Functionals in DFT Miguel A. L. Marques 1LPMCN, Universite´ Claude Bernard Lyon 1 and CNRS, France 2European Theoretical Spectroscopy Facility Les Houches 2012 M. A. L. Marques (Lyon) XC functionals Les Houches 2012 1 / 63 Overview Hybrids 1 Introduction Orbital functionals 2 Jacob’s ladder 3 What functional to use LDA 4 Functional derivatives GGA 5 Functionals for vxc metaGGA 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches 2012 2 / 63 Outline Hybrids 1 Introduction Orbital functionals 2 Jacob’s ladder 3 What functional to use LDA 4 Functional derivatives GGA 5 Functionals for vxc metaGGA 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches 2012 3 / 63 What do we need to approximate? In DFT the energy is written as Z 3 E = Ts + d r vext(r)n(r) + EHartree + Ex + Ec In Kohn-Sham theory we need to approximate Ex[n] and Ec[n] In orbital-free DFT we also need Ts[n] The questions I will try to answer in these talks are: Which functionals exist and how are they divided in families? How to make a functional? Which functional should I use? M. A. L. Marques (Lyon) XC functionals Les Houches 2012 4 / 63 What do we need to approximate? In DFT the energy is written as Z 3 E = Ts + d r vext(r)n(r) + EHartree + Ex + Ec In Kohn-Sham theory we need to approximate Ex[n] and Ec[n] In orbital-free DFT we also need Ts[n] The questions I will try to answer in these talks are: Which functionals exist and how are they divided in families? How to make a functional? Which functional should I use? M. A. L. Marques (Lyon) XC functionals Les Houches 2012 4 / 63 What do we need to approximate? In DFT the energy is written as Z 3 E = Ts + d r vext(r)n(r) + EHartree + Ex + Ec In Kohn-Sham theory we need to approximate Ex[n] and Ec[n] In orbital-free DFT we also need Ts[n] The questions I will try to answer in these talks are: Which functionals exist and how are they divided in families? How to make a functional? Which functional should I use? M. A. L. Marques (Lyon) XC functionals Les Houches 2012 4 / 63 Outline Hybrids 1 Introduction Orbital functionals 2 Jacob’s ladder 3 What functional to use LDA 4 Functional derivatives GGA 5 Functionals for vxc metaGGA 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches 2012 5 / 63 The families — Jacob’s ladder Marc Chagall – Jacob’s dream M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63 The families — Jacob’s ladder C˙mical Heaffin Marc Chagall – Jacob’s dream M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63 The families — Jacob’s ladder C˙mical Heaffin LDA Marc Chagall – Jacob’s dream M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63 The families — Jacob’s ladder C˙mical Heaffin GGA LDA Marc Chagall – Jacob’s dream M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63 The families — Jacob’s ladder C˙mical Heaffin mGGA GGA LDA Marc Chagall – Jacob’s dream M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63 The families — Jacob’s ladder C˙mical Heaffin Occ. orbitals mGGA GGA LDA Marc Chagall – Jacob’s dream M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63 The families — Jacob’s ladder C˙mical Heaffin All orbitals Occ. orbitals mGGA GGA LDA Marc Chagall – Jacob’s dream M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63 The families — Jacob’s ladder C˙mical Heaffin Many-body All orbitals Occ. orbitals mGGA GGA LDA Semi-empirical Marc Chagall – Jacob’s dream M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63 The true ladder! (M. Escher – Relativity) M. A. L. Marques (Lyon) XC functionals Les Houches 2012 7 / 63 Let’s start from the bottom: the LDA In the original LDA from Kohn and Sham, one writes the xc energy as Z LDA 3 HEG Exc = d r n(r)exc (n(r)) HEG The quantity exc (n), exchange-correlation energy per unit particle, is a function of n. Sometimes you can see appearing HEG xc (n(r)), which is the energy per unit volume. They are related HEG HEG xc (n) = n exc (n) The exchange part of eHEG is simple to calculate and gives 2=3 HEG 3 3 1 ex = − 4 2π rs with rs the Wigner-Seitz radius 3 1=3 r = s 4πn M. A. L. Marques (Lyon) XC functionals Les Houches 2012 8 / 63 What about the correlation It is not possible to obtain the correlation energy of the HEG analytically, but we can calculate it to arbitrary precision numerically using, e.g., Quantum Monte-Carlo. