density functional theory - quantum chemistry...
TRANSCRIPT
Density Functional Theory
Łukasz Rajchel
Interdisciplinary Centerfor Mathematical and Computational Modeling
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
Outline of the Lecture
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
Outline of the Lecture
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
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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
1 DFT — A Real Celebrity
2 PreliminariesBasic Concepts of Quantum ChemistryElectronic DistributionApproximate Methods
3 Hartree-Fock
4 Fermi and Coulomb Holes
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),
ri =
xiyizi
, Rα =
Xα
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 8 / 101
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),
ri =
xiyizi
, Rα =
Xα
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 8 / 101
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),
ri =
xiyizi
, Rα =
Xα
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 8 / 101
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),
ri =
xiyizi
, Rα =
Xα
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 8 / 101
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),
ri =
xiyizi
, Rα =
Xα
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 8 / 101
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),
ri =
xiyizi
, Rα =
Xα
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 8 / 101
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),
ri =
xiyizi
, Rα =
Xα
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 8 / 101
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),
ri =
xiyizi
, Rα =
Xα
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 8 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 9 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 9 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 9 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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
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
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 9 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 10 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 10 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 10 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 10 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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 ).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 11 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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 ).
more on Pij
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 11 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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 ).
This yields the two possibilities:
Pijψ =
ψ → bosons: photons, gluons, W, Z, Higgs?, . . .
−ψ → fermions: electrons, protons, neutrons, quarks, . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 11 / 101
Preliminaries Basic Concepts of Quantum Chemistry
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 ).
For electrons, the wavefunctions must be antisymmetric with respect toelectron permutation:
Pijψ = −ψ.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 11 / 101
Preliminaries Basic Concepts of Quantum Chemistry
Some remarks on the wavefunction
Wavefunction is a fairly complicated object! For N -electron system itdepends on 4N variables. For systems of biological importance thismay boil down to several thousand variables . . .
But Hamiltionian contains only one- and two-electron operators, sinceelectrons don’t have internal structure (no many-body contributions).
So, do we really need the state of the art, but incredibly expensivewavefunction? Is there something cheaper that would do the job wewant, i.e. yield the energy and other properties?
Fortunately, the answer is yes. It’s the electron density — it’ssimple, cheap, and you can buy it in Walmart.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 12 / 101
Preliminaries Basic Concepts of Quantum Chemistry
Some remarks on the wavefunction
Wavefunction is a fairly complicated object! For N -electron system itdepends on 4N variables. For systems of biological importance thismay boil down to several thousand variables . . .
But Hamiltionian contains only one- and two-electron operators, sinceelectrons don’t have internal structure (no many-body contributions).
So, do we really need the state of the art, but incredibly expensivewavefunction? Is there something cheaper that would do the job wewant, i.e. yield the energy and other properties?
Fortunately, the answer is yes. It’s the electron density — it’ssimple, cheap, and you can buy it in Walmart.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 12 / 101
Preliminaries Basic Concepts of Quantum Chemistry
Some remarks on the wavefunction
Wavefunction is a fairly complicated object! For N -electron system itdepends on 4N variables. For systems of biological importance thismay boil down to several thousand variables . . .
But Hamiltionian contains only one- and two-electron operators, sinceelectrons don’t have internal structure (no many-body contributions).
So, do we really need the state of the art, but incredibly expensivewavefunction? Is there something cheaper that would do the job wewant, i.e. yield the energy and other properties?
Fortunately, the answer is yes. It’s the electron density — it’ssimple, cheap, and you can buy it in Walmart.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 12 / 101
Preliminaries Basic Concepts of Quantum Chemistry
Some remarks on the wavefunction
Wavefunction is a fairly complicated object! For N -electron system itdepends on 4N variables. For systems of biological importance thismay boil down to several thousand variables . . .
But Hamiltionian contains only one- and two-electron operators, sinceelectrons don’t have internal structure (no many-body contributions).
So, do we really need the state of the art, but incredibly expensivewavefunction? Is there something cheaper that would do the job wewant, i.e. yield the energy and other properties?
Fortunately, the answer is yes. It’s the electron density — it’ssimple, cheap, and you can buy it in Walmart.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 12 / 101
Preliminaries Electronic Distribution
Electron density
The probability of finding any electron anywhere must be 1, so a properwavefunction should be normalized, giving 1 upon full integration:∑
σ1,...,σN
∫R3
. . .
∫R3
|ψ(q1; . . . ; qN )|2 d3r1 . . . d3rN = 1.
more on integration
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 13 / 101
Preliminaries Electronic Distribution
Electron density
The probability of finding any electron anywhere must be 1, so a properwavefunction should be normalized, giving 1 upon full integration:∑
σ1,...,σN
∫R3
. . .
∫R3
|ψ(q1; . . . ; qN )|2 d3r1 . . . d3rN = 1.
If we perform the integration over all the spatial coordinates but one (arbi-trarily chosen, say one) and over all spin variables, we get the well-knowndensity distribution (the quantity measured in crystallography!):
ρ(r) = N∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
|ψ(r;σ1; q2; . . . ; qN )|2 d3r2 . . . d3rN .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 13 / 101
Preliminaries Electronic Distribution
Electron density
The probability of finding any electron anywhere must be 1, so a properwavefunction should be normalized, giving 1 upon full integration:∑
σ1,...,σN
∫R3
. . .
∫R3
|ψ(q1; . . . ; qN )|2 d3r1 . . . d3rN = 1.
ρ(r) is a 3D function and as such can’t be presented by a 3D graph. How-ever, its isosurfaces, i.e. implicit functions ρ(r) = const > 0 may be plotted,e.g. for water:
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 13 / 101
Preliminaries Electronic Distribution
Ground-state electron density:
vanishes at infinity: limr→∞
ρ(r) = 0,
integrates to the number of electrons,∫
R3
ρ(r) d3r = N ,
has a finite value at nuclei positions and cusps in their neigbourhood(r→ Rα),
the cusp steepness keeps the information on nuclear charge:∂
∂rρ(r)|Rα
= −2Zαρ(Rα).
