Plane waves, pseudopotentials and PAW. X. Gonze Université catholique de Louvain, Louvain-la-neuve, Belgium

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1 Plane waves, pseudopotentials and PAW X. Gonze Université catholique de Louvain, Louvain-la-neuve, Belgium 1

2 Basic equations in DFT Solve self-consistently the Kohn-Sham equation H ψ n = ε n ψ n!!! ρ(r ) = ψ *n (r )ψ n (r ) occ n ˆ V +V [ρ] H =T+ Hxc or minimize ψ n (r) H ρ(r) δ mn = ψ m ψ n for m, n occupied set occ ˆ V ψ +E [ρ] Eel {ψ } = ψ n T+ n Hxc n with! Z V (r) = -!!κ a aκ r-r κ How to represent these quantities? Let s try plane waves

3 Prerequisites of plane waves Plane waves e ikr : simple but infinite spatial extent Cannot use a finite set of planewaves for finite systems! Need periodic boundary conditions. Primitive vectors R j, primitive cell volume Ω0 R3 R1 R OK for crystalline solids But : finite systems, surfaces, defects, polymers, nanosystems...? 3

4 The supercell technique Molecule, cluster Surface : treatment of a slab Interface Point defect in a bulk solid The supercell must be sufficiently big : convergence study 4

5 Plane wave basis set : the maths ψ k (r) = ( NΩ0 ) -1/ i(k +G)r u (G) e k G 5

6 Periodic system : wavevectors For a periodic Hamiltonian : wavefunctions characterized by a wavevector k (crystal momentum) in Brillouin Zone ψ m,k (r+r j ) = e Bloch s theorem ψ m,k (r) = ( NΩ0 ) -1/ ψ m,k (r) ik.r j e ik.r u m,k (r) u m,k (r+r j )=u m,k (r) u = periodic part of the Bloch wavefunction Normalization? Born-von Karman supercell ψ m,k (r+n jr j ) = ψ m,k (r) supercell vectors N jr j with N=N1N N 3 1 1= Ωo Ωo u*k (r) u k (r) dr 6

7 Planewave basis set Reciprocal lattice : set of G vectors such that e e igr has the periodicity of the real lattice u k (r) = u k (G) e igr ψ k (r) = ( NΩ0 ) -1/ igr j u k =1 (G) e i(k +G)r G G 1 -igr (Fourier transform) u k (G) = e u k (r) dr Ω Ωo o (k+g) Kinetic energy of a plane wave The coefficients u k (G) for the lowest eigenvectors ( k + G ) decrease fast with the kinetic energy Selection of plane waves determined by a cut-off energy Ecut (k + G) < E Plane wave sphere cut 7

8 Plane waves : the density and potential Fourier transform of a periodic function 1 f(g) = Ωo!r Poisson equation VH (r) = Ωo!r f(r) f(r) = e igr f(g) e-igr f(r) dr! G n (G) and VH (G) n(r') dr' VH r-r' r = -4π n(r) Relation between Fourier coefficients: G VH (G) = 4π n(g) VH (G) = 4π n(g) G For G =0 ( G=0 ) divergence of VH ( G=0 ) 1 n(g=0) = n(r) dr Average Ω Ωor or Compensation with ionic potential 8

9 Representation of the density Density associated with one eigenfunction : n nk (r) = u*nk (r) u nk (r) Computation of u*nk (r) u nk (r) * -igr -ig'r = u nk (G) e u nk (G') e G G' = u*nk (G) u nk (G') e i(g'-g)r GG' Non-zero coefficients for k+g sphere k+g' sphere k+g k+g' G'-G The sphere for n(g) has a doubled radius 9

10 From real space to reciprocal space n(r) = n(g) e igr G sphere() Use of the discrete Fourier transform {r i } {G} 1 -igri n(g) = n(r ) e i N ri {ri } Reciprocal lattice Fast Fourier Transform algorithm Real lattice: original cell FFT grid in real space used to evaluate the XC contribution occ ˆ V ψ +E [ρ] Eel {ψ } = ψ n T+ n Hxc n 10

