Key Concepts in Density Functional Theory from the Many Body Problem to the Kohn-Sham Scheme

The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary

Key concepts in Density Functional Theory From the many body problem to the Kohn-Sham scheme

Silvana Botti

ILM (LPMCN) CNRS, Universit´eLyon 1 - France European Theoretical Spectroscopy Facility (ETSF)

December 12, 2012 – Lyon

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Outline

1 The many-body problem

2 Earlier solutions: Hartree, Hartree-Fock, Thomas-Fermi

3 The solution: density functional theory

4 Hohenberg-Kohn theorems

5 Practical implementations: the Kohn-Sham scheme

6 Summary

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Outline

1 The many-body problem

2 Earlier solutions: Hartree, Hartree-Fock, Thomas-Fermi

3 The solution: density functional theory

4 Hohenberg-Kohn theorems

5 Practical implementations: the Kohn-Sham scheme

6 Summary

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary The many-body problem

Schr¨odingerequation for a quantum system of N interacting particles:

How to deal with N ≈ 1023 particles?

Hˆ Ψ({R} , {r}) = EΨ({R} , {r})

ˆ ˆ ˆ Ne electrons H = Tn ({R}) + Vnn ({R}) + Nn nuclei Tê ({r}) + Vêe ({r}) + Uên ({R} , {r})

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary The many-body Hamiltonian

ˆ ˆ ˆ ˆ ˆ ˆ H = Tn ({R}) + Vnn ({R}) + Te ({r}) + Vee ({r}) + Uen ({R} , {r})

Nn 2 Ne 2 X ∇I X ∇i Tˆn = − , Tˆe = − , 2MI 2m I =1 i=1

Nn Ne 1 X ZI ZJ 1 X 1 Vˆnn = , Vˆee = , 2 |RI − RJ | 2 |ri − rj | I ,J,I 6=J i,j,i6=j

Ne,Nn X ZJ Uˆen = − |RJ − rj | j,J

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Starting approximations

Born-Oppenheimer separation In the adiabatic approximation the nuclei are frozen in their equilibrium positions.

Example of equilibrium geometries

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Starting approximations

Pseudopotential and pseudowavefunction Concept of pseudopotentials The chemically intert core electrons are frozen in their atomic configuration and their effect on chemically active valence electrons is incorporated in an effective potential.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Pseudopotentials: generation criteria

A pseudopotential is not unique, several methods of generation also exist.

1 The pseudo-electron eigenvalues must be the same as the valence eigenvalues obtained from the atomic wavefunctions. 2 Pseudo-wavefunctions must match the all-electron wavefunctions outside the core (plus continuity conditions). 3 The core charge produced by the pseudo-wavefunctions must be the same as that produced by the atomic wavefunctions (for norm-conserving pseudopotentials). 4 The logaritmic derivatives and their ﬁrst derivatives with respect to the energy must match outside the core radius (scattering properties). 5 Additional criteria for diﬀerent recipes.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Pseudopotentials: quality assessment

It is important to find a compromise between 1 Transferability: ability to describe the valence electrons in different environments. 2 Efficiency: softness – few plane waves basis functions.

Moreover: Which states should be included in the valence and which states in the core? Problem of semicore states.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary The many-body Hamiltonian

Applying the Born-Oppenheimer separation...

Nn 2 Ne 2 X ∇I X ∇i 0 = Tˆn = − , Tˆe = − , 2MI 2m I =1 i=1

Nn Ne 1 X ZI ZJ 1 X 1 constant→Vˆnn = , Vˆee = , 2 |RI − RJ | 2 |ri − rj | I ,J,I 6=J i,j,i6=j

Ne,Nn Ne X ZJ X Uˆen = − = v (rj ) |RJ − rj | j,J j

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary The many-body problem

Schr¨odingerEquation for a quantum-system of Ne interacting electrons:

23 Still, how to deal with Ne ≈ 10 particles?

Hˆ Ψ({r}) = EeΨ({r})

Ne 2 Ne X ∇i 1 X 1 Hˆ = − + v (ri ) + Ne electrons 2m 2 |ri − rj | i=1 i,j,i6=j

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Why we don’t like the electronic wavefunction

How many DVDs are necessary to store a wavefunction? Classical example: Oxygen atom (8 electrons)

Ψ(r1,..., r8) depends on 24 coordinates

Rough table of the wavefunction:

