Pseudopotentials (Part II) and PAW - VASP

1 Pseudopotentials ( part II) and paw . Georg KRESSE. Materialphysik and Center for Computational Materials Science Institut fur Universit at Wien, Sensengasse 8, A-1090 Wien, Austria b-initio ackage ienna imulation G. K RESSE , P SEUDOPOTENTIALS ( part II) and paw 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.

2 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.

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 r .. replace exact r inside pseudization radius by .. a suitable soft pseudo wave function r must fulfill some continuity conditions .. i i i r r rc . r .. r . r rc .. rc n rc n for n 0 2.. possibly impose normconservation condition rc rc 4 r r dr 2 2. 4 r 2 r2 dr .. 0 0. G. K RESSE , P SEUDOPOTENTIALS ( part II) and paw Page 4. Which expansion set should one use?

4 Many different basis sets have been proposed in the literature presently the two most prominent ones are polynomials (Troullier and Martins). r c0 c2 r 2 c4 r 4 c6 r 6 c8 r 8 c10 r10 c12 r12.. spherical Bessel-functions (RRKJ Rappe, Rabe, et. al.). 34. j l q i rc rc i j l .. r . qi r with qi such that .. rc .. j l q i rc .. i 1.. 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.

5 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. r i jl .. qi r .. i 1.. h 2 2. find maximum qi Ecut . 2me max qi 15.. 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.

6 K RESSE , P SEUDOPOTENTIALS ( part II) and paw Page 6. Factorization required to speed up the calculations Bylander, et al., Phys. Rev. B 46, 13756 (1992). chose local reference potential Vloc construct a projector such that p 1.. h 2. p Vloc . 2me .. the factorized Hamiltonian is given by h 2. H Vloc pD p . 2me .. h 2. with D 2me Vloc .. ! . ". h 2. one recognizes immediately that: 2me Vloc pD p .. ! . ".. G. K RESSE , P SEUDOPOTENTIALS ( part II) and paw Page 7. wave-function s : E= R c= What have we acchived at this point p : E= R c= 0 1 2 3 4.

7 R ( ). the exact wavefunction has been replaced by it's pseudo counterpart and a pseudo Hamiltonian has been constructed h 2. VAE .. 2me .. h 2. Vloc pD p .. 2me .. at the energy , and are identical outside of the cutoff radius and have the same norm inside the cutoff radius at the energy and #.. are identical outside of the cutoff radius G. K RESSE , P SEUDOPOTENTIALS ( part II) and paw Page 8. Two reference energies pseudize at two reference energies: i i 12.. $.. %. construct two projectors such that pi j i j for all i j.

8 H 2. pi i j Vloc j j . 2me .. j factorized Hamiltonian is given by h 2. H Vloc p i Di j p j . 2me .. ij h 2. Di j i 2me Vloc j j .. ! . ". one recognizes immediately that: i H j j i j .. 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 rc 4 i r j r r dr 2. 4 i r j r r2 dr i j . &.. 0 0. in the VASP PP generation program only rc rc i r i r r dr 2. i r i r r2 dr .. 0 0. is enforced to correct for this error, augmentation charges would be required, but these are neglected as a result Di j is not Hermitian h 2.

9 Di j i 2me Vloc j 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 construct two projectors such that pi j i j for all i j .. h 2. pi i j Vloc j j . 2me .. j the factorized Hamiltonian and overlap operator are given by h 2. H Vloc p i Di j p j S 1 p i Qi j p j .. 2me .. ij ij h 2. Di j i 2me Vloc j j j Qi j Qi j i j i j .. ! .. ". one can show that: i H j i S j j.

10 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 h 2. Di j i Vloc j j j Qi j 2me .. 2. h . i Vloc j j i j Qi j . 2me .. h 2. i Vloc j j i j i j i j . 2me .. h 2. i Vloc j i j j . 2me .. h 2 h 2. i Vloc j i VAE j . 2me 2me .. '. *. '. *. (). (). energy pseudo onsite energy AE onsite G. K RESSE , P SEUDOPOTENTIALS ( part II) and paw Page 12. Page 13. / ?> ?> ?> ?> ?> ?> ?>. ;: ;: ;: ;: ;: ;: ;: ?> ?> ?> ?> ?> ?> ?>. VAE j . ;: ;: ;: ;: ;: ;: ;: ?