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The Nuts and Bolts of First-Principles Simulation

The Nuts and Bolts of First-Principles Simulation. 6: Plane waves, unit cells, k-points and all that…. Durham, 6th-13th December 2001. CASTEP Developers’ Group with support from the ESF  k Network. Overview. Start here Basis set – plane waves, the discerning choice!

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The Nuts and Bolts of First-Principles Simulation

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  1. The Nuts and Bolts of First-Principles Simulation 6: Plane waves, unit cells, k-points and all that… Durham, 6th-13th December 2001 CASTEP Developers’ Groupwith support from the ESF k Network

  2. Overview • Start here • Basis set – plane waves, the discerning choice! • Grids, grids everywhere… • Unit cells – when is a crystal not a crystal…? • K-points and symmetry Lