1. Numerical Aspects of Many-Body Theory Choice of basis for crystalline solids Local orbital versus Plane wave Plane waves e i(q+G).r Complete (in practice for valence space) No all electron treatment (PAW?) Large number of functions x.10 4 Slow for HF exchange Straightforward to code (abundance of Dirac delta’s) Local orbital (x - A x ) i (y - A y ) j e –  (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,..) G q q+Gq+G IBZ

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

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

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

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

6. 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 r r’r’ r r’r’

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

8. Quasiparticle energies in solid Ne and Ar h(1) One-body Hamiltonian V(1) Hartree potential  (1,2) Self energy G o Non-interacting GF G Interacting GF H(1) Non-interacting Hamiltonian  m QP Quasiparticle amplitude  m Quasiparticle energy Quasiparticle equation Dyson equation Dyson and Quasiparticle equations F125

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

10. RPA Polarisability and Dielectric Function Projection of  o onto plane wave basis

11. RPA Polarisability and Dielectric Function Projection of  o onto plane wave basis

12. Dielectric bandstructure  (  ) expanded in eigenfunctions of static inverse dielectric function Plasmon pole approximation for  -1 (q,  ) Pole strength z q and plasmon frequency  q fitted at  = 0 and several imaginary frequencies Baldereschi and Tossatti, Sol. St. Commun. (1979)

13. Ar  15v Ar  1c Energy dependence of self-energies in Ar Nicastro, Galamic-Mulaomerovic and Patterson, J. Phys. Cond. Matt. (2001) Dielectric bandstructure and self energy

14. Self-energy operator matrix elements Rohlfing, Kruger and Pollmann, Phys. Rev. B (1993) HF exchange - looks like dynamically screened HFT Self-energy calculated from dielectric bandstructure Gq 1

15. fcc Ne DFT & GW bandstructures Ne DFT PP GW PP DFT AE GW AE Expt.  15 -13.14-19.37-13.18-19.10-20.21  1c -1.350.86-1.421.031.3 WvWv 0.710.930.790.931.3 EgEg 11.7920.2311.7620.1321.51 Expt. Runne and Zimmerer, Nucl. Instrum. Methods Phys. Res. B (1995) DFT/GW Galamic-Mulaomerovic and Patterson, Phys. Rev. B (2005) Ne

16. fcc Ar DFT & GW bandstructures Ar DFT PP GW PP DFT AE GW AE Expt.  15 -9.74-13.15-10.27-13.00-13.75  1c -0.600.72-0.760.810.4 WvWv 1.351.731.321.851.7 EgEg 9.1413.879.5113.8114.15 Expt. Runne and Zimmerer, Nucl. Instrum. Methods Phys. Res. B (1995) DFT/GW Galamic-Mulaomerovic and Patterson, Phys. Rev. B (2005) Ar

17. Bethe-Salpeter Equation (F 558) i  o  G(1,2)G(2,1) i.e. dressed Green’s function product K* proper part of electron/hole scattering kernel  o is a special case of the particle-hole Green’s function 4-index function  (1,1,2,2) =  o (1,1,2,2) +  o (1,1,3,4) K*(3,4,5,6)  (5,6,2,2) Bethe-Salpeter Equation     K*  =+ 1 46 53 2

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

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

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

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

22. Bethe-Salpeter Equation: Expansion of functions of 2 or 4 variables Need all 4 arguments of  o Bethe-Salpeter Equation 1,t 1 2,t 2   (1,2) 1,t 1 2,t 2 (1,2,3,4)(1,2,3,4) 3,t 3 4,t 4

23. Bethe-Salpeter Equation: Solution as an eigenvalue problem  o and  o -1 are diagonal in the basis of single particle states Bethe-Salpeter Equation 1,t 1 2,t 2   (1,2) 1,t 1 2,t 2 (1,2,3,4)(1,2,3,4) 3,t 3 4,t 4

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

25. Bethe-Salpeter Equation: Solution as an eigenvalue problem Bethe-Salpeter Equation Ne v(q)     W(q)     v(q)

26. Bethe-Salpeter Equation: numerical calculation of matrix elements Bethe-Salpeter Equation Direct term -W(1,2,  ) 1,t 1 3,t 3 2,t 2 4,t 4 1,t 1 3,t 3 2,t 2 4,t 4 Exchange term (singlet excitons only) v(1,2)

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) nE n BSEE n EXPTE B BSEE B EXPT 117.2517.364.444.22 219.9020.251.791.33 320.5520.941.140.64 420.9521.190.740.39 521.1521.320.540.26 E g 21.69 E g 21.58  LT 0.30  LT 0.25

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) nE n BSEE n EXPTE B BSEE B EXPT 111.6012.102.092.06 213.0513.580.640.58 313.4513.900.240.26 E g 13.69 E g 14.25  LT 0.36  LT 0.15