The electron density already provides all the information on the molecule!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 14 / 101
Preliminaries Electronic Distribution
Ground-state electron density:
vanishes at infinity: limr→∞
ρ(r) = 0,
integrates to the number of electrons,∫
R3
ρ(r) d3r = N ,
has a finite value at nuclei positions and cusps in their neigbourhood(r→ Rα),
the cusp steepness keeps the information on nuclear charge:∂
∂rρ(r)|Rα
= −2Zαρ(Rα).
The electron density already provides all the information on the molecule!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 14 / 101
Preliminaries Electronic Distribution
Ground-state electron density:
vanishes at infinity: limr→∞
ρ(r) = 0,
integrates to the number of electrons,∫
R3
ρ(r) d3r = N ,
has a finite value at nuclei positions and cusps in their neigbourhood(r→ Rα),
the cusp steepness keeps the information on nuclear charge:∂
∂rρ(r)|Rα
= −2Zαρ(Rα).
The electron density already provides all the information on the molecule!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 14 / 101
Preliminaries Electronic Distribution
Ground-state electron density:
vanishes at infinity: limr→∞
ρ(r) = 0,
integrates to the number of electrons,∫
R3
ρ(r) d3r = N ,
has a finite value at nuclei positions and cusps in their neigbourhood(r→ Rα),
the cusp steepness keeps the information on nuclear charge:∂
∂rρ(r)|Rα
= −2Zαρ(Rα).
The electron density already provides all the information on the molecule!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 14 / 101
Preliminaries Electronic Distribution
Ground-state electron density:
vanishes at infinity: limr→∞
ρ(r) = 0,
integrates to the number of electrons,∫
R3
ρ(r) d3r = N ,
has a finite value at nuclei positions and cusps in their neigbourhood(r→ Rα),
the cusp steepness keeps the information on nuclear charge:∂
∂rρ(r)|Rα
= −2Zαρ(Rα).
The electron density already provides all the information on the molecule!Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 14 / 101
Preliminaries Electronic Distribution
Expectation values
Dynamical variable A (energy, momentum, velocity, time, . . . ) →operator A → expectation (mean) value of that operator:
〈A〉 = 〈ψ|A|ψ〉
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) . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 15 / 101
Preliminaries Electronic Distribution
Expectation values
Dynamical variable A (energy, momentum, velocity, time, . . . ) →operator A → expectation (mean) value of that operator:
〈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) . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 15 / 101
Preliminaries Electronic Distribution
Expectation values
Dynamical variable A (energy, momentum, velocity, time, . . . ) →operator A → expectation (mean) value of that operator:
〈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) . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 15 / 101
Preliminaries Electronic Distribution
Expectation values
Dynamical variable A (energy, momentum, velocity, time, . . . ) →operator A → expectation (mean) value of that operator:
〈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) . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 15 / 101
Preliminaries Electronic Distribution
One-matrix
Density function (one-matrix) is a generalization of the electronic density:
ρ(r; r′) = N∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
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
∫R3
. . .
∫R3
ψ∗(q1; . . . ; qN )×
×
(−1
2
N∑i=1
∆ri
)ψ(q1; . . . ; qN ) d3r1 . . . d
3rN =
= −12
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
Electronic density is the diagonal part of one-matrix: ρ(r) = ρ(r; r).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 16 / 101
Preliminaries Electronic Distribution
One-matrix
Density function (one-matrix) is a generalization of the electronic density:
ρ(r; r′) = N∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
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
∫R3
. . .
∫R3
ψ∗(q1; . . . ; qN )×
×
(−1
2
N∑i=1
∆ri
)ψ(q1; . . . ; qN ) d3r1 . . . d
3rN =
= −12
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
Electronic density is the diagonal part of one-matrix: ρ(r) = ρ(r; r).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 16 / 101
Preliminaries Electronic Distribution
One-matrix
Density function (one-matrix) is a generalization of the electronic density:
ρ(r; r′) = N∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
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
∫R3
. . .
∫R3
ψ∗(q1; . . . ; qN )×
×
(−1
2
N∑i=1
∆ri
)ψ(q1; . . . ; qN ) d3r1 . . . d
3rN =
= −12
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
Electronic density is the diagonal part of one-matrix: ρ(r) = ρ(r; r).Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 16 / 101
Preliminaries Electronic Distribution
Pair density
γ(r1; r2) = N(N − 1)×
×∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
|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).
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 17 / 101
Preliminaries Electronic Distribution
Pair density
γ(r1; r2) = N(N − 1)×
×∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
|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).
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 17 / 101
Preliminaries Electronic Distribution
Pair density
γ(r1; r2) = N(N − 1)×
×∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
|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).
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 17 / 101
Preliminaries Electronic Distribution
Pair density
γ(r1; r2) = N(N − 1)×
×∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
|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).
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 17 / 101
Preliminaries Electronic Distribution
Pair density
γ(r1; r2) = N(N − 1)×
×∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
|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).
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 17 / 101
Preliminaries Electronic Distribution
Pair density
γ(r1; r2) = N(N − 1)×
×∑
σ1,σ2,...,σN
∫R3
. . .
∫R3
|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).
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 17 / 101
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〉 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 18 / 101
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 = min
ψ→N〈ψ|H|ψ〉 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 18 / 101
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〉 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 18 / 101
Outline of the Talk
1 DFT — A Real Celebrity
2 Preliminaries
3 Hartree-FockVariation in HFEquationsCorrelation and exchangeSelf-Interaction in HF
4 Fermi and Coulomb Holes
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 20 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 20 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 20 / 101
Hartree-Fock Variation in HF
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′).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 21 / 101
Hartree-Fock Variation in HF
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′).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 21 / 101
Hartree-Fock Variation in HF
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)
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 22 / 101
Hartree-Fock Variation in HF
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)
.
what is a functional?
how do these terms look like?
