Second Les Houches school in computational physics «Ab initio quantum simulations for the condensed matter» June 19-June 29, 2012

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1 Second Les Houches school in computational physics «Ab initio quantum simulations for the condensed matter» June 19-June 29, 2012 An introduction to Density Functional Theory Xavier Blase, CNRS/UJF, Institut Néel, Grenoble

Question: can we «predict» from basic quantum mechanics the structural, dynamical, electronic, optical, transport, superconducting, properties of real systems without any adjustable parameter of

2 Question: can we «predict» from basic quantum mechanics the structural, dynamical, electronic, optical, transport, superconducting, properties of real systems without any adjustable parameter of input from experiment? Yes, a very few exemples: O-O pair distribution in water (T=300 K) Electronic band structure of Si Phonons band structure in GaAs (points=experiment; Rev. Mod. Phys.73, 515 (2001)

Superconductivity Predict new materials For a given stoichiometry

Random search (120000 trials) Gaps supra π and σ in MgB 2 : theory and

Search guided by some «genetic» Algorythm (1000 steps) (PRB 83, 115108,

3 Superconductivity Predict new materials For a given stoichiometry (perovskite MgSiO3), what is the most stable (enthalpy) compound? Random search ( trials) Gaps supra π and σ in MgB 2 : theory and experiment (PRL 94, , 2005) Electronic transport: conductance Search guided by some «genetic» Algorythm (1000 steps) (PRB 83, , 2011) (page web code USPEX: Universal Structure Predictor: Evolutionary Xtallography)

4 Quantum mechanics Mécanique quantique, Basdevant & Dalibard, Eds Ecole Polytechnique; Cohen-Tannoudji, Diu, Laloë, Eds Hermann. Electronic quantum state of a N-electrons system described by wavefunction: Ψ(r 1 σ 1,...,r N σ N ) Ψ({r i }) such that : Ψ({r i }) 2 dr 1...dr N = probability to find N electrons in an elementary volume: dr 1.dr N centered on {r 1,, r N } For fermions, the wavefunction is totaly anti-symmetric! By definition, the charge density can be written as: n(r) = N dr 2...dr N Ψ(r,r 2 r N ) 2 with : dr n(r) = N since Ψ Ψ n(r)dr =number of electrons in volume dr centred on r = 1 (normalization) The electronic stationary eigenstates are solutions of the eigenvalue equation: bra - ket notation : ˆ H Ψ n = E n Ψ n ˆ H = Kinetic energy T ˆ 2 N N i vext (r i ) +, v ext (r i ) = i i=1 Ionic potentiel i< j r ij Electron-electron interaction V ˆ ee Z I I RI r i (Hamiltonien operator) (atomic units: e=m e = 4πε 0 =1)

5 Quantum mechanics (2) Mécanique quantique, Basdevant & Dalibard, Eds Ecole Polytechnique; Cohen-Tannoudji, Diu, Laloë, Eds Hermann. Variational principle: the ground-state energy E 0 is given by: E 0 = min < Ψ H ˆ Ψ > under the constraint that Ψ is normalised. Ψ => Minimisation under constraint: Lagrance multipliers» technique Minimize : Ω(Ψ,λ) =< ΨH ˆ Ψ > λ ( < Ψ Ψ > 1) ), λ = Lagrange multiplier Ω λ = 0 Ψ Ψ =1 and Ω Ψ = 0 H ˆ Ψ = λ Ψ Complexity of the N-body problem for the ground-state energy We know the Hamiltonien. One «just need» to get low lying eigenvalue or minimize the energy functional (Eigenvalue equation) < Ψ ˆ H Ψ >= dr 1...dr N Ψ * ({r i }) ˆ H ({ i,r i })Ψ({r i }) BUT: this is a 3N-dimension correlated integral. If every direction discretized with n grid points => (n grid ) 3N operations!! With (tera) flops (floating-point operation per second).. A long long time!!

