Title: Introduction to PAW method
1Introduction 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
??? TKU
2 Density Functional Theory and Pseudopotential
- Kohn Sham energy functional
- 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- 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- 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- 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
(
within rc , with identical ci)
10sum 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 12- Derivation of the PAW method is straightforward
13- 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
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
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
- - a2.464A c6.711A (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- The PAW potentials
- three different flavors, one LDA and two GGAs
- 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. arXivcond-mat/0201015 v2
12 Jul (2002)