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 22 / 101
Hartree-Fock Variation in HF
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)
.
Goal: minimize HF energy varying the orbitals ϕi (spatial parts of spinor-bitals φi) while keeping them orthonormal.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 22 / 101
Hartree-Fock Variation in HF
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)
.
Result: HF equations for the best orbitals, i.e. orbitals yielding minimumHF energy.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 22 / 101
Hartree-Fock Equations
Fock operator
HF equations for best orbitals:
fϕi = εiϕi.
Fock operator:
f(r) = −12
∆r + vne(r) + vHF(r).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 23 / 101
Hartree-Fock Equations
Fock operator
HF equations for best orbitals:
fϕi = εiϕi.
Fock operator:
f(r) = −12
∆r + vne(r) + vHF(r).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 23 / 101
Hartree-Fock Equations
Fock operator
HF equations for best orbitals:
fϕi = εiϕi.
Fock operator:
f(r) = −12
∆r + vne(r) + vHF(r).
Nuclear potential:
vne(r) = −M∑α=1
Zα|r−Rα|
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 23 / 101
Hartree-Fock Equations
Fock operator
HF equations for best orbitals:
fϕi = εiϕi.
Fock operator:
f(r) = −12
∆r + vne(r) + vHF(r).
HF potential: the average repulsive potential experienced by the electronfrom to the remaining N − 1 electrons.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 23 / 101
Hartree-Fock Equations
Fock operator
HF equations for best orbitals:
fϕi = εiϕi.
Fock operator:
f(r) = −12
∆r + vne(r) + vHF(r).
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 23 / 101
Hartree-Fock Equations
Coulomb and echange operators
HF potential: Coulomb− exchange,
vHF = ︸︷︷︸Coulomb
− k︸︷︷︸exchange
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 24 / 101
Hartree-Fock Equations
Coulomb and echange operators
HF potential: Coulomb− exchange,
vHF = ︸︷︷︸Coulomb
− k︸︷︷︸exchange
.
(r)f(r) =(∫
R3
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 24 / 101
Hartree-Fock Equations
Coulomb and echange operators
HF potential: Coulomb− exchange,
vHF = ︸︷︷︸Coulomb
− k︸︷︷︸exchange
.
(r)f(r) =(∫
R3
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 24 / 101
Hartree-Fock Equations
Coulomb and echange operators
HF potential: Coulomb− exchange,
vHF = ︸︷︷︸Coulomb
− k︸︷︷︸exchange
.
(r)f(r) =(∫
R3
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 24 / 101
Hartree-Fock Equations
Coulomb and echange operators
HF potential: Coulomb− exchange,
vHF = ︸︷︷︸Coulomb
− k︸︷︷︸exchange
.
k(r)f(r) =12
∫R3
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 24 / 101
Hartree-Fock Equations
Coulomb and echange operators
HF potential: Coulomb− exchange,
vHF = ︸︷︷︸Coulomb
− k︸︷︷︸exchange
.
k(r)f(r) =12
∫R3
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 24 / 101
Hartree-Fock Equations
Coulomb and echange operators
HF potential: Coulomb− exchange,
vHF = ︸︷︷︸Coulomb
− k︸︷︷︸exchange
.
k(r)f(r) =12
∫R3
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 24 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 25 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 25 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 25 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 25 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 25 / 101
Hartree-Fock Correlation and exchange
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
EHF
E
6?−Ecor
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 26 / 101
Hartree-Fock Correlation and exchange
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
EHF
E
6?−Ecor
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 26 / 101
Hartree-Fock Correlation and exchange
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
EHF
E
6?−Ecor
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 26 / 101
Hartree-Fock Correlation and exchange
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
EHF
E
6?−Ecor
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 26 / 101
Hartree-Fock Correlation and exchange
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
EHF
E
6?−Ecor
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 26 / 101
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 27 / 101
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 27 / 101
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 27 / 101
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 27 / 101
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 27 / 101
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 27 / 101
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 27 / 101
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 27 / 101
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 27 / 101
Outline of the Talk
1 DFT — A Real Celebrity
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
1rij.
Exact wavefunction ψ → exact pair density γ → exact e-e repulsion:
Eee = 〈ψ|Vee|ψ〉 =12
∫R3
∫R3
γ(r1; r2)r12
d3r1 d3r2.
In HF the e-e repulsion is
J [ρ] + Ex[ρ] =12
∫R3
∫R3
(ρ(r1)ρ(r2)
r12− 1
2ρ(r1; r2)ρ(r2; r1)
r12
)d3r1 d
3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 29 / 101
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
1rij.
Exact wavefunction ψ → exact pair density γ → exact e-e repulsion:
Eee = 〈ψ|Vee|ψ〉 =12
∫R3
∫R3
γ(r1; r2)r12
d3r1 d3r2.
In HF the e-e repulsion is
J [ρ] + Ex[ρ] =12
∫R3
∫R3
(ρ(r1)ρ(r2)
r12− 1
2ρ(r1; r2)ρ(r2; r1)
r12
)d3r1 d
3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 29 / 101
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
1rij.
Exact wavefunction ψ → exact pair density γ → exact e-e repulsion:
Eee = 〈ψ|Vee|ψ〉 =12
∫R3
∫R3
γ(r1; r2)r12
d3r1 d3r2.