10 entries per coordinate: =⇒ 1024 entries 1 byte per entry : =⇒ 1024 bytes 5 × 109 bytes per DVD: =⇒ 2 × 1014 DVDs 10 g per DVD: =⇒ 2 × 1015 g DVDs = ×109 t DVDs

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Outline

1 The many-body problem

2 Earlier solutions: Hartree, Hartree-Fock, Thomas-Fermi

3 The solution: density functional theory

4 Hohenberg-Kohn theorems

5 Practical implementations: the Kohn-Sham scheme

6 Summary

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary The Hartree equations: Hartree potential

Hartree introduced in 1927 a procedure, which he called the self-consistent ﬁeld method, to approximate the Schr¨odingerequation variational principle to an ansatz (trial wave function) as a product of single-particle functions:ΨH = φ1 (r1) φ2 (r2) . . . φN (rN )

  Z 0 2 1 X |φj (r )| − ∇2 + v (r) + d3r 0 φ (r) = ε φ (r)  2 i ext |r − r0|  i i i j

If we do not restrict the sum to j 6= i: self-interaction problem and the antisymmetry of the wavefunction?

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary The Hartree equations: Hartree potential

  Z 0 2 1 X |φj (r )| − ∇2 + v (r) + d3r 0 φ (r) = ε φ (r)  2 i ext |r − r0|  i i i j

If we do not restrict the sum to j 6= i: self-interaction problem and the antisymmetry of the wavefunction?

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary The approach of the “best wavefunction”

In 1930 Slater and Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry. A Slater determinant trivially satisﬁes the antisymmetry of the exact solution and hence is a suitable ansatz for applying the variational principle The original Hartree method can then be viewed as an approximation to the Hartree-Fock method by neglecting exchange. The Hartree-Fock method was little used until the advent of electronic computers in the 1950s (need of self-consistency!).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary The Hartree-Fock equations: exchange potential

The Hartree-Fock method determines the set of (spin) orbitals which minimizes the energy and give us the best single-determinant: ˛ ˛ ˛φ1(x1) φ2(x1) ··· φN (x1)˛ ˛ ˛ 1 ˛φ1(x2) φ2(x2) ··· φN (x2)˛ Ψ (x , x ,..., x ) = √ ˛ ˛ HF 1 2 N ˛ . . . ˛ N! ˛ . . . ˛ ˛ ˛ ˛φ1(xN ) φ2(xN ) ··· φN (xN )˛

  Z 0 2 1 X |φj (r )| − ∇2 + v (r) + d3r 0 φ (r)  2 i ext |r − r0|  i j Z ∗ 0 X φj (r) φi (r ) − δ d3r 0 = ε φ (r) σi σj |r − r0| i i j

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Some remarks on Hartree-Fock

The HF potential is self-interaction free. Ignoring relaxation (i.e. the change in the self-consistent potential) the first ionization energy is equal to the negative of the energy of the HOMO: Koopman’s theorem. The approximate solution of the HF equations cannot be exact because the true many-body wavefunction is not a single-determinant. In quantum chemistry the correlation energy is defined as the difference between the HF energy and the exact ground-state energy. To go beyond HF a trial wavefunction can be built as a linear combination of HF orbitals, including excited orbitals: configuration interaction (CI).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Thomas-Fermi theory

Formulated in 1927 in terms of the electronic density alone, the TF theory is viewed as a precursor to density functional theory.

3 2 Z T TF[ρ] = 3π2 3 ρ5/3(r)d3r 10

The power of ρ can be deduced by dimensional analysis, while the coeﬃcient is chosen to agree with the uniform electron gas. Adding the classical expressions for the nuclear-electron and electron-electron interactions we obtain the original TF functional:

TF TF F [ρ] = T [ρ] + VHartree[ρ] + V [ρ]

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Problems of Thomas-Fermi theory

The kinetic energy is a sizable part of the total energy and it is here described by a too poor approximation. The original formulation did not include the exchange energy (Pauli principle): an exchange energy functional was added by Dirac in 1928. However, the Thomas-Fermi-Dirac theory remained rather inaccurate for most applications. The largest source of error was in the representation of the kinetic energy, followed by the errors in the exchange energy, and to the neglect of electron correlation. In 1962, Edward Teller showed that TF theory cannot describe molecular bonding – the energy of any molecule calculated with TF theory is higher than the sum of the energies of the constituent atoms. More generally, the total energy of a molecule decreases when the bond lengths are uniformly increased.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Outline

1 The many-body problem

2 Earlier solutions: Hartree, Hartree-Fock, Thomas-Fermi

3 The solution: density functional theory

4 Hohenberg-Kohn theorems

5 Practical implementations: the Kohn-Sham scheme

6 Summary

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Ground state densities vs potentials

Question at the heart of DFT Is there a 1-to-1 mapping between diﬀerent external potentials v(r) and their corresponding ground state densities ρ(r)?

