introduction to dft part 1

XXXVIII ENFMC Brazilian Physical Society Meeting

Introduction todensity functional theory

Mariana M. Odashima

Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham

Today’s task:

is it ... a program ?

a method?

obscure

theory?

Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 1/64

Today’s task:

a method?

obscure

theory?

Today’s task:

is it ...

a program ?

a method?

obscure

theory?

Today’s task:

a method?

obscure

theory?

Today’s task:

a method?

obscure

theory?

Today’s task:

a method?

obscure

theory?

Some perspective

(adapted from PhDComics)

Calm down!

Some perspective

Calm down!

Some perspective

Calm down!

Some perspective

Calm down!Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 2/64

Some perspective

(adapted from Original “DFT song” written by V. Blum and K. Burke)

When I find my model’s unpredictive,Walter Kohn just comes to me,

speaking words of wisdom, DFT. �

♩ And in my hour of code-debugging,he stands right in front of me,

saying ”you just gotta learnyour chemistree”. �

� LSD, PBE,B3LYP, HSE.

I thought you were first principles, DFT... �

Some perspective

� When I find my model’s unpredictive,Walter Kohn just comes to me,

Some perspective

Adapted from Original “DFT song” written by V. Blum and K. Burke

Some perspective

Walter Kohn Computer simulations

Molecules, nanostructures Acronyms of functionals

Some perspective

Walter Kohn

Computer simulations

Some perspective

Molecules, nanostructures

Acronyms of functionals

Some perspective

Today’s task:

Introduction to density-functional theory

I Context and key concepts (1927-1930)

I Fundamentals (1964-1965)

I Approximations (≈ 1980-2010)

Today’s task:

Disclaimer

“Density-functional theory isa subtle,

seductive,provocative business.”

A. Becke

Disclaimer

“Density-functional theory is

a subtle,seductive,

provocative business.”

A. Becke

Disclaimer

A. Becke

Disclaimer

seductive,

provocative business.”

A. Becke

Disclaimer

A. Becke

Outline

1 Electronic structure

2 Hartree and Hartree-Fock methods

3 Thomas-Fermi model

4 Hohenberg-Kohn Theorem

5 Kohn-Sham Scheme

Outline

5 Kohn-Sham Scheme

General context

I Quantum method to describe properties of materials

I Chemistry, Physics, Material science

General context

Modelling

I Consider N electrons, P nuclei. Schrodinger Equation:

HΨ = EΨ

Ψ(r1, r2, ..., rN ,R1,R2, ...,RP)

H =N∑i− ~2

2m∇2ri +

P∑i− ~2

2Mi∇2

Ri +N∑

|ri − rj |

−N∑i

|ri −Rj |+

P∑i<j

ZiZje2

|Ri −Rj |,

Modelling

HΨ = EΨ

Ψ(r1, r2, ..., rN ,R1,R2, ...,RP)

H =N∑i− ~2

2m∇2ri +

P∑i− ~2

2Mi∇2

Ri +N∑

|ri − rj |

−N∑i

|ri −Rj |+

P∑i<j

ZiZje2

|Ri −Rj |,

Modelling

HΨ = EΨ

Ψ(r1, r2, ..., rN ,R1,R2, ...,RP)

H =N∑i− ~2

2m∇2ri +

P∑i− ~2

2Mi∇2

Ri +N∑

|ri − rj |

−N∑i

|ri −Rj |+

P∑i<j

ZiZje2

|Ri −Rj |,

Modelling

HΨ = EΨ

Ψ(r1, r2, ..., rN ,R1,R2, ...,RP)

H =N∑i− ~2

2m∇2ri +

P∑i− ~2

2Mi∇2

Ri +N∑

|ri − rj |

−N∑i

|ri −Rj |+

P∑i<j

ZiZje2

|Ri −Rj |,

Modelling

HΨ = EΨ

Ψ(r1, r2, ..., rN ,R1,R2, ...,RP)

H =N∑i− ~2

2m∇2ri +

P∑i− ~2

2Mi∇2

Ri +N∑

|ri − rj |

−N∑i

|ri −Rj |+

P∑i<j

ZiZje2

|Ri −Rj |,

Born-Oppenheimer Approximation

I Simplification: separation of the nuclear and electronic scales

I Born-Oppenheimer: nuclei fixed

H =N∑i− ~2

2m∇2ri +

P∑i− ~2

2Mi∇2

Ri + +N∑

|ri − rj |

−N∑i

|ri −Rj |+

P∑i<j

ZiZje2

|Ri −Rj |,

I Simplification: separation of the nuclear and electronic scalesI Born-Oppenheimer: nuclei fixed

H =N∑i− ~2

2m∇2ri +

P∑i− ~2

2Mi∇2

Ri + +N∑

|ri − rj |

−N∑i

|ri −Rj |+

P∑i<j

ZiZje2

|Ri −Rj |,

I Simplification: separation of the nuclear and electronic scalesI Born-Oppenheimer: nuclei fixed

H =N∑i− ~2

2m∇2ri +

P∑i− ~2

2Mi∇2

Ri + +N∑

|ri − rj |

−N∑i

|ri −Rj |+

P∑i<j

ZiZje2

|Ri −Rj |,

Born-Oppenheimer approximation

H = Hel + Hnuc

Hel =N∑i− ~2

2m∇2ri +

N∑i<j

|ri − rj |−

|ri −Rj |

= T + U + V

Hnuc =P∑

ZiZje2

|Ri −Rj |= Vnuc .

Ψ(ri ; Ri) = Ψel(ri ; Ri) Φnuc(Ri) .

H = Hel + Hnuc

Hel =N∑i− ~2

2m∇2ri +

N∑i<j

|ri − rj |−

|ri −Rj |

= T + U + V

Hnuc =P∑

ZiZje2

|Ri −Rj |= Vnuc .

