Functionals in DFT. Miguel A. L. Marques. Les Houches Universite Claude Bernard Lyon 1 and CNRS, France Theoretical Spectroscopy Facility

Transcription

1 Functionals in DFT Miguel A. L. Marques 1 LPMCN, Universite Claude Bernard Lyon 1 and CNRS, France Theoretical Spectroscopy Facility 2 European Les Houches 2012 M. A. L. Marques (Lyon) XC functionals Les Houches / 63

2 Overview 1 Introduction 2 Jacob s ladder LDA GGA metagga Hybrids Orbital functionals 3 What functional to use 4 Functional derivatives 5 Functionals for v xc 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches / 63

3 Outline 1 Introduction 2 Jacob s ladder LDA GGA metagga Hybrids Orbital functionals 3 What functional to use 4 Functional derivatives 5 Functionals for v xc 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches / 63

4 What do we need to approximate? In DFT the energy is written as E = T s + d 3 r v ext (r)n(r) + E Hartree + E x + E c In Kohn-Sham theory we need to approximate E x [n] and E c [n] In orbital-free DFT we also need T s [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 / 63

5 What do we need to approximate? In DFT the energy is written as E = T s + d 3 r v ext (r)n(r) + E Hartree + E x + E c In Kohn-Sham theory we need to approximate E x [n] and E c [n] In orbital-free DFT we also need T s [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 / 63

6 What do we need to approximate? In DFT the energy is written as E = T s + d 3 r v ext (r)n(r) + E Hartree + E x + E c In Kohn-Sham theory we need to approximate E x [n] and E c [n] In orbital-free DFT we also need T s [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 / 63

7 Outline 1 Introduction 2 Jacob s ladder LDA GGA metagga Hybrids Orbital functionals 3 What functional to use 4 Functional derivatives 5 Functionals for v xc 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches / 63

8 The families Jacob s ladder Marc Chagall Jacob s dream M. A. L. Marques (Lyon) XC functionals Les Houches / 63

9 The families Jacob s ladder C mical Heaffin Marc Chagall Jacob s dream M. A. L. Marques (Lyon) XC functionals Les Houches / 63

10 The families Jacob s ladder C mical Heaffin LDA Marc Chagall Jacob s dream M. A. L. Marques (Lyon) XC functionals Les Houches / 63

11 The families Jacob s ladder C mical Heaffin GGA LDA Marc Chagall Jacob s dream M. A. L. Marques (Lyon) XC functionals Les Houches / 63

12 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 / 63

13 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 / 63

14 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 / 63

15 The families Jacob s ladder C mical Heaffin All orbitals Many-body Occ. orbitals mgga GGA LDA Semi-empirical Marc Chagall Jacob s dream M. A. L. Marques (Lyon) XC functionals Les Houches / 63

16 The true ladder! (M. Escher Relativity) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

17 Let s start from the bottom: the LDA In the original LDA from Kohn and Sham, one writes the xc energy as Exc LDA = d 3 r n(r)exc HEG (n(r)) The quantity exc HEG (n), exchange-correlation energy per unit particle, is a function of n. Sometimes you can see appearing (n(r)), which is the energy per unit volume. They are related (n) = n e HEG (n) ɛ HEG xc ɛ HEG xc The exchange part of e HEG is simple to calculate and gives ex HEG = 3 ( ) 3 2/ π with r s the Wigner-Seitz radius ( 3 r s = 4πn xc ) 1/3 M. A. L. Marques (Lyon) XC functionals Les Houches / 63 r s

18 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 / 63

19 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 / 63

20 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 / 63

21 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 / 63

22 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 / 63

23 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 / 63

24 An example: Perdew & Wang 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) = 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 { A(β 1 r 1/2 s + β 2 r s + β 3 r 3/2 s + β 4 r 2 s ) } M. A. L. Marques (Lyon) XC functionals Les Houches / 63

25 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 / 63

26 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 / 63

27 The wrong asymptotics For a finite system, the electronic density decays asymptotically (when one moves away from the system) as n(r) e αr where α is related to the ionization potential of the system. As most LDA are simple rational functions of n, also exc LDA and the decay exponentially. v LDA xc However, one knows from very simple arguments that the true e xc (r) 1 2r Note that most of the more modern functionals do not solve this problem. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

28 The wrong asymptotics For a finite system, the electronic density decays asymptotically (when one moves away from the system) as n(r) e αr where α is related to the ionization potential of the system. As most LDA are simple rational functions of n, also exc LDA and the decay exponentially. v LDA xc However, one knows from very simple arguments that the true e xc (r) 1 2r Note that most of the more modern functionals do not solve this problem. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