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Density functional theory (DFT): the essence

If we can give a positive answer, then it can be proved that (i) all observable quantities of a quantum system are completely determined by the density. (ii) which means that the basic variable is no more the many-body wavefunction Ψ({r)} but the electron density ρ(r).

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

You can ﬁnd all details in R. M. Dreizler and E.K.U. Gross, Density Functional Theory, Springer (Berlin, 1990).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Density functional theory (DFT): the essence

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

You can ﬁnd all details in R. M. Dreizler and E.K.U. Gross, Density Functional Theory, Springer (Berlin, 1990).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Outline

1 The many-body problem

2 Earlier solutions: Hartree, Hartree-Fock, Thomas-Fermi

3 The solution: density functional theory

4 Hohenberg-Kohn theorems

5 Practical implementations: the Kohn-Sham scheme

6 Summary

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Density functional theory (DFT)

Hohenberg-Kohn (HK) theorem – I The expectation value of any physical observable of a many-electron system is a unique functional of the electron density ρ.

Hohenberg-Kohn (HK) theorem – II

The total energy functional has a minimum, the ground state energy E0, at the ground state density ρ0.

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Density functional theory (DFT)

Restrictions: In practice, only ground state properties. The original proof is valid for local, spin-independent external potential, non-degenerate ground state. There exist extensions to degenerate ground states, spin-dependent, magnetic systems, etc.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Hohenberg-Kohn theorem – I

~ A A

ρ v(r)ψ ({r}) (r)

ground−state single−particle ground−state densities potentials having wavefunctions a nondegenerate ground state G : v (r) → ρ (r) is obvious. HK theorem states that G is invertible.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Hohenberg-Kohn theorem – I

Proof: 1 A is invertible: the Schr¨odingerequation can be always solved for the external potential, yielding the potential as a unique function of Ψ.

ˆ ˆ X E − T − Vee Ψ Tˆ Ψ Vˆ = v (r ) = = − − Vˆ + const. i Ψ Ψ ee i

2 A˜ is invertible (proof for non-degenerate ground state): Tˆ + Vˆee + Vˆ Ψ = EΨ

0 0 0 0 Tˆ + Vˆee + Vˆ Ψ = E Ψ

Now what is left to show is that Ψ 6= Ψ0 ⇒ ρ 6= ρ0

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Hohenberg-Kohn theorem – I

Applying the variational principle (Rayleigh-Ritz): Z E = hΨ|Hˆ |Ψi < hΨ0|Hˆ |Ψ0i = E 0 + d3r ρ0(r) v(r) − v 0(r) Z E 0 = hΨ0|Hˆ 0|Ψ0i < hΨ|Hˆ 0|Ψi = E + d3r ρ(r) v 0(r) − v(r)

Proof by contradiction: If ρ = ρ0 it has to be E + E 0 < E + E 0, which is absurd. Therefore, we deduce that Ψ 6= Ψ0 ⇒ ρ 6= ρ0

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Hohenberg-Kohn theorem – I

Direct consequence of the 1st HK theorem The expectation value of any physical observable of a many-electron system is a unique functional of the electron density ρ.

G −1 solving S.E. Proof: ρ −→ v [ρ] −→ Ψ0 [ρ] ˆ ˆ Then an observable O [ρ] = hΨ0 [ρ] |O|Ψ0 [ρ]i is a functional of ρ.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Reminder: what is a functional?

A functional maps a function to a number

E[ρ] ρ(r) R functional

set of functions set of real numbers vr [ρ] = v [ρ](r) is a functional that depends parametrically on r

Ψr1...rN [ρ] = Ψ [ρ](r1 ... rN ) is a functional that depends parametrically on r1 ... rN

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Hohenberg-Kohn (HK) theorem – II

2nd HK theorem: Variational principle

The total energy functional has a minimum, the ground state energy E0, at the ground state density ρ0.

hΨ0|Hˆ |Ψ0i = min {EHK [ρ]} = E0 [ρ0]

Euler-Lagrange equation:

δ Z E [ρ] − µ d3r ρ(r) = 0 δρ(r) HK

∂E where µ is the chemical potential since µ = ∂N .