H = Hel + Hnuc

Hel =N∑i− ~2

2m∇2ri +

N∑i<j

|ri − rj |−

|ri −Rj |

= T + U + V

Hnuc =P∑

ZiZje2

|Ri −Rj |= Vnuc .

H = Hel + Hnuc

Hel =N∑i− ~2

2m∇2ri +

N∑i<j

|ri − rj |−

|ri −Rj |

= T + U + V

Hnuc =P∑

ZiZje2

|Ri −Rj |= Vnuc .

The electronic structure problem

I Many interacting electrons: Ψel(~r1,~r2, ...,~rN )

I Paradigms: Hydrogen atom (single particle problem);uniform electron gas

(by K. Capelle)

I Many interacting electrons: Ψel(~r1,~r2, ...,~rN )

I Paradigms: Hydrogen atom (single particle problem);uniform electron gas

(by K. Capelle)

I Many interacting electrons: Ψel(~r1,~r2, ...,~rN )I Paradigms: Hydrogen atom (single particle problem);

uniform electron gas

(by K. Capelle)

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

under the static nuclei external potential.

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)

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

under the static nuclei external potential.

I Quantum many-body problemof N interacting electrons: Ψel(~r1,~r2, ...,~rN )under the static nuclei external potential.

Parenthesis

Timeline

Dirac’s impressions (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.”

Timeline

DFT predecessors

I Hartree

I Hartree-Fock

I Thomas-Fermi-Dirac

DFT predecessors

I Hartree

I Hartree-Fock

I Thomas-Fermi-Dirac

Outline

5 Kohn-Sham Scheme

Hartree’s Method (1928)

In the late 1920’s, Douglas Hartree (phy-sicist, mathematician, pioneer compu-ter scientist) developed a self-consistentapproach to solve the single-electronSchrodinger equation; each electronwould be under a mean field that accoun-ted for the other electrons.

Hartree’s Method (1928)I Schrodinger’s equation for independent electrons

(− ~2

2m∇2 + V (r)

)ϕi(r) = εiϕi(r) ,

I Wave function: product of monoelectronic orbitalsΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .

I Remaining electrons: effective field

V (r) = −P∑j

|r−Rj |+ e2

∫d3r ′ |ϕi(r′)|2

|r− r′|

= Vion(r) + VH (r) . n(r) =N∑i|ϕi(r)|2

I Self-consistent solution

Hartree’s Method (1928)I Schrodinger’s equation for independent electrons(

− ~2

2m∇2 + V (r)

V (r) = −P∑j

|r−Rj |+ e2

∫d3r ′ |ϕi(r′)|2

|r− r′|

− ~2

2m∇2 + V (r)

I Wave function: product of monoelectronic orbitals

ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .

V (r) = −P∑j

|r−Rj |+ e2

∫d3r ′ |ϕi(r′)|2

|r− r′|

− ~2

2m∇2 + V (r)

V (r) = −P∑j

|r−Rj |+ e2

∫d3r ′ |ϕi(r′)|2

|r− r′|

− ~2

2m∇2 + V (r)

V (r) = −P∑j

|r−Rj |+ e2

∫d3r ′ |ϕi(r′)|2

|r− r′|

− ~2

2m∇2 + V (r)

V (r) = −P∑j

|r−Rj |+ e2

∫d3r ′ |ϕi(r′)|2

|r− r′|

− ~2

2m∇2 + V (r)

V (r) = −P∑j

|r−Rj |+ e2

∫d3r ′ |ϕi(r′)|2

|r− r′|

I Self-consistent solutionMariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 19/64

Hartree’s Method (1928)Wave function

ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .

Total energy:

〈ΨH |H |ΨH 〉 =N∑i

∫d3r ϕ∗i (r)

(− ~2

2m∇2 + Vion(r)

)ϕi(r) +

∫d3r

∫d3r ′ |ϕi(r)|2|ϕj(r′)|2

|r− r′| .

“Hartree functional”: n(r) =N∑i

|ϕi(r)|2

〈ΨH |U |ΨH 〉 = e2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′| = UH [n] .

ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .

Total energy:

∫d3r ϕ∗i (r)

(− ~2

2m∇2 + Vion(r)

)ϕi(r) +

∫d3r

∫d3r ′ |ϕi(r)|2|ϕj(r′)|2

|r− r′| .

|ϕi(r)|2

〈ΨH |U |ΨH 〉 = e2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′| = UH [n] .

ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .

Total energy:

∫d3r ϕ∗i (r)

(− ~2

2m∇2 + Vion(r)

)ϕi(r) +

∫d3r

∫d3r ′ |ϕi(r)|2|ϕj(r′)|2

|r− r′| .

|ϕi(r)|2

〈ΨH |U |ΨH 〉 = e2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′| = UH [n] .

ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .

Total energy:

∫d3r ϕ∗i (r)

(− ~2

2m∇2 + Vion(r)

)ϕi(r) +

∫d3r

∫d3r ′ |ϕi(r)|2|ϕj(r′)|2

|r− r′| .

|ϕi(r)|2

〈ΨH |U |ΨH 〉 = e2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′| = UH [n] .

ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .

Total energy:

∫d3r ϕ∗i (r)

(− ~2

2m∇2 + Vion(r)

)ϕi(r) +

∫d3r

∫d3r ′ |ϕi(r)|2|ϕj(r′)|2

|r− r′| .

|ϕi(r)|2

〈ΨH |U |ΨH 〉 = e2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′| = UH [n] .

ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .

Total energy:

∫d3r ϕ∗i (r)

(− ~2

2m∇2 + Vion(r)

)ϕi(r) +

∫d3r

∫d3r ′ |ϕi(r)|2|ϕj(r′)|2

|r− r′| .

|ϕi(r)|2

〈ΨH |U |ΨH 〉 = e2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′|

= UH [n] .

ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .

