Projector-Augmented Wave Method:

Transcription

1 Projector-Augmented Wave Method: An introduction Peter E. Blöchl Clausthal University of Technology Germany Juli 2003

2 Why PAW all-electron wave functions (EFG s, etc.) ab-initio molecular dynamics (Fictitious Lagrangian) exact when converged (no transferability problems) computationally efficient provides theoretical basis for pseudopotentials PAW unifies all-electron and pseudopotential approaches

3 PAW Transformation theory Transform physical wave functions Ψ(r) onto auxilary wave functions Ψ(r) Ψ(r) = ÛΨ(r) Goal: Smooth auxilary wave functions Ψ(r) that can be represented in a plane wave expansion

4 PAW Transformation theory start with auxilary wave functions Ψ n (r) define transformation operator ˆT = Û 1 Ψ n (r) = ˆT Ψ n (r) Ψ n (r) = ÛΨ n (r) that maps the auxiliary wave functions Ψ n (r) onto true wave functions Ψ n (r) express total energy by auxiliary wave functions E = E[Ψ n (r)] = E[ ˆT Ψ n (r)] Schrödinger-like equation for auxiliary functions E ( ) Ψ n (r) = T HT T T ɛ n Ψ n (r) = 0 Expectation values A = n Ψ n A Ψ n = n Ψ n T AT Ψ n Find a transformation ˆT so, that the auxiliary wave function are well behaved

5 Requirements for a suitable transformation operator the relevant wave functions shall be transformed onto numerically convenient auxiliary wave functions Ψ n ( r) = G e i G r Ψ n ( G) linear (algebraic operations) local (no interaction between sites) T = 1 + R S R

6 PAW Transformation operator I How to define the operator T? define a complete set of initial and final states for the transformation i φ i = T φ i T 1. Final states: All-electron valence partial waves φ i }{{} solve Schrödinger equation for the i= R,l,m,α all-electron atomic potential for a set of energies 2. Initial states: auxiliary partial waves φ i true and auxiliary wave function are pairwise identical outside some augmentation radius r c φ i ( r) = φ i ( r) for r R > r c auxiliary partial waves are smooth s-type p-type d-type

7 PAW Transformation operator II Find a closed expression for the transformation operator T = 1 + R S R Derivation: true {}}{ φ i = aux. {}}{ aux. φ {}}{ i +S Ri φ i }{{} T φ i S φ i = φ i φ i = j T = 1 + j ( φ j φ ) j p j φ i }{{} δ i,j ( ) φ j φ j p j } {{ } S R Projector functions p i must be localized within its own augmentation region obey bi-orthogonality condition p i φ j = δ i,j projector functions p are not yet uniquely determined: closure relation will be explained later

8 PAW Projector functions p i s-type projector functions p-type projector functions d-type projector function Projector functions probe the character of the wave function

9 Reconstruction of the true wave function Using the transformation operator T = 1 + ( φ j φ ) j p j, j the all-electron wave function obtaines the form: Ψ n = Ψ n + j ( φ j φ ) j p j Ψ n the projector functions probe the character of the auxiliary wave function, this wrong character is replaced by the correct character

10 PAW Augmentation Example: p-σ orbital of Cl 2 Ψ = Ψ + Ψ 1 Ψ 1 = Ψ + i ( φ i φ ) i p i Ψ Ψ, Ψ Ψ, Ψ 1 Ψ, Ψ 1 Ψ 1, Ψ 1

11 Numerical representation auxiliary wave function Ψ n is represented as plane wave expansion partial waves φ(r), φ(r) are represented as radial functions on a logarithmic radial grid r i = r 1 α i 1 r i+1 = αr i multiplied with real spherical harmonics. projector functions are Bessel-transformed from a radial grid into Fourier space projector functions are smooth and local scalar product p Ψ n evaluated in Fourier space

12 PAW Expectation values Expectation value for a sufficiently local one-particle operator A = n f n Ψ n A Ψ n = n f n Ψ n A Ψ n plane wave part + i,j D i,j φ i A φ j one-center exps. of true WF i,j D i,j φ i A φ j one-center exps. of aux. WF + n Ψ c n A Ψc n core contribution with a one-center density matrix D i,j = n p j Ψ n f n Ψ n p i and the core states Ψ c n! Similarity to pseudopotentials: A = n à = A + i,j Ψ n Ã Ψ n ( p i φ i A φ j φ i A φ ) j p j Pseudo-operator has the form of a separable pseudopotential. PAW provides a rule to obtain expectation values in a pseudopotential calculation.

