Post on 24-Oct-2020
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
Physical approximations
Numerical approximations and tricks
Software packages
Quantum mechanics and electrodynamics as it applies to many particle systems
9/29/2018 HybriD3 2018 4
Fundamental Theory
Physical approximations
Numerical approximations and tricks
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
Physical approximations
Numerical approximations and tricks
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
***Practical Density Functional Theory with Plane Waves)***
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
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
***Practical Density Functional Theory with Plane Waves)***
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
9/29/2018 HybriD3 2018 19
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
***Practical Density Functional Theory with Plane Waves)***
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
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
***Practical Density Functional Theory with Plane Waves)***
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
Basic ideas of the Projector Augmented Wave (PAW) method
• 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
Basic ideas of the Projector Augmented Wave (PAW) method
• 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
( )ˆa an− −r R( )ˆa 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[ ( ) ( ) ( )]
ˆand [ ( ) ( ) ( )]
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
−
+
+
+
+
=
=
%
%
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
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
***Practical Density Functional Theory with Plane Waves)***
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
9/29/2018 HybriD3 2018 43
Webpage for standardized comparison of codes
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
9/29/2018 HybriD3 2018 46
Various code packages
https://www.abinit.org/
ABINIT -- Electron structure code package mainly based on plane waves
9/29/2018 HybriD3 2018 47
Various code packages
http://www.quantum-espresso.org/
Quantum Espresso – Electron structure code package mainly based on plane waves
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.
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.
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.