Physics 250-06 “Advanced Electronic Structure” Lecture 3. Improvements of DFT Contents: 1. LDA+U.

1. Physics 250-06 “Advanced Electronic Structure” Lecture 3. Improvements of DFT Contents: 1. LDA+U. 2. LDA+DMFT. 3. Supplements: Self-interaction corrections, GW

2. Concept of delocalized and localized states Systems with d and f electrons show localized (atomic like) behavior. Examples: cuprates, manganites, lanthanides, actinides, transition metal oxides, etc. LDA is a static mean field theory and cannot describe many-body features in the spectrum: example: atomic limit is with multiplets is missing in LDA. When magnetic order exists, LSDA frequently helps!

3. (Anti)ferromagnets Splitting Vup-Vdn between up and down bands can be calculated in LSDA. It always comes out small (~1 eV). In many systems, it is of the order of 5-10 eV.

4. LDA+U In LSDA splitting Vup-Vdn is controlled by Stoner parameter I while on-site Coulomb interaction U can be much larger than that: In simplest Hartree-Fock approximation: LDA+U functional with built-in Hubbard parameter U:

5. Paramagnetic Mott Insulators LDA/LDA+U, other static mean field theories, cannot access paramagnetic insulating state because spin up and spin down solutions become degenerate How to recover the gap in the spectrum? Frequency dependence in self-energy is required:

6. Concept of Spectral Functions Effective (DFT-like) single particle spectrum always consists of delta like peaks Real excitational spectrum can be quite different

7. Localized electrons: LDA+DMFT Electronic structure is composed from LDA Hamiltonian for sp(d) electrons and dynamical self-energy for (d)f-electrons extracted from solving impurity problem Poles of the Green function have information about atomic multiplets and other many body effects. N(w) dn->dn+1 dn->dn-1 w 0 Better description compared to LDA is obtained

8. Spectral Density Functional Theory A functional method where electronic spectral function is a variable would predict both energetics and spectra. A DMFT based electronic structure method - an approach where local spectral function (density of states) is at the center of interest. Can be entitled as Spectral density functional theory (Kotliar et.al, RMP 2006) • Total Energy and local excitational spectrum are accessed • Good approximation to exchange-correlation functional • is provided bylocal dynamical mean field theory. • Role of Kohn-Sham potential is played by amanifestly local self-energy operator M(r,r’,w). • Generalized Kohn Sham equations for continuous distribution of spectral weight to be solved self-consistently.

9. Local Green function Functionals Family of Functionals

10. Generalization of Kohn Sham Idea To obtain kinetic functional: introduce fictious particles which describe local Green function: Exactly as in DFT:

11. Local Self-Energy of Spectral Density Functional • Spectral Density Functional looks similar to DFT • Effective mass operator is local by construction and plays • auxiliary role exactly like Kohn-Sham potential in DFT • Energy dependent Kohn-Sham (Dyson) equations give • rise to energy-dependent band structure • have physical meaning in contrast to Kohn-Sham spectra. are designed to reproduce local spectral density

12. Self-Interactions LDA is not self-interaction free theory. Simplest example: electron in hydrogen atom produces charge density cloud And would have excnage correlation potential according to LDA. Perdew and Zunger (1984) proposed to subtract spirituous self-interaction energy for each orbital from LDA total energy by introducing Self-Interaction Corrections (SIC) LDA-SIC theory produces orbital-dependent potential since one needs to define orbitals which self-interact. SIC theory produces better total energies but wrong spectra in many cases.

13. GW Theory of Hedin In GW (Hedin, 1965) spectrum is deduced from Dyson equation with approximate self-energy: GW theory can be viewed as perturbation theory with respect to Coulomb interaction. It produces correct energy gap in semiconductors which is an improvement on top of LDA Being a weakly coupled pertrurbation theory it also has wrong atomic limit and does not produce atomic multiplets

14. Computing GW Self-Energy Solve LDA equations and construct LDA Green functions and GW self-energy Here, dynamically screened Coulomb interaction is calculated from the Knowledge of the dielectric function of the material: