density functional theory - quantum chemistry...

Density Functional Theory

Łukasz Rajchel

Interdisciplinary Centerfor Mathematical and Computational Modeling

[email protected]

Warsaw, 2010

Lecture available online:http://tiger.chem.uw.edu.pl/staff/lrajchel/

Questions, comments, mistakes in the Lecture — don’t hesitate to write me!

Outline of the Lecture

Part I

1 DFT — A Real Celebrity2 Preliminaries

Basic Concepts of Quantum ChemistryElectronic DistributionApproximate Methods

3 Hartree-FockVariation in HFEquationsCorrelation and exchangeSelf-Interaction in HF

4 Fermi and Coulomb HolesDefinitions

Part II

5 Density and EnergyRemarks and ProblemsHistorical ModelsResults

6 Hohenberg-Kohn TheoremsDefinitionsThe TheoremsRepresentability of the Density

7 Kohn-Sham ApproachIntroductory RemarksKS Determinant and KS Energy

8 xc FunctionalsIs There a Road Map?Adiabatic ConnectionKohn-Sham Machinery

Part III

9 Approximate xc FunctionalsIntroductionLDA and LSDGGAHybrid FunctionalsBeyond GGAProblems of Approximate Functionals

Part I

The Road to DFT. Recapitulation of Basic Concepts ofQuantum Chemistry

Outline of the Talk

1 DFT — A Real Celebrity

2 Preliminaries

3 Hartree-Fock

4 Fermi and Coulomb Holes

DFT — A Real Celebrity

DFT vs. CC vs. nano

density functional theorycoupled clusternanotechnology

Number of publications returned by the Web of Science for the respectivetopics

Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 6 / 101

Outline of the Talk

2 PreliminariesBasic Concepts of Quantum ChemistryElectronic DistributionApproximate Methods

3 Hartree-Fock

Preliminaries Basic Concepts of Quantum Chemistry

Symbols used in the lecture:

system composed of N electrons and M nuclei,

atomic units used (melectron = 1, ~ = 1, e = 1),

xiyizi

, Rα =

YαZα

— positions electrons, nuclei,

rij = |ri − rj | — interelectron distance,

Zα — nuclear charge,

q = (r;σ) — spatial (r ∈ R3) and spin (σ = ±12) variable,

∆ = ∇2 = ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2— Laplacian,

Pij — operator exchanging particles i and j (permutator).

xiyizi

, Rα =

YαZα

∆ = ∇2 = ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2— Laplacian,

xiyizi

, Rα =

YαZα

∆ = ∇2 = ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2— Laplacian,

xiyizi

, Rα =

YαZα

∆ = ∇2 = ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2— Laplacian,

xiyizi

, Rα =

YαZα

∆ = ∇2 = ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2— Laplacian,

xiyizi

, Rα =

YαZα

∆ = ∇2 = ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2— Laplacian,

xiyizi

, Rα =

YαZα

∆ = ∇2 = ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2— Laplacian,

xiyizi

, Rα =

YαZα

∆ = ∇2 = ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2— Laplacian,

Born-Oppenheimer approximation

We can separate nuclear and electronic motions because

mproton ≈ mneutron ≈ 1836×melectron.

We restrict our attention to electronic Hamiltonian only,

Hel = T︸︷︷︸kinetic energy

+ Vne︸︷︷︸nuclear-electron attraction

+ Vee.︸︷︷︸electron-electron repulsion

T = −12

N∑i=1

∆ri , Vne = −N∑i=1

M∑α=1

Zα|ri −Rα|

, Vee =N−1∑i=1

N∑j=i+1

r−1ij .

Clamped nuclei ⇒ nuclear-nuclear repulsion is a constant, so we can skip itnow, but remember to add it to the result:

H = Hel + Vnn = Hel +M−1∑α=1

M∑β=α+1

ZαZβRαβ

Schrodinger equation

Spectrum of the Hamiltonian — wavefunctions and energies for theelectronic states:

Hψk(q1; q2; . . . ; qN ) = Ekψk(q1; q2; . . . ; qN ).

The Schrodinger equation is an eigeinequation in which the input is theHamiltonian itself. As output, we obtain ψk’s and Ek’s. Solving theSchrodinger equation is not a trivial issue → analytical solution knownonly for H and H-like systems.From now on we are interested in the ground state only:

ψk → ψ0 .

The Schrodinger equation is an eigeinequation in which the input is theHamiltonian itself. As output, we obtain ψk’s and Ek’s.

Solving theSchrodinger equation is not a trivial issue → analytical solution knownonly for H and H-like systems.From now on we are interested in the ground state only:

ψk → ψ0 .

The Schrodinger equation is an eigeinequation in which the input is theHamiltonian itself. As output, we obtain ψk’s and Ek’s. Solving theSchrodinger equation is not a trivial issue → analytical solution knownonly for H and H-like systems.

From now on we are interested in the ground state only:

ψk → ψ0 .

The Schrodinger equation is an eigeinequation in which the input is theHamiltonian itself. As output, we obtain ψk’s and Ek’s. Solving theSchrodinger equation is not a trivial issue → analytical solution knownonly for H and H-like systems.From now on we are interested in the ground state only:

ψk → ψ0 .

Wavefunction

ψ: a complex function of 4N variables, no physical meaning. But:

P (q1; q2; . . . ; qN ) = |ψ(q1; q2; . . . ; qN )|2

is the density probability of finding the electrons at positionsq1,q2, . . . ,qN .

Indistinguishable particles → the exchange of any two particles can’tchange the density probabily, so

PijP (q1; . . . ; qi; . . . ; qj ; . . . ; qN ) = P (q1; . . . ; qj ; . . . ; qi; . . . ; qN ).

Wavefunction

P (q1; q2; . . . ; qN ) = |ψ(q1; q2; . . . ; qN )|2

is the density probability of finding the electrons at positionsq1,q2, . . . ,qN .Indistinguishable particles → the exchange of any two particles can’tchange the density probabily, so

more on Dirac braket notation

ψ eigenfunction of A ⇒ Aψ = Aψ ⇒ 〈A〉 = A.Energy (E) operator → Hamiltonian (H). In practice we don’t know ψsatisfying Schrodinger equation, Hψ = Eψ, we only know H!But there are cures for that (more — later on) . . .

Expectation values

〈A〉 = 〈ψ|A|ψ〉

ψ eigenfunction of A ⇒ Aψ = Aψ ⇒ 〈A〉 = A.

Energy (E) operator → Hamiltonian (H). In practice we don’t know ψsatisfying Schrodinger equation, Hψ = Eψ, we only know H!But there are cures for that (more — later on) . . .

Expectation values

〈A〉 = 〈ψ|A|ψ〉

ψ eigenfunction of A ⇒ Aψ = Aψ ⇒ 〈A〉 = A.Energy (E) operator → Hamiltonian (H). In practice we don’t know ψsatisfying Schrodinger equation, Hψ = Eψ, we only know H!

But there are cures for that (more — later on) . . .

Expectation values

〈A〉 = 〈ψ|A|ψ〉

ψ eigenfunction of A ⇒ Aψ = Aψ ⇒ 〈A〉 = A.Energy (E) operator → Hamiltonian (H). In practice we don’t know ψsatisfying Schrodinger equation, Hψ = Eψ, we only know H!But there are cures for that (more — later on) . . .

One-matrix

Density function (one-matrix) is a generalization of the electronic density:

ρ(r; r′) = N∑

σ1,σ2,...,σN

ψ(r;σ1; q2; . . . ; qN )×

× ψ∗(r′;σ1; q2; . . . ; qN ) d3r2 . . . d3rN .

We need it to calculate expectation values of operators which are notsimply multiplicative, e.g. kinetic energy:

T = 〈ψ|T |ψ〉 =∑

σ1,σ2,...,σN

ψ∗(q1; . . . ; qN )×

N∑i=1

)ψ(q1; . . . ; qN ) d3r1 . . . d

= −12

[∆rρ(r; r′)

]r′=r

Electronic density is the diagonal part of one-matrix: ρ(r) = ρ(r; r).

One-matrix

ρ(r; r′) = N∑

σ1,σ2,...,σN

ψ(r;σ1; q2; . . . ; qN )×

× ψ∗(r′;σ1; q2; . . . ; qN ) d3r2 . . . d3rN .

T = 〈ψ|T |ψ〉 =∑

σ1,σ2,...,σN

ψ∗(q1; . . . ; qN )×

N∑i=1

)ψ(q1; . . . ; qN ) d3r1 . . . d

= −12

[∆rρ(r; r′)

]r′=r

Electronic density is the diagonal part of one-matrix: ρ(r) = ρ(r; r).

One-matrix

ρ(r; r′) = N∑

σ1,σ2,...,σN

ψ(r;σ1; q2; . . . ; qN )×

× ψ∗(r′;σ1; q2; . . . ; qN ) d3r2 . . . d3rN .

T = 〈ψ|T |ψ〉 =∑

σ1,σ2,...,σN

ψ∗(q1; . . . ; qN )×

N∑i=1

)ψ(q1; . . . ; qN ) d3r1 . . . d

= −12

[∆rρ(r; r′)

]r′=r

Electronic density is the diagonal part of one-matrix: ρ(r) = ρ(r; r).Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 16 / 101

Pair density

γ(r1; r2) = N(N − 1)×

σ1,σ2,...,σN

|ψ(r1;σ1; r2;σ2; q3; . . . ; qN )|2 d3r3 . . . d3rN :

gives the probabilty distribution of any of two electrons beingat r1 and r2 at the same time.normalized to the number of non-distinct pairs, N(N − 1).yields density if integrated over one variable:∫

γ(r1; r2) d3r2 = (N − 1)ρ(r1).

is a measure of electron correlation, i.e. mutual interaction betweenelectrons.but don’t confuse γ(r1; r2) (pair density) with ρ(r; r′) (one-matrix).

Pair density

γ(r1; r2) = N(N − 1)×

σ1,σ2,...,σN

|ψ(r1;σ1; r2;σ2; q3; . . . ; qN )|2 d3r3 . . . d3rN :

gives the probabilty distribution of any of two electrons beingat r1 and r2 at the same time.

normalized to the number of non-distinct pairs, N(N − 1).yields density if integrated over one variable:∫

γ(r1; r2) d3r2 = (N − 1)ρ(r1).

Pair density

γ(r1; r2) = N(N − 1)×

σ1,σ2,...,σN

|ψ(r1;σ1; r2;σ2; q3; . . . ; qN )|2 d3r3 . . . d3rN :

gives the probabilty distribution of any of two electrons beingat r1 and r2 at the same time.normalized to the number of non-distinct pairs, N(N − 1).

yields density if integrated over one variable:∫R3

γ(r1; r2) d3r2 = (N − 1)ρ(r1).

Pair density

γ(r1; r2) = N(N − 1)×

σ1,σ2,...,σN

|ψ(r1;σ1; r2;σ2; q3; . . . ; qN )|2 d3r3 . . . d3rN :

γ(r1; r2) d3r2 = (N − 1)ρ(r1).

Pair density

γ(r1; r2) = N(N − 1)×

σ1,σ2,...,σN

|ψ(r1;σ1; r2;σ2; q3; . . . ; qN )|2 d3r3 . . . d3rN :

γ(r1; r2) d3r2 = (N − 1)ρ(r1).

is a measure of electron correlation, i.e. mutual interaction betweenelectrons.

but don’t confuse γ(r1; r2) (pair density) with ρ(r; r′) (one-matrix).

Pair density

γ(r1; r2) = N(N − 1)×

σ1,σ2,...,σN

|ψ(r1;σ1; r2;σ2; q3; . . . ; qN )|2 d3r3 . . . d3rN :

γ(r1; r2) d3r2 = (N − 1)ρ(r1).

Preliminaries Approximate Methods

Variational principle

We take any function ψ depending on the same variables that the functionψ we are looking for and satisfying the usual boundary and antisymmetryconditions. Then

〈ψ|H|ψ〉 = E ≥ E0 ,

where E0 — ground-state energy.

This is the recipe for the quest for our wavefunction — find the functionyielding the smallest energy.Schematically,

E0 = minψ→N

〈ψ|H|ψ〉 .

〈ψ|H|ψ〉 = E ≥ E0 ,

where E0 — ground-state energy.This is the recipe for the quest for our wavefunction — find the functionyielding the smallest energy.

Schematically,E0 = min

ψ→N〈ψ|H|ψ〉 .

〈ψ|H|ψ〉 = E ≥ E0 ,

where E0 — ground-state energy.This is the recipe for the quest for our wavefunction — find the functionyielding the smallest energy.Schematically,

E0 = minψ→N

〈ψ|H|ψ〉 .

Outline of the Talk

2 Preliminaries

3 Hartree-FockVariation in HFEquationsCorrelation and exchangeSelf-Interaction in HF

Hartree-Fock Variation in HF

Hartree-Fock wavefunction

The wavefunction assumed to be a single determinant built of one-electronfunctions (spinorbitals, φi):

ψHF(q1; q2; . . . ; qN ) =1√N !

∣∣∣∣∣∣∣∣∣φ1(q1) φ2(q1) . . . φN (q1)φ1(q2) φ2(q2) . . . φN (q2)

......

. . ....

φ1(qN ) φ2(qN ) . . . φN (qN )

∣∣∣∣∣∣∣∣∣ .

Constraint: spinorbitals orthonormal, i.e.

〈φi|φj〉 = δij =

1, i = j,

0, i 6= j.

Then, the wavefunction is antisymmetrical: P12ψHF = −ψHF, andnormalised: 〈ψHF|ψHF〉 = 1.

ψHF(q1; q2; . . . ; qN ) =1√N !

∣∣∣∣∣∣∣∣∣φ1(q1) φ2(q1) . . . φN (q1)φ1(q2) φ2(q2) . . . φN (q2)

......

. . ....

φ1(qN ) φ2(qN ) . . . φN (qN )

∣∣∣∣∣∣∣∣∣ .Constraint: spinorbitals orthonormal, i.e.

1, i = j,

0, i 6= j.

ψHF(q1; q2; . . . ; qN ) =1√N !

∣∣∣∣∣∣∣∣∣φ1(q1) φ2(q1) . . . φN (q1)φ1(q2) φ2(q2) . . . φN (q2)

......

. . ....

φ1(qN ) φ2(qN ) . . . φN (qN )

∣∣∣∣∣∣∣∣∣ .Constraint: spinorbitals orthonormal, i.e.

1, i = j,

0, i 6= j.

Orbitals and spinorbitals

Spinorbitals and orbitals in the closed-shell restricted HF (RHF):

spinorbital = orbital× spin function.φ2i−1(r;σ) = ϕi(r)α(σ)φ2i(r;σ) = ϕi(r)β(σ)

Density function for ψ = ψHF:

ρ(r; r′) = 2N/2∑i=1

ϕi(r)ϕ∗i (r′).

Orbitals and spinorbitals

Spinorbitals and orbitals in the closed-shell restricted HF (RHF):

spinorbital = orbital× spin function.φ2i−1(r;σ) = ϕi(r)α(σ)φ2i(r;σ) = ϕi(r)β(σ)

Density function for ψ = ψHF:

ρ(r; r′) = 2N/2∑i=1

ϕi(r)ϕ∗i (r′).

Optimization of Hartree-Fock energy

HF energy of a system:

EHF = 〈ψHF|H|ψHF〉 ,

and being a very diligent audience, we remember very well that

H = T + Vne + Vee + Vnn.

HF energy in terms of density and density function:

EHF[ρ] = T [ρ]︸︷︷︸kineticenergy

+ Vne[ρ]︸︷︷︸nuclear-electron

attraction

+ J [ρ]︸︷︷︸classical electrostatic

electron-electron repulsion

− K[ρ]︸︷︷︸non-classical electron-electron

exchange interaction

+ Vnn︸︷︷︸nuclear-nuclear

repulsion (constant)

EHF = 〈ψHF|H|ψHF〉 ,and being a very diligent audience, we remember very well that

attraction

what is a functional?

how do these terms look like?

attraction

Goal: minimize HF energy varying the orbitals ϕi (spatial parts of spinor-bitals φi) while keeping them orthonormal.

attraction

Result: HF equations for the best orbitals, i.e. orbitals yielding minimumHF energy.

Hartree-Fock Equations

Fock operator

HF equations for best orbitals:

fϕi = εiϕi.

Fock operator:

f(r) = −12

∆r + vne(r) + vHF(r).

Fock operator

fϕi = εiϕi.

Fock operator:

f(r) = −12

Fock operator

fϕi = εiϕi.

Fock operator:

f(r) = −12

Nuclear potential:

vne(r) = −M∑α=1

Zα|r−Rα|

Fock operator

fϕi = εiϕi.

Fock operator:

f(r) = −12

HF potential: the average repulsive potential experienced by the electronfrom to the remaining N − 1 electrons.

Fock operator

fϕi = εiϕi.

Fock operator:

f(r) = −12

The complicated two-electron repulsion operator r−1ij in the Hamiltonian is

replaced by the simple one-electron operator vHF(r), but now the electron-electron repulsion is taken into account only in an average way.

Coulomb and echange operators

HF potential: Coulomb− exchange,

vHF = ︸︷︷︸Coulomb

− k︸︷︷︸exchange

(r)f(r) =(∫

ρ(r′)|r− r′|

d3r′)f(r) :

the classical electrostatic interaction between electron at postion rwith the charge density ρ.

its action on f(r) requires the knowledge of f value at r only ⇒(r) is local .

(r)f(r) =(∫

ρ(r′)|r− r′|

d3r′)f(r) :

(r)f(r) =(∫

ρ(r′)|r− r′|

d3r′)f(r) :

k(r)f(r) =12

ρ(r; r′)|r− r′|

f(r′) d3r′ :

non-classical and entirely due to the antisymmetry of the Slaterdeterminant.

its action on f(r) requires the knowledge of f value at all points in

space (because of the integration) ⇒ k(r) is non-local .

k(r)f(r) =12

ρ(r; r′)|r− r′|

f(r′) d3r′ :

k(r)f(r) =12

ρ(r; r′)|r− r′|

f(r′) d3r′ :

Hartree-Fock Correlation and exchange

Pair densities and exchange in Hartree-Fock

HF pair density function for the electron with opposite spins:

γαβ(r1; r2) = ρα(r1)ρβ(r2),

and for the electron with the same spin:

γαα(r1; r2) = ρα(r1)ρα(r2)− ρα(r1; r2)ρα(r2; r1),

Conclusions:

probability density of two electrons with opposite spins occupyingsome regions in space is just the product of probability densities ofeach of the two events occuring independently: no correlation.

but probability density of two electrons with same spins occupyingsome regions in space is correlated and that density vanishes forr2 → r1 — this prevents the two electrons with like spins occupy thesame region of space: it’s called Fermi correlation or exchange.

Conclusions:

Electron correlation

ψHF is the best function within the one-electron approximation (eachelectron is ascribed to a spinorbital).

But it is not the true wavefunction of the system (i.e. the onefrom Hψ = Eψ), and in most cases we don’t know that true function.Thus, due to the variational principle, HF energy of the system isalways higher than its true energy, EHF > E.The error introduced throgh the HF scheme is called the correlationenergy: Ecor = E − EHF, and is always negative. We assume HFmethod misses any electron correlation (though it properly accountsfor the exchange).

6energy

6?−Ecor

ψHF is the best function within the one-electron approximation (eachelectron is ascribed to a spinorbital).But it is not the true wavefunction of the system (i.e. the onefrom Hψ = Eψ), and in most cases we don’t know that true function.

Thus, due to the variational principle, HF energy of the system isalways higher than its true energy, EHF > E.The error introduced throgh the HF scheme is called the correlationenergy: Ecor = E − EHF, and is always negative. We assume HFmethod misses any electron correlation (though it properly accountsfor the exchange).

6energy

6?−Ecor

ψHF is the best function within the one-electron approximation (eachelectron is ascribed to a spinorbital).But it is not the true wavefunction of the system (i.e. the onefrom Hψ = Eψ), and in most cases we don’t know that true function.Thus, due to the variational principle, HF energy of the system isalways higher than its true energy, EHF > E.

The error introduced throgh the HF scheme is called the correlationenergy: Ecor = E − EHF, and is always negative. We assume HFmethod misses any electron correlation (though it properly accountsfor the exchange).

6energy

6?−Ecor

ψHF is the best function within the one-electron approximation (eachelectron is ascribed to a spinorbital).But it is not the true wavefunction of the system (i.e. the onefrom Hψ = Eψ), and in most cases we don’t know that true function.Thus, due to the variational principle, HF energy of the system isalways higher than its true energy, EHF > E.The error introduced throgh the HF scheme is called the correlationenergy: Ecor = E − EHF, and is always negative. We assume HFmethod misses any electron correlation (though it properly accountsfor the exchange).

6energy

6?−Ecor

ψHF is the best function within the one-electron approximation (eachelectron is ascribed to a spinorbital).But it is not the true wavefunction of the system (i.e. the onefrom Hψ = Eψ), and in most cases we don’t know that true function.Thus, due to the variational principle, HF energy of the system isalways higher than its true energy, EHF > E.The error introduced throgh the HF scheme is called the correlationenergy: Ecor = E − EHF, and is always negative. We assume HFmethod misses any electron correlation (though it properly accountsfor the exchange).

6energy

6?−Ecor

Hartree-Fock Self-Interaction in HF

Self-interaction problem

Hydrogen atom has only one electron, so obviously there is noelectron-electron interaction of any kind.

Now, we all remember (because we are a diligent audience!)that EHF[ρ] = T [ρ] + Vne[ρ] + J [ρ] + Ex[ρ] + Vnn.

Thus, the electron-electron repulsion in HF is J [ρ] + Ex[ρ]: Coulomband exchange. What happens in hydrogen atom in HF picture?

Let’s look at the numbers (from Molpro, QChem, Gaussian, . . . ):

T [ρ] + Vne[ρ] = −0.49999,

J [ρ] = 0.31250,

Ex[ρ] = −0.31250,

so, J [ρ] + Ex[ρ] = 0.

There is no self-interaction in HF! The unphysical self-interaction ofelectron with itself contained in J [ρ] is removed by Ex[ρ].

T [ρ] + Vne[ρ] = −0.49999,

J [ρ] = 0.31250,

Ex[ρ] = −0.31250,

so, J [ρ] + Ex[ρ] = 0.

T [ρ] + Vne[ρ] = −0.49999,

J [ρ] = 0.31250,

Ex[ρ] = −0.31250,

so, J [ρ] + Ex[ρ] = 0.

T [ρ] + Vne[ρ] = −0.49999,

J [ρ] = 0.31250,

Ex[ρ] = −0.31250,

so, J [ρ] + Ex[ρ] = 0.

T [ρ] + Vne[ρ] = −0.49999,

J [ρ] = 0.31250,

Ex[ρ] = −0.31250,

so, J [ρ] + Ex[ρ] = 0.

T [ρ] + Vne[ρ] = −0.49999,

J [ρ] = 0.31250,

Ex[ρ] = −0.31250,

so, J [ρ] + Ex[ρ] = 0.

T [ρ] + Vne[ρ] = −0.49999,

J [ρ] = 0.31250,

Ex[ρ] = −0.31250,

so, J [ρ] + Ex[ρ] = 0.

T [ρ] + Vne[ρ] = −0.49999,

J [ρ] = 0.31250,

Ex[ρ] = −0.31250,

so, J [ρ] + Ex[ρ] = 0.

T [ρ] + Vne[ρ] = −0.49999,

J [ρ] = 0.31250,

Ex[ρ] = −0.31250,

so, J [ρ] + Ex[ρ] = 0.

Outline of the Talk

2 Preliminaries

3 Hartree-Fock

4 Fermi and Coulomb HolesDefinitions

Fermi and Coulomb Holes Definitions

Electron-electron repulsion

The source of all troubles and misfortunes (but also a nice grant-generator)in quantum chemistry: electron-electron repulsion operator,

Vee =N−1∑i=1

N∑j=i+1

Exact wavefunction ψ → exact pair density γ → exact e-e repulsion:

Eee = 〈ψ|Vee|ψ〉 =12

γ(r1; r2)r12

d3r1 d3r2.

In HF the e-e repulsion is

J [ρ] + Ex[ρ] =12

(ρ(r1)ρ(r2)

r12− 1

2ρ(r1; r2)ρ(r2; r1)

)d3r1 d

Vee =N−1∑i=1

N∑j=i+1

γ(r1; r2)r12

d3r1 d3r2.

J [ρ] + Ex[ρ] =12

(ρ(r1)ρ(r2)

r12− 1

2ρ(r1; r2)ρ(r2; r1)

)d3r1 d

Vee =N−1∑i=1

N∑j=i+1

γ(r1; r2)r12

d3r1 d3r2.

J [ρ] + Ex[ρ] =12

(ρ(r1)ρ(r2)

r12− 1

2ρ(r1; r2)ρ(r2; r1)

)d3r1 d

Correlation factor

So, in HF we reduce the Devil (e-e repulsion) as:

γHF(r1; r2) = ρ(r1)ρ(r2)− 12ρ(r1; r2)ρ(r2; r1).

Clearly, in HF there’s some correlation included, otherwise γ(r1; r2)would simply decompose to ρ(r1)ρ(r2). We already know HF accounts forthe Fermi correlation.Let’s generalize and write

γ(r1; r2) = ρ(r1)ρ(r2)(

1 + f(r1; r2)),

thus f(r1; r2) = 0 refers to uncorrelated case. f(r1; r2) — correlationfactor.For HF we easily get

f(r1; r2) = −12ρ(r1; r2)ρ(r2; r1)

ρ(r1)ρ(r2).

Correlation factor

Clearly, in HF there’s some correlation included, otherwise γ(r1; r2)would simply decompose to ρ(r1)ρ(r2). We already know HF accounts forthe Fermi correlation.

Let’s generalize and write

γ(r1; r2) = ρ(r1)ρ(r2)(

1 + f(r1; r2)),

f(r1; r2) = −12ρ(r1; r2)ρ(r2; r1)

ρ(r1)ρ(r2).

Correlation factor

γ(r1; r2) = ρ(r1)ρ(r2)(

1 + f(r1; r2)),

thus f(r1; r2) = 0 refers to uncorrelated case. f(r1; r2) — correlationfactor.

For HF we easily get

f(r1; r2) = −12ρ(r1; r2)ρ(r2; r1)

ρ(r1)ρ(r2).

Correlation factor

γ(r1; r2) = ρ(r1)ρ(r2)(

1 + f(r1; r2)),

f(r1; r2) = −12ρ(r1; r2)ρ(r2; r1)

ρ(r1)ρ(r2).

Conditional probability

The probability of A under the condition B:

P (A|B) =P (A ∩B)P (B)

P (A ∩B) is the probability of both events together.

Eo ipso,

Ω(r2|r1) =γ(r1; r2)ρ(r1)

is the probability density of finding any electron at r2 if there is onealready known to be at r1.If we integrate over all coordinates of electron 2, we get∫

Ω(r2|r1) d3r2 =(N − 1)ρ(r1)

ρ(r1)= N − 1,

the number of all electrons of the systems but our reference one (whichsits at r1).

P (A|B) =P (A ∩B)P (B)

P (A ∩B) is the probability of both events together.Eo ipso,

Ω(r2|r1) =γ(r1; r2)ρ(r1)

is the probability density of finding any electron at r2 if there is onealready known to be at r1.

If we integrate over all coordinates of electron 2, we get∫R3

Ω(r2|r1) d3r2 =(N − 1)ρ(r1)

ρ(r1)= N − 1,

P (A|B) =P (A ∩B)P (B)

P (A ∩B) is the probability of both events together.Eo ipso,

Ω(r2|r1) =γ(r1; r2)ρ(r1)

is the probability density of finding any electron at r2 if there is onealready known to be at r1.If we integrate over all coordinates of electron 2, we get∫

Ω(r2|r1) d3r2 =(N − 1)ρ(r1)

ρ(r1)= N − 1,

Exchange-correlation hole

Short recapitulation:

exact pair density, γ(r1; r2), would include all previously mentionedeffects: self-interaction, correlation, exchange;

but we don’t know it. For instance, HF pair density takes care ofself-interaction and exchange, but misses any correlation.

Let’s now introduce the quantity:

hxc(r1; r2) = Ω(r2|r1)− ρ(r2) = ρ(r2)f(r1; r2).

It’s obviously the difference between conditional probability density offinding any electron at r2 if there is one already known to be at r1 and theuncorrelated probabily density of finding any electron at r2.

The conditional probability density is likely lower than the independent one,so hxc(r1; r2) is called the exchange-correlation hole (xc hole).

Because we love integrals, let’s integrate:∫

hxc(r1; r2) d3r2 = −1: xc

hole contains exactly the charge of one electron.Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 32 / 101

Fermi and Coulomb holes

The xc hole can be split into the Fermi and Coulomb holes:

hxc(r1; r2) = hx(r1; r2) + hc(r1; r2).

Fermi hole:

applies to electrons with thesame spin.

integrates to −1.

takes care of theself-interaction problem.

ensures the Pauli principle isfulfilled (no two electronwith the same spin in thesame point of space).

Coulomb hole:

applies to all electrons.

integrates to 0.

ensures the cusp condition isfulfilled.

is dominated by the Fermihole.

hxc(r1; r2) = hx(r1; r2) + hc(r1; r2).

Fermi hole:

integrates to −1.

Coulomb hole:

integrates to 0.

hxc(r1; r2) = hx(r1; r2) + hc(r1; r2).

Fermi hole:

integrates to −1.

Coulomb hole:

integrates to 0.

xc hole for H2

Pictorially, we can imagine that electron digs a hole around itself so thatthe probability of finding another electron around it is diminished.

The reference electron is 0.3 bohr to the left from the right proton. Onlythe total xc hole has a physical sense.

xc hole for H2

Pictorially, we can imagine that electron digs a hole around itself so thatthe probability of finding another electron around it is diminished.

The reference electron is 0.3 bohr to the left from the right proton. Onlythe total xc hole has a physical sense.

Back to electron-electron repulsion again

Using the xc hole we’ve just made friends with, we can write

γ(r1; r2) = ρ(r1)ρ(r2) + ρ(r1)hxc(r1; r2),

so the e-e repulsion becomes

Eee =12

γ(r1; r2)r12

d3r1 d3r2 =

ρ(r1)ρ(r2)r12

d3r1 d3r2 +

ρ(r1)hxc(r1; r2)r12

d3r1 d3r2 = J [ρ] + Encl[ρ] :

J [ρ]: classical electrostatic energy of a charge distribution with itself,it contains unphysical self-interaction (remember — H atom).Encl[ρ]: interaction between the charge density and the chargedistribution of the xc hole. It includes the correction for theself-interaction and all contributions of quantum-mechanical(non-classical) correlation effects.

γ(r1; r2) = ρ(r1)ρ(r2) + ρ(r1)hxc(r1; r2),

Eee =12

γ(r1; r2)r12

d3r1 d3r2 =

ρ(r1)ρ(r2)r12

d3r1 d3r2 +

d3r1 d3r2 = J [ρ] + Encl[ρ] :

γ(r1; r2) = ρ(r1)ρ(r2) + ρ(r1)hxc(r1; r2),

Eee =12

γ(r1; r2)r12

d3r1 d3r2 =

ρ(r1)ρ(r2)r12

d3r1 d3r2 +

d3r1 d3r2 = J [ρ] + Encl[ρ] :

J [ρ]: classical electrostatic energy of a charge distribution with itself,it contains unphysical self-interaction (remember — H atom).

Encl[ρ]: interaction between the charge density and the chargedistribution of the xc hole. It includes the correction for theself-interaction and all contributions of quantum-mechanical(non-classical) correlation effects.

γ(r1; r2) = ρ(r1)ρ(r2) + ρ(r1)hxc(r1; r2),

Eee =12

γ(r1; r2)r12

d3r1 d3r2 =

ρ(r1)ρ(r2)r12

d3r1 d3r2 +

d3r1 d3r2 = J [ρ] + Encl[ρ] :

Summary

Points to remember:

The density gives all the information on the molecule.

Hartree-Fock method treats electron-electron repulsion in a verysimplified manner: it properly accounts for the exchange, but lacksany Coulomb correlation.

But Hartree-Fock properly deals with the self-interaction problem.

xc hole is a nice concept allowing for the separation ofelectron-electron repulsion into the classical and non-classical parts.

But the non-classical part takes a lot of responsibility: it has toaccount for Coulomb and Fermi types of correlation and to removethe unphysical self-interaction included in the classical part.

Summary

Points to remember:

Summary

Points to remember:

Summary

Points to remember:

Summary

Points to remember:

Summary

Points to remember:

The End (for today)

Part II

DFT: How It’s Made

Outline of the Talk

5 Density and EnergyRemarks and ProblemsHistorical ModelsResults

6 Hohenberg-Kohn Theorems

7 Kohn-Sham Approach

8 xc Functionals

Density and Energy Remarks and Problems

Energy of a molecule

How to get the energy of a molecule — a kosher recipe:

specify molecule’s geometry: positions of nuclei RαMα=1,

specify molecular charge: N — number of electrons, the totalcharge Q =

∑Mα=1 Zα −N ,

write the total hamiltonian: H = T + Vne + Vee + Vnn,

solve the Schrodinger equation, Hψ0 = E0ψ0: we’ve got the energy.

∑Mα=1 Zα −N ,

But we don’t need ψ — one-matrix ρ and pair density γ suffice:

E = 〈ψ|H|ψ〉 = T [ρ] + Vne[ρ] + Eee[γ] + Vnn =

= −12

[∆rρ(r; r′)

]r′=r

d3r +∫

vne(r)ρ(r) d3r +

γ(r1; r2)r12

d3r1 d3r2 +

M−1∑α=1

M∑β=α+1

ZαZβRαβ

∑Mα=1 Zα −N ,

e-e repulsion can be separated into classical interaction of charge densitywith itself (with unphysical self-interaction) and the interaction of chargedensity with the xc hole, containing all non-classical effects (correlation,exchange, correction for self-interaction):

Eee = J [ρ] + Encl[ρ] =12

ρ(r1)ρ(r2)r12

d3r1 d3r2 +

d3r1 d3r2.

Wavefunction and density

Question : can we replace ψ(q1; . . . ; qN ), depending on 4N variables,with ρ(r), depending on just 3 variables?

Hamiltonian uniquely defined by:

the number of electrons,

the position of the nuclei,and

the charges of the nuclei.

Ground-state density:

integrates to the number ofelectrons.

has cusps at the position ofthe nuclei.

the cusp steepness isintimately related to thecharge of the nucleus.

Answer : yes! The ground-state density provides us with all theinformation we need to solve the Schrodinger equation.

Problems with energy

Problems with calculation of the energy of the system,

E = T [ρ] + Vne[ρ] + Eee[γ] + Vne :

the rigorous expression for the kinetic energy employs the one-matrix

instead of the density: T [ρ] = −12

[∆rρ(r; r′)

]r′=r

the notorious e-e repulsion term which on the pair density instead of

the density alone: Eee[γ] =12

γ(r1; r2)r12

d3r1 d3r2.

So, formally density is not enough — we need the one-matrix andpair-density to calculate E. . . On the other hand, we’ve already learnt thatthe density yields all the information on the system.

[∆rρ(r; r′)

]r′=r

γ(r1; r2)r12

d3r1 d3r2.

[∆rρ(r; r′)

]r′=r

γ(r1; r2)r12

d3r1 d3r2.

[∆rρ(r; r′)

]r′=r

γ(r1; r2)r12

d3r1 d3r2.

Density and Energy Historical Models

Thomas-Fermi model

Thomas and Fermi (1920s) were the first to give an approximateexpression of the energy in therms of the density only.

Assumptions of the Thomas-Fermi model:

the kinetic energy functional is taken from the theory of uniformnoninteracting electron gas.

the electronic exchange and correlations effects are completelyneglected, the electron-electron repulsion and electron-nucleusattraction are treated in a classical way only.

The energy functional in TF model reads

E[ρ] = TTF[ρ] + Vne[ρ] + J [ρ] + Vnn,

TTF[ρ] = CF

ρ5/3(r) d3r.

Thomas-Fermi model

Thomas and Fermi (1920s) were the first to give an approximateexpression of the energy in therms of the density only.Assumptions of the Thomas-Fermi model:

TTF[ρ] = CF

ρ5/3(r) d3r.

Thomas-Fermi model

TTF[ρ] = CF

ρ5/3(r) d3r.

Thomas-Fermi model

TTF[ρ] = CF

ρ5/3(r) d3r.

Thomas-Fermi model

TTF[ρ] = CF

ρ5/3(r) d3r.

Thomas-Fermi model

TTF[ρ] = CF

ρ5/3(r) d3r.

Thomas-Fermi-Dirac model

In the Thomas-Fermi-Dirac model the Thomas-Fermi energy functional

issupplemented with the exchange energy taken from the theory of uniformnoninteracting electron gas, as was the case for the kinetic energy:

E[ρ] = TTF[ρ] + Vne[ρ] + J [ρ] +

Ex[ρ]

Ex[ρ] = −Cx

ρ4/3(r) d3r.

The same formula for the exchange was derived by Slater in 1950s basedon the assumption of the Fermi hole being spherically symmetric aroundthe reference electron. That expression, depending only on local values ofelectron density, replaced the original non-local Hartree-Fock formula:

In the Thomas-Fermi-Dirac model the Thomas-Fermi energy functional issupplemented with the exchange energy taken from the theory of uniformnoninteracting electron gas, as was the case for the kinetic energy:

E[ρ] = TTF[ρ] + Vne[ρ] + J [ρ] + Ex[ρ]+Vnn.

Ex[ρ] = −Cx

ρ4/3(r) d3r.

Ex[ρ] = −Cx

ρ4/3(r) d3r.

Ex[ρ] = −Cx

ρ4/3(r) d3r.

EHFx [ρ] =

ρ(r1)hx(r1; r2)r12

d3r1 d3r2.

Ex[ρ] = −Cx

ρ4/3(r) d3r.

EHFx [ρ] = −1

ρ(r1; r1)ρ(r2; r1)r12

d3r1 d3r2.

Density and Energy Results

Some results . . .

Both TF and TFD models are based on the theory of uniformnoninteracting electron gas.

But the electronic density in an atom or molecule is obviously notuniform at all.That has drastic consequences — for instance, TF and TFD modelsdo not allow for any chemical bonding!

For atoms the TFD model results are not too bad:

Atom −EHF −ETFD

He 2.8615 2.2159Ne 128.5551 124.1601Ar 526.7942 518.8124Kr 2752.0164 2755.4398Xe 7232.4982 7273.2788Rn 21866.2779 22019.7140

Source: [Parr and Yang(1989)]

Some results . . .

Both TF and TFD models are based on the theory of uniformnoninteracting electron gas.But the electronic density in an atom or molecule is obviously notuniform at all.

That has drastic consequences — for instance, TF and TFD modelsdo not allow for any chemical bonding!

