Practical Density Functional Theory with Plane Waves) · 2018. 10. 8. · approximated by single particle approximation using a self-consistent mean field. Frozen core approximation[von

9/29/2018 HybriD3 2018 1

HybriD3 Theory Training Workshop-- organized by Y. Kanai (UNC) and V. Blum (Duke)

Acknowledgements:• Cameron Kates, Jamie Drewery, Hannah Zhang, Zachary Pipkorn

(former WFU undergrads)• Zachary Hood (WFU chemistry alum, Ga Tech Ph. D)• Jason Howard, Ahmad Al-Qawasmeh, Yan Li, Larry E. Rush,

Nicholas Lepley (current and former WFU grad students)• NSF grant DMR-1507942

***Practical Density Functional Theory with Plane Waves)***

Natalie A. W. Holzwarth, Wake Forest University, Department of Physics, Winston-Salem, NC, USA

9/29/2018 HybriD3 2018 2

input parameters useful results

quantum-espresso (http://www.quantum-espresso.org/)

abinit (https://www.abinit.org/)

Qbox (http://qboxcode.org/)

FHI-aims (https://aimsclub.fhi-berlin.mpg.de/)

Perspectives on Materials Simulations

It is important to know what is inside the box!

Fundamental Theory

Physical approximations

Numerical approximations and tricks

Software packages

9/29/2018 HybriD3 2018 3

Fundamental Theory



Software packages

Quantum mechanics and electrodynamics as it applies to many particle systems

9/29/2018 HybriD3 2018 4

Fundamental Theory



Software packages

Born-Oppenheimer approximation [Born & Huang, Dynamical

Theory of Crystal Lattices , Oxford (1954)]: Nuclear motions treated classically while electronic motions treated quantum mechanically because MN>>me

Density functional theory [Kohn, Hohenberg, Sham, PR 136, B864

(1964), PR 140, A1133 (1965)]: Many electron system approximated by single particle approximation using a self-consistent mean field.

Frozen core approximation[von Barth, Gelatt, PRB 21, 2222

(1980)]: Core electrons assumed to be “frozen” at their atomic values; valence electrons evaluated variationally.

9/29/2018 HybriD3 2018 5

Fundamental Theory



Software packages

Plane wave representations of electronic wavefunctions and densities

Pseudopotential formulations

9/29/2018 HybriD3 2018 6

Numerical methods more generally --

Nu

me

rica

l met

ho

ds

Mu

ffin

tin

s

Density Functional Theory

Plane waves or Grids

9/29/2018 HybriD3 2018 7

Outline• Treatment of core and valence electrons;

frozen core approximation• Use of plane wave expansions in materials

simulations• Pseudopotentials

• Norm conserving pseudopotentials• Projector augmented wave formalism

• Assessment of the calculations


9/29/2018 HybriD3 2018 8

Summary and conclusions:• Materials simulations is a mature field; there are many great

ideas to use, but there still is plenty of room for innovation.• Maintain a skeptical attitude to the literature and to your

own results.• Introduce checks into your work. For example, perform at

least two independent calculations for a representative sample.

• On balance, static lattice results seem to be under good control. The next frontier is more accurate treatment of thermal effects and other aspects of representing macroscopic systems.

• Developing first-principles models of real materials to understand and predict their properties continues to challenge computational scientists.

9/29/2018 HybriD3 2018 9







9/29/2018 HybriD3 2018 10

9/29/2018 HybriD3 2018 11

Two examples --

Partitioning electrons into core and valence contributions

( ) ( ) ( )core valen r n r n r= +

9/29/2018 HybriD3 2018 12

Example for carbon

( )

( )

2

22 2

22 2

2

1 2 2

1 2 2

=4 2 2 2

4 =

( ) ( )

( ) ( )

2

( )

( ) (2 ) ( )2

i i i i

s s p

s s

n l

p

l n

i

n r w r

r r r

P r P rr

P r

+

+

=

+

+

ˆ(

For

)

spherically symmetric a

( ) ( )

( )

:

