introduction to dft part 2

XXXVIII ENFMC Brazilian Physical Society Meeting

Introduction todensity functional theory

Mariana M. Odashima

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http://sites.google.com/site/mmodashima/

Problem HK-KS xc LDA Construction Challenges Final Remarks

This tutorial

Introduction to density-functional theory

X Context and key concepts (1927-1930)(Born-Oppenheimer, Hartree, Hartree-Fock, Thomas-Fermi)

X Fundamentals (1964-1965)(Hohenberg-Kohn theorem, Kohn-Sham scheme)

I Approximations (≈ 1980-2010)(local density and generalized gradient approximations (LDA andGGA), construction of functionals)

Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 1/76

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http://www.sbfisica.org.br/~enfmc/xxxviii/



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Dirac (1929)

“The general theory of quantum mechanics isnow almost complete (...) The underlying physi-cal laws necessary for the mathematical theory ofa large part of physics and the whole of chemistryare thus completely known, and the difficulty isonly that the exact application of these laws leadsto equations much too complicated to be soluble.(...) It therefore becomes desirable that approxi-mate practical methods of applying quantum me-chanics should be developed, which can lead toan explanation of the main features of complexatomic systems without too much computation.”


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The electronic structure problem

I Quantum many-body problemof N interacting electrons: Ψel(~r1,~r2, ...,~rN )

I Paradigms: atom / electron gas

I Methods based on the wavefunction(Hartree-Fock, CI, Coupled Cluster, MP2, QMC)

I Methods based on the Green’s function, reduced densitymatrix, density (density functional theory)


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Hartree’s method

I Single-particle Schrodinger equation(− ~2

2m∇2 + vext(r) + vH (r)

)ϕi(r) = εiϕi(r) ,

I Mean field potential

vH (r) = e2∫

d3r ′ n(r′)|r− r′|

I Hartree energy

UH [n] = 〈ΨH |U |ΨH 〉 = e2

2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′| .


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Hartree’s method


2m∇2 + vext(r) + vH (r)



vH (r) = e2∫

d3r ′ n(r′)|r− r′|

I Hartree energy

UH [n] = 〈ΨH |U |ΨH 〉 = e2

2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′|

.


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Hartree’s method


2m∇2 + vext(r) + vH (r)



vH (r) = e2∫

d3r ′ n(r′)|r− r′|

I Hartree energy

UH [n] = 〈ΨH |U |ΨH 〉 = e2

2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′| .


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Hartree-Fock

I Antisymmetrization in a Slater determinant

ΨHF (r) = 1√N !

∣∣∣∣∣∣∣∣∣∣ϕ1(x1) ϕ1(x2) · · · ϕ1(xN )ϕ2(x1) ϕ2(x2) · · · ϕ2(xN )

...... . . . ...

ϕN (x1) ϕN (x2) · · · ϕN (xN )

∣∣∣∣∣∣∣∣∣∣I Fock exchange energy (indirect)

Ex = 〈ΨHF |U |ΨHF〉 = −e2

2∑i,j,σ

∫dr∫

drϕ∗iσ(r)ϕ∗jσ(r′)ϕiσ(r′)ϕjσ(r)

|r− r′|


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Thomas-Fermi model

I Use the infinite gas of non-interacting electrons with auniform density n = n(r) to evaluate the kinetic energy ofatoms, molecules

TTF [n] =∫

tgas(n(r))n(r)d3r


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Our tutorial

Introduction to density-functional theory

X Context and key concepts (1927-1930)(Born-Oppenheimer, Hartree, Hartree-Fock, Thomas-Fermi)

X Fundamentals (1964-1965)(Hohenberg-Kohn theorem, Kohn-Sham scheme)

I Approximations (≈ 1980-2010)(local density and generalized gradient approximations (LDA andGGA), construction of functionals)


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Back to our question

a program ? a method?

some

obscure

theory?


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Density functional theory (DFT)

I Quantum theory based on the density n(r)

wave functions Ψ(r1, r2, ...rN )

I Single-particle Kohn-Sham equationsI Electronic structure boom: Nobel Prize to

W.Kohn/J.Pople

Hohenberg-Kohn theorem: Ψ(r1, r2, ..., rN )⇔ n(r)

Which means,I Ψ(r) = Ψ[n(r)]I observables = observables[n(r)]


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wave functions Ψ(r1, r2, ...rN )I Single-particle Kohn-Sham equations

I Electronic structure boom: Nobel Prize toW.Kohn/J.Pople




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wave functions Ψ(r1, r2, ...rN )I Single-particle Kohn-Sham equationsI Electronic structure boom: Nobel Prize to

W.Kohn/J.Pople




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Hohenberg-Kohn (1964)

Phys. Rev. 136 B864 (1964).