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980) M. A. L. Marques (Lyon) XC functionals Les Houches 2012 9 / 63 The fits you should know about 1980: Vosko, Wilk & Nusair 1981: Perdew & Zunger 1992: Perdew & Wang (do not mix with the GGA from ’91) These are all fits to the correlation energy of Ceperley-Alder. They differ in some details, but all give more or less the same results. There are also versions of PZ and PW fitted to the more recent (and precise) Monte-Carlo results of Ortiz & Ballone (1994). But how does one make such fits? M. A. L. Marques (Lyon) XC functionals Les Houches 2012 10 / 63 The fits you should know about 1980: Vosko, Wilk & Nusair 1981: Perdew & Zunger 1992: Perdew & Wang (do not mix with the GGA from ’91) These are all fits to the correlation energy of Ceperley-Alder. They differ in some details, but all give more or less the same results. There are also versions of PZ and PW fitted to the more recent (and precise) Monte-Carlo results of Ortiz & Ballone (1994). But how does one make such fits? M. A. L. Marques (Lyon) XC functionals Les Houches 2012 10 / 63 The fits you should know about 1980: Vosko, Wilk & Nusair 1981: Perdew & Zunger 1992: Perdew & Wang (do not mix with the GGA from ’91) These are all fits to the correlation energy of Ceperley-Alder. They differ in some details, but all give more or less the same results. There are also versions of PZ and PW fitted to the more recent (and precise) Monte-Carlo results of Ortiz & Ballone (1994). But how does one make such fits? M. A. L. Marques (Lyon) XC functionals Les Houches 2012 10 / 63 The fits you should know about 1980: Vosko, Wilk & Nusair 1981: Perdew & Zunger 1992: Perdew & Wang (do not mix with the GGA from ’91) These are all fits to the correlation energy of Ceperley-Alder. They differ in some details, but all give more or less the same results. There are also versions of PZ and PW fitted to the more recent (and precise) Monte-Carlo results of Ortiz & Ballone (1994). But how does one make such fits? M. A. L. Marques (Lyon) XC functionals Les Houches 2012 10 / 63 An example: Perdew & Wang The strategy: The spin[ ζ = (n" − n#)=n] dependence is taken from VWN, that obtained it from RPA calculations. The 3 different terms of this expression are fit using an “educated” functional form that depends on several parameters Some of the coefficients are chosen to fulfill some exact conditions. The high-density limit (RPA). The low-density expansion. The rest of the parameters are fitted to Ceperley-Alder numbers. M. A. L. Marques (Lyon) XC functionals Les Houches 2012 11 / 63 An example: Perdew & Wang Perdew and Wang parametrized the correlation energy per unit particle: f (ζ) e (r ; ζ) = e (r ; 0) + α (r ) (1 − ζ4) + [e (r ; 1) − e (r ; 0)]f (ζ)ζ4 c s c s c s f 00(0) c s c s The function f (ζ) is [1 + ζ]4=3 + [1 − ζ]4=3 − 2 f (ζ) = ; 24=3 − 2 00 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 ( ) 1 g = −2A(1 + α1rs) log 1 + 1=2 3=2 2 2A(β1rs + β2rs + β3rs + β4rs ) M. A. L. Marques (Lyon) XC functionals Les Houches 2012 12 / 63 How good are the LDAs In spite of their simplicity, the LDAs yield extraordinarily good results for many cases, and are still currently used. However, they also fail in many cases Reaction energies are not to chemical accuracy (1 kcal/mol). Tends to overbind (bonds too short). Electronic states are usually too delocalized. Band-gaps of semiconductors are too small. Negative ions often do not bind. No van der Waals. etc. Many of these two problems are due to: The LDAs have the wrong asymptotic behavior. The LDAs have self-interaction. M. A. L. Marques (Lyon) XC functionals Les Houches 2012 13 / 63 How good are the LDAs In spite of their simplicity, the LDAs yield extraordinarily good results for many cases, and are still currently used. However, they also fail in many cases Reaction energies are not to chemical accuracy (1 kcal/mol). Tends to overbind (bonds too short). Electronic states are usually too delocalized. Band-gaps of semiconductors are too small. Negative ions often do not bind. No van der Waals. etc. Many of these two problems are due to: The LDAs have the wrong asymptotic behavior.