11 Simplicity of PW requires psps The Fourier transform theory teaches us : -details in real space are described if their characteristic length is larger than the inverse of the largest wavevector norm (roughly speaking) -quality of a plane wave basis set can be systematically increased by increasing the cut-off energy Problem : huge number of PWs is required to describe localized features (core orbitals, oscillations of other orbitals close to the nucleus) Pseudopotentials (or, in general, «pseudization») to eliminate the undesirable small wavelength features (will be the subject of a later section) 11

12 Convergence wrt to kinetic energy cutoff Total energy (Ha) bulk silicon Cut-off energy (Ha) Courtesy of F. Bruneval 1

13 Number of planewaves Number of plane waves = function of the kinetic energy cut-off not continuous d N PW 4 0 Also, a (discontinuous) function of lattice parameter at fixed kinetic energy ( E cut )1 13

14 Discontinuities in energy and pressure => Energy (and pressure) also (discontinuous) functions of lattice parameter at fixed kinetic energy Energy Pressure Lattice parameter Lattice parameter 14

15 Removing discontinuities Kinetic energy Kinetic energy u(g) 0 u(g)=0 u(g) 0 u(g) weak u(g)=0 k+g k+g ecutsm 15

16 Representation : wrap-up Choice of a basis (e.g. Plane waves) Truncating of the basis -> finite basis ( k+g ) < E cut Sphere of plane waves Discontinuous increase of the number of plane waves? Smearing of u(g) => Progressive incorporation of new G vectors Representation of the density Sphere with a double radius in the reciprocal space Going from the real space to reciprocal space Discrete Fourier transform Grid of points + Fast Fourier Transform {r i } {G} 16

17 Pseudopotentials 17

18 Core and valence electrons (I) Core electrons occupy orbitals that are "the same" in the atomic environment or in the bonding environment "the same" depends on the accuracy of the calculation! Separation between core and valence orbitals : the density... N n(r) = ψ i* (r)ψ i (r) i = N core * ψ i (r)ψ i (r) + i core N val * ψ i (r)ψ i (r) = ncore (r) + nval (r) i val atom «Frozen core» for i core : ψ i = ψ i 18

19 Separation between core/valence Depends on the target accuracy of the calculation! (remark also valid for pseudopotentials, with similar cores) For some elements, core/valence partitioning is obvious, for some others, it is not. F atom : IP Ti atom : (1s) + 1keV (s) ( p) ev (1s) (s) ( p) (3s) (3p) (4s) (3d) small core (1s) (s) ( p) (3s) (3p) (4s) (3d) large core 6 6 IP Gd atom : ev 43.3eV small core with n=1,,3 shells, might include 4s, 4p, and 4d in the core. 4f partially filled 19

20 Energy : core and valence 1 1 n(r )n(r ) EKS {ψ i } = ψ i ψ i + Vext (r)n(r) dr + 1 dr1dr + Exc [ n ] r1 - r i EKS {ψ i } = N core i core N val + i val 1 1 n (r )n (r ) ψ i ψ i + Vext (r)ncore (r) dr + core 1 core dr1dr r1 - r 1 1 nval (r1 )nval (r ) ψ i ψ i + Vext (r)nval (r) dr + dr1dr r1 - r nval (r1 )ncore (r ) + dr1dr + Exc [ ncore + nval ] r1 - r 0

21 Valence electrons in screened potential The potential of the nuclei κ is screened by the core electrons ncore,κ (r1 ) Zκ Vion,κ (r) = + dr1 r Rκ r - r1 Z core,κ Z val,κ ncore,κ (r1 ) Vion,κ (r) = + + dr1 r Rκ r Rκ r - r1 The total energy becomes Z val,κ Z val,κ ' 1 E = Eval + Ecore,κ + (κ,κ ') Rκ Rκ ' κ κ κ ' with 1 ψ ψ + V (r) i i ion,κ nval (r) dr i val κ Non-linear XC 1 nval (r1 )nval (r ) core correction + dr1dr + Exc [ ncore + nval ] r1 - r Eval,KS {ψ i } = N val 1