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After THK

I From the ground-state density it is possible, in principle, tocalculate the corresponding wave functions and all itsobservables.

I However: the Hohenberg-Kohn theorem does notprovide any means to actually calculate them.

I We have DFT in theory, now, in practice?...


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After THK

arXiv:1403.5164

“By the late fall of 1964, Kohn was thinking about alternativeways to transform the theory he and Hohenberg had developedinto a practical scheme for atomic, molecular, and solid statecalculations. Happily, he was very well acquainted with anapproximate approach to the many-electron problem that wasnotably superior to the Thomas-Fermi method, at least for thecase of atoms. This was a theory proposed by Douglas Hartree in1923 which exploited the then just-published Schrodinger equationin a heuristic way to calculate the orbital wave functions φk(r), theelectron binding energies εk , and the charge density n(r) of anN -electron atom. Hartree’s theory transcended Thomas-Fermitheory primarily by its use of the exact quantum-mechanicalexpression for the kinetic energy of independent electrons.”


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After THK

I Kohn believed the Hartree equations could be an example ofthe HK variational principle.

I He knew the self-consistent scheme and that it could give anapproximate density

I So he suggested to his new post-doc, Lu Sham, that he try toderive the Hartree equations from the Hohenberg-Kohnformalism.


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Kohn-Sham approach/scheme

I Auxiliary non-interacting systemI Single-particle equations(

−~2∇2

2m + vKS(r))ϕk(r) = εkϕk(r)

I Effective potential

vKS(r) = vext(r) + vH (r) + vxc(r)

I Formally: constraint on the ground-state density


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Kohn-Sham kindergarden

Interacting

(complicated)

Ficticious non-interacting

under effective field


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Outline

1 Review of our problem

2 Review of HK-KS

3 Exchange-correlation

4 LDA and GGA

5 Construction

6 Challenges

7 Final Remarks


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Exchange-correlation


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arXiv:1403.5164

“As trained solid-state physicists, Hohenberg and Kohn knewthat the entire history of research on the quantum mechanicalmany-electron problem could be interpreted as attempts toidentify and quantify the physical effects described by thisuniversal density functional.” For example, many years ofapproximate quantum mechanical calculations for atoms andmolecules had established that the phenomenon of exchange -a consequence of the Pauli exclusion principle - contributessignificantly to the potential energy part of U[n]. Exchangereduces the Coulomb potential energy of the system by tendingto keep electrons with parallel spin spatially separated.”.


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Universal functional

Energy functional: Kinetic + Coulomb + External

E [n] = T [n] + U [n] + V [n]

We can define a universal F[n]

F [n] = T [n] + U [n]

which is the same independent of your system. Our task isapproximate U[n], the many-particle problem.


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arXiv:1403.5164

“As trained solid-state physicists, Hohenberg and Kohn knewthat the entire history of research on the quantum mechanicalmany-electron problem could be interpreted as attempts toidentify and quantify the physical effects described by thisuniversal density functional. For example, many years ofapproximate quantum mechanical calculations for atoms andmolecules had established that the phenomenon of exchange -a consequence of the Pauli exclusion principle - contributessignificantly to the potential energy part of U[n].Exchangereduces the Coulomb potential energy of the system by tendingto keep electrons with parallel spin spatially separated.”


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arXiv:1403.5164

“As trained solid-state physicists, Hohenberg and Kohn knewthat the entire history of research on the quantum mechanicalmany-electron problem could be interpreted as attempts toidentify and quantify the physical effects described by thisuniversal density functional. For example, many years ofapproximate quantum mechanical calculations for atoms andmolecules had established that the phenomenon of exchange -a consequence of the Pauli exclusion principle - contributessignificantly to the potential energy part of U[n]. Exchangereduces the Coulomb potential energy of the system by tendingto keep electrons with parallel spin spatially separated.”


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Coulomb energy

Coulomb energy

U [n] = UH [n] + Ex [n] + �

whereUH [n] = e2

2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′|.

is the electrostatic, mean field repulsion, and

Ex [ϕ[n]] = −e2

2∑i,j,σ

∫d3r

∫d3r ′

ϕ∗iσ(r)ϕ∗jσ(r′)ϕiσ(r′)ϕjσ(r)|r− r′|

is the exchange energy due to the Pauli principle.


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On correlation

arXiv:1403.5164

Coulomb energy

U [n] = UH [n] + Ex [n] + �

“The remaining potential energy part of U[n] takes account ofshort-range correlation effects.

