Numerical Aspects of Many-Body Theory

1. q G q+G IBZ Numerical Aspects of Many-Body Theory • Choice of basis for crystalline solids • Local orbital versus Plane wave • Plane waves ei(q+G).r • Complete (in practice for valence space) • No all electron treatment (PAW?) • Large number of functions x.104 • Slow for HF exchange • Straightforward to code (abundance of Dirac delta’s) • Local orbital (x - Ax)i(y - Ay)j e–a(x - A)2 • Incomplete (needs care in choice of basis) • All electron possible and relatively inexpensive • Relatively small number of functions permits large unit cells to be treated • Relatively fast for HF exchange in gapped materials • Difficult to code (lattice sum convergence, exploitation of symmetry, ..)

2. r’ r r’ r Numerical Aspects of Many-Body Theory • Coulomb Energy in real and reciprocal spaces • Coulomb interaction • Ewald form of Coulomb interaction

3. r Numerical Aspects of Many-Body Theory • Density Matrix Representation of Charge Density

4. r’ r Numerical Aspects of Many-Body Theory • Coulomb Energy with real space representation of charge density

5. r’ r Numerical Aspects of Many-Body Theory • Coulomb Energy with reciprocal space representation of interaction

6. r’ r r r’ Numerical Aspects of Many-Body Theory • Exchange Energy with real space representation of interaction • No Ewald transformation possible since h sum is split • 3 lattice sums instead of 2 • Absolute convergence neither guaranteed nor rapid

7. r r’ Numerical Aspects of Many-Body Theory • Exchange Energy with reciprocal space representation of interaction • q + G lattice sum instead of just G • Absolute convergence not guaranteed nor rapid

8. Quasiparticle energies in solid Ne and Ar • Dyson and Quasiparticle equations F125 h(1) One-body Hamiltonian V(1) Hartree potential S(1,2) Self energy Go Non-interacting GF G Interacting GF H(1) Non-interacting Hamiltonian YmQP Quasiparticle amplitude em Quasiparticle energy Dyson equation Quasiparticle equation

9. RPA Polarisability and Dielectric Function • Projection of functions onto orthogonal bases

10. RPA Polarisability and Dielectric Function • Projection of Po onto plane wave basis

11. RPA Polarisability and Dielectric Function • Projection of Po onto plane wave basis

12. Plasmon pole approximation for e-1(q,w) • Dielectric bandstructure • e(w) expanded in eigenfunctions of static inverse dielectric function Pole strength zqand plasmon frequencywqfitted atw = 0 and several imaginary frequencies Baldereschi and Tossatti, Sol. St. Commun. (1979)

13. Energy dependence of self-energies in Ar • Dielectric bandstructure and self energy ArG15v ArG1c Nicastro, Galamic-Mulaomerovic and Patterson, J. Phys. Cond. Matt. (2001)

14. S y y = ( E ) k k n n 2 p 1 4 e å - + + y y y y (q G).r (q G' ).r i i e e + + k k q k q k n n ' n ' n W + + q G' qGG' n ' ì ü é ù w z 1 - q - q i i i i * ï ï ê ú f f - + Î 1 n ' VB - qG - qG' - + w ï ï LDA ê ú 2 E E q + ë û k q n ' ï ï ï ï å í é ù ý w z 1 - q - q i i i i * ï ê ú ï f f Î i n ' CB - qG - qG' - - w LDA ï ê ú ï 2 E E q + ë û k q n ' ï ï ï ï î þ Self-energy operator matrix elements • Self-energy calculated from dielectric bandstructure 1 q G HF exchange - looks like dynamically screened HFT Rohlfing, Kruger and Pollmann, Phys. Rev. B (1993)

15. fcc Ne DFT & GW bandstructures Ne Expt. Runne and Zimmerer, Nucl. Instrum. Methods Phys. Res. B (1995) DFT/GW Galamic-Mulaomerovic and Patterson, Phys. Rev. B (2005)

