Key concepts in Density Functional Theory

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1 From the many body problem to the Kohn-Sham scheme ILM (LPMCN) CNRS, Université Lyon 1 - France European Theoretical Spectroscopy Facility (ETSF) December 12, 2012 Lyon

2 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

3 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

ĤΨ ({R}, {r}) = EΨ ({R}, {r}) N e electrons N n nuclei Ĥ = ˆT n

4 The many-body problem Schrödinger equation for a quantum system of N interacting particles: How to deal with N particles? ĤΨ ({R}, {r}) = EΨ ({R}, {r}) N e electrons N n nuclei Ĥ = ˆT n ({R}) + ˆV nn ({R}) + ˆT e ({r}) + ˆV ee ({r}) + Û en ({R}, {r})

5 The many-body Hamiltonian Ĥ = ˆT n ({R}) + ˆV nn ({R}) + ˆT e ({r}) + ˆV ee ({r}) + Ûen ({R}, {r}) ˆV nn = 1 2 N n ˆT n = 2 I, 2M ˆT e = I N n I,J,I J I =1 N e i=1 Z I Z J R I R J, ˆV ee = 1 2 N e,n n Z J Û en = R J r j j,j 2 i 2m, N e i,j,i j 1 r i r j,

6 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

7 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.

8 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.

9 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.

10 The many-body Hamiltonian Applying the Born-Oppenheimer separation... N n ˆT n = 2 I, 2M ˆT N e e = 2 i 0 = I 2m, constant ˆV nn = 1 2 N n I,J,I J I =1 i=1 Z I Z J R I R J, ˆV ee = 1 2 j,j j N e i,j,i j N e,n n Ne Z J Û en = R J r j = v (r j ) 1 r i r j,

11 The many-body problem Schrödinger Equation for a quantum-system of N e interacting electrons: Still, how to deal with N e particles? ĤΨ ({r}) = E e Ψ ({r}) N e electrons N e [ ] Ĥ = 2 i 2m + v (r i) i=1 N e i,j,i j 1 r i r j

12 Why we don t like the electronic wavefunction How many DVDs are necessary to store a wavefunction? Classical example: Oxygen atom (8 electrons) Ψ (r 1,..., r 8 ) depends on 24 coordinates Rough table of the wavefunction: 10 entries per coordinate: = entries 1 byte per entry : = bytes bytes per DVD: = DVDs 10 g per DVD: = g DVDs = 10 9 t DVDs

13 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

14 The Hartree equations: Hartree potential Hartree introduced in 1927 a procedure, which he called the self-consistent field method, to approximate the Schrödinger equation variational principle to an ansatz (trial wave function) as a product of single-particle functions: Ψ H = φ 1 (r 1 ) φ 2 (r 2 )... φ N (r N ) i + v ext (r) + j d 3 r φ j (r ) 2 r r φ i (r) = ε i φ i (r) If we do not restrict the sum to j i: self-interaction problem and the antisymmetry of the wavefunction?

15 The Hartree equations: Hartree potential Hartree introduced in 1927 a procedure, which he called the self-consistent field method, to approximate the Schrödinger equation variational principle to an ansatz (trial wave function) as a product of single-particle functions: Ψ H = φ 1 (r 1 ) φ 2 (r 2 )... φ N (r N ) i + v ext (r) + j d 3 r φ j (r ) 2 r r φ i (r) = ε i φ i (r) If we do not restrict the sum to j i: self-interaction problem and the antisymmetry of the wavefunction?

16 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!).

17 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: φ 1(x 1) φ 2(x 1) φ N (x 1) Ψ HF(x 1, x 2,..., x N ) = 1 φ 1(x 2) φ 2(x 2) φ N (x 2) N!... φ 1(x N ) φ 2(x N ) φ N (x N ) i + v ext (r) + j d 3 r φ j (r ) 2 r r φ i (r) j δ σi σ j d 3 r φ j (r) φ i (r ) r r = ε i φ i (r)

18 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).

19 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. T TF [ρ] = 3 10 ( 3π 2 ) 2 3 ρ 5/3 (r)d 3 r 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: F TF [ρ] = T TF [ρ] + V Hartree [ρ] + V [ρ]

20 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 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.

21 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

22 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)?

23 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).

24 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).

25 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

26 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 E 0, at the ground state density ρ 0. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

27 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.

28 Hohenberg-Kohn theorem I G A A ~ v(r) ψ ({r}) ρ(r) single particle potentials having a nondegenerate ground state ground state wavefunctions ground state densities G : v (r) ρ (r) is obvious. HK theorem states that G is invertible.