6 The many-electron problem: a large variety of approaches An important historical approximation: Hartree-Fock, the exact Hamiltonian with an approximate many-body wavefunction (a single Slater determinant) (Lecture V. Robert) Variational Monte Carlo: modeling the many-body wavefunction with a Slater determinant times a «correlation» Jastrow factor (Lecture M. Holzmann, Monday 25) Many-body wavefunction based quantum chemistry approaches: modeling the many-body wavefunction as a sum of Slater determinant Many-body perturbation theory And many-other variations (Lectures E. Fromager, Wednesday-Thursday 27-28) (Lectures V. Olévano and C. Attaccalite, Thursday/Friday 28-29)

An introduction to mean-field: electron-electron energy and the pair distribution function E e e = Ψ i< j 1 Ψ = dr 1 r i r j 1 dr N Ψ(r 1 r N ) 2 i< j r i r j By anti-symmetry of wavefunction and

7 An introduction to mean-field: electron-electron energy and the pair distribution function E e e = Ψ i< j 1 Ψ = dr 1 r i r j 1 dr N Ψ(r 1 r N ) 2 i< j r i r j By anti-symmetry of wavefunction and with (r i,r j ) => (r,r ): Ψ(r dr 1 dr 1 r N ) 2 N r i r j = drdr' r r' dr 3 dr N Ψ(r,r',r 3 r N ) 2 E e e = drdr' ρ 2(r,r') with : ρ 2 (r,r') = r r' N(N 1) 2 dr 3 dr N Ψ(r,r',r 3 r N ) 2 ρ 2 (r,r ) is the pair density integrating to N(N-1)/2 You do not need to know the wavefunction, but a (very) integrated quantity! Exercise: Ψ V (r i ) Ψ = drv (r)n(r) i

The density functional theory (DFT) The central idea is to show that the ground-state total energy is a functional of the charge density n(r) (scalar field and physical observable) instead of the

8 The density functional theory (DFT) The central idea is to show that the ground-state total energy is a functional of the charge density n(r) (scalar field and physical observable) instead of the many-body wavefunction: Theorem HK1 (Hohenberg+Kohn, Phys Rev 136, B864 (1964)) Given n(r) the charge density, there exists only one external potential V ext (r) (up to a constant) that can realize n(r) (the reverse statement is obvious!). By absurdum, let s assume that there exist 2 external potentials realizing the same charge density:

9 Note: demo for non-degenerate ground-state. Further problem of existence of v ext for n(r) => Levy constrained search formulation Therefore if n(r) is given then the external potential v ext is unique => the eigenvalue equation and thus its ground-state wavefunction in uniquely defined! n(r) v ext ˆ H Ψ 0

11 DFT without orbitals. From this variational principle derives a Euler-Lagrange equation which in principle allows to find n(r): n(r) [ E[n] µ[ dr n(r) N] ] = 0 v ext (r) + F HK[n] A 3D ordinary differential equation! n(r) with µ the chemical potential associated with the conservation of number of electrons. However, F HK [n]=t[n]+v ee [n] is still unknown!! = µ An interesting formal result: Convexity of the F HK functional F HK [n] Given n 0 associated with v 0 ext and Ψ 0 n 1 v 1 ext Ψ 1 We build: n λ = (1-λ) n 0 + λ n 1 (By HK1, corresponds a unique v λ ext and Ψ λ ) Then: F HK [n λ ] (1 λ) F HK [n 0 ] + λ F HK [n 1 ] Local minima forbiden The non-existence of local minima is an important property in the search of the ground-state by minimization scheme starting from a «guess» density.