Atom −EHF −ETFD

He 2.8615 2.2159Ne 128.5551 124.1601Ar 526.7942 518.8124Kr 2752.0164 2755.4398Xe 7232.4982 7273.2788Rn 21866.2779 22019.7140

Some results . . .

Both TF and TFD models are based on the theory of uniformnoninteracting electron gas.But the electronic density in an atom or molecule is obviously notuniform at all.That has drastic consequences — for instance, TF and TFD modelsdo not allow for any chemical bonding!

Atom −EHF −ETFD

He 2.8615 2.2159Ne 128.5551 124.1601Ar 526.7942 518.8124Kr 2752.0164 2755.4398Xe 7232.4982 7273.2788Rn 21866.2779 22019.7140

Some results . . .

Both TF and TFD models are based on the theory of uniformnoninteracting electron gas.But the electronic density in an atom or molecule is obviously notuniform at all.That has drastic consequences — for instance, TF and TFD modelsdo not allow for any chemical bonding!

Atom −EHF −ETFD

He 2.8615 2.2159Ne 128.5551 124.1601Ar 526.7942 518.8124Kr 2752.0164 2755.4398Xe 7232.4982 7273.2788Rn 21866.2779 22019.7140

Source: [Parr and Yang(1989)]Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 44 / 101

Outline of the Talk

5 Density and Energy

6 Hohenberg-Kohn TheoremsDefinitionsThe TheoremsRepresentability of the Density

8 xc Functionals

Hohenberg-Kohn Theorems Definitions

External potential

The external potential — potential vext acting on electrons the source ofwhich are not electrons themselves.

H = T + Vext + Vee + Vnn,

Vext =N∑i=1

vext(ri).

Without any external (electric, magnetic) fields it’s just the nuclearpotential of the system:

vext(r) = vne(r) = −M∑α=1

Zα|r−Rα|

External potential

Vext =N∑i=1

vext(ri).

Zα|r−Rα|

External potential

Vext =N∑i=1

vext(ri).

Zα|r−Rα|

External potential

Vext =N∑i=1

vext(ri).

Zα|r−Rα|

Hohenberg-Kohn functional

We assume that the kinetic energy and e-e repulsion may be representedas a functional of the density only:

E[ρ] = T [ρ] + Vext[ρ] + Vee[ρ] + Vnn.

Let’s now regroup the energy functional a little bit:

E[ρ] = Vext[ρ] + FHK[ρ],

Vext[ρ] =∫

vext(r)ρ(r) d3r — system-dependent part (vext changes

with the system).

FHK[ρ] = T [ρ] + J [ρ] + Encl[ρ] — Hohenberg-Kohn functional:universal for all systems. But we don’t know T [ρ] nor Encl[ρ].

Vext[ρ] =∫

with the system).

Vext[ρ] =∫

with the system).

Vext[ρ] =∫

with the system).

Hohenberg-Kohn Theorems The Theorems

Hohenberg-Kohn Theorems

E[ρ] =∫

vext(r)ρ(r) d3r + FHK[ρ]

Theorem (One, HK1)

The external potential vext(r) and hence the total energy, is a uniquefunctional of the electron density ρ(r). So, there is one-to-one mapping

vext ↔ ρ.

Theorem (Two, HK2)

The density ρ0 minimizing the total energy is the exact ground-statedensity. So, given a trial density ρ (non-negative and integrating to N) weget

E[ρ] ≥ E[ρ0] = E0.

E[ρ] =∫

Theorem (One, HK1)

vext ↔ ρ.

Theorem (Two, HK2)

E[ρ] ≥ E[ρ0] = E0.

E[ρ] =∫

Theorem (One, HK1)

vext ↔ ρ.

Theorem (Two, HK2)

E[ρ] ≥ E[ρ0] = E0.

A few remaks on Hohenberg-Kohn theorems:

if we knew FHK[ρ], we would get gorund-state density and energy.FHK[ρ] is the Holy Grail of DFT!

HK theorems prove that there is indeed one-to-one mapping betweenground-state density and energy: ρ↔ E, but give us no clue how toconstruct the functional yielding the ground-state density.

the variational principle introduced by HK2 applies to the exactfunctional only! And we don’t know it — we use only someapproximations. That means variational principle doesn’t work inpractice — we can get energies lower than the true ones.

example: using BPW91 functional in cc-pV5Z basis set, for H atomwe get E = −0.5042, the true energy being E = −0.5:−0.5042 < −0.5.

Hohenberg-Kohn Theorems Representability of the Density

v-representability and N -representability

The density ρ is:v-representable if it is associated with ground-state antisymmetric ψ0

satisfying Hψ0 = E0ψ0, where H contains the external potential vext:

T +N∑i=1

vext(ri) + Vee + Vnn → Hψ = Eψ → ψ → ρ .

N -representable if it can be obtained from some antisymmetric ψ:

ψ → ρ .

All v-representable ρ’s are N -representable.HK2 originally deals only with v-representable ρ’s.But the conditions of ρ’s v-representability are yet unknown.Fortunately, it turns out we can lift that condition and extend ourvariational search on all N -representable ρ’s, without the explicitconnection to an external potential.

T +N∑i=1

ψ → ρ .

T +N∑i=1

ψ → ρ .

All v-representable ρ’s are N -representable.

HK2 originally deals only with v-representable ρ’s.But the conditions of ρ’s v-representability are yet unknown.Fortunately, it turns out we can lift that condition and extend ourvariational search on all N -representable ρ’s, without the explicitconnection to an external potential.

T +N∑i=1

ψ → ρ .

All v-representable ρ’s are N -representable.HK2 originally deals only with v-representable ρ’s.

But the conditions of ρ’s v-representability are yet unknown.Fortunately, it turns out we can lift that condition and extend ourvariational search on all N -representable ρ’s, without the explicitconnection to an external potential.

T +N∑i=1

ψ → ρ .

All v-representable ρ’s are N -representable.HK2 originally deals only with v-representable ρ’s.But the conditions of ρ’s v-representability are yet unknown.

Fortunately, it turns out we can lift that condition and extend ourvariational search on all N -representable ρ’s, without the explicitconnection to an external potential.

T +N∑i=1

ψ → ρ .

Non-interacting v-representability

Suppose we have a system described with hamiltonian containing onlyone-body electron operators (no e-e interaction):

HS = T +N∑i=1

v(ri) + Vnn.

We solve the Schrodinger equation, HSψS = EψS and obtain thedensity (through the integration):

ψS → ρ.

Such a density is said to be non-interacting v-representable, becauseit refers to the system of non-interacting electrons.

HS = T +N∑i=1

v(ri) + Vnn.

ψS → ρ.

HS = T +N∑i=1

v(ri) + Vnn.

ψS → ρ.

Constrained-search approach

Variational principle in the quest for ground-state energy: minimize energyexpectation value over all antisymmetric N -electron ψ’s:

E0 = minψ→N

⟨ψ∣∣∣ T + Vext + Vee + Vnn

∣∣∣ψ⟩ .

Constrained-search approach is performed in the two steps:

given a particular ρi integrating to N , find the ψi yielding ρi thatgives the minimal energy: in result we get ψmin

from the set of the densities ρiMi=1 and corresponding wavefunctionsψmin

i Mi=1 choose the one which yields the smallest energy.

Schematically,

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) .

E0 = minψ→N

∣∣∣ψ⟩ .Constrained-search approach is performed in the two steps:

Schematically,

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) .

E0 = minψ→N

Schematically,

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) .

E0 = minψ→N

Schematically,

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) .

E0 = minψ→N

Schematically,

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) .Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101

To identify the tallest child in aschool, we don’t need to line all the children up inthe schoolyard. Simply choose the tallest child in each classroom and ask thoseto come to the schoolyard, where the final search is performed.

Each striped area represents ψ’s giving the particular ρi.

minψ→ρ

: we constrain our search to the particular striped area and find ψmini

yielding the smallest energy, represented by the point •.minρ

: we minimize over all points (•) to find E0.

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩)

= minρ

(minψ→ρ

⟨ψ∣∣∣ T + Vee

∣∣∣ψ⟩+∫

vext(r)ρ(r) d3r)

+ Vnn =

= minρ

(F [ρ] +

vext(r)ρ(r) d3r)

+ Vnn,

F [ρ] = minψ→ρ

⟨ψ∣∣∣ T + Vee

∣∣∣ψ⟩ .We’ve already introduced HK functional,

FHK[ρ] = T [ρ] + Vee[ρ].

Clearly, for the ground-state density we have

F [ρ0] = FHK[ρ0].

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) =

= minρ

(minψ→ρ

⟨ψ∣∣∣ T + Vee

∣∣∣ψ⟩+∫

vext(r)ρ(r) d3r)

= minρ

(F [ρ] +

vext(r)ρ(r) d3r)

+ Vnn,

F [ρ] = minψ→ρ

⟨ψ∣∣∣ T + Vee

F [ρ0] = FHK[ρ0].

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) =

= minρ

(minψ→ρ

⟨ψ∣∣∣ T + Vee

∣∣∣ψ⟩+∫

vext(r)ρ(r) d3r)

+ Vnn =

= minρ

(F [ρ] +

vext(r)ρ(r) d3r)

+ Vnn,

F [ρ] = minψ→ρ

⟨ψ∣∣∣ T + Vee

F [ρ0] = FHK[ρ0].

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) =

= minρ

(minψ→ρ

⟨ψ∣∣∣ T + Vee

∣∣∣ψ⟩+∫

vext(r)ρ(r) d3r)

+ Vnn =

= minρ

(F [ρ] +

vext(r)ρ(r) d3r)

+ Vnn,

F [ρ] = minψ→ρ

⟨ψ∣∣∣ T + Vee

∣∣∣ψ⟩ .

We’ve already introduced HK functional,

F [ρ0] = FHK[ρ0].

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) =

= minρ

(minψ→ρ

⟨ψ∣∣∣ T + Vee

∣∣∣ψ⟩+∫

vext(r)ρ(r) d3r)

+ Vnn =

= minρ

(F [ρ] +

vext(r)ρ(r) d3r)

+ Vnn,

F [ρ] = minψ→ρ

⟨ψ∣∣∣ T + Vee

F [ρ0] = FHK[ρ0].

E0 = minρ

(minψ→ρ

∣∣∣ψ⟩) =

= minρ

(minψ→ρ

⟨ψ∣∣∣ T + Vee

∣∣∣ψ⟩+∫

vext(r)ρ(r) d3r)

+ Vnn =

= minρ

(F [ρ] +

vext(r)ρ(r) d3r)

+ Vnn,

F [ρ] = minψ→ρ

⟨ψ∣∣∣ T + Vee

F [ρ0] = FHK[ρ0].

Outline of the Talk

7 Kohn-Sham ApproachIntroductory RemarksKS Determinant and KS Energy

8 xc Functionals

Kohn-Sham Approach Introductory Remarks

A few remaks on the Hartree-Fock model

It lacks any correlation, but it properly describes the exchange.

So, the electrons described by HF function may be viewed asuncharged fermions: particles obeying the Pauli principle andneglecting the Coulomb repulsion.In this sense the HF function, ψHF, can be considered as the exactwavefunction of a fictitious system of N non-interacting electrons.Each electron is described by the orbital ϕi, which is the solution ofthe HF equation, fϕi = εiϕi, with the Fock operator

f(r) = −12

∆r + vne(r) + vHF(r),

so each electron moves in the effective potential veff = vne + vHF.The electronic kinetic energy reads

T [ρ] = −12

[∆rρ(r; r′)

]r′=r

d3r = −N/2∑i=1

〈ϕi|∆r|ϕi〉 .

It lacks any correlation, but it properly describes the exchange.So, the electrons described by HF function may be viewed asuncharged fermions: particles obeying the Pauli principle andneglecting the Coulomb repulsion.

In this sense the HF function, ψHF, can be considered as the exactwavefunction of a fictitious system of N non-interacting electrons.Each electron is described by the orbital ϕi, which is the solution ofthe HF equation, fϕi = εiϕi, with the Fock operator

f(r) = −12

T [ρ] = −12

[∆rρ(r; r′)

]r′=r

d3r = −N/2∑i=1

It lacks any correlation, but it properly describes the exchange.So, the electrons described by HF function may be viewed asuncharged fermions: particles obeying the Pauli principle andneglecting the Coulomb repulsion.In this sense the HF function, ψHF, can be considered as the exactwavefunction of a fictitious system of N non-interacting electrons.

Each electron is described by the orbital ϕi, which is the solution ofthe HF equation, fϕi = εiϕi, with the Fock operator

f(r) = −12

T [ρ] = −12

[∆rρ(r; r′)

]r′=r

d3r = −N/2∑i=1

It lacks any correlation, but it properly describes the exchange.So, the electrons described by HF function may be viewed asuncharged fermions: particles obeying the Pauli principle andneglecting the Coulomb repulsion.In this sense the HF function, ψHF, can be considered as the exactwavefunction of a fictitious system of N non-interacting electrons.Each electron is described by the orbital ϕi, which is the solution ofthe HF equation, fϕi = εiϕi, with the Fock operator

f(r) = −12

so each electron moves in the effective potential veff = vne + vHF.