13 PAW Electron Density Electron density n(r) turns into a plane wave part ñ(r) and two one-center components n 1 (r) and ñ 1 (r) n(r) = ñ(r) + n 1 (r) ñ 1 (r) = n f n Ψ n (r) Ψ n (r) + ñ c + i,j φ i (r)d i,jφ j (r) + n c i,j φ i (r)d i,j φ j (r) ñ c D i,j = n p i Ψ n f n Ψ p j Electron density divided (like the wave function) into plane wave part two expansions per atom in radial functions times spherical harmonics

14 PAW Total energy Total energy divided into plane wave integral and two one-center expansions per atom Plane wave part: E([ Ψ n ], R i ) = Ẽ + E 1 Ẽ 1 Ẽ = f n Ψ n Ψ n n + E H [ñ(r) + ˆn(r)] + E xc [ñ(r)] + One-center expansion of plane wave part d 3 r v(r)ñ(r) Ẽ 1 = n D i,j φ i φ j + E H [ñ 1 (r) + ˆn(r)] + E xc [ñ 1 (r)] + d 3 r v(r)ñ 1 (r) One-center expansion of true density E 1 = n D i,j φ i φ j + E H [n 1 (r) + Z(r)] + E xc [n 1 (r)] Everything else (Hamiltonian, Forces) follows from this total energy functional

15 PAW Approximations The following approximations have been made frozen core approximation (can be overcome) truncate plane wave expansion (basisset) truncate partial wave expansion (augmentation) Convergence: plane wave convergence comparable to ultrasoft pseudopotentials (E P W =30 Ry) 1-2 partial waves per site and angular momentum sufficient Charge and energy transferability problems of the pseudopotential approach are under control

16 The auxiliary Hamiltonian effective Schrödigner-like equation for auxiliary wave functions ( H ɛ n Õ) Ψ n = 0 where H = T HT = ṽ + i,j p i h i,j p j Õ = T T = 1 + i,j p i o i,j p j have the form of a separable pseudopotential h i,j and o i,j have closed expressions h i,j = φ i v φ j φ i ṽ φ j o i,j = φ i φ j φ i φ j are evaluated atom-by-atom on radial grids with spherical harmonics depend on the actual, non-spherical potential

17 Closure relation for projector functions Closure: Auxiliary partial waves obey the effective PAW Schrödinger Eq. for the atom ṽ ɛ k + i,j p i (h i,j ɛ k o i,j ) p j φ k = 0 This restricts the Hilbert space for the projector functions [ 12 ] 2 + ṽ ɛ j p i = j φ j c j,i Coefficients c i,j are determined from the biorthogonality condition. p i φ j = δ i,j Converging set: Additional degree of freedom ( φ i φ i ) p i = ( φ i φ 1 i )A }{{ i,k } k,j p j }{{} i,j,k i φ k φ k A p k is used to provide a set that can be truncated for any number of partial waves. Projector functions are uniquely defined with the definition of partial waves

18 Comparison with pseudopotentials PAW Total energy: E = Ẽ + R (E 1 R Ẽ R ) = Ẽ + E with therefore E = E(D i,j ) E(D i,j ) = E(D at i,j ) + i,j E ( ) D i,j D at i,j D i,j +... = E self + n f n Ψ v ps Ψ +... PAW calculates this expression exactly the PP method truncates the Taylor expansion after the linear term charge transferability problem! PAW creates pseudopotentials that adjust to the instantaneous electronic structure

