projector-augmented w ave method · 2006-08-14 · reminder: car-parrinello method generalization...
TRANSCRIPT
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
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��� � � �� ����� �� � � �� � �� � �� �� � �� ��� �� � �� � �� ����� ����������������� � ��� "! � �$# ! �&% � �# ! � � �' ! �Propertiesof PAW( ab-initio molecular dynamics( full all-electron wave functions
(accuratehyperfine parameters)( simple forcecalculations( rigor ousmathematical basis(exactwhen converged)( avoids transferability problemsof pseudopotentials( providestheoretical basisfor pseudopotentials
Reminder: Car-Parrinello Method
Generalizationof first principles electronic structur emethodsinvented1985by R. Car and M. Parrinello
Goals( study molecular dynamicson the basisoffirst principles electronic structur emethods( makestructural optimization efficient( handle largebasissetsefficiently
Principle( solveclassicalequationsof motion for nuclei) !+*� ! � , � % -/. �0� ! �21 � �43 �- � !
DensityFunctional Theory insteadof empirical forcefield( avoid selfconsistencyiterations in eachstepby intr oducinga fictitious dynamicsfor the wave functions.
5 6 *� � �879� � % -/. ��� ! �:1 � ��3 �- �<;� �879�
PAW Transformation theory
( start with auxilary wave functions�� � �=79�
( definean transformation operator�>
,� � �=7?� � �> �� � �=79�@�that mapsthe auxiliary wave functions
�� � �879�onto true wave functions
� � �879�( expresstotal energy by auxiliary wave functions
. � . 1 � �9�879� 3 � . 1 �> �� �A�879� 3( Schrodinger-lik eequation for auxiliary functions� > �0B > % > � > C � � �� � �879� � D( Expectationvalues � � � � E� � � � ��� � ��� � �� � � > � � > � �� � �
Find a transformation FG so,that the auxiliary wave function are well behaved
PAW Transformation operator
Definea transformation fr om auxiliary (pseudo)wavefunctions�H���to the true all-electron wave functions
��� �.��� � � � � �� � �I� ! � �$# ! �&% � �# ! � � �' ! � �� � �
( 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% � �# ! � � �' ! �
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
PAW Projector functions ~��� ���s-typeprojector functions
p-type projector functions
d-type projector function
Projector functions probethe character of the wave function
PAW Expectationvalues
Expectationvaluefor a “sufficiently local” one-particleoperator � � � � �� � � � � �� � �I� ! a ]Z�! a ] 0# ! � � �$# ] �&% ! a ]U�
! a ] �# ! � � � �# ] �� � E� V� � � ��� V� �
with a one-centerdensitymatrix
� ! a ] � � �' ][� �� � �� �� � � �' ! �and the corestates
��� V� � !Similarity to pseudopotentials: � � � � �� � � �� � �� � ��� � � � ! a ] � �'
! � � 0# ! � � ��#^]u�p% �# ! � � � �#^]�� � �' ]��Pseudo-operatorhasthe form of a separablepseudopotential.� PAW providesa rule to obtain expectationvalues
in a pseudopotentialcalculation.
PAW Electron Density
Electron density � �879� tur ns into a plane wave part �� �879� andtwo one-centercomponents� � �879� and �� � �879�
� �=79� � �� �879� � � � �=79� % �� � �=79�� � � � �� ;� �=79� �� �?�879� � �� V� ! a ] # ;! �879� � ! a ]�#^] �=7?� � � V% ! a ] �# ;! �879� � ! a ] �# ] �=7?� % �� V
� ! a ] � � �' ! � �� � � � � �� � �' ] �
Electron densitydivided (lik e the wave function) into( plane wavepart( two expansionsper atom in radial functionstimesspherical harmonics
PAW Total energy
Total energy divided into plane wave integraland two one-centerexpansionsper atom. ��1 �� � 3 �_� ! � � �. � . � % �. �Planewavepart:�. � � � � �� � ��% ��+� f � �� � �
� ������N� 7 ���N� 7}� � �� �879� � �� �=7?����� �� �=7 � � � �� �=7 � �H�� 7 % 7 � �� ��� � 7 �� �879� C�� V �=7��}1 �� 3 � � ��� � 7��� �=7?� �� �879�One-centerexpansionof plane wavepart�. � � � � ! a ]4 �# ! ��% �� � f � �#^]_�
� �� � �N� 7 � �N� 7}� � ��� �879� � �� �=7?����� �� � �=7 � � � �� �=7 � �H�� 7 % 7 � �� �c�9� 7 �� � �879� C � V �=7��}1 �� � 3 � � ���N� 7��� �=7?� �� � �=79�
One-centerexpansionof true density
. � � � � ! a ][�# ! �2% �� � f �$#�]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
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)
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
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.
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