In HF the e-e repulsion is
J [ρ] + Ex[ρ] =12
∫R3
∫R3
(ρ(r1)ρ(r2)
r12− 1
2ρ(r1; r2)ρ(r2; r1)
r12
)d3r1 d
3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 29 / 101
Fermi and Coulomb Holes Definitions
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 30 / 101
Fermi and Coulomb Holes Definitions
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 30 / 101
Fermi and Coulomb Holes Definitions
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 30 / 101
Fermi and Coulomb Holes Definitions
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 30 / 101
Fermi and Coulomb Holes Definitions
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∫
R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 31 / 101
Fermi and Coulomb Holes Definitions
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∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 31 / 101
Fermi and Coulomb Holes Definitions
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∫
R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 31 / 101
Fermi and Coulomb Holes Definitions
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 32 / 101
Fermi and Coulomb Holes Definitions
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 32 / 101
Fermi and Coulomb Holes Definitions
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 32 / 101
Fermi and Coulomb Holes Definitions
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 32 / 101
Fermi and Coulomb Holes Definitions
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.
Because we love integrals, let’s integrate:∫
R3
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 Definitions
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 33 / 101
Fermi and Coulomb Holes Definitions
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 33 / 101
Fermi and Coulomb Holes Definitions
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 33 / 101
Fermi and Coulomb Holes Definitions
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 34 / 101
Fermi and Coulomb Holes Definitions
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 34 / 101
Fermi and Coulomb Holes Definitions
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
∫R3
∫R3
γ(r1; r2)r12
d3r1 d3r2 =
12
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2 +
+12
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 35 / 101
Fermi and Coulomb Holes Definitions
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
∫R3
∫R3
γ(r1; r2)r12
d3r1 d3r2 =
12
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2 +
+12
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 35 / 101
Fermi and Coulomb Holes Definitions
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
∫R3
∫R3
γ(r1; r2)r12
d3r1 d3r2 =
12
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2 +
+12
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 35 / 101
Fermi and Coulomb Holes Definitions
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
∫R3
∫R3
γ(r1; r2)r12
d3r1 d3r2 =
12
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2 +
+12
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 35 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 36 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 36 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 36 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 36 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 36 / 101
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.
The End (for today)
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 36 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 39 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 39 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 39 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 39 / 101
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.
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r +∫
R3
vne(r)ρ(r) d3r +
+12
∫R3
∫R3
γ(r1; r2)r12
d3r1 d3r2 +
M−1∑α=1
M∑β=α+1
ZαZβRαβ
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 39 / 101
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.
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
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2 +
12
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
d3r1 d3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 39 / 101
Density and Energy Remarks and Problems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 40 / 101
Density and Energy Remarks and Problems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 40 / 101
Density and Energy Remarks and Problems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 40 / 101
Density and Energy Remarks and Problems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 40 / 101
Density and Energy Remarks and Problems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 40 / 101
Density and Energy Remarks and Problems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 40 / 101
Density and Energy Remarks and Problems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 40 / 101
Density and Energy Remarks and Problems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 40 / 101
Density and Energy Remarks and Problems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 40 / 101
Density and Energy Remarks and Problems
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
the notorious e-e repulsion term which on the pair density instead of
the density alone: Eee[γ] =12
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 41 / 101
Density and Energy Remarks and Problems
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
the notorious e-e repulsion term which on the pair density instead of
the density alone: Eee[γ] =12
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 41 / 101
Density and Energy Remarks and Problems
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
the notorious e-e repulsion term which on the pair density instead of
the density alone: Eee[γ] =12
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 41 / 101
Density and Energy Remarks and Problems
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
the notorious e-e repulsion term which on the pair density instead of
the density alone: Eee[γ] =12
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 41 / 101
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
∫R3
ρ5/3(r) d3r.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 42 / 101
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
∫R3
ρ5/3(r) d3r.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 42 / 101
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
∫R3
ρ5/3(r) d3r.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 42 / 101
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
∫R3
ρ5/3(r) d3r.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 42 / 101
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
∫R3
ρ5/3(r) d3r.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 42 / 101
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
∫R3
ρ5/3(r) d3r.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 42 / 101
Density and Energy Historical Models
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]
+Vnn.
Ex[ρ] = −Cx
∫R3
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:
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 43 / 101
Density and Energy Historical Models
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]+Vnn.
Ex[ρ] = −Cx
∫R3
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:
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 43 / 101
Density and Energy Historical Models
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]+Vnn.
Ex[ρ] = −Cx
∫R3
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:
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 43 / 101
Density and Energy Historical Models
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]+Vnn.
Ex[ρ] = −Cx
∫R3
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:
EHFx [ρ] =
12
∫R3
∫R3
ρ(r1)hx(r1; r2)r12
d3r1 d3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 43 / 101
Density and Energy Historical Models
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]+Vnn.
Ex[ρ] = −Cx
∫R3
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:
EHFx [ρ] = −1
4
∫R3
∫R3
ρ(r1; r1)ρ(r2; r1)r12
d3r1 d3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 43 / 101
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 44 / 101
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 44 / 101
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 44 / 101
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)]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
7 Kohn-Sham Approach
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|
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 46 / 101
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|
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 46 / 101
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|
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 46 / 101
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|
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 46 / 101
Hohenberg-Kohn Theorems Definitions
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] =∫
R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 47 / 101
Hohenberg-Kohn Theorems Definitions
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] =∫
R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 47 / 101
Hohenberg-Kohn Theorems Definitions
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] =∫
R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 47 / 101
Hohenberg-Kohn Theorems Definitions
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] =∫
R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 47 / 101
Hohenberg-Kohn Theorems The Theorems
Hohenberg-Kohn Theorems
E[ρ] =∫
R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 48 / 101
Hohenberg-Kohn Theorems The Theorems
Hohenberg-Kohn Theorems
E[ρ] =∫
R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 48 / 101
Hohenberg-Kohn Theorems The Theorems
Hohenberg-Kohn Theorems
E[ρ] =∫
R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 48 / 101
Hohenberg-Kohn Theorems The Theorems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 49 / 101
Hohenberg-Kohn Theorems The Theorems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 49 / 101
Hohenberg-Kohn Theorems The Theorems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 49 / 101
Hohenberg-Kohn Theorems The Theorems
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 49 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 50 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 50 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 50 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 50 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 50 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 50 / 101
Hohenberg-Kohn Theorems Representability of the Density
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 51 / 101
Hohenberg-Kohn Theorems Representability of the Density
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 51 / 101
Hohenberg-Kohn Theorems Representability of the Density
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 51 / 101
Hohenberg-Kohn Theorems Representability of the Density
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
i .
