Functionals in DFT
Miguel A. L. Marques
1LPMCN, Universite Claude Bernard Lyon 1 and CNRS, France2European Theoretical Spectroscopy Facility
Les Houches 2012
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 1 / 63
Overview
1 Introduction
2 Jacob’s ladderLDAGGAmetaGGA
HybridsOrbital functionals
3 What functional to use4 Functional derivatives5 Functionals for vxc6 Availability
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 2 / 63
Outline
1 Introduction
2 Jacob’s ladderLDAGGAmetaGGA
HybridsOrbital functionals
3 What functional to use4 Functional derivatives5 Functionals for vxc6 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
E = Ts +
∫d3r vext(r)n(r) + EHartree + Ex + Ec
In Kohn-Sham theory we need to approximate Ex[n] andEc[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 infamilies?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
E = Ts +
∫d3r vext(r)n(r) + EHartree + Ex + Ec
In Kohn-Sham theory we need to approximate Ex[n] andEc[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 infamilies?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
E = Ts +
∫d3r vext(r)n(r) + EHartree + Ex + Ec
In Kohn-Sham theory we need to approximate Ex[n] andEc[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 infamilies?How to make a functional?Which functional should I use?
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 4 / 63
Outline
1 Introduction
2 Jacob’s ladderLDAGGAmetaGGA
HybridsOrbital functionals
3 What functional to use4 Functional derivatives5 Functionals for vxc6 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
Marc Chagall – Jacob’s dream
C˙mical Heaffin
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63
The families — Jacob’s ladder
Marc Chagall – Jacob’s dream
C˙mical Heaffin
LDA
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63
The families — Jacob’s ladder
Marc Chagall – Jacob’s dream
C˙mical Heaffin
LDA
GGA
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63
The families — Jacob’s ladder
Marc Chagall – Jacob’s dream
C˙mical Heaffin
LDA
GGA
mGGA
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63
The families — Jacob’s ladder
Marc Chagall – Jacob’s dream
C˙mical Heaffin
LDA
GGA
mGGA
Occ. orbitals
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63
The families — Jacob’s ladder
Marc Chagall – Jacob’s dream
C˙mical Heaffin
LDA
GGA
mGGA
Occ. orbitals
All orbitals
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 6 / 63
The families — Jacob’s ladder
Marc Chagall – Jacob’s dream
C˙mical Heaffin
LDA
GGA
mGGA
Occ. orbitals
All orbitalsMany-body
Semi-empirical
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 xcenergy as
ELDAxc =
∫d3r n(r)eHEG
xc (n(r))
The quantity eHEGxc (n), exchange-correlation energy per unit
particle, is a function of n. Sometimes you can see appearingεHEGxc (n(r)), which is the energy per unit volume. They are
relatedεHEGxc (n) = n eHEG
xc (n)
The exchange part of eHEG is simple to calculate and gives
eHEGx = −3
4
(3
2π
)2/3 1rs
with rs the Wigner-Seitz radius
rs =
(3
4πn
)1/3
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 HEGanalytically, but we can calculate it to arbitrary precisionnumerically 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 & Nusair1981: Perdew & Zunger1992: 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 sameresults.
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 & Nusair1981: Perdew & Zunger1992: 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 sameresults.
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 & Nusair1981: Perdew & Zunger1992: 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 sameresults.
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 & Nusair1981: Perdew & Zunger1992: 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 sameresults.
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 severalparametersSome of the coefficients are chosen to fulfill some exactconditions.
The high-density limit (RPA).The low-density expansion.
The rest of the parameters are fitted to Ceperley-Aldernumbers.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 11 / 63
An example: Perdew & Wang
Perdew and Wang parametrized the correlation energy per unitparticle:
ec(rs, ζ) = ec(rs,0) + αc(rs)f (ζ)
f ′′(0)(1− ζ4) + [ec(rs,1)− ec(rs,0)]f (ζ)ζ4
The function f (ζ) is
f (ζ) =[1 + ζ]4/3 + [1− ζ]4/3 − 2
24/3 − 2,
while its second derivative f ′′(0) = 1.709921. The functions ec(rs,0),ec(rs,1), and −αc(rs) are all parametrized by the function
g = −2A(1 + α1rs) log
{1 +
1
2A(β1r1/2s + β2rs + β3r3/2
s + β4r2s )
}
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 goodresults 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 goodresults 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. Asmost LDA are simple rational functions of n, also eLDA
xc and thevLDA
xc decay exponentially.