( )

tom

i i i i i i i

i i

i i

n l m n l l m

n l

n l

r Y

P rr

r

=

=

r r

9/29/2018 HybriD3 2018 13

Radial wavefunctions for carbon

r (Bohr)

P nl(r

)

1s

2s

2p

9/29/2018 HybriD3 2018 14

ncore(r) [1s2]

nval(r) [2s22p2]

r (Bohr)

nr2

/4

Electron density of C atom

9/29/2018 HybriD3 2018 15

Example for Cu

1s22s22p63s23p63d104s1

r (Bohr)

ncore(r)

nvale(r)

nr2

/4

Electron density of copper

9/29/2018 HybriD3 2018 16

r (Bohr)

4s

4p3d

3s

3p

Pn

l(r)

Radial wavefunctions for Cu

Note that in this case, the “semi-core” states 3s23p63d10

are included as “valence”

9/29/2018 HybriD3 2018 17

Frozen core approximation

Example for Cu

1s22s22p63s23p63d104s1

r (Bohr)

ncore(r)

nvale(r)

nr2

/4

core vale

vale

( ) ( )

Variationally optimize energy wrt

)

(

(

)

r n r

n r

n r n= +

9/29/2018 HybriD3 2018 18

Systematic study of frozen core approximation in DFT

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.21.2222

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.21.2222

9/29/2018 HybriD3 2018 19







9/29/2018 HybriD3 2018 20

a1

a2

a3

Consider a periodic material with a unit cell described by primitive lattice vectors a1, a2, and a3:

Multiple unit cells

1 1 2 2 3 3

Any position in the material is related to an infinite number of

other points in the material because

of its periodic symmetry.

n n n+ + + +

r

r T r a a a

i

Because of Bloch's theorem, any wavefunction of the system having wavevector ,

has the property: ( ) ( ). This means that ( ) ( ),

where ( ) is a periodic functio

n.

All d

i ie e u

u

+ = =k T k r

k k k k

k

k

r T r r r

r

rectly measurable properties of the system also reflect the periodic nature of

the system. For example, the electron density: ( ) ( )n n+ =r T r

9/29/2018 HybriD3 2018 21

Mathematical relationships –The Fourier transform of a periodic function results in a discrete summation based on the reciprocal lattice.

32 2 31 1For the electron density: ( ) ( ) , where

The reciprocal lattice vectors are related to the primitive translation

vectors according to =2 ; 2

i

k

i

j

i j ij i

n n e m m m

= = + +

=

G r

G

b

a

r G G b b

b ba

aa

%

unit cell

3

1Here ( ) ( ).

( )

k

i

j

n d r e n−

=

G rG

a

r

a

%

( )

Fourier transforms are a natural basis for periodic functions.

For example, the periodic part of the Bloch wave function:

( ) and ( ) ( )

This works well, provided the

( )iiu u e u e

+

= = k G rG r

G G

k k k kGr r G% %

max .

summation converges in the sen

t foa r

se

h t ( )u K k k + GG% ò

9/29/2018 HybriD3 2018 22







9/29/2018 HybriD3 2018 23

The notion of pseudopotential has been attributedTo Enrico Fermi in the 1930’s. “First principles” pseudopotentials were developed by Hamann, Schlüter, and Chiang, PRL 43, 1494 (1979) and J. Kerker, J. Phys. C 13, L189 (1980).

r (bohr) r (bohr)

Po

ten

tial

(R

y)

Vall-Electron

VPseudo

Cs effective potential

all-Electron

Pseudo

Cs 5s orbitalrc rc

Pseudo all-Electron( ) ( )cr r

r r

=

Pseudo all-Electron( ) ( )cr r

V r V r=

9/29/2018 HybriD3 2018 24

Justification for pseudopotential formalism

Valence electron orthogonality to core electrons provide a repulsive effective potential, resulting in an effective smooth “pseudopotential” for valence electrons.

