Transcript Document 7437940

Introduction to PAW method
( Report on VASP workshop in Vienna )
 Density Functional Theory and Pseudopotential
 Basic Concept of Projector Augmented Wave
 Transformation theory
 Partial Waves and Projectors
 An Example to Show How the PAW Method Works
 Compare the Results with US-PP and AE
 Conclusion
李啟正 TKU
 Density Functional Theory and Pseudopotential
• Kohn Sham energy functional
• Kohn Sham equation
• An large plane wave basis sets are required to expand the electric
wave functions.
• Frozen core approximation
• The valence electrons is important outside the core region.
• The nucleus and its core orbitals are replaced by a pseudo potential.
It should reproduce the exact valence orbitals outside the core region.
 Density Functional Theory and Pseudopotential
• Schematic illustration of Pseudopotential
• No core orbitals are taken into account in the whole calculation.
• The valence wavefunctions have an incorrect shape in the core region.
• Sometimes the wavefunction close to the nucleus is important.
• We need to find an more efficient (PP) and accurate (AE) method.
 Basic Concept of Projector Augmented Wave
AE
Pseudo
• same trick works for
- wavefunctions
- charge density
- kinetic energy
- exchange correlation energy
- Hartree energy
Pseudo-onsite
AE-onsite
 Basic Concept of Projector Augmented Wave
 Transformation theory
• We need to find a transformation T from the auxiliary (pseudo) to the
physical (all electron, true) wave functions.
~
• Ψn  T Ψn
Ψn
is one particle wave functions
n is the label for a band index, a K-point and a spin index
• The electronic ground state is determined by minimizing a total
energy functional E[Ψn] of the density functional theory.
The one-particle wave functions have to be orthogonal.
( Kohn Sham equations)
 Transformation theory
• Express the functional F in terms of auxiliary wave functions
(Schrodinger-like equation)
• The expectation values of an operator A can be expressed in terms
of the true or the auxiliary wave functions.
• In the representation of auxiliary wave functions we need to use
transformed operators
 Transformation theory
• T has to modify the smooth auxiliary valence wave function in
each atomic region.
• The local terms SR are defined in terms of solutions i of the
Schrodinger equation for the isolated atoms.
i atomic partial waves , serve as a basis set near the nucleus
orthogonal to the core wave functions
 Transformation theory
• All relevant valence wave functions near nucleus can be expressed as
~
• For each of the partial waves we choose an auxiliary partial wave  i .
and require
•
projector function
•
is valid within rc
(
~
~
~
  T   T (i Ci i )  i CiT i  i Ci i
within rc , with identical ci)
 Transformation theory
•
•
sum over all partial waves of all atoms
•
with
• All partial waves and projector functions need to be determined before
doing calculation.
• We can derive the forms for expectation values, electron density,
total energy functional, and everything else from the form of T now.
 Transformation theory
 Transformation theory
• Derivation of the PAW method is straightforward
•
 Transformation theory
~
~
• PAW energy functional E  E  E1  E 1
(
)
 Partial Waves and Projectors
• The basic ingredients of the PAW method are partial waves and
projectors. There is an infinite number of ways to construct them.
• Although the PAW method works using any of a variety of basis
and projector functions, the efficiency and accuracy of the calculation
are affected by this choice.
• some way to get all-electron, pseudo partial waves and projectors
i are found by solving the Schrodinger equation for the isolated atom
~
 i - first select a PS potential
- choose
using a cutoff function of the form
- define for each AE partial wave a PS potential of the form
- the PS partial wave obtained from
the energy is from AE results and wave coincides outside rc
~
pi choose
; if zero, set equal to k(r)
 Partial Waves and Projectors
• Gram-Schmidt orthogonalization procedure
~
~
p1
1
~
p2
2
~
p3
3
~
p4
4
~
p5
5
~
p6
6
:
:
~
~
~
~
~
 Partial Waves and Projectors
• Gram-Schmidt orthogonalization procedure
~
~
p1
1
~
p2
2
~
p3
3
~
p4
4
~
p5
5
~
p6
6
:
:
~
~
~
~
~
 An Example to Show How the PAW Method Works
• goal
AE p-s orbital of Cl2
 An Example to Show How the PAW Method Works
• construct AE partial waves, PS partial waves, and projector functions
in the augmented region
• projector waves of Cl
 An Example to Show How the PAW Method Works
• solve the self-consistent Schrodinger equation
~
E
[