Correlation also reduces theCoulomb potential energy by tending to keep all pairs ofelectrons spatially separated.”

Correlation energy: Ec < 0


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On correlation

arXiv:1403.5164

Coulomb energy

U [n] = UH [n] + Ex [n] + �

“The remaining potential energy part of U[n] takes account ofshort-range correlation effects. Correlation also reduces theCoulomb potential energy by tending to keep all pairs ofelectrons spatially separated.”



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On correlation

arXiv:1403.5164

Coulomb energy

U [n] = UH [n] + Ex [n] + Ec[n]

“The remaining potential energy part of U[n] takes account ofshort-range correlation effects. Correlation also reduces theCoulomb potential energy by tending to keep all pairs ofelectrons spatially separated.”



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On correlation

arXiv:1403.5164

Coulomb energy

U [n] = UH [n] + Ex [n] + Ec[n]

“Note for future reference that the venerable Hartree-Fockapproximation takes account of the kinetic energy and theexchange energy exactly but (by definition) takes no accountof the correlation energy”.

Hartree-Fock energy

EHF [n] = Ts[ϕ[n]] + V [n] + UH [n] + Ex [ϕ[n]]


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Exchange-correlation in DFT

Kohn-Sham effective potential:

vKS(r) = vext(r) + vH (r) + vxc(r)

Our task is to find vxc, preferrably as a functional of the density.

Orbital functionals bring non-locality (integrals over r and r′).

So, in the Kohn-Sham DFT, we recast the many-particle problemin finding xc potentials.


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Exchange-correlation in DFT

Total energy

E [n] = T [n] + V [n] + U [n]= Ts[ϕi [n]] + V [n] + UH [n] + Exc[n]

Some approximations: single-particle kinetic and Hartree.

Leave the corrections (T − Ts and U −UH ) to the Exc.

Ts[ϕi [n]] = − ~2

2m

N∑i

∫d3rϕ∗i (r)∇2ϕi(r)

UH [n] = e2

2

∫d3r

∫d3r ′n(r)n(r′)

| r− r′ |


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Exchange-correlation energy

The exchange-correlation energy Exc is the new clothing of themany-body problem

I exchange: Pauli principleI correlation: kinetic and Coulombic contributions beyond

single-particle (one Slater determinant)I xc = “nature’s glue” that binds matter together (Exc < 0)


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“Electrons moving through the densityswerve to avoid one another, like shoppersin a mall.”

“The resulting reduction of the potential energy of mutualCoulomb repulsion is the main contribution to the negativeexchange-correlation energy. The swerving motion also makes asmall positive kinetic energy contribution to the correlation energy”

J.Perdew et al. in J. Chem. Theory Comput. 5, 902 (2009).


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In Kohn-Sham DFT, the exchange-correlation energy Exc[n] holdsthe main difficulty of the many-body problem.

Now, how to construct an approximate Exc[n]?


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Outline


2 Review of HK-KS


4 LDA and GGA

5 Construction

6 Challenges

7 Final Remarks


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State-of-the-art


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Back in 65


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Back in 65

I Introduce KS equationsI Explore possible Exc


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Density functional

I Starting point: electron gas

Exc =∫

d3rexc[n]n(r) (exc: energy density per particle)


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Thomas-Fermi-Dirac spirit

I Using the paradigm of an uniform, homogeneous system tohelp with inhomogeneous problems

E ≈ ETFD[n] = TLDAs [n] + UH [n] + ELDA

x + V [n] .


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Local density approximation (LDA)

ELDAxc [n] =

∫d3r ehom

xc (n(r))

ehomxc (n) = ehom

x (n) + ehomc (n)

For the homogeneous electron gas, we have the expression of theDirac exchange energy

ehomx (n) = −3

4

( 3π

)1/3n4/3 ,

For ehomc ? Monte Carlo Quantico → parametrizations

ePW 92c = −2c0(1+α1rs)ln

[1 + 1

2c1(β1r1/2s + β2rs + β3r3/2

s + β4r2s )

]


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ELDAxc [n] =

∫d3r ehom

xc (n(r))

ehomxc (n) = ehom

x (n) + ehomc (n)


ehomx (n) = −3

4

( 3π

)1/3n4/3 ,

For ehomc ?

