1. 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

2. 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.

3. 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.

4. Basic Concept of Projector Augmented Wave
AE Pseudo Pseudo-onsite AE-onsite
same trick works for
- wavefunctions
- charge density
- kinetic energy
- exchange correlation energy
- Hartree energy

5. Basic Concept of Projector Augmented Wave

6. Transformation theory
We need to find a transformation T from the auxiliary (pseudo) to the
physical (all electron, true) wave functions.
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[ ] of the density functional theory.
The one-particle wave functions have to be orthogonal.
( Kohn Sham equations)

7. 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

8. 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 of the
Schrodinger equation for the isolated atoms.
atomic partial waves , serve as a basis set near the nucleus
orthogonal to the core wave functions

9. 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
and require
projector function
is valid within rc
( within rc , with identical ci)

10. 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.

11. Transformation theory

12. Transformation theory
Derivation of the PAW method is straightforward

13. Transformation theory
PAW energy functional ( )

14. 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
are found by solving the Schrodinger equation for the isolated atom
- 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
choose ; if zero, set equal to k(r)

15. Partial Waves and Projectors
Gram-Schmidt orthogonalization procedure : :

16. Partial Waves and Projectors
Gram-Schmidt orthogonalization procedure : :

17. An Example to Show How the PAW Method Works
goal
AE p-s orbital of Cl2

18. 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

19. An Example to Show How the PAW Method Works
solve the self-consistent Schrodinger equation
to get the PS wave function to minimize the total energy functional

20. An Example to Show How the PAW Method Works
projector functions probe the character of the PS wavefunction

21. An Example to Show How the PAW Method Works+ - =

22. An Example to Show How the PAW Method Works

23. Compare the Results with US-PP and AE

24. Compare the Results with US-PP and AE

25. Compare the Results with US-PP and AE

26. Compare the Results with US-PP and AE

27. Compare the Results with US-PP and AE

28. Compare the Results with US-PP and AE

29. Compare the Results with US-PP and AE

30. Some phonon test by myself for graphite sheet

31. 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

32. Some phonon test by myself for graphite sheet
* Chem. Phys. Lett. R.A. Jishi 209, p77 (1993)

33. Some phonon test by myself for graphite sheet (frequency unit : 1/cm)
- CASTEP VASP
- fit-exp (not AE)

34. Some phonon test by myself for graphite sheet (frequency unit : 1/cm)
- CASTEP VASP
- fit-exp (not AE)

35. 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.

36. three different flavors, one LDA and two GGA’s
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 (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/ v2 12 Jul (2002)