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Hohenberg-Kohn (HK) theorem – II

2nd HK theorem: Variational principle

The total energy functional has a minimum, the ground state energy E0, at the ground state density ρ0.

hΨ0|Hˆ |Ψ0i = min {EHK [ρ]} = E0 [ρ0]

Euler-Lagrange equation: δE [ρ] HK = µ δρ(r)

It yields the exact ground-state energy E0 and density ρ0(r).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Hohenberg-Kohn (HK) theorem – II

Formal construction of EHK [ρ]:

˜−1 For an arbitrary ground state density it is always true ρ(r) −→A Ψ[ρ] ⇒ we can deﬁne the functional of the density:

EHK [ρ] = hΨ[ρ] |Tˆ + Vˆee + Vˆ |Ψ[ρ]i

EHK [ρ] >E0 for ρ6=ρ0 EHK [ρ]= E0 for ρ=ρ0 R 3 EHK [ρ] = FHK [ρ] + d r v(r)ρ(r), where FHK [ρ] is universal, as it does not depend on the external potential. Euler-Lagrange equation:

δFHK[ρ] δρ(r) + vext(r) = µ

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Hohenberg-Kohn (HK) theorem – II

In principle: the Euler-Lagrange equation allows to calculate ρ0(r) without introducing a Schr¨odingerequation. The HK theorem proves the existence of the universal functional Z 3 EKS [ρ] = FHK [ρ] + d r v(r)ρ(r)

FHK [ρ] = hΨ|Tˆ + Vˆee |Ψi but it does not tell us how to determine it.

In practice: FHK [ρ] needs to be approximated and approximations of T [ρ] lead to large errors in the total energy. (See again the same problem as in Thomas-Fermi theory).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Outline

1 The many-body problem

2 Earlier solutions: Hartree, Hartree-Fock, Thomas-Fermi

3 The solution: density functional theory

4 Hohenberg-Kohn theorems

5 Practical implementations: the Kohn-Sham scheme

6 Summary

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Reformulation: Kohn-Sham scheme

HK 1−1 mapping for HK 1−1 mapping interacting particles non−interacting particles

v [ρ](r) ρ [ρ] ext (r) vKS (r)

Essence of the mapping The density of a system of interacting particles can be calculated exactly as the density of an auxiliary system of non-interacting particles

=⇒ Reformulation in terms of single-particle orbitals!

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Back to the Hohenberg-Kohn variational principle

For the non-interacting system: Z 3 EKS [ρ] = hΨ[ρ]|Tˆs + VˆKS|Ψ[ρ]i = Ts[ρ] + d r ρ(r)vKS(r)

Euler-Lagrange equation for the non-interacting system δ Z E [ρ] − µ d3r ρ(r) = 0 δρ(r) KS δT [ρ] s + v (r) = µ δρ(r) KS

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Using a one-particle Schr¨odingerequation

Kohn-Sham equations ∇2 − + v (r) φ (r) = ε φ (r) 2 KS i i i

X 2 ρ0 (r) = |φi (r) |

i, lowest εi

εi = KS eigenvalues, φi (r) = KS single-particle orbitals

Can we always build vKS for the non-interacting electron system?

Uniqueness of vKS follows from HK 1-1 mapping.

Existence of vKS is guaranteed by V-representability theorem.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Problem of V-representability

Deﬁnition ρ(r) is V-representable if it is the ground-state density of some potential V .

Question Are all reasonable functions ρ(r) V-representable?

Answer: V-representability theorem On a lattice (ﬁnite or inﬁnite) any normalizable positive function ρ(r), that is compatible with the Pauli principle, is both interacting and non-interacting V-representable. For degenerate ground states such a ρ(r) is ensemble V-representable, i.e. representable as a linear combination of the degenerate ground-states densities.