The electronic kinetic energy reads

T [ρ] = −12

[∆rρ(r; r′)

]r′=r

d3r = −N/2∑i=1

It lacks any correlation, but it properly describes the exchange.So, the electrons described by HF function may be viewed asuncharged fermions: particles obeying the Pauli principle andneglecting the Coulomb repulsion.In this sense the HF function, ψHF, can be considered as the exactwavefunction of a fictitious system of N non-interacting electrons.Each electron is described by the orbital ϕi, which is the solution ofthe HF equation, fϕi = εiϕi, with the Fock operator

f(r) = −12

T [ρ] = −12

[∆rρ(r; r′)

]r′=r

d3r = −N/2∑i=1

Kohn-Sham Approach KS Determinant and KS Energy

Non-interacting reference system

We now set up a system described by the HamiltonianHS = T + VS + Vnn with

VS =N∑i=1

vS(ri).

HS contains no e-e interaction, so it obviously describes thenon-interacting system! Its wavefunction is then a single Slaterdeterminant:

ψS = |ϕ1αϕ1β . . . ϕN/2αϕN/2β〉 .

ψS → ρS: non-interacting v-representable (no e-e interaction in HS).Each electron of this system moves in the effective potential veff = vS,so the orbitals are obtained from HF-like equations: fKSϕi = εiϕiwith the operator (called Kohn-Sham operator) being

fKS = −12

∆r + vS(r).

VS =N∑i=1

vS(ri).

ψS = |ϕ1αϕ1β . . . ϕN/2αϕN/2β〉 .

ψS → ρS: non-interacting v-representable (no e-e interaction in HS).

Each electron of this system moves in the effective potential veff = vS,so the orbitals are obtained from HF-like equations: fKSϕi = εiϕiwith the operator (called Kohn-Sham operator) being

fKS = −12

∆r + vS(r).

VS =N∑i=1

vS(ri).

ψS = |ϕ1αϕ1β . . . ϕN/2αϕN/2β〉 .

ψS → ρS: non-interacting v-representable (no e-e interaction in HS).Each electron of this system moves in the effective potential veff = vS,so the orbitals are obtained from HF-like equations: fKSϕi = εiϕiwith the operator (called Kohn-Sham operator) being

fKS = −12

∆r + vS(r).

Kohn-Sham model

We’ve already set up the non-interaction reference system and come upwith the orbital equations. But what is vS and how do we get it?

The recipe = Kohn-Sham model:

we require that the density resulting from KS determinant ψKS:

ρS(r) = 2N/2∑i=1

|ϕi(r)|2

is the same as the density of the real target system of interactingelectrons: ρS = ρ. So, ρ is also non-interacting v-representable.since we don’t know the explicit T [ρ] functional, the kinetic energy iscomputed using HF-like expression:

TS[ρ] = −N/2∑i=1

〈ϕi|∆r|ϕi〉 ,

and the remainder is shifted to the exchange-correlation energy.

Kohn-Sham model

We’ve already set up the non-interaction reference system and come upwith the orbital equations. But what is vS and how do we get it?The recipe = Kohn-Sham model:

ρS(r) = 2N/2∑i=1

|ϕi(r)|2

TS[ρ] = −N/2∑i=1

Kohn-Sham model

ρS(r) = 2N/2∑i=1

|ϕi(r)|2

is the same as the density of the real target system of interactingelectrons: ρS = ρ. So, ρ is also non-interacting v-representable.

since we don’t know the explicit T [ρ] functional, the kinetic energy iscomputed using HF-like expression:

TS[ρ] = −N/2∑i=1

Kohn-Sham model

ρS(r) = 2N/2∑i=1

|ϕi(r)|2

TS[ρ] = −N/2∑i=1

and the remainder is shifted to the exchange-correlation energy.Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 56 / 101

Kohn-Sham energy functional

The total energy of a real system in Kohn-Sham model:

E[ρ] = T [ρ] + Vne[ρ] + Eee[ρ] + Vnn

== T [ρ] + Vne[ρ] + J [ρ] + Encl[ρ] + Vnn == TS[ρ] + T [ρ]− TS[ρ] + Vne[ρ] + J [ρ] + Encl[ρ] + Vnn == TS[ρ] + Vne[ρ] + J [ρ] + T [ρ]− TS[ρ] + Encl[ρ]︸︷︷︸

Exc[ρ]: exchange-correlation energy

Finally, the famous exchange-correlation (xc) energy functional is

Exc[ρ] = T [ρ]− TS[ρ] + Encl[ρ].

Apparently, the Exc[ρ]’s responsibility is enormous: it containsnon-classical effects of self-interaction correction, exchange andcorrelation, plus portion of kinetic energy not present in thenon-interacting reference system!

E[ρ] = T [ρ] + Vne[ρ] + Eee[ρ] + Vnn == T [ρ] + Vne[ρ] + J [ρ] + Encl[ρ] + Vnn

== TS[ρ] + T [ρ]− TS[ρ] + Vne[ρ] + J [ρ] + Encl[ρ] + Vnn == TS[ρ] + Vne[ρ] + J [ρ] + T [ρ]− TS[ρ] + Encl[ρ]︸︷︷︸

E[ρ] = T [ρ] + Vne[ρ] + Eee[ρ] + Vnn == T [ρ] + Vne[ρ] + J [ρ] + Encl[ρ] + Vnn == TS[ρ] + T [ρ]− TS[ρ] + Vne[ρ] + J [ρ] + Encl[ρ] + Vnn

== TS[ρ] + Vne[ρ] + J [ρ] + T [ρ]− TS[ρ] + Encl[ρ]︸︷︷︸

E[ρ] = T [ρ] + Vne[ρ] + Eee[ρ] + Vnn == T [ρ] + Vne[ρ] + J [ρ] + Encl[ρ] + Vnn == TS[ρ] + T [ρ]− TS[ρ] + Vne[ρ] + J [ρ] + Encl[ρ] + Vnn == TS[ρ] + Vne[ρ] + J [ρ] + T [ρ]− TS[ρ] + Encl[ρ]︸︷︷︸

Apparently, the Exc[ρ]’s responsibility is enormous: it containsnon-classical effects of self-interaction correction, exchange andcorrelation,

plus portion of kinetic energy not present in thenon-interacting reference system!

Kohn-Sham equations

Thus, we have established the total energy of a systems as a functional ofdensity:

E[ρ] = TS[ρ] + Vne[ρ] + J [ρ] + Exc[ρ] + Vnn.

Task : minimze E[ρ] with the constraint on the density integration:∫R3

ρ(r) d3r = N,

and the calculus of variations comes with the proper tools to do it!Result : Kohn-Sham equations for optimal orbitals:

fKSϕi = εiϕi.

Kohn-Sham equations

ρ(r) d3r = N,

and the calculus of variations comes with the proper tools to do it!

Result : Kohn-Sham equations for optimal orbitals:

fKSϕi = εiϕi.

Kohn-Sham equations

ρ(r) d3r = N,

and the calculus of variations comes with the proper tools to do it!Result : Kohn-Sham equations for optimal orbitals:

fKSϕi = εiϕi.

Kohn-Sham operator

Based on the considerations about the non-interacting reference systemwe’ve arrived at

fKS(r) = −12

∆r + vS(r).

And now, based on the total Kohn-Sham energy functional minimizationwe find the effective potential we’ve been looking for:

vS(r) = vne(r) + (r) + vxc(r),

but since we don’t know the explicit form of xc energy, we don’t knowhow xc potential looks either, so we can only put it as the functionalderivative of the xc energy with respect to the density:

vxc(r) =δExc[ρ]δρ(r)

Kohn-Sham operator

fKS(r) = −12

∆r + vS(r).

Kohn-Sham operator

fKS(r) = −12

∆r + vS(r).

Hartree-Fock and Kohn-Sham models

Hartree-Fock operator: f(r) = −12∆r + vne(r) + (r)− k(r),

Kohn-Sham operator: fKS(r) = −12∆r + vne(r) + (r) + vxc(r).

Hartree-Fock:

contains non-local exchangeoperator.

takes no parameters, theenergy is well-defined.

is purely variational, theenergy is always higher thanits true value.

yields the best energy withinone-electron approximation.

Kohn-Sham:

all operators are local.

the energy depends on theapproximation to xc energy.

variational method worksonly for exact xc functional,in practice it does not apply.

is potentially exact — oncewe knew exact xc functional,we would get the exactenergy.

Hartree-Fock:

Kohn-Sham:

Hartree-Fock:

Kohn-Sham:

Outline of the Talk

8 xc FunctionalsIs There a Road Map?Adiabatic ConnectionKohn-Sham Machinery

xc Functionals Is There a Road Map?

Some remarks on xc functionals

Exc[ρ] is the central object in DFT and KS.

Exact Exc[ρ] gives exact energy, i.e. energy strictly satisfyingSchrodinger equation.

But no one knows the exact Exc[ρ]!So, we must make explicit approximations to this functional,otherwise KS model makes no sense and is practically useless!

DFT mission: the never-ending quest for better and better xcfunctionals . . .

But no one knows the exact Exc[ρ]!

So, we must make explicit approximations to this functional,otherwise KS model makes no sense and is practically useless!

The Holy Grail of DFT: exact xc functional

There is no systematic strategy how to get closer to the exactxc functional, as is the case in wavefunction-based approaches, wherethe the variational method is the cornerstone.

The explicit form of the exact functional remains a total mystery tous.

The attempts to find better functionals rely to a large extent onphysical and mathematical intuition, and have strong trial and errorcomponent.

There are some physical constraints that the reasonable functionalsshould obey: sum rules for the xc holes, cusp condition, asymptoticproperties of the resulting xc potentials, etc.

Nevertheless, it turns out that some successful functionals obeyseveral of these conditions and are still better than some kosherones . . .

xc Functionals Adiabatic Connection

xc holes and xc functionals

We remember that the non-classical e-e repulsion in terms of xc hole is

Encl[ρ] =12

d3r1 d3r2.

But the xc functional of KS scheme includes also the kinetic energycorrelation contribution:

Exc[ρ] = T [ρ]− TS[ρ] + Encl[ρ] = Tcor[ρ] + Encl[ρ].

We also know that KS model uses two crucial objects:

non-interacting reference system (density: ρS),

the real system with fully interacting electrons (density: ρ).

The two systems share the same density: ρS = ρ.

Encl[ρ] =12

d3r1 d3r2.

Encl[ρ] =12

d3r1 d3r2.

Encl[ρ] =12

d3r1 d3r2.

Encl[ρ] =12

d3r1 d3r2.

Encl[ρ] =12

d3r1 d3r2.

Coupling the two systems of KS model

We introduce the coupling parameter λ ∈ 〈0; 1〉:

H(λ) = T + Vext(λ) + λVee + Vnn.

Vext(λ) changes with λ so that the density of the system describedwith H(λ) equals the density of the real system.Boundary cases:

λ = 0: non-interacting system Hamiltonian, Vext(0) = VS.

λ = 1: real system Hamiltonian, Vext(1) = Vne for isolated molecule.

Vext(λ) changes with λ so that the density of the system describedwith H(λ) equals the density of the real system.

Boundary cases:

Adiabatic connection formula

Through the artificial and smooth coupling of the two systems thefollowing energy expression is derived:

E[ρ] = TS[ρ]+Vne[ρ]+Jne[ρ]+12

d3r1 d3r2 +Vnn,

whereas the equivalent expression, which we already know very well, reads

E[ρ] = T [ρ] +Vne[ρ] +Jne[ρ] +12

d3r1 d3r2+, Vnn.

d3r1 d3r2 +Vnn,

E[ρ] = T [ρ] +Vne[ρ] +Jne[ρ] +12

d3r1 d3r2+, Vnn.

d3r1 d3r2 +Vnn,

E[ρ] = T [ρ] +Vne[ρ] +Jne[ρ] +12

d3r1 d3r2+, Vnn.

hxc(r1; r2) =∫ 1

0hxc(r1; r2;λ) dλ :

coupling-strength integrated xc hole: it has the same formal properties asthe standard xc hole (sum rules, cusp conditions).

d3r1 d3r2 +Vnn,

E[ρ] = T [ρ] +Vne[ρ] +Jne[ρ] +12

d3r1 d3r2+, Vnn.

Finally, the xc energy in the adiabatic connection approach reads

Exc[ρ] =ρ(r1)hxc(r1; r2)

r12d3r1 d

xc Functionals Kohn-Sham Machinery

How KS method works

KS orbitals satisfy KS equations:

fKS(r)ϕi(r) =(−1

2∆r + vne(r) + j(r) + vxc(r)

)ϕi(r) = εiϕi(r).

We expand the molecular orbitals (MOs) in the atomic orbitals (AOs):,

ϕi(r) =M∑j=1

cjiχj(r).

Now the KS equations can be cast into a nice M ×M matrix form:

FKSC = SCε,

(FKS)ij = 〈χi|fKS|χj〉, (C)ij = cij ,

(S)ij = 〈χi|χj〉, ε =M∑i=1

εiIM .

