Many electrons: Density functional theory Part II. Bedřich Velický VI.

Many electrons: Density functional theory Part II. Bedřich Velický velicky@karlov.mff.cuni.cz VI. NEVF 514 Surface Physics Winter Term 013-014 Troja 1 st November 013

This class is the second devoted to many-electron aspects of the solid state (computational) theory it gives examples of the reliable DFT results introduces the problem of KS orbitals and their possible physical meaning outlines the relationship between the Kohn-Sham Density functional theory (DFT) and the Green s function method of many-body perturbation theory shows the results of DFT treatment of the surface electronic structure of semi-infinite jellium

Using DFT within and outside of its nominal validity range

Where DFT/ DFT LDA works DFT and its approximate implemetations were designed to optimize the particle density distribution by minimizing total energy. 1. Charge distribution in various compounds... compare semiconductor with a metal chemical bond. Total energies and equilibrium geometry of atomic stability of 3 d transition metal structures all but one correctly predicted straightforward use of LDA, fully ab initio systems 3. Static susceptibilities magnetic susceptibilities of alkali use of LSDA metals

Where DFT works 1: Electron density maps SILICON CRYSTAL (110) PLANE density contours of valence electrons ALUMINUM CRYSTAL (110) PLANE density contours of valence electrons Si nuclei and bond connecting lines

Where DFT works : Stability of 3d metal crystals d-cubic s-cubic s-hex hcp bcc fcc diamond cubic struct. simple cubic struct. simple hexagonal struct. hexagonal close packed struct. body centered cubic struct. face centered cubic struct. Ω Ω = 0 unit cell volume experimental value CRYSTAL STRUCTURE X exp. comp. Sc hcp hcp Ti hcp hcp V bcc bcc Co hcp fcc Ni fcc fcc Cu fcc fcc

Where DFT works 3: Paramagnetic susceptibilities of alkali metals

Problem: KS orbitals, KS orbital energies 1 Essence of the problem with orbitals in general: In interacting systems they are not supposed to exist. But they are seen in the experiments. Think of band structure mapping in PES ACCORDING TO THE CONSTRUCTION OF THE ONE-ELECTRON POTENTIAL methods V ( r) note eff IMPORTANT ORBITAL THEORIES problem I. semi-empirical parametrized "ideal" orbital theory unjustified II. ab initio often not respected one-electron V + VH + Vx podle Vx: Hartree, HF, Xα one-particle, but approximate KS DFT V + VH + Vxc approx.: LDA, GGA, fictitious KS particles quasi-particle V + V + Σ ( E) approx.: RPA, GW, very costly in computations H III. natural undefined eigenfunctions of the one- nearly observable, but not orbitals particle density matrix easy to obtain directly The Kohn-Sham orbitals and energies are definitely fictitious, yet very appealing: - are obtained from a Schrödinger-like equation so similar to Hartree or Xα - look so neat and useful and physical - you have them in heaps as an unavoidable by-product at a low-cost effort - you have nothing else at hand to calculate conductance, optical absorption

Problem: KS orbitals, KS orbital energies Some exact results for the Kohn-Sham orbitals: o They compose together the particle density: n( r) = n α ( r) KS The same is true for natural orbitals: n( r) = n κ ϕ κ ( r) n.o. α ψ For jellium, KS orbitals and natural orbitals are the same, plane waves. But the occupation numbers are different n k n k KS kf k n.o. o The highest occupied KS level has the energy equal to the ionization potential: EHOMO = I o More generally, there holds the Janak s theorem, an analog of the Koopmans theorem: Eα = δ δ n E v [ n] α kf k

Problem: KS orbitals, KS orbital energies 3 Experimenting numerically (Stowasser and Hoffman): o Calculate water by different approximations method details KS Kohn Sham hybrid functionals HF Hartree Fock restricted HF, Gaussian EH Extended Hückel semi-empirical method NOT REALLY CONCLUSIVE

Problem: KS orbitals, KS orbital energies 4 A. Savin, C.J. Umrigar, Xavier Two electron atoms, exact KS potential constructed from the exact density n( r)... known from exact calculation for He v H 3 e r r ( r) = d r n( r) KS KS 1s ψ1s n( r) = ψ ( r) ( r) = n( r) ( e r m + e H + xc = m e r 1 υ υ ) ψ ψ KS KS 1s E1s 1s two unknowns... by iteration H KS n For higher (empty) ψ use as S R: ( + υ + υ ) ψ = ψ e xc KS KS n En n 1,3 1 KS= En E1s E(1sn ) L E(1 s ) S single KS "particle" single particle excitation (many body) ε The exact KS energies close to the true excitation energies Explanation by the asymptotically equivalent differential equations