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∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
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
i .
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∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
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
i .
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∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
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
i .
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∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
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
i .
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∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) .Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
Constrained-search approach
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
Constrained-search approach
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
Constrained-search approach
E0 = minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩)
=
= minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vee
∣∣∣ψ⟩+∫
R3
vext(r)ρ(r) d3r)
+ Vnn =
= minρ
(F [ρ] +
∫R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
Constrained-search approach
E0 = minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) =
= minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vee
∣∣∣ψ⟩+∫
R3
vext(r)ρ(r) d3r)
+ Vnn
=
= minρ
(F [ρ] +
∫R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
Constrained-search approach
E0 = minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) =
= minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vee
∣∣∣ψ⟩+∫
R3
vext(r)ρ(r) d3r)
+ Vnn =
= minρ
(F [ρ] +
∫R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
Constrained-search approach
E0 = minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) =
= minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vee
∣∣∣ψ⟩+∫
R3
vext(r)ρ(r) d3r)
+ Vnn =
= minρ
(F [ρ] +
∫R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
Constrained-search approach
E0 = minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) =
= minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vee
∣∣∣ψ⟩+∫
R3
vext(r)ρ(r) d3r)
+ Vnn =
= minρ
(F [ρ] +
∫R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Hohenberg-Kohn Theorems Representability of the Density
Constrained-search approach
E0 = minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vext + Vee + Vnn
∣∣∣ψ⟩) =
= minρ
(minψ→ρ
⟨ψ∣∣∣ T + Vee
∣∣∣ψ⟩+∫
R3
vext(r)ρ(r) d3r)
+ Vnn =
= minρ
(F [ρ] +
∫R3
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].
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 52 / 101
Outline of the Talk
5 Density and Energy
6 Hohenberg-Kohn Theorems
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r = −N/2∑i=1
〈ϕi|∆r|ϕi〉 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 54 / 101
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r = −N/2∑i=1
〈ϕi|∆r|ϕi〉 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 54 / 101
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r = −N/2∑i=1
〈ϕi|∆r|ϕi〉 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 54 / 101
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r = −N/2∑i=1
〈ϕi|∆r|ϕi〉 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 54 / 101
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r = −N/2∑i=1
〈ϕi|∆r|ϕi〉 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 54 / 101
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 55 / 101
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 55 / 101
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 55 / 101
Kohn-Sham Approach KS Determinant and KS 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:
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 56 / 101
Kohn-Sham Approach KS Determinant and KS 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:
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 56 / 101
Kohn-Sham Approach KS Determinant and KS 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:
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 56 / 101
Kohn-Sham Approach KS Determinant and KS 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:
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.Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 56 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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
+Vnn
.
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!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 57 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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
+Vnn
.
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!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 57 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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
+Vnn
.
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!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 57 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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
+Vnn.
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!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 57 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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
+Vnn.
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!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 57 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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
+Vnn.
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!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 57 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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
+Vnn.
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!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 57 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 58 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 58 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 58 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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)
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 59 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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)
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 59 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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)
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 59 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 60 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 60 / 101
Kohn-Sham Approach KS Determinant and KS Energy
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 60 / 101
Outline of the Talk
5 Density and Energy
6 Hohenberg-Kohn Theorems
7 Kohn-Sham Approach
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 62 / 101
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 62 / 101
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 62 / 101
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 62 / 101
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 62 / 101
xc Functionals Is There a Road Map?
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 63 / 101
xc Functionals Is There a Road Map?
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 63 / 101
xc Functionals Is There a Road Map?
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 63 / 101
xc Functionals Is There a Road Map?
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 63 / 101
xc Functionals Is There a Road Map?
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 . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 63 / 101
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 64 / 101
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 64 / 101
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 64 / 101
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 64 / 101
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 64 / 101
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 64 / 101
xc Functionals Adiabatic Connection
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 65 / 101
xc Functionals Adiabatic Connection
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 65 / 101
xc Functionals Adiabatic Connection
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 65 / 101
xc Functionals Adiabatic Connection
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 65 / 101
xc Functionals Adiabatic Connection
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 65 / 101
xc Functionals Adiabatic Connection
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
d3r1 d3r2 +Vnn,
whereas the equivalent expression, which we already know very well, reads
E[ρ] = T [ρ] +Vne[ρ] +Jne[ρ] +12
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
d3r1 d3r2+, Vnn.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 66 / 101
xc Functionals Adiabatic Connection
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
d3r1 d3r2 +Vnn,
whereas the equivalent expression, which we already know very well, reads
E[ρ] = T [ρ] +Vne[ρ] +Jne[ρ] +12
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
d3r1 d3r2+, Vnn.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 66 / 101
xc Functionals Adiabatic Connection
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
d3r1 d3r2 +Vnn,
whereas the equivalent expression, which we already know very well, reads
E[ρ] = T [ρ] +Vne[ρ] +Jne[ρ] +12
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 66 / 101
xc Functionals Adiabatic Connection
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
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
d3r1 d3r2 +Vnn,
whereas the equivalent expression, which we already know very well, reads
E[ρ] = T [ρ] +Vne[ρ] +Jne[ρ] +12
∫R3
∫R3
ρ(r1)hxc(r1; r2)r12
d3r1 d3r2+, Vnn.
Finally, the xc energy in the adiabatic connection approach reads
Exc[ρ] =ρ(r1)hxc(r1; r2)
r12d3r1 d
3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 66 / 101
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 67 / 101
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 67 / 101
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 67 / 101
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 .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 67 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 68 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 68 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 68 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 68 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 68 / 101
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.