29 The self-interaction problem For a system composed of a single electron (like the hydrogen atom), the total energy has to be equal to E = T s + d 3 r v ext (r)n(r) which means that E Hartree + E x + E c = 0 In particular, it is the exchange term that has to cancel the spurious Hartree contribution. The first rung where it is possible to cancel the self-interaction is the meta-gga. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

30 Beyond the LDA: the GEA To go beyond the LDA, for many years people tried the so-called Gradient Expansion Approximations. It is a systematic expansion of e xc in terms of derivatives of the density. In lowest order we have exc GEA (n, n, ) = exc LDA + a 1 (n) n 2 + Using different approaches, people went painfully to sixth order in the derivatives. Results were, however, much worse than the LDA. The reason was, one knows now, sum rules! M. A. L. Marques (Lyon) XC functionals Les Houches / 63

31 Beyond the LDA: the GEA To go beyond the LDA, for many years people tried the so-called Gradient Expansion Approximations. It is a systematic expansion of e xc in terms of derivatives of the density. In lowest order we have exc GEA (n, n, ) = exc LDA + a 1 (n) n 2 + Using different approaches, people went painfully to sixth order in the derivatives. Results were, however, much worse than the LDA. The reason was, one knows now, sum rules! M. A. L. Marques (Lyon) XC functionals Les Houches / 63

32 the GGAs The solution to this dilemma was the Generalized Gradient Approximation: Soften the requirement of having rigorous derivations and controlled approximations, and dream up a some more or less justified expression that depends on n and some free parameters. Or, mathematically E GGA xc = d 3 r n(r)exc GGA (n(r), n) Probably the first modern GGA for the xc was by Langreth & Mehl in M. A. L. Marques (Lyon) XC functionals Les Houches / 63

33 the GGAs The solution to this dilemma was the Generalized Gradient Approximation: Soften the requirement of having rigorous derivations and controlled approximations, and dream up a some more or less justified expression that depends on n and some free parameters. Or, mathematically E GGA xc = d 3 r n(r)exc GGA (n(r), n) Probably the first modern GGA for the xc was by Langreth & Mehl in M. A. L. Marques (Lyon) XC functionals Les Houches / 63

34 How to design an GGAs Write down an expression that obey some exact constrains such as reduces to the LDA when n = 0. is exact for some reference system like the He atom has some known asymptotic limits, for small gradients, large gradients, etc. obeys some known inequalities like the Lieb-Oxford bound etc. E x [n] [n] λ E LDA x M. A. L. Marques (Lyon) XC functionals Les Houches / 63

35 Exchange functionals Exchange functionals are almost always written as Ex GGA [n] = d 3 r n(r)e LDA (n(r))f(x(r)) with the reduced gradient x(r) = n(r) n(r) 4/3 Furthermore, they obey the spin-scaling relation for exchange E x [n, n ] = 1 2 (E x[2n ] + E x [2n ]) It is relatively simple to come up with and exchange GGA, so it is not surprising that there are more than 50 different versions in the literature. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

36 Example: B88 exchange Becke s famous 88 functional reads where F B88 x (x σ ) = A x βx 2 σ 1 + 6βx σ arcsinh(x σ ), For small x fulfills the gradient expansion. The energy density has the right asymptotics. The parameter β was fitted to the exchange energies of noble gases. By far the most used exchange functional in quantum chemistry. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

37 Example: PBE exchange The exchange part of the Perdew-Burke-Erzernhof functional reads: ( ) Fx PBE κ (x σ ) = 1 + κ 1 κ + µsσ 2, where The parameter s = n /2k F n. To recover the LDA response, µ = βπ 2 / Obeys the local version of the Lieb-Oxford bound. F PBE x (s) (Note that Becke 88 violates strongly and shamelessly this requirement.) By far the most used exchange functional in physics. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

38 Local behavior Unfortunately, locally most GGA exchange functionals are completely wrong M. A. L. Marques (Lyon) XC functionals Les Houches / 63

39 Correlation functionals Correlation functionals are much harder to design, so there are many less in the literature (around 20). For correlation there is no spin sum-rule, so the spin dependence is much more complicated. Even if the correlation energy is 5 smaller than exchange, it is important as energy differences are of the same order of magnitude. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