Monte Carlo Quantico → parametrizations


[1 + 1

2c1(β1r1/2s + β2rs + β3r3/2

s + β4r2s )

]


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ELDAxc [n] =

∫d3r ehom

xc (n(r))

ehomxc (n) = ehom

x (n) + ehomc (n)


ehomx (n) = −3

4

( 3π

)1/3n4/3 ,

For ehomc ? Monte Carlo Quantico → parametrizations


[1 + 1

2c1(β1r1/2s + β2rs + β3r3/2

s + β4r2s )

]


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Parametrizations of the correlation energyE.g.: low-density limit of the electron gas

ec(rs) = −e2(

d0

rs+ d1

r3/2s

+ d2

r4s

+ ...

)rs →∞ ,

Wigner’s parametrization (1934):

eWc (rs) = − 0.44e2

7.8 + rs.

I W (Wigner-1934)I BR (Brual Rothstein-1978)I vBH (von Barth e

Hedin-1972)I GL (Gunnarson e

Lundqvist-1976)

I VWN (Vosko, Wilk eNusair-1980)

I PZ (Perdew e Zunger-1981)I PW92 (Perdew e

Wang-1992)I EHTY (Endo et al-1999)


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Next step: Inhomogeneities, gradient of the density


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Gradient expansion approximation (GEA)

I Systematic corrections to LDA for slowly varying densitiesI Inhomogeneities captured by “reduced density gradients”

Ex [n] = Ax

∫d3r n4/3[1+µs2+...]

Ec[n] =∫

d3r n[ec(n)+β(n)t2+...]

where s = |∇n|2kFn e t = |∇n|

2ksn

I Truncated expansion leads to violation of sum rulesI For atoms, exchange improves over LDA, but not correlation (gets

even positive)


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Generalized gradient approximation (GGA)

I GEA successor; widened the applications of DFT in quantumchemistry

EGGAxc [n] =

∫d3r f (n(r),∇n(r))

I Ma e Brueckner (1968): first GGA, empirical parameter corrects positivecorrelation energies

I Langreth e Mehl (1983): random-phase approximation helps corrections;correlation cutoff; semiempirical

I Perdew e Wang (PW86): LM83 extended without empiricism, lowerexchange errors of LDA to 1-10%

I Becke (B88): correct assintotic behavior of exchange energy; fittedparameter from atomic energies

I PW91: same Becke’s Fxc idea, impose correlation cutoff, and a goodparametrization of correlation (PW92). Attempts to obey as manyuniversal constraints as possible. No empirical parameters.

I PBE GGA was announced as “GGA made simple”, PW91 substitute


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Generalized gradient approximation (GGA)

I GEA successor; widened the applications of DFT in quantumchemistry

EGGAxc [n] =

∫d3r f (n(r),∇n(r))

I Ma e Brueckner (1968): first GGA, empirical parameter corrects positivecorrelation energies

I Langreth e Mehl (1983): random-phase approximation helps corrections;correlation cutoff; semiempirical

I Perdew e Wang (PW86): LM83 extended without empiricism, lowerexchange errors of LDA to 1-10%

I Becke (B88): correct assintotic behavior of exchange energy; fittedparameter from atomic energies

I PW91: same Becke’s Fxc idea, impose correlation cutoff, and a goodparametrization of correlation (PW92). Attempts to obey as manyuniversal constraints as possible. No empirical parameters.

I PBE GGA was announced as “GGA made simple”, PW91 substituteMariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 34/76

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State-of-the-art


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Perdew-Burke-Ernzerhof GGA (1996)


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Visualizing GGAs non-localityEnhancement factor Fxc:

EGGAxc [n] ≈

∫d3r n Fxc(rs, ζ, s) ex(rs, ζ = 0)

Captures the effects ofI correlation (through rs)I spin polarization (ζ)I density inhomogeneity (through the reduced density gradient s).


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Example: PBE exchange

FPBEx (s) = 1 + κ− κ

1 + µκs2 ,

I µ = π2βGE/3, so that there will be a cancellation of the exchangeand correlation gradients, and the jellium result is recovered.

I βGE comes from the second-order gradient expansion in the limit ofslowly-varying densities

I κ is fixed by the Lieb-Oxford bound

s is the “reduced density gradient”

s = |∇n|2(3π2)1/3n4/3 = |∇n|

2kFn ,

which corresponds to a inhomogeneity parameter, measuring how fast thedensity changes in the scale of the Fermi wavelength 2π/kF .