29 Hohenberg-Kohn theorem I Proof: 1 A is invertible: the Schrödinger equation can be always solved for the external potential, yielding the potential as a unique function of Ψ. ) (E ˆV = ˆT ˆV ee Ψ v (r i ) = = ˆT Ψ Ψ Ψ ˆV ee + const. i 2 Ã is invertible (proof for non-degenerate ground state): ( ) ˆT + ˆV ee + ˆV Ψ = EΨ ( ˆT + ˆVee + ˆV ) Ψ = E Ψ Now what is left to show is that Ψ Ψ ρ ρ

30 Hohenberg-Kohn theorem I Applying the variational principle (Rayleigh-Ritz): E = Ψ Ĥ Ψ < Ψ Ĥ Ψ = E + d 3 r ρ (r) [ v(r) v (r) ] E = Ψ Ĥ Ψ < Ψ Ĥ Ψ = E + d 3 r ρ(r) [ v (r) v(r) ] Proof by contradiction: If ρ = ρ it has to be E + E < E + E, which is absurd. Therefore, we deduce that Ψ Ψ ρ ρ

31 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. Proof: ρ G 1 solving S.E. v [ρ] Ψ 0 [ρ] Then an observable Ô [ρ] = Ψ 0 [ρ] Ô Ψ 0 [ρ] is a functional of ρ.

32 Reminder: what is a functional? A functional maps a function to a number ρ(r) E[ρ] functional R set of functions set of real numbers v r [ρ] = v [ρ] (r) is a functional that depends parametrically on r Ψ r1...r N [ρ] = Ψ [ρ] (r 1... r N ) is a functional that depends parametrically on r 1... r N

33 Hohenberg-Kohn (HK) theorem II 2nd HK theorem: Variational principle The total energy functional has a minimum, the ground state energy E 0, at the ground state density ρ 0. Ψ 0 Ĥ Ψ 0 = min {E HK [ρ]} = E 0 [ρ 0 ] Euler-Lagrange equation: δ δρ(r) [ E HK [ρ] µ ( )] d 3 r ρ(r) = 0 where µ is the chemical potential since µ = E N.

34 Hohenberg-Kohn (HK) theorem II 2nd HK theorem: Variational principle The total energy functional has a minimum, the ground state energy E 0, at the ground state density ρ 0. Ψ 0 Ĥ Ψ 0 = min {E HK [ρ]} = E 0 [ρ 0 ] Euler-Lagrange equation: δe HK [ρ] δρ(r) = µ It yields the exact ground-state energy E 0 and density ρ 0 (r).

35 Hohenberg-Kohn (HK) theorem II Formal construction of E HK [ρ]: For an arbitrary ground state density it is always true ρ(r) Ã 1 Ψ [ρ] we can define the functional of the density: E HK [ρ] = Ψ [ρ] ˆT + ˆV ee + ˆV Ψ [ρ] E HK [ρ] >E 0 for ρ ρ 0 E HK [ρ] =E 0 for ρ=ρ 0 E HK [ρ] = F HK [ρ] + d 3 r v(r)ρ(r), where F HK [ρ] is universal, as it does not depend on the external potential. Euler-Lagrange equation: δf HK [ρ] δρ(r) + v ext (r) = µ

36 Hohenberg-Kohn (HK) theorem II In principle: the Euler-Lagrange equation allows to calculate ρ 0 (r) without introducing a Schrödinger equation. The HK theorem proves the existence of the universal functional E KS [ρ] = F HK [ρ] + d 3 r v(r)ρ(r) F HK [ρ] = Ψ ˆT + ˆV ee Ψ but it does not tell us how to determine it. In practice: F HK [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).

37 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

38 Reformulation: Kohn-Sham scheme HK 1 1 mapping for interacting particles HK 1 1 mapping non interacting particles v ext [ρ](r) ρ(r) v KS [ρ](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).

39 Back to the Hohenberg-Kohn variational principle For the non-interacting system: E KS [ρ] = Ψ[ρ] ˆT s + ˆV KS Ψ[ρ] = T s [ρ] + d 3 r ρ(r)v KS (r) Euler-Lagrange equation for the non-interacting system [ ] δ E KS [ρ] µ d 3 r ρ(r) = 0 δρ(r) δt s [ρ] δρ(r) + v KS(r) = µ

40 Using a one-particle Schrödinger equation Kohn-Sham equations ] [ v KS (r) φ i (r) = ε i φ i (r) ρ 0 (r) = i, lowest ε i φ i (r) 2 ε i = KS eigenvalues, φ i (r) = KS single-particle orbitals Can we always build v KS for the non-interacting electron system? Uniqueness of v KS follows from HK 1-1 mapping. Existence of v KS is guaranteed by V-representability theorem.

41 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).

42 Reformulation: Kohn-Sham scheme Kohn-Sham one-particle equations ] [ v KS (r) φ i (r) = ε i φ i (r) to be solved self-consistenlty with ρ 0 (r) = i φ i (r) 2 ε i = KS eigenvalues, φ i (r) = KS single-particle orbitals Which is the form of v KS for the non-interacting electrons? Hartree potential v KS (r) = v (r) + v H (r) + v xc (r) unknown exchange-correlation (xc) potential

43 Kohn-Sham scheme: Hartree and xc potentials Hartree potential v H [ρ] (r) = v H describes classic electrostatic interaction Exchange-correlation (xc) potential v xc encompasses many-body effects d 3 r ρ (r ) r r v xc [ρ] (r) = δ E xc [ρ] δ ρ (r)