Total energy:

∫d3r ϕ∗i (r)

(− ~2

2m∇2 + Vion(r)

)ϕi(r) +

∫d3r

∫d3r ′ |ϕi(r)|2|ϕj(r′)|2

|r− r′| .

|ϕi(r)|2

〈ΨH |U |ΨH 〉 = e2

∫d3r

∫d3r ′ n(r)n(r′)

|r− r′| = UH [n] .Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 20/64

Hartree-Fock Method (1930)

I John Slater, Vladimir FockI Fermionic wave function: antisimetrized product

Ψ(x1,x2, ...,xi , ...,xj , ...,xN ) = −Ψ(x1,x2, ...,xj , ...,xi , ...,xN ) ,

where x = (r, σ).

I Slater determinant

ΨHF (r) = 1√N !

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

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

ϕN (x1) ϕN (x2) · · · ϕN (xN )

∣∣∣∣∣∣∣∣∣∣

Ψ(x1,x2, ...,xi , ...,xj , ...,xN ) =

−Ψ(x1,x2, ...,xj , ...,xi , ...,xN ) ,

where x = (r, σ).

ΨHF (r) = 1√N !

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

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

ϕN (x1) ϕN (x2) · · · ϕN (xN )

∣∣∣∣∣∣∣∣∣∣

where x = (r, σ).

ΨHF (r) = 1√N !

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

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

ϕN (x1) ϕN (x2) · · · ϕN (xN )

∣∣∣∣∣∣∣∣∣∣

where x = (r, σ).

ΨHF (r) = 1√N !

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

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

ϕN (x1) ϕN (x2) · · · ϕN (xN )

∣∣∣∣∣∣∣∣∣∣Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 21/64

Example

For two orbitals:

I Slater determinant:

ΨHF (x1,x2) = 1√2

∣∣∣∣∣ ϕ1(x1) ϕ1(x2)ϕ2(x1) ϕ2(x2)

∣∣∣∣∣

= 1√2

[ϕ1(x1)ϕ2(x2)− ϕ1(x2)ϕ2(x1)]

〈ΨHF |H |ΨHF〉 = 1/2∫

dx1∫

[ϕ∗1(x1)ϕ∗2(x2)− ϕ∗1(x2)ϕ∗2(x1)][Σhi + U

][ϕ1(x1)ϕ2(x2)− ϕ1(x2)ϕ2(x1)]

Example

For two orbitals:

ΨHF (x1,x2) = 1√2

∣∣∣∣∣ ϕ1(x1) ϕ1(x2)ϕ2(x1) ϕ2(x2)

∣∣∣∣∣= 1√

2[ϕ1(x1)ϕ2(x2)− ϕ1(x2)ϕ2(x1)]

〈ΨHF |H |ΨHF〉 = 1/2∫

dx1∫

[ϕ∗1(x1)ϕ∗2(x2)− ϕ∗1(x2)ϕ∗2(x1)][Σhi + U

][ϕ1(x1)ϕ2(x2)− ϕ1(x2)ϕ2(x1)]

Example

For two orbitals:

ΨHF (x1,x2) = 1√2

∣∣∣∣∣ ϕ1(x1) ϕ1(x2)ϕ2(x1) ϕ2(x2)

∣∣∣∣∣= 1√

2[ϕ1(x1)ϕ2(x2)− ϕ1(x2)ϕ2(x1)]

〈ΨHF |H |ΨHF〉 = 1/2∫

dx1∫

[ϕ∗1(x1)ϕ∗2(x2)− ϕ∗1(x2)ϕ∗2(x1)][Σhi + U

][ϕ1(x1)ϕ2(x2)− ϕ1(x2)ϕ2(x1)]

Hartree-Fock methodI Total energy:

〈ΨHF |H |ΨHF〉 =

∫d3r ϕ∗iσ(r)

(− ~2

2m∇2 + Vext(r)

)ϕiσ(r) +

2∑i,j

∑σi ,σj

∫d3r

∫d3r ′|ϕiσi (r)|2|ϕjσj (r′)|2

|r− r′| −

2∑i,j

∫d3r

∫d3r ′

ϕ∗iσ(r)ϕ∗jσ(r′)ϕiσ(r′)ϕjσ(r)|r− r′|

= T + Vext + UH + Ex

I Coulomb energy, direct and indirect:

〈ΨH |U |ΨH 〉 = UH > 0〈ΨHF |U |ΨHF 〉 = UH + Ex (Ex < 0)

〈ΨHF |H |ΨHF〉 =∑

∫d3r ϕ∗iσ(r)

(− ~2

2m∇2 + Vext(r)

)ϕiσ(r) +

2∑i,j

∑σi ,σj

∫d3r

|r− r′| −

2∑i,j

∫d3r

∫d3r ′

∫d3r ϕ∗iσ(r)

(− ~2

2m∇2 + Vext(r)

)ϕiσ(r) +

2∑i,j

∑σi ,σj

∫d3r

|r− r′| −

2∑i,j

∫d3r

∫d3r ′

∫d3r ϕ∗iσ(r)

(− ~2

2m∇2 + Vext(r)

)ϕiσ(r) +

2∑i,j

∑σi ,σj

∫d3r

|r− r′| −

2∑i,j

∫d3r

∫d3r ′

∫d3r ϕ∗iσ(r)

(− ~2

2m∇2 + Vext(r)

)ϕiσ(r) +

2∑i,j

∑σi ,σj

∫d3r

|r− r′| −

2∑i,j

∫d3r

∫d3r ′

∫d3r ϕ∗iσ(r)

(− ~2

2m∇2 + Vext(r)

)ϕiσ(r) +

2∑i,j

∑σi ,σj

∫d3r

|r− r′| −

2∑i,j

∫d3r

∫d3r ′

〈ΨH |U |ΨH 〉 = UH > 0

〈ΨHF |U |ΨHF 〉 = UH + Ex (Ex < 0)