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Exchange enhancement factors


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PBE: “GGA made simple”


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Outline


2 Review of HK-KS


4 LDA and GGA

5 Construction

6 Challenges

7 Final Remarks


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Two construction approaches

I Fitting empirical parametersE.g.: B3LYP (A. Becke on the right)

I Inserting exact constraints (↔ J. Perdew)n = uniform → LDAn ≈ uniform → LDA + O(5) = GEAEx < 0, Ec 6 0Uniform density scalingSpin scalingOne-electron limitDerivative discontinuityLower bounds

Ex.: PW86, PW91, PBE


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I Inserting exact constraints (↔ J. Perdew)

n = uniform → LDAn ≈ uniform → LDA + O(5) = GEAEx < 0, Ec 6 0Uniform density scalingSpin scalingOne-electron limitDerivative discontinuityLower bounds



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I Inserting exact constraints (↔ J. Perdew)n = uniform → LDAn ≈ uniform → LDA + O(5) = GEAEx < 0, Ec 6 0Uniform density scalingSpin scalingOne-electron limitDerivative discontinuityLower bounds



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Constraint satisfaction


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State-of-the-art


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Beyond LDA and GGA

Meta-GGA: + non-interacting kinetic energy density τ . Ex: TPSS, PKZB

EMGGAxc [n] =

∫d3rf (n(r),∇n(r), τ [n])

Hiper-GGA: + exact exchange energy density ex

EHGGAxc [n] =

∫d3rf (n(r),∇n(r), τ [n], ex [n]) ,

Hybrids: mix of exact exchange Ex with ELDAx and Eaprox

c . Ex: B3LYP

Ehibxc [n] = aEexact

x + (1− a)ELDAx [n] + Eaprox

c


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Beyond LDA and GGA functionals


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Systematic improvement?


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Systematic trends?

ConsiderI Localized vs extended densities; covalent and ionic bonds

I Systematic trends between LDA e PBE; between GGAs ehybrids

I Example: lattice constants


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Systematic trends?

ConsiderI Localized vs extended densities; covalent and ionic bondsI Systematic trends between LDA e PBE; between GGAs e

hybrids

I Example: lattice constants


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Systematic trends?

ConsiderI Localized vs extended densities; covalent and ionic bondsI Systematic trends between LDA e PBE; between GGAs e

hybridsI Example: lattice constants


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Outline


2 Review of HK-KS


4 LDA and GGA

5 Construction

6 Challenges

7 Final Remarks


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DFT downsides

I DFT is variational, not perturbative: no systematicimprovement

I Kohn-Sham quantities lack physical meaning

I In principle, everything can be extracted from the density;however, there is no prescription for building the HK or xcdensity functional


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DFA downsides (density-functional approximations)

I No prescription for building the xc density functional

I Combining exact constraints: arbitrary forms

I Single-particle and electron gas paradigm may not beenough

I Often we miss the condensed-matter richness: strongcorrelations, excitations, dispersion forces, relativisticeffects


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What typical functionals miss

Strong correlations

Dispersion forces

Band gaps Charge-transfer


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Strong correlations

Dispersion forces

Band gaps

Charge-transfer


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Strong correlations

Dispersion forces

Band gaps Charge-transferMariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 51/76

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What is wrong in our approximations?


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There are different problems that arise in common densityfunctional approximations.

I will quickly comment two of them.

I Self-interaction error and delocalization errorI Derivative discontinuity


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Self-interaction error

Take your functional and evaluate it for a one-electron density.

In principle, if you have one electron, there is no Coulombinteraction and you should have

U [n(1)] = 0

this means that

UH [n(1)] + Ex [n(1)] + Ec[n(1)] = 0

However, many common functionals have a spurious error, calledself-interaction, leaving a small amount of extra charge. This is aproblem that affects strongly correlated systems.


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Delocalization error

Consider a system of N electrons.If I add or remove one electron, it was proved [Perdew et al 1982]that the total energy behaves linearly with N:

However, common density functionals behave concavely,sometimes favoring fractional configurations. This affects problemsof charge transfer in molecules or electronic transport.


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Delocalization errorConsider a system of N electrons.

If I add or remove one electron, it was proved [Perdew et al 1982]that the total energy behaves linearly with N:



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Delocalization errorConsider a system of N electrons.If I add or remove one electron, it was proved [Perdew et al 1982]that the total energy behaves linearly with N:



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There are several illnesses that arise from the KS picture andcommon density functional approximations.

I will quickly comment two of them.

I Self-interaction error and delocalization errorI Derivative discontinuity


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Derivative discontinuity (I)

As we observed, the derivative of energy changes discontinuoslywhen we change the particle number:


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Derivative discontinuity and the fundamental gap

The fundamental gap in solid-state physics (photoemission gap, 2xchemical hardness) is defined by

Fundamental gap: Ionization potential - Electron affinity

I Ionization potential:

I = E(N−1)−E(N ) = − ∂E∂N

∣∣∣∣N−δN

I Electron affinity:

A = E(N )−E(N+1) = − ∂E∂N

∣∣∣∣N+δN


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Derivative discontinuity in our energy functional

In our density functional, the discontinuity will also appear

E [n] = Ts[n] + UH [n] + V [n] + Exc[n]

The discontinuous kinetic part is called Kohn-Sham non-interacinggap, and the xc part is the derivative discontinuity, the many-bodycorrection to the Kohn-Sham non-interacting gap.