40 Example: LYP correlation The starting point is the Colle-Salvetti correlation functional From an ansatz to the many-body wave-function. Approximate the one- and two-particle density matrices. Approximate the Coulomb hole. Fudge the resulting formula and perform a dubious fit to the He atom. The results is a meta-gga (i.e. it depends on τ). Lee-Yang-Parr transformed the meta-gga of Colle-Salvetti by using the gradient expansion of the kinetic energy density leading to a functional depending on 2 n. Later it was found that the 2 n term could be rewritten by integrating by parts, leading to the current LYP GGA functional. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

41 Example: LYP correlation The starting point is the Colle-Salvetti correlation functional From an ansatz to the many-body wave-function. Approximate the one- and two-particle density matrices. Approximate the Coulomb hole. Fudge the resulting formula and perform a dubious fit to the He atom. The results is a meta-gga (i.e. it depends on τ). Lee-Yang-Parr transformed the meta-gga of Colle-Salvetti by using the gradient expansion of the kinetic energy density leading to a functional depending on 2 n. Later it was found that the 2 n term could be rewritten by integrating by parts, leading to the current LYP GGA functional. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

42 Example: LYP correlation The starting point is the Colle-Salvetti correlation functional From an ansatz to the many-body wave-function. Approximate the one- and two-particle density matrices. Approximate the Coulomb hole. Fudge the resulting formula and perform a dubious fit to the He atom. The results is a meta-gga (i.e. it depends on τ). Lee-Yang-Parr transformed the meta-gga of Colle-Salvetti by using the gradient expansion of the kinetic energy density leading to a functional depending on 2 n. Later it was found that the 2 n term could be rewritten by integrating by parts, leading to the current LYP GGA functional. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

43 Example: LYP correlation The starting point is the Colle-Salvetti correlation functional From an ansatz to the many-body wave-function. Approximate the one- and two-particle density matrices. Approximate the Coulomb hole. Fudge the resulting formula and perform a dubious fit to the He atom. The results is a meta-gga (i.e. it depends on τ). Lee-Yang-Parr transformed the meta-gga of Colle-Salvetti by using the gradient expansion of the kinetic energy density leading to a functional depending on 2 n. Later it was found that the 2 n term could be rewritten by integrating by parts, leading to the current LYP GGA functional. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

44 Example: LYP correlation The starting point is the Colle-Salvetti correlation functional From an ansatz to the many-body wave-function. Approximate the one- and two-particle density matrices. Approximate the Coulomb hole. Fudge the resulting formula and perform a dubious fit to the He atom. The results is a meta-gga (i.e. it depends on τ). Lee-Yang-Parr transformed the meta-gga of Colle-Salvetti by using the gradient expansion of the kinetic energy density leading to a functional depending on 2 n. Later it was found that the 2 n term could be rewritten by integrating by parts, leading to the current LYP GGA functional. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

45 Example: LYP correlation The starting point is the Colle-Salvetti correlation functional From an ansatz to the many-body wave-function. Approximate the one- and two-particle density matrices. Approximate the Coulomb hole. Fudge the resulting formula and perform a dubious fit to the He atom. The results is a meta-gga (i.e. it depends on τ). Lee-Yang-Parr transformed the meta-gga of Colle-Salvetti by using the gradient expansion of the kinetic energy density leading to a functional depending on 2 n. Later it was found that the 2 n term could be rewritten by integrating by parts, leading to the current LYP GGA functional. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

46 Example: LYP correlation The starting point is the Colle-Salvetti correlation functional From an ansatz to the many-body wave-function. Approximate the one- and two-particle density matrices. Approximate the Coulomb hole. Fudge the resulting formula and perform a dubious fit to the He atom. The results is a meta-gga (i.e. it depends on τ). Lee-Yang-Parr transformed the meta-gga of Colle-Salvetti by using the gradient expansion of the kinetic energy density leading to a functional depending on 2 n. Later it was found that the 2 n term could be rewritten by integrating by parts, leading to the current LYP GGA functional. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

47 Example: LYP correlation The starting point is the Colle-Salvetti correlation functional From an ansatz to the many-body wave-function. Approximate the one- and two-particle density matrices. Approximate the Coulomb hole. Fudge the resulting formula and perform a dubious fit to the He atom. The results is a meta-gga (i.e. it depends on τ). Lee-Yang-Parr transformed the meta-gga of Colle-Salvetti by using the gradient expansion of the kinetic energy density leading to a functional depending on 2 n. Later it was found that the 2 n term could be rewritten by integrating by parts, leading to the current LYP GGA functional. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

