Advancements in Self-Consistent Meta-GGA PAW Datasets

Progress on self-consistent meta-gga PAW datasets from ATOMPAW Natalie Holzwarth, Physics Dept., Wake Forest U. Winston-Salem, NC, USA Collaborators: Marc Torrent, CEA also Michel C t and Surya Timilsina, U. Montreal Acknowledgements: NSF DMR-1940324, WFU DEAC cluster Outline 1. Motivation and brief history 2. Numerical issues associated with SCAN functional 3. Self-consistent atomic solver and results for r2SCAN 4. Pseudopotential construction methods for meta-GGA and results for r2SCAN 5. Summary and outlook 6/1/2021 ABINIT Developer Workshop 2021 1

To do list Develop a self-consistent all electron atomic solver for the generalized Kohn-Sham equations needed to evaluate exchange-correlation functionals having the meta-GGA form. Adapt pseudization schemes for basis and projector functions for use in the generalized Kohn-Sham formalism. Test and evaluate. 6/1/2021 ABINIT Developer Workshop 2021 2

Motivation and brief history As density functional theory develops, new exchange correlation functionals are frequently proposed and, thanks to Libxc (Marques et al. Comp. Phys. Comm. 183, 2272 (2012), Lehtola et al. Software X 7, 1 (2018)), they can be incorporated into various codes. The meta-GGA functional form adds the kinetic energy density into the functional; considerable improvements in DFT prediction of materials properties has been reported within the context of generalized DFT (Yang et al. PRB 93, 205205 (2016)) SCAN form of meta-GGA -- Strongly Constrained and Appropriately Normed Semilocal Density Functional (Sun et al. PRL 115, 036402 (2015)) demonstrates improved materials prediction for a variety of systems; >950 citations as of 5/2021. 6/1/2021 ABINIT Developer Workshop 2021 3

Motivation and brief history continued -- However, most of the SCAN results were obtained using codes with localized basis sets or VASP which appears to use a special form of SCAN functional. Several authors reported numerical difficulties in using plane wave codes with the SCAN functional (Yao et al. JCP 146, 224105 (2017), Bart k et al. JCP 150, 161101 (2019) & rSCAN) We found that the effective potentials for the SCAN form analytically diverges in regions of space where the radial wavefunction decreases exponentially. A revised functional was introduced by the Tulane and Temple groups ( Accurate and Numerically Efficient r2SCAN Meta- Generalized Gradient Approximation , Furness et al., J. Phys. Chem. Lett 11, 8208 (2020)). 6/1/2021 ABINIT Developer Workshop 2021 4

Functional form of Vxc(r) for textbook model of He atom thanks to L. Schiff, Quantum Mechanics (1955) 3 2 27 16 A = = 2 1 Ar ( ) n r Bohr e A 2 3 2 Ae = 2 2 Ar ( ) r A 2 m Unphysical behavior of Vxc(r) for SCAN discouraged further development. Better behavior of Vxc(r) for r2SCAN allowed for resumption of project. 6/1/2021 ABINIT Developer Workshop 2021 5

With the introduction of the r2SCAN functional, work on developing a self-consistent atomic solver for meta-GGA formalism resumed. General form of the exchange-correlation functional: = ( ), ( )) r 3 2 ( ( ), ( ), n r r r E d r f n x c xc ( ) having occupancy r In ter s m of single par ticle sta tes , w i i 2 2 2 = ( ) r r r r ( ) r ( ) r ( ) r ( ) ( ) ( ) n w n n w i i i i 2 m i i The generalized Kohn-Sham equations take the form 2 f ( ) ( ) = + + 2 ( ) r ( ) r ( ) where r r ( ) dimensionless kinetic potential xc H V V V eff 2 m + + ( ) = r ( ) r ( ) r ( ) r V electron-nucleus V V V e f f Hart r e e xc f f f = + 2 ( ) r 2 xc n xc xc V n ( ) xc 2 n 6/1/2021 ABINIT Developer Workshop 2021 6

Some details for spherical atom ( ) r r n l = r ( ) r ( ) occupancy for state nili Y i i i l m i i 2 1 1 2 = = ( ) n r ( ) ( ) r ( ) r w r w n l n l n l n l 2 2 4 2 4 r m r i i i i i i i i n l n l i i i i 2 2 ( ) r ( ) r r ( ) r r d n l dr n l n l = + + ( ) r ( 1) l l i i i i i i n l i i i i Self-consistent generalized Kohn-Sham equations: ( ) = ( ) ( ) r 0 H r n i i l n i i l + 2 2 ( 1) 1 r dV dr d dr l l d dr ( ) = + + + ( ) 1 ( ) r H r V V eff 2 2 2 m r 6/1/2021 ABINIT Developer Workshop 2021 7

