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. The LDAs have self-interaction.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 13 / 63 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 LDA most LDA are simple rational functions of n, also exc and the LDA vxc decay exponentially.
However, one knows from very simple arguments that the true
1 e (r) ∼ − xc 2r Note that most of the more modern functionals do not solve this problem.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 14 / 63 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 LDA most LDA are simple rational functions of n, also exc and the LDA vxc decay exponentially.
However, one knows from very simple arguments that the true
1 e (r) ∼ − xc 2r Note that most of the more modern functionals do not solve this problem.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 14 / 63 The self-interaction problem
For a system composed of a single electron (like the hydrogen atom), the total energy has to be equal to Z 3 E = Ts + d r vext(r)n(r)
which means that
EHartree + Ex + Ec = 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 2012 15 / 63 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 exc in terms of derivatives of the density. In lowest order we have
GEA LDA 2 exc (n, ∇n, ··· ) = exc + a1(n)|∇n| + ···
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 2012 16 / 63 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 exc in terms of derivatives of the density. In lowest order we have
GEA LDA 2 exc (n, ∇n, ··· ) = exc + a1(n)|∇n| + ···
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 2012 16 / 63 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 Z GGA 3 GGA Exc = d r n(r)exc (n(r), ∇n)
Probably the first modern GGA for the xc was by Langreth & Mehl in 1981.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 17 / 63 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 Z GGA 3 GGA Exc = d r n(r)exc (n(r), ∇n)
Probably the first modern GGA for the xc was by Langreth & Mehl in 1981.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 17 / 63 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
Ex [n] LDA ≤ λ Ex [n] etc.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 18 / 63 Exchange functionals
Exchange functionals are almost always written as Z GGA 3 LDA Ex [n] = d r n(r)e (n(r))F(x(r))
with the reduced gradient
|∇n(r)| x(r) = n(r)4/3
Furthermore, they obey the spin-scaling relation for exchange
1 E [n , n ] = (E [2n ] + E [2n ]) x ↑ ↓ 2 x ↑ x ↓ 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 2012 19 / 63 Example: B88 exchange
Becke’s famous ’88 functional reads
2 B88 1 βxσ Fx (xσ) = 1 + , Ax 1 + 6βxσ arcsinh(xσ) where 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 2012 20 / 63 Example: PBE exchange
The exchange part of the Perdew-Burke-Erzernhof functional reads: κ PBE( ) = + κ − , Fx xσ 1 1 2 κ + µsσ where
The parameter s = |∇n|/2kF n. To recover the LDA response, µ = βπ2/3 ≈ 0.21951. Obeys the local version of the Lieb-Oxford bound.
PBE Fx (s) ≤ 1.804.
(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 2012 21 / 63 Local behavior
Unfortunately, locally most GGA exchange functionals are completely wrong
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 22 / 63 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 2012 23 / 63 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 ∇2n. Later it was found that the ∇2n term could be rewritten by integrating by parts, leading to the current LYP GGA functional.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 24 / 63 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 ∇2n. Later it was found that the ∇2n term could be rewritten by integrating by parts, leading to the current LYP GGA functional.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 24 / 63 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 ∇2n. Later it was found that the ∇2n term could be rewritten by integrating by parts, leading to the current LYP GGA functional.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 24 / 63 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 ∇2n. Later it was found that the ∇2n term could be rewritten by integrating by parts, leading to the current LYP GGA functional.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 24 / 63 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 ∇2n. Later it was found that the ∇2n term could be rewritten by integrating by parts, leading to the current LYP GGA functional.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 24 / 63 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 ∇2n. Later it was found that the ∇2n term could be rewritten by integrating by parts, leading to the current LYP GGA functional.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 24 / 63 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 ∇2n. Later it was found that the ∇2n term could be rewritten by integrating by parts, leading to the current LYP GGA functional.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 24 / 63 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 ∇2n. Later it was found that the ∇2n term could be rewritten by integrating by parts, leading to the current LYP GGA functional.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 24 / 63 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. Z PBE 3 h HEG i Ec = d r n(r) ec + H
where β 1 + At2 H = γφ3 log 1 + t2 γ 1 + At2 + A2t4 and β h i−1 A = exp{−eHEG/(γφ3)} − 1 γ c
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 25 / 63 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 ∇2n and the kinetic energy density
occ. X 1 τ = |∇ϕ|2 2 i 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 − τW , |∇n|2 where τW = 8n is the von Weizsacker¨ kinetic energy. This is also the main quantity entering the electron localization function (ELF).