48 Example: PBE correlation Conditions: Obeys the second-order gradient expansion. In the rapidly varying limit correlation vanishes. Correct density scaling to the high-density limit. Ec PBE = [ ] d 3 r n(r) ec HEG + H where and H = γφ 3 log {1 + βγ [ 1 + At 2 ]} t2 1 + At 2 + A 2 t 4 A = β γ [ ] 1 exp{ ec HEG /(γφ 3 )} 1 M. A. L. Marques (Lyon) XC functionals Les Houches / 63

49 The metaggas To go beyond the GGAs, one can try the same trick and increase the number of arguments of the functional. In this case, we use both the Laplacian of the density 2 n and the kinetic energy density τ = occ. i 1 2 ϕ 2 Note that there are several other possibilities to define τ that lead to the same (integrated) kinetic energy, but to different local values. Often, the variables appear in the combination τ τ W, where τ W = n 2 8n is the von Weizsäcker kinetic energy. This is also the main quantity entering the electron localization function (ELF). M. A. L. Marques (Lyon) XC functionals Les Houches / 63

50 The electron localization function A. Savin, R. Nesper, S. Wengert, and T, F. Fässler, Angew. Chem. Int. Ed. Engl. 36, 1808 (1997) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

51 The most used metaggas TPSS (Tao, Perdew, Staroverov, Scuseria) John Perdew s school of functionals, i.e, many sum-rules and exact conditions. It is based on the PBE. M06L (Zhao and Truhlar) This comes from Don Truhlar s group, and it was crafted for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. VSXC (Van Voorhis and Scuseria) Based on a density matrix expansion plus fitting procedure. etc. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

52 Hybrid functionals The experimental values of some quantities lie often between they Hartree-Fock and DFT (LDA or GGA) values. So, we can try to mix, or to hybridize both theories. 1 Write an energy functional: E xc = ae Fock [ϕ i ] + (1 a)e DFT [n] 2 Minimize energy functional w.r.t. to the orbitals: v xc (r, r ) = av Fock (r, r ) + (1 a)v DFT (r) Note: for pure density functionals, minimizing w.r.t. the orbitals or w.r.t. the density gives the same, as: δf[n] δf[n] δn δϕ = δn δϕ = δf[n] δn ϕ M. A. L. Marques (Lyon) XC functionals Les Houches / 63

53 A short history of hybrid functionals 1993: The first hybrid functional was proposed by Becke, the B3PW91. It was a mixture of Hartree-Fock with LDA and GGAs (Becke 88 and PW91). The mixing parameter is 1/ : The famous B3LYP appears, replacing PW91 with LYP in the Becke functional. 1999: PBE0 proposed. The mixing was now 1/ : The screened hybrid HSE06 was proposed. It gave much better results for the band-gaps of semiconductors and allowed the calculation of metals. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

54 What is the mixing parameter? Let us look at the quasi-particle equation: ] [ v ext(r) + v H (r) φ QP i (r)+ d 3 r Σ(r, r ; ε QP i And now let us look at the different approximations: COHSEX: occ Σ = Hybrids occ Σ = i i φ QP i φ QP i So, we infer that a 1/ɛ! )φ QP i (r ) = ε QP (r)φ QP (r )W (r, r ; ω = 0) + δ(r r )Σ COH (r) i i φ QP (r)φ QP (r )a v(r r ) + δ(r r )(1 a) v DFT (r) i i (r ) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

55 Does it work (a = 1/ɛ )? Theoretical gap (ev) y=x PBE PBE0 PBE0ε Experimental gap (ev) Errors: PBE (46%), Hartree-Fock (230%), PBE0 (27%), PBE0ɛ (16.53%) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

56 Problems with traditional hybrids Hybrids certainly improve some properties of both molecules and solids, but a number of important problems do remain. For example: For metals, the long-range part of the Coulomb interaction leads to a vanishing density of states at the Fermi level due to a logarithmic singularity (as Hartree-Fock). For semiconductors, the quality of the gaps varies very much with the material and the mixing. For molecules, the asymptotics of the potential are still wrong, which leads to problems, e.g. for charge transfer states. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

57 Splitting of the Coulomb interaction The solution is to split the Coulomb interaction in a short-range and a long-range part: 1 = 1 erf(µr 12) + erf(µr 12) r 12 r }{{ 12 r }}{{ 12 } short range long range We now treat the one of the terms by a standard DFT functional and make a hybrid out of the other. There are two possibilities 1 DFT: long-range; Hybrid: short-range. Such as HSE, good for metals. 2 DFT: short-range; Hybrid: long-range. The LC functionals for molecules. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