∆L = δExc[n]δn(r)

∣∣∣∣N+δN

− δExc[n]δn(r)

∣∣∣∣N−δN

The fundamental gap (I-A) is given by the sum

∆fund = ∆KS + ∆L


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Kohn-Sham gap vs fundamental gap

Therefore the Kohn-Sham gap is not equal to the fundamental gap.

Most functionals show no derivative discontinuity jump.Ex. LDA:

adapted from PRL 107, 183002 (2011).


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Kohn-Sham gap vs fundamental gapTherefore the Kohn-Sham gap is not equal to the fundamental gap.


adapted from PRL 107, 183002 (2011).


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Most functionals show no derivative discontinuity jump.

Ex. LDA:

adapted from PRL 107, 183002 (2011).


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adapted from PRL 107, 183002 (2011).Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 60/76

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PRL 96, 226402 (2006).


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Some observations on KS quantities

The price for the simplification of the problem is that Kohn-Shamis an auxiliary tool.

The KS mapping gives you the energy and ground-state density.

There is no proof that the KS quantities have a physical meaning,with few exceptions.

The KS gap is not equal to the fundamental gap, and theeigenvalues are not quasiparticle spectra.


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Some observations on KS quantities

Nonetheless, the KS eigenvalues can be a very good approximationto the quasiparticle spectrum.


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Some observations on KS quantitiesNonetheless, the KS eigenvalues can be a very good approximationto the quasiparticle spectrum.


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General recommendations

Functionals families (LDA,GGA,MGGA,hybrids):

I Important to know the functional proposal and itsimprovements

I Check previous literature on the atomic, bulk trends, theircharacter and problems

I When possible, confrontation with experimental or highlyaccurate methods


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Outline


2 Review of HK-KS


4 LDA and GGA

5 Construction

6 Challenges

7 Final Remarks


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Timeline


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DFT Impact

Citation Statistics from 110 Years of Physical Review (1893 - 2003)

(Physics Today, p.49 Junho 2005)


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Back to the electronic structure spirit

“Where solid-state physics hasFermi energy, chemical potential,band gap, density of states, andlocal density of states, quantumchemistry has ionization potential,electron affinity, hardness, softness,and local softness. Much more too.DFT is a single language that coversatoms, molecules, clusters, surfaces,and solids.”

Robert Parr


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1964/65-2015

Hohenberg-Kohn ’64:

Kohn-Sham ’65:


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Walter Kohn

I Born in 1923, in a jew middle-class familyI World War II: fled to England with help of

family friends -wishing to become a farmerI First interned in British camps for “enemy

aliens”I In Canadian camps, supported by Red Cross, studies mathI Working as lumberjacks, earning 20 cents per day, buys

Slater’s book “Chemical Physics”I Joins the Canadian army and gets a BS degree in Applied

MathematicsI Finishes a crash master’s course and applies for PhDsI Awarded a scholarship for Harvard; becomes PhD student of

Julian Schwinger


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Walter Kohn

I Born in 1923, in a jew middle-class family

I World War II: fled to England with help offamily friends -wishing to become a farmer

I First interned in British camps for “enemyaliens”

I In Canadian camps, supported by Red Cross, studies mathI Working as lumberjacks, earning 20 cents per day, buys



Julian Schwinger


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Walter Kohn


family friends -wishing to become a farmer

I First interned in British camps for “enemyaliens”




Julian Schwinger


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Walter Kohn



aliens”




Julian Schwinger


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Walter Kohn



aliens”I In Canadian camps, supported by Red Cross, studies math

I Working as lumberjacks, earning 20 cents per day, buysSlater’s book “Chemical Physics”

I Joins the Canadian army and gets a BS degree in AppliedMathematics

I Finishes a crash master’s course and applies for PhDsI Awarded a scholarship for Harvard; becomes PhD student of

Julian Schwinger


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Walter Kohn




Slater’s book “Chemical Physics”

I Joins the Canadian army and gets a BS degree in AppliedMathematics


Julian Schwinger


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Walter Kohn





Mathematics


Julian Schwinger


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Walter Kohn





MathematicsI Finishes a crash master’s course and applies for PhDs

I Awarded a scholarship for Harvard; becomes PhD student ofJulian Schwinger


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Walter Kohn






Julian Schwinger


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Walter Kohn and Julian Schwinger

Kohn met Schwinger only “a few times a year”.

