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Pseudopotentials (Part I): Georg KRESSE Institut für Materialphysik and Center for Computational Materials Science Universität Wien, Sensengasse 8, A-1090 Wien, Austria b-initio ackage ienna G. K RESSE , P SEUDOPOTENTIALS (PART I) imulation Page 1 Overview the very basics – periodic boundary conditions – the Bloch theorem – plane waves – pseudopotentials determining the electronic groundstate effective forces on the ions general road-map to the things you will hear in more detail later G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 2 Periodic boundary conditions as almost all plane wave codes VASP uses always periodic boundary conditions the interaction between repeated images must be handled by a sufficiently large vacuum region sounds disastrous for the treatment of molecules but large molecules can be handled with a comparable or even better performance than by e.g. Gaussian G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 3 The Bloch theorem the translational invariance implies that a good quantum number exists, which is usually termed k k corresponds to a vector in the Brillouin zone all electronic states can be indexed by this quantum number Ψk in a one-electron theory, one can introduce a second index, corresponding to the one electron band n ψn k r the Bloch theorem implies that the (single electron) wavefunctions observe the equations ψn k r eikτ τ ψn k r where τ is any translational vector leaving the Hamiltonian invariant G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 4 The DFT Hamiltonian the charge density is determined by integrating over the entire Brillouin zone and summing over the filled bands 3 ∑∞ n d k f n k ψn k r ψ n k r ρe r where the charge density is cell periodic (can be seen by inserting the Bloch theorem) 1 are the Fermi-weights 1 exp β εn k εFermi and fn k the KS-DFT equations (Schrödinger like) are given by ε n k ψn k r Vxc ρe r ρion r 3 d r r r ρe r e2 V eff r ρe r ψn k r V eff r ρe r h̄2 2me ∆ ρion is the ionic charge distribution G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 5 Plane waves introduce the cell periodic part un k of the wavefunctions ψn k r un k r eikr un k r is cell periodic (insert into Bloch theorem) all cell periodic functions are now written as a sum of plane waves kr iG C e Gnk ∑ 2 1 Ω1 ψn k r un k r 1 iGr C e Gnk ∑ Ω1 2 G G iGr ρ e G ∑ ρr G ∑ VG eiGr V r G in practice only those plane waves G k are included which satisfy k2 h̄2 G 2me Ecutoff G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 6 Fast Fourier transformation FFT real space τ reciprocal space 2 G cut b2 0 1 2 3 N−1 0 τ 1 0 1 2 b1 g = n 2π / τ 1 1 1 1 ikr C e rnk Ω1 2 G. K RESSE , P SEUDOPOTENTIALS (PART I) r ψn k r NFFT iGr G ∑ Crnk e 1 ∑ CGnk eiGr CGnk 5 −4 −3 −2 −1 0 N/2 −N/2+1 x =n /N τ 1 1 1 Crnk 3 4 Page 7 Why are plane waves so convenient historical reason: many elements exhibit a band-structure that can be interpreted in a free electron picture (metallic s and p elements) the pseudopotential theory was initially developed to cope with these elements (pseudopotential perturbation theory) practicle reason: the total energy expressions and the Hamiltonian H are dead simple to implement a working pseudopotential program can be written in a few weeks using a modern rapid prototyping language computational reason: because of it’s simplicity the evaluations of Hψ is exceedingly efficient using FFT’s G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 8 Computational reason evaluation of Hψn k r kr G ! k ψn k Ω1 2 ei G 1 k and using the convention r G ψn k r V r h̄2 ∆ 2me CGnk kinetic energy: h̄2 G k 2 CGnk 2me k G h̄2 ∆ ψn k 2me Nplanewaves local potential: VrCrnk e NFFT ∑ k V ψn k G 1 iGr NFFT log NFFT r " if H would be stored as a matrix with Nplanewaves Nplanewaves components Nplanewaves Nplanewaves operations would be required " G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 9 Local part of Hamiltonian Gcut FFT ψ ψ G+k V r r FFT <k+G V ψ > ψ V r r G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 10 Pseudopotential approximation the number of plane waves would exceed any practicle limits except for H and Li # pseudopotentials instead of exact potentials must be applied three different types of potentials are supported by VASP – norm-conserving pseudopotentials – ultra-soft pseudopotentials – PAW potentials they will be discussed in more details in later sessions all three methods have in common that they are presently frozen core methods i.e. the core electrons are pre-calculated in an atomic environment and kept frozen in the course of the remaining calculations