Problem: KS orbitals, KS orbital energies 5: DFT vs. Green s functions (MBPT) DFT has a primary competitor: many body perturbation theory in the form of the Green s functions They run in parallel from the outset: Kohn Sham DFT LDA paper 1965 Lars Hedin GF GW paper 1965 description of excitations disputable description of excitations reliable but... a by-product of calculation but... requires heavy computation commonly used, reasonable success spreading in practice more recently difficult problem: semiconductor gap gap problem non-existent basic object: Kohn Sham particle basic object: quasi-particle effective potential veff ( r) self-energy Σ ( r, E)... true Hermitian operator... non-hermitian, energy dependent real one particle energies quasi-particle energy + life-time

Problem: KS orbitals, KS orbital energies 5: DFT vs. Green s functions (MBPT) Fundamental difference between DFT and the Green s function method In DFT, exact or approximate alike, the true many-body ground state is represented as a Slater determinant of N spin-orbitals occupied by the Kohn- Sham particles (structure akin to the Hartree-Fock approximate ground state) In the Green s function method, true many-body ground state is probed by injecting an electron and letting the N+1 particle state propagate. This electron excitation behaves like a quasi-particle (electron+ xc hole) extracting an electron and letting the N--1 particle state propagate. This is a hole excitation, it behaves like a quasi-particle again. The formal parallel: ( + + + ) = E ( + e e + + ( )) = E equation for KS particle state equation for quasi-particle state Link by the 1 st order perturbation expression: qp E E ( E ) E υ υ υ ψ ψ υ υ Σ ψ ψ qp qp qp qp m H xc α α α m H Eα α α α + ψ Σ + i0 υ ψ = + Λ + iγ α α α α xc α α α α E Λ + quasi-particle energy / Γ α α α quasi-particle life-time

Problem: KS orbitals, KS orbital energies 6: DFT vs. Green s functions (MBPT) About the gap problem LDA band structures of semiconductors are not bad, except for too small values of the gap. The source of the error has been found, but not eliminated. Green s functions in the GW approximation give much better results.

Surface electronic structure of semi-infinite jellium

Recapitulation of DFT-LDA 3 For jellium, KS theory is exact ε KS XC LDA Ansatz ε υ KS XC xc δe δ n( r) J XC (, r n) = ε () n J XC ( r,[ n]) = ε ( n( r)) ( r) XC = n LDA KS equations ( nε ) XC ( m ( r) H ( r) xc( r) e defines the LDA XC energy + υ + υ + υ ) ψ = E ψ (1) As easy to solve as Hartree or Slater equations () Of course, by iteration in a self-consistent cycle J α α α

Clean semi-infinite jellium semi-infinite jellium far more realistic than truncated Sommerfeld + simplest theory of surface ever -- a single parameter n or r s treated in DFT-LDA means fully ab initio, no additional fudging with parameters etc. the calculation permitted to find also total energies, in particular the surface energy here, we first inspect a few figures of surface charge distribution and related quantities ( N.D. Lang and W. Kohn, PRB 1(1970), 4555) µ = v 1 eff ( ) + k veff ( ) = vh ( ) + vxc( ) Φ = v ( + ) µ H F

Clean semi-infinite jellium here, we first inspect a few figures of surface charge distribution and related quantities ( N.D. Lang and W. Kohn, PRB 1( 1970), 4555) the first figure shows the surface dipole due to electron "spilling out"+ the Friedel oscillations; the neutrality condition is satisfied automatically the second figure shows the total effective potential and its elst. comp. There is no image force, problem of all local approximations

Clean semi-infinite jellium The calculation permitted to find also total energies, in particular the surface energy = of the cleavage energy (a subtle consideration) 1 Jellium surface energies failed for dense systems and it was necessary to invoke the crystal pseudopotential as a perturbation Considering the primitive assumptions concerning the pseudopotential and the correlation energy, the agreement with experiment was excellent

Clean semi-infinite jellium In a second paper, the focus was on the work function The table gives details of the jellium calculation, the figure inclusion of the pseudopotential and comparison with the experiment r s k 1 F v XC k +v 1 F XC v ( + ) H v H ( ) Φ

Clean semi-infinite jellium and image potential Various attempts were undertaken to obtain the image potential. Here, we show an application of the so-called Sham-Schlüter equation, in conjunction with a GW calculation. It yields an "exact" non-local veff. The figure shows the image potential obtained at large distances. The 1/ r part of the effective potential is of the Coulombic origin, the exchange contribution decays as 1/ r. Good physics in classical limit.

Epilogue The DFT technique is still expanding, but shifting in three (well, more) directions: TDDFT in the form of NGF functionals of the third generation merger with the Green s functions

The end