22 Removing core electrons (I) From the previous construction : valence orbitals must still be orthogonal to core orbitals ( => oscillations, slope at the nucleus...) Pseudopotentials try to remove completely the core orbitals from the simulation Problem with the number of nodes This is a strong modification of the system... Pseudopotentials confine the strong changes within a «cut-off radius»

23 Removing core electrons (II) Going from 1 + v 1 + vps To ψ ψ i > = εi ψ i > ps,i > = ε ps,i ψps, i > Possible set of conditions (norm-conserving pseudopotentials) NCPP - Hamann D.R., Schlüter M., Chiang C, Phys.Rev.Lett. 43, 1494 (1979) ε i = εps,i ψi (r) = ψps,i(r) for r > rc r < rc ψi (r) dr = r < rc ψps,i(r) dr For the lowest angular momentum channels (s + p... d...f) Generalisation : ultra-soft pseudopotentials (USPP), projector-augmented plane waves (PAW) 3

24 Example of NC pseudopotential 3s Radial wave function of Si s 0.5 p Vps(r) [Ha] Amplitude All-electron wave function Pseudo-wave function Radial distance [a.u.] Z val r d Radial distance [a.u.] 4

25 Forms of pseudopotentials Must be a linear, hermitian operator General form : (V ψ ) (r) = V ps kernel ps (r, r')ψ (r')dr' Vpskernel (r, r') = Vloc (r)δ (r-r') + Vnloc (r, r') includes a "non-local" part + Spherically symmetric! Vloc (r) = V (r) Vnloc (r, r') = Yℓ*m (θ, ϕ )Vℓ (r, r ')Yℓm (θ ', ϕ ') ℓm Semi-local psp Vℓ (r, r ') = Vℓ (r)δ (r r ') see Bachelet, Hamann and Schlüter, Phys.Rev.B 6, 4199 (198) Separable pspvℓ (r, r ') = ξℓ* (r) fℓξℓ (r ') Kleinman L., Bylander D.M., Phys.Rev.Lett. 48, 145 (198) 5

26 Ultrasoft Pseudopotentials and Projector-Augmented Waves (PAW) 6

27 Ultra-soft pseudopotentials : the idea Problem with NC pseudopotentials : norm conservation limits the softness! When orbitals without nodes (1s, p, 3d, 4f) treated as valence => small characteristic length energy cut-off large. Idea (Vanderbilt, Phys. Rev. B 41, 789 (1990)) Suppress norm-conservation condition : - modify normalization, to keep correct scattering properties - introduce charge density corrections, For selected elements, can decrease number of PW/FFT Grid points by a factor of two or three, with even larger speed up. More difficult to implement than norm-conserving PPs. Can be obtained as a particular case of PAW construction... 7

28 Projector-Augmented Waves : the idea Idea P. Blöchl Phys. Rev. B 50, (1994) The true wave-function and a well-behaving pseudowavefunction are linked by a linear transformation Strong oscillations near the nucleus Ψ = TΨ True wave function (all electrons) No oscillation near the nucleus Auxiliary wave function (soft pseudofunction) More rigorous than USPP 8

29 USPP and PAW : common features Generalized Schrödinger Eq., with overlap operator S. + v ps ψ ps,i = ε i Sψ ps,i Charge density of each state to be corrected for the missing norm. 9

30 Projector-Augmented Waves : the math Ψ Ψ Ψ = TΨ True wave-function Well-behaving pseudo-wavefunction Linked by a linear transformation Physical quantities like computed in the pseudo representation ΨAΨ A Ψ, Ψ A = T + AT with Similarly, variational principle for total energy gives E T Ψ = εt +T Ψ Ψ [ ] equivalent to Kohn-Sham equation, for pseudowavefunctions. Search for ground state can also be done in the pseudo space. 30

31 Transformation operator Operator T has to modify the smooth pseudowavefunction in each atomic region, to give it the correct nodal structure. Identity + sum of atomic contributions Ψ = TΨ with T = 1+ SR R Choose : Partial waves (R=atomic site label) φi = basis set, solutions of the Schrödinger Eq. for the isolated atoms within some cut-off radius rc,r Pseudo partial waves φ i Define S such as : φ i = (1+ SR ) φ i, identical to the partial waves beyond the cut-off radius, but smoother inside 31