G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 11 '& '& '& '& '& '& '& '& '& '& '& &' '& '& '& '& '& '& '& '& -, -, -, -, +* +* +* +* +* +* +* +* +* +* +* *+ 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 89 )( )( )( )( )( )( )( )( 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 89 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 89 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 89 23 23 23 32 32 32 32 32 32 32 32 32 32 32 32 32 32 23 23 23 32 32 32 32 32 32 32 32 32 32 98 98 98 3p 3s 98 2p 1s 98 98 98 98 3p 3s 76 76 76 76 76 76 76 76 76 76 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 76 54 45 54 54 54 54 54 54 54 54 54 Page 12 G. K RESSE , P SEUDOPOTENTIALS (PART I) 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 89 23 23 23 32 32 32 32 32 32 32 32 32 32 32 32 32 32 23 10 10 10 01 01 01 01 01 01 01 01 01 01 01 01 10 10 10 01 01 01 01 01 01 01 01 01 01 01 01 10 10 01 01 01 01 01 01 01 01 01 01 01 01 01 )( )( )( )( )( )( )( )( )( )( )( () +* +* +* +* +* +* +* +* +* +* *+ +* +* +* +* /. /. /. ./ ./ ./ ./ ./ ./ ./ ./ ./ ./ +* +* *+ +* +* /. /. /. ./ ./ ./ ./ ./ ./ ./ ./ ./ ./ /. /. ./ ./ ./ ./ ./ ./ ./ ./ ./ ./ ./ )( )( )( )( )( )( )( )( )( )( )( () nodal structure is retained 2p and 1s are nodeless !!!! 2p 2s 1s %$ %$ %$ %$ %$ %$ %$ %$ PAW Al atom : E= -0.205 R c=1.9 %$ %$ %$ %$ %$ %$ %$ %$ %$ %$ %$ $% effectiv Al atom 4 3 2 R (a.u.) 1 0 0.0 p 0.5 Al '& '& '& '& '& '& '& '& '& '& '& &' %$ %$ %$ %$ %$ %$ %$ %$ %$ %$ %$ $% Al 1.0 : E= -0.576 R c=1.9 1.5 s Al pseudopotential exact potential (interstitial region) 2.0 wave-function Pseudopotential: essential idea 2.5 2.5 [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> >? >? >? >? >? >? >? >? Al wave-function =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< =< ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: Scattering approach Al 2.0 1.5 s : E= -0.576 R c=1.9 p : E= -0.205 R c=1.9 1.0 0.5 0.0 0 1 2 R (a.u.) 3 4 solve the Schrödinger equation in the interstitial region only, using energy and angular momentum dependent boundary conditions at the spheres C C C C rc ∂ log φl r ε ∂r B B C C C C B 1 ∂ φl r ε ∂r φl r ε rc φl are the regular solutions of the radial Schrödinger equation inside the spheres for the angular momentum quantum number l and the energy ε details of the wavefunctions (number of nodes in the spheres) do not enter pseudopotential: select one specific energy ε in the centre of the valence band and replace the exact wavefunction φl by a pseudo-wavefunction with φ̃l C C C C rc ∂ log φ̃l r ε ∂r B B C C C C ∂ log φl r ε ∂r G. K RESSE , P SEUDOPOTENTIALS (PART I) rc Page 13 2.5 wave-function KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH IH Al GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED Norm-conserving PP Al 2.0 1.5 s : E= -0.576 R c=1.9 p : E= -0.205 R c=1.9 1.0 0.5 0.0 0 1 2 R (a.u.) 3 4 ! obviously one would like to get the right distribution of charge between the spheres and the interstitial region as well norm-conserving pseudopotentials rc 0 φ̃l r ε φ̃l r ε 4πr2 dr 0 φl r ε φl r ε 4πr dr rc 2 norm-conservation has another important consequence the scattering properties are not only correct at the reference energy ε but also in a small energy interval around ε G. K RESSE , P SEUDOPOTENTIALS (PART I) l M L L L L rc ∂ ∂ log φ̃l r ε ∂ε ∂r L L L L ∂ ∂ log φl r ε ∂ε ∂r rc Page 14 Pseudopotential generation all-electron calculation for a reference atom chose energies εl at which the pseudisation is performed usually these are simply the eigen-energies of the bound valence states, but in principle any energy can be chosen (centre of valence band) replace the exact wavefunction by a node less pseudo-wavefunction observing the following four requirements: n n is the n th derivative N N 2 rc 4π norm-conservation condition φ r 2 r2 dr φ̃ r r dr 2 2 0 0 ONN for n rc 4π φ rc n φ̃ rc 0 this pseudopotential conserves exactly the scattering properties of the original atom in the atomic configuration but it is an approximation in another environment G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 15 Projector augmented wave method P.E. Blöchl, Phys. Rev. B50, 17953 (1994); G. Kresse, and J. Joubert, Phys. Rev. B 59, 1758 (1999). wave function (and energy) are decomposed into three terms: atoms atoms YX YX YX YX YX YX YX YX YX YX YX XY [Z [Z [Z Z[ Z[ Z[ [Z [Z [Z [Z [Z [Z YX YX YX YX YX YX YX YX YX YX YX XY SR SR SR SR SR SR SR SR SR SR SR SR QP QP QP QP QP QP QP QP QP QP QP PQ [Z Z[ [Z [Z SR SR SR SR YX YX YX YX QP QP QP QP [Z [Z [Z Z[ Z[ Z[ [Z [Z [Z Z[ [Z [Z ]\ ]\ ]\ ]\ ]\ ]\ ]\ ]\ ]\ ]\ ]\ \] SR SR SR SR SR SR SR SR SR RS SR SR UT UT UT UT UT UT UT UT UT UT UT TU _^ _^ _^ _^ _^ _^ _^ _^ _^ _^ _^ _^ ]\ ]\ ]\ ]\ ]\ ]\ ]\ ]\ ]\ ]\ ]\ \] WV WV WV WV WV WV WV WV WV WV WV WV UT UT UT UT UT UT UT UT UT UT UT TU _^ _^ _^ _^ _^ _^ _^ _^ ]\ \] ]\ ]\ ]\ ]\ ]\ ]\ WV WV WV WV WV WV WV WV UT TU UT UT UT UT UT UT ` – kinetic energy – charge densities – Hartree.