projector-augmented w ave method · 2006-08-14 · reminder: car-parrinello method generalization...

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Pr ojector -A ugmented W a v e Method: ab-initio MD with full wa v e functions P eter E. Bl ¨ ochl Clausthal Uni v ersity of T echnology Germany http://www .pt.tu-clausthal .de/a tp/ 25. November 2001

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Page 1: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

Projector-AugmentedWaveMethod:

ab-initio MD with full wave functions

Peter E. BlochlClausthal University of Technology

Germany

http://www.pt.tu-clausthal.de/atp/25.November2001

Page 2: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

Historical context

Historical contextof electronic structur e methodsscatteredWaves empirical(KKR, APW) pseudopotentials

energy independent ab-initiobasisfunctions(LMT O,LAPW) pseudopotentials

energy- and potential independentaugmentedwaves(PAW)

PAW joins all-electron and pseudopotentialmethods

PAW built around a consistenttransformation theory��� � � �� ����� �� � � �� � �� � �� �� � �� ��� �� � �� � �� ����� ����������������� � ��� "! � �$# ! �&% � �# ! � � �' ! �Propertiesof PAW( ab-initio molecular dynamics( full all-electron wave functions

(accuratehyperfine parameters)( simple forcecalculations( rigor ousmathematical basis(exactwhen converged)( avoids transferability problemsof pseudopotentials( providestheoretical basisfor pseudopotentials

Page 3: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

Reminder: Car-Parrinello Method

Generalizationof first principles electronic structur emethodsinvented1985by R. Car and M. Parrinello

Goals( study molecular dynamicson the basisoffirst principles electronic structur emethods( makestructural optimization efficient( handle largebasissetsefficiently

Principle( solveclassicalequationsof motion for nuclei) !+*� ! � , � % -/. �0� ! �21 � �43 �- � !

DensityFunctional Theory insteadof empirical forcefield( avoid selfconsistencyiterations in eachstepby intr oducinga fictitious dynamicsfor the wave functions.

5 6 *� � �879� � % -/. ��� ! �:1 � ��3 �- �<;� �879�

Page 4: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

PAW Transformation theory

( start with auxilary wave functions�� � �=79�

( definean transformation operator�>

,� � �=7?� � �> �� � �=79�@�that mapsthe auxiliary wave functions

�� � �879�onto true wave functions

� � �879�( expresstotal energy by auxiliary wave functions

. � . 1 � �9�879� 3 � . 1 �> �� �A�879� 3( Schrodinger-lik eequation for auxiliary functions� > �0B > % > � > C � � �� � �879� � D( Expectationvalues � � � � E� � � � ��� � ��� � �� � � > � � > � �� � �

Find a transformation FG so,that the auxiliary wave function are well behaved

Page 5: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

PAW Transformation operator

Definea transformation fr om auxiliary (pseudo)wavefunctions�H���to the true all-electron wave functions

��� �.��� � � � � �� � �I� ! � �$# ! �&% � �# ! � � �' ! � �� � �

( index J � K� ��L2� 5 ��M( all-electron partial waves

�$# ! �areconstructed

fr om true atomic potential( pseudopartial waves� �# ! �

fr om a pseudopotentialwith�# ! � K7N� � # ! � K7N� OQPSR � K7 % K� �UT 7WV( projector functions

� �' ! � arechosensuchthat������� "!X� �# ! �� �' ! �H��� OQPSR � K7 % K� �ZY 7[V( \ �' ! � �#^]_��� ` ! a ] )

Transformation operator>

\ �b� � � � > � �� � �> � �c� "! � �$# ! �d% � �# ! � � �' ! �

Page 6: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

PAW Augmentation

Example: p- e orbital of Cl f

g�h i j g kh iml g h n ipo g kh n i�j g kh iql r"s+tug v s ido g kv s ixwzy k{ s gHkhi

gbh iu|}g kh i g�h iu|:g h n i

g kh iu|}g kh n i g h n iu|:g kh n i

Page 7: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

PAW Projector functions ~��� ���s-typeprojector functions

p-type projector functions

d-type projector function

Projector functions probethe character of the wave function

Page 8: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

PAW Expectationvalues

Expectationvaluefor a “sufficiently local” one-particleoperator � � � � �� � � � � �� � �I� ! a ]Z�! a ] 0# ! � � �$# ] �&% ! a ]U�

! a ] �# ! � � � �# ] �� � E� V� � � ��� V� �

with a one-centerdensitymatrix

� ! a ] � � �' ][� �� � �� �� � � �' ! �and the corestates

��� V� � !Similarity to pseudopotentials: � � � � �� � � �� � �� � ��� � � � ! a ] � �'

! � � 0# ! � � ��#^]u�p% �# ! � � � �#^]�� � �' ]��Pseudo-operatorhasthe form of a separablepseudopotential.� PAW providesa rule to obtain expectationvalues

in a pseudopotentialcalculation.