Norm-conserving pseudopotential construction schemes

9/29/2018 HybriD3 2018 25

Norm-conserving pseudopotentials -- continued

2all-Electron all-Electron

Pseudo Ps2

eu o

2

2 d

( ) 0

( ) 0

Constructed from all-electron treatments of spherical atoms or ions:

( )2

( )2

nl nl

nl nl

V rm

r

V r rm

− + −

=− + −

=

h

h

ò

ò

Pseudo all-Electron

Pseudo all-Electr

2

o

3

n

Pseudo all-Electron2

3

Require:

( ) ( ) for

( ) ( ) for

Also require:

( ) ( )

c c

c

nl nl c

nl n

r r rr

lr

r

V r V r r r

r r r

d r d rr

Norm conservation condition; has several benefits

Pseudo Pseudo Pseudo

loc

Procedure can be carried out for one orbital at

( ) ( ) ( )

a timenl

nl nl

nl

r V VV r r

= + P Non-local projector operator

9/29/2018 HybriD3 2018 26

Recent improvements to norm-conserving pseudopotentials

➔Improves the accuracy of the norm conserving formulationand allows for accurate plane wave representations of thewavefunctions:

( )

max for

( ) ( )K

iu e

+ =

k G r

k k

k+GG

r G%

9/29/2018 HybriD3 2018 27

References:1. P. E. Blöchl, PRB 50, 17953 (1994)2. D. Vanderbilt, PRB 41, 7892 (1990); K.

Laasonen, A. Pasquarello, R. Car, C. Lee, D. Vanderbilt, PRB 47, 10142 (1993)

3. D. R. Hamann, M. Schlüter, C. Chiang, PRL 43, 1494 (1979); D. R. Hamann, PRB 88, 085117 (2013)

Nu

me

rica

l met

ho

ds

Mu

ffin

tin

s

Density Functional Theory

(1) (2)

(3)

Other plane wave compatible schemes --

9/29/2018 HybriD3 2018 28

Basic ideas of the Projector Augmented Wave (PAW) method

Blöch presented his ideas at ES93 --“PAW: an all-electron method for first-principles molecular dynamics”Reference: P. E. Blöchl, PRB 50, 17953 (1994)

Peter Blöchl, Institute of Theoretical Physics TU Clausthal, Germany

Features• Operationally similar to other pseudopotential methods,

particularly to the ultra-soft pseudopotential method of D. Vanderbilt; often run within frozen core approximation

• Can retrieve approximate ‘’all-electron” wavefunctions from the results of the calculation; useful for NMR analysis for example

• May have additional accuracy controls particularly of the higher multipole Coulombic contributions.

Atom-centered functions:All electron basis functionsPseudo basis functionsProjector functions

9/29/2018 HybriD3 2018 29


• Valence electron wavefunctions are approximated by the form

( ) ( )( ) ( )( ) ( ) ( )a

a a a

b b b

a

n a

b

n a np + − − − − k k kr r r R r R r R r% %%%

All-electron wavefunction

Pseudowavefunction, optimized in solving Kohn-Sham equations

( ) : determined self-consistently within calculation

( ), ( ), ( ) : part of pseudopotential construction; stored in PAW dataset

n

a a a

b b bp

k r

r r r

%

% %

9/29/2018 HybriD3 2018 30


• Evaluation of the total electronic energy:

total total a

aE E E= + %

Pseudoenergy(evaluated in plane wave basis or on regular grid)

One-center atomic contributions (evaluated within augmentation spheres)

9/29/2018 HybriD3 2018 31

Comment on one center energy contributions• Norm-conserving pseudopotential scheme using the

Kleinman-Bylander method (PRL 48, 1425 (1982)):

• PAW and USPS :

,

( ) are

fixed functions depending on the non-local pseudopotentials

and corresponding pseudobasis functi

The non-local pseudopotential contributions for site :

,where a a a a

n n n

n b

a b b bE

a

W = − k k k

k

r R% %% % %

ons; are occupancy

and sampling weights.

nW k

'

' '

'

,

( ) are

projector functions, are matrix elements depending on

all-electron and pseudobasis functions, and are

occupancy and Brillouin zone

,where a a a a a

a b bb b b

a

n n n

n bb

n

bb

E p M p p

M

W

W −= k k k

k

k

r R% %% % %

sampling weights.