- and exchange correlation energy _^ _^ _^ _^ _^ _^ _^ _^ WV WV WV WV WV WV WV WV – wave functions Page 16 G. K RESSE , P SEUDOPOTENTIALS (PART I) efficient no interaction between different spheres and plane waves QP QP QP QP QP QP QP QP QP QP QP PQ pseudo (node less) pseudo−onsite exact onsite plane waves radial grids radial grids exact + − = φlmε clmε ∑ φ̃lmε clmε ∑ ψ̃n ψn decomposition into three terms holds for Determining the electronic groundstate by iteration – self consistency (old fashioned) start with a trial density ρe , set up the Schrödinger equation, and solve the Schrödinger equation to obtain wavefunctions ψn r Ne 2 ONN a 2 and a new one ∑n ψn r Vxc ρe r ρion r 3 d r r r e ρe r r ρe r V 2 1 as a result one obtains a new charge density ρe r electron potential: eff n N ε n ψn r ψn r V eff r ρe r h̄2 2 ∇ 2me # new Schrödinger equation iteration G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 17 Self-consistency scheme b b b b trial-charge ρin and trial-wavevectors ψn f f c set up Hamiltonian H ρin g c iterative refinements of wavefunctions ψn g c new charge density ρout 2 ∑n fn ψn r c refinement of density ρin ρout # new ρin refinement of density: DIIS algorithm P. Pulay, Chem. Phys. Lett. 73, 393 (1980). g refinement of wavefunctions: blocked Davidson like algorithm d ce e d ee dd e d Ebreak e e d ∆E d e dd ee no i h two subproblems optimization of ψn and ρin d c calculate forces, update ions G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 18 What have all iterative matrix diagonalisation schemes in common ? one usually starts with a set of trial vectors (wavefunctions) representing the filled states and a few empty one electron states 1 Nbands NN ψn n these are initialised using a random number generator then the wavefunctions are improved by adding a certain amount of the residual vector to each the residual vector is defined as ψn H ψ n εapp εapp S ψn H R ψn Hψn is exactly operation discussed before (efficient) adding a small amount of the residual vector ! λR ψn ψn ψn is in the spirit of the steepest descent approach (“Jacobi relaxation”) G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 19 We know the groundstate, what now ? Hellman-Feynmann theorem allows to calculate the ionic forces and the stress tensor by moving along the ionic forces (steepest descent) the ionic groundstate can be calculated we can displace ions from the ionic groundstate, and determine the forces on all other ions # effective inter-atomic force constants and vibrational frequencies molecular dynamics by using Newtons equation of motion G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 20 Vibrational properties Frequency cm -1 the vibrational frequencies are calculated using a brute force method: in the supercell all “selected atoms” are displaced from LO 1600 their groundstate position SH 1400 1200 1000 800 600 400 200 0 Γ LA ZO SH ZA M K Γ not possible presently # forces force constants symmetry is not used !!! works well for molecules, but has to be applied with great care to solids G. K RESSE , P SEUDOPOTENTIALS (PART I) Page 21 Pseudopotentials (Part II) and PAW Georg KRESSE Institut für Materialphysik and Center for Computational Materials Science Universität Wien, Sensengasse 8, A-1090 Wien, Austria b-initio ackage ienna G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW imulation Page 1 Overview pseudopotential basics normconserving pseudopotentials adopted pseudization strategy G. Kresse, and J. Hafner, J. Phys.: Condens. Matter 6 (1994) from normconserving to ultrasoft pseudopotentials the PAW method G. Kresse, and J. Joubert, Phys. Rev. B 59, 1758 (1999). where to be careful ? – local pseudopotentials – simultaneous representation of valence and semi-core states – magnetic calculations G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 2 Normconserving pseudopotentials: General strategy all-electron calculation for a reference atom (rhfsps) pseudization of valence wave functions (rhfsps) chose local pseudopotential and factorize (fourpot3) un-screening of atomic potential to obtained ionic pseudopotential (fourpot3) G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 3 Pseudization of valence wave functions different schemes have been proposed in literature, but the general strategy is always similar calculate exact all-electron wave function φ r replace exact φ r inside pseudization radius by a suitable “soft” pseudo