1. Introduction to the SIESTA method SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Some technicalities

2. Born-Oppenheimer DFT Pseudopotentials Numerical atomic orbitals Numerical evaluation of matrix elements relaxations, MD, phonons. LDA, GGA norm conserving, factorised. finite range P. Ordejon, E. Artacho & J. M. Soler, Phys. Rev. B 53, R10441 (1996) J. M.Soler et al, J. Phys.: Condens. Matter 14, 2745 (2002) SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Linear-scaling DFT based on Numerical Atomic Orbitals (NAOs)

3. SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Expand in terms of a finite set of known wave-functions unknown unknown Def. and ~ - ~- HC n =ɛ n SC n Matrix equations:

4. Self-consistent iterations Initial guess Total energy Charge density Forces Mixing

5. Basis Set: Atomic Orbitals s p d f SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Strictly localised (zero beyond a cut-off radius)

6. Long-range potentials H = T + V ion (r) + V nl + V H (r) + V xc (r) Long range H = T + V nl + V na (r) +  V H (r) + V xc (r) Two-center integrals Grid integrals SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005  V H (r) = V H [  SCF (r)] - V H [  atoms (r)] V na (r) = V ion (r) + V H [  atoms (r)] Neutral-atom potential

7. Sparsity KB pseudopotential projector Basis orbitals Non-overlap interactions 1 2 3 4 5 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  and H  are sparse   is not strictly sparse but only a sparse subset is needed SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005

8. Two-center integrals SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Convolution theorem

9. Grid work FFT SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005

10. Egg-box effect SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Affects more to forces than to energy Grid-cell sampling E x Grid points Orbital/atom

11. Grid smoothness: energy cutoff  x  k c =  /  x  E cut =h 2 k c 2 /2m e SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005

12. Grid smoothness: convergence SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005

13.  Isolated object (atom, molecule, cluster): Open boundary conditions (defined at infinity) Open boundary conditions (defined at infinity)  3D Periodic object (crystal): Periodic Boundary Conditions Periodic Boundary Conditions  Mixed: 1D periodic (chains) 1D periodic (chains) 2D periodic (slabs) 2D periodic (slabs) Boundary conditions Unit cell SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005