9/29/2018 HybriD3 2018 32

Comment on one center energy contributions -- continued for PAW and USPS

' '

'

2

' ''

matrix elements (different for USPS and PAW) are evaluated

within the augmentation spheres. For example, the kinetic energy term:

( ) ( ) ( ) ( )

2bb bb

a

bb

a a a aa b b b bb l m mb l

M

d r d r d r d rK dr

m dr dr dr dr

= −

% %h

( )

0

' '2

0

( 1) ( ) ( ) ( ) ( )

( ) ( )ˆ ˆwhere ( ) ( ) and ( ) ( )

+

c

c

b b b b

r

r

a a a a

b b b b b b

a aa ab bb l m b l m

drl r r r r

r

r rY Y

r r

l

+ −

r r r r

% %

%%

( )' '' ( ) ( ) ( ) ( )

is pseudized, while for PAW it is evaluated within matrix elements and

"compensation charges" are added. In both cases, multipole

Note that for USPS, the operator ( ) a a a a

b b b b

a

bb r r r rQ r − % %

moments

are conserved.

9/29/2018 HybriD3 2018 33

Summary of properties of norm-conserving (NC),ultra-soft-pseudopotential (USPS) and projector augmented wave (PAW) methods

NC USPS PAWConservation of charge

Multipole momentsin Hartree interaction

Retrieve all-electron wavefunction

9/29/2018 HybriD3 2018 34

Some details – use of “compensation charge”

( ) ( )( ) ( )

2

valence

2

PAW approximation to valence all-electron wave function

( ) ( ) ( )

PAW approximation to all-electron density

( ) ( )

( )

n n a a a n

n n

n

n n

n

n n

a a a

b b b

ab

W

p

W

W

n

+ − − − −

+

k k k

k k

k

k k

k

k

r r r R r R r R r

r r

r

% % %

%

%

%

( ) ( ) ( ) ( )( )

( )

( ) ( )( )

( ) ( )

2

' ' '

, '

' '

, '

( )

( )

= ( )

a a a a a a

b b b b b b

a bb

a a a

b b bb

a b

n a a a a

n

n n n n a

n

a a

a a

a a

a

b

a

a

p p

pW

n

Q

n

p

n n

n n

− − − − −

+ −

+ − − −

+ −

− −

k k

k

k k k k

k

r R r R r R r R

r r R

r r R r R

r r R r R

%

% % %

%

%

%%% %

% %

%

%( ) a

( )â an− −r R( )â a

a

n+ − r R

9/29/2018 HybriD3 2018 35

Some details – use of “compensation charge”-- continued

( )3 3

Compensation charge is designed to have the same

multipole moments of one-center charge differences:

ˆ ˆˆ( ) ( ) ( ) ( ) ( ) a ac c

LM LM

r

L a L a a

r r r

d Yr n dr nr Yr n

= − r r r r r%

0L =Typical shape of compensation charge for L=0 component --

rc

9/29/2018 HybriD3 2018 36

Some details – use of “compensation charge”-- continued

The inclusion of the "compensation" charge ensures

1. Hartree energy of smooth charge density represents correct charge

2. Hartree energy contributions of one-center charge is confined within

a

3 Hartree

o

ˆ( ) ( ) (

ugmentation sphere:

( ) for

0 f r'

'

)

ac

a

r

a a a a

c

a

cr

n V r rd

r r

n nr

=

− −

−r r r

r r

r%

9/29/2018 HybriD3 2018 37

Some details – form of exchange-correlation contributions

core

core core

3For [ ( )] ( )) :

Smooth contribution: [ ( ) ( )]

One-center contributions: = [ ( ) ( )] [ ( ) ( )]

(xc

xc xc

a a

xc x

c

c xc xc

x

a a a a

E n d n

E E n n

E E E n n E n n

r K

= +

− + − +

r r

r r

r r r r

% % %

% % %

core

core

Note that VASP and Quantum-Espresso use

ˆ[ ( ) ( ) ( )]

