Calculation of matrix elements

1. Calculation of matrix elements José M. Soler Universidad Autónoma de Madrid

2. Schrödinger equation Hi(r) = Eii(r) i(r) =  ci (r) (H-EiS)ci = 0 H = <  | H |  > S = <  |  >

3. Kohn-Sham hamiltonian H = T + VPS+ VH(r) + Vxc(r) T = -(1/2)2 VPS = Vion(r) + Vnl Vion(r)  - Zval / r Local pseudopotentiual Vnl =  |>  <| Kleinman-Bylander VH(r) =  dr’ (r’) / |r-r’| Hartree potential Vxc(r) = vxc((r)) Exchange & correlation

4. H = T + Vion(r) + Vnl+ VH(r) + Vxc(r) Long range Two-center integrals Grid integrals H = T + Vnl + Vna(r) + dVH (r)+ Vxc(r) Long-range potentials Vna(r) = Vion(r) + VH[ratoms(r)] Neutral-atom potential dVH(r) = VH[rSCF(r)] - VH[ratoms(r)]

5. 5 4 1 2 3 Non-overlap interactions Basis orbitals KB pseudopotential projector Sparsity 1 with 1 and 2 2 with 1,2,3, and 5 3 with 2,3,4, and 5 4 with 3,4 and 5 5 with 2,3,4, and 5 SandHare sparse is not strictly sparse but only a sparse subset is needed

6. Two-center integrals Convolution theorem

7. Atomic orbitals

8. Overlap and kinetic matrix elements • K2max= 2500 Ry • Integrals by special radial-FFT • Slm(R) and Tlm (R)calculated and tabulated once and for all

9. Real spherical harmonics

10. FFT Grid work

11. 2VH(r) = - 4 (r) (r) = G G eiGr  VH(r) = G VG eiGr VG = - 4 G / G2 (r)  G  VG  VH(r) FFT FFT Poisson equation

12. GGA

13. Grid points Orbital/atom E x Egg-box effect • Affects more to forces than to energy • Grid-cell sampling

14. x  kc=/x  Ecut=h2kc2/2me Grid fineness: energy cutoff

15. Grid fineness convergence

16. Points (index and value stored) Subpoints (value only) x  Ecut Points and subpoints Sparse storage: (i,i )

17. 3 1 2 3 1 16 17 18 19 20 6 4 5 6 4 11 12 13 14 15 3 1 2 3 1 6 7 8 9 10 6 4 5 6 4 1 2 3 4 5 +1,+5,+6 1  7  8,12,13  2,4,5 6  14  15,19,20  4,3,1 +1,+5,+6 Extended mesh

18. Forces and stress tensor Analytical: e.g. Calculated only in the last SCF iteration