12 A few simple functionals for the kinetic energy: T=T[n] Thomas-Fermi: Take free electrons in a box of volume V: eigen-energies: ħk 2 /2m quantization of k-vector => Two quantum state (with spin) in k-space unitary volume (2π) 3 /V In the ground state, with N electrons: k F3 ~ (N/V) and E/V ~ (N/V) 5/3 Suggests: von Weizsäcker E[n] = T 0 [n] = C dr n 5 / 3 (r) Local density of kinetic energy Take the hydrogen atom with ϕ 0 (r) e r / a 0 With only one orbital everywhere positive: ϕ 0 (r) = n(r) T[n] = 1 2 drϕ 0 (r) 2 ϕ 0 (r) = 1 8 dr ( n(r))2 Involves the gradient! n(r) The results are very poor. And which one to choose?

13 The Kohn and Sham equivalence idea The «true» system? An equivalent «non-interacting electrons» fictitious system in an effective V eff mean external potential Same energy Same density 2 n(r) = N dr 2 dr N Ψ(r,r 2 r N many-body wavefunction n(r) = φ i (r) N i=1 one-body wavefunctions 2 N non-interacting electrons in ground state under V eff T[n] =? T 0 = N φ i 2 i=1 2 φ i Kinetic energy of non-interacting system { ε i } H ˆ Ψ = E Ψ V eff (r) φ i (r) = ε i φ i (r) In a non-interacting electronic system, everything is «easy» to calculate (n,t 0,etc.)

14 The Kohn and Sham equivalence idea (II) Does the non-interacting equivalent system exists? What is the effective potential v eff? Start from: E[n] = T[n] + dr v ext (r)n(r) + V ee [n] E[n] = T 0 + dr v ext rearrange (r)n(r) + J[n] + [ T[n] T 0 + V ee [n] J[n] ] (Hartree classical) J[n] = 1 2 drdr' ρ(r)ρ(r') r r' E xc [n] by definition! (Exchange+Correlation) Now equivalence of Euler equations: [T + V ee ] n(r) + v ext (r) = µ = T 0 n(r) + v eff (r) v eff (r) = v ext (r) + n(r')dr' r r' + E xc [n] n(r) Hartree potential V H (r) Exchange-correlation potential: V xc [n](r)

15 The Kohn and Sham equivalence idea (III) 1) This was just a rearrangement of energy terms => the energy is the same! 2) Since the energy is the same, the Euler-Lagrange equation that gives the charge density is the same : [T + V ee ] n(r) + v ext (r) = µ = T 0 n(r) + v eff (r) 3) By the first Hohenberg and Kohn theorem, the V eff potential is unique. 4) E xc is related to differences: (T-T 0 ) and (V ee -J) => the unknown part is pushed in a «smaller term». This is all formal and exact! But we do not know E XC [n] since we do not know T[n] and V ee [n]

16 Exchange and correlation Hartree approximation: Ψ(x 1, x N ) = φ 1 (x 1 ) φ N (x N ), with x = (r,σ) Uncorrelated probabilities : large Coulomb penalty since the wavefunction does not allow electrons to avoid each other! No exchange and no correlation. Hartree-Fock approximation: Ψ(x 1, x N ) = 1 N ( 1) σ (P ) φ P(1) (x 1 ) φ P(N ) (x N ) Fermi repulsion built between electrons of same spin from the exchange term. Electrons of different spin remain uncorrelated => pay Coulomb repulsion energy. ρ 2 (x, x') n(x)n(x') P The correlation energy is in general defined as the difference between the exact energy and the Hartree-Fock energy: E C =E exact -E HF < 0 It is a negative energy since the Hartree-Fock wave function is «constrained». Within DFT, what we call exchange and correlation also includes the (T-T 0 ) term!!