The End (for today)
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 68 / 101
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 71 / 101
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 71 / 101
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 71 / 101
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 71 / 101
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 71 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 72 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 72 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 72 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 72 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 72 / 101
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 ( Thomas-Fermi(-Dirac) models ) and, to very highaccuracy, correlation.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 72 / 101
Approximate xc Functionals LDA and LSD
Local density approximation (LDA)
In the LDA the xc energy is assumed to be
ELDAxc [ρ] =
∫R3
ρ(r)ε0xc
(ρ(r)
)d3r
ε0xc
(ρ(r)
)— xc energy density in uniform electron gas model (depends
only on the density). It splits into exchange and correlation parts
ε0xc
(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
∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 73 / 101
Approximate xc Functionals LDA and LSD
Local density approximation (LDA)
In the LDA the xc energy is assumed to be
ELDAxc [ρ] =
∫R3
ρ(r)ε0xc
(ρ(r)
)d3r
ε0xc
(ρ(r)
)— xc energy density in uniform electron gas model (depends
only on the density).
It splits into exchange and correlation parts
ε0xc
(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
∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 73 / 101
Approximate xc Functionals LDA and LSD
Local density approximation (LDA)
In the LDA the xc energy is assumed to be
ELDAxc [ρ] =
∫R3
ρ(r)ε0xc
(ρ(r)
)d3r
ε0xc
(ρ(r)
)— xc energy density in uniform electron gas model (depends
only on the density). It splits into exchange and correlation parts
ε0xc
(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
∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 73 / 101
Approximate xc Functionals LDA and LSD
Local density approximation (LDA)
In the LDA the xc energy is assumed to be
ELDAxc [ρ] =
∫R3
ρ(r)ε0xc
(ρ(r)
)d3r
ε0xc
(ρ(r)
)— xc energy density in uniform electron gas model (depends
only on the density). It splits into exchange and correlation parts
ε0xc
(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
∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 73 / 101
Approximate xc Functionals LDA and LSD
Local density approximation (LDA)
In the LDA the xc energy is assumed to be
ELDAxc [ρ] =
∫R3
ρ(r)ε0xc
(ρ(r)
)d3r
ε0xc
(ρ(r)
)— xc energy density in uniform electron gas model (depends
only on the density). It splits into exchange and correlation parts
ε0xc
(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 ⇒ ELDA
x [ρ] = −Cx
∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 73 / 101
Approximate xc Functionals LDA and LSD
Local density approximation (LDA)
In the LDA the xc energy is assumed to be
ELDAxc [ρ] =
∫R3
ρ(r)ε0xc
(ρ(r)
)d3r
ε0xc
(ρ(r)
)— xc energy density in uniform electron gas model (depends
only on the density). It splits into exchange and correlation parts
ε0xc
(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 ⇒ ELDA
x [ρ] = −Cx
∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 73 / 101
Approximate xc Functionals LDA and LSD
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] =
∫R3
ρ(r)εxc
(ρ(r); ζ(r)
)d3r.
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
).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 74 / 101
Approximate xc Functionals LDA and LSD
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] =
∫R3
ρ(r)εxc
(ρ(r); ζ(r)
)d3r.
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
).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 74 / 101
Approximate xc Functionals LDA and LSD
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] =
∫R3
ρ(r)εxc
(ρ(r); ζ(r)
)d3r.
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
).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 74 / 101
Approximate xc Functionals LDA and LSD
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] =
∫R3
ρ(r)εxc
(ρ(r); ζ(r)
)d3r.
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
).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 74 / 101
Approximate xc Functionals LDA and LSD
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 75 / 101
Approximate xc Functionals LDA and LSD
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 75 / 101
Approximate xc Functionals LDA and LSD
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 75 / 101
Approximate xc Functionals LDA and LSD
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 75 / 101
Approximate xc Functionals LDA and LSD
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 75 / 101
Approximate xc Functionals LDA and LSD
Remarks on LDA and LSD
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 75 / 101
Approximate xc Functionals LDA and LSD
Remarks on LDA and LSD
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 75 / 101
Approximate xc Functionals LDA and LSD
Remarks on LDA and LSD
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 75 / 101
Approximate xc Functionals LDA and LSD
Remarks on LDA and LSD
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 75 / 101
Approximate xc Functionals GGA
Beyond LSD
The situation of people is totally different when they are on a steady, plainterrain with (almost) uniform density. . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 76 / 101
Approximate xc Functionals GGA
Beyond LSD
. . . then when they are put in a region with very rapidly changing density!
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 76 / 101
Approximate xc Functionals GGA
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 76 / 101
Approximate xc Functionals GGA
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 76 / 101
Approximate xc Functionals GGA
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 76 / 101
Approximate xc Functionals GGA
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 76 / 101
Approximate xc Functionals GGA
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] =
∫R3
fGGAxc
(ρα(r); ρβ(r);∇ρα(r);∇ρβ(r)
)d3r.
As usual, we split the energy: EGGAxc = EGGA
x + EGGAc .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 77 / 101
Approximate xc Functionals GGA
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] =
∫R3
fGGAxc
(ρα(r); ρβ(r);∇ρα(r);∇ρβ(r)
)d3r.
As usual, we split the energy: EGGAxc = EGGA
x + EGGAc .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 77 / 101
Approximate xc Functionals GGA
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] =
∫R3
fGGAxc
(ρα(r); ρβ(r);∇ρα(r);∇ρβ(r)
)d3r.
As usual, we split the energy: EGGAxc = EGGA
x + EGGAc .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 77 / 101
Approximate xc Functionals GGA
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] =
∫R3
fGGAxc
(ρα(r); ρβ(r);∇ρα(r);∇ρβ(r)
)d3r.