44 Kohn-Sham scheme: xc potential Proof: Knowing that we want the total energy of the real system, we rewrite F HK [ρ] = T s [ρ] + E H [ρ] + E xc [ρ], after defining E xc [ρ] = F HK [ρ] E H [ρ] T s [ρ]. We use the variational principle (and δ E H[ρ] δ ρ(r) δf HK [ρ] δρ(r) + v(r) = µ δt s [ρ] δρ(r) + v KS(r) = µ = v H (r)) To obtain v KS (r) = v (r) + v H (r) + v xc (r), δ Exc[ρ] δ ρ(r) = v xc (r)

45 Approximations for the xc potential LDA: LSDA: GGA: Exc LDA [ρ] = Exc LSDA [ρ, ρ ] = Exc GGA [ρ, ρ ] = d 3 r ρ (r) ɛ HEG xc (ρ (r)) d 3 r ρ (r) ɛ HEG xc (ρ, ρ ) d 3 r ρ (r) ɛ GGA xc (ρ, ρ, ρ, ρ ) meta-gga: E MGGA xc [ρ, ρ ] = d 3 r ρ (r) ɛ MGGA ( xc ρ, ρ, ρ, ρ, 2 ρ, 2 ) ρ, τ, τ EXX, SIC-LDA, hybrid Hartree-Fock/DFT functionals,...

46 Approximations for the xc potential LDA: LSDA: GGA: Exc LDA [ρ] = Exc LSDA [ρ, ρ ] = Exc GGA [ρ, ρ ] = d 3 r ρ (r) ɛ HEG xc (ρ (r)) d 3 r ρ (r) ɛ HEG xc (ρ, ρ ) d 3 r ρ (r) ɛ GGA xc (ρ, ρ, ρ, ρ ) meta-gga: E MGGA xc [ρ, ρ ] = d 3 r ρ (r) ɛ MGGA ( xc ρ, ρ, ρ, ρ, 2 ρ, 2 ) ρ, τ, τ EXX, SIC-LDA, hybrid Hartree-Fock/DFT functionals,...

47 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...

48 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...

49 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...

50 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%).

51 The many-body problem From Hartree to Thomas Fermi DFT HK theorems KS scheme and bands Summary Discontinuity in V xc Band gap error not due to LDA, but to the discontinuity in the exact V xc. ε.6 ε.6 E g = (E (N+1) E (N) ) (E (N) E (N 1) ).6 ε 1 1 E g = ε KS N+1 (N + 1) εks N.6 ε 1 ε 1.6 ;& ( JDS.6 ( JDS E KS g = ε KS N+1 εks N 1HOHFWURQV N 1HOHFWURQV N xc = E g E KS g = V + xc(r) V xc(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. Schlüter and L. J. Sham, Phys. Rev. Lett. 56, 2415 (1986).

52 GaAs band structure 8 6 L 4 3,c Γ 15,c 2 L 1,c 0 X 3,c X 1,c Γ 1,c Γ 15,v Experimental gap: 1.53 ev DFT-LDA gap: 0.57 ev Energy (ev) L 3,v L 2,v X 5,v X 3,v X 1,v GaAs Applying a scissor operator (0.8 ev) we can correct the band structure. L 1,v -12 Γ 1,v L Γ X K Γ

53 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

54 Software supporting DFT Abinit ADF AIMPRO Atomistix Toolkit CADPAC CASTEP CPMD CRYSTAL06 DACAPO DALTON demon2k DFT++ DMol3 EXCITING Fireball FSatom - list of codes GAMESS (UK) GAMESS (US) GAUSSIAN JAGUAR MOLCAS MOLPRO MPQC NRLMOL NWChem OCTOPUS OpenMX ORCA ParaGauss PLATO PWscf (Quantum-ESPRESSO) Q-Chem SIESTA Spartan S/PHI/nX TURBOMOLE VASP WIEN2k

The code ABINIT http://www.abinit.org First-principles computation of material properties : the ABINIT software project. X.

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

56 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

57 Equilibrium geometry of silicon Our DFT-LDA lattice parameter: a = Bohr = Å Exp. value: a = Å at 25.

58 Kohn-Sham band structure of silicon 10 8 Energy [ev] ~ 0.5 ev Indirect gap Good dispersion of bands close to the gap Exp. gap = 1.17 ev Scissor operator = ev -12 U W Γ X W L Γ

59 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.

60 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

61 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 E HK [ρ] has a minimum, the ground state energy E 0, in correspondence to the ground state density ρ 0. The universal functional F HK [ρ] is hard to approximate. The Kohn-Sham scheme allows a reformulation in terms of one-particle orbitals.

62 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, (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,

63 Suggestion of essential bibliography Some additional items: R. M. Martin, Electronic structure: Basic Theory and Practical Methods, Cambridge University Press (2004). and references there.

Key concepts in Density Functional Theory (I) Silvana Botti

Key concepts in Density Functional Theory (I) Silvana Botti From the many body problem to the Kohn-Sham scheme European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Temporary Address: Centre