Progress on self-consistent meta-gga PAW datasets from …

6/1/2021 ABINIT Developer Workshop 2021 1

Progress on self-consistentmeta-gga PAW datasets from ATOMPAW

Natalie Holzwarth, Physics Dept., Wake Forest U. Winston-Salem, NC, USA

Collaborators: Marc Torrent, CEA also Michel Côté and Surya Timilsina, U. Montreal

Acknowledgements: NSF DMR-1940324, WFU DEAC cluster

Outline1. Motivation and brief history2. Numerical issues associated with SCAN functional3. Self-consistent atomic solver and results for r2SCAN4. Pseudopotential construction methods for meta-GGA

and results for r2SCAN5. Summary and outlook


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.


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.


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), Bartók 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)).


Functional form of Vxc(r) for textbook model of He atomthanks to L. Schiff,Quantum Mechanics(1955)

32 1

22 3

2

2 27( )

(

Bohr16

22

)

Ar

Ar

An r e A

A er Am

π

τπ

− −

−

= =

=

Unphysical behavior of Vxc(r) for SCAN discouraged further development.

Better behavior of Vxc(r) for r2SCAN allowed for resumption of project.


With the introduction of the r2SCAN functional, work on developing a self-consistent atomic solver for meta-GGA formalism resumed.

3 2

2

General form of the exchange-correlation functional:

In ter s

( ( ), ( ), ( ), ( ))

( ) having occupancy

( ) ( (

m of single par

(

ticle sta

tes

) ) ) ( )

,xc

i i

c

ii

x

i

r f n n

w

n w n

E d

n

σ τ

σ

∇

Φ

≡ Φ ≡ ∇

=

⋅∇∑

∫ r r r r

r

r r r r r2

2( ) ( )2 i i

iw

mτ = ∇Φ∑r r

( )( )

( )

e

22

e

2

eff

electron-nucleus Hartf r

2

ef

The generalized Kohn-Sham equations take the form

( ) ( ) ( ) where ( )2

( ) = ( ) ( ) ( )

( ) 2

xc

xc xc xc

xc

xc

fV V

V

f f f

H Vm

V

nn n

V V

V

τ τ τ

σ

∂∇ +∇⋅ ≡

∂

∂ ∂ ∂ ∇ ⋅ ∇ +∇

= − ∇ +

=∂ ∂ ∂ ∇

+

−

+

r r r r

r r r r

r

dimensionless“kinetic potential”


Some details for spherical atom

( )2 2

2 2 eff( 1) 1( ) 1 ( )

2dVd l l dH r V V r

m dr r dr dr rτ

τ

+ = − + − + − +

22

2 2

2 2

( )ˆ( ) ( )

1 1( ) ( ) ( ) ( ) 4 2 4

( ) ( ) ( ) ( ) ( 1)

i i

i i

i i i i i i i ii i i i

i i i i i i

i i

n li l m

n l n l n l n ln l n l

n l n l n ln l i i

rY

r

n r w r r w rr m r

d r r rr l

dr r rl

ϕ

ϕ τ τπ π

ϕ ϕ ϕτ +

Φ =

= =

= − +

∑ ∑

r r

( )Self-consistent generalized Kohn-Sham equations:

( ) ( ) 0i i i in nl lH r rϕ− =

occupancy for state nili


Differential equations to solve numerically

( )

( ) ( )

112 2 21 1

1

12

2

21 2

2

( ) ( )( ) ( ) ( ) ( )

( )with ( ) ( ) ( ) 1 ( )

( )1 ( 1) 1

Method 1: Coupled first order equations

( ) ( ) 1 ( )

-

1

-

( )

nlnl

dy dyz zd

r rr y r r y r

d ry r r y r V rdr

dV rl lz r z r

r dr

V rV r r dr

τ

ττ

τ

ϕϕ

=

= = +

+=

+

=

= + + ( )eff2

2 ( ) nlm V r

r+ −

( )( )

22 2eff

2 2 2 2

M

)

ethod 2: Tra

1

nsformed seco

(

n( ) 1( ) ( ) whe

d n

re ( ln ( )21 ( )

( ) ( )( ) 2 ) ( ) ( 1

o

(

rder equatio --

Le

)

t

) 21

x rnlnl nl

nl nl

y rr y r e x r V rV r

d y r V rd x r dx r dx r l l mdr dr r dr dr r V r

ττ

τ

ϕ = ≡ = − ++

−+ + + − − − +

( ) 0nl ry

=

in ATOMPAW


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

2/32unif

unif

3 310

Original choice: =0.001Better numerics: 0.01

8

WW

W

n nn

nτ τα τ τ πτ ητ

ηη

+∇−

≡ =

=

=


Comparison of self-consistent r2SCAN potentialsfor atomic S with η=0.01 and η=0.001.


Some results for Vτ(r) comparing r2SCAN η=0.01 (Furness, 2020)with rSCAN (Bartók, 2019)

BNe Ar

Al


Some results for Vxc(r) comparing r2SCAN η=0.01 (Furness, 2020)with rSCAN (Bartók, 2019)

BNe Ar

Al


Self-consistent total energy results

Find that self-consistency is hard to achieve for some atoms with the η=0.001 version of r2SCAN.


How does the r2scan Vxc(r) compare with other functionals?


Construction of PAW datasetsFrozen core approximation: All electron treatment Pseudo electron treatment ( ) ( ) ( ) ( ) ( )

core vale core vale

core

n r n r n r n rrτ τ+ ++

( ) ( ) ( )vale core valer r rτ τ+

rc

Core functions for Si atom --

20

0

core2

core

2core

re

2

co

4 ( )4 ( )

4; 4

4 ( )4 ( )

2; 4

MP m

m cm

MP m

m c

c

c

m

r T

r

r C

r

r r rr r

r r

r

r r

rn r

n r rP

r

M

P M

r

π τπ τ

ππ

=

=

≤= >

=

≤

=

= >

= =

∑

∑


Construction of PAW datasets -- continuedThe 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).

rc

2vale vale4 / (Ry/Bohr)rπ τ τ

2vale vale4 / (1/Bohr)r n nπ

Valence functions for Si atom

Vτ

Vτ


Dimensionless kinetic potential and kinetic pseudopotential for Si for r2SCAN η=0.01

Construction of PAW datasets -- continued


xcrV

xcrV

Exchange-correlation potential and pseudopotential for Si for r2SCAN η=0.01




Various schemes to constructed screened pseudopotential –http://pwpaw.wfu.edu

http://pwpaw.wfu.edu/



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 “ultrasoft” with or without norm conservation.

http://pwpaw.wfu.edu/


Ionic pseudopotential for Si for r2SCAN η=0.01in comparison with LDA and GGA

Si


Determine ionic pseudopotential by unscreening VPS(r)

( )2 2vale

( ) ( )

ˆ( ) 4 ( )

( ) ( )

( )c H

PSxcZ

H r

r V

n

r

r e Q n r

V r V r V

V π

= −

= +

−

∇ ∆

( )3vale vale )

ˆ( ) unit compensation char

) (

ge

(Q d r

r

n r

n

n r−

≡

∆ ≡ ∫

VPSMATCHNC option


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.

34

Still to do


Additional considerations

How sensitive are the results to the dataset?

Given apparently reasonable results for r2SCAN η=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?

Documents

Progress on self-consistent meta-gga PAW datasets from …