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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

Aug 22, 2023 202 likes •621 views

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

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G

  • i

n g b e y

  • n

d L

  • c

a l D e n s i t y a n d G r a d i e n t C

  • r

r e c t e d X C f u n c t i

  • n

a l s i n Q u a n t u m- E S P R E S S O

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SLIDE 2

Jacob's ladder of Density Functional Theory

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SLIDE 3

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

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SLIDE 4

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

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SLIDE 5

simple approximations can work reasonably

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SLIDE 6

simple approximations can work reasonably L(S)DA GGA

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SLIDE 7
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SLIDE 8
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SLIDE 9
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SLIDE 10
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SLIDE 11
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SLIDE 12
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SLIDE 13

P r

  • b

l e ms w i t h L D A / G G A f u n c t i

  • 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

  • l

) i s f a r .

  • t

r e n d s a r e

  • f

t e n a c c u r a t e f

  • r

s t r

  • n

g b

  • n

d s ( c

  • v

a l e n t , i

  • n

i c , m e t a l l i c )

  • w

e a k b

  • n

d s / s m a l l

  • v

e r l a p s a r e p r

  • b

l e m a t i c

S e l f i n t e r a c t i

  • n

c a n c e l l a t i

  • n

i s

  • n

l y a p p r

  • 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
  • l

e c u l a r d i s s

  • c

i a t i

  • n

l i m i t , T M O & R E a n d

  • t

h e r a t

  • m
  • i

n

  • s
  • 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

  • n

s a r e n

  • t

t a k e n i n t

  • a

c c

  • u

n t

  • c

c a s i

  • n

a l a g r e e m e n t w i t h e x p . f r

  • m

c

  • m

p e n s a t i

  • n
  • f

e r r

  • r

s

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SLIDE 14

P r

  • b

l e ms w i t h L D A / G G A f u n c t i

  • 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

  • l

) i s f a r .

  • t

r e n d s a r e

  • f

t e n a c c u r a t e f

  • r

s t r

  • n

g b

  • n

d s ( c

  • v

a l e n t , i

  • n

i c , m e t a l l i c )

  • w

e a k b

  • n

d s / s m a l l

  • v

e r l a p s a r e p r

  • b

l e m a t i c

S e l f i n t e r a c t i

  • n

c a n c e l l a t i

  • n

i s

  • n

l y a p p r

  • 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
  • l

e c u l a r d i s s

  • c

i a t i

  • n

l i m i t , T M O & R E a n d

  • t

h e r a t

  • m
  • i

n

  • s
  • 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

  • n

s a r e n

  • t

t a k e n i n t

  • a

c c

  • u

n t

  • c

c a s i

  • n

a l a g r e e m e n t w i t h e x p . f r

  • m

c

  • m

p e n s a t i

  • n
  • f

e r r

  • r

s

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SLIDE 15

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.

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SLIDE 16
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SLIDE 17
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SLIDE 18

H a r t r e e

  • F
  • c

k e n e r g y

  • Hartree-Fock
  • Exact Exchange (OEP)
  • Hybrid Functionals: HH, B3LYP

, PBE0 (range separated) HSE

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SLIDE 19

H F V x u s i n g P Ws

F F T c h a r g e t

  • r

e c i p . s p a c e a n d s

  • l

v e P

  • i

s s

  • n

e q . F F T b a c k t

  • 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

  • r

e s u l t F F T p s e u d

  • w

f c t

  • r

e a l s p a c e F

  • r

e a c h q p

  • i

n t a n d e a c h

  • 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 ”

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SLIDE 20

T h e q + G = 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 5 ( 1 9 8 6 )

integrable divergence

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SLIDE 21

T h e q + G = 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 5 ( 1 9 8 6 )

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SLIDE 22

T h e q + G = 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 5 ( 1 9 8 6 )

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SLIDE 23

S i l i c

  • n

B u l k

Fernandez, Dal Corso, Baldereschi, PRB 58, R7480 (1998) F .Gygi, EPFL PhD thesis (1988)

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SLIDE 24

S i mp l e M

  • l

e c u l e s

Energies in kcal/mol = 43.3 meV

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S c a l i n g

  • Kinetic energy and local Potential
  • Non local potential
  • Fock operator
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SLIDE 26

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

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SLIDE 27

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)

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SLIDE 28

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

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SLIDE 29

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

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SLIDE 30

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

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SLIDE 31

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 delta_exx > 0 !

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SLIDE 32

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

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SLIDE 33

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

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SLIDE 34

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

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SLIDE 35

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

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SLIDE 36

A d a p t i v e l y C

  • 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 !

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SLIDE 37

A d a p t i v e l y C

  • 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.

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SLIDE 38

A d a p t i v e l y C

  • 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.

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SLIDE 39

A d a p t i v e l y C

  • 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.

  • n the fully self-consistent wfcs the ACE operator is exact !

Lin Lin. Adaptively Compressed Exchange operator. arXiv, 2016.

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SLIDE 40

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