wave function φ̃ r must fulfill some continuity conditions rc n 2 0 rc for n φ rc r φr n φ̃ rc r ∑i αi βi r φ̃ r possibly impose normconservation condition rc rc 4π 0 φ r 2 r2 dr φ̃ r r dr 4π 2 2 0 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 4 Which expansion set should one use? many different basis sets have been proposed in the literature presently the two most prominent ones are – polynomials (Troullier and Martins) c10 r10 c8 r 8 c6 r 6 c4 r 4 c2 r 2 c0 φ̃ r c12 r12 – spherical Bessel-functions (RRKJ—Rappe, Rabe, et. al.) i 1 φ rc φ rc with qi such that qi r j l q i rc j l q i rc ∑ αi j l φ̃ r 34 the last one is the standard scheme for VASP pseudopotentials the basis set I use is generally minimal (3 or sometimes 4 Bessel-functions) for PAW and US pseudopotentials only 2 spherical Bessel-functions are required G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 5 Why are spherical Bessel-functions so convenient close analogy between plane waves and spherical Bessel-functions the required cutoff can be calculated directly from the expansion set 34 ∑ αi jl φ̃ r qi r i 1 2 15 qi h̄2 2me max Ecut find maximum qi I always use a minimal basis set in the original RRKJ scheme, more spherical Bessel-functions were used, and the wave functions were optimized for a selected cutoff our tests indicate that this is contra-productive G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 6 Factorization required to speed up the calculations D.M. Bylander, et al., Phys. Rev. B 46, 13756 (1992) chose local reference potential Vloc p φ̃ construct a projector such that 1 Vloc ε φ̃ p ∝ h̄2 ∆ 2me the factorized Hamiltonian is given by H Vloc Vloc pD p " ε φ̃ φ̃ G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW φ̃ ! h̄2 2me ∆ one recognizes immediately that: φ̃ ε φ̃ " ! h̄2 2me ∆ φ̃ pD p D with Vloc h̄2 ∆ 2me Page 7 wave-function 2.5 What have we acchived at this point 2.0 1.5 s : E= -0.576 R c=1.9 p : E= -0.205 R c=1.9 1.0 0.5 0.0 1 0 2 R (a.u.) 3 4 the exact wavefunction has been replaced by it’s pseudo counterpart and a “pseudo” Hamiltonian has been constructed φ̃ pD p εφ ε φ̃ Vloc φ VAE h̄2 ∆ 2me h̄2 ∆ 2me at the energy ε, φ and φ̃ are identical outside of the cutoff radius # G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW at the energy ε φ and φ̃ have the same norm inside the cutoff radius φ̃ are identical outside of the cutoff radius δε φ and Page 8 Two reference energies pseudize at two reference energies: φi i % δi j pi φ̃ j φ̃ j j εj Vloc ∑ αi j pi h̄2 ∆ 2me for all i j $ construct two projectors such that 12 factorized Hamiltonian is given by p i Di j p j ∑ ij φ̃ j εj " ! ε j φ̃i φ̃ j one recognizes immediately that: φ̃i H φ̃ j Vloc h̄2 2me ∆ φ̃i Di j Vloc H h̄2 ∆ 2me G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 9 Two reference energies: practical considerations the pseudo wavef. must fulfill a generalized normconserv. condition: rc & 0 i j φi r φ j r r2 dr 0 4π φ̃i r φ̃ j r r dr 4π rc 2 in the VASP PP generation program only rc 0 0 φi r φi r r2 dr φ̃i r φ̃i r r dr rc 2 is enforced to correct for this error, augmentation charges would be required, but these are neglected as a result Di j is not Hermitian φ̃ j εj Vloc " ! h̄2 2me ∆ φ̃i Di j off-diagonal elements are averaged to make the matrix D symmetric G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 10 US pseudopotentials, very similar to NC pseudopotentials pseudize at two reference energies εj Vloc φ̃ j j for all i j ∑ αi j pi h̄2 ∆ 2me δi j pi φ̃ j construct two projectors such that the factorized Hamiltonian and overlap operator are given by S 1 φ̃i φ̃ j φi φ j Qi j ε j Qi j " ! φ̃ j εj Vloc p i Qi j p j ij h̄2 2me ∆ ∑ p i Di j p j ∑ ij φ̃i Vloc Di j H h̄2 ∆ 2me φ̃i S φ̃ j ε j one can show that: φ̃i H φ̃ j G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 11 What does all that mean? let us look again at the definition of Di j ε j Qi j φ̃ j Qi j ε j φ̃i φ̃ j Vloc φ̃ j εj h̄2 ∆ φ̃i 2me Vloc φ̃i Di j h̄2 ∆ 2me φ̃i φ̃ j VAE φ j h̄2 ∆ 2me φi Vloc φ̃ j φ̃i h̄2 ∆ 2me φi ε j φ j Vloc φ̃ j φi φ j φ̃i h̄2 ∆ 2me ε j φ̃i φ̃ j Vloc φ̃ j φ̃i h̄2 ∆ 2me * ' * ' () () energy pseudo onsite energy AE onsite G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 12 US-PP: what they really do , / . / . - + . , , + . + + , - 98 98 98 98 98 98 98 98 98 98 98 89 10 10 10 10 10 10 10 10 10 10 10 01 ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: ;: 98 98 98 98 98 98 98 98 98 98 98 89 32 32 32 32 32 32 32 32 32 32 32 32 10 10 10 10 10 10 10 10 10 10 10 01 ;: ;: ;: ;: 32 32 32 32 98 98 98 98 10 10 10 10 ;: ;: ;: ;: ;: ;: ;: ;: ;: :; ;: ;: =< =< =< =< =< =< =< =< =< =< =< <= 32 32 32 32 32 32 32 32 32 23 32 32 54 54 54 54 54 54 54 54 54 54 54 45 ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> ?