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 26 / 63 The electron localization function
A. Savin, R. Nesper, S. Wengert, and T,F. Fassler,¨ Angew. Chem. Int. Ed. Engl. 36, 1808 (1997)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 27 / 63 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 2012 28 / 63 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:
Fock DFT Exc = aE [ϕi ] + (1 − a)E [n]
2 Minimize energy functional w.r.t. to the orbitals:
0 Fock 0 DFT vxc(r, r ) = av (r, r ) + (1 − a)v (r)
Note: for pure density functionals, minimizing w.r.t. the orbitals or w.r.t. the density gives the same, as:
δF[n] Z δF[n] δn δF[n] = = ϕ δϕ∗ δn δϕ∗ δn
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 29 / 63 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 1/5. 1994: The famous B3LYP appears, replacing PW91 with LYP in the Becke functional. 1999: PBE0 proposed. The mixing was now 1/4. 2003: 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 2012 30 / 63 What is the mixing parameter?
Let us look at the quasi-particle equation:
∇2 Z − + v (r) + v (r) φQP(r)+ d3r 0 Σ(r, r 0; εQP)φQP(r 0) = εQPφQP(r 0) 2 ext H i i i i i
And now let us look at the different approximations: COHSEX:
occ X QP QP 0 0 0 Σ = − φi (r)φi (r )W (r, r ; ω = 0)+ δ(r − r )ΣCOH(r) i
Hybrids
occ X QP QP 0 0 0 DFT Σ = − φi (r)φi (r )a v(r − r )+ δ(r − r )(1 − a) v (r) i
So, we infer that a ∼ 1/∞!
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 31 / 63 Does it work (a = 1/∞)?
y=x PBE 20 PBE0 PBE0ε∞ 15 10 Theoretical gap (eV) 5 0 0 5 10 15 20 Experimental gap (eV)
Errors: PBE (46%), Hartree-Fock (230%), PBE0 (27%), PBE0∞ (16.53%)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 32 / 63 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 2012 33 / 63 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 ) erf(µr ) = 12 + 12 r12 r12 r12 | {z } | {z } 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 2012 34 / 63 A screened hybrid: the HSE
The Heyd-Scuseria-Ernzerhof functional is written as
HSE HF, SR PBE, SR PBE, LR PBE Exc = αEx (µ) + (1 − α)Ex (µ) + Ex (µ) + Ec
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
PBE HF+c PBE0 HSE06 G0W0 47% 250% 29% 17% 11%
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 35 / 63 Gaps with the HSE
y=x y=x PBE HSE06 20 20 PBE0 HSE06mix PBE0ε∞ Ar, LiF Ar, LiF
PBE0 Xe Xe
mix 2 2 Kr Kr
15 TB09 15 SiO SiO LiCl LiCl 2 2 MgO MgO ZnO ZnO BN, AlN BN, AlN 10 10 C C Ne Ne Si, MoS Si, MoS ZnS ZnS GaN GaN Theoretical gap (eV) Theoretical gap (eV) SiC,CdS,AlP SiC,CdS,AlP GaAs GaAs 5 5 Ge Ge 0 0 0 5 10 15 20 0 5 10 15 20 Experimental gap (eV) Experimental gap (eV)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 36 / 63 Parenthesis: Mind the GAP!