However, one knows from very simple arguments that the true
exc(r) ∼ − 12r
Note that most of the more modern functionals do not solve thisproblem.
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. Asmost LDA are simple rational functions of n, also eLDA
xc and thevLDA
xc decay exponentially.
However, one knows from very simple arguments that the true
exc(r) ∼ − 12r
Note that most of the more modern functionals do not solve thisproblem.
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 hydrogenatom), the total energy has to be equal to
E = Ts +
∫d3r vext(r)n(r)
which means that
EHartree + Ex + Ec = 0
In particular, it is the exchange term that has to cancel thespurious Hartree contribution.
The first rung where it is possible to cancel the self-interactionis 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 theso-called Gradient Expansion Approximations. It is asystematic expansion of exc in terms of derivatives of thedensity. In lowest order we have
eGEAxc (n,∇n, · · · ) = eLDA
xc + a1(n)|∇n|2 + · · ·
Using different approaches, people went painfully to sixth orderin the derivatives.
Results were, however, much worse than the LDA. The reasonwas, 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 theso-called Gradient Expansion Approximations. It is asystematic expansion of exc in terms of derivatives of thedensity. In lowest order we have
eGEAxc (n,∇n, · · · ) = eLDA
xc + a1(n)|∇n|2 + · · ·
Using different approaches, people went painfully to sixth orderin the derivatives.
Results were, however, much worse than the LDA. The reasonwas, 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 GradientApproximation:
Soften the requirement of having “rigorous derivations” and“controlled approximations”, and dream up a some more orless justified expression that depends on ∇n and somefree parameters.
Or, mathematically
EGGAxc =
∫d3r n(r)eGGA
xc (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 GradientApproximation:
Soften the requirement of having “rigorous derivations” and“controlled approximations”, and dream up a some more orless justified expression that depends on ∇n and somefree parameters.
Or, mathematically
EGGAxc =
∫d3r n(r)eGGA
xc (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 constrainssuch as
reduces to the LDA when ∇n = 0.is exact for some reference system like the He atomhas some known asymptotic limits, for small gradients,large gradients, etc.obeys some known inequalities like the Lieb-Oxford bound
Ex [n]
ELDAx [n]
≤ λ
etc.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 18 / 63
Exchange functionals
Exchange functionals are almost always written as
EGGAx [n] =
∫d3r n(r)eLDA(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
Ex[n↑,n↓] =12
(Ex[2n↑] + Ex[2n↓])
It is relatively simple to come up with and exchange GGA, so itis not surprising that there are more than 50 different versionsin the literature.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 19 / 63
Example: B88 exchange
Becke’s famous ’88 functional reads
F B88x (xσ) = 1 +
1Ax
βx2σ
1 + 6βxσ arcsinh(xσ),
whereFor small x fulfills the gradient expansion.The energy density has the right asymptotics.The parameter β was fitted to the exchange energies ofnoble gases.By far the most used exchange functional in quantumchemistry.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 20 / 63
Example: PBE exchange
The exchange part of the Perdew-Burke-Erzernhof functionalreads:
F PBEx (xσ) = 1 + κ
(1− κ
κ+ µs2σ
),
whereThe parameter s = |∇n|/2kF n.To recover the LDA response, µ = βπ2/3 ≈ 0.21951.Obeys the local version of the Lieb-Oxford bound.
F PBEx (s) ≤ 1.804.
(Note that Becke 88 violates strongly and shamelessly thisrequirement.)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 arecompletely wrong
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 22 / 63
Correlation functionals
Correlation functionals are much harder to design, so thereare many less in the literature (around 20).For correlation there is no spin sum-rule, so the spindependence is much more complicated.Even if the correlation energy is ∼ 5 smaller thanexchange, it is important as energy differences are of thesame 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 theHe atom.The results is a meta-GGA (i.e. it depends on τ ).
Lee-Yang-Parr transformed the meta-GGA of Colle-Salvettiby using the gradient expansion of the kinetic energydensity leading to a functional depending on ∇2n.Later it was found that the ∇2n term could be rewritten byintegrating by parts, leading to the current LYP GGAfunctional.
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 theHe atom.The results is a meta-GGA (i.e. it depends on τ ).