> =< =< =< =< =< =< =< =< =< =< =< <= 76 76 76 76 76 76 76 76 76 76 76 76 54 54 54 54 54 54 54 54 54 54 54 45 ?> ?> ?> ?> ?> ?> ?> ?> =< <= =< =< =< =< =< =< 76 76 76 76 76 76 76 76 54 45 54 54 54 54 54 54 ?> ?> ?> ?> ?> ?> ?> ?> 76 76 76 76 76 76 76 76 Page 13 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW AE-onsite pseudo-onsite pseudo AE + = VAE φ j ∆ ρi j φ i Vloc φ̃ j ∆ ρi j φ̃i Vloc ∆ E Ψ̃n pi p j Ψ̃n onsite occupancy matrix (or density matrix): ρi j δi j pi φ̃ j p̃i Ψ̃n character of wave function: ci energy is the sum of three terms US-PP method is in principle an exact frozen core all-electron method Mixed basis set with an implicit dependency US-PP’s carry a small rucksack, with two additional sets of basis functions defined around each atomic site % $ – one for the soft pseudo-wave functions φ̃i % $ – one for the AE wave functions φi for each atomic sphere the energy is evaluated using these two sets and the calculated energy is subtracted and added, respectively the onsite occupancy matrix (density matrix) for these two sets is calculated from the plane wave coefficients Ψ̃n pi p j Ψ̃n ρi j PAW inspired formulation G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 14 Practical considerations the US-PP method can not be implemented exactly, since currently no method exists to handle the rapid variations of the all-electron wave functions a regular grid does not work, maybe wavelets would be an option in practice, I therefore adopt a modified prescription for US-PP’s: exact AE wave functions conserving r φ̃i r φ̃ j r nc φnc i r φj r augmentation charge: Qi j r φnorm @ φr norm-conserving wave functions these US-PP’s yield exactly the same results as the corresponding NC-PP’s, but at much lower cutoffs G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 15 Why are US-PP’s softer pseudo wave function is represented as a sum of two spherical Bessel functions instead of three φ rc φ rc i 1 j l q i rc j l q i rc with qi such that qi r ∑ αi j l φ̃ r 2 additionally rc can be increased compared to NC potentials there is no need to represent the charge distribution of the AE wave function, since this is done by the augmentation the pseudo wave functions follow remarkably well the AE wave functions, even for values much smaller than rc basis sets are roughly a factor 2-3 smaller G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 16 Example for this behavior Cu (US) 3.5 3.5 3.0 3.0 2.5 s wave-function wave-function e.g. Cu (NC) :E=-0.357 Rc=2.5 2.0 1.5 p : E=-0.058 Rc=2.5 1.0 s : E=-0.357 Rc=2.5 2.0 1.5 p : E=-0.058 Rc=2.5 1.0 0.5 0.0 0 2.5 0.5 d : E=-0.393 Rc=2.0 1 2 R (a.u.) 3 4 0.0 0 d : E=-0.393 Rc=2.0 1 2 3 4 R (a.u.) G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 17 The role of the local potential the local potential needs to describe scattering properties for radial quantum numbers not included in the projectors – much underestimated problem – resulting errors can be 1-2 % in the lattice constant the tails of d electrons (transition metals) or p electrons (oxygen) overlap into the pseudization region they are picked up as high l components (FLAPW) ideally one would like to use very attractive local potentials # but ghost-state problem in most cases, compromises must be made G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 18 Ghost-state problems particularly severe for alkali, alkali-earth and early transition metals the more attractive the local potential and the smaller rc , the more likely it is to have a ghost-state; Zr example 4 4 2 g p f 0 g f s xl (E) xl (E) 2 s d -2 0 d -2 p -4 -4 -3 -2 -1 0 E (Ry) 1 2 -3 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW -2 -1 0 E (Ry) 1 2 Page 19 solution: treat semi-core states as valence Pseudopotential generation in practice: the PSCTR file TITEL = US O LULTRA = T use ultrasoft PP ? RWIGS = 1.40 nn distance ICORE NE LCOR QCUT l 0 0 1 1 2 = = = = 2 100 .TRUE. -1 Description E TYP 0 15 0 15 0 15 0 15 0.0 7 RCUT 1.13 1.13 1.13 1.13 1.55 TYP 23 23 23 23 7 RCUT 1.40 1.40 1.55 1.55 1.55 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 20 PAW: basic idea P.E. Blöchl, Phys. Rev. B50, 17953 (1994) Kohn-Sham equation B Exc n A nZ B EH n A n Ψn ∑ fn E 1 ∆ Ψn 2 frozen core approximation for the valence electrons, a transformation from the pseudo to the AE wavefunction is defined: p̃lmε Ψ̃n φ̃lmε ∑sites lmε φlmε Ψ̃n Ψn lm is an index for the angular and magnetic quantum numbers ε refers to a particular reference energy p̃lmε projector function φ̃lmε partial wave G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 21 PAW: basic idea p̃lmε Ψ̃n φ̃lmε ∑ φlmε Ψ̃n Ψn transformation: the “character” of an arbitrary pseudo-wavefunction Ψ̃n at one site can be calculated by multiplication with the projector function at that site p̃lmε Ψ̃n clmε inside each sphere the wavefunctions can be determined: sphere clmε C ∑ φl m ε C clmε C ∑ φ̃l m ε Ψ̃n lmε sphere C C C Ψn lmε the projector functions must be dual to the pseudo-wavefunction C D C D δl l δm m δε ε C D C C C p̃lmε φ̃l m ε G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 22 PAW: addidative augmentation PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO PO NM NM NM NM NM NM NM NM NM NM NM NM NM NM NM NM NM NM NM NM NM NM NM MN NM NM NM NM HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG HG FE FE FE FE FE FE FE FE FE FE FE FE FE FE FE FE FE FE FE FE FE FE FE EF FE FE FE FE TS TS TS TS TS TS TS TS TS TS TS TS TS RQ RQ RQ RQ RQ RQ RQ RQ RQ RQ RQ QR PO PO TS PO PO TS PO PO TS OP PO RQ RQ RQ RQ RQ RQ RQ RQ RQ RQ RQ QR LK LK LK LK LK LK LK LK LK LK LK LK LK JI JI JI JI JI JI JI JI JI JI JI IJ HG HG LK HG HG LK HG HG LK GH HG JI JI JI JI JI JI JI JI JI JI JI IJ RQ RQ RQ RQ JI JI JI JI TS TS TS TS TS TS TS TS TS TS TS ST LK LK LK LK LK LK LK LK LK LK LK KL Page 23 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW – kinetic energy – Hartree energy – charge density – exchange correlation energy – wavefunctions AE-onsite pseudo-onsite pseudo AE + = ∑ φlmε clmε ∑ φ̃lmε clmε Ψ̃n Ψn p̃lmε Ψ̃n character of wavefunction: clmε same trick works for Derivation of the PAW method is straightforward for instance, the kinetic energy is given by ∆ Ψn φi φ̃i Ψn ∑ fn Ekin n by inserting the transformation (i p̃i Ψ̃n ∑ Ψ̃n Ψn lmε) i Ẽ 1 / - . ∆ φj Y site i j Y site i j φi + . ∑ ∑ ρi j ,/ ∆ φ̃ j . φ̃i + . ∑ ∑ ρi j - / n E 1 (assuming completeness) Ẽ + . ∆ Ψ̃n + . Ψ̃n - ∑ fn into Ekin one obtains: Ekin X U VW X U VW VW Ẽ 1 E1 ∑n f n ci c j Z ρi j is an on-site density matrix: ρi j X U Ẽ G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 24 Hartree energy the pseudo-wavefunctions do not have the same norm as the AE wavefunctions inside the spheres to deal with long range electrostatic interactions between spheres a soft compensation charge n̂ is introd. (similar to FLAPW) ji ji ji ji ji ji ji ji ji ji ji ji ji hg hg hg hg hg hg hg hg hg hg hg gh hg hg hg hg `_ `_ `_ `_ ji ji ji ji ji ji ji ji ji ji ji ij ba ba ba ba ba ba ba ba ba ba ba ab B A B A B A Page 25 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW n̂1 ∑sites EH n1 fe fe ji fe fe ji fe fe ji ef fe hg hg hg hg hg hg hg hg hg hg hg gh ba ba ba ba ba ba ba ba ba ba ba ba ba `_ `_ `_ `_ `_ `_ `_ `_ `_ `_ `_ _` ^] ^] ba ^] ^] ba ^] ^] ba ]^ ]^ `_ `_ `_ `_ `_ `_ `_ `_ `_ `_ `_ _` n̂1 compensation charge at site ñ1 pseudo-charge at one site E1 Ẽ 1 Ẽ Hartree energy becomes: EH n̂1 ∑sites EH ñ1 n̂ EH ñ fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe fe dc dc dc dc dc dc dc dc dc dc dc dc dc dc dc dc dc dc dc dc dc cd cd cd cd cd cd cd ^] ^] ^] ^] ^] ^] ^] ^] ^] ^] ^] ^] ^] ^] ^] ^] ^] ^] ]^ ]^ ]^ ]^ ]^ ]^ \[ \[ \[ \[ \[ \[ \[ [\ [\ [\ [\ [\ [\ [\ \[ \[ \[ \[ \[ \[ \[ \[ \[ [\ \[ \[ \[ \[ pseudo + compens. pseudo+comp. onsite AE-onsite AE + = PAW energy functional P.E. Blöchl, Phys. Rev. B50, 17953 (1994). d3r A ñc B n̂ Exc ñ1 U R Zion D i j sites 1 ∆ φ̃ j 2 ñc n̂ r A φ̃i ∑ ∑ ρi j B vH ñZc ñ r B Ẽ 1 n̂ Ẽ 1 B n̂ A EH ñ E1 n Exc ñ A 1 ∆ Ψ̃n 2 ∑ fn Ψ̃n Ẽ Ẽ total energy becomes a sum of three terms E l k A nc B Exc n1 D i j sites 1 ∆ φj 2 d3r n̂ r B ∑ ∑ ρi j φi vH ñZc ñ1 r A A E1 Ωr B n̂ EH ñ1 B Ωr vH nZc n1 r d 3 r A A B E H n1 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 26 Ẽ is evaluated on a regular grid Kohn Sham functional evaluated in a plane wave basis set with additional compensation charges to account for the incorrect norm of the pseudo-wavefunction (very similar to ultrasoft pseudopotentials) n̂ ∑n fn Ψ̃n Ψ̃n Z ñ pseudo charge density compensation charge E 1 and Ẽ 1 are evaluated on radial grids centered around each ion % $ % $ Kohn-Sham energy evaluated for basis sets ψ̃i and ψi these terms correct for the shape difference between the pseudo and AE wavefunctions no cross-terms between plane wave part and radial grids exist G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 27 General scheme xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw xw vu vu vu vu vu vu vu vu vu vu vu vu vu vu vu vu vu vu vu vu vu uv uv uv uv uv uv uv po po po po po po po po po po po po po po po po po po op op op op op op nm nm nm nm nm nm nm mn mn mn mn mn mn mn nm nm nm nm nm nm nm nm nm mn nm nm nm nm |{ |{ |{ |{ |{ |{ |{ |{ |{ |{ |{ |{ |{ zy zy zy zy zy zy zy zy zy zy zy yz zy zy zy zy rq rq rq rq |{ |{ |{ |{ |{ |{ |{ |{ |{ |{ |{ {| ts ts ts