The Self Interaction Correction revisited
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1 The Self Interaction Correction revisited Explicit dynamics of clusters and molecules under irradiation Spectroscopic accuracy at low energy SIC problem : one electron interacts with its own mean-field! Density Functional Theory (DFT) for electrons, classical ions From Local Density Approximation (LDA) to SIC and beyond Toulouse-Erlangen collaboration (P. G. Reinhard)
2 S u m m a r y Electronic systems DFT and Local Density Approximation (LDA) From Density Functional Theory (DFT) to LDA LDA Self Interaction problem Ionization potential problem Self Interaction Correction (SIC) Optimized Effective Potential (OEP) Simple approximations, Slater, ADSIC Examples of application Defects of simple approximations SIC revisited General SIC formulation Double set formulation and Generalized Slater Time Dependent SIC Problem Solution Problem Solution
3 S u m m a r y Electronic systems DFT and Local Density Approximation (LDA) From Density Functional Theory (DFT) to LDA LDA Self Interaction problem Ionization potential problem Self Interaction Correction (SIC) Optimized Effective Potential (OEP) Simple approximations, Slater, ADSIC Examples of application Defects of simple approximations SIC revisited General SIC formulation Double set formulation and Generalized Slater Time Dependent SIC Problem Solution Problem Solution
4 About electronic systems (Quantum) electrons and (classical) ions in molecules «Who»follows«whom» Ex: Born Oppenheimer surface Large «phase» space for deformations Ex: degeneracy lifting Well defined center of mass (ions) Key role of electron exchange (and correlation) neutral systems most binding comes from exchange (~3/4) and correlation (~1/4) Ex: infinite jellium metal Ex: also in finite systems
5 Some observables Intrinsic single particle properties Density of states (DOS), Band Gap, Ionization Potential (IP) Ex: solids (DOS, gap), molecules, chemistry (IP) Global structure properties Bond length, Dissociation energy, Potential barriers Ex: potential energy surfaces in chemistry Response to electromagnetic fields Static and dynamical responses Polarizability, Magnetic moment, Optical response Ex: physics and chemistry of irradiation Typical ingredients for irradiation dynamics
6 S u m m a r y Electronic systems DFT and Local Density Approximation (LDA) From Density Functional Theory (DFT) to LDA LDA Self Interaction problem Ionization potential problem Self Interaction Correction (SIC) Optimized Effective Potential (OEP) Simple approximations, Slater, ADSIC Examples of application Defects of simple approximations SIC revisited General SIC formulation Double set formulation and Generalized Slater Time Dependent SIC Problem Solution Problem Solution
7 From DFT to LDA Electrons (Time Dependent) Density Functional Theory (TD) DFT Ensemble of (orthonormal) orbitals (1 electron) One body density Effective mean field theory Local Density Approximation Hartree Exch. + Corr. LDA and TDLDA Exchange-correlation function of density ρ
8 Local Density Approximation (LDA) Plasmon response Local exchange-correlation taken from infinite E electron gas U xc assumed to be a function of ρ (r) Ex: pure exchange ρ 1/3 ω e d A priori not valid for rapidly varying ρ (r) 24% Much better than expected Error cancellations between exchange3% and correlations Ex: problem with exact exchange functionals r e D e CI In general correct 1% for global energies Vlasov Ex: IP from E(N+1) E(N), dissociation energies within 10-20% Exhibits good formal features Many sucessful applications (solid state, molecules, clusters) LDA tends to delocalize wavefunctions metals (simple ones!!)
9 S u m m a r y Electronic systems DFT and Local Density Approximation (LDA) From Density Functional Theory (DFT) to LDA LDA Self Interaction problem Ionization potential problem Self Interaction Correction (SIC) Optimized Effective Potential (OEP) Simple approximations, Slater, ADSIC Examples of application Defects of simple approximations SIC revisited General SIC formulation Double set formulation and Generalized Slater Time Dependent SIC Problem Solution Problem Solution
10 From DFT to LDA But Electrons (Time Dependent) Density Functional Theory (TD) DFT Ensemble of (orthonormal) orbitals (1 electron) One body density Effective mean field theory Local Density Approximation Hartree Exch. + Corr. LDA and TDLDA Exchange-correlation function of density ρ Energy Potential Self Interaction Ionization Potential Problem
11 The Ionization Potential Problem Potential and level scheme IP Example of anion clusters : K 7 - (8 electrons) Exp: Little bound system Ionization Potential about 1 ev LDA : Almost no binding Totally wrong IP (reasonable ΔE) dynamics?
