Presentation on theme: "Introduction to PAW method"— Presentation transcript:
1 Introduction to PAW method ( Report on VASP workshop in Vienna )Density Functional Theory and PseudopotentialBasic Concept of Projector Augmented WaveTransformation theoryPartial Waves and ProjectorsAn Example to Show How the PAW Method WorksCompare the Results with US-PP and AEConclusion李啟正 TKU
2 Density Functional Theory and Pseudopotential Kohn Sham energy functionalKohn Sham equationAn large plane wave basis sets are required to expand the electricwave functions.Frozen core approximationThe 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 PseudopotentialNo 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-onsitesame trick works for- wavefunctions- charge density- kinetic energy- exchange correlation energy- Hartree energy
6 Transformation theory We need to find a transformation T from the auxiliary (pseudo) to thephysical (all electron, true) wave functions.is one particle wave functionsn is the label for a band index, a K-point and a spin indexThe electronic ground state is determined by minimizing a totalenergy 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 termsof the true or the auxiliary wave functions.In the representation of auxiliary wave functions we need to usetransformed operators
8 Transformation theory T has to modify the smooth auxiliary valence wave function ineach atomic region.The local terms SR are defined in terms of solutions of theSchrodinger equation for the isolated atoms.atomic partial waves , serve as a basis set near the nucleusorthogonal to the core wave functions
9 Transformation theory All relevant valence wave functions near nucleus can be expressed asFor each of the partial waves we choose an auxiliary partial waveand requireprojector functionis valid within rc( within rc , with identical ci)
10 Transformation theory sum over all partial waves of all atomswithAll partial waves and projector functions need to be determined beforedoing calculation.We can derive the forms for expectation values, electron density,total energy functional, and everything else from the form of T now.
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 andprojectors. There is an infinite number of ways to construct them.Although the PAW method works using any of a variety of basisand projector functions, the efficiency and accuracy of the calculationare affected by this choice.some way to get all-electron, pseudo partial waves and projectorsare found by solving the Schrodinger equation for the isolated atom- first select a PS potential- chooseusing a cutoff function of the form- define for each AE partial wave a PS potential of the form- the PS partial wave obtained fromthe energy is from AE results and wave coincides outside rcchoose ; 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 goalAE p-s orbital of Cl2
18 An Example to Show How the PAW Method Works construct AE partial waves, PS partial waves, and projector functionsin the augmented regionprojector waves of Cl
19 An Example to Show How the PAW Method Works solve the self-consistent Schrodinger equationto 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 +-=
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- RPBEfor CASTEP- USPfor 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 ConclusionThe transformation should be considered merely as change of representationanalogous to a coordinate transform. If the total energy functional istransformed consistently, its minimum will yield an auxiliary wave functionthat 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 actinidescompare 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 FLAPWThe pseudopotential approach can actually be derived from the PAW methodby making some approximation.
36 three different flavors, one LDA and two GGA’s The PAW potentialsthree different flavors, one LDA and two GGA’sdownload location of LDA potentials: paw/potcar.date.tardownload location of PW91 potentials: paw_GGA/potcar.date.tardownload location of PBE potentials: paw_PBE/potcar.date.tarreference- Projector augmented-wave methodP.E. Blochl PRB. V50 N24 p (1994)Comparison of the projector augmented-wave, pseudopotential, andlinearized augmented-plane-wave formalisms for density-functionalcalculations of solidsN.A.W. Holzwarth, et al. PRB. V55 N4 p.2005 (1997)From ultrasoft pseudopotential to the projector augmented-wave methodG. Kresse, et al. PRB. V59 N3 p.1758 (1999)The projector augmented wave method: ab-initio molecular dynamics withfull wave functionsP.E. Blochl, et al. arXiv:cond-mat/ v2 12 Jul (2002)