58 A screened hybrid: the HSE The Heyd-Scuseria-Ernzerhof functional is written as E HSE xc = αe HF, SR x (µ) + (1 α)e PBE, SR x (µ) + E PBE, LR x (µ) + E PBE The most common version of the HSE chooses µ = 0.11 and α = 1/4. Mind that basically every code has a different version of the HSE. For comparison, here are the average percentual errors for the gaps of a series of semiconductors and insulators c PBE HF+c PBE0 HSE06 G 0 W 0 47% 250% 29% 17% 11% M. A. L. Marques (Lyon) XC functionals Les Houches / 63

59 Gaps with the HSE Theoretical gap (ev) Si, MoS 2 Ge GaAs y=x PBE PBE0 PBE0ε PBE0 mix TB09 SiC,CdS,AlP GaN ZnS ZnO C BN, AlN MgO Xe LiCl SiO 2 Kr Ar, LiF Ne Theoretical gap (ev) Si, MoS 2 Ge GaAs y=x HSE06 HSE06 mix SiC,CdS,AlP GaN ZnS ZnO C BN, AlN MgO Xe LiCl SiO 2 Kr Ar, LiF Ne Experimental gap (ev) Experimental gap (ev) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

60 Parenthesis: Mind the GAP! M. A. L. Marques (Lyon) XC functionals Les Houches / 63

61 The band gap of CuAlO 2 The agreement of LDA+U and HSE06 hybrid functional to the experiment is accidental ScGW shows that the band gaps are much higher Experimental data are for optical gap: exciton binding energy 0.5 ev Agreement with experiment can only be achieved by the addition of phonons. F. Trani et al, PRB 82, (2010); J. Vidal et al, PRL 104, (2010) E g [ev] E g indirect E g direct =E g direct -Eg indirect exp. direct gap exp. indirect gap LDA LDA+U B3LYP HSE03 HSE06 G 0 W 0 scgw scgw+p 3.5 ev (exp) = 5 ev (el. QP) ev (excitons) - 1 ev (phonons) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

62 The band gap of CuAlO 2 The agreement of LDA+U and HSE06 hybrid functional to the experiment is accidental ScGW shows that the band gaps are much higher Experimental data are for optical gap: exciton binding energy 0.5 ev Agreement with experiment can only be achieved by the addition of phonons. F. Trani et al, PRB 82, (2010); J. Vidal et al, PRL 104, (2010) E g [ev] E g indirect E g direct =E g direct -Eg indirect exp. direct gap exp. indirect gap LDA LDA+U B3LYP HSE03 HSE06 G 0 W 0 scgw scgw+p 3.5 ev (exp) = 5 ev (el. QP) ev (excitons) - 1 ev (phonons) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

63 CAM functionals For LC functionals to have the right asymptotics they need α = 1. This value is however too large in order to obtain good results for several molecular properties. To improve this behavior one needs more flexibility 1 = 1 [α + βerf(µr 12)] + α + βerf(µr 12) r 12 r }{{ 12 r }}{{ 12 } short range long range The asymptotics are now determined by α + β. Note that this form leads to a normal hybrid for β = 0 and to a screened hybrid for α = 0. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

64 CAM-B3LYP The most used CAM functional is probably CAM-B3LYP that is constructed in a similar way to B3LYP, but with α = 0.19 β = 0.46 µ = 0.33 This functional gives very much improved charge transfer excitations. Note that in any case α + β = , which means that the asymptotics are still wrong. The problem, as it often happens in functional development, is that CAM-B3LYP is better for change transfer, but worse for many other properties... M. A. L. Marques (Lyon) XC functionals Les Houches / 63

65 CAM-B3LYP The most used CAM functional is probably CAM-B3LYP that is constructed in a similar way to B3LYP, but with α = 0.19 β = 0.46 µ = 0.33 This functional gives very much improved charge transfer excitations. Note that in any case α + β = , which means that the asymptotics are still wrong. The problem, as it often happens in functional development, is that CAM-B3LYP is better for change transfer, but worse for many other properties... M. A. L. Marques (Lyon) XC functionals Les Houches / 63

66 Orbital functionals Self-interaction correction: E SIC xc [ϕ] = Exc LDA [n, n ] i 1 2 E LDA xc [ ϕ i (r) 2, 0] i d 3 r d 3 r ϕ i(r) 2 ϕ i (r ) 2 r r Exact-exchange: E exact x [n, ϕ] = 1 2 jk d 3 r d 3 r ϕ j (r)ϕ k (r )ϕ k (r)ϕ j (r ) r r M. A. L. Marques (Lyon) XC functionals Les Houches / 63