Lee-Yang-Parr transformed the meta-GGA of Colle-Salvettiby using the gradient expansion of the kinetic energydensity leading to a functional depending on ∇2n.Later it was found that the ∇2n term could be rewritten byintegrating by parts, leading to the current LYP GGAfunctional.
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 theHe atom.The results is a meta-GGA (i.e. it depends on τ ).
Lee-Yang-Parr transformed the meta-GGA of Colle-Salvettiby using the gradient expansion of the kinetic energydensity leading to a functional depending on ∇2n.Later it was found that the ∇2n term could be rewritten byintegrating by parts, leading to the current LYP GGAfunctional.
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 theHe atom.The results is a meta-GGA (i.e. it depends on τ ).
Lee-Yang-Parr transformed the meta-GGA of Colle-Salvettiby using the gradient expansion of the kinetic energydensity leading to a functional depending on ∇2n.Later it was found that the ∇2n term could be rewritten byintegrating by parts, leading to the current LYP GGAfunctional.
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 theHe atom.The results is a meta-GGA (i.e. it depends on τ ).
Lee-Yang-Parr transformed the meta-GGA of Colle-Salvettiby using the gradient expansion of the kinetic energydensity leading to a functional depending on ∇2n.Later it was found that the ∇2n term could be rewritten byintegrating by parts, leading to the current LYP GGAfunctional.
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 theHe atom.The results is a meta-GGA (i.e. it depends on τ ).
Lee-Yang-Parr transformed the meta-GGA of Colle-Salvettiby using the gradient expansion of the kinetic energydensity leading to a functional depending on ∇2n.Later it was found that the ∇2n term could be rewritten byintegrating by parts, leading to the current LYP GGAfunctional.
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 theHe atom.The results is a meta-GGA (i.e. it depends on τ ).
Lee-Yang-Parr transformed the meta-GGA of Colle-Salvettiby using the gradient expansion of the kinetic energydensity leading to a functional depending on ∇2n.Later it was found that the ∇2n term could be rewritten byintegrating by parts, leading to the current LYP GGAfunctional.
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 theHe atom.The results is a meta-GGA (i.e. it depends on τ ).
Lee-Yang-Parr transformed the meta-GGA of Colle-Salvettiby using the gradient expansion of the kinetic energydensity leading to a functional depending on ∇2n.Later it was found that the ∇2n term could be rewritten byintegrating by parts, leading to the current LYP GGAfunctional.
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.
EPBEc =
∫d3r n(r)
[eHEG
c + H]
where
H = γφ3 log{
1 +β
γt2[
1 + At2
1 + At2 + A2t4
]}and
A =β
γ
[exp{−eHEG
c /(γφ3)} − 1]−1
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 andincrease the number of arguments of the functional. In thiscase, we use both the Laplacian of the density ∇2n andthe kinetic energy density
τ =occ.∑
i
12|∇ϕ|2
Note that there are several other possibilities to define τthat lead to the same (integrated) kinetic energy, but todifferent local values.Often, the variables appear in the combination τ − τW ,where τW = |∇n|2
8n is the von Weizsacker kinetic energy.This is also the main quantity entering the electronlocalization 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) — JohnPerdew’s school of functionals, i.e, many sum-rules andexact conditions. It is based on the PBE.M06L (Zhao and Truhlar) — This comes from Don Truhlar’sgroup, and it was crafted for main-group thermochemistry,transition metal bonding, thermochemical kinetics, andnoncovalent interactions.VSXC (Van Voorhis and Scuseria) — Based on a densitymatrix 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 betweenthey Hartree-Fock and DFT (LDA or GGA) values. So, we cantry to mix, or to “hybridize” both theories.