19 Comparison with Ultrasoft Pseudopotentials For an identical implementation both methods should require same computational effort same plane wave cutoff for the wave functions Implementations differ: USPP: Plane wave cutoff for the density equivalent to that of norm-conserving pseudopotentials support grids required PAW uses one-center expansions with radial grids and a smooth compensation density lower cutoff for the density than USPP

20 Where do we need PAW? [Kresse,Joubert PRB59,1759(1999)] Magnetization energies Pseudopotentials introduce errors of order 0.06 ev/µ B Energy difference between magnetic and non-magnetic iron LAPW PAW US-PP Fe(bcc) ev ev ev Fe(fcc) ev ev ev Fe(hcp) Surface magnetism of V(001): LAPW PAW US-PP magnetic moment V(001) 0µ B 0 µ B 0.75 µ B alkali, alkaline earth, and early transition metals Lattice constant of CaF 2 too small by 2% in PSP calculations Summary: Pseudopotentials are accurate in most cases Pseudopotentials can be made accurate, but at a tremendous computational cost (hard pseudopotentials) PAW is comparable to other all-electron schemes (LAPW)

21 Electric field gradients Comparison with LAPW H. Petrilli, P.E. Blöchl, P. Blaha, K. Schwarz, PRB 57,14690 (1998) V zz [V/m 2 ] PAW-LAPW expt-lapw Cl % 2.8 % Br % -0.9 % I % -4.9 % Li 3 N:Li(1) % % Li 3 N:Li(2) % % Li 3 N:N % -8.0 % Fe(C 5 H 5 ) % 23.5 % FeS 2 (pyrite) % 5.5 % FeS(marcasite) % % FeSi % -9.6 % Fe 2 O % 8.5 % TiO 2 :Ti % 9.1 % TiO 2 :O % 11.7 % Cu 2 O % 22.9 % average 4.4 % 12.0 % PAW is three times better than experiment!

22 Hyperfine parameters PAW expt PP Si Si(1a) Si(1b) H 0 in SiO2 H 0 in vacuum PAW expt H PAW exact H

23 PAW options (CP-PAW Code) ab-initio molecular dynamics (Car-Parrinello) all-electron wave functions and densities electric field gradients(efg), hyperfine parameters gradient corrected density functionals (various forms) spin unrestricted, non-collinear spin isolated molecules and extended crystals QM-MM coupling activation energies crystal orbital populations (local chemical bond analysis) general k-points variable occupations and finite electron-temperature variable cell shape (Parrinello-Rahman) LDA+U GW approximation object oriented program architecture (Fortran 90) efficiently parallelized (MPI) portable (Intel-Linux, IBM-AIX, Dec-Alpha-Linux) ) Strasbourg version

24 Implementations of PAW CP-PAW; P. Blöchl, Clausthal University of Technology PWPAW; A. Tackett, N. Holzwarth, Wake Forest U., USA NWChem; M. Valiev, J.H. Weare. UC San Diego VASP Code; G. Kresse et al., Vienna University F. Mauri, Princeton University *EStCoMPP; S. Blügel, Osnabrück University AbInit, X. Gonze, PCPM Louvain-la-Neuve, Belgium *DFT++; S. Ismail-Beigi, T. Arias, UC Berkeley, Cornell U. *SFHIngX (Planned), S. Boeck+Neugebauer+Co., Berlin

25 Conclusion all-electron method for ab-initio molecular dynamics rigorous theoretical basis accurate and efficient extends and joins all-electron methods with the pseudopotential approach pseudopotentials as approximation of PAW extends the tradition of Linear Methods (LMTO,LAPW) Projector Augmented Wave Method P.E. Blöchl, Phys. Rev. B 50, (1994) The Projector Augmented Wave Method: Ab-initio Molecular Dynamics Simulations with Full Wave Functions P.E. Blöchl, C. Först and J. Schimpl, Bull. Mater. Sci. 26, 33 (2003).