“It was during these meetings, sometimesmore than 2 hours long, that I learned themost from him. (...) to dig for the essential;to pay attention to the experimental facts;to try to say something precise and operati-onally meaningful, even if one cannot calcu-late everything a priori; not to be satisfied un-til one has embedded his ideas in a coherent,logical, and aesthetically satisfying structure.(...) I cannot even imagine my subsequent sci-entific life without Julian’s example and tea-ching.”


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Kohn’s scientific background

I Schwinger: Green’s functions, variational principles, scattering

I Van Vleck: entered solid-state physicsI Rostocker: Green’s functions to solve the electron band

structure problem (KKR)I Bell Labs: semiconductor physics (transistor rush)I Luttinger; effective mass equation for the energy levels of

impurity states in Silicon: “one-particle method”I ... electronic transport; phonons; insulating state;I Mott: Thomas-Fermi for screeningI de Gennes, Friedel: metals and alloys;


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I Schwinger: Green’s functions, variational principles, scatteringI Van Vleck: entered solid-state physics

I Rostocker: Green’s functions to solve the electron bandstructure problem (KKR)

I Bell Labs: semiconductor physics (transistor rush)I Luttinger; effective mass equation for the energy levels of



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I Schwinger: Green’s functions, variational principles, scatteringI Van Vleck: entered solid-state physicsI Rostocker: Green’s functions to solve the electron band

structure problem (KKR)

I Bell Labs: semiconductor physics (transistor rush)I Luttinger; effective mass equation for the energy levels of



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structure problem (KKR)I Bell Labs: semiconductor physics (transistor rush)

I Luttinger; effective mass equation for the energy levels ofimpurity states in Silicon: “one-particle method”

I ... electronic transport; phonons; insulating state;I Mott: Thomas-Fermi for screeningI de Gennes, Friedel: metals and alloys;


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impurity states in Silicon: “one-particle method”

I ... electronic transport; phonons; insulating state;I Mott: Thomas-Fermi for screeningI de Gennes, Friedel: metals and alloys;


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impurity states in Silicon: “one-particle method”I ... electronic transport; phonons; insulating state;

I Mott: Thomas-Fermi for screeningI de Gennes, Friedel: metals and alloys;


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impurity states in Silicon: “one-particle method”I ... electronic transport; phonons; insulating state;I Mott: Thomas-Fermi for screening

I de Gennes, Friedel: metals and alloys;


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“Kohn’s seminal papers (...) are all most notable fortheir clarity and the simplicity of the mathematics oneencounters. On many occasions, after reading throughthe material, I found myself saying something like “ofcourse things go that way, I could have written thismyself”. (...) It is the case that the most importantand fundamental new ideas and concepts in our fieldare very simple and obvious, once they have been setforth for the first time. I am reminded of remarks Ihave read recently in an essay by Steven Weinberg,who states that the very important and fundamentalpapers in physics are notable for their clarity. The newideas are applied quickly because of this.”

Douglas L. Mills


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“Kohn’s seminal papers (...)

are all most notable fortheir clarity and the simplicity of the mathematics oneencounters. On many occasions, after reading throughthe material, I found myself saying something like “ofcourse things go that way, I could have written thismyself”. (...) It is the case that the most importantand fundamental new ideas and concepts in our fieldare very simple and obvious, once they have been setforth for the first time. I am reminded of remarks Ihave read recently in an essay by Steven Weinberg,who states that the very important and fundamentalpapers in physics are notable for their clarity. The newideas are applied quickly because of this.”

Douglas L. Mills


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“Kohn’s seminal papers (...) are all most notable fortheir clarity and the simplicity of the mathematics oneencounters.

On many occasions, after reading throughthe material, I found myself saying something like “ofcourse things go that way, I could have written thismyself”. (...) It is the case that the most importantand fundamental new ideas and concepts in our fieldare very simple and obvious, once they have been setforth for the first time. I am reminded of remarks Ihave read recently in an essay by Steven Weinberg,who states that the very important and fundamentalpapers in physics are notable for their clarity. The newideas are applied quickly because of this.”

Douglas L. Mills


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“Kohn’s seminal papers (...) are all most notable fortheir clarity and the simplicity of the mathematics oneencounters. On many occasions, after reading throughthe material, I found myself saying something like “ofcourse things go that way, I could have written thismyself”. (...)