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 37 / 63 The band gap of CuAlO2
The agreement of 6 indirect Eg direct LDA+U and HSE06 E 5 g ∆ direct indirect hybrid functional to the =Eg -Eg exp. direct gap experiment is accidental 4 ScGW shows that the 3 [eV] exp. indirect g gap band gaps are much E higher 2
Experimental data are 1 for optical gap: exciton 0 LDA LDA+U B3LYP HSE03 HSE06 G W scGW scGW+P binding energy ∼0.5 eV 0 0 Agreement with experiment can only be 3.5 eV (exp) = 5 eV (el. QP) achieved by the addition - 0.5 eV (excitons) of phonons. - 1 eV (phonons) F. Trani et al, PRB 82, 085115 (2010); J. Vidal et al, PRL 104, 136401 (2010)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 38 / 63 The band gap of CuAlO2
The agreement of 6 indirect Eg direct LDA+U and HSE06 E 5 g ∆ direct indirect hybrid functional to the =Eg -Eg exp. direct gap experiment is accidental 4 ScGW shows that the 3 [eV] exp. indirect g gap band gaps are much E higher 2
Experimental data are 1 for optical gap: exciton 0 LDA LDA+U B3LYP HSE03 HSE06 G W scGW scGW+P binding energy ∼0.5 eV 0 0 Agreement with experiment can only be 3.5 eV (exp) = 5 eV (el. QP) achieved by the addition - 0.5 eV (excitons) of phonons. - 1 eV (phonons) F. Trani et al, PRB 82, 085115 (2010); J. Vidal et al, PRL 104, 136401 (2010)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 38 / 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 )] α + βerf(µr ) = 12 + 12 r12 r12 r12 | {z } | {z } 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 2012 39 / 63 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 = 0.65 6= 1, 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 2012 40 / 63 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 = 0.65 6= 1, 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 2012 40 / 63 Orbital functionals
Self-interaction correction:
SIC LDA X LDA 2 Exc [ϕ] = Exc [n↑, n↓] − Exc [|ϕi (r)| , 0] i 2 1 X Z Z |ϕ (r)|2 |ϕ (r 0)| − d3r d3r 0 i i 2 |r − r 0| i Exact-exchange: Z Z ∗ ∗ 0 0 1 X ϕj (r)ϕk (r )ϕk (r)ϕj (r ) E exact[n, ϕ] = − d3r d3r 0 x 2 |r − r 0| jk
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 41 / 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 42 / 63 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 toGW 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 2012 43 / 63 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 2012 44 / 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 45 / 63 What do we need for a Kohn-Sham calculation?
The energy is usually written as: Z Z 3 3 Exc = d r xc(r) = d r n(r)exc(r)
and the xc potential that enters the Kohn-Sham equations is defined as δE v (r) = xc xc δn(r) if we are trying to solve response equations then also the following quantities may appear
δ2E δ3E f (r, r 0) = xc k (r, r 0, r 00) = xc xc δn(r)δn(r 0) xc δn(r)δn(r 0)δn(r 00)
And let’s not forget spin...
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 46 / 63 What do we need for a Kohn-Sham calculation?
The energy is usually written as: Z Z 3 3 Exc = d r xc(r) = d r n(r)exc(r)
and the xc potential that enters the Kohn-Sham equations is defined as δE v (r) = xc xc δn(r) if we are trying to solve response equations then also the following quantities may appear
δ2E δ3E f (r, r 0) = xc k (r, r 0, r 00) = xc xc δn(r)δn(r 0) xc δn(r)δn(r 0)δn(r 00)
And let’s not forget spin...