Chayes, Chayes, Ruskai, J Stat. Phys. 38, 497 (1985).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Reformulation: Kohn-Sham scheme

Kohn-Sham one-particle equations ∇2 − + v (r) φ (r) = ε φ (r) 2 KS i i i

P 2 to be solved self-consistenlty with ρ0 (r) = i |φi (r) |

εi = KS eigenvalues, φi (r) = KS single-particle orbitals

Which is the form of vKS for the non-interacting electrons? Hartree potential . vKS (r) = v (r) + vH (r) + vxc (r) % unknown exchange-correlation (xc) potential

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Kohn-Sham scheme: Hartree and xc potentials

Hartree potential Z ρ (r0) v [ρ](r) = d3r 0 H |r − r0|

vH describes classic electrostatic interaction

Exchange-correlation (xc) potential δ E [ρ] v [ρ](r) = xc xc δ ρ (r)

vxc encompasses many-body eﬀects

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Kohn-Sham scheme: xc potential

Proof: Knowing that we want the total energy of the real system, we rewrite FHK[ρ] = Ts [ρ] + EH [ρ] + Exc [ρ], after deﬁning Exc [ρ] = FHK [ρ] − EH [ρ] − Ts [ρ]. δ EH[ρ] We use the variational principle (and δ ρ(r) = vH (r))

δF [ρ] HK + v(r) = µ δρ(r)

δT [ρ] s + v (r) = µ δρ(r) KS

δ Exc[ρ] To obtain vKS (r) = v (r) + vH (r) + vxc (r), δ ρ(r) = vxc (r)

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Approximations for the xc potential

LDA: Z LDA 3 HEG Exc [ρ] = d r ρ (r) xc (ρ (r))

LSDA: Z LSDA 3 HEG Exc [ρ↑, ρ↓] = d r ρ (r) xc (ρ↑, ρ↓)

GGA: Z GGA 3 GGA Exc [ρ↑, ρ↓] = d r ρ (r) xc (ρ↑, ρ↓, ∇ρ↑, ∇ρ↓)

meta-GGA: Z MGGA 3 MGGA 2 2 Exc [ρ↑, ρ↓] = d r ρ (r) xc ρ↑, ρ↓, ∇ρ↑, ∇ρ↓, ∇ ρ↑, ∇ ρ↓, τ↑, τ↓

EXX, SIC-LDA, hybrid Hartree-Fock/DFT functionals, ...

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Approximations for the xc potential

LDA: Z LDA 3 HEG Exc [ρ] = d r ρ (r) xc (ρ (r))

LSDA: Z LSDA 3 HEG Exc [ρ↑, ρ↓] = d r ρ (r) xc (ρ↑, ρ↓)

GGA: Z GGA 3 GGA Exc [ρ↑, ρ↓] = d r ρ (r) xc (ρ↑, ρ↓, ∇ρ↑, ∇ρ↓)

meta-GGA: Z MGGA 3 MGGA 2 2 Exc [ρ↑, ρ↓] = d r ρ (r) xc ρ↑, ρ↓, ∇ρ↑, ∇ρ↓, ∇ ρ↑, ∇ ρ↓, τ↑, τ↓

EXX, SIC-LDA, hybrid Hartree-Fock/DFT functionals, ...

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Can we calculate excited states within static DFT?

Density functional theory in the Kohn-Sham scheme gives an eﬃcient and accurate description of GROUND STATE properties (total energy, lattice constants, atomic structure, elastic constants, phonon spectra ...) is not designed to access EXCITED STATES however ...

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Kohn-Sham band structure: some facts

One-electron band structure is the dispersion of the energy levels n as a function of k in the Brillouin zone. The Kohn-Sham eigenvalues and eigenstates are not one-electron energy states for the electron in the solid. However, it is common to interpret the solutions of Kohn-Sham equations as one-electron states: the result is often a good representation, especially concerning band dispersion. Gap problem: the KS band structure underestimates systematically the band gap (often by more than 50%).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary

Discontinuity in Vxc

Band gap error not due to LDA, but to the discontinuity in the exact Vxc.

.6 .6 ε ε Eg = (E(N+1) − E(N)) − (E(N) − E(N−1))

KS KS ε .6 1 1 Eg = εN+1(N + 1) − εN ∆ ;& .6 (

ε JDS 1 .6 ( ε .6 JDS 1 KS KS KS Eg = εN+1 − εN

N N 1HOHFWURQV 1HOHFWURQV KS + − ∆xc = Eg − Eg = Vxc(r) − Vxc(r)

L. J. Sham and M. Schlter, Phys. Rev. Lett. 51, 1888 (1983); L. J. Sham and M. Schlter, Phys. Rev. B 32, 3883 (1985).

J. P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 (1983).

R. W. Godby, M. Schl¨uterand L. J. Sham, Phys. Rev. Lett. 56, 2415 (1986).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary GaAs band structure