∫d3r ϕ∗iσ(r)

(− ~2

2m∇2 + Vext(r)

)ϕiσ(r) +

2∑i,j

∑σi ,σj

∫d3r

|r− r′| −

2∑i,j

∫d3r

∫d3r ′

Exchange energy (Fock)

I Antisimetrized wave function: Pauli principle pushes samespin electrons apart,

reducing the Coulomb repulsionI Exchange contribution Ex < 0I Fock exchange depends on the orbitals:

Ex [ϕ] = −e2

2∑i,j

∫d3r

∫d3r ′

I Antisimetrized wave function: Pauli principle pushes samespin electrons apart, reducing the Coulomb repulsion

I Exchange contribution Ex < 0I Fock exchange depends on the orbitals:

Ex [ϕ] = −e2

2∑i,j

∫d3r

∫d3r ′

I Exchange contribution Ex < 0

I Fock exchange depends on the orbitals:

Ex [ϕ] = −e2

2∑i,j

∫d3r

∫d3r ′

Ex [ϕ] = −e2

2∑i,j

∫d3r

∫d3r ′

Ex [ϕ] = −e2

2∑i,j

∫d3r

∫d3r ′

Ex [ϕ] = −e2

2∑i,j

∫d3r

∫d3r ′

Ex [ϕ] = −e2

2∑i,j

∫d3r

∫d3r ′

Correlation energy

I Hartree-Fock: a single Slater determinant

I The real electronic repulsion is more correlated than themean-field description

I In quantum chemistry and many-body perturbation theory wecall “correlation” the contributions beyond a single Slaterdeterminant

I Coulomb energy: Hartree + Exchange + Correlation

〈Ψ|U |Ψ〉 = EH + Ex + Ec

Tipically in atoms and molecules, Ec ≈ 0.1Ex ...

Correlation energy

I Hartree-Fock: a single Slater determinantI The real electronic repulsion is more correlated than the

mean-field description

I In quantum chemistry and many-body perturbation theory wecall “correlation” the contributions beyond a single Slaterdeterminant

〈Ψ|U |Ψ〉 = EH + Ex + Ec

Correlation energy

mean-field descriptionI In quantum chemistry and many-body perturbation theory we

call “correlation” the contributions beyond a single Slaterdeterminant

〈Ψ|U |Ψ〉 = EH + Ex + Ec

Correlation energy

〈Ψ|U |Ψ〉 = EH + Ex + Ec

Correlation energy

〈Ψ|U |Ψ〉 = EH + Ex + Ec

Outline

5 Kohn-Sham Scheme

Thomas-Fermi model (1927)

In 1927, Llewellyn Thomas and EnricoFermi proposed independently a methodbased on semiclassical and statisticalideas to determine the ground-state ofmany-electron atoms. The N electronsare treated as a Fermi gas on its groundstate, confined by an effective potentialthat goes to zero in the infinty.

Thomas-Fermi model (1927)

In 1927, Llewellyn Thomas and EnricoFermi proposed independently a methodbased on semiclassical and statisticalideas to determine the ground-state ofmany-electron atoms. The N electronsare treated as a Fermi gas on its groundstate, confined by an effective potentialthat goes to zero in the infinty.

Thomas-Fermi model

The total energy E in the TF model is the sum of the Fermi gasenergy

EF = ~2

2m(3π2 n

with the effective potential

Veff (r) = Vext(r) + e2∫

d3r ′ n(r′)|r− r′| .

E = EF + Veff

The density can be written in terms of these contributions

n(r) = 13π2

)3/2(E −Vef (r))3/2 ,

Thomas-Fermi model

With Poisson’s equation,

∇2ϕ(r) = 4πe n(r) , ϕ(r) = −Vef (r)/e

we can obtain a self-consistent solution, iterating the TF densityand the Poisson’s equation density.

n(r) = 13π2

)3/2(E −Vef (r))3/2 ,

Thomas-Fermi model

I Ground-state energy of atoms via Fermi gas

I Approximates the kinetic energy of atoms by use of thedensity of the uniform electron gas

Rewriting the total energy as a functional of the density

E = TTF [n] +∫

d3r n(r)vext(r) + e2

xd3rd3r ′ n(r)n(r′)

|r− r′|

we obtain the Thomas-Fermi approximation to the kinetic energy,

TTF [n] =∫

ts(n(r))n(r)d3r = 310

m(3π2)2/3

∫d3r n5/3(r)

known also as the first local density approximation (LDA).

Thomas-Fermi model

I Ground-state energy of atoms via Fermi gasI Approximates the kinetic energy of atoms by use of the

density of the uniform electron gas

E = TTF [n] +∫

|r− r′|

TTF [n] =∫

ts(n(r))n(r)d3r = 310

m(3π2)2/3

∫d3r n5/3(r)

Thomas-Fermi model

E = TTF [n] +∫

|r− r′|

TTF [n] =∫

ts(n(r))n(r)d3r = 310

m(3π2)2/3

∫d3r n5/3(r)

Thomas-Fermi model

E = TTF [n] +∫

|r− r′|

TTF [n] =∫

ts(n(r))n(r)d3r = 310

m(3π2)2/3

∫d3r n5/3(r)

Thomas-Fermi model

I Qualitative trends of total energies

I No chemical binding, and exact only in the Z →∞ limit

I Fermi energy sphere is purely kinetic

I Absence of quantum correlations (xc)

Thomas-Fermi model

I Qualitative trends of total energies

I No chemical binding, and exact only in the Z →∞ limit

I Fermi energy sphere is purely kinetic

I Absence of quantum correlations (xc)

Dirac’s approximation (1929)

I Dirac derives the exchange energy density of the electron gas

ELDAx [n] = −Axe2/3

∫d3rn4/3(r) ,

I Thomas-Fermi-Dirac modelE ≈ ETFD[n] = TLDA

s [n] + UH [n] + ELDAx + V [n] .