1 Write an energy functional:
Exc = aEFock[ϕi ] + (1− a)EDFT[n]
2 Minimize energy functional w.r.t. to the orbitals:
vxc(r , r ′) = avFock(r , r ′) + (1− a)vDFT(r)
Note: for pure density functionals, minimizing w.r.t. the orbitalsor w.r.t. the density gives the same, as:
δF [n]
δϕ∗=
∫δF [n]
δnδnδϕ∗
=δF [n]
δnϕ
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 LDAand GGAs (Becke 88 and PW91). The mixing parameter is1/5.1994: The famous B3LYP appears, replacing PW91 withLYP in the Becke functional.1999: PBE0 proposed. The mixing was now 1/4.2003: The screened hybrid HSE06 was proposed. It gavemuch better results for the band-gaps of semiconductorsand 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
2+ vext(r) + vH(r)
]φQP
i (r)+
∫d3r ′ Σ(r , r ′; εQP
i )φQPi (r ′) = εQP
i φQPi (r ′)
And now let us look at the different approximations:
COHSEX:
Σ = −occ∑
i
φQPi (r)φQP
i (r ′)W (r , r ′;ω = 0) + δ(r − r ′)ΣCOH(r)
Hybrids
Σ = −occ∑
i
φQPi (r)φQP
i (r ′)a v(r − r ′) + δ(r − r ′)(1− a) vDFT(r)
So, we infer that a ∼ 1/ε∞!
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 31 / 63
Does it work (a = 1/ε∞)?
0 5
10
15
20
0 5 10 15 20
The
oret
ical
gap
(eV
)
Experimental gap (eV)
y=xPBEPBE0PBE0ε∞
Errors: PBE (46%), Hartree-Fock (230%), PBE0 (27%), PBE0ε∞(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 moleculesand solids, but a number of important problems do remain. Forexample:
For metals, the long-range part of the Coulomb interactionleads to a vanishing density of states at the Fermi level dueto a logarithmic singularity (as Hartree-Fock).For semiconductors, the quality of the gaps varies verymuch with the material and the mixing.For molecules, the asymptotics of the potential are stillwrong, which leads to problems, e.g. for charge transferstates.
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-rangeand a long-range part:
1r12
=1− erf(µr12)
r12︸ ︷︷ ︸short range
+erf(µr12)
r12︸ ︷︷ ︸long range
We now treat the one of the terms by a standard DFT functionaland make a hybrid out of the other. There are two possibilities
1 DFT: long-range; Hybrid: short-range. Such as HSE, goodfor metals.
2 DFT: short-range; Hybrid: long-range. The LC functionalsfor 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
EHSExc = αEHF, SR
x (µ) + (1− α)EPBE, SRx (µ) + EPBE, LR
x (µ) + EPBEc
The most common version of the HSE chooses µ = 0.11 andα = 1/4. Mind that basically every code has a different“version” of the HSE.
For comparison, here are the average percentual errors for thegaps of a series of semiconductors and insulators
PBE HF+c PBE0 HSE06 G0W047% 250% 29% 17% 11%
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 35 / 63
Gaps with the HSE
0 5
10
15
20
0 5 10 15 20
The
oret
ical
gap
(eV
)
Experimental gap (eV)
Ne
Ar,
LiF
Kr
Xe
C
Si,
MoS
2G
e
LiC
l
MgO
BN
, AlN
GaN
GaA
sS
iC,C
dS,A
lP
ZnS
ZnO
SiO
2
y=xPBEPBE0PBE0ε∞PBE0mixTB09
0 5
10
15
20
0 5 10 15 20
The
oret
ical
gap
(eV
)Experimental gap (eV)
Ne
Ar,
LiF
Kr
Xe
C
Si,
MoS
2G
e
LiC
l
MgO
BN
, AlN
GaN
GaA
sS
iC,C
dS,A
lP
ZnS
ZnO
SiO
2
y=xHSE06HSE06mix
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 36 / 63
The band gap of CuAlO2
The agreement ofLDA+U and HSE06hybrid functional to theexperiment is accidentalScGW shows that theband gaps are muchhigherExperimental data arefor optical gap: excitonbinding energy ∼0.5 eVAgreement withexperiment can only beachieved by the additionof phonons.
LDA LDA+U B3LYP HSE03 HSE06 G0W
0scGW scGW+P0
1
2
3
4
5
6
Eg [
eV]
Eg
indirect
Eg
direct
∆=Eg
direct-E
g
indirect
exp. direct gap
exp. indirectgap
3.5 eV (exp) = 5 eV (el. QP)- 0.5 eV (excitons)- 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 ofLDA+U and HSE06hybrid functional to theexperiment is accidentalScGW shows that theband gaps are muchhigherExperimental data arefor optical gap: excitonbinding energy ∼0.5 eVAgreement withexperiment can only beachieved by the additionof phonons.
