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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


Outline

1 Introduction





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?


Outline

1 Introduction





The families — Jacob’s ladder

Marc Chagall – Jacob’s dream




C˙mical Heaffin




C˙mical Heaffin

LDA




C˙mical Heaffin

LDA

GGA




C˙mical Heaffin

LDA

GGA

mGGA




C˙mical Heaffin

LDA

GGA

mGGA

Occ. orbitals




C˙mical Heaffin

LDA

GGA

mGGA

Occ. orbitals

All orbitals




C˙mical Heaffin

LDA

GGA

mGGA

Occ. orbitals

All orbitalsMany-body

Semi-empirical


The true ladder!

(M. Escher – Relativity)


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


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)


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?


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.


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 )

}


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.


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.


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.


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!


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.


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.


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.


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.


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.


Local behavior

Unfortunately, locally most GGA exchange functionals arecompletely wrong


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.


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.


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


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).


The electron localization function

A. Savin, R. Nesper, S. Wengert, and T, F. Fassler, Angew. Chem. Int. Ed. Engl. 36, 1808 (1997)


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.


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


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.


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∞!


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%)


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.


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.


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%


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


Parenthesis: Mind the GAP!


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)


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.


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...


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 ′|


Outline

1 Introduction





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.


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.


Outline

1 Introduction





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...


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)


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.


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)


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:


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 ′)


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.


Outline

1 Introduction





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.


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)


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)


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)


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)


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)


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.


Outline

1 Introduction





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


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!


functionals in dft - tddft. · pdf filefunctionals in dft miguel a.l. marques 1lpmcn,...

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