Introduction to PAW method

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

??? 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
• - 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)