17 The local density approximation (LDA) DFT is a rigorous theory, but E xc is unknown. We need to do an approximation. An historical and fruitful approximation is the Local density Approximation (LDA) which consists in introducing a local density of exchange and correlation (XC) energy at point r which only depends on the value of the charge density n at r: To go further, the XC energy density ε XC is obtained thanks to very accurate Quantum Monte Carlo simulations (QMC) on interacting but homogeneous electron gas (Ceperley + Alder, 1986) of density n hom =n(r): ε LDA XC (n(r)) = ε QMC XC (n hom ), with : n hom = n(r)

18 The local density approximation and the QMC data on the homogeneous electron gas For an homogeneous electron gas, T 0 (n hom ), J(n hom ) and the exchange energy E X (n hom ) are analytic. By subtraction from total energy E QMC (n hom ) as given by accurate QMC calculations, the correlation energy E C (n hom ) can be obtained. ε C LDA (n(r)) = E C QMC (n hom = n(r)) /N The QMC numerical values are fitted by some functional form (e.g. Perdew+Zunger, PRB 23, 5048 (1981) Figure: density of correlation energy as a function of the Wigner-Seitz electronic effective radius r s /a 0. 1/n e = (4 /3)π r s 3 avec n e average electron density. The QMC data are compared to high and low analytic calculations within perturbation theory. The LDA is exact for an homogeneous electron gas. What about highly inhomogeneous (real) systems?

19 A crucial aspect: self-consistency V eff depends on n(r) which depends on the unknown Kohn-Sham eigenstates: N n(r) = φ i (r) 2 i=1 Initial charge density n 0 (r) One needs a self-consistent cycle The convergence and speed of convergence of this self-consistent loop is a difficult numerical problem. You do not need the full electronic spectrum The charge density requires only occupied states => no need for «full diagonalization» Construct effective potential V eff (n (i) (r)) and Hamiltonian Solve Kohn-Sham equation and build: n(r) = φ i (r) 2 i=1 Obtain new wavefunctions and charge density n (i+1) (r) N unconverged Compare n (i) and n (i+1) convergency

20 DFT-LDA for covalent semiconductors with very inhomogeneous charge densities a 0 (a.u.) B 0 (kbar) E C (ev) ω TO (cm-1) Si LDA (10.22) 941 (960) 5.23 (5.28) 517 Si BP* (10.46) 830 (800) Si PW* (10.39) 850 (830) Exp GaAs LDA (10.62) 760 (740) 8.04 (7.99) 277 GaAs BP* (10.88) 617 (600) GaAs PW* (10.87) 621 (650) Exp Calculations with planewaves and the VBC pseudos from PRB 53, 1180 (1996). In parenthesis, all-electron calculations from PRB 50, (1994). PW/BP: Perdew-Wang/Becke-Perdew generalized gradient corrected (GGA) functional. LDA calculations yield in general excellent lattice constant (bond lengths), bulk moduli, phonon frequencies. Cohesive energies suffer from the calculation on the isolated atom reference. Gradient corrected functionals do not improve systematically all properties. Xavier Blase, Les Houches 2012

21 The DFT/LDA charge density in inhomogeneous systems Density matrix in silicon As shown here for solid argon and silicon, with very inhomogeneous charge densities, the DFT/LDA charge density is in excellent agreement with higher level approaches. (QMC) Comparing DFT/LDA with a self-consistent many-body approach (QPscGW) for solid Argon (PRB 74, ) (LDA) (PRB 57, (1998))

22 The exchange-correlation hole and the electron-electron interaction Remember: V e e = drdr' ρ 2 (r,r') r r' with : drdr' ρ 2 (r,r') = N(N 1) /2 number of pairs Define: h(r,r') such that : ρ 2 (r,r') = ρ(r)ρ(r') 2 [ 1+ h(r,r') ] deviation from classical (Hartree) term Then: V e e = 1 2 drdr' ρ(r)ρ(r') [r r' drdr' ρ(r)ρ xc (r,r'), with : ρ xc (r,r') = ρ(r')h(r,r') [r r' Classic term (Hartree) Interaction of an electron with its «exchange-correlation hole» SUM RULE: dr' ρ xc (r,r') = 1 Exact result (Exercise). Si