32 Representation of the wavefunctions ( T = 1+ φ i φ i Ψ = TΨ i ) p i Explicitly, ( + ΨR1 Ψ R1 Ψ=Ψ R ) + = Plane wave part All electron spherical part ΨR1 = φ i p i Ψ i R Pseudized spherical part R1 = φ i p i Ψ Ψ i R 3

33 Wavefunctions, density, energy represented by plane waves (might use other representations) Ψ represented on a radial grid, centered on R, times spherical harmonics R1 ΨR1 and Ψ Outside of the spheres, Note : Inside one sphere, R1 ΨR1 = Ψ 1 = Ψ R Ψ = φ i p i Ψ i R Density: Energy: n(r ) = n~ (r ) + å n1r (r ) - n~r1 (r ) ~ ~1 1 E = E + å ER - ER ( R ) ( ) R 33

34 Approximations (1) Core electrons : usually treated in the frozen-core approximation, and treated on radial grid (spherical harmonics). () Finite PW basis set (same as PPs) (3) The partial wave expansion is truncated : only one or two partial wave(s), for each atom R, and each l,m channel φ i p i 1 i Inside one sphere, = Ψ R1 Ψ is only approximately true! Ψ = φ i p i Ψ i R contributes inside the atomic spheres, and corrects for the missing terms due to truncation of partial wave expansion 34

35 Advantages of PW+PP or PAW method? (1) The basis set does not depend on atomic coordinates : easy computation of forces as numerically exact derivatives of the total energy with respect to atomic coordinates (no Pulay forces). Easy structural optimisation, or MD. This leads also to Car-Parrinello technique. () Systematic way to complete the basis set For PW-PP : simple implementation For PW-PAW : can be a numerically accurate implementation of DFT, including properties related to cores ; usually faster than PW-PP Disadvantages : cannot lead to Order(N) implementation, does not treat efficiently finite systems (vacuum!). Higher lying states (very high in energy) cannot be trusted... 35

36 Pseudopotential and PAW atomic data availability and reliability 36

37 Generators (non-complete list) Norm-conserving pseudos :!ONCVPSP (Don Hamann, (M. Oliveira (A. Rappe USPP : Vanderbilt USPP generator ( PAW ATOMPAW (N. Holzwarth GPAW generator ( + converter USPPPAW ( 37

38 Implementations Plane Waves need pseudopotentials (or PAW), but pseudopotentials can be used with many basis sets! Norm-conserving pseudos :!with planewave basis set : ABINIT, QE, CASTEP!with localized orbitals : SIESTA...!with wavelets : BigDFT... USPP : with planewave basis set : QE, DaCapo, CASTEP PAW with planewave basis set : ABINIT, QE, GPAW, VASP Same pseudopotentials can sometimes be used by different codes. 38

39 Pseudopotentials/PAW data " Example of PAW atomic dataset table : JTH v1.0 Δ = 0.4 mev Jollet, Torrent, Holzwarth, Computer Physics Comm. 185, 146 (014) Also : GPAW table, GBRV v1.0 table, norm-conserving pseudopotential table (e.g. ONCVPSP pseudo generator), or many other pseudos see 39

40 Testing pseudopotentials 40

41 Comparing code/pseudopotential Recently, large effort to improve uncertainty related to psps : «Delta-factor» collaboration Lejaeghere... Cottenier, Science 351, aad3000 (016) Specification of 71 elemental solids for different volumes. 41

42 More about pseudopotentials / PAW datasets Review 4

43 More about pseudopotentials / PAW datasets Review 43

44 More about pseudopotentials / PAW datasets Review 44

45 Improvement with time Review JTH 1.0 => 0.4 mev 45

46 Other comparison with All-Electron data 46

47 Summary - Plane waves basis set - PW need pseudopotentials - Easy computation of forces relaxation of geometry, or molecular dynamics - PAW (also USPP) : more accurate, faster for cases with 3d and 4f orbitals 47