As usual, we split the energy: EGGAxc = EGGA
x + EGGAc .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 77 / 101
Approximate xc Functionals GGA
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] =
∫R3
fGGAxc
(ρα(r); ρβ(r);∇ρα(r);∇ρβ(r)
)d3r.
As usual, we split the energy: EGGAxc = EGGA
x + EGGAc .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 77 / 101
Approximate xc Functionals GGA
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
∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 78 / 101
Approximate xc Functionals GGA
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
∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 78 / 101
Approximate xc Functionals GGA
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
∫R3
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 78 / 101
Approximate xc Functionals GGA
Exchange in GGA — examples:
FB =βs2
σ
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 79 / 101
Approximate xc Functionals GGA
Exchange in GGA — examples:
FPW91 =1 + 0.19645sσ sinh−1 7.7956sσ + (0.2743− 0.1508e−100s2σ)s2
σ
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 79 / 101
Approximate xc Functionals GGA
Exchange in GGA — examples:
FPBE = κ− κ
1 + µκs
2σ
,
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 79 / 101
Approximate xc Functionals GGA
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 80 / 101
Approximate xc Functionals GGA
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 80 / 101
Approximate xc Functionals GGA
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 80 / 101
Approximate xc Functionals GGA
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 80 / 101
Approximate xc Functionals GGA
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 80 / 101
Approximate xc Functionals 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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 81 / 101
Approximate xc Functionals GGA
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 81 / 101
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
4
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 82 / 101
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
4
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 82 / 101
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
4
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 82 / 101
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
4
∫R3
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 82 / 101
Approximate xc Functionals Hybrid Functionals
Exact exchange problems
So, why not just mix the exact exchange with the GGA correlation:
Exc = Eexactx + EGGA
c .
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 83 / 101
Approximate xc Functionals Hybrid Functionals
Exact exchange problems
So, why not just mix the exact exchange with the GGA correlation:
Exc = Eexactx + EGGA
c .
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 83 / 101
Approximate xc Functionals Hybrid Functionals
Exact exchange problems
So, why not just mix the exact exchange with the GGA correlation:
Exc = Eexactx + EGGA
c .
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 83 / 101
Approximate xc Functionals Hybrid Functionals
Exact exchange problems
So, why not just mix the exact exchange with the GGA correlation:
Exc = Eexactx + EGGA
c .
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 83 / 101
Approximate xc Functionals Hybrid Functionals
Exact exchange problems
So, why not just mix the exact exchange with the GGA correlation:
Exc = Eexactx + EGGA
c .
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 83 / 101
Approximate xc Functionals Hybrid Functionals
Exact exchange problems
So, why not just mix the exact exchange with the GGA correlation:
Exc = Eexactx + EGGA
c .
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 83 / 101
Approximate xc Functionals Hybrid Functionals
Exact exchange problems
So, why not just mix the exact exchange with the GGA correlation:
Exc = Eexactx + EGGA
c .
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 83 / 101
Approximate xc Functionals Hybrid Functionals
Exact exchange problems
So, why not just mix the exact exchange with the GGA correlation:
Exc = Eexactx + EGGA
c .
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 83 / 101
Approximate xc Functionals Hybrid Functionals
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 84 / 101
Approximate xc Functionals Hybrid Functionals
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 84 / 101
Approximate xc Functionals Hybrid Functionals
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 84 / 101
Approximate xc Functionals Hybrid Functionals
Examples of hybrid functionals
EB3xc = aEexact
x + (1− a)ELSDx + bEB88
x + ELSDc + EPW91
c ,
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 85 / 101
Approximate xc Functionals Hybrid Functionals
Examples of hybrid functionals
EB3LYPxc = aEexact
x +(1−a)ELSDx +bEB88
x +cELYPc +(1−c)ELSD
c ,
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 85 / 101
Approximate xc Functionals Hybrid Functionals
Examples of hybrid functionals
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 85 / 101
Approximate xc Functionals Hybrid Functionals
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)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 86 / 101
Approximate xc Functionals Hybrid Functionals
Hybrid functionals results
Dipole moment for different molecules: calculations vs. experiment
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
CO
H2 O
H2 S
HCl
HF
LiH
LiF
NH3
PH3
SO2
(µca
lcu
late
d -
µexp)/
au
molecule
HFMP2
BLYPHCTHB3LYP
Source: [Cohen and Tantirungrotechai(1999)]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 86 / 101
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] =
∫R3
fMGGAxc
(ρα; ρβ;∇ρα;∇ρβ; ∆ρα; ∆ρβ; τα; τβ
)d3r,
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 87 / 101
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] =
∫R3
fMGGAxc
(ρα; ρβ;∇ρα;∇ρβ; ∆ρα; ∆ρβ; τα; τβ
)d3r,
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 87 / 101
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] =
∫R3
fMGGAxc
(ρα; ρβ;∇ρα;∇ρβ; ∆ρα; ∆ρβ; τα; τβ
)d3r,
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 87 / 101
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] =
∫R3
fMGGAxc
(ρα; ρβ;∇ρα;∇ρβ; ∆ρα; ∆ρβ; τα; τβ
)d3r,
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 87 / 101
Approximate xc Functionals Beyond GGA
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]
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 88 / 101
Approximate xc Functionals Beyond GGA
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
)d3r,
εxα(r) = − 12ρσ(r)
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 89 / 101
Approximate xc Functionals Beyond GGA
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
)d3r,
εxα(r) = − 12ρσ(r)
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 89 / 101
Approximate xc Functionals Beyond GGA
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
)d3r,
εxα(r) = − 12ρσ(r)
∫R3
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 89 / 101
Approximate xc Functionals Beyond GGA
Jacob’s Ladder
LSDρ
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:
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 90 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 91 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 91 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 91 / 101
Approximate xc Functionals Problems of Approximate Functionals
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
r.