ts ts ts ts ts ts ts ts st Page 28 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW xw xw |{ xw xw |{ xw xw |{ wx xw zy zy zy zy zy zy zy zy zy zy zy yz ts ts ts ts ts ts ts ts ts ts ts ts ts rq rq rq rq rq rq rq rq rq rq rq qr po po ts po po ts po po ts op op rq rq rq rq rq rq rq rq rq rq rq qr pseudo + compens. pseudo+comp. onsite AE-onsite AE + = applies to all quantities US-PP D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). the original derivation of US-PP is somewhat “problematic” it’s more like accepting things than understanding them in fact, the equations for US-PP’s can be derived rigidly from the PAW functional by linearisation of the on-site terms E 1 and Ẽ 1 around the atomic reference configuration this shows the close relation between both approaches but it also indicates when US-PP’s might be problematic: the more the environment differs from the reference state the less accurate US-PP are our tests indicate that magnetism is the strongest perturbation in other cases, US-PP and the PAW yield almost identical results G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 29 Construction of PAW potentials first an AE calculation for a reference atom is performed the AE wavefunctions of the valence states are pseudised projectors are constructed as s : E= -0.296 R c=2.6 j p1/2: E= -2.537 R c=2.6 1 0 1 2 R (a.u.) 3 4 φ̃ j δi n D p̃i φ̃n 0 εj they must obey d3/2: E= -0.228 R c=2.6 Vloc ∑ αi j h̄2 ∆ 2me pi 2 wave-function 3 projectors are dual to the pseudo wavefunction to have a rather complete set of projectors two partial waves for each quantum channel lm are constructed G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 30 PAW — US-PP method: molecules results for the bond length of several molecules obtained with the US-PP, PAW and AE approaches plane wave cutoffs were around 200-400 eV US-PP and the PAW method give the same results within 0.1-0.3% well converged relaxed core AE calculations yield identical results G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 31 US-PP(data base) US-PP(special) PAW AE H2 1.447 1.447 1.446a Li2 5.127 5.120 5.120a Be2 4.524 4.520 4.521a Na2 5.667 5.663 5.67 a 2.163 2.141 (2.127) 2.141 (2.128) 2.129a N2 2.101 2.077 (2.066) 2.076 (2.068) 2.068a F2 2.696 2.640 (2.626) 2.633 (2.621) 2.615a P2 3.576 3.570 3.570 3.572a H2 O 1.840 (1.834) 1.839 (1.835) 1.833a α(H2 O)( ) 105.3 (104.8) 105.3 (104.8) 105.0a BF3 2.476 (2.470) 2.476 (2.470) 2.464b SiF4 2.953 (2.948) 2.953 (2.948) 2.949b } CO values in paranthese were obtained with hard potentials at 700 eV a NUMOL, R.M. Dickson, A.D. Becke, J. Chem. Phys. 99,3898 (1993), b Gaussian94 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 32 PAW — AE methods: molecules - energetics 400 eV plane wave cutoff xc PBE PBE rPBEb rPBEb meth PAW AEa PAW AEa CO 11.65 11.66 11.15 11.18 11.24 N2 10.39 10.53 10.09 10.09 9.91 NO 7.31 7.45 6.95 7.01 6.63 O2 6.17 6.14-6.24 5.75 5.78 5.22 exp a S. Kurth, J.P. Perdew, P. Blaha, Int. J. Quantum Chem. 75, 889 (1999). b revised Perdew Burke Ernzerhof functional, B. Hammer, L.B. Hansen, J.K. Norskov, Phys. Rev. B 59, 7413 (1999). G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 33 PAW — US-PP method: bulk, semiconductors a Å3 Ecoh (eV) B (MBar) diamond results for the equilibrium lattice constant a, cohesive energy Ecoh (with respect to non spin polarised atoms) and bulk modulus B for several materials calculated with the the PAW, USPP, and the FLAPW approach US-PP(current) 3.53 -10.15 4.64 PAW(current) 3.53 -10.13 4.63 LAPWa 3.54 -10.13 4.70 PAWa 3.54 -10.16 4.60 US-PP(current) 5.39 -5.96 0.95 PAW(current) 5.39 -5.96 0.95 LAPWa 5.41 -5.92 0.98 PAWa 5.38 -6.03 0.98 silicon G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 34 PAW — US-PP method: bulk, metals a Å3 Ecoh (eV) B (MBar) bcc V results for the equilibrium lattice constant a, cohesive energy Ecoh (with respect to non spin polarised atoms) and bulk modulus B for several materials calculated with the the PAW, USPP, and the FLAPW approach US-PP(current) 2.93 -9.41 2.02 PAW(current) 2.93 -9.39 2.09 LAPWa 2.94 -9.27 2.00 PAWa 2.94 -9.39 2.00 US-PP(current) 5.34 -2.20 0.0181 PAW(3s3p val) 5.34 -2.19 0.0187 PAW(3p val) 5.34 -2.20 0.0187 LAPWa 5.33 -2.20 0.019 PAWa 5.32 -2.24 0.019 fcc Ca G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 35 PAW — US-PP method: bulk, ionic compounds a Å3 results for the equilibrium lattice constant a, cohesive energy Ecoh (with respect to non spin polarised atoms) and bulk modulus B for several materials calculated with the the PAW, USPP, and the FLAPW approach Ecoh (eV) B (MBar) CaF2 a US-PP(current) 5.36 -6.32 0.97 PAW(3s3p val) 5.35 -6.32 1.01 PAW(3p val) 5.31 -6.36 