I First density functionals

∫d3rn4/3(r) ,

s [n] + UH [n] + ELDAx + V [n] .

∫d3rn4/3(r) ,

s [n] + UH [n] + ELDAx + V [n] .

Timeline

Remark

“In the 20s and early 30s there was a flush ofsuccesses in establishing the ability of quantummechanics to describe the simplest moleculesaccurately:

the Born-Oppenheimer approximation,the nature of chemical bonding, and thefundamentals of molecular spectroscopy. But thenthe quantitative theory of molecular structure, whichwe call quantum chemistry, was stymied, by thedifficulty of solving the Schrodinger equation formolecules. The senior chemical physicists of the 30spronounced the problem unsolvable.” But the youngertheoreticians in the period coming out of WWIIthought otherwise. (...) It would not be as easy ashandling an infinite periodic solid, but a number of usset to work.” Robert Parr

Remark

“In the 20s and early 30s there was a flush ofsuccesses in establishing the ability of quantummechanics to describe the simplest moleculesaccurately:

the Born-Oppenheimer approximation,the nature of chemical bonding, and thefundamentals of molecular spectroscopy. But thenthe quantitative theory of molecular structure, whichwe call quantum chemistry, was stymied, by thedifficulty of solving the Schrodinger equation formolecules. The senior chemical physicists of the 30spronounced the problem unsolvable.” But the youngertheoreticians in the period coming out of WWIIthought otherwise. (...) It would not be as easy ashandling an infinite periodic solid, but a number of usset to work.” Robert Parr

Remark

“In the 20s and early 30s there was a flush ofsuccesses in establishing the ability of quantummechanics to describe the simplest moleculesaccurately: the Born-Oppenheimer approximation,the nature of chemical bonding, and thefundamentals of molecular spectroscopy.

But thenthe quantitative theory of molecular structure, whichwe call quantum chemistry, was stymied, by thedifficulty of solving the Schrodinger equation formolecules. The senior chemical physicists of the 30spronounced the problem unsolvable.” But the youngertheoreticians in the period coming out of WWIIthought otherwise. (...) It would not be as easy ashandling an infinite periodic solid, but a number of usset to work.” Robert Parr

Remark

“In the 20s and early 30s there was a flush ofsuccesses in establishing the ability of quantummechanics to describe the simplest moleculesaccurately: the Born-Oppenheimer approximation,the nature of chemical bonding, and thefundamentals of molecular spectroscopy. But thenthe quantitative theory of molecular structure, whichwe call quantum chemistry, was stymied, by thedifficulty of solving the Schrodinger equation formolecules.

The senior chemical physicists of the 30spronounced the problem unsolvable.” But the youngertheoreticians in the period coming out of WWIIthought otherwise. (...) It would not be as easy ashandling an infinite periodic solid, but a number of usset to work.” Robert Parr

Remark

“In the 20s and early 30s there was a flush ofsuccesses in establishing the ability of quantummechanics to describe the simplest moleculesaccurately: the Born-Oppenheimer approximation,the nature of chemical bonding, and thefundamentals of molecular spectroscopy. But thenthe quantitative theory of molecular structure, whichwe call quantum chemistry, was stymied, by thedifficulty of solving the Schrodinger equation formolecules. The senior chemical physicists of the 30spronounced the problem unsolvable.”

But the youngertheoreticians in the period coming out of WWIIthought otherwise. (...) It would not be as easy ashandling an infinite periodic solid, but a number of usset to work.” Robert Parr

Remark

“In the 20s and early 30s there was a flush ofsuccesses in establishing the ability of quantummechanics to describe the simplest moleculesaccurately: the Born-Oppenheimer approximation,the nature of chemical bonding, and thefundamentals of molecular spectroscopy. But thenthe quantitative theory of molecular structure, whichwe call quantum chemistry, was stymied, by thedifficulty of solving the Schrodinger equation formolecules. The senior chemical physicists of the 30spronounced the problem unsolvable.” But the youngertheoreticians in the period coming out of WWIIthought otherwise. (...) It would not be as easy ashandling an infinite periodic solid, but a number of usset to work.”

Robert Parr

Remark

“In the 20s and early 30s there was a flush ofsuccesses in establishing the ability of quantummechanics to describe the simplest moleculesaccurately: the Born-Oppenheimer approximation,the nature of chemical bonding, and thefundamentals of molecular spectroscopy. But thenthe quantitative theory of molecular structure, whichwe call quantum chemistry, was stymied, by thedifficulty of solving the Schrodinger equation formolecules. The senior chemical physicists of the 30spronounced the problem unsolvable.” But the youngertheoreticians in the period coming out of WWIIthought otherwise. (...) It would not be as easy ashandling an infinite periodic solid, but a number of usset to work.” Robert Parr

Timeline

Warning

“Density-functional theory (DFT) is asubtle, seductive, provocative business.

Its basic premise, that all the intricatemotions and pair correlations in amany-electron system are somehowcontained in the total electron densityalone, is so compelling it can drive onemad.”

Axel D. Becke

Warning

“Density-functional theory (DFT) is asubtle, seductive, provocative business.”

Its basic premise, that all the intricatemotions and pair correlations in amany-electron system are somehowcontained in the total electron densityalone, is so compelling it can drive onemad.”

Axel D. Becke

Warning

“Density-functional theory (DFT) is asubtle, seductive, provocative business.Its basic premise, that all the intricatemotions and pair correlations in amany-electron system

are somehowcontained in the total electron densityalone, is so compelling it can drive onemad.”

Axel D. Becke

Warning

“Density-functional theory (DFT) is asubtle, seductive, provocative business.Its basic premise, that all the intricatemotions and pair correlations in amany-electron system are somehowcontained in the total electron densityalone,

is so compelling it can drive onemad.”