Page 9: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

PAW Electron Density

Electron density � �879� tur ns into a plane wave part �� �879� andtwo one-centercomponents� � �879� and �� � �879�

� �=79� � �� �879� � � � �=79� % �� � �=79�� � � � �� ;� �=79� �� �?�879� � �� V� ! a ] # ;! �879� � ! a ]�#^] �=7?� � � V% ! a ] �# ;! �879� � ! a ] �# ] �=7?� % �� V

� ! a ] � � �' ! � �� � � � � �� � �' ] �

Electron densitydivided (lik e the wave function) into( plane wavepart( two expansionsper atom in radial functionstimesspherical harmonics

Page 10: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

PAW Total energy

Total energy divided into plane wave integraland two one-centerexpansionsper atom. ��1 �� � 3 �_� ! � � �. � . � % �. �Planewavepart:�. � � � � �� � ��% ��+� f � �� � �

� ������N� 7 ���N� 7}� � �� �879� � �� �=7?����� �� �=7 � � � �� �=7 � �H�� 7 % 7 � �� ��� � 7 �� �879� C�� V �=7��}1 �� 3 � � ��� � 7��� �=7?� �� �879�One-centerexpansionof plane wavepart�. � � � � ! a ]4 �# ! ��% �� � f � �#^]_�

� �� � �N� 7 � �N� 7}� � ��� �879� � �� �=7?����� �� � �=7 � � � �� �=7 � �H�� 7 % 7 � �� �c�9� 7 �� � �879� C � V �=7��}1 �� � 3 � � ���N� 7��� �=7?� �� � �=79�

One-centerexpansionof true density

. � � � � ! a ][�# ! �2% �� � f �$#�]u�� �� � �9� 7 � �9� 7}� � �

� �=79� � � �=7?����� � � �=7 � � � � �87 � ���� 7 % 7 � �� � �9� 7 � � �=79� C � V �=72�}1 � � 3 �Everything else(Hamiltonian, Forces)followsfr om this total energy functional

Page 11: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

PAW Approximations

The following approximationshavebeenmade( fr ozencoreapproximation (canbe overcome)( truncate planewaveexpansion(basisset)( truncate partial waveexpansion(augmentation)

Convergence:( plane waveconvergencecomparabletoultrasoft pseudopotentials( .���� =30Ry)( 1-2partial wavesper siteand angular momentumsufficient

Chargeand energy transferability problemsof the pseudopotentialapproachareunder control(High-spin atoms)

Page 12: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

PAW options (CP-PAW Code)

( ab-initio molecular dynamics(Car-Parrinello)( all-electron wave functions and densities( electric field gradients(EFG),hyperfine parameters( gradient correcteddensity functionals (various forms)( spin unrestricted,non-collinear spin( isolatedmoleculesand extendedcrystals( QM-MM coupling( activation energies( crystal orbital populations (local chemicalbond analysis)( generalk-points( variable occupationsand finite electron-temperature( variable cell shape(Parrinello-Rahman)( LDA+U;

( GW approximation;

( object oriented program architecture (Fortran 90)( efficiently parallelized (MPI)( portable (Intel-Linux, IBM-AIX, Dec-Alpha-Linux)�) Strasbourg version

Page 13: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

Implementationsof PAW

( CP-PAW; P. Blochl, Clausthal University of Technology( PWPAW; A. Tackett, N. Holtzwarth, WakeForestU.( M. Valiev, J.H. Weare. UC SanDiego( VASPCode;G. Kr esseet al. ViennaUniversity( EStCoMPP; S.Blugel,Osnabruck University( F. Mauri, Princeton University( DFT++; S. Ismail-Beigi, T. Arias, UC Berkeley, Cornell U.

Page 14: Projector-Augmented W ave Method · 2006-08-14 · Reminder: Car-Parrinello Method Generalization of first principles electronic structure methods invented 1985 by R. Car and M

Conclusion

( all-electron method for ab-initio molecular dynamics( rigor oustheoretical basis( accurate,efficient, and simple( extendsand joins all-electron methodswith the pseudopotentialapproach

– Linear Methodsasvery “special case”of PAW

– Pseudopotentialsasapproximation of PAW

Projector AugmentedWaveMethodP. Bochl, Phys. Rev. B 50,17953(1994)

Thanks to ClemensForst, JohannesSchimpl