12 The Ionization Potential Problem (2) Example of organic molecules: Benzene-like cyclic structures: H N, O, S Neutral species Covalent bond Ionization Potential [ ev] Exp LDA C 6 H 6 C 5 H 5 N C 4 H 4 N 2 C 4 H 4 O C 4 H 4 S 50% error on IP! 0 Benzene Fulvene Pyridine Pyrimidine Pyrazine Furane Thiophene LDA : Totally wrong IP
13 Self Interaction Correction (SIC) SIC energy functional Subtract explicitely self interaction One-body equations, α Orbital dependent field Loss of orthonormality Time dependent formulation? Approximation by One potential: Optimized Effective Potential NB: No SIC problem in HF due to exact cancellation by exchange
14 S u m m a r y Electronic systems DFT and Local Density Approximation (LDA) From Density Functional Theory (DFT) to LDA LDA Self Interaction problem Ionization potential problem Self Interaction Correction (SIC) Optimized Effective Potential (OEP) Simple approximations, Slater, ADSIC Examples of application Defects of simple approximations SIC revisited General SIC formulation Double set formulation and Generalized Slater Time Dependent SIC Problem Solution Problem Solution
15 Optimized Effective Potential (OEP) SIC and beyond Try to restore a common 1-body potential by «optimal» choice of a unique common V 0 : «parametrize» wavefunctions ϕ i by V 0 Full OEP «painful»
16 Optimized Effective Potential (OEP) SIC and beyond Try to restore a common potential by «optimal» choice of a unique common V 0 : «parametrize» wavefunctions ϕ i by V 0 Practically Almost only approximate versions of OEP are numerically tractable KLI, Slater, ADSIC
17 From full OEP to Krieger Li Iafrate (KLI) Approx. OEP «Local» Average (Slater) «Global» Average Averaged Density SIC (ADSIC)
18 Back to anion clusters Potential and level scheme IP Example of anion clusters : K 7 - (8 electrons) Exp: Little bound system Ionization Potential about 1 ev Correct IP, restoration Koopmann s theorem Access to 1-photon processes (anions)
19 Back to carbon cycles Benzene Fulvene Pyridine Ionization Potential [ ev] Pyrimidine Pyrazine Furane Thiophene Case of Organic molecules Good reproduction of IP with ADSIC ADSIC : Correct IP Exp LDA C 6 H 6 C 5 H 5 N ADSIC C 4 H 4 N 2 C 4 H 4 O C 4 H 4 S
20 New bad surprises Orthonormality violation Potential energy surface Localization/delocalization? Time variation of energy C 2 Na 5 exact exch. Dissociation problem Polarizability problem Violation of Energy conservation Momentum conservation
21 S u m m a r y Electronic systems DFT and Local Density Approximation (LDA) From Density Functional Theory (DFT) to LDA LDA Self Interaction problem Ionization potential problem Self Interaction Correction (SIC) Optimized Effective Potential (OEP) Simple approximations, Slater, ADSIC Examples of application Defects of simple approximations SIC revisited General SIC formulation Double set formulation and Generalized Slater Time Dependent SIC Problem Solution Problem Solution
22 SIC revisited New variational principle One-body equations (Key) Symetry condition Orthonormality Double set of wavefunctions Idea: Exploit a left over degree of freedom : unitary transform Localized vs physical wavefunctions Symetrizing vs diagonalizing wavefunctions Several new possibilities Application to OEP Generalized Slater becomes accurate Time dependent calculations in full SIC possible
23 OEP with double set: Generalized Slater Single electron density «Diagonalizing set» «Natural» Slater approximation on «localized» wavefunctions ψ α Generalized Slater (GSlat) Total XY «Localized set»
24 Generalized Slater Potential energy surface Energy and polarizability of Carbon atom Exchange only calculation Hartree-Fock exact «0%» GSlater full SIC HF C 2 GSlater and SIC restore dissociation properties of C 2 Energy deviation [%] Polarizability deviation LDA r Slater C atom exchange only x,y z GSLAT SIC
25 Application to Polarizabilities H 4 chains Polyacetylene Standard DFT test case Hydrogen chains H 4 Longitudinal Transverse GSlater full SIC H 2 -H 2 distance
26 S u m m a r y Electronic systems DFT and Local Density Approximation (LDA) From Density Functional Theory (DFT) to LDA LDA Self Interaction problem Ionization potential problem Self Interaction Correction (SIC) Optimized Effective Potential (OEP) Simple approximations, Slater, ADSIC Examples of application Defects of simple approximations SIC revisited General SIC formulation Double set formulation and Generalized Slater Time Dependent SIC Problem Solution Problem Solution
27 Time Dependent SIC (TDSIC) Variational principle TDSIC equation Orthonormality Symetry condition Propagation, puzzle
28 TDSIC made practical Propagation scheme 2 sets of wavefunctions linked by a unitary transform Choice : Propagating set Symetrizing set An example from molecular physics Simple 1D dimer Benchmark SIC calculation
29 Some conclusions and perspectives From LDA to SIC and beyond SIC schemes solve the electronic IP problem Several difficulties remain: formal and numerical challenges Localized vs delocalized wavefunctions GSlater provides an appealing alternative to OEP Full TDSIC provides the first benchmak calculation Applications cluster dynamics from linear to non linear domain Ex: optical response, photoelectrons spectra, pump/probe dynamics dynamics of organic molecules Ex: irradiation of biological molecules
30 Thank you for your attention
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