LDA LDA+U B3LYP HSE03 HSE06 G0W
0scGW scGW+P0
1
2
3
4
5
6
Eg [
eV]
Eg
indirect
Eg
direct
∆=Eg
direct-E
g
indirect
exp. direct gap
exp. indirectgap
3.5 eV (exp) = 5 eV (el. QP)- 0.5 eV (excitons)- 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 needα = 1. This value is however too large in order to obtain goodresults for several molecular properties. To improve thisbehavior one needs more flexibility
1r12
=1− [α + βerf(µr12)]
r12︸ ︷︷ ︸short range
+α + βerf(µr12)
r12︸ ︷︷ ︸long range
The asymptotics are now determined by α + β. Note that thisform leads to a normal hybrid for β = 0 and to a screenedhybrid 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 isconstructed in a similar way to B3LYP, but with
α = 0.19 β = 0.46 µ = 0.33
This functional gives very much improved charge transferexcitations. Note that in any case α + β = 0.65 6= 1, whichmeans that the asymptotics are still wrong.
The problem, as it often happens in functional development, isthat CAM-B3LYP is better for change transfer, but worse formany 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 isconstructed in a similar way to B3LYP, but with
α = 0.19 β = 0.46 µ = 0.33
This functional gives very much improved charge transferexcitations. Note that in any case α + β = 0.65 6= 1, whichmeans that the asymptotics are still wrong.
The problem, as it often happens in functional development, isthat CAM-B3LYP is better for change transfer, but worse formany other properties...
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 40 / 63
Orbital functionals
Self-interaction correction:
ESICxc [ϕ] = ELDA
xc [n↑,n↓]−∑
i
ELDAxc [|ϕi(r)|2 ,0]
− 12
∑i
∫d3r∫
d3r ′|ϕi(r)|2 |ϕi(r ′)|2
|r − r ′|
Exact-exchange:
Eexactx [n, ϕ] = −1
2
∑jk
∫d3r∫
d3r ′ϕ∗j (r)ϕ∗k (r ′)ϕk (r)ϕj(r ′)
|r − r ′|
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 41 / 63
Outline
1 Introduction
2 Jacob’s ladderLDAGGAmetaGGA
HybridsOrbital functionals
3 What functional to use4 Functional derivatives5 Functionals for vxc6 Availability
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 42 / 63
If you are a physicist!
You are runningSolids
If problem is small enough and code available use HSEOtherwise use PBE SOL or AM05However, whenever you can just stick to GW and BSE
Moleculesvan der Waals: use the Langreth-Lundqvist functional (or avariant)Charge transfer: no good alternatives hereIf problem is small enough and code available use PBE0Time-dependent problem try LB94Otherwise use PBE
Note that if you want to calculate response, you are basicallystuck with standard GGA functionals. In any case, stick tofunctionals from the J. Perdew family.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 43 / 63
If you are a chemist!
You are runningSolids
You are a physicist, so go back to the previous slideMolecules
van der Waals: you might escape with Grimme’s trickCharge transfer: CAM-B3LYPIf problem is small enough use B3LYPOtherwise use BLYP
Note that you also have a chance of getting your paperaccepted if you use a functional by G. Scuseria or D. Truhlar.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 44 / 63
Outline
1 Introduction
2 Jacob’s ladderLDAGGAmetaGGA
HybridsOrbital functionals