It is the case that the most importantand fundamental new ideas and concepts in our fieldare very simple and obvious, once they have been setforth for the first time. I am reminded of remarks Ihave read recently in an essay by Steven Weinberg,who states that the very important and fundamentalpapers in physics are notable for their clarity. The newideas are applied quickly because of this.”

Douglas L. Mills


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“Kohn’s seminal papers (...) are all most notable fortheir clarity and the simplicity of the mathematics oneencounters. On many occasions, after reading throughthe material, I found myself saying something like “ofcourse things go that way, I could have written thismyself”. (...) It is the case that the most importantand fundamental new ideas and concepts in our fieldare very simple and obvious, once they have been setforth for the first time.

I am reminded of remarks Ihave read recently in an essay by Steven Weinberg,who states that the very important and fundamentalpapers in physics are notable for their clarity. The newideas are applied quickly because of this.”

Douglas L. Mills


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“Kohn’s seminal papers (...) are all most notable fortheir clarity and the simplicity of the mathematics oneencounters. On many occasions, after reading throughthe material, I found myself saying something like “ofcourse things go that way, I could have written thismyself”. (...) It is the case that the most importantand fundamental new ideas and concepts in our fieldare very simple and obvious, once they have been setforth for the first time. I am reminded of remarks Ihave read recently in an essay by Steven Weinberg,who states that the very important and fundamentalpapers in physics are notable for their clarity.

The newideas are applied quickly because of this.”

Douglas L. Mills


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“Kohn’s seminal papers (...) are all most notable fortheir clarity and the simplicity of the mathematics oneencounters. On many occasions, after reading throughthe material, I found myself saying something like “ofcourse things go that way, I could have written thismyself”. (...) It is the case that the most importantand fundamental new ideas and concepts in our fieldare very simple and obvious, once they have been setforth for the first time. I am reminded of remarks Ihave read recently in an essay by Steven Weinberg,who states that the very important and fundamentalpapers in physics are notable for their clarity. The newideas are applied quickly because of this.”

Douglas L. Mills


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Acknowledgements (I)

Klaus Capelle, UFABC, Brazil

E.K.U. Gross, MPI-Halle,Germany

Sam Trickey, QTP-Univ.Florida

Caio Lewenkopf, UFF, Brazil


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References

I Kohn’s Nobel lecture, Electronic structure of matter—wave functions anddensity functionals, (http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/kohn-lecture.html)

I A. Becke, Perspective: Fifty years of density-functional theory in chemicalphysics, (http://www.ncbi.nlm.nih.gov/pubmed/24832308)

I K. Capelle, A bird’s-eye view of density-functional theory,(http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332006000700035)

I Perdew and Kurth, A Primer in Density Functional Theory,(http://www.physics.udel.edu/˜bnikolic/QTTG/NOTES/DFT/BOOK=primer_dft.pdf)

I Perdew et al., Some Fundamental Issues in Ground-State Density FunctionalTheory: A Guide for the Perplexedhttp://pubs.acs.org/doi/full/10.1021/ct800531s

I Zangwill, The education of Walter Kohn and the creation of density functionaltheory, (http://arxiv.org/abs/1403.5164)

I M. M. Odashima, PHD Thesis(http://www.teses.usp.br/teses/disponiveis/76/76131/tde-14062010-164125/pt-br.php)


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http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/kohn-lecture.html




http://www.ncbi.nlm.nih.gov/pubmed/24832308

http://www.ncbi.nlm.nih.gov/pubmed/24832308

http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332006000700035




http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/DFT/BOOK=primer_dft.pdf




http://pubs.acs.org/doi/full/10.1021/ct800531s

http://arxiv.org/abs/1403.5164

http://arxiv.org/abs/1403.5164

http://www.teses.usp.br/teses/disponiveis/76/76131/tde-14062010-164125/pt-br.php

http://www.teses.usp.br/teses/disponiveis/76/76131/tde-14062010-164125/pt-br.php




References

I Electronic Structure Basic - Theory and Practical Methods. Richard M Martin,Cambridge (2008)

I Atomic and Electronic Structure of Solids. Efthimios Kaxiras, Cambridge(2003).

I Density Functional Theory - An Advanced Course. Eberhard Engel and ReinerM. Dreizler, Springer (2011).

I Many-Electron Approaches in Physics, Chemistry and Mathematics: AMultidisciplinary View. Eds. Volker Bach, Luigi Delle Site, Springer (2014).

I Many-Body Approach to Electronic Excitations - Concepts and Applications.Friedhelm Bechstedt, Springer (2015).


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Acknowledgements

To all ENFMC organizers and FAPERJ.

Thank you for your attention!

https://sites.google.com/site/mmodashima/


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introduction to dft part 2

Education