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XXXVIII ENFMC Brazilian Physical Society Meeting
Introduction todensity functional theory
Mariana M. Odashima
ENFMC
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
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 2/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 3/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 4/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 4/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 4/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 4/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 4/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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′| .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 5/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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′|
.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 5/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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′| .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 5/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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′|
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 6/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 7/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 7/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 7/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Back to our question
a program ? a method?
some
obscure
theory?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 7/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Back to our question
a program ? a method?
some
obscure
theory?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 6/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 7/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Density functional theory (DFT)
I Quantum theory based on the density n(r)
wave functions Ψ(r1, r2, ...rN )I Single-particle Kohn-Sham equations
I Electronic structure boom: Nobel Prize toW.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)]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 7/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 7/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 7/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 7/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Hohenberg-Kohn (1964)
Phys. Rev. 136 B864 (1964).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 8/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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?...
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 9/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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?...
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 9/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 10/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 11/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 11/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 11/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 12/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn-Sham kindergarden
Interacting
(complicated)
Ficticious non-interacting
under effective field
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 13/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 13/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Exchange-correlation
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 13/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Exchange-correlation
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.”.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 14/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 15/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Exchange-correlation
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.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 16/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Exchange-correlation
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.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 16/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 17/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 17/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 18/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 18/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.”
Correlation energy: Ec < 0
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 18/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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]]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 19/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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]]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 19/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 20/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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′ |
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 21/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 22/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 22/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Exchange-correlation energy
“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).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 23/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Exchange-correlation energy
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]?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 24/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Exchange-correlation energy
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]?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 24/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 24/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
State-of-the-art
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 25/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Back in 65
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 26/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Back in 65
I Introduce KS equationsI Explore possible Exc
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 27/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Density functional
I Starting point: electron gas
Exc =∫
d3rexc[n]n(r) (exc: energy density per particle)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 28/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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] .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 29/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 )
]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 30/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 )
]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 30/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 )
]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 30/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 )
]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 30/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 )
]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 30/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 )
]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 30/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 )
]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 30/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 31/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Next step: Inhomogeneities, gradient of the density
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 32/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 33/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 34/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 34/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 34/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 34/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 34/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 34/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
State-of-the-art
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 35/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Perdew-Burke-Ernzerhof GGA (1996)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 36/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 37/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 38/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Exchange enhancement factors
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 39/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
PBE: “GGA made simple”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 40/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 40/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 41/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 41/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 41/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Constraint satisfaction
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 42/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Constraint satisfaction
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 43/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
State-of-the-art
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 44/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
State-of-the-art
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 44/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 45/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 45/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 45/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Beyond LDA and GGA functionals
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 46/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Systematic improvement?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 47/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Systematic improvement?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 47/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 48/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 48/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Systematic trends?
ConsiderI Localized vs extended densities; covalent and ionic bondsI Systematic trends between LDA e PBE; between GGAs e
hybridsI Example: lattice constants
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 48/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Systematic trends?
ConsiderI Localized vs extended densities; covalent and ionic bondsI Systematic trends between LDA e PBE; between GGAs e
hybridsI Example: lattice constants
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 48/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Systematic trends?
ConsiderI Localized vs extended densities; covalent and ionic bondsI Systematic trends between LDA e PBE; between GGAs e
hybridsI Example: lattice constants
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 48/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 48/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 49/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 49/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 49/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 49/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 50/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 50/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 50/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 50/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 50/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
What typical functionals miss
Strong correlations
Dispersion forces
Band gaps Charge-transfer
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 51/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
What typical functionals miss
Strong correlations
Dispersion forces
Band gaps Charge-transfer
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 51/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
What typical functionals miss
Strong correlations
Dispersion forces
Band gaps
Charge-transfer
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 51/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
What typical functionals miss
Strong correlations
Dispersion forces
Band gaps Charge-transferMariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 51/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
What is wrong in our approximations?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 52/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 53/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 53/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 53/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 54/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 54/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 54/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 54/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 54/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 55/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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:
However, common density functionals behave concavely,sometimes favoring fractional configurations. This affects problemsof charge transfer in molecules or electronic transport.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 55/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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:
However, common density functionals behave concavely,sometimes favoring fractional configurations. This affects problemsof charge transfer in molecules or electronic transport.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 55/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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:
However, common density functionals behave concavely,sometimes favoring fractional configurations. This affects problemsof charge transfer in molecules or electronic transport.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 55/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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:
However, common density functionals behave concavely,sometimes favoring fractional configurations. This affects problemsof charge transfer in molecules or electronic transport.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 55/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 56/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Derivative discontinuity (I)
As we observed, the derivative of energy changes discontinuoslywhen we change the particle number:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 57/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Derivative discontinuity (I)
As we observed, the derivative of energy changes discontinuoslywhen we change the particle number:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 57/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Derivative discontinuity (I)
As we observed, the derivative of energy changes discontinuoslywhen we change the particle number:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 57/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 58/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 58/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 58/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 58/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 59/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 59/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 59/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 59/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 60/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn-Sham gap vs fundamental gapTherefore 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).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 60/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn-Sham gap vs fundamental gapTherefore 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).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 60/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn-Sham gap vs fundamental gapTherefore 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).Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 60/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn-Sham gap vs fundamental gapTherefore the Kohn-Sham gap is not equal to the fundamental gap.