67 Outline 1 Introduction 2 Jacob s ladder LDA GGA metagga Hybrids Orbital functionals 3 What functional to use 4 Functional derivatives 5 Functionals for v xc 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches / 63

68 If you are a physicist! You are running Solids If problem is small enough and code available use HSE Otherwise use PBE SOL or AM05 However, whenever you can just stick to GW and BSE Molecules van der Waals: use the Langreth-Lundqvist functional (or a variant) Charge transfer: no good alternatives here If problem is small enough and code available use PBE0 Time-dependent problem try LB94 Otherwise use PBE Note that if you want to calculate response, you are basically stuck with standard GGA functionals. In any case, stick to functionals from the J. Perdew family. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

69 If you are a chemist! You are running Solids You are a physicist, so go back to the previous slide Molecules van der Waals: you might escape with Grimme s trick Charge transfer: CAM-B3LYP If problem is small enough use B3LYP Otherwise use BLYP Note that you also have a chance of getting your paper accepted if you use a functional by G. Scuseria or D. Truhlar. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

70 Outline 1 Introduction 2 Jacob s ladder LDA GGA metagga Hybrids Orbital functionals 3 What functional to use 4 Functional derivatives 5 Functionals for v xc 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches / 63

71 What do we need for a Kohn-Sham calculation? The energy is usually written as: E xc = d 3 r ɛ xc (r) = d 3 r n(r)e xc (r) and the xc potential that enters the Kohn-Sham equations is defined as v xc (r) = δe xc δn(r) if we are trying to solve response equations then also the following quantities may appear f xc (r, r ) = δ 2 E xc δn(r)δn(r ) k xc (r, r, r ) = δ 3 E xc δn(r)δn(r )δn(r ) And let s not forget spin... M. A. L. Marques (Lyon) XC functionals Les Houches / 63

72 What do we need for a Kohn-Sham calculation? The energy is usually written as: E xc = d 3 r ɛ xc (r) = d 3 r n(r)e xc (r) and the xc potential that enters the Kohn-Sham equations is defined as v xc (r) = δe xc δn(r) if we are trying to solve response equations then also the following quantities may appear f xc (r, r ) = δ 2 E xc δn(r)δn(r ) k xc (r, r, r ) = δ 3 E xc δn(r)δn(r )δn(r ) And let s not forget spin... M. A. L. Marques (Lyon) XC functionals Les Houches / 63

73 Derivatives for the LDA For LDA functionals, it is trivial to calculate these functional derivatives. For example δn( r)e HEG xc (n( r)) xc (r) = d 3 r δn(r) = d 3 r d dn nelda xc (n) δ(r r) n=n( r) = d dn nelda xc (n) v LDA n=n(r) n=n(r) Higher derivatives are also simple: fxc LDA (r) = d 2 d 2 n nelda xc (n) kxc LDA (r) = d 3 d 3 n nelda xc (n) n=n(r) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

74 Derivatives for the GGA For the GGAs it is a bit more complicated vxc GGA GGA δn( r)e xc (n( r), n( r)) (r) = d 3 r δn(r) = d 3 r n negga xc (n, n) δ(r r) n=n( r) + n n egga xc (n, n) δ(r r) n=n( r) = xc (n, n) xc (n, n) n=n(r) n nelda with similar expressions for f xc and k xc. ( n) nelda n=n(r) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

75 Derivatives for the meta-ggas Meta-GGAs are technically orbital functions due to the functional dependence on τ. Therefore, to calculate correctly v xc within DFT one has to resort to the OEP procedure (see next slide). However, the expression that one normally uses is v mgga xc,i (r) = 1 ϕ i (r) δe xc δϕ i (r) This definition gives the correct potentials for the case of an LDA or a GGA, as 1 δe xc [n] ϕ i (r) δϕ i (r) = 1 ϕ i (r) d 3 r δe xc[n] δn( r) δn( r) δϕ i (r) = 1 ϕ i (r) d 3 r δe xc[n] δn( r) ϕ i ( r)δ(r r) = δe xc[n] δn(r) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

76 The optimized effective method If E xc depends on the KS orbitals, we have to use the chain-rule δe xc δn(r) = d 3 r δe xc δv KS (r) δv KS (r) δn(r) The second term is the inverse non-interacting density response function. Using again the chain-rule δe xc δn(r) = d 3 r d 3 r j δe xc δϕ j (r ) δϕ j (r ) δv KS (r) δv KS (r) δn(r) The second term can be calculated with perturbation theory. Now, multiplying by χ and after some algebra, we arrive at: M. A. L. Marques (Lyon) XC functionals Les Houches / 63

