin zone of the reciprocal space lattice. • The band indeX n is of the order if the number of electrons per unit cell. Reciprocal space & the first Brillouin zone z L 4pi/L z z b b b1 2 1 b2 a2 y y a1 x x a3 y x b3 b3 A B C 2⇡ 2⇡ 2⇡ b = a a b = a a b = a a 1 ⌦ 2 ⇥ 3 2 ⌦ 3 ⇥ 1 3 ⌦ 1 ⇥ 2 ⌦ = a a a a b =2⇡ 1 · 2 ⇥ 3 i · j ij Sampling the 1st BZ The evaluation of many key quantities involves an integral over the 1st BZ. For instance the charge density: 1 ⇢(r)= f (r) 2dk ⌦ nk| nk | BZ n BZ X Z We eXploit the fact that the orbitals at Bloch vectors k that are close together are almost identical and approXimate the integral over the 1st BZ by a weighted sum over a discrete set of k-points: ⇢(r)= w f (r) 2dk, k nk| nk | Xnk 23 Fazit: the intractable task of determining with N=10 (r1,...,rN ) , has been st reduced to calculating at a discrete set of nk(r) k-points in the 1 BZ, for a number of bands that is of the order if the number of electrons in the unit cell. Idea: equally spaced mesh in the 1st Brillouin zone Example: a quadratic 2D lattice r • q1=q2=4, i.e., 16 points in total b • Only 3 symmetry inequivalent points: 44 k1 1 1 1 k2 4 k1 =( , ) !1 = k ⇥ 8 8 ) 4 3 r 3 3 1 b 4 k =( , ) ! = 0 ⇥ 2 8 8 ) 2 4 3 1 1 IBZ 8 k =( , ) ! = ⇥ 3 8 8 ) 3 2 BZ 1 1 1 1 F (k)dk F (k1)+ F (k2)+ F (k3) ⌦BZ ) 4 4 2 BZZ Algorithm: • Calculate equally spaced mesh. • Shift the mesh if desired. • Apply all symmetry operations of the Bravais lattice to all k-points. • Extract the irreducible k-points (IBZ). • Calculate the proper weighting. Common meshes: Two different choices for the center of the mesh. • Centered on Γ • Centered around Γ (can break the symmetry!) before after shifted to symmetrization r r • In certain cell geometries (e.g. heXagonal cells) even meshes break the symmetry. • Symmetrization results in non-uniform distributions of k-points. • Γ-point centered meshes preserve the symmetry. AB C The total energy E[⇢, R,Z ]=T [ [⇢] ]+E [⇢, R,Z ]+E [⇢]+U( R,Z ) { } s { nk } H { } xc { } The kinetic energy 1 T [ [⇢] ]= w f ∆ s { nk } k nkh nk| − 2 | nki Xnk The Hartree energy 1 ⇢eZ (r)⇢eZ (r0) E [⇢, R,Z ]= dr0dr H { } 2 r r ZZ | − 0| where 2 ⇢ (r)=⇢(r)+ Z δ(r R ) ⇢(r)= wkfnk nk(r) dk eZ i − i | | i nk X X The Kohn-Sham equations 1 ∆ + V [⇢ ](r)+V [⇢](r) (r)=✏ (r) −2 H eZ xc nk nk nk ⇣ ⌘ The Hartree potential ⇢eZ (r0) V [⇢ ](r)= dr0 H eZ r r Z | − 0| A plane wave basis set ikr nk(r)=unk(r)e unk(r + R)=unk(r) All cell-periodic functions are eXpanded in plane waves (Fourier analysis): 1 iGr 1 i(G+k)r unk(r)= CGnke (r)= C e ⌦1/2 nk 1/2 Gnk G ⌦ X XG iGr iGr ⇢(r)= ⇢Ge V (r)= VGe XG XG The basis set includes all plane waves for which 1 G + k 2 <E 2| | cuto↵ Transformation by means of FFT between “real” space and “reciprocal” space: iGr FFT 1 iGr C = C e C = C e− rnk Gnk ! Gnk N rnk FFT r XG X Why use plane waves? • Historical reason: Many elements eXhibit a band-structure that can be interpreted in a free electron picture (metallic s and p elements). Pseudopotential theory was initially developed to cope with these elements (pseudopotential perturbation theory). • Practical reason: The total energy eXpressions and the Hamiltonian are easy to implement. • Computational reason: The action of the Hamiltonian on the orbitals can be efficiently evaluated using FFTs. FFT real space reciprocal space G cut τ 2 b2 01 23N−1 0 5 −4012345−3 −2 −1 τ 1 b1 −N/2+1 N/2 x = n / N τ g = n 2 π / τ 1 1 1 1 1 1 iGr FFT 1 iGr C = C e C = C e− rnk Gnk ! Gnk N rnk FFT