How KS method works

fKS(r)ϕi(r) =(−1

2∆r + vne(r) + j(r) + vxc(r)

ϕi(r) =M∑j=1

cjiχj(r).

FKSC = SCε,

εiIM .

How KS method works

fKS(r)ϕi(r) =(−1

2∆r + vne(r) + j(r) + vxc(r)

ϕi(r) =M∑j=1

cjiχj(r).

FKSC = SCε,

εiIM .

How KS method works

fKS(r)ϕi(r) =(−1

2∆r + vne(r) + j(r) + vxc(r)

ϕi(r) =M∑j=1

cjiχj(r).

FKSC = SCε,

εiIM .

Summary

Points to remember:

HK theorems state that there is indeed one-to-one mapping betweenthe external potential and the ground-state density.

Although HF theorems state that there is variational principle, inpractice we can’t make any use of it since we don’t know the exact xcfunctional.

KS method is the central to DFT, like HF to wavefunction theory. Itis potentially exact and all the operators it uses are local.

KS introduces the xc energy which contains the correlation kineticenergy, self-interaction correction, correlation and exchange.

But we don’t know how the exact xc functional looks like, it remainsa complete mystery to us.

Summary

Points to remember:

Summary

Points to remember:

Summary

Points to remember:

Summary

Points to remember:

Summary

Points to remember:

The End (for today)

Part III

DFT in Real Life: Defective Functional Theory

Outline of the Talk

9 Approximate xc FunctionalsIntroductionLDA and LSDGGAHybrid FunctionalsBeyond GGAProblems of Approximate Functionals

Approximate xc Functionals Introduction

The desired features of an approximate xc energyfunctional:

a non-empirical derivation, since the principles of quantum mechanicsare well-known and sufficient.

universality, since in principle one functional should work for diversesystems (atoms, molecules, solids) with different bonding characters(covalent, ionic, metallic, hydrogen, and van der Waals).

simplicity, since this is our only hope for intuitive understanding andour best hope for practical calculation.

accuracy enough to be useful in calculations for real systems.

Source: [Perdew and Kurt(2003)]

Approximate xc Functionals LDA and LSD

Uniform electron gas model

Uniform electron gas (jellium) is the central model on which almost allapproximate xc functionals are based.

Electrons move in the external potential from uniformly distributedbackground positive charge (positive jelly background).

Number of electrons N and the volume of electron gas V are infinite,but ∀r : ρ(r) = N

V = const.

It is quite a good model of metals with positive cores smeared out toobtain the uniform background positive charge.

Of course, the electron density in atoms and molecules can changedrastically with r and is far from being homogeneous.

But it’s the only system for which we know the explicit functionals forkinetic energy, exchange () and, to very high accuracy, correlation.

V = const.

But it’s the only system for which we know the explicit functionals forkinetic energy, exchange ( Thomas-Fermi(-Dirac) models ) and, to very highaccuracy, correlation.

Local density approximation (LDA)

In the LDA the xc energy is assumed to be

ELDAxc [ρ] =

ρ(r)ε0xc

(ρ(r)

)— xc energy density in uniform electron gas model (depends

only on the density). It splits into exchange and correlation parts

(ρ(r)

)= ε0x

(ρ(r)

)+ ε0c

(ρ(r)

The ε0x in uniform electron gas model was given by Dirac in late 1920s:

ε0x(ρ) = −Cxρ1/3

⇒ ELDAx [ρ] = −Cx

ρ4/3(r) d3r.

But we don’t know the explicit form for εc(ρ(r)

). However, sophisticated

analytical fits to the numerical Monte Carlo simulations results areavailable and are termed as VWN (for Vosko, Wilk and Nusair whoobtained the fits).

ELDAxc [ρ] =

ρ(r)ε0xc

(ρ(r)

only on the density).

It splits into exchange and correlation parts

(ρ(r)

)= ε0x

(ρ(r)

)+ ε0c

(ρ(r)

ε0x(ρ) = −Cxρ1/3

ρ4/3(r) d3r.

ELDAxc [ρ] =

ρ(r)ε0xc

(ρ(r)

)= ε0x

(ρ(r)

)+ ε0c

(ρ(r)

ε0x(ρ) = −Cxρ1/3

ρ4/3(r) d3r.

ELDAxc [ρ] =

ρ(r)ε0xc

(ρ(r)

)= ε0x

(ρ(r)

)+ ε0c

(ρ(r)

ε0x(ρ) = −Cxρ1/3

ρ4/3(r) d3r.

ELDAxc [ρ] =

ρ(r)ε0xc

(ρ(r)

)= ε0x

(ρ(r)

)+ ε0c

(ρ(r)

ε0x(ρ) = −Cxρ1/3 ⇒ ELDA

x [ρ] = −Cx

ρ4/3(r) d3r.

ELDAxc [ρ] =

ρ(r)ε0xc

(ρ(r)

)= ε0x

(ρ(r)

)+ ε0c

(ρ(r)

ε0x(ρ) = −Cxρ1/3 ⇒ ELDA

x [ρ] = −Cx

ρ4/3(r) d3r.

Local spin-density approximation (LSD)

In the unrestricted version of KS model there two densities, for spin-up andspin-down electrons, respectively, summing to the total electron density:

ρ(r) = ρα(r) + ρβ(r).

In the LSD the xc energy depends on the two densities

ELSDxc [ρα; ρβ] =

ρ(r)εxc

(ρ(r); ζ(r)

Spin-polarization parameter:

ζ(r) =ρα(r)− ρβ(r)

ρ(r)=

0, spin-compenstated case (closed-shell).

1, completely spin-polarized ferromagnetic case.

Again, we only know the explicit expression for the exchange energydensity:

εx(ρ; ζ)

= ε0x(ρ) +Ax

(εx(ρ; 1)− ε0x(ρ)

)((1 + ζ)4/3 + (1− ζ)4/3 − 2

ρ(r) = ρα(r) + ρβ(r).

ρ(r)εxc

(ρ(r); ζ(r)

ρ(r)=

εx(ρ; ζ)

= ε0x(ρ) +Ax

(εx(ρ; 1)− ε0x(ρ)

)((1 + ζ)4/3 + (1− ζ)4/3 − 2

ρ(r) = ρα(r) + ρβ(r).

ρ(r)εxc

(ρ(r); ζ(r)

ρ(r)=

εx(ρ; ζ)

= ε0x(ρ) +Ax

(εx(ρ; 1)− ε0x(ρ)

)((1 + ζ)4/3 + (1− ζ)4/3 − 2

ρ(r) = ρα(r) + ρβ(r).

ρ(r)εxc

(ρ(r); ζ(r)

ρ(r)=

εx(ρ; ζ)

= ε0x(ρ) +Ax

(εx(ρ; 1)− ε0x(ρ)

)((1 + ζ)4/3 + (1− ζ)4/3 − 2

Remarks on LDA and LSD

The separation of the total density into spin-up and spin-downcomponents is somewhat artificial as the exact xc functional willdepend on the total density only.

However, the division of ρ into ρα and ρβ:

I is necessary for spin-dependent external potential (e.g. magnetic fieldcoupling to electronic spin).

I is needed if we are interested in the physical spin magnetization (e.g.in magnetic materials).

I allows for more flexibility in the approximate functionals which typicallyperform better when we use the two densities instead of just one.

The separation of the total density into spin-up and spin-downcomponents is somewhat artificial as the exact xc functional willdepend on the total density only.However, the division of ρ into ρα and ρβ:

The way we calculate the xc energy in LDA/LSD means we assumethat the xc potentials depend only on the local values of density. Butthe density in real systems, atoms and molecules, often variesdrastically with r. /

It turns out that the xc hole in the uniform electron gas model, onwich LDA/LSD is based, satisfies the formal properties of the exactxc hole. ,

Since LDA is a special case of LSD for spin-compensated cases, fromnow on we will refer to both methods as LSD.

LSD has been extensively popular in the solid state physics. But forthe sparse matter which we have to do with in chemistry there was aneed to go beyond the local approximation.

Approximate xc Functionals GGA

Beyond LSD

The situation of people is totally different when they are on a steady, plainterrain with (almost) uniform density. . .

Beyond LSD

. . . then when they are put in a region with very rapidly changing density!

Beyond LSD

In the LSD we used only the information on the density at the specificpoint to calculate the contribution to xc energy. . .

so the obvious next step is to supplement that with the informationon how density changes in that point.That information is stored in the density gradient:

∇ρ =

∂ρ∂x∂ρ∂y∂ρ∂z

ρ′xρ′yρ′z

vector pointing in the direction

of the greatest rate

of the increase of the density.

The gradient magnitude (scalar!) gives the rate of the greatestchange of the density:

|∇ρ| =√∇ρ · ∇ρ =

√(ρ′x)2 + (ρ′y)2 + (ρ′z)2.

Beyond LSD

In the LSD we used only the information on the density at the specificpoint to calculate the contribution to xc energy. . .so the obvious next step is to supplement that with the informationon how density changes in that point.

That information is stored in the density gradient:

∇ρ =

ρ′xρ′yρ′z

|∇ρ| =√∇ρ · ∇ρ =

√(ρ′x)2 + (ρ′y)2 + (ρ′z)2.

Beyond LSD

In the LSD we used only the information on the density at the specificpoint to calculate the contribution to xc energy. . .so the obvious next step is to supplement that with the informationon how density changes in that point.That information is stored in the density gradient:

∇ρ =

ρ′xρ′yρ′z

|∇ρ| =√∇ρ · ∇ρ =

√(ρ′x)2 + (ρ′y)2 + (ρ′z)2.

Beyond LSD

In the LSD we used only the information on the density at the specificpoint to calculate the contribution to xc energy. . .so the obvious next step is to supplement that with the informationon how density changes in that point.That information is stored in the density gradient:

∇ρ =

ρ′xρ′yρ′z

|∇ρ| =√∇ρ · ∇ρ =

√(ρ′x)2 + (ρ′y)2 + (ρ′z)2.

Generalized gradient approximation (GGA)

After insertion of gradients into the xc functional it turned out thatthe results are even worse than for LSD!

This is because in such an approach the xc holes no longer satisfyformal properties as was the case for LSD.

So, let’s be brutal: enforce the resulting holes to satisfy the formalproperties by truncating them in the regions where they misbehave.

With the hope to correct the LSD we now introduce the gradient into thexc functional and correct the xc holes where necessary — this way weobtain the generalized gradient approximation to the xc energy:

EGGAxc [ρα; ρβ] =

fGGAxc

(ρα(r); ρβ(r);∇ρα(r);∇ρβ(r)

As usual, we split the energy: EGGAxc = EGGA

x + EGGAc .

fGGAxc

x + EGGAc .

fGGAxc

x + EGGAc .

fGGAxc

x + EGGAc .

fGGAxc

x + EGGAc .

Exchange in GGA

The exchange energy in GGA is usually assumed to be composed of LSDpart plus some correction:

EGGAx [ρα; ρβ] = ELSD

x [ρα; ρβ]−∑σ

F(sσ(r)

)ρ4/3(r) d3r,

where the reduced density gradient is a measure of local densityinhomogeneity:

sσ(r) =|∇ρ(r)|ρ4/3(r)

it is large for large density gradients

(regions of rapidly changing density)

and for small densities

(tails of density far from nuclei).

Exchange in GGA

F(sσ(r)

)ρ4/3(r) d3r,

sσ(r) =|∇ρ(r)|ρ4/3(r)

Exchange in GGA

F(sσ(r)

)ρ4/3(r) d3r,

sσ(r) =|∇ρ(r)|ρ4/3(r)

Exchange in GGA — examples:

FB =βs2

1 + 6βsσ sinh−1 sσ, β = 4.2 · 10−3

[Becke(1988)]

β obtained by a least-squares fit to the exactly known exchangeenergies of the rare gas atoms He through Rn.

The functional designed to recover the exchange energy densityasymptotically far from a finite system.

Sum rules for the exchange hole fulfilled.

Empirical.

Similar functionals: PW91, CAM(A), CAM(B), FT97.

FPW91 =1 + 0.19645sσ sinh−1 7.7956sσ + (0.2743− 0.1508e−100s2σ)s2

1 + 0.19645sσ sinh−1 7.7956sσ + 0.004s4σ

[Perdew et al.(1992)Perdew, Chevary, Vosko, Jackson, Pederson, Singh, and Fiolhais]

The analytical fit to the second-order density-gradient expansion forthe xc hole surrounding the electron in a system of slowly varyingdensity.

The spurious long-range parts of the xc hole cut off to satisfy sumrules on the exact hole.

According to Perdew, overparametrized.

Non-empirical.

FPBE = κ− κ

1 + µκs

κ = 0.804µ = 0.21951

[Perdew et al.(1996)Perdew, Burke, and Ernzerhof]

κ set to the maximum value allowed by the local Lieb-Oxford bound.

µ set to recover the linear response of the uniform electron gas.

Non-empirical.

Correlation in GGA

EGGAc ’s have a very complicated analytical form and cannot be

understood by simple physically motivated reasonings.

Some examples:

P86C: includes parameter fitted to the correlation energy of Ne.

PW91C: based on xc hole investigation.

LYP: derived from an expression for the correlation energy of He fromaccurate ab initio calculations.