23 A possible explanation for the success of LDA: a) E xc involves the spherically averaged XC-hole b) The XC-hole sum-rule is exactly satisfied E xc = dr n( r ) 4πs ds ρ SA xc ( r,s) with the spherical averaged hole : ρ SA xc ( r,s) = 0 dr ' ρ xc ( r, r ') r r ' =s Spherically averaged XC hole in Si around an Electron centered at middile of Si-Si bond: LDA Vs VMC (variational Monte Carlo) Exchange hole for Neon atom along some specific (Hood et al., PRB 1998). direction (left) and spherically averaged (right) (Gunnarsson et al. PRB 1979) Xavier Blase, Les Houches 2012

A first well identified problem with the LDA: the self-interaction The self-consistent potential (Hartree+exchange-correlation) acting on an occupied state depends on the charge density build

? This interaction is repulsive: pushes occupied states higher in energy!

24 A first well identified problem with the LDA: the self-interaction The self-consistent potential (Hartree+exchange-correlation) acting on an occupied state depends on the charge density build with the contribution from this occupied state. V SCF LDA (n(r))φ i (r) = V LDA SCF ( N j =1 φ j (r) 2 )φ i (r) What happens when i=j?? This interaction is repulsive: pushes occupied states higher in energy! A good example: the Hydrogen atom: the electron acts on itself => self-interaction error of the order of half a Rydberg! E 0 LDA (Hydrogen) 6 to 7eV instead of 13.6 ev Hydrogen atom n(r) Hartree-Fock is self-interaction free V SCF (n(r)) < r ( ˆ J ˆ K ) φ i >= N j =1 dr' φ * j(r')φ j (r') φ r r' i (r) N j φ * j (r')φ j (r)φ i (r') dr' r r' Xavier Blase, Les Houches 2012

25 Another deficiency: the image charge potential The long range behavior of the potential seen by an electron outside a surface should be the classical (electrostatic) image charge potential in 1/z (z «distance» to the surface). In DFT/LDA, the effective potential decays exponentially with the charge density. [PRL 80, 4265 (1998)] For occupied states, one needs the true non-local exchange to get the correct long-range behavior. What about van der Waals interactions? What interaction can you get within DFT-LDA between two neutral objects in vacuum separated by a distance larger than the decay length of the wavefunctions in the vacuum?

26 Improving the exchange-correlation functional (see Lecture by Miguel Marques) This is an important and difficult field. Several ideas in several directions improving some properties for some systems. Generalized gradient corrections: look for functionals that also depend on the gradient of the charge density (remember von Weizsäcker) to get information on the inhomogeneities of charge. Make sure that sum rules are satisfied. Examples: PBE (Perdew, Becke, Ernzerhof), PW (Perdew, Wang), etc. Self-interaction corrected functionals (SIC): cure the self-interaction problem by removing the contribution of an orbital from the charge density used to build the effective potential acting on this orbital ( )φ i (r) " " V eff ( n(r) )φ i (r) V eff n(r) φ i (r) 2 Hybrid and range-separated hybrid functionals: mixing local exchange and correlation with exact (non-local) exchange with coefficients that can depend on the range. Examples: B3LYP (Becke, three-parameter, Lee-Yang-Parr), HSE (Heyd-Scuseria-Ernzerhof), etc. The parameters are «educated» on a family of systems and properties. Etc.

27 The meaning of the Kohn-Sham eigenvalues («the band structure»)? Photoemission measure differences of energy hν + E(N) = E(N 1) + KE(e ) IP = ε HOMO = E(N) E(N 1) AE = ε LUMO = E(N +1) E(N) E Gap = E(N +1) + E(N 1) 2E(N) The ionization potential (IP), electronic affinity (AE) and gap of finite size systems can be obtained as differences of ground state energies (ΔSCF technique). This works quite well but cannot be used for extended systems and the full energy spectrum. The Kohn-Sham equation provides «energy levels and wavefunctions» : 2 +V eff (r) φ i (r) = ε i φ i (r) - 2 V eff (r) = V ext (r) + n(r')dr' r r' What is their meaning? + E XC [n] n(r)

