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Advanced Density Functional Theory.Jesus M. Ugalde

Kimika Fakultatea, Euskal Herriko Unibertsitatea and DonostiaInternational Physics Center (DIPC); P.K. 1072, 20080 Donostia,

Euskadi (Spain)http://www.ehu.es/chemistry/theory

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Advanced Density Functional Theory for

Electronic Structure Calculations.

Jesus M. Ugalde

Kimika Fakultatea, Euskal Herriko Unibertsitatea, and DonostiaInternational Physics Center (DIPC); P.K. 1072, 20080 Donostia,

Euskadi (Spain)http://www.ehu.es/chemistry/theory

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The Hohenberg-Kohn Theorem

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The Hohenberg-Kohn Theorem

Let the ground states of H and H be nondegenerate,E =

| H

| [] | T + U + V | [] = E []

The functional E v [] achieves its minimum value for the true ground

state electron density associated with the external potential v.6

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The Euler Equation of DFT

Variational functional:

= E []

dr ( r )

N

Euler equation:

E []( r ) = 0

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The Chemical Potential:

E [N, ( r ) ] E [N, v ( r ) ]Consequently,

dE =E N v

dN + E v( r ) v( r ) drUse

dr ( r ) = dN ; = E N v ; E v( r ) = ( r )to obtain the fundamental equation for the chemical reactivity:

dE = dN + ( r ) v( r ) drentry point to Conceptual DFT

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The Kohn-Sham formulation

Recall the Euler equation:

E []( r ) = 0 ; E [] = T [] + dr ( r ) v( r ) + 12 dr dr ( r ) ( r )| r r | + E xc []

Theorem. (T. L. Gilbert, Phys. Rev. B 12, 2111 (1975))

(r ) 0 , ( r ) dr = N ; ( i) N i=1 / i | j = ij / ( r ) =

N

i=1i ( r ) i( r )

Namely, ( r ) 0, ( r ) dr = N is N-representable

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Consider the non-interacting system described by the single Slater de-terminant made by the {i}N i=1 orbitals and estimate its kinetic energy ,namely:

T s =N

i=1i | 122 | i

and express the energy of the real system as:

E [] = T s [] + (r ) v( r ) dr +

12 d

r dr ( r ) ( r

)

| r r | + E xc []

E xc [] = E xc [] + ( T [] T s [])The exchange-correlation functional contains someundetermined kinetic energy

Its non-interacting potential will be determined later10

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The expression for the Euler equation is:

T s []( r )

+ v( r ) + (r

) dr

| r r | + E xc []

( r )

veff ( r )= s

The Kohn-Sham presciption: Determine E xc [] to obtain a workable ex-pression for veff ( r ). Then, the ( r ) can be obtained by solving thenon-interacting system:

12

2i + veff ( r ) i = ii ; i = 1 , N

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The Kohn-Sham Implementation

1. Devise: E xc []

vxc ( r ) = E xc []

(r

)

2. Make a guess for the Kohn-Sham orbitals {i}N i=1

Build: veff ( r ) = v( r ) + (r ) d r |r r |

+ vxc ( r )

Solve: 122i + veff ( r ) i = ii , i = 1 , N ; until consistency

3. Calculate: E [] = T s [] + ( r ) v( r ) dr + 12 dr dr (r ) ( r )|r r |

+ E xc []

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The exchangecorrelation functional

E ee = dr dr 2 ( r , r )|

r

r |

2 ( r 1 , r 2 ) = N ( N 1)

2 |( r 1 , r 2 , . . . , r N ) |2 dr 3 . . . d r N

2 ( r 1 , r 2 ) = 12

( r 1 ) [( r 2 ) + xc ( r 1 , r 2 ) ]

E ee [] = 12 dr dr ( r ) ( r )

|r

r |

+12 dr dr ( r ) xc ( r , r )

|r

r |

= J []+ E xc []

[E xc []( T []T s [])]13

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H = T + V + U , 0 1

V

/

||

= ( r ) ,. Hence : V =0

= veff ( r ) , V =1

= v( r )

1

0

E

d = E =1

E =0

dH = H

d =

V

d + Ud

E

d =

dr ( r )

V

d +

1

2dJ [] +

1

2d

( r 1 ) xc ( r 1 , r 2 )

|r

1 r

2 | dr 1 r 2

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E =1 E =0 = dr ( r ) 10 V

d +

12

J []+12 ( r 1 )

10 xc ( r 1 , r 2 ) d

|r 1 r 2 | dr 1 r 2

E =0 = T S + dr ( r ) V =0

E =1 = T S + dr ( r ) V =1 ( r ) + 12 J [] + ( r 1 ) xc ( r 1 , r 2 )|r 1 r 2 | dr 1 r 2But, recall that the original expresson for E =1 E

E = T + dr ( r ) v( r ) + 12 J [] + ( r 1 ) xc ( r 1 , r 2 )|r 1 r 2 | dr 1 r 216

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Same expressions upon consideration of the following equivalencies:

T S xc ( r 1 , r 2 ) = 10 xc ( r 1 , r 2 ) dT xc (

r

1 ,r

2 )The adiabatic connection has adsorbed the excess kinetic en-ergy term into an exchangecorrelation hole description.

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The Local Density ApproximationE xc [( r )] =

( r )

xc [( r )] dr

xc [( r 1 )] = 12

10 xc ( r 1 , r 2 ) d

|r 1 r 2 | dr 2

xc [( r )] = x [( r )] + c[( r )]

LDAx [] = 34

3 ( r )

1 / 3

V W N c [] = F []

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Practical Approximate Exchange-Correlation Functionals

E xc = dr ( r ) xc , , , , , , . . .Jacobs ladder

DFT heaven . . .

. . . . . . meta-GGA: . . . , ( r ) GGA: . . . ,( r ) LSDA: ( r ) Hartree world

And he dreamed, and behold a ladder set up on the earth, and

the top of it reached to heaven... (Gen 28:12)19

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The Hybrid Approximate Functionals

E xc = dr 1 ( r 1 ) xc ( r 1 , r 2 )

|r 1

r 2

|

dr 2

E xc = 10 E xc d20

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The Crudest Hybrid Functional:

E xc 12

E =0xc + E =1xc

E =0xc Hartree-Fock like exchange with the Kohn-Sham orbitalsE =1xc Exchange-correlation functional

Better Approximate Hybrid Functionals:E Hybridxc = (1 ax ) E DF T x + a xE HF x + E DF T c

B3LYP Approximate Hybrid Functional:E B 3 LY P xc = E

LDAxc + a 0 ( E

HF x E LDAx )+ ax ( E B 88x E LDAx )+ a c( E LY P c E LDAc )

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Doubly Hybrid Functionals: (nal) fth-rung func-tionals because they add information about theunoccupied Kohn-Sham orbitals.

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Doubly hybrid density functional for accurate descriptions of

nonbond interactions, thermochemistry, and thermochemicalkinetics

E R 5xc [] = E LDAxc

+ c1 E HF x E LDAx + c2 E GGAx+ c3 E MP 2c E LDAc + c4 E GGAc

Y. Zhang, X. Xu and W. A. Goddard III; PNAS, 106, 4963 (2009)

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