to get the PS wave function to minimize the total energy functional
n]
 An Example to Show How the PAW Method Works
•
projector functions probe the character of the PS wavefunction
 An Example to Show How the PAW Method Works
•
+
=
-
 An Example to Show How the PAW Method Works
•
 Compare the Results with US-PP and AE
 Compare the Results with US-PP and AE
 Compare the Results with US-PP and AE
 Compare the Results with US-PP and AE
 Compare the Results with US-PP and AE
 Compare the Results with US-PP and AE
 Compare the Results with US-PP and AE
• Some phonon test by myself for graphite sheet
• Some phonon test by myself for graphite sheet
• CASTEP and VASP
- a=2.464A c=6.711A (primitive)
- 3x3x1 supercell
- single point energy
- move red atom x,-x,y,-y,z,-z 0.02A
- Ecut 400 eV
- K-points 5x5x5
- RPBE
for CASTEP
- USP
for VASP
- PAW
• Some phonon test by myself for graphite sheet
(eV/A)
CASTEP
0.02
1C
0.493
1Ap
0.225
1Az
0.129
2C
0.098
2Ap
-0.058
2Az
-0.013
VASP
0.02
1C
0.549
1Ap
0.216
1Az
0.121
2C
0.099
2Ap
-0.057
2Az
-0.008
-0.02
-0.421
-0.226
-0.129
-0.087
0.062
0.013
-0.02
-0.472
-0.218
-0.121
-0.091
0.054
0.008
average
0.457
0.226
0.129
0.093
-0.060
-0.013
average
0.511
0.217
0.121
0.095
-0.056
-0.008
(eV/A^2)
force constant
22.849
11.283
6.465
4.635
-2.995
-0.641
force constant
25.530
10.846
6.030
4.745
-2.779
-0.422
* Chem. Phys. Lett. R.A. Jishi 209, p77 (1993)
*
fit(not-AE)
1C
1Ap
1Az
2C
2Ap
2Az
(eV/A^2)
force constant
22.784
15.293
6.130
5.493
-2.016
-0.250
• Some phonon test by myself for graphite sheet
- CASTEP
- VASP
1600
1600
1400
1400
1200
1200
1000
1000
800
800
600
600
400
400
200
200
0

M

K
0

M
- fit-exp (not AE)
1600
1400
1200
1000
800
600
400
200
0

M
K
(frequency unit : 1/cm)

K

• Some phonon test by myself for graphite sheet
- CASTEP
- VASP
0.0030
0.0030
0.0025
0.0025
0.0020
0.0020
0.0015
0.0015
0.0010
0.0010
0.0005
0.0005
(frequency unit : 1/cm)
0.0000
0.0000
0
200
400
600
800
1000
1200
1400
1600
0
200
400
600
1000
1200
1400
1600
- fit-exp (not AE)
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
0
200
400
600
800
800
1000
1200
1400
1600
 Conclusion
• The transformation should be considered merely as change of representation
analogous to a coordinate transform. If the total energy functional is
transformed consistently, its minimum will yield an auxiliary wave function
that produces a correct wave function.
• PAW method is in an efficient way to get AE wavefunction.
• improved accuracy for
- magnetic materials
- alkali and alkali earth elements, 3d elements
- lanthanides and actinides
• compare to other methods :
- all test indicate the accuracy is as good as for other all electron methods
(FLAPW, NUMOL, Gaussian)
- efficiency for large system should be significantly better than with FLAPW
• The pseudopotential approach can actually be derived from the PAW method
by making some approximation.
• The PAW potentials
three different flavors, one LDA and two GGA’s
- download location of LDA potentials: paw/potcar.date.tar
- download location of PW91 potentials: paw_GGA/potcar.date.tar
- download location of PBE potentials: paw_PBE/potcar.date.tar
• reference
- Projector augmented-wave method
P.E. Blochl PRB. V50 N24 p.17953 (1994)
- Comparison of the projector augmented-wave, pseudopotential, and
linearized augmented-plane-wave formalisms for density-functional
calculations of solids
N.A.W. Holzwarth, et al. PRB. V55 N4 p.2005 (1997)
- From ultrasoft pseudopotential to the projector augmented-wave method
G. Kresse, et al. PRB. V59 N3 p.1758 (1999)
- The projector augmented wave method: ab-initio molecular dynamics with
full wave functions
P.E. Blochl, et al. arXiv:cond-mat/0201015 v2 12 Jul (2002)