r XG X FFT real space reciprocal space G cut τ 2 b2 01 23N−1 0 5 −4012345−3 −2 −1 τ 1 b1 −N/2+1 N/2 x = n / N τ g = n 2 π / τ 1 1 1 1 1 1 iGr FFT 1 iGr C = C e C = C e− rnk Gnk ! Gnk N rnk FFT r XG X iGr FFT 1 iGr C = C e C = C e− rnk Gnk ! Gnk N rnk FFT r XG X FFT real space reciprocal space G cut τ 2 b2 01 23N−1 0 5 −4012345−3 −2 −1 τ 1 b1 −N/2+1 N/2 x = n / N τ g = n 2 π / τ 1 1 1 1 1 1 iGr FFT 1 iGr C = C e C = C e− rnk Gnk ! Gnk N rnk FFT r XG X The charge density Gcut FFT ψ ψ ψr G 2 G r cut FFT ρ G ρ r The action of the Hamiltonian The action H | nki 1 ∆ + V (r) (r) −2 nk ✓ ◆ Using the convention 1 r G + k = ei(G+k)r G + k = C h | i ⌦1/2 ! h | nki Gnk • Kinetic energy: 1 1 2 G + k ∆ = G + k C NNPLW h | − 2 | nki 2| | Gnk • Local potential: V = VH[⇢]+Vxc[⇢]+Vext • Exchange-correlation: easily obtained in real space Vxc,r = Vxc[⇢r] • FFT to reciprocal space V V { xc,r} ! { xc,G} 4⇡ • Hartree potential: solve Poisson eq. in reciprocal space VH,G = ⇢G G 2 • Add all contributions VG = VH,G + Vxc,G + Vext,G | | • FFT to real space V V { G} ! { r} The action 1 iGr G + k V = V C e− N log N h | | nki N r rnk FFT FFT FFT r X The action of the local potential 2Gcut 4π e 2 FFT G 2 V V ρ G G r ψr 3G Gcut cut add FFT R R (residual vector) ψG G r The Projector-Augmented-Wave method The number of plane waves needed to describe • tightly bound (spatially strongly localized) states, • and rapid oscillations (nodal features) of the orbitals near the nucleus exceeds any practical limit, except maybe for Li and H. The common solution: • Introduce the frozen core approXimation: Core electrons are pre-calculated in an atomic environment and kept frozen in the course of the remaining calculations. • Use of pseudopotentials instead of eXact potentials: • Norm-conserving pseudopotentials • Ultra-soft pseudopotentials • The Projector-Augmented-Wave (PAW) method [P. E. Blöchl, Phys. Rev. B 50, 17953 (1994)] Pseudopotentials: the general idea ~ Ψ Ψ rc ~ V V Pseudopotentials: cont. 2.5 2.0 1.5 s : E= -0.576 R c=1.9 1.0 wave-function 0.5 Al Al p : E= -0.205 R c=1.9 0.0 0 1 2 3 4 R (a.u.) Exact potential (interstitial) Pseudopotential Al effective Al atom PAW Al atom 3p 2p 3p 3s 1s 3s 2p 2p and 1s are nodeless nodal structure retained 2s 12 The PAW orbitals = + ( φ φ ) p | ni | ni | ii| ii h i| ni i X e e e e φ p φ p ˜ | ni | ni− | iih i| ni | ni | iih i| ni i i X X+ φ p e e e e e e e | ieih i| ni i X e e PW PS-LCAO AE-LCAO = - + AE pseudopseudo-onsite AE-onsite The PAW orbitals (cont.) = + ( φ φ ) p | ni | ni | ii| ii h i| ni i X e e e • n is the pseudo (PS) orbital, eXpressed in a plane wave basis sete | i • φei , , and are atomφ˜ p˜i -centered localized functions | i | ii | i • The all-electron partial waves are solutions to the radial scalar relativistic non-spinpolarized Schrödinger equation: 1 ( ∆ + v ) φ = ✏ φ −2 e↵ | ii i| ii • A pseudization procedure yields: φ φ v v p φ = δ | ii!| ii e↵ ! e↵ h i| ji ij e e e e The PAW orbitals (cont.) • The pseudo partial waves obey: 1 ∆ + v + p D p φ = ✏ 1+ p Q p φ − 2 e↵ | ii ijh j| | ki k | ii ijh j| | ki ij ij ⇣ X ⌘ ⇣ X ⌘ e e e e e e e • with the so-called PAW strength parameters and augmentation charges: 1 1 D = φ ∆ + v φ φ ∆ + v φ ij h i| − 2 e↵ | jih i| − 2 e↵ | ji e e e Q = φ φ φ φ ij h i| jih i| ji e e The all-electron and pseudo eigenvalue spectrum is identical! All-electron scattering properties are reproduced over a wide energy-range.
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