Introduction to PAW method ( Report on VASP workshop in Vienna ) Density Functional Theory and...

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

• Kohn Sham energy functional

• Kohn Sham equation

• Frozen core approximation

• An large plane wave basis sets are required to expand the electric wave functions.

• 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.

Density Functional Theory and Pseudopotential

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

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.

nn Ψ~

Ψ T is one particle wave functionsnΨ•

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.

nΨ

( 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 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

i

i

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

( within rc , with identical ci) i i i iiiiii CTCCTT ~

)~

(~

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 ( ) 11 ~~

EEEE

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

choose ; if zero, set equal to k(r)ip~

Partial Waves and Projectors• Gram-Schmidt orthogonalization procedure

1~p

2~p

3~p

4~p

5~p

6~p

:

1

~

2

~

3

~

4

~

5

~

6

~:

Partial Waves and Projectors• Gram-Schmidt orthogonalization procedure

1~p

2~p

3~p

4~p

5~p

6~p

:

1

~

2

~

3

~

4

~

5

~

6

~:

An Example to Show How the PAW Method Works

• goal

AE p- 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 to get the PS wave function to minimize the total energy functional ]

~[ nE

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

* Chem. Phys. Lett. R.A. Jishi 209, p77 (1993)

(eV/A) (eV/A^2) * (eV/A^2)CASTEP 0.02 -0.02 average force constant fit(not-AE) force constant

1C 0.493 -0.421 0.457 22.849 1C 22.7841Ap 0.225 -0.226 0.226 11.283 1Ap 15.2931Az 0.129 -0.129 0.129 6.465 1Az 6.1302C 0.098 -0.087 0.093 4.635 2C 5.493

2Ap -0.058 0.062 -0.060 -2.995 2Ap -2.0162Az -0.013 0.013 -0.013 -0.641 2Az -0.250

VASP 0.02 -0.02 average force constant1C 0.549 -0.472 0.511 25.530

1Ap 0.216 -0.218 0.217 10.8461Az 0.121 -0.121 0.121 6.0302C 0.099 -0.091 0.095 4.745

2Ap -0.057 0.054 -0.056 -2.7792Az -0.008 0.008 -0.008 -0.422

• Some phonon test by myself for graphite sheet (frequency unit : 1/cm)

- CASTEP - VASP

- fit-exp (not AE)

• Some phonon test by myself for graphite sheet (frequency unit : 1/cm)

- CASTEP - VASP

- fit-exp (not AE)

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)