3 What functional to use4 Functional derivatives5 Functionals for vxc6 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:
Exc =
∫d3r εxc(r) =
∫d3r n(r)exc(r)
and the xc potential that enters the Kohn-Sham equations isdefined as
vxc(r) =δExc
δn(r)
if we are trying to solve response equations then also thefollowing quantities may appear
fxc(r , r ′) =δ2Exc
δn(r)δn(r ′)kxc(r , r ′, r ′′) =
δ3Exc
δn(r)δn(r ′)δn(r ′′)
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:
Exc =
∫d3r εxc(r) =
∫d3r n(r)exc(r)
and the xc potential that enters the Kohn-Sham equations isdefined as
vxc(r) =δExc
δn(r)
if we are trying to solve response equations then also thefollowing quantities may appear
fxc(r , r ′) =δ2Exc
δn(r)δn(r ′)kxc(r , r ′, r ′′) =
δ3Exc
δn(r)δn(r ′)δn(r ′′)
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 functionalderivatives. For example
vLDAxc (r) =
∫d3r
δn(r)eHEGxc (n(r))
δn(r)
=
∫d3r
ddn
neLDAxc (n)
∣∣∣∣n=n(r)
δ(r − r)
=ddn
neLDAxc (n)
∣∣∣∣n=n(r)
Higher derivatives are also simple:
f LDAxc (r) =
d2
d2nneLDA
xc (n)
∣∣∣∣n=n(r)
kLDAxc (r) =
d3
d3nneLDA
xc (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
vGGAxc (r) =
∫d3r
δn(r)eGGAxc (n(r),∇n(r))
δn(r)
=
∫d3 r
∂
∂nneGGA
xc (n,∇n)
∣∣∣∣n=n(r)
δ(r − r)
+ n∂
∂∇neGGA
xc (n,∇n)
∣∣∣∣n=n(r)
∇δ(r − r)
=∂
∂nneLDA
xc (n,∇n)
∣∣∣∣n=n(r)
−∇ ∂
∂(∇n)neLDA
xc (n,∇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 thefunctional dependence on τ . Therefore, to calculate correctlyvxc within DFT one has to resort to the OEP procedure (seenext slide). However, the expression that one normally uses is
vmGGAxc,i (r) =
1ϕ∗i (r)
δExc
δϕi(r)
This definition gives the correct potentials for the case of anLDA or a GGA, as
1ϕ∗i (r)
δExc[n]
δϕi(r)=
1ϕ∗i (r)
∫d3r
δExc[n]
δn(r)
δn(r)
δϕi(r)
=1
ϕ∗i (r)
∫d3r
δExc[n]
δn(r)ϕ∗i (r)δ(r − r)
=δExc[n]
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
δExc
δn(r)=
∫d3r ′
δExc
δvKS(r)
δvKS(r)
δn(r)
The second term is the inverse non-interacting densityresponse function. Using again the chain-rule
δExc
δn(r)=
∫d3r ′∫
d3r ′′∑
j
δExc
δϕj(r ′′)δϕj(r ′′)δvKS(r)
δvKS(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 2012 50 / 63
The OEP equation
The OEP integral equation is then written as∫d3r ′Q(r , r ′)vOEP
xc = Λ(r)
where
Q(r , r ′) =N∑
j=1
ϕ∗j (r ′)Gj(r ′, r)ϕj(r) + c.c
Λ(r) =N∑
j=1
∫d3r ′ ϕ∗j (r ′)uxc,j(r ′)ϕj(r) + c.c
and
Gj(r ′, r) =∑k 6=j
ϕk (r ′)ϕ∗k (r)
εj − εkuxc,j(r ′) =
1ϕ∗j (r ′)
δExc
δϕj(r ′)
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 51 / 63
The KLI approximation
One way of performing this approximation consists inapproximating
Gj(r ′, r) ≈∑k 6=j
ϕk (r ′)ϕ∗k (r)
∆ε=
1∆ε
[δ(r − r ′)− ϕj(r)ϕj(r ′)
]which leads to a very simple expression for the xc potential
vKLIxc =
∑j
nj(r)
n(r)
[uxc,j(r) + vKLI
xc,j − uKLIxc,j
]The KLI approximation is often an excellent approximation tothe OEP potential.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 52 / 63
Outline
1 Introduction
2 Jacob’s ladderLDAGGAmetaGGA
HybridsOrbital functionals
3 What functional to use4 Functional derivatives5 Functionals for vxc6 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. Mostof the functionals are concerned by the energy, but it is alsopossible to write down directly a functional for vxc.