28 A few formal things we know about Kohn-Sham eigenvalues A) Kohn-Sham eigenvalues cannot be identified formally to electronic energy levels and do not represent differences of total energies due to double counting terms associated with the self-consistent nature of the Kohn-Sham equations: N E[n] = ε i + E I [n] i=1 dr E I [n] n(r) n(r), with : E I [n] = J[n] + E XC [n] B) They satisfy however the Janak theorem related to the variation of the total energy with respect to an infinitesimal variation of the energy levels population : ε i = E[n], with f f i the fractionnal occupation of level i i { ε i, f i } C) Kohn-Sham eigenvalues are formally the «Lagrange multipliers» associated with the orthonormality constraint between Kohn-Sham orbitals when minimizing the ground-state total energy.

29 A few practical things we know about Kohn-Sham eigenvalues (I) The band structure of silicon Band gaps of some semiconductors For a large family of standard semiconductors, the band gap is too small but the dispersion of the bands is very reasonable!

30 The DFT/LDA Kohn-Sham single particle states Even though the band gap is usually wrong within DFT/LDA Kohn-Sham, the wavefunctions can be excellent! This is consistent with the «rigid opening of the band gap observation», equivalent to a «more or less» constant negative/ positive potential correction for the occupied/unoccupied states. More sophisticated approaches (many-body perturbation theory) aiming at correcting electron/hole energy levels usually assume that the DFT-LDA wavefunctions are correct (see Lecture V. Olévano, Thursday 28 th ).

The self-interaction affects mainly occupied localized orbitals The self-interaction problem affects more strongly localized states as compared to delocalized states => induce a differential of

31 The self-interaction affects mainly occupied localized orbitals The self-interaction problem affects more strongly localized states as compared to delocalized states => induce a differential of errors that affects the relative position of states which are different in nature. Cytosine 3d vs (s,p) levels ZnS with its Zn 3d Surface vs bulk states H-Si(111)1x1 surface LDA 3d LDA 3d G 0 W 0 3d exp. Exp. GW [PRB 83, (2012)] [PRB 57, 6485 (1998)] [PRL 72, 1878 (1994)] Xavier Blase, Les Houches 2012

32 Demo 1: the total energy and the Kohn-Sham eigenvalues 2 +V eff (r) φ i (r) = ε i φ i (r) - 2 drφ * i (r) 2 ε i = φ i 2 φ i + dr v eff (r)φ i (r) 2 V eff (r) = V ext (r) + n(r')dr' r r' + E XC [n] n(r) n(r) = N i=1 φ i (r) 2 N ε i = T 0 + i=1 drdr' n(r)n(r') + dr v ext (r)n(r) + dr E XC [n] n(r) r r' n(r) 2J[n] NOT E XC [n] Compare to : E[n] = T 0 + J[n] + drv ext (r)n(r) + E XC [n] With : E I [n] = J[n] + E XC [n] dr E I [n] n(r) n(r) = n(r)n(r') drdr' + dr E XC [n] n(r) r r' n(r) E[n] = N ε i + E I [n] i=1 dr E I [n], CQFD n(r)

33 Demo 2: Kohn-Sham eigenvalues are Lagrange multipliers In the Kohn-Sham formalism, we have re-introduce (one-body) eigentates. We use the variational principle with respect to these wavefunctions imposing orthonormality: Minimize : Ω[{φ i,λ ij }] = E[{φ i }] λ ij φ i φ j δ ij E φ i * (r) λ ijφ j (r) = 0 j i< j ( ) T 0 φ * i (r) + (E T ) 0 n(r) n(r) φ * i (r) = N N With : n(r) = drφ * i (r) φ i (r), and : T 0 = drφ * i (r) 2 φ 2 i (r), i=1 and : E T 0 = 1 2 drdr' n(r)n(r') r r' i=1 + drn(r)v ext (r) + E XC [n] v ext (r) + v Hartree (r) + E XC [n] φ n(r) i (r) = λ ij φ j (r) V eff (r) j λ ijφ j (r) j (We have rederived the expression for V eff ). A unitary transformation that preserves the charge density and energy can diagonalize the λ ij matrix => λ i =ε i are the «Kohn-Sham eigenvalues». Xavier Blase, Les Houches 2012

