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

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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? H^ Ψ(fRg ; frg) = EΨ(fRg ; frg) ^ ^ ^ Ne electrons H = Tn (fRg) + Vnn (fRg) + Nn nuclei T^e (frg) + V^ee (frg) + U^en (fRg ; frg) 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 ^ ^ ^ ^ ^ ^ H = Tn (fRg) + Vnn (fRg) + Te (frg) + Vee (frg) + Uen (fRg ; frg) Nn 2 Ne 2 X rI X ri T^n = − ; T^e = − ; 2MI 2m I =1 i=1 Nn Ne 1 X ZI ZJ 1 X 1 V^nn = ; V^ee = ; 2 jRI − RJ j 2 jri − rj j I ;J;I 6=J i;j;i6=j Ne;Nn X ZJ U^en = − jRJ − 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 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 rI X ri 0 = T^n = − ; T^e = − ; 2MI 2m I =1 i=1 Nn Ne 1 X ZI ZJ 1 X 1 constant!V^nn = ; V^ee = ; 2 jRI − RJ j 2 jri − rj j I ;J;I 6=J i;j;i6=j Ne;Nn Ne X ZJ X U^en = − = v (rj ) jRJ − rj j 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? H^ Ψ(frg) = EeΨ(frg) Ne 2 Ne X ri 1 X 1 H^ = − + v (ri ) + Ne electrons 2m 2 jri − rj j 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 ) 2 3 Z 0 2 1 X jφj (r )j − r2 + v (r) + d3r 0 φ (r) = " φ (r) 4 2 i ext jr − r0j 5 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 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 ) 2 3 Z 0 2 1 X jφj (r )j − r2 + v (r) + d3r 0 φ (r) = " φ (r) 4 2 i ext jr − r0j 5 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 ) = p ˛ ˛ HF 1 2 N ˛ . ˛ N! ˛ . ˛ ˛ ˛ ˛φ1(xN ) φ2(xN ) ··· φN (xN )˛ 2 3 Z 0 2 1 X jφj (r )j − r2 + v (r) + d3r 0 φ (r) 4 2 i ext jr − r0j 5 i j Z ∗ 0 X φj (r) φi (r ) − δ d3r 0 = " φ (r) σi σj jr − r0j 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.
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