9/29/2018 HybriD3 2018 1 HybriD 3 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
57
Embed
***Practical Density Functional Theory with Plane Waves)***hybrid3.duke.edu/.../files/u63/HybriD3Holzwarth.pdf · Theory Physical approximations Numerical approximations and tricks
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
9/29/2018 HybriD3 2018 1
HybriD3 Theory Training Workshop-- organized by Y. Kanai (UNC) and V. Blum (Duke)
***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
***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/
• 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.
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.