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 46 / 63 Derivatives for the LDA
For LDA functionals, it is trivial to calculate these functional derivatives. For example
Z δn(˜r)eHEG(n(˜r)) v LDA(r) = d3˜r xc xc δn(r) Z 3 d LDA = d ˜r nexc (n) δ(r − ˜r) dn n=n(˜r)
d LDA = nexc (n) dn n=n(r)
Higher derivatives are also simple:
d 2 d 3 LDA( ) = LDA( ) LDA( ) = LDA( ) fxc r 2 nexc n kxc r 3 nexc n d n n=n(r) d n n=n(r)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 47 / 63 Derivatives for the GGA
For the GGAs it is a bit more complicated
Z δn(˜r)eGGA(n(˜r), ∇n(˜r)) v GGA(r) = d3˜r xc xc δn(r) Z 3 ∂ GGA = d ˜r nexc (n, ∇n) δ(r − ˜r) ∂n n=n(˜r)
∂ GGA + n exc (n, ∇n) ∇δ(r − ˜r) ∂∇n n=n(˜r)
∂ LDA ∂ LDA = nexc (n, ∇n) − ∇ nexc (n, ∇n) ∂n n=n(r) ∂(∇n) n=n(r)
with similar expressions for fxc and kxc.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 48 / 63 Derivatives for the meta-GGAs
Meta-GGAs are technically orbital functions due to the functional dependence on τ. Therefore, to calculate correctly vxc within DFT one has to resort to the OEP procedure (see next slide). However, the expression that one normally uses is
mGGA 1 δExc vxc,i (r) = ∗ ϕi (r) δϕi (r) This definition gives the correct potentials for the case of an LDA or a GGA, as 1 δE [n] 1 Z δE [n] δn(˜r) xc = d3˜r xc ∗ ∗ ˜ ϕi (r) δϕi (r) ϕi (r) δn(r) δϕi (r) 1 Z δE [n] = d3˜r xc ϕ∗(˜r)δ(r − ˜r) ∗ ˜ i ϕi (r) δn(r) δE [n] = xc δn(r)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 49 / 63 The optimized effective method
If Exc depends on the KS orbitals, we have to use the chain-rule δE Z δE δv (r) xc = d3r 0 xc KS δn(r) δvKS(r) δn(r) The second term is the inverse non-interacting density response function. Using again the chain-rule
Z Z 00 δExc 3 0 3 00 X δExc δϕj (r ) δvKS(r) = d r d r 00 δn(r) δϕj (r ) δv (r) δn(r) j KS
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 2012 50 / 63 The OEP equation
The OEP integral equation is then written as Z 3 0 0 OEP d r Q(r, r )vxc = Λ(r)
where N 0 X ∗ 0 0 Q(r, r ) = ϕj (r )Gj (r , r)ϕj (r) + c.c j=1 N Z X 3 0 ∗ 0 0 Λ(r) = d r ϕj (r )uxc,j (r )ϕj (r) + c.c j=1
and 0 ∗ 0 X ϕk (r )ϕk (r) 0 1 δExc Gj (r , r) = uxc,j (r ) = ∗ 0 0 j − k ϕ (r ) δϕj (r ) k6=j j
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 51 / 63 The KLI approximation
One way of performing this approximation consists in approximating
0 ∗ X ϕk (r )ϕ (r) 1 G (r 0, r) ≈ k = δ(r − r 0) − ϕ (r)ϕ (r 0) j ∆ ∆ j j k6=j
which leads to a very simple expression for the xc potential
X nj (r) h i v KLI = u (r) + v¯KLI − u¯KLI xc n(r) xc,j xc,j xc,j j
The KLI approximation is often an excellent approximation to the OEP potential.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 52 / 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 53 / 63 The van Leeuwen-Baerends GGA
It can be proved that it is impossible to get, at the same time, the correct asymptotics for Ex and vx 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 vxc. This was done by van Leeuwen and Baerends in 1994 that used a form similar to Becke 88
2 LB94 LDA 1/3 xσ ∆vxc (xσ) = vxc − βnσ , 1 + 3βxσ arcsinh(xσ) 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.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 54 / 63 The Becke-Roussel functional
Meta-GGA energy functional (depends on n, ∇n, ∇2n, 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 2012 55 / 63 The Becke-Johnson functional
At this point, it is useful to write the (KS) exchange potential as a sum SL OEP vxσ(r) = vxσ (r) + ∆vxσ (r) The BJ potential is a simple approximation to the OEP contribution s OEP BJ τσ(r) ∆vxσ (r) ≈ ∆vxσ (r) = C∆v nσ(r)
