Libxc a library of exchange and correlation functionals

1. Libxc a library of exchange and correlation functionals Miguel A. L. Marques 1LPMCN, Universit´e Claude Bernard Lyon 1 and CNRS, France 2European Theoretical Spectroscopy Facility March 2009 – ABINIT Workshop M. A. L. Marques Libxc

2. 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 150–200) 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 M. A. L. Marques Libxc

9. Kohn-Sham equations The main equations of DFT are the Kohn-Sham equations: − 1 2 2 + vext(r) + vH(r) + vxc(r) ϕi(r) = iϕi(r) where the exchange-correlation potential is deﬁned as vxc(r) = δExc δn(r) In any practical application of the theory, we have to use an approximation to Exc, or vxc(r). M. A. L. Marques Libxc

10. Kohn-Sham equations The main equations of DFT are the Kohn-Sham equations: − 1 2 2 + vext(r) + vH(r) + vxc(r) ϕi(r) = iϕi(r) where the exchange-correlation potential is deﬁned as vxc(r) = δExc δn(r) In any practical application of the theory, we have to use an approximation to Exc, or vxc(r). M. A. L. Marques Libxc

11. Jacob’s ladder Local density approximation: ELDA xc (r) = ELDA xc [n] n=n(r) Generalized gradient approximation: EGGA xc (r) = EGGA xc [n, n] n=n(r) Meta-generalized gradient approximation: EmGGA xc (r) = EmGGA xc [n, n, τ] n=n(r),τ=τ(r) And more: orbital functionals, hybrid functionals, hyper-GGAs, etc. M. A. L. Marques Libxc

15. What do we need? - I The energy is usually written as: Exc = d3 r exc(r) = d3 r n(r) xc(r) The potential in the LDA is: vLDA xc (r) = d dn eLDA xc (n) n=n(r) In the GGA: vGGA xc (r) = ∂ ∂n eLDA xc (n, n) n=n(r) − ∂ ∂( n) eLDA xc (n, n) n=n(r) M. A. L. Marques Libxc

16. What do we need? - II For response properties we also need higher derivatives of exc 1st-order response (polarizabilities, phonon frequencies, etc.): fLDA xc (r) = d2 d2n eLDA xc (n) n=n(r) 2st-order response (hyperpolarizabilities, etc.): kLDA xc (r) = d3 d3n eLDA xc (n) n=n(r) And let’s not forget spin... M. A. L. Marques Libxc

17. What do we need? - II For response properties we also need higher derivatives of exc 1st-order response (polarizabilities, phonon frequencies, etc.): fLDA xc (r) = d2 d2n eLDA xc (n) n=n(r) 2st-order response (hyperpolarizabilities, etc.): kLDA xc (r) = d3 d3n eLDA xc (n) n=n(r) And let’s not forget spin... M. A. L. Marques Libxc

18. An example: Perdew & Wang 91 (an LDA) Perdew and Wang parametrized the correlation energy per unit particle: ec(rs, ζ) = ec(rs, 0) + αc(rs) f(ζ) f (0) (1 − ζ4 ) + [ec(rs, 1) − ec(rs, 0)]f(ζ)ζ4 The function f(ζ) is f(ζ) = [1 + ζ]4/3 + [1 − ζ]4/3 − 2 24/3 − 2 , while its second derivative f (0) = 1.709921. The functions ec(rs, 0), ec(rs, 1), and −αc(rs) are all parametrized by the function g = −2A(1 + α1rs) log 1 + 1 2A(β1r 1/2 s + β2rs + β3r 3/2 s + β4r2 s ) M. A. L. Marques Libxc

19. Libxc 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 19 LDA functionals, 55 GGA functionals, 24 hybrids, and 7 mGGAs Contains functionals for 1D, 2D, and 3D calculations Returns εxc, vxc, fxc, and kxc Quite mature: included in octopus, APE, GPAW, ABINIT, and in the GW code of Murilo Tiago M. A. L. Marques Libxc

27. What is working! εxc vxc fxc kxc LDA OK OK OK OK GGA OK OK PARTIAL NO HYB GGA OK OK PARTIAL NO mGGA TEST TEST NO NO M. A. L. Marques Libxc

28. 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 . functional , 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 . functional , xc . nspin ) ; xc gga vxc (& gga func , xc . rho , xc . sigma , &xc . zk , xc . vrho , xc . vsigma xc gga end (& gga func ) ; break ; default : f p r i n t f ( stderr , "Functional ’%d’ not foundn" , xc . f u n c t i o n a l ) ; e x i t ( 1 ) ; } M. A. L. Marques Libxc

29. 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 real (8) : : rho , e c , v c TYPE( xc func ) : : xc c func 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 , xc c info , & XC LDA C VWN, XC UNPOLARIZED) CALL xc f90 lda vxc ( 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 M. A. L. Marques Libxc

30. 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 functional , e . g . ”PBE” ∗/ i n t family ; /∗ type of the functional , e . g . XC FAMILY GGA ∗/ char ∗ refs ; /∗ 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 . info−>name, b88 . info−>refs ) ; xc gga end (&b88 ) ; M. A. L. Marques Libxc

31. The future More functionals! More derivatives! More codes using it! M. A. L. Marques Libxc

34. Where to ﬁnd us! http://www.tddft.org/programs/octopus/wiki/ index.php/Libxc Comput. Phys. Commun. 151, 60–78 (2003) Phys. Stat. Sol. B 243, 2465–2488 (2006) M. A. L. Marques Libxc

Libxc a library of exchange and correlation functionals

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Libxc a library of exchange and correlation functionals