34 Demo 3: Reduced density matrix and the Janak theorem (I) The reduced density matrix and the natural orbitals γ(r,r') = N dr 2 dr N Ψ * ( r,r 2 r N )Ψ r',r 2 r N n(r) = γ(r,r) et Trace(ˆ γ ) = dr γ(r,r) = N ( ) It is a positive semi-definite hermitian operator with real positive eigenvalues dr γ(r,r')ϕ i (r') = f i ϕ i (r) with f i 0 and Trace(ˆ γ ) = f i = N i The orbital ϕ i are named the Löwdin natural orbitals. ˆ γ = f i ϕ i ϕ i r ˆ i De même (Exercice) : Ψ ˆ T Ψ = 1 2 γ r = γ(r,r) = n(r) = f i ϕ i (r) 2 i dr( r γ(r,r') ) r= r' = 2 f i ϕ i 2 ϕ i The charge density and kinetic energy can be written exactly as a sum over one-body orbital contributions but with a fractional occupation factor f! i

35 Fractional occupancy generalization of Kohn-Sham approach. Janak theorem (II) We follow Kohn-Sham and come back to a formulation in function of one-body orbitals but with fractional occupation f. E[n] E[ { f i,ϕ i }] with n(r) = f i ϕ i (r) 2 and T = f i ϕ i i i 2 /2 ϕ i Then it follows (Exercice): E f i = ε i (Janak s theorem) The variation of the total energy with respect to infinitesimal variation of population is NOT what we measure in photoemission where an entire electron is removed/added. Reminder: In Hartree-Fock theory, Koopmans theorem establishes that the Hartree-Fock eigenvalues are related to electron/hole removal energies provided that the other orbitals do not relax (see lecture Vincent Robert).

36 A shamefuly short bibliography A very general book for DFT-based simulations in condensed-matter physics: "Electronic structure : Basic theory and practical methods", Richard Martin, Cambridge University Press, DFT books (theory oriented): "DFT of atoms and molecules", Parr and Yang, International Series and Monographs on Chemistry-16, Oxford University Press (1989). «Electron density functional theory (Lecture notes, rough draft), "Electron correlations in molecules and solids", Peter Fulde, Springer Verlag Series on Solid-State Sciences, Vol.100, Editions Springer Verlag (Second edition 1993). Review articles: "The DFT, its applications and prospects", Jones, Gunnarsson, Rev. Mod. Phys. 61, No.3, July "Nobel Lecture: Electronic structure of matter-wave functions and density functionals, W. Kohn, Rev. Mod. Phys. 71, (1999).

37 "Iterative minimization techniques for ab initio total energy calculations", Payne, Teter, Allan, Arias, Joanopoulos, Rev. Mod. Phys. 64, No.4, octobre «Historical» DFT articles: "Inhomogeneous electron gaz", Hohenberg and Kohn, Phys. Rev. 136, B864 (1964). "Self-consistent equations including exchange and correlation effects", Kohn and Sham, Phys. Rev. 140, A1133 (1965). "Description of exchange and correlation effects in inhomogeneous electron systems", Gunnarsson, Jonson, Lundqvist, Phys. Rev. B 20, 3136 (1979). "Self-interaction correction to density functional approximations for manyelectron systems", Perdew and Zunger, Phys. Rev. B 23, 5048 (1981). ""Density-functional thermochemistry. III. The role of exact exchange". Becke, Axel D. (1993). The Journal of Chemical Physics 98 (7): 5648.

Density Functional Theory for Electrons in Materials

Density Functional Theory for Electrons in Materials Richard M. Martin Department of Physics and Materials Research Laboratory University of Illinois at Urbana-Champaign 1 Density Functional Theory for