ând [ ( ) ( ) ( )]

which can cause trouble occasionally.

a

xc

xc

a a

E n n n

E n n n

+ +

+ +

r r r

r r r

% % %

% % %

non-linear core correction (S. G. Louie et al. PRB 26, 1738 (1982))

9/29/2018 HybriD3 2018 38

Pseudopotential schemes enable the accurateuse of plane wave and regular grid based numerical methods

( ) ( )max

C

(

onve

)

rgence of plane wave expansions

( )

:

i

Kn nu e+

+ =k G r

k Gk k

G

r G%

( )

max max

2

(occ)

2

(occ)

2

Electron density: ( ) ( )

( ) ( ) ( )

=

n n

n

i i

n n

n

K K

n w

n w u e n e+

=

=

k k

k

k G r G r

k k

k G G

k+G G

r r

r G G% %

9/29/2018 HybriD3 2018 39

Some convenient numerical “tricks” involved with Fourier transforms using discrete Fourier transforms

( )

( )max

( ) ( )

i

n n

K

u e+

+

=k G r

k k

G

k G

r G%

Discrete summation due to lattice periodicity

Discrete Fourier transforms ➔ Fast Fourier transformsFFT equations http://www.fftw.org/

( ) ( )

( ) ( )

3 32 2

2 3

1 1

1

3 3

1 2 3

1 1 2 2

2 31

1 2 3

2

1 2 3 1 2 3

, ,

2

1 2 3 1 2 3

, ,1 2 3

, , , ,

1, , , ,

n mn m n m

N N N

n mn

i

m m m

m n m

Ni

n

N

n n

N

f n n n f m m m e

f m m m f n n eN N

nN

−

+

+

+

+

=

=

%

%

http://www.fftw.org/

9/29/2018 HybriD3 2018 40

FFT grid size Reciprocal space

maxKG

Enclosing parallelepiped

2 3 31 21

0 i i

n n n

n N

= + +

G b b b

322 3

2 3

3 32 2

2 3

11

1

1 1

1

Real space grid poin s

2

t

mm m

N N N

n mn m n m

N N N

= +

= +

+

+r a

G r

a a

9/29/2018 HybriD3 2018 41







9/29/2018 HybriD3 2018 42

Assessment of the calculationsScience 25 MARCH 2016 VOL 351 ISSUE 6280

http://science.sciencemag.org/content/sci/351/6280/aad3000.full.pdf

http://science.sciencemag.org/content/sci/351/6280/aad3000.full.pdf

9/29/2018 HybriD3 2018 43

Webpage for standardized comparison of codes

https://molmod.ugent.be/deltacodesdft

https://molmod.ugent.be/deltacodesdft

9/29/2018 HybriD3 2018 44

Assessment of your specific calculations

• You can expect that even well-converged calculations will differ between pseudopotential datasets and code packages. It is incumbent on us to trace and document these differences.

• There is some error cancelation in a set of calculations using a given set of pseudopotential datasets and a single code package.

• On the other hand, the best way to validate your results, is to compare two or more independent calculations for a representative sample.

9/29/2018 HybriD3 2018 45

Various code packages

http://pwpaw.wfu.edu

Code for generating PAW datasets

http://pwpaw.wfu.edu/

9/29/2018 HybriD3 2018 46


https://www.abinit.org/

ABINIT -- Electron structure code package mainly based on plane waves

https://www.abinit.org/

9/29/2018 HybriD3 2018 47


http://www.quantum-espresso.org/

Quantum Espresso – Electron structure code package mainly based on plane waves

http://www.quantum-espresso.org/

9/29/2018 HybriD3 2018 48

General advice about generating PAW datasets

• ATOMPAW code* available at http://pwpaw.wfu.edu• Develop and test atomic datasets for the full scope of your

project ➔ determine rca, define frozen core

• Determine local pseudopotential from self-consistent all-electron potential

• Determine basis functions for valence (and perhaps semicore) states; usually 2 sets of basis functions and projectors for each l channel.

• Test binding energy curves for a few binary compounds related to your project.

