libxc a library of exchange and correlation functionals

Libxca library of exchange and correlation functionals

Miguel A. L. Marques

1LPMCN, Universite Claude Bernard Lyon 1 and CNRS, France2European Theoretical Spectroscopy Facility

March 2009 ABINIT Workshop

M. A. L. Marques Libxc

Why the need for libxc?

The xc functional is at the heart of DFTThere are many approximations for the xc (probably of theorder of 150200)Most computer codes only include a very limited quantityof functionals, typically around 1015Chemist and Physicists do not use the same functionals!

Difficult to reproduce older calculations with olderfunctionalsDifficult to reproduce calculations performed with othercodesDifficult to perform calculations with the newest functionals

Kohn-Sham equations

The main equations of DFT are the Kohn-Sham equations:[1

22 + vext(r) + vH(r) + vxc(r)

]i(r) = ii(r)

where the exchange-correlation potential is defined as

vxc(r) =Excn(r)

In any practical application of the theory, we have to use anapproximation to Exc, or vxc(r).

Kohn-Sham equations

The main equations of DFT are the Kohn-Sham equations:[1

22 + vext(r) + vH(r) + vxc(r)

]i(r) = ii(r)

where the exchange-correlation potential is defined as

vxc(r) =Excn(r)

In any practical application of the theory, we have to use anapproximation to Exc, or vxc(r).

Jacobs ladder

Local density approximation:

ELDAxc (r) = ELDAxc [n]

n=n(r)

Generalized gradient approximation:

EGGAxc (r) = EGGAxc [n,n]

n=n(r)

Meta-generalized gradient approximation:

EmGGAxc (r) = EmGGAxc [n,n, ]

n=n(r),=(r)

And more: orbital functionals, hybrid functionals, hyper-GGAs,etc.

Jacobs ladder

n=n(r)

n=n(r),=(r)

Jacobs ladder

n=n(r)

n=n(r),=(r)

Jacobs ladder

n=n(r)

n=n(r),=(r)

What do we need? - I

The energy is usually written as:

Exc =

d3r exc(r) =

d3r n(r)xc(r)

The potential in the LDA is:

vLDAxc (r) =ddn

eLDAxc (n)n=n(r)

In the GGA:

vGGAxc (r) =

neLDAxc (n,n)

n=n(r)

(n)

eLDAxc (n,n)n=n(r)

What do we need? - II

For response properties we also need higher derivatives of exc

1st-order response (polarizabilities, phonon frequencies,etc.):

f LDAxc (r) =d2

d2neLDAxc (n)

n=n(r)

2st-order response (hyperpolarizabilities, etc.):

kLDAxc (r) =d3

d3neLDAxc (n)

n=n(r)

And lets not forget spin...

What do we need? - II

For response properties we also need higher derivatives of exc

1st-order response (polarizabilities, phonon frequencies,etc.):

f LDAxc (r) =d2

d2neLDAxc (n)

n=n(r)

2st-order response (hyperpolarizabilities, etc.):

kLDAxc (r) =d3

d3neLDAxc (n)

n=n(r)

And lets not forget spin...

An example: Perdew & Wang 91 (an LDA)

Perdew and Wang parametrized the correlation energy per unitparticle:

ec(rs, ) = ec(rs,0) + c(rs)f ()f (0)

(1 4) + [ec(rs,1) ec(rs,0)]f ()4

The function f () is

f () =[1 + ]4/3 + [1 ]4/3 2

24/3 2,

while its second derivative f (0) = 1.709921. The functions ec(rs,0),ec(rs,1), and c(rs) are all parametrized by the function

g = 2A(1 + 1rs) log

{1 +

1

2A(1r1/2s + 2rs + 3r

3/2s + 4r2s )

}

Libxc

Written in C from scratchBindings both in C and in FortranLesser GNU general public license (v. 3.0)Automatic testing of the functionalsContains at the moment 19 LDA functionals, 55 GGAfunctionals, 24 hybrids, and 7 mGGAsContains functionals for 1D, 2D, and 3D calculationsReturns xc, vxc, fxc, and kxcQuite mature: included in octopus, APE, GPAW, ABINIT,and in the GW code of Murilo Tiago

Libxc

What is working!

xc vxc fxc kxc

LDA OK OK OK OKGGA OK OK PARTIAL NOHYB GGA OK OK PARTIAL NOmGGA TEST TEST NO NO

An example in C

swi tch ( x c f a m i l y f r o m i d ( xc . f u n c t i o n a l ) ){case XC FAMILY LDA :

i f ( xc . f u n c t i o n a l == XC LDA X )x c l d a x i n i t (& lda func , xc . nspin , 3 , 0 ) ;

e lsex c l d a i n i t (& lda func , xc . f u n c t i o n a l , xc . nspin ) ;

xc lda vxc (& lda func , xc . rho , &xc . zk , xc . vrho ) ;xc lda end (& lda func ) ;break ;

case XC FAMILY GGA :x c g g a i n i t (& gga func , xc . f u n c t i o n a l , xc . nspin ) ;xc gga vxc (& gga func , xc . rho , xc . sigma , &xc . zk , xc . vrho , xc . vsigma ) ;xc gga end (& gga func ) ;break ;

d e f a u l t :f p r i n t f ( s tde r r , "Functional %d not found\n" , xc . f u n c t i o n a l ) ;e x i t ( 1 ) ;

}

Another example in Fortran

program l x c t e s tuse l i b x c

i m p l i c i t none

r e a l ( 8 ) : : rho , e c , v c

TYPE( xc func ) : : xc c funcTYPE( x c i n f o ) : : x c c i n f o

CALL x c f 9 0 l d a i n i t ( xc c func , x c c i n f o , &XC LDA C VWN, XC UNPOLARIZED)

CALL x c f 90 l d a v xc ( xc c func , rho , e c , v c )CALL xc f90 lda end ( xc c func )

end program l x c t e s t

The info structure

typedef s t r u c t {i n t number ; / i n d e n t i f i e r number /i n t k ind ; / XC EXCHANGE or XC CORRELATION /

char name ; / name of the f u n c t i o n a l , e . g . PBE /i n t f a m i l y ; / type o f the f u n c t i o n a l , e . g . XC FAMILY GGA /char r e f s ; / re ferences /

i n t prov ides ; / e . g . XC PROVIDES EXC | XC PROVIDES VXC /. . .

} x c f u n c i n f o t y p e ;

This is an example on how you can use it:

xc gga type b88 ;

x c g g a i n i t (&b88 , XC GGA X B88 , XC UNPOLARIZED ) ;p r i n t f ("The functional %s is defined in the reference(s):\n%s" ,

b88 . i n fo>name, b88 . i n fo>r e f s ) ;xc gga end (&b88 ) ;

The future

More functionals!More derivatives!More codes using it!

The future

Where to find us!

http://www.tddft.org/programs/octopus/wiki/index.php/Libxc

Comput. Phys. Commun. 151, 6078 (2003)

Phys. Stat. Sol. B 243, 24652488 (2006)

http://www.tddft.org/programs/octopus/wiki/index.php/Libxchttp://www.tddft.org/programs/octopus/wiki/index.php/Libxc

libxc a library of exchange and correlation functionals

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