p 2 where C∆v = 5/(12π ) 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, 221101 (2006)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 56 / 63 The Becke-Johnson functional
At this point, it is useful to write the (KS) exchange potential as a sum SL OEP vxσ(r) = vxσ (r) + ∆vxσ (r) The BJ potential is a simple approximation to the OEP contribution s OEP BJ τσ(r) ∆vxσ (r) ≈ ∆vxσ (r) = C∆v nσ(r)
p 2 where C∆v = 5/(12π ) 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, 221101 (2006)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 56 / 63 Extensions – the Tran and Blaha potential
By looking at band gaps of solids, Tran and Blaha proposed s TB BR τσ(r) vxσ (r) = cvxσ (r) + (3c − 2)C∆v nσ(r)
where c is obtained from 1 Z |∇n(r)|1/2 c = α + β d3r Vcell cell n(r)
Band-gaps are of similar quality as G0W0, 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, 226401 (2009)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 57 / 63 Extensions – the Tran and Blaha potential
By looking at band gaps of solids, Tran and Blaha proposed s TB BR τσ(r) vxσ (r) = cvxσ (r) + (3c − 2)C∆v nσ(r)
where c is obtained from 1 Z |∇n(r)|1/2 c = α + β d3r Vcell cell n(r)
Band-gaps are of similar quality as G0W0, 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, 226401 (2009)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 57 / 63 Some solutions - the RPP functional
Ras¨ anen,¨ Pittalis, and Proetto (RPP) proposed the following correction to the BJ potential s RPP BR Dσ(r) vxσ (r) = vxσ (r) + C∆v nσ(r)
where the function D is
2 2 1 |∇nσ(r)| jσ(r) Dσ(r)= τσ(r) − − 4 nσ(r) nσ(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.
ERas¨ anen,¨ S Pittalis, and C Proetto, J. Chem. Phys. 132, 044112 (2010)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 58 / 63 Some solutions - the RPP functional
Ras¨ anen,¨ Pittalis, and Proetto (RPP) proposed the following correction to the BJ potential s RPP BR Dσ(r) vxσ (r) = vxσ (r) + C∆v nσ(r)
where the function D is
2 2 1 |∇nσ(r)| jσ(r) Dσ(r)= τσ(r) − − 4 nσ(r) nσ(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.
ERas¨ anen,¨ S Pittalis, and C Proetto, J. Chem. Phys. 132, 044112 (2010)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 58 / 63 Benchmark of the mGGAs
Test-set composed of 17 atoms, 19 molecules, 10 H2 chains, and 20 solids.
LDA PBE LB94 BJ RPP TB Ionization potentials atoms 41 42 3.7 14.4 7.4 molecules 35 36 8.0 19 5.7 Polarizabilities molecules 6.1 5.3 9.8 2.0 8.9 H2 chains 56 46 54 36 28 Band gaps 52 47 35 33 7.6 (mean average relative error in %)
M. Oliveira et al, JCTC 6, 3664-3670 (2010)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 59 / 63 Problems with these functionals
It may seem like a very nice idea to model directly vxc (or fxc). 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 2012 60 / 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 61 / 63 Availability of functionals
The problem of availability: There are many approximations for the xc (probably of the order of 200–250) Most computer codes only include a very limited quantity of functionals, typically around 10–15 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 2012 62 / 63 Availability of functionals
The problem of availability: There are many approximations for the xc (probably of the order of 200–250) Most computer codes only include a very limited quantity of functionals, typically around 10–15 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 2012 62 / 63 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 εxc, vxc, fxc, and kxc 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 2012 63 / 63 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 εxc, vxc, fxc, and kxc 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 2012 63 / 63