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Introduction to PAW method

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Title: 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)
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