Axel D. Becke

Warning

“Density-functional theory (DFT) is asubtle, seductive, provocative business.Its basic premise, that all the intricatemotions and pair correlations in amany-electron system are somehowcontained in the total electron densityalone, is so compelling it can drive onemad.”

Axel D. Becke

Outline

5 Kohn-Sham Scheme

Hohenberg-Kohn theorem (1964)

“There seems to be a misguided beliefthat a one-particle density can determinethe exact N-body ground state.”

Criticism on the Hohenberg-Kohn theorem, (1980).

n(r) Ψ(r1, r2, ..., rN )

n(r) · · · · · · · · · · · · Ψ(r1, r2, ..., rN )

I An N-electron system has a complex multi-dimensionalwavefunction Ψ(r1, r2, ..., rN ) that depends on thecoordinates of all of its electrons

I Integrating out N-1 degrees of freedom of |Ψ|2, we obtain theprobability of finding one electron in the volume element d3rat r

n(r) = N∫

d3r2 · · ·∫

d3rN |Ψ(r1, r2, ..., rN )|2

⇒ Can we eliminate all reference to N-electron wavefunction,working entirely in terms of 1-electron density?

n(r) = N∫

d3r2 · · ·∫

d3rN |Ψ(r1, r2, ..., rN )|2

n(r) = N∫

d3r2 · · ·∫

d3rN |Ψ(r1, r2, ..., rN )|2

The 1964-65 papers

I A 40-year-old researcher doing a sabbatical in Paris, interestedin the electronic structure of alloys, publishes two seminalpapers.

I In the first, he prooves by reductio ad absurdum, that therelation

Ψ(r1, r2, ..., rN )⇒ n(r)

can be inverted:

Given a ground-state density it is possible, in principle, tocalculate the corresponding ground-state wave function.

The 1964-65 papers

Ψ(r1, r2, ..., rN )⇒ n(r)

can be inverted:

The 1964-65 papers

Ψ(r1, r2, ..., rN )⇒ n(r)

can be inverted:

The 1964 paper

General formulation: density as basic variable

“We shall be considering a collection of arbitrary number of electrons,enclosed in a large box and moving under the influence of an externalpotential v(r) and the mutual Coulomb repulsion.”

The Hamiltonian has the form H = T + V + U , where

T = −12

∫ψ∗(r)∇2ψ(r)d3r (2)

V =∫

v(r)ψ∗(r)ψ(r)d3r (3)

U = 12

x 1|r− r′|ψ

∗(r)ψ∗(r′)ψ(r′)ψ(r)d3r d3r ′ (4)

We consider T and U “universal” terms, whereas the external potentialv(r) determines the specificities of the system/Hamiltonian.

T = −12

∫ψ∗(r)∇2ψ(r)d3r (2)

V =∫

U = 12

x 1|r− r′|ψ

∗(r)ψ∗(r′)ψ(r′)ψ(r)d3r d3r ′ (4)

T = −12

∫ψ∗(r)∇2ψ(r)d3r (2)

V =∫

U = 12

x 1|r− r′|ψ

∗(r)ψ∗(r′)ψ(r′)ψ(r)d3r d3r ′ (4)

T = −12

∫ψ∗(r)∇2ψ(r)d3r (2)

V =∫

U = 12

x 1|r− r′|ψ

∗(r)ψ∗(r′)ψ(r′)ψ(r)d3r d3r ′ (4)

Phys. Rev. 136 B864 (1964).

We shall in all that follows assume for simplicity that we are only dealingwith situations in which the ground state is nondegenerate.

We denote the electronic density in the ground state Ψ by n(r),

n(r) = 〈Ψ|n|Ψ〉 = 〈Ψ|ψ〉〈ψ|Ψ〉 (5)

which is clearly a functional of v(r) (v → H → Ψ→ n).

We shall now show that conversely v(r) is a unique functional of n(r),apart from a trivial additive constant.”

Phys. Rev. 136 B864 (1964).

n(r) = 〈Ψ|n|Ψ〉 = 〈Ψ|ψ〉〈ψ|Ψ〉 (5)

Phys. Rev. 136 B864 (1964).

n(r) = 〈Ψ|n|Ψ〉 = 〈Ψ|ψ〉〈ψ|Ψ〉 (5)

Phys. Rev. 136 B864 (1964).

The proof proceeds by reductio ad absurdum.

Assume that another potential v′(r), different by more than a constant,with ground state Ψ′ gives rise to the same density n(r).

Now clearly Ψ′ cannot be equal to Ψ since they satisfy differentSchrodinger equations:

H |Ψ 〉 = (T + U + V ) |Ψ 〉 = E |Ψ 〉H ′ |Ψ′〉 = (T + U + V ′) |Ψ′〉 = E ′ |Ψ′〉 .

Phys. Rev. 136 B864 (1964).

Hence, if we denote the Hamiltonian and ground-state energiesassociated with Ψ and Ψ′ by H , H ′ and E , E ′,

we have by the minimal property of the ground state,

E ′ = 〈Ψ′|H ′|Ψ′〉 < 〈Ψ|H ′|Ψ〉< 〈Ψ|T + U + V ′|Ψ〉< 〈Ψ|T + U + V −V + V ′|Ψ〉< 〈Ψ|H −V + V ′|Ψ〉

E ′ < E + 〈Ψ|V ′ −V |Ψ〉 ,

so that

E ′ < E +∫

d3r n(r) [v′(r)− v(r)] .

Hence, if we denote the Hamiltonian and ground-state energiesassociated with Ψ and Ψ′ by H , H ′ and E , E ′,we have by the minimal property of the ground state,

E ′ < E + 〈Ψ|V ′ −V |Ψ〉 ,

so that

E ′ < E +∫

d3r n(r) [v′(r)− v(r)] .