Correlation in GGA

understood by simple physically motivated reasonings.Some examples:

Correlation in GGA

LSD/GGA results

Exc for atoms

Atom LSD GGA exactH −0.29 −0.31 −0.31He −1.00 −1.06 −1.09Li −1.69 −1.81 −1.83Be −2.54 −2.72 −2.76N −6.32 −6.73 −6.78Ne −11.78 −12.42 −12.50

LSD: VWN for correlation, GGA: PBE for correlation and exchangeSource: [Perdew and Kurt(2003)]

LSD/GGA results

Atomization energies for molecules

Molecule LSD GGA exactH2 0.18 0.169 0.173CH4 0.735 0.669 0.669NH3 0.537 0.481 0.474H2O 0.426 0.371 0.371CO 0.478 0.43 0.412O2 0.279 0.228 0.191

LSD: VWN for correlation, GGA: PBE for correlation and exchangeSource: [Perdew and Kurt(2003)]

Approximate xc Functionals Hybrid Functionals

Exact exchange

Although we don’t know exact xc functional, it’s clear from numericalexperience that the exchange dominates the correlation:

|Ex| >> |Ec|.

Thus, designing appropriate exchange functional is crucial to gettingmeaningful results from KS method. From HF theory we know the exactexpression for the exchange resulting from single Slater determinant:

Eexactx [ρ] = −1

ρ(r1; r2)ρ(r2; r1)r12

d3r1 d3r2.

This exchange is termed exact in DFT jargon, though it’s different thanthe exchange in HF model as the one-matrix ρ(r; r′) used here is that ofKS model, which doesn’t equal that of HF model. Also, as we remember,that exchange is non-local.

Exact exchange

|Ex| >> |Ec|.

Thus, designing appropriate exchange functional is crucial to gettingmeaningful results from KS method.

From HF theory we know the exactexpression for the exchange resulting from single Slater determinant:

Eexactx [ρ] = −1

ρ(r1; r2)ρ(r2; r1)r12

d3r1 d3r2.

Exact exchange

|Ex| >> |Ec|.

Eexactx [ρ] = −1

ρ(r1; r2)ρ(r2; r1)r12

d3r1 d3r2.

Exact exchange

|Ex| >> |Ec|.

Eexactx [ρ] = −1

ρ(r1; r2)ρ(r2; r1)r12

d3r1 d3r2.

Exact exchange problems

So, why not just mix the exact exchange with the GGA correlation:

Exc = Eexactx + EGGA

This at first nice idea proves to yield results even worse than HF!That’s because full exact exchange is incompatible with GGA correlation:

the exact exchange hole in a molecule usually has a highly nonlocal,multi-center character.this is cancelled by an almost equal, but opposite, nonlocal andmulticenter character in the exact correlation hole.so, the exact xc hole is localized around the reference electron.the GGA-approximated exchange and correlation holes are morelocalized around the reference electron. . .and finally, mixing the full exact exchange hole with the local GGAcorrelation hole results in non-local xc hole, which can’t model thelocality of the exact xc hole.

This at first nice idea proves to yield results even worse than HF!

That’s because full exact exchange is incompatible with GGA correlation:

the exact exchange hole in a molecule usually has a highly nonlocal,multi-center character. xc hole for H2

this is cancelled by an almost equal, but opposite, nonlocal andmulticenter character in the exact correlation hole.so, the exact xc hole is localized around the reference electron.the GGA-approximated exchange and correlation holes are morelocalized around the reference electron. . .and finally, mixing the full exact exchange hole with the local GGAcorrelation hole results in non-local xc hole, which can’t model thelocality of the exact xc hole.

this is cancelled by an almost equal, but opposite, nonlocal andmulticenter character in the exact correlation hole.

so, the exact xc hole is localized around the reference electron.the GGA-approximated exchange and correlation holes are morelocalized around the reference electron. . .and finally, mixing the full exact exchange hole with the local GGAcorrelation hole results in non-local xc hole, which can’t model thelocality of the exact xc hole.

this is cancelled by an almost equal, but opposite, nonlocal andmulticenter character in the exact correlation hole.so, the exact xc hole is localized around the reference electron.

the GGA-approximated exchange and correlation holes are morelocalized around the reference electron. . .and finally, mixing the full exact exchange hole with the local GGAcorrelation hole results in non-local xc hole, which can’t model thelocality of the exact xc hole.

this is cancelled by an almost equal, but opposite, nonlocal andmulticenter character in the exact correlation hole.so, the exact xc hole is localized around the reference electron.the GGA-approximated exchange and correlation holes are morelocalized around the reference electron. . .

and finally, mixing the full exact exchange hole with the local GGAcorrelation hole results in non-local xc hole, which can’t model thelocality of the exact xc hole.

this is cancelled by an almost equal, but opposite, nonlocal andmulticenter character in the exact correlation hole.so, the exact xc hole is localized around the reference electron.the GGA-approximated exchange and correlation holes are morelocalized around the reference electron. . .and finally, mixing the full exact exchange hole with the local GGAcorrelation hole results in non-local xc hole, which can’t model thelocality of the exact xc hole.

Hybrid functionals

Adding the full exact exchange doesn’t work well, but we know that thatexchange properly describes the non-interacting system. So, instead of fullexact exchange, let’s just combine some fraction of it with the GGAcounterparts.

That’s how we obtain the hybrid functionals. Generally,

Ehybxc = a Eexact

x︸︷︷︸exact

non-local exchange

+ (1− a) EGGAx︸︷︷︸

GGAlocal exchange

+ EGGAc︸︷︷︸

GGAlocal correlation

, a < 1.

There are now plenty of hybrid functionals available. Some variationsinvolve mixtures of three kinds of exchange: exact one, LSD one (calledSlater exchange), and GGA local one. Accordingly, they involve moreparameters than just one (a).

Hybrid functionals

Adding the full exact exchange doesn’t work well, but we know that thatexchange properly describes the non-interacting system. So, instead of fullexact exchange, let’s just combine some fraction of it with the GGAcounterparts.That’s how we obtain the hybrid functionals. Generally,

Ehybxc = a Eexact

x︸︷︷︸exact

non-local exchange

+ (1− a) EGGAx︸︷︷︸

GGAlocal exchange

+ EGGAc︸︷︷︸

, a < 1.

Hybrid functionals

Adding the full exact exchange doesn’t work well, but we know that thatexchange properly describes the non-interacting system. So, instead of fullexact exchange, let’s just combine some fraction of it with the GGAcounterparts.That’s how we obtain the hybrid functionals. Generally,

Ehybxc = a Eexact

x︸︷︷︸exact

non-local exchange

+ (1− a) EGGAx︸︷︷︸

GGAlocal exchange

+ EGGAc︸︷︷︸

, a < 1.

Examples of hybrid functionals

EB3xc = aEexact

x + (1− a)ELSDx + bEB88

x + ELSDc + EPW91

a = 0.20b = 0.72c = 0.81

[Becke(1993)]

Parameters a, b and c chosen to optimally reproduce the atomizationand ionization energies and proton affinities from the G2thermochemical database.

EB3LYPxc = aEexact

x +(1−a)ELSDx +bEB88

x +cELYPc +(1−c)ELSD

a = 0.20b = 0.72c = 0.81

[Stephens et al.(1994)Stephens, Devlin, Chabalowski, and Frisch]

Parameters a, b and c take from the B3 functional.

Particularly good results for vibrational spectra.

Undeniably the most popular and widely used functional in DFT.

EPBE0xc = aEexact

x + (1− a)EPBEx + EPBE

c , a = 0.25

[Adamo and Barone(1999)]

The value of a deducted from perturbation theory.

Promising performance for all important properties.

Competitive with the most reliable, empirically parameterizedfunctionals.

Hybrid functionals results

Properties of H2O molecule:experimental values and deviation from experiment

for different levels of theory

Property Exp. HF MP2Functionals

SVWN BLYP SLYP BVWN B3LYPROH/A 0.957 −0.016 0.004 0.013 0.015 0.019 0.010 0.005νs/cm−1 3832 288 −9 −106 −177 −155 −132 −33νas/cm−1 3943 279 5 −107 −186 −156 −142 −42µ/D 1.854 0.084 0.006 0.005 −0.051 0.007 −0.052 −0.006〈α〉/A3 1.427 −0.207 −0.004 0.109 0.143 0.179 0.075 0.026

MP2 — Møller–Plesset perturbation theory

Source: [Koch and Holthausen(2001)]

Hybrid functionals results

Dipole moment for different molecules: calculations vs. experiment

µexp)/

molecule

BLYPHCTHB3LYP

Source: [Cohen and Tantirungrotechai(1999)]

Approximate xc Functionals Beyond GGA

Meta-generalized gradient approximation (MGGA)

The next step to improve functionals is to introduce the Laplacians andkinetic energy density into the functional — this is the meta-generalizedgradient approximation scheme:

EMGGAxc [ρα; ρβ] =

fMGGAxc

(ρα; ρβ;∇ρα;∇ρβ; ∆ρα; ∆ρβ; τα; τβ

τσ =12

N/2∑i=1

|∇ϕiσ|2 — kinetic energy density of the σ-occupied orbitals.

Several meta-GGA’s have been constructed by a combination of theoreticalconstraints and fitting to chemical data. Some of them contain as manyas 20 parameters! Examples: PKZB (only one empirical parameter),TPSS (fully non-empirical). They usually perform better than LSD’s andGGA’s, but there are exceptions (e.g., surface energies and latticeconstants are less correct).

fMGGAxc

τσ =12

N/2∑i=1

Several meta-GGA’s have been constructed by a combination of theoreticalconstraints and fitting to chemical data. Some of them contain as manyas 20 parameters!

Examples: PKZB (only one empirical parameter),TPSS (fully non-empirical). They usually perform better than LSD’s andGGA’s, but there are exceptions (e.g., surface energies and latticeconstants are less correct).

fMGGAxc

τσ =12

N/2∑i=1

Several meta-GGA’s have been constructed by a combination of theoreticalconstraints and fitting to chemical data. Some of them contain as manyas 20 parameters! Examples: PKZB (only one empirical parameter),TPSS (fully non-empirical).

They usually perform better than LSD’s andGGA’s, but there are exceptions (e.g., surface energies and latticeconstants are less correct).

fMGGAxc

τσ =12

N/2∑i=1

Several meta-GGA’s have been constructed by a combination of theoreticalconstraints and fitting to chemical data. Some of them contain as manyas 20 parameters! Examples: PKZB (only one empirical parameter),TPSS (fully non-empirical). They usually perform better than LSD’s andGGA’s, but there are exceptions (e.g., surface energies and latticeconstants are less correct).

MGGA results

Statistical summary of the errors of density functionalsfor various properties of molecules and solids

Property Test set LSDGGA MGGA

PBE PBE0 PKZB TPSSAtomization en./(kcal/mol) G2 (148 mols.) 83.8 17.1 5.1 4.4 6.2Ionization en./eV G2 (86 species) 0.22 0.22 0.20 0.29 0.23Electron affinity/eV G2 (58 species) 0.26 0.12 0.17 0.14 0.14Bond length/A 96 molecules 0.013 0.016 0.010 0.027 0.014Harmonic frequency 82 diatomics 48.9 42.0 43.6 51.7 30.4

Source: [Tao et al.(2003)Tao, Perdew, Staroverov, and Scuseria]

Hyper-GGA and beyond

In hyper-GGA the MGGA functional is appended with the exact exchangeenergy densities:

EHGGAxc [ρα; ρβ] =∫R3

fHGGAxc

(ρα; ρβ;∇ρα;∇ρβ; ∆ρα; ∆ρβ; τα; τβ; εxα; εxβ

εxα(r) = − 12ρσ(r)

ρσ(r; r′)|r− r′|

d3r′.

Semiempirical hyper-GGAs include the widely used global hybridfunctionals such as B3LYP, B3PW91, or PBE0 that mix a fixed fraction ofexact exchange with GGA exchange, and the local hybrids, though thesefunctionals do not use all the ingredients prescribed above. Finally, toobtain the chemical accuracy, we can incorporate all the Kohn-Shamorbitals (occupied and virtual) into the functional. That requires hugebasis sets and is not yet ready for practical use.

fHGGAxc

εxα(r) = − 12ρσ(r)

ρσ(r; r′)|r− r′|

d3r′.

Semiempirical hyper-GGAs include the widely used global hybridfunctionals such as B3LYP, B3PW91, or PBE0 that mix a fixed fraction ofexact exchange with GGA exchange, and the local hybrids, though thesefunctionals do not use all the ingredients prescribed above.

Finally, toobtain the chemical accuracy, we can incorporate all the Kohn-Shamorbitals (occupied and virtual) into the functional. That requires hugebasis sets and is not yet ready for practical use.

fHGGAxc

εxα(r) = − 12ρσ(r)

ρσ(r; r′)|r− r′|

d3r′.

Semiempirical hyper-GGAs include the widely used global hybridfunctionals such as B3LYP, B3PW91, or PBE0 that mix a fixed fraction ofexact exchange with GGA exchange, and the local hybrids, though thesefunctionals do not use all the ingredients prescribed above. Finally, toobtain the chemical accuracy, we can incorporate all the Kohn-Shamorbitals (occupied and virtual) into the functional. That requires hugebasis sets and is not yet ready for practical use.