77 The OEP equation The OEP integral equation is then written as d 3 r Q(r, r )vxc OEP = Λ(r) where Q(r, r ) = Λ(r) = N ϕ j (r )G j (r, r)ϕ j (r) + c.c j=1 N d 3 r ϕ j (r )u xc,j (r )ϕ j (r) + c.c j=1 and G j (r, r) = k j ϕ k (r )ϕ k (r) u xc,j (r ) = 1 ɛ j ɛ k ϕ j (r ) δe xc δϕ j (r ) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

78 The KLI approximation One way of performing this approximation consists in approximating G j (r, r) k j ϕ k (r )ϕ k (r) ɛ = 1 ɛ [ δ(r r ) ϕ j (r)ϕ j (r ) ] which leads to a very simple expression for the xc potential v KLI xc = j n j (r) [ ] u xc,j (r) + v xc,j KLI ūxc,j KLI n(r) The KLI approximation is often an excellent approximation to the OEP potential. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

79 Outline 1 Introduction 2 Jacob s ladder LDA GGA metagga Hybrids Orbital functionals 3 What functional to use 4 Functional derivatives 5 Functionals for v xc 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches / 63

80 The van Leeuwen-Baerends GGA It can be proved that it is impossible to get, at the same time, the correct asymptotics for E x and v x using a GGA form. Most of the functionals are concerned by the energy, but it is also possible to write down directly a functional for v xc. This was done by van Leeuwen and Baerends in 1994 that used a form similar to Becke 88 vxc LB94 (x σ ) = vxc LDA βnσ 1/ βx σ arcsinh(x σ ), This functional is particularly useful when calculating, e.g., ionization potentials from the value of the HOMO, or when performing time-dependent simulations with laser fields. x 2 σ M. A. L. Marques (Lyon) XC functionals Les Houches / 63

81 The Becke-Roussel functional Meta-GGA energy functional (depends on n, n, 2 n, τ). Models the exchange hole of hydrogenic atoms. Correct asymptotic 1/r behavior for finite systems. Excellent description of the Slater part of the EXX potential. Exact for the hydrogen atom AD Becke and MR Roussel, Phys. Rev. A 39, 3761 (1989) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

82 The Becke-Johnson functional At this point, it is useful to write the (KS) exchange potential as a sum v xσ (r) = vxσ SL (r) + vxσ OEP (r) The BJ potential is a simple approximation to the OEP contribution vxσ OEP (r) vxσ BJ τ σ (r) (r) = C v n σ (r) where C v = 5/(12π 2 ) Exact for the hydrogen atom and for the HEG. Yields the atomic step structure in the exchange potential very accurately. It has the derivative discontinuity for fractional particle numbers. Goes to a finite constant at. Not gauge-invariant. AD Becke and ER Johnson, J. Chem. Phys. 124, (2006) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

83 The Becke-Johnson functional At this point, it is useful to write the (KS) exchange potential as a sum v xσ (r) = vxσ SL (r) + vxσ OEP (r) The BJ potential is a simple approximation to the OEP contribution vxσ OEP (r) vxσ BJ τ σ (r) (r) = C v n σ (r) where C v = 5/(12π 2 ) Exact for the hydrogen atom and for the HEG. Yields the atomic step structure in the exchange potential very accurately. It has the derivative discontinuity for fractional particle numbers. Goes to a finite constant at. Not gauge-invariant. AD Becke and ER Johnson, J. Chem. Phys. 124, (2006) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

84 Extensions the Tran and Blaha potential By looking at band gaps of solids, Tran and Blaha proposed vxσ TB (r) = cvxσ BR τ σ (r) (r) + (3c 2)C v n σ (r) where c is obtained from ( 1 c = α + β V cell cell d 3 r n(r) n(r) ) 1/2 Band-gaps are of similar quality as G 0 W 0, but at the computational cost of an LDA! Value of c is always larger than one, as the BJ gaps are too small. α and β are fitted parameters. Parameter c creates problems of size-consistency. F Tran and P Blaha, Phys. Rev. Lett. 102, (2009) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

