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. 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.

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 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.

3 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? 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.

4 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. 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).

5 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. 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 #.

6 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 .. 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.

7 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. 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.

8 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 . ;: ;: ;: ;: ;: ;: ;: ?> ?> ?> ?> ?> ?> ?>. AE-onsite ;: ;: ;: ;: ;: ;: ;: ?> ?> ?> ?> ?> ?> ?>. , ;: ;: ;: ;: :; ;: ;: =< =< =< =< =< =< =< <=. 98 98 98 98 98 98 98 =< =< =< =< =< =< =< =<.

9 + . 98 98 98 98 98 98 98 =< =< =< =< =< =< =< =<. US-PP method is in principle an exact frozen core all-electron method 98 98 98 98 98 98 98 =< =< <= =< =< <= =< =<. i j i - 98 98 89 98 98 89 98. n n pi p j .. +. , i j /. pseudo-onsite . Vloc j . pi j US-PP: what they really do .. , G. K RESSE , P SEUDOPOTENTIALS ( part II) and paw .. + . onsite occupancy matrix (or density matrix): i j i j i - . - n +. p i .. pseudo energy is the sum of three terms Vloc character of wave function: ci , . +. =.. 76 76 76 76 76 76 76. 32 32 32 32 32 32 32 76 76 76 76 76 76 76. 32 32 32 32 32 32 32 76 76 76 76 76 76 76. 32 32 32 32 32 32 32 76 76 76 76 76 76 76. 32 32 32 32 23 32 32 54 54 54.

10 AE. 54 54 54 54 45. 10 10 10 10 10 10 10 54 54 54 54 54 54 54 54. 10 10 10 10 10 10 10 54 54 54 54 54 54 54 54. E. 10 10 10 10 10 10 10 54 54 45 54 54 45 54 54. 10 10 01 10 10 01 10. 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 i $. %. one for the AE wave functions i $. %. 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 i j . n pi p j . n.