• Check plane wave (or grid spacing) convergence of your data sets before starting production runs.

*With major modification by Marc Torrent and other Abinitdevelopers.

http://pwpaw.wfu.edu/

9/29/2018 HybriD3 2018 49

Recipes for constructing projector and basis functions

( ) ( ) ( )ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )

Constraints: ( ) ( ) for

( ) 0 for

b b b b b b

a a aa a ab b bb l b l b l

a a a

b b c

a a

b

m m m

c

r r p rY Y p Y

r r r

r r r r

p r r r

= = =

=

=

r r r r r r% %

% %

%

%

' ' = bb

a a

b bp %%

Peter Blöchl’s scheme (set #1) David Vanderbilt’s scheme (set #2)

Choose projectors ( )

Derive ( )

a

b

a

b

p r

r

%

%

Choose pseudo bases ( )

Derive ( )

a

b

a

b

r

p r

%

%

*

*Polynomial or Bessel function form following RRKJ, PRB 41, 1227 (1990)

*

*Typically Bessel-like function with zero value and derivative at rc

a

9/29/2018 HybriD3 2018 50

( )a

b r

( )a

b r%

( )a

bp r%

Br 4s orbital

From set #1 From set #2

( )a

b r

( )a

b r%

( )a

bp r%

Example projector and basis functions

9/29/2018 HybriD3 2018 51

Br

4s

4p

3dx1/2 ed

ep

es

Cs

x1/2

5s 6s

5p ep

4d ed

Set of basis and projector functions for set #1

9/29/2018 HybriD3 2018 52

Sources of pseudopotential and paw datasets on the web

https://www.abinit.org/psp-tables http://www.quantum-espresso.org/pseudopotentials

If you use datasets from the web, it is still your responsibility to test them for accuracy wrt to your project.

https://www.abinit.org/psp-tables

http://www.quantum-espresso.org/pseudopotentials

9/29/2018 HybriD3 2018 53

Measure of accuracySometimes, calculations can surprise!!

Binding energy curve for CsBr

Set #2 fails to converge in QE!

9/29/2018 HybriD3 2018 54

Mystery –Why does the Cs dataset #1 do well in abinit but fail in espresso??

Binding energy curves for CsBr

core

core vale

Treatments of the exchange-correlation energies

Abinit : [ ( ) ( )]

ˆQE & VASP: [ ( ) ( ) ( )]

xc xc

xc xc

E E n n

E E n n n

= +

= + +

r r

r r r

% % %

% % %

Answer --

Compensation charge; does not logically belong in this expression and can cause argument to be negative.

Set #2’ now converges in QE and abinit!

9/29/2018 HybriD3 2018 55

Binding energy curve for CsBr

rad

ial d

ensi

ty

r (bohr)

Set #2’

Set #2

coren%

core valeˆn n+%

coren%

core valeˆn n+%

9/29/2018 HybriD3 2018 56

Other surprises due to the same issueFor NaCl, the electronic structure calculations performed with both QE and Abinit agreed well, but the density functional perturbation theory step resulted in incorrect phonon densities of states.

n (cm)-1 n (cm)-1

Phonon densities of states for NaCl

Original dataset for Cl Corrected dataset for Cl

Abinit

QEQE

Abinit

9/29/2018 HybriD3 2018 57

Summary and conclusions:• Materials simulations is a mature field; there are many great

ideas to use, but there still is plenty of room for innovation• Maintain a skeptical attitude to the literature and to your

own results• Introduce checks into your work. For example, perform at

least two independent calculations for a representative sample.

• On balance, static lattice results seem to be under good control. The next frontier is more accurate treatment of thermal effects and other aspects of representing macroscopic systems.

• Developing first-principles models of real materials to understand and predict their properties continues to challenge computational scientists.

Documents

***Practical Density Functional Theory with Plane Waves)*** · 2018. 10. 8. · approximated by single particle approximation using a self-consistent mean field. Frozen core approximation[von

Practical Density Functional Theory with Plane Waves) · 2018. 10. 8. · approximated by single particle approximation using a self-consistent mean field. Frozen core approximation[von