16. fcc Ar DFT & GW bandstructures Ar Expt. Runne and Zimmerer, Nucl. Instrum. Methods Phys. Res. B (1995) DFT/GW Galamic-Mulaomerovic and Patterson, Phys. Rev. B (2005)

17. Po K* P Po K* P P Po 3 5 2 1 4 6 Bethe-Salpeter Equation • Bethe-Salpeter Equation (F 558) iPo=G(1,2)G(2,1) i.e. dressed Green’s function product • K* proper part of electron/hole scattering kernel • Po is a special case of the particle-hole Green’s function • 4-index function • P(1,1,2,2) = Po(1,1,2,2) + Po(1,1,3,4) K*(3,4,5,6)P(5,6,2,2) = +

18. i k i k j j ℓ ℓ k i j ℓ Bethe-Salpeter Equation • Electron-hole scattering kernel K* k i ℓ j Time flows from left to right here

19. Can’t have dangling ends Bethe-Salpeter Equation • Electron-hole scattering Lego • Electron-hole pair scattering (summed in BSE) • Electron-hole scattering (summed in screened electron-hole interaction)

20. 3 5 + + + … 4 6 Bethe-Salpeter Equation • Electron-hole scattering kernel K* • K*(3,4,5,6) = • Iteration of the Bethe-Salpeter equation leads to a series of the form • P = Po + PoK*Po + PoK*PoK*Po + PoK*PoK*PoK*Po + … • Generates sums of ring and screened ladder diagrams

21. Bethe-Salpeter Equation • Bethe-Salpeter Equation: Solution as an eigenvalue problem • P = Po + Po K* P • (1 - Po K* ) P = Po • P = (1 - Po K* ) -1 Po • P = (1 - Po K* )-1 (Po-1)-1 • P = (Po-1 - K* )-1 • P -1= Po-1 - K* Look for zeros of P -1 equivalent to poles of P P -1= Po-1 - K* = 0 an eigenvalue equation

22. 1,t1 3,t3 Po(1,2,3,4) 4,t4 2,t2 1,t1 Po(1,2) 2,t2 Bethe-Salpeter Equation • Bethe-Salpeter Equation: Expansion of functions of 2 or 4 variables • Need all 4 arguments of Po

23. 1,t1 3,t3 Po(1,2,3,4) 4,t4 2,t2 1,t1 Po(1,2) 2,t2 Bethe-Salpeter Equation • Bethe-Salpeter Equation: Solution as an eigenvalue problem • Po and Po-1 are diagonalin the basis of single particle states

24. 3,t3 1,t1 4,t4 2,t2 Bethe-Salpeter Equation • Bethe-Salpeter Equation: Solution as an eigenvalue problem • K* in the basis of single particle states 1,t1 3,t3 4,t4 2,t2 Direct term -W(1,2,e) Exchange term (singlet excitons only) v(1,2)

25. g b b g v(q) W(q) a a d d Bethe-Salpeter Equation • Bethe-Salpeter Equation: Solution as an eigenvalue problem Ne v(q)

26. 3,t3 1,t1 1,t1 3,t3 4,t4 4,t4 2,t2 2,t2 Direct term -W(1,2,e) Exchange term (singlet excitons only) v(1,2) Bethe-Salpeter Equation • Bethe-Salpeter Equation: numerical calculation of matrix elements

27. Excitons in solid Ne Expt. Runne and Zimmerer Nucl. Instrum. Methods Phys. Res. B (1995). DFT/GW Galamic-Mulaomerovic and Patterson Phys. Rev. B (2005).

28. Singlet Ne energy levels, band gaps, binding energies (eV)

29. Excitons in solid Ar Expt. Runne and Zimmerer Nucl. Instrum. Methods Phys. Res. B (1995). GW/BSE Galamic-Mulaomerovic and Patterson Phys. Rev. B (2005).

30. Singlet Ar energy levels, band gaps, binding energies (eV)