The asymptotics of n-e and Coulomb potentials:
I limr→∞
vne(r) = − limr→∞
M∑α=1
Zα|r−Rα|
= −Zr
,
I limr→∞
(r) = limr→∞
∫R3
ρ(r′)|r− r′|
d3r′ =N
r.
So, the correct asymptotics of xc potential is limr→∞
vxc(r) = −1r
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
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
r.
The asymptotics of n-e and Coulomb potentials:
I limr→∞
vne(r) = − limr→∞
M∑α=1
Zα|r−Rα|
= −Zr
,
I limr→∞
(r) = limr→∞
∫R3
ρ(r′)|r− r′|
d3r′ =N
r.
So, the correct asymptotics of xc potential is limr→∞
vxc(r) = −1r
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
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
r.
The asymptotics of n-e and Coulomb potentials:
I limr→∞
vne(r) = − limr→∞
M∑α=1
Zα|r−Rα|
= −Zr
,
I limr→∞
(r) = limr→∞
∫R3
ρ(r′)|r− r′|
d3r′ =N
r.
So, the correct asymptotics of xc potential is limr→∞
vxc(r) = −1r
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
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
r.
The asymptotics of n-e and Coulomb potentials:
I limr→∞
vne(r) = − limr→∞
M∑α=1
Zα|r−Rα|
= −Zr
,
I limr→∞
(r) = limr→∞
∫R3
ρ(r′)|r− r′|
d3r′ =N
r.
So, the correct asymptotics of xc potential is limr→∞
vxc(r) = −1r
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
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
r.
The asymptotics of n-e and Coulomb potentials:
I limr→∞
vne(r) = − limr→∞
M∑α=1
Zα|r−Rα|
= −Zr
,
I limr→∞
(r) = limr→∞
∫R3
ρ(r′)|r− r′|
d3r′ =N
r.
So, the correct asymptotics of xc potential is limr→∞
vxc(r) = −1r
.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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).
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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):
1r12
=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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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):
1r12
=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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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):
1r12
=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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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):
1r12
=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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
xc potential asymptotics
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 92 / 101
Approximate xc Functionals Problems of Approximate Functionals
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 93 / 101
Approximate xc Functionals Problems of Approximate Functionals
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 93 / 101
Approximate xc Functionals Problems of Approximate Functionals
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 93 / 101
Approximate xc Functionals Problems of Approximate Functionals
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.
The End (for today)
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 93 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 94 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 94 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 94 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 94 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 94 / 101
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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 94 / 101
Supplement
back to electron density
∑σ
→
12∑
σ=− 12
.
∫R3
f(r) d3r =∫ ∞−∞
∫ ∞−∞
∫ ∞−∞
f(r) dx dy dz =
=∫ ∞
0
∫ π
0
∫ 2π
0f(r)r2 sin θ dr dθ dϕ =
= (or other coordinates) . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 95 / 101
Supplement
back to electron density
∑σ
→
12∑
σ=− 12
.
∫R3
f(r) d3r =∫ ∞−∞
∫ ∞−∞
∫ ∞−∞
f(r) dx dy dz =
=∫ ∞
0
∫ π
0
∫ 2π
0f(r)r2 sin θ dr dθ dϕ =
= (or other coordinates) . . .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 95 / 101
Supplement
back to expectation values
Brakets are shorthand notation for the integration over all electroniccoordinates:
〈ψ|A|ψ〉 =
=∑
σ1...σN
∫R3
. . .
∫R3
ψ(r1; . . . ; qN )∗Aψ(r1; . . . ; qN )d3r1 . . . d3rN .
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 96 / 101
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
a
√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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 97 / 101
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
a
√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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 97 / 101
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
a
√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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 97 / 101
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
a
√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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 97 / 101
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
a
√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.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 97 / 101
Supplement
back to Fock operator
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
n-e attraction: Vne[ρ] =∫
R3
vne(r)ρ(r) d3r.
Coulomb e-e repulsion: J [ρ] =12
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2.
e-e exchange:
K[ρ] = −Ex[ρ] =14
∫R3
∫R3
ρ(r1; r2)ρ(r2; r1)r12
d3r1 d3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 98 / 101
Supplement
back to Fock operator
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
n-e attraction: Vne[ρ] =∫
R3
vne(r)ρ(r) d3r.
Coulomb e-e repulsion: J [ρ] =12
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2.
e-e exchange:
K[ρ] = −Ex[ρ] =14
∫R3
∫R3
ρ(r1; r2)ρ(r2; r1)r12
d3r1 d3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 98 / 101
Supplement
back to Fock operator
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
n-e attraction: Vne[ρ] =∫
R3
vne(r)ρ(r) d3r.
Coulomb e-e repulsion: J [ρ] =12
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2.
e-e exchange:
K[ρ] = −Ex[ρ] =14
∫R3
∫R3
ρ(r1; r2)ρ(r2; r1)r12
d3r1 d3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 98 / 101
Supplement
back to Fock operator
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
n-e attraction: Vne[ρ] =∫
R3
vne(r)ρ(r) d3r.
Coulomb e-e repulsion: J [ρ] =12
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2.
e-e exchange:
K[ρ] = −Ex[ρ] =14
∫R3
∫R3
ρ(r1; r2)ρ(r2; r1)r12
d3r1 d3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 98 / 101
Supplement
back to Fock operator
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
∫R3
[∆rρ(r; r′)
]r′=r
d3r.
n-e attraction: Vne[ρ] =∫
R3
vne(r)ρ(r) d3r.
Coulomb e-e repulsion: J [ρ] =12
∫R3
∫R3
ρ(r1)ρ(r2)r12
d3r1 d3r2.
e-e exchange:
K[ρ] = −Ex[ρ] =14
∫R3
∫R3
ρ(r1; r2)ρ(r2; r1)r12
d3r1 d3r2.
Łukasz Rajchel (University of Warsaw) DFT Warsaw, 2010 98 / 101
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Thank you for your attention . . .