Libxc - a library of exchange and correlation functionals - TDDFT.org

Libxc 

a library of exchange and correlation functionals 

Miguel A. L. Marques 

1 LPMCN, Université Claude Bernard Lyon 1 and CNRS, France 

2 European Theoretical Spectroscopy Facility 

Octopus Workshop 

M. A. L. Marques Libxc

Why the need for libxc 

The xc functional is at the heart of DFT 

There are many approximations for the xc (probably of the 

order of 250–300) 

Most computer codes only include a very limited quantity 

of functionals, typically around 10–15 

Chemist and Physicists do not use the same functionals! 

Difficult to reproduce older calculations with older 

functionals 

Difficult to reproduce calculations performed with other 

codes 

Difficult to perform calculations with the newest functionals 












codes 













codes 













codes 













codes 













codes 













codes 



Kohn-Sham equations 

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

[ 

− 1 ] 

2 ∇2 + v ext (r) + v H (r) + v xc (r) ϕ i (r) = ɛ i ϕ i (r) 

where the exchange-correlation potential is defined as 

v xc (r) = δE xc 

δn(r) 

In any practical application of the theory, we have to use an 

approximation to E xc , or v xc (r). 


Kohn-Sham equations 

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

[ 

− 1 ] 

2 ∇2 + v ext (r) + v H (r) + v xc (r) ϕ i (r) = ɛ i ϕ i (r) 

where the exchange-correlation potential is defined as 

v xc (r) = δE xc 

δn(r) 

In any practical application of the theory, we have to use an 

approximation to E xc , or v xc (r). 


Jacob’s ladder 

Local density approximation: 

E LDA 

xc 

(r) = Exc 

LDA [n] ∣ n=n(r) 

Generalized gradient approximation: 

E GGA 

xc 

(r) = Exc 

GGA [n, ∇n] ∣ n=n(r) 

Meta-generalized gradient approximation: 

E mGGA 

xc 

(r) = Exc 

mGGA [n, ∇n, τ] ∣ n=n(r),τ=τ(r) 

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

etc. 




E LDA 

xc 

(r) = Exc 



E GGA 

xc 

(r) = Exc 

GGA [n, ∇n] ∣ n=n(r) 


E mGGA 

xc 

(r) = Exc 



etc. 




E LDA 

xc 

(r) = Exc 



E GGA 

xc 

(r) = Exc 

GGA [n, ∇n] ∣ n=n(r) 


E mGGA 

xc 

(r) = Exc 



etc. 




E LDA 

xc 

(r) = Exc 



E GGA 

xc 

(r) = Exc 

GGA [n, ∇n] ∣ n=n(r) 


E mGGA 

xc 

(r) = Exc 



etc. 


What do we need - I 

The energy is usually written as: 

∫ 

∫ 

E xc = d 3 r e xc (r) = 

d 3 r n(r)ɛ xc (r) 

The potential in the LDA is: 

(r) = d dn eLDA (n) ∣ 

v LDA 

xc 

xc 

∣ 

n=n(r) 

In the GGA: 

v GGA 

xc (r) = ∂ 

∂n eLDA xc 

(n, ∇n) 

∣ − ∇ 

n=n(r) 

∂ 

∂(∇n) eLDA xc (n, ∇n) ∣ 

∣ 

n=n(r) 


What do we need - II 

For response properties we also need higher derivatives of e xc 

1st-order response (polarizabilities, phonon frequencies, 

etc.): 

fxc LDA (r) = d 2 

d 2 n eLDA xc (n) ∣ 

∣ 

n=n(r) 

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

kxc LDA (r) = d 3 


And let’s not forget spin... 

∣ 

n=n(r) 


What do we need - II 

For response properties we also need higher derivatives of e xc 

1st-order response (polarizabilities, phonon frequencies, 

etc.): 

fxc LDA (r) = d 2 


∣ 

n=n(r) 

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

kxc LDA (r) = d 3 


And let’s not forget spin... 

∣ 

n=n(r) 


An example: Perdew & Wang 91 (an LDA) 

Perdew and Wang parametrized the correlation energy per unit 

particle: 

e c (r s , ζ) = e c (r s , 0) + α c (r s ) f (ζ) 

f ′′ (0) (1 − ζ4 ) + [e c (r s , 1) − e c (r s , 0)]f (ζ)ζ 4 

The function f (ζ) is 

f (ζ) = [1 + ζ]4/3 + [1 − ζ] 4/3 − 2 

, 

2 4/3 − 2 

while its second derivative f ′′ (0) = 1.709921. The functions e c (r s , 0), 

e c (r s , 1), and −α c (r s ) are all parametrized by the function 

g = −2A(1 + α 1 r s ) log 

{ 

1 + 

1 

2A(β 1 r 1/2 

s + β 2 r s + β 3 r 3/2 

s + β 4 r 2 s ) 

} 



Written in C from scratch 

Bindings both in C and in Fortran 

Lesser GNU general public license (v. 3.0) 

Automatic testing of the functionals 

Contains at the moment 33 LDA functionals, 142 GGA 

functionals, 36 hybrids, and 14 mGGAs 

Contains functionals for exchange, correlation, and kinetic 

energy 

Contains functionals for 1D, 2D, and 3D calculations 

Returns ε xc , v xc , f xc , and k xc 

Quite mature: included in 16 codes (including abinit, 

espresso, cp2k, etc.) 










energy 














energy 














energy 














energy 














energy 














energy 














energy 














energy 






What is working! 

ε xc v xc f xc k xc 

LDA OK OK OK OK 

GGA OK OK OK NO 

HYB GGA OK OK OK NO 

mGGA TEST TEST TEST NO 


An example in C 

switch ( 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 ) ; 

else 

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

x c l d a v x c (& lda func , xc . rho , &xc . zk , xc . vrho ) ; 

xc lda end (& l d a f u n c ) ; 

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 t d e 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 t 

use 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 ) : : x c c f u n c 

TYPE( 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 9 0 l d a v x c ( xc c func , rho , e c , v c ) 

CALL x c f 9 0 l d a e n d ( x c c f u n c ) 

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 kind ; / ∗ 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 of the f u n c t i o n a l , e . g . XC FAMILY GGA ∗ / 

char ∗ r e f s ; / ∗ references ∗ / 

i n t provides ; / ∗ 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 f o −>name , b88 . i n f o −>r e f s ) ; 

xc gga end (&b88 ) ; 


The future 

More functionals! 

More derivatives! 

More codes using it! 


The future 





The future 





Where to find us! 

http://www.tddft.org/programs/octopus/wiki/ 

index.php/Libxc 

Libxc: a library of exchange and correlation functionals for DFT 

M.A.L. Marques, M.J.T. Oliveira, and T. Burnus 

Comput. Phys. Commun. 183, 2272-2281 (2012)

Libxc - a library of exchange and correlation functionals - TDDFT.org

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