This was done by van Leeuwen and Baerends in 1994 thatused a form similar to Becke 88
∆vLB94xc (xσ) = vLDA
xc − βn1/3σ
x2σ
1 + 3βxσ arcsinh(xσ),
This functional is particularly useful when calculating, e.g.,ionization potentials from the value of the HOMO, or whenperforming 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 onn, ∇n, ∇2n, τ ).Models the exchange hole of hydrogenicatoms.Correct asymptotic −1/r behavior for finitesystems.Excellent description of the Slater part ofthe 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 asa sum
vxσ(r) = vSLxσ (r) + ∆vOEP
xσ (r)
The BJ potential is a simple approximation to the OEPcontribution
∆vOEPxσ (r) ≈ ∆vBJ
xσ (r) = C∆v
√τσ(r)
nσ(r)
where C∆v =√
5/(12π2)
Exact for the hydrogen atom and for the HEG.Yields the atomic step structure in the exchange potentialvery accurately.It has the derivative discontinuity for fractional particlenumbers.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 asa sum
vxσ(r) = vSLxσ (r) + ∆vOEP
xσ (r)
The BJ potential is a simple approximation to the OEPcontribution
∆vOEPxσ (r) ≈ ∆vBJ
xσ (r) = C∆v
√τσ(r)
nσ(r)
where C∆v =√
5/(12π2)
Exact for the hydrogen atom and for the HEG.Yields the atomic step structure in the exchange potentialvery accurately.It has the derivative discontinuity for fractional particlenumbers.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
vTBxσ (r) = cvBR
xσ (r) + (3c − 2)C∆v
√τσ(r)
nσ(r)
where c is obtained from
c = α + β
(1
Vcell
∫cell
d3r|∇n(r)|
n(r)
)1/2
Band-gaps are of similar quality as G0W0, but at thecomputational cost of an LDA!Value of c is always larger than one, as the BJ gaps aretoo 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
vTBxσ (r) = cvBR
xσ (r) + (3c − 2)C∆v
√τσ(r)
nσ(r)
where c is obtained from
c = α + β
(1
Vcell
∫cell
d3r|∇n(r)|
n(r)
)1/2
Band-gaps are of similar quality as G0W0, but at thecomputational cost of an LDA!Value of c is always larger than one, as the BJ gaps aretoo 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
Rasanen, Pittalis, and Proetto (RPP) proposed the followingcorrection to the BJ potential
vRPPxσ (r) = vBR
xσ (r) + C∆v
√Dσ(r)
nσ(r)
where the function D is
Dσ(r) = τσ(r)− 14|∇nσ(r)|2
nσ(r)− j2σ(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.
E Rasanen, 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
Rasanen, Pittalis, and Proetto (RPP) proposed the followingcorrection to the BJ potential
vRPPxσ (r) = vBR
xσ (r) + C∆v
√Dσ(r)
nσ(r)
where the function D is
Dσ(r) = τσ(r)− 14|∇nσ(r)|2
nσ(r)− j2σ(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.
E Rasanen, 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 TBIonization potentials
atoms 41 42 3.7 14.4 7.4molecules 35 36 8.0 19 5.7Polarizabilitiesmolecules 6.1 5.3 9.8 2.0 8.9H2 chains 56 46 54 36 28Band 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 thefunctional derivative of an energy functional. This opens atheoretical Pandora’s box.
No unique way of calculating the energy by integration.The energy depends on the path used. Results can varydramatically.Energy is not conserved when performing a TD simulationZero-force and zero-torque theorems broken. Spuriousforces and torques appear during a TD simulation....
In any case, and even if we don’t have the energy, we can haveaccess to all derivatives of the energy, i.e., all responseproperties of the system.
M. A. L. Marques (Lyon) XC functionals Les Houches 2012 60 / 63
Outline
1 Introduction
2 Jacob’s ladderLDAGGAmetaGGA
HybridsOrbital functionals
3 What functional to use4 Functional derivatives5 Functionals for vxc6 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 theorder of 200–250)Most computer codes only include a very limited quantityof functionals, typically around 10–15Chemists and Physicists do not use the same functionals!
It is therefore difficult to:Reproduce older calculations with older functionalsReproduce calculations performed with other codesPerform 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 theorder of 200–250)Most computer codes only include a very limited quantityof functionals, typically around 10–15Chemists and Physicists do not use the same functionals!
It is therefore difficult to:Reproduce older calculations with older functionalsReproduce calculations performed with other codesPerform 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 mGGAsfor the exchange, correlation, and the kinetic energyFunctionals for 1D, 2D, and 3DReturns εxc, vxc, fxc, and kxc
Quite mature: in 14 different codes including OCTOPUS,APE, GPAW, ABINIT, etc.
The technicalities:Written in C from scratchBindings both in C and in FortranLesser 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 mGGAsfor the exchange, correlation, and the kinetic energyFunctionals for 1D, 2D, and 3DReturns εxc, vxc, fxc, and kxc
Quite mature: in 14 different codes including OCTOPUS,APE, GPAW, ABINIT, etc.
The technicalities:Written in C from scratchBindings both in C and in FortranLesser 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