Jacob’s Ladder

GGA∇ρ

meta-GGA∇2ρ, τ

hyper-GGAεx

full orbital-based DFTvirtual ϕa

HARTREE WORLD

HEAVEN OF CHEMICAL ACCURACY

The xc functional approximations were arranged by J. P. Perdew withgrowing accuracy as rungs of a ladder. We can climb that ladder to get tothe heaven of chemical accuracy, an analogy to biblical Jacob’s Ladder:

Approximate xc Functionals Problems of Approximate Functionals

Self-interaction

In the HF model the non-physical self-interaction of the Coulomb e-erepulsion is removed by the exchange, so for hydrogen atom we always get

J [ρ] + Ex[ρ] = 0.

But it’s not the case for most of the approximate xc functionals: here arethe results for hydrogen atom for several functionals:

Functional J [ρ] Ex[ρ] Ec[ρ] J [ρ] + Exc[ρ]SVWN 0.29975 −0.25753 −0.03945 0.00277BLYP 0.30747 −0.30607 0.0 0.00140B3LYP 0.30845 −0.30370 −0.00756 −0.00281BP86 0.30653 −0.30479 −0.00248 −0.00074BPW91 0.30890 −0.30719 −0.00631 −0.00460HF 0.31250 −0.31250 0.0 0.0

To alleviate the problem, several solutions have been proposed, e.g. theself-interaction corrected KS in which the self-interaction is subtracteddemanding that J [ρ] = −Exc[ρ] for one-electron systems.

Self-interaction

J [ρ] + Ex[ρ] = 0.

Self-interaction

J [ρ] + Ex[ρ] = 0.

xc potential asymptotics

An electron in infinite distance r from other N − 1 electrons and Mnuclei of total charge Z =

∑Mα=1 Zα sees the potential

v(r) =N − 1− Z

The asymptotics of n-e and Coulomb potentials:

I limr→∞

vne(r) = − limr→∞

M∑α=1

Zα|r−Rα|

= −Zr

I limr→∞

(r) = limr→∞

ρ(r′)|r− r′|

d3r′ =N

So, the correct asymptotics of xc potential is limr→∞

vxc(r) = −1r

v(r) =N − 1− Z

I limr→∞

M∑α=1

Zα|r−Rα|

= −Zr

I limr→∞

(r) = limr→∞

ρ(r′)|r− r′|

d3r′ =N

vxc(r) = −1r

v(r) =N − 1− Z

I limr→∞

M∑α=1

Zα|r−Rα|

= −Zr

I limr→∞

(r) = limr→∞

ρ(r′)|r− r′|

d3r′ =N

vxc(r) = −1r

v(r) =N − 1− Z

I limr→∞

M∑α=1

Zα|r−Rα|

= −Zr

I limr→∞

(r) = limr→∞

ρ(r′)|r− r′|

d3r′ =N

vxc(r) = −1r

v(r) =N − 1− Z

I limr→∞

M∑α=1

Zα|r−Rα|

= −Zr

I limr→∞

(r) = limr→∞

ρ(r′)|r− r′|

d3r′ =N

vxc(r) = −1r

But that’s true for functionals satisfying the derivative discontinuitybehaviour, i.e. potentials which are not continous for the integerelectron numbers and continuous for fractional ones.

The correct asymptotics of the continuous xc potential is

limr→∞

vxc(r) = −1r

+ I + εHOMO,

I — first ionization energy, εHOMO — energy of the highest occupiedKS orbital.

Approximate xc functionals vanish exponentially which is too fast.

That’s why they need the asymptotic correction to properly describethe properties depending on long-range parts of xc potentials(electron affinities, polarizabilities, excitation energies).

limr→∞

vxc(r) = −1r

+ I + εHOMO,

limr→∞

vxc(r) = −1r

+ I + εHOMO,

limr→∞

vxc(r) = −1r

+ I + εHOMO,

Since vexactx has a correct long-range behaviour, the hybrid functionals

(with Eexactx ) have better asymptotics than the pure local functionals.

But the inclusion of too much exact exchange leads to problems. . .

Now, the idea is simple: preserve the GGA exchange at short-rangeand activate the exact exchange asymptotically through therange-separated Coulomb operator (ω — switching parameter):

=1− erf(ωr12)

r12︸︷︷︸short-range

Coulomb-like potential

+erf(ωr12)

r12︸︷︷︸nonsingular long-rangebackground potential

The functionals using this or similar ansatz are termed aslong-range-corrected (e.g. CAM-B3LYP). They are often used inTDDFT to calculate excited states related properties.

=1− erf(ωr12)

+erf(ωr12)

=1− erf(ωr12)

+erf(ωr12)

=1− erf(ωr12)

+erf(ωr12)

Long-range corrected functionals improve over:

linear and nonlinear optical properties of long-chain molecules,

the poor description of van der Waals bonds,

barrier heights,

charge transfer and Rydberg excitation energies,

and the corresponding oscillator strengths in time-dependent DFTcalculations.

barrier heights,

Points to remember:

There is a clear gradation of approximate xc functionals, from LSD tohyper-GGA.

Approximate xc functionals include unphysical self-interactioncontribution.

Also, their asymptotics is not correct and needs some patches iflong-range properties are of interest.

Points to remember:

The End (for today)

Supplement

back to symbols

Permutation examples:

f(x1;x2) = cos (x1 − x2),

P12f(x1;x2) = cos (x2 − x1) = cos (x1 − x2),

P12f(x1;x2) = f(x1;x2) : symmetric.

g(x1;x2) = sin (x1 − x2),

P12g(x1;x2) = sin (x2 − x1) = − sin (x1 − x2),

P12g(x1;x2) = −g(x1;x2) : antisymmetric.

Supplement

back to symbols

f(x1;x2) = cos (x1 − x2),

P12f(x1;x2) = cos (x2 − x1) = cos (x1 − x2),

g(x1;x2) = sin (x1 − x2),

P12g(x1;x2) = sin (x2 − x1) = − sin (x1 − x2),

Supplement

back to symbols

f(x1;x2) = cos (x1 − x2),

P12f(x1;x2) = cos (x2 − x1) = cos (x1 − x2),

g(x1;x2) = sin (x1 − x2),

P12g(x1;x2) = sin (x2 − x1) = − sin (x1 − x2),

Supplement

back to symbols

f(x1;x2) = cos (x1 − x2),

P12f(x1;x2) = cos (x2 − x1) = cos (x1 − x2),

g(x1;x2) = sin (x1 − x2),

P12g(x1;x2) = sin (x2 − x1) = − sin (x1 − x2),

Supplement

back to symbols

f(x1;x2) = cos (x1 − x2),

P12f(x1;x2) = cos (x2 − x1) = cos (x1 − x2),

g(x1;x2) = sin (x1 − x2),

P12g(x1;x2) = sin (x2 − x1) = − sin (x1 − x2),

Supplement

back to symbols

f(x1;x2) = cos (x1 − x2),

P12f(x1;x2) = cos (x2 − x1) = cos (x1 − x2),

g(x1;x2) = sin (x1 − x2),

P12g(x1;x2) = sin (x2 − x1) = − sin (x1 − x2),

Supplement

back to electron density

σ=− 12

f(r) d3r =∫ ∞−∞

∫ ∞−∞

f(r) dx dy dz =

=∫ ∞

∫ π

∫ 2π

0f(r)r2 sin θ dr dθ dϕ =

= (or other coordinates) . . .

Supplement

back to electron density

σ=− 12

f(r) d3r =∫ ∞−∞

∫ ∞−∞

f(r) dx dy dz =

=∫ ∞

∫ π

∫ 2π

0f(r)r2 sin θ dr dθ dϕ =

= (or other coordinates) . . .

Supplement

back to expectation values

Brakets are shorthand notation for the integration over all electroniccoordinates:

〈ψ|A|ψ〉 =

σ1...σN

ψ(r1; . . . ; qN )∗Aψ(r1; . . . ; qN )d3r1 . . . d3rN .

Supplement

back to Fock operator

Functional F maps a function f to a number α:

f 7→ F [f ] = α ∈ C.

Examples of functionals:

F [f ] =∫ b

a|f(x)| dx — area under the curve f(x) for x ∈ 〈a; b〉

F [f ] =∫ b

√1 + [f ′(x)]2 dx — length of curve f(x) for x ∈ 〈a; b〉.

A[ψ] =〈ψ|A|ψ〉〈ψ|ψ〉

= 〈A〉 — every physical observable is a functional of

the wavefunction.

Supplement

f 7→ F [f ] = α ∈ C.

F [f ] =∫ b

A[ψ] =〈ψ|A|ψ〉〈ψ|ψ〉

the wavefunction.

Supplement

f 7→ F [f ] = α ∈ C.

F [f ] =∫ b

A[ψ] =〈ψ|A|ψ〉〈ψ|ψ〉

the wavefunction.

Supplement

f 7→ F [f ] = α ∈ C.

F [f ] =∫ b

A[ψ] =〈ψ|A|ψ〉〈ψ|ψ〉

the wavefunction.

Supplement

f 7→ F [f ] = α ∈ C.

F [f ] =∫ b

A[ψ] =〈ψ|A|ψ〉〈ψ|ψ〉

the wavefunction.

Supplement

Density function and the density in HF:

ρ(r; r′) = 2N/2∑i=1

ϕi(r)ϕ∗i (r′), ρ(r) = ρ(r; r) = 2

N/2∑i=1

|ϕi(r)|2.

Kinetic energy: T [ρ] = −12

[∆rρ(r; r′)

]r′=r

n-e attraction: Vne[ρ] =∫

vne(r)ρ(r) d3r.

Coulomb e-e repulsion: J [ρ] =12

ρ(r1)ρ(r2)r12

d3r1 d3r2.

e-e exchange:

K[ρ] = −Ex[ρ] =14

ρ(r1; r2)ρ(r2; r1)r12

d3r1 d3r2.

Supplement

ρ(r; r′) = 2N/2∑i=1

ϕi(r)ϕ∗i (r′), ρ(r) = ρ(r; r) = 2

N/2∑i=1

|ϕi(r)|2.

[∆rρ(r; r′)

]r′=r

vne(r)ρ(r) d3r.

ρ(r1)ρ(r2)r12

d3r1 d3r2.

e-e exchange:

K[ρ] = −Ex[ρ] =14

ρ(r1; r2)ρ(r2; r1)r12

d3r1 d3r2.

Supplement

ρ(r; r′) = 2N/2∑i=1

ϕi(r)ϕ∗i (r′), ρ(r) = ρ(r; r) = 2

N/2∑i=1

|ϕi(r)|2.

[∆rρ(r; r′)

]r′=r

vne(r)ρ(r) d3r.

ρ(r1)ρ(r2)r12

d3r1 d3r2.

e-e exchange:

K[ρ] = −Ex[ρ] =14

ρ(r1; r2)ρ(r2; r1)r12

d3r1 d3r2.

Supplement

ρ(r; r′) = 2N/2∑i=1

ϕi(r)ϕ∗i (r′), ρ(r) = ρ(r; r) = 2

N/2∑i=1

|ϕi(r)|2.

[∆rρ(r; r′)

]r′=r

vne(r)ρ(r) d3r.

ρ(r1)ρ(r2)r12

d3r1 d3r2.

e-e exchange:

K[ρ] = −Ex[ρ] =14

ρ(r1; r2)ρ(r2; r1)r12

d3r1 d3r2.

Supplement

ρ(r; r′) = 2N/2∑i=1

ϕi(r)ϕ∗i (r′), ρ(r) = ρ(r; r) = 2

N/2∑i=1

|ϕi(r)|2.

[∆rρ(r; r′)

]r′=r

vne(r)ρ(r) d3r.

ρ(r1)ρ(r2)r12

d3r1 d3r2.

e-e exchange:

K[ρ] = −Ex[ρ] =14

ρ(r1; r2)ρ(r2; r1)r12

d3r1 d3r2.

Bibliography I

R. G. Parr and W. Yang, Density-Functional Theory of Atoms andMolecules (Oxford University Press, 1989).

J. P. Perdew and S. Kurt, A Primer in Density Functional Theory(Springer Berlin / Heidelberg, 2003), vol. 620 of Lecture Notes inPhysics, chap. Density Functionals for Non-relativistic CoulombSystems in the New Century.

A. D. Becke, Phys. Rev. A 38, 3098 (1988).

J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R.Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992).

J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865(1996).

A. Becke, J. Chem. Phys. 98, 5648 (1993).

Bibliography II

P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, J.Phys. Chem. 98, 11623 (1994).

C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 (1999).

W. Koch and M. C. Holthausen, A Chemist’s Guide to DensityFunctional Theory (Wiley-VCH Verlag GmbH, Weinheim, 2001).

A. J. Cohen and Y. Tantirungrotechai, Chem. Phys. Lett. 299, 465(1999).

J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys. Rev.Lett. 91, 146401 (2003).

Thank you for your attention . . .

density functional theory - quantum chemistry...

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