E ′ = 〈Ψ′|H ′|Ψ′〉

< 〈Ψ|H ′|Ψ〉< 〈Ψ|T + U + V ′|Ψ〉< 〈Ψ|T + U + V −V + V ′|Ψ〉< 〈Ψ|H −V + V ′|Ψ〉

E ′ < E + 〈Ψ|V ′ −V |Ψ〉 ,

so that

E ′ < E +∫

d3r n(r) [v′(r)− v(r)] .

E ′ = 〈Ψ′|H ′|Ψ′〉 < 〈Ψ|H ′|Ψ〉

< 〈Ψ|T + U + V ′|Ψ〉< 〈Ψ|T + U + V −V + V ′|Ψ〉< 〈Ψ|H −V + V ′|Ψ〉

E ′ < E + 〈Ψ|V ′ −V |Ψ〉 ,

so that

E ′ < E +∫

d3r n(r) [v′(r)− v(r)] .

E ′ = 〈Ψ′|H ′|Ψ′〉 < 〈Ψ|H ′|Ψ〉< 〈Ψ|T + U + V ′|Ψ〉

< 〈Ψ|T + U + V −V + V ′|Ψ〉< 〈Ψ|H −V + V ′|Ψ〉

E ′ < E + 〈Ψ|V ′ −V |Ψ〉 ,

so that

E ′ < E +∫

d3r n(r) [v′(r)− v(r)] .

E ′ = 〈Ψ′|H ′|Ψ′〉 < 〈Ψ|H ′|Ψ〉< 〈Ψ|T + U + V ′|Ψ〉< 〈Ψ|T + U + V −V + V ′|Ψ〉

< 〈Ψ|H −V + V ′|Ψ〉E ′ < E + 〈Ψ|V ′ −V |Ψ〉 ,

so that

E ′ < E +∫

d3r n(r) [v′(r)− v(r)] .

E ′ < E + 〈Ψ|V ′ −V |Ψ〉 ,

so that

E ′ < E +∫

d3r n(r) [v′(r)− v(r)] .

E ′ < E + 〈Ψ|V ′ −V |Ψ〉 ,

so that

E ′ < E +∫

d3r n(r) [v′(r)− v(r)] .

E ′ < E + 〈Ψ|V ′ −V |Ψ〉 ,

so that

E ′ < E +∫

d3r n(r) [v′(r)− v(r)] .

...E ′ < E +

∫d3r n(r) [v′(r)− v(r)] . (6)

Interchanging primed and unprimed quantities, we find in exactly thesame way that

E < E ′ +∫

d3r n(r) [v(r)− v′(r)] . (7)

Addition of 6 and 7 leads to the inconsistency

E ′ + E < E + E ′ . (8)

Thus v(r) is (to within a constant) a unique functional of n(r).

Since, in turn, v(r) fixes H we see that the full many-particle groundstate is a unique functional of n(r).

...E ′ < E +

∫d3r n(r) [v′(r)− v(r)] . (6)

E < E ′ +∫

d3r n(r) [v(r)− v′(r)] . (7)

E ′ + E < E + E ′ . (8)

...E ′ < E +

∫d3r n(r) [v′(r)− v(r)] . (6)

E < E ′ +∫

d3r n(r) [v(r)− v′(r)] . (7)

E ′ + E < E + E ′ . (8)

...E ′ < E +

∫d3r n(r) [v′(r)− v(r)] . (6)

E < E ′ +∫

d3r n(r) [v(r)− v′(r)] . (7)

E ′ + E < E + E ′ . (8)

...E ′ < E +

∫d3r n(r) [v′(r)− v(r)] . (6)

E < E ′ +∫

d3r n(r) [v(r)− v′(r)] . (7)

E ′ + E < E + E ′ . (8)

...E ′ < E +

∫d3r n(r) [v′(r)− v(r)] . (6)

E < E ′ +∫

d3r n(r) [v(r)− v′(r)] . (7)

E ′ + E < E + E ′ . (8)

...E ′ < E +

∫d3r n(r) [v′(r)− v(r)] . (6)

E < E ′ +∫

d3r n(r) [v(r)− v′(r)] . (7)

E ′ + E < E + E ′ . (8)

The original proof

Hohenberg-Kohn theorem

Since n(r) determines v(r), it gives us the full Hamiltonian.

Hence n(r) determines implicitly all properties derivable from H throughthe solution of the Schrodinger equation, such as:

I the many-body eigenstates Ψ0(r1, ..., rN ), Ψ1(r1, ..., rN ), ...I the Green’s functions G(r1t1; ...; rN tN )I the response functions χ(r, r′, ω)

I all observables 〈Ψ[n]| O |Ψ[n]〉 = O[n]

I the many-body eigenstates Ψ0(r1, ..., rN ), Ψ1(r1, ..., rN ), ...I the Green’s functions G(r1t1; ...; rN tN )I the response functions χ(r, r′, ω)I all observables 〈Ψ[n]| O |Ψ[n]〉 = O[n]

“Finally it occurred to me that for a single particle there isan explicit elementary relation between the potential v(r)and the density, n(r), of the groundstate. Taken together,these provided strong support for the conjective that thedensity n(r) completely determines the external potentialv(r). This would imply that n(r) which integrates to N, thetotal number of electrons, also determines the totalHamiltonian H and hence all properties derivable from Hand N, e.g. the wavefunction of the 17th excited state! (...)Could this be true? And how could it be decided? Couldtwo different potentials, v1(r) and v2(r), with associateddifferent groundstates Ψ1(r1, ...rN ) and Ψ2(r1, ...rN ) giverise to the same density distribution? It turned out that asimple 3-line argument, using my beloved Rayleigh Ritzvariational principle, confirmed the conjecture. It seemedsuch a remarkable result that I did not trust myself.”