L3,c 4 Γ 15,c X 2 3,c Experimental gap: 1.53 eV X L 1,c Γ 1,c 1,c 0 Γ 15,v DFT-LDA gap: 0.57 eV

L3,v -2 X5,v

Energy (eV) -4 Applying a scissor operator -6 L 2,v X3,v (0.8 eV) we can correct the -8 band structure. GaAs

-10 X1,v

L1,v

-12 Γ 1,v L Γ XK Γ

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary DFT in practice

1 Pseudopotential or all-electron? 2 Represent Kohn-Sham orbitals on a basis (plane waves, atomic orbitals, gaussians, LAPW, real space grid,..) 3 Calculate the total energy for trial orbitals. For plane waves: 1 kinetic energy, Hartree potential in reciprocal space, 2 xc potential, external potential in real space 3 FFTs! 4 Sum over states = BZ integration for solids: special k-points 5 Iterate or minimize to self-consistency

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Software supporting DFT

Abinit EXCITING OpenMX ADF Fireball ORCA AIMPRO FSatom - list of codes ParaGauss Atomistix Toolkit GAMESS (UK) PLATO CADPAC GAMESS (US) PWscf CASTEP GAUSSIAN (Quantum-ESPRESSO) CPMD JAGUAR Q-Chem CRYSTAL06 MOLCAS SIESTA DACAPO MOLPRO Spartan DALTON MPQC S/PHI/nX deMon2K NRLMOL TURBOMOLE DFT++ NWChem VASP DMol3 OCTOPUS WIEN2k http://en.wikipedia.org/wiki/Density_functional_theory

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary The code ABINIT

http://www.abinit.org

”First-principles computation of material properties : the ABINIT software project.” X. Gonze et al, Computational Materials Science 25, 478-492 (2002). ”A brief introduction to the ABINIT software package.” X. Gonze et al, Zeit. Kristallogr. 220, 558-562 (2005).

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Tutorial III

Ground state geometry and band structure of bulk silicon 1 Determination of the total energy 2 Determination of the lattice parameter a 3 Computation of the Kohn-Sham band structure

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Equilibrium geometry of silicon

Our DFT-LDA lattice parameter: a = 10.217 Bohr = 5.407 A˚ Exp. value: a = 5.431 A˚ at 25◦.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Kohn-Sham band structure of silicon

4 Indirect gap 2 Good dispersion of 0 ~ 0.5 eV bands close to the -2 gap Energy [eV] -4 Exp. gap = 1.17 eV -6 Scissor operator = -8 0.65 – 0.7 eV -10

-12 UW Γ XWL Γ

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Kohn-Sham band structures

Application of standard DFT to solids: band structure calculations (Kohn-Sham bands) We know that the Kohn-Sham bands are not quasiparticle states. However they turn out to give a qualitative picture in many cases. When they are completely wrong we need to go beyond standard DFT.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Outline

1 The many-body problem

2 Earlier solutions: Hartree, Hartree-Fock, Thomas-Fermi

3 The solution: density functional theory

4 Hohenberg-Kohn theorems

5 Practical implementations: the Kohn-Sham scheme

6 Summary

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Summary

The electron density is the key-variable to study ground-state properties of an interacting electron system. The ground state expectation value of any physical observable of a many-electron system is a unique functional of the electron density ρ.

The total energy functional EHK [ρ] has a minimum, the ground state energy E0, in correspondence to the ground state density ρ0.

The universal functional FHK [ρ] is hard to approximate. The Kohn-Sham scheme allows a reformulation in terms of one-particle orbitals.

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Suggestion of essential bibliography

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).

W. Kohn, Rev. Mod. Phys. 71, 1253 - 1266 (1999).

R. M. Dreizler and E.K.U. Gross, Density Functional Theory, Springer (Berlin, 1990).

R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, (Oxford, New York, 1989).

Eds. C. Fiolhais, F. Nogueira, M. A. L. Marques, A primer in Density Functional Theory, Springer-Verlag (Berlin, 2003).

K. Burke, Lecture Notes in Density Functional Theory, http://dft.rutgers.edu/kieron/beta/

Key concepts in Density Functional Theory Silvana Botti The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Suggestion of essential bibliography

Some additional items: R. M. Martin, Electronic structure: Basic Theory and Practical Methods, Cambridge University Press (2004).

http://www.abinit.org and references there.

Key concepts in Density Functional Theory Silvana Botti