85 Extensions the Tran and Blaha potential By looking at band gaps of solids, Tran and Blaha proposed vxσ TB (r) = cvxσ BR τ σ (r) (r) + (3c 2)C v n σ (r) where c is obtained from ( 1 c = α + β V cell cell d 3 r n(r) n(r) ) 1/2 Band-gaps are of similar quality as G 0 W 0, but at the computational cost of an LDA! Value of c is always larger than one, as the BJ gaps are too small. α and β are fitted parameters. Parameter c creates problems of size-consistency. F Tran and P Blaha, Phys. Rev. Lett. 102, (2009) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

86 Some solutions - the RPP functional Räsänen, Pittalis, and Proetto (RPP) proposed the following correction to the BJ potential vxσ RPP (r) = vxσ BR D σ (r) (r) + C v n σ (r) where the function D is D σ (r) = τ σ (r) 1 n σ (r) 2 j2 σ(r) 4 n σ (r) n σ (r) It is exact for all one-electron systems (and for the e-gas). It is gauge-invariant. It has the correct asymptotic behavior for finite systems. E Räsänen, S Pittalis, and C Proetto, J. Chem. Phys. 132, (2010) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

87 Some solutions - the RPP functional Räsänen, Pittalis, and Proetto (RPP) proposed the following correction to the BJ potential vxσ RPP (r) = vxσ BR D σ (r) (r) + C v n σ (r) where the function D is D σ (r) = τ σ (r) 1 n σ (r) 2 j2 σ(r) 4 n σ (r) n σ (r) It is exact for all one-electron systems (and for the e-gas). It is gauge-invariant. It has the correct asymptotic behavior for finite systems. E Räsänen, S Pittalis, and C Proetto, J. Chem. Phys. 132, (2010) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

88 Benchmark of the mggas Test-set composed of 17 atoms, 19 molecules, 10 H 2 chains, and 20 solids. LDA PBE LB94 BJ RPP TB Ionization potentials atoms molecules Polarizabilities molecules H 2 chains Band gaps (mean average relative error in %) M. Oliveira et al, JCTC 6, (2010) M. A. L. Marques (Lyon) XC functionals Les Houches / 63

89 Problems with these functionals It may seem like a very nice idea to model directly v xc (or f xc ). However, it can be proved that these functionals are not the functional derivative of an energy functional. This opens a theoretical Pandora s box. No unique way of calculating the energy by integration. The energy depends on the path used. Results can vary dramatically. Energy is not conserved when performing a TD simulation Zero-force and zero-torque theorems broken. Spurious forces and torques appear during a TD simulation.... In any case, and even if we don t have the energy, we can have access to all derivatives of the energy, i.e., all response properties of the system. M. A. L. Marques (Lyon) XC functionals Les Houches / 63

90 Outline 1 Introduction 2 Jacob s ladder LDA GGA metagga Hybrids Orbital functionals 3 What functional to use 4 Functional derivatives 5 Functionals for v xc 6 Availability M. A. L. Marques (Lyon) XC functionals Les Houches / 63

91 Availability of functionals The problem of availability: There are many approximations for the xc (probably of the order of ) Most computer codes only include a very limited quantity of functionals, typically around Chemists and Physicists do not use the same functionals! It is therefore difficult to: Reproduce older calculations with older functionals Reproduce calculations performed with other codes Perform calculations with the newest functionals M. A. L. Marques (Lyon) XC functionals Les Houches / 63

92 Availability of functionals The problem of availability: There are many approximations for the xc (probably of the order of ) Most computer codes only include a very limited quantity of functionals, typically around Chemists and Physicists do not use the same functionals! It is therefore difficult to: Reproduce older calculations with older functionals Reproduce calculations performed with other codes Perform calculations with the newest functionals M. A. L. Marques (Lyon) XC functionals Les Houches / 63

93 Our solution: LIBXC The physics: Contains 27 LDAs, 123 GGAs, 25 hybrids, and 13 mggas for the exchange, correlation, and the kinetic energy Functionals for 1D, 2D, and 3D Returns ε xc, v xc, f xc, and k xc Quite mature: in 14 different codes including OCTOPUS, APE, GPAW, ABINIT, etc. The technicalities: 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 Just type LIBXC in google! M. A. L. Marques (Lyon) XC functionals Les Houches / 63

94 Our solution: LIBXC The physics: Contains 27 LDAs, 123 GGAs, 25 hybrids, and 13 mggas for the exchange, correlation, and the kinetic energy Functionals for 1D, 2D, and 3D Returns ε xc, v xc, f xc, and k xc Quite mature: in 14 different codes including OCTOPUS, APE, GPAW, ABINIT, etc. The technicalities: 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 Just type LIBXC in google! M. A. L. Marques (Lyon) XC functionals Les Houches / 63