W.KohnMariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguacu 46/64

Parenthesis

“The approach Hohenberg and Kohn took (...) reflects theway Kohn chose to frame the final published paper. Thereis no mention of the alloy problem or even of any desire tore-formulate the electronic structure problem for solids.Instead, the title of the HK paper is simply“Inhomogeneous electron gas” and the first line of theabstract announces that “this paper deals with the groundstate of an interacting electron gas in an external potentialv(r).” The Introduction goes on (...) HK then remind thereader about the Thomas-Fermi method.”

Andrew Zangwill, arxiv:1403:5164

Parenthesis

“The approach Hohenberg and Kohn took (...) reflects theway Kohn chose to frame the final published paper. Thereis no mention of the alloy problem or even of any desire tore-formulate the electronic structure problem for solids.Instead, the title of the HK paper is simply“Inhomogeneous electron gas” and the first line of theabstract announces that “this paper deals with the groundstate of an interacting electron gas in an external potentialv(r).” The Introduction goes on (...) HK then remind thereader about the Thomas-Fermi method”

Outline

5 Kohn-Sham Scheme

Kohn Sham (1965)

I We have seen that from a ground-state density it is possible,in principle, to calculate the corresponding wave functions andall its observables.

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

I We have seen that from a ground-state density it is possible,in principle, to calculate the corresponding wave functions andall its observables.

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

1964 paper’s concerns

I HK mentions an universal functional (F [n] = T [n] + U [n])and recognize the necessity to determine it

1964 paper’s concerns

I They distinguish the Hartree term (classical Coulomb energy)

and separate it from the functional to be approximated

I HK knew that the Thomas-Fermi model follows from (15) byapproximating the kinetic energy

E =∫

d3r n(r)vext(r) + 12

|r− r′| + TTF [n]

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.”

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.”

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.”

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.”

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.

After THK

I First Kohn and Sham observed that in the Hartree method,each electron moves independently in an effective potentialwhich does not recognize the individual identity of the otherelectrons.(

− ~2

2m∇2 + veff (r)

I The kinetic energy for independent (non-interacting) electronsis:

TS [n] =N∑

∫ϕ∗(r)

[− ~2

2m∇2]ϕ(r)d3r

(S: single-particle)I OBS: The true kinetic energy of an interacting system is not

TS , we miss a term that describes the correlated motion

− ~2

2m∇2 + veff (r)

TS [n] =N∑

∫ϕ∗(r)

[− ~2

2m∇2]ϕ(r)d3r

− ~2

2m∇2 + veff (r)

TS [n] =N∑

∫ϕ∗(r)

[− ~2

2m∇2]ϕ(r)d3r

− ~2

2m∇2 + veff (r)

TS [n] =N∑

∫ϕ∗(r)

[− ~2

2m∇2]ϕ(r)d3r

− ~2

2m∇2 + veff (r)

TS [n] =N∑

∫ϕ∗(r)

[− ~2

2m∇2]ϕ(r)d3r

Now Kohn and Sham knew how to proceed with the generalmany-electron problem.

They wrote the total energy as

E [n] = TS [n]+∫

vext(r) n(r) dr+e2

∫ ∫ n(r) n(r′)|r− r′| dr dr′+Exc[n] ,

(9)where they defined the functional Exc.

Exc[n] should account for the quantum many-body corrections tothe kinetic term and the Coulomb contributions missing, exchangeand correlation.

E [n] = TS [n]+∫

vext(r) n(r) dr+e2

∫ ∫ n(r) n(r′)|r− r′| dr dr′+Exc[n] ,

E [n] = TS [n]+∫

vext(r) n(r) dr+e2

∫ ∫ n(r) n(r′)|r− r′| dr dr′+Exc[n] ,

E [n] = TS [n]+∫

vext(r) n(r) dr+e2

∫ ∫ n(r) n(r′)|r− r′| dr dr′+Exc[n] ,

I Minimizing the energy functional with respect to the density,keeping fixed the number of particles via the Lagrangemultiplier µ, we obtain

µ = veff (r) + δTs[n]δn ,

veff (r) = vext(r) + e2∫

d3r n(r)|r− r′| + vxc(r)

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

andvxc(r) = δExc[n]

This equation

veff(r) = vext(r) + vH (r) + vxc(r) , (10)

has the same form of the potential for non-interacting particles.I Therefore, the minimizing density n(r) is given by solving the

single-particle equation(− ~2

2m∇2 + veff (r)

n(r) =N∑i|ϕi(r)|2

These self-consistent equations are now called the Kohn-Shamequations.

This equation

2m∇2 + veff (r)

This equation

2m∇2 + veff (r)

Kohn-Sham approach

I Auxiliary, ficticious 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 Constraint on the ground state density

nKS0 (r) =

occ∑kϕ∗k(r)φk(r)

== n0(r)

Kohn-Sham approach

−~2∇2

nKS0 (r) =

== n0(r)

Kohn-Sham approach

−~2∇2

nKS0 (r) =

== n0(r)

Kohn-Sham approach

−~2∇2

nKS0 (r) =

== n0(r)

Kohn-Sham approach

I Mapping of an interacting system in a non-interacting systemI Typically a local potencial ((r))I Correlated via vxc

Kohn-Sham approach

Illustration

Interacting

(complicated)

Ficticious non-interacting

under effective field

Illustration

Interacting

(complicated)

Illustration

Interacting

(complicated)

Illustration

Interacting

(complicated)

Self-consistent cycle

Our tutorial

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)

Our tutorial

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)

Acknowledgements (I)

Klaus Capelle, UFABC, Brazil

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

Sam Trickey, QTP-Univ.Florida

Caio Lewenkopf, UFF, Brazil

Acknowledgements (I)

Klaus Capelle, UFABC, Brazil

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

Sam Trickey, QTP-Univ.Florida

Caio Lewenkopf, UFF, Brazil

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)

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).

Acknowledgements

To all ENFMC organizers and FAPERJ.

Thank you for your attention!

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

introduction to dft part 1

Education