G o i n g b e y o n d L o c a l D e n s i t y - PowerPoint PPT Presentation

G o i n g b e y o n d L o c a l D e n s i t y a n d G r a d i e n t C o r r e c t e d X C f u n c t i o n a l s i n Q u a n t u m- E S P R E S S O

Jacob's ladder of Density Functional Theory

LDA and LSDA GGA : PW91, PBE, revPBE, RPBE, BLYP META-GGA: PKZB, TPSS, SIC, DFT+U, hybrids van der Waals functionals ... exact DFT

LDA and LSDA GGA : PW91, PBE, revPBE, RPBE, BLYP META-GGA: PKZB, TPSS, SIC, DFT+U, hybrids Van der Waals functionals ... exact DFT

simple approximations can work reasonably

simple approximations can work reasonably L(S)DA GGA

P r o b l e ms w i t h L D A / G G A f u n c t i o n a l s C h e m i c a l a c c u r a c y ( 1 k c a l / m o l ) i s f a r . - t r e n d s a r e o f t e n a c c u r a t e f o r s t r o n g b o n d s ( c o v a l e n t , i o n i c , m e t a l l i c ) - w e a k b o n d s / s m a l l o v e r l a p s a r e p r o b l e m a t i c S e l f i n t e r a c t i o n c a n c e l l a t i o n i s o n l y a p p r o x i m a t e l y v e r i fj e d i n L D A a n d G G A . - m o l e c u l a r d i s s o c i a t i o n l i m i t , T M O & R E a n d o t h e r a t o m - i n - s o l i d s y s t e m . v a n d e r Wa a l s i n t e r a c t i o n s a r e n o t t a k e n i n t o a c c o u n t - o c c a s i o n a l a g r e e m e n t w i t h e x p . f r o m c o m p e n s a t i o n o f e r r o r s

P r o b l e ms w i t h L D A / G G A f u n c t i o n a l s C h e m i c a l a c c u r a c y ( 1 k c a l / m o l ) i s f a r . - t r e n d s a r e o f t e n a c c u r a t e f o r s t r o n g b o n d s ( c o v a l e n t , i o n i c , m e t a l l i c ) - w e a k b o n d s / s m a l l o v e r l a p s a r e p r o b l e m a t i c S e l f i n t e r a c t i o n c a n c e l l a t i o n i s o n l y a p p r o x i m a t e l y v e r i fj e d i n L D A a n d G G A . - m o l e c u l a r d i s s o c i a t i o n l i m i t , T M O & R E a n d o t h e r a t o m - i n - s o l i d s y s t e m . v a n d e r Wa a l s i n t e r a c t i o n s a r e n o t t a k e n i n t o a c c o u n t - o c c a s i o n a l a g r e e m e n t w i t h e x p . f r o m c o m p e n s a t i o n o f e r r o r s

SIC, DFT+U, hybrids Self interaction correction was proposed as early as in 1981 by Perdew-Zunger. Conceptually important but not widely used. DFT+U has been introduced by Anisimov, Zaanen and Andersen as an approximation to treat strongly correlated materials. It has been more recently been applied also in more normal system with encouraging results. Hybrid functionals (like PBE0, B3LYB) mix a fraction of Self-interaction-free HF with LDA/GGA functionals. Is the method preferred by chemists. It is very expensive in a plane-wave basis.

H a r t r e e - F o c k e n e r g y ● Hartree-Fock ● Exact Exchange (OEP) ● Hybrid Functionals: HH, B3LYP , PBE0 (range separated) HSE

H F V x u s i n g P Ws F F T p s e u d o w f c t o r e a l s p a c e F o r e a c h q p o i n t a n d e a c h o c c u p i e d b a n d b u i l d “ c h a r g e d e n s i t y ” F F T c h a r g e t o r e c i p . s p a c e a n d s o l v e P o i s s o n e q . F F T b a c k t o r e a l s p a c e , mu l t i p l y b y w f c a n d a d d t o r e s u l t

T h e q + G = 0 d i v e r g e n c e G y g i - B a l d e r e s c h i P R B 3 4 , 4 4 0 5 ( 1 9 8 6 ) integrable divergence

T h e q + G = 0 d i v e r g e n c e G y g i - B a l d e r e s c h i P R B 3 4 , 4 4 0 5 ( 1 9 8 6 )

T h e q + G = 0 d i v e r g e n c e G y g i - B a l d e r e s c h i P R B 3 4 , 4 4 0 5 ( 1 9 8 6 )

S i l i c o n B u l k Fernandez, Dal Corso, Baldereschi, PRB 58, R7480 (1998) F .Gygi, EPFL PhD thesis (1988)

S i mp l e M o l e c u l e s Energies in kcal/mol = 43.3 meV

S c a l i n g ● Kinetic energy and local Potential ● Non local potential ● Fock operator

S c a l i n g ● Kinetic energy and local Potential ● Non local potential ● Fock operator From 10 to 100 times slower than standard case

S c a l i n g ● Kinetic energy and local Potential ● Non local potential ● Fock operator From 10 to 100 times slower than standard case Moore's law: computer power doubles every 18 months ( a factor of 10 in 5 yrs)

S c a l i n g ● Kinetic energy and local Potential ● Non local potential ● Fock operator From 10 to 100 times slower than standard case Separation of long- and short-range part in X can help

t h e mo d i fj e d s c f c y c l e The HF energy is with The HF equations are therefore

t h e mo d i fj e d s c f c y c l e

t h e mo d i fj e d s c f c y c l e Let's introduce an auxiliary set of functions delta_exx > 0 ! with

t h e mo d i fj e d s c f c y c l e Let's introduce an auxiliary set of functions with

t h e mo d i fj e d s c f c y c l e Let's introduce an auxiliary set of functions with

t h e mo d i fj e d s c f c y c l e Let's introduce an auxiliary set of functions with The minimizing equations become and

t h e mo d i fj e d s c f c y c l e

A d a p t i v e l y C o mp r e s s e d E x c h e n g e ( A C E ) Applying the Fock operator is estremely expensive !

A d a p t i v e l y C o mp r e s s e d E x c h e n g e ( A C E ) Applying the Fock operator is estremely expensive ! One can try to approximate it via a KB-type factorization in the inner loop of the nested scf-cycle (ACE) such that it works exactly on the reference wfc Lin Lin. Adaptively Compressed Exchange operator. arXiv, 2016.

A d a p t i v e l y C o mp r e s s e d E x c h e n g e ( A C E ) Applying the Fock operator is estremely expensive ! One can try to approximate it via a KB-type factorization in the inner loop of the nested scf-cycle (ACE) such that it works exactly on the reference wfc Lin Lin. Adaptively Compressed Exchange operator. arXiv, 2016.

A d a p t i v e l y C o mp r e s s e d E x c h e n g e ( A C E ) Applying the Fock operator is estremely expensive ! One can try to approximate it via a KB-type factorization in the inner loop of the nested scf-cycle (ACE) such that it works exactly on the reference wfcs in this way the calculation of H_psi in the inner loop is comparable to a non-hybrid functional. on the fully self-consistent wfcs the ACE operator is exact ! Lin Lin. Adaptively Compressed Exchange operator. arXiv, 2016.

LDA and LSDA GGA : PW91, PBE, revPBE, RPBE, BLYP META-GGA: PKZB, TPSS, SIC, DFT+U, hybrids van der Waals functionals ... exact DFT … to be continued