1.00 LAPWa 5.33 -6.30 1.10 PAWa 5.34 -6.36 1.00 N.A.W. Holzwarth, et al.; Phys. Rev. B 55, 2005 (1997) G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 36 Semi-core states; alkali and alkali earth metals from a practicle point of view, an accurate treatment of these elements in ionic compounds is very important: oxides e.g. perovskites strongly ionized, and small core radii around 2.0 a.u. (1 Å) are desirable e.g. Ca: one would like to treat 3s, 3p, 4s states as valence states wave-function 3 3s : E= -3.437 R c=2.3 2 3p : E= -2.056 R c=2.3 it is very difficult to represent the charge distribution of the 3s and 4s states equally well in a pseudopotential approaches 1 4s : E= -0.284 R c=2.3 0 0 1 2 R (a.u.) 3 4 general problem for pseudopotentials G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 37 Semi core states in VASP, NC wavefunctions describe the augmentation charges φ̃i r φ̃ j r nc φnc i r φj r Qi j r it is very difficult to construct accurate NC-PP for 3s and 4s (mutual orthogonality) wave-function 3 3s : E= -3.437 R c=2.3 2 3p : E= -2.056 R c=2.3 1 4s : E= -0.284 R c=2.3 0 0 1 2 R (a.u.) 3 node in 4s must be included in some way if one succeeds, the augmentation charges become quite hard, and require fine regular grids PAW method is the best solution to this problem 4 G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 38 PAW — US-PP method: atoms O N ∆Em 1.55 1.40 1.41 gs 2s2 2p3 2s2 2p3 2s2 2p3 ∆Em 3.14 2.88 2.89 gs 3d 6 2 4s1 8 3d 6 2 4s1 8 3d 6 2 4s1 8 ∆Em 2.95 2.77 2.76 gs 3d 7 7 4s1 3 3d 7 7 4s1 3 3d 7 7 4s1 3 ∆Em 1.40 1.32 1.31 gs 3d 9 4s1 3d 9 4s1 3d 9 4s1 ∆Em 0.54 0.52 0.52 2s2 2p4 2s2 2p4 2s2 2p4 gs AE + PAW Fe ~ comparison of GGA PAW, US-PP and scalar relativistic all-electron calculations for O, N, Fe, Co and Ni magnetic energy: EM gs ENM 4s1 3d n 1 (in ∆Em eV) US-PP Ni Co for atoms the magnetisation is wrong by 5-10% with US-PP G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 39 Bulk properties of Fe energy differences between different phases of Fe FLAPWa bcc Fe NM US-AE US-PP 373 372 369 bcc Fe FM -73 -73 -73 -191 fcc Fe NM 78 61 62 62 0 0 0 hcp Fe NM a PAW FLAPW, L. Stixrude and R.E. Cohen, Geophys. Res. Lett. 22, 125, (1995). PAW and FLAPW give almost identical results US-PP overestimates the magnetisation energy by around 5-10 % calculations for other systems indicate that the accuracy of the PAW method for magnetic systems is comparable to other AE methods – α-Mn, bulk Cr and Cr surfaces, LaMnO3 (perovskites) G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 40 Why do pseudopotentials fail in spin-polarised calculations non linear core corrections were included in the PP’s ! pseudo-wavefunction for a normconserving pseudopotentials wave-function 3 — all electron s : E= -0.297 R c=2.2 – – pseudo 2 the peak in the d-wavefunction is shifted outward to make the PP softer p1/2: E= -0.094 R c=2.5 1 d3/2: E= -0.219 R c=2.0 0 0 1 2 3 4 R (a.u.) similar compromises are usually made in US-pseudopotentials as a result, the valence-core overlap is artifically reduced and the spin enhancement factor ξ r is overestimated mr nvalence r ncore r ξr G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 41 Oxides CeO2 and UO2 CeO2 PAW FLAPW Exp 5.47 Å 5.47 Å 5.41 Å 172 GPa i 176 GPa 236 GPa UO2 PAW FLAPW Exp a (Å3 ) 5.425 Å 5.46 Å B 200 GPa 209 GPa a (Å3 ) B G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 42 Computational costs, efficiency code complexity of the kernel of course increases with PAW local pseudopotentials low 2000 lines NC-PP pseudopotential low-medium 7000 lines US-PP medium 10 000 lines PAW medium-high 15 000 lines parallelisation or error removal becomes progressively difficult computational efficiency: Ge 64 atoms, 1 k-point (Γ), Alpha ev6 (500 MHz), 14 electronic cycles type cutoff time total error per atom NC-PP 140 eV 514 sec 400 meV NC-PP 240 eV 1030 sec 10 meV US-PP 140 eV 522 sec 10 meV PAW 140 eV 528 sec 10 meV the extra costs for PAW or US-PP’s at a fixed cutoff are small PAW method is particularly good for transition metals and oxides G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 43 PAW advantages formal justification is very sound improved accuracy for: – magnetic materials – alkali and alkali earth elements, 3d elements (left of PT) – lathanides and actinides generation of datasets is fairly simple (certainly easier than for US-PP) AE wavefunction are available comparison to other methods: – all test indicate that the accuracy is as good as for other all electron methods (FLAPW, NUMOL, Gaussian) – efficiency for large system should be significantly better than with FLAPW G. K RESSE , P SEUDOPOTENTIALS (PART II) AND PAW Page 44