Most functionals show no derivative discontinuity jump.Ex. LDA:
PRL 96, 226402 (2006).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 61/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 62/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 62/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 62/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 62/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 62/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Some observations on KS quantities
Nonetheless, the KS eigenvalues can be a very good approximationto the quasiparticle spectrum.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 63/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Some observations on KS quantitiesNonetheless, the KS eigenvalues can be a very good approximationto the quasiparticle spectrum.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 63/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Some observations on KS quantitiesNonetheless, the KS eigenvalues can be a very good approximationto the quasiparticle spectrum.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 63/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 64/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 64/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 64/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 64/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 64/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 64/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 64/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 65/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 65/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
DFT Impact
Citation Statistics from 110 Years of Physical Review (1893 - 2003)
(Physics Today, p.49 Junho 2005)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 66/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
DFT Impact
Citation Statistics from 110 Years of Physical Review (1893 - 2003)
(Physics Today, p.49 Junho 2005)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 66/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 67/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 67/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 67/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 67/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 67/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
1964/65-2015
Hohenberg-Kohn ’64:
Kohn-Sham ’65:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 68/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
1964/65-2015
Hohenberg-Kohn ’64:
Kohn-Sham ’65:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 68/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 69/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 69/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 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
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 69/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 69/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 69/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 AppliedMathematics
I Finishes a crash master’s course and applies for PhDsI Awarded a scholarship for Harvard; becomes PhD student of
Julian Schwinger
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 69/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mathematics
I Finishes a crash master’s course and applies for PhDsI Awarded a scholarship for Harvard; becomes PhD student of
Julian Schwinger
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 69/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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 PhDs
I Awarded a scholarship for Harvard; becomes PhD student ofJulian Schwinger
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 69/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 69/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 70/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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;
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 71/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn’s scientific background
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
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;
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 71/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn’s scientific background
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
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;
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 71/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn’s scientific background
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 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;
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 71/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn’s scientific background
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
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;
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 71/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn’s scientific background
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
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;
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 71/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn’s scientific background
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
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;
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 71/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Kohn’s scientific background
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
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;
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 71/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
“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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 72/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
“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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 72/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
“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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 72/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
“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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 72/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
“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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 72/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
“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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 72/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
“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
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 72/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Acknowledgements (I)
Klaus Capelle, UFABC, Brazil
E.K.U. Gross, MPI-Halle,Germany
Sam Trickey, QTP-Univ.Florida
Caio Lewenkopf, UFF, Brazil
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 73/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
Acknowledgements (I)
Klaus Capelle, UFABC, Brazil
E.K.U. Gross, MPI-Halle,Germany
Sam Trickey, QTP-Univ.Florida
Caio Lewenkopf, UFF, Brazil
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 73/76
ENFMC
Problem HK-KS xc LDA Construction Challenges Final Remarks
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)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 74/76
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Problem HK-KS xc LDA Construction Challenges Final Remarks
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).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 75/76
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Problem HK-KS xc LDA Construction Challenges Final Remarks
Acknowledgements
To all ENFMC organizers and FAPERJ.
Thank you for your attention!
https://sites.google.com/site/mmodashima/
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 76/76
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