Differential equations to solve numerically Method 1: Coupled first order equations dy z d - - ( ) r r ( ) r dy = = ( ) r y r ( ) ( ) ( ) r y r 1 2 z 12 2 21 1 dr in ATOMPAW ( ) r d ( ) = = + with ( ) ( ) ( ) r 1 ( ) nl dr y r y r V r 1 2 nl + ( ) 1 V r ( 1) 1 r+ 2 dV r dr l l mV ( ) ( ) = = + + ( ) r ( ) r 1 ( ) ( ) r z z V r ( ) 12 21 eff nl + 2 2 1 ( ) r M ethod 2: Tra nsformed seco ( ) 1 ( ) V r + n d o rder equatio -- n 1 2 y r ( ) = = + ( ) x r Le t ( ) r ( ) r e whe re ( ) ln 1 ( ) nl y x r V r nl nl 2 + 2 2 ( ) r ( ) r V r + ( ) 2 r ( ) ( ) dr ( 1 ) 2 d y V d x r dr dx r dr dx r l l m + + = ( ) 0 eff 1 nl nl y nlr 2 2 2 2 ( ) dr r 6/1/2021 ABINIT Developer Workshop 2021 8

Some further details In developing the self-consistent atomic solver, we found some further numerical issues with r2SCAN which could be ameliorated by making one further small change in the formulation. = 2 n n 3 10 n ( ) 2/3 = = 2 3 W n unif W + 8 unif W Original choice: =0.001 Better numerics: 0.01 6/1/2021 ABINIT Developer Workshop 2021 9

Comparison of self-consistent r2SCAN potentials for atomic S with =0.01 and =0.001. 6/1/2021 ABINIT Developer Workshop 2021 10

Some results for V (r) comparing r2SCAN =0.01 (Furness, 2020) with rSCAN (Bart k, 2019) Al B Ne Ar 6/1/2021 ABINIT Developer Workshop 2021 11

Some results for Vxc(r) comparing r2SCAN =0.01 (Furness, 2020) with rSCAN (Bart k, 2019) Ar Ne B Al 6/1/2021 ABINIT Developer Workshop 2021 12

Self-consistent total energy results Find that self- consistency is hard to achieve for some atoms with the =0.001 version of r2SCAN. 6/1/2021 ABINIT Developer Workshop 2021 13

How does the r2scan Vxc(r) compare with other functionals? 6/1/2021 ABINIT Developer Workshop 2021 14

Construction of PAW datasets Frozen core approximation: All electron treatment Pseudo electron treatment ( ) ( ) ( ) core r + ( ) vale r + + + ( ) r ( ) r ( ) r ( ) r core n r vale n r core n vale n core vale Core functions for Si atom -- M P m r C r r r m c = 2 4 ( ) r r core n = 0 n m r 2 4 ( ) r r r co re c = = rc 2; 4 P M M P m r T r r r m c = 2 4 ( ) r r = 0 m r core 2 4 ( ) r r r core c = = 4; 4 P M 6/1/2021 ABINIT Developer Workshop 2021 15

Construction of PAW datasets -- continued The datasets are focused on the valence electrons and there are many formulations which can be adopted for use with the generalized Kohn-Sham PAW equations (although some schemes may be more difficult). Valence functions for Si atom 2 4 / (Ry/Bohr) r vale vale rc 2 4 / (1/Bohr) r n n vale vale 6/1/2021 ABINIT Developer Workshop 2021 16

Construction of PAW datasets -- continued Dimensionless kinetic potential and kinetic pseudopotential for Si for r2SCAN =0.01 V V 6/1/2021 ABINIT Developer Workshop 2021 17

Construction of PAW datasets -- continued Exchange-correlation potential and pseudopotential for Si for r2SCAN =0.01 rV xc rV xc 6/1/2021 ABINIT Developer Workshop 2021 18

Construction of PAW datasets -- continued Various schemes to constructed screened pseudopotential http://pwpaw.wfu.edu 6/1/2021 ABINIT Developer Workshop 2021 19

Construction of PAW datasets -- continued Various schemes to constructed screened pseudopotential VPS(r) http://pwpaw.wfu.edu specifically from Marc Torrent s usersguide For now, added VPSMATCHNC/VPSMATCHNNC similar to 6/1/2021 ultrasoft with or without norm conservation. ABINIT Developer Workshop 2021 20

Construction of PAW datasets -- continued Determine ionic pseudopotential by unscreening VPS(r) = PS ( ) r n ( ) r ( ) r ( ) r ( ) r V V V V unit compensation char ge Z H xc c ( ) ( ) = + ( ) Q n r 2 2 ( ) r 4 ( ) r V e n 3 ( ) ( ) Q d r n r n r vale H vale vale Ionic pseudopotential for Si for r2SCAN =0.01 in comparison with LDA and GGA Si VPSMATCHNC option 6/1/2021 ABINIT Developer Workshop 2021 21

To do list Develop a self-consistent all electron atomic solver for the generalized Kohn-Sham equations needed to evaluate exchange-correlation functionals having the meta-GGA form. 3 4 Adapt pseudization schemes for basis and projector functions for use in the generalized Kohn-Sham formalism. Test and evaluate. Still to do 6/1/2021 ABINIT Developer Workshop 2021 22

Additional considerations How sensitive are the results to the dataset? Given apparently reasonable results for r2SCAN =0.01, should we stick with that form or work harder on the solver? (Perhaps should also consult with Tulane/Temple developers.) Not all pseudization schemes are easily converted for use in generalized density functional formulation due to the dimensionless kinetic energy potential term. Which ones should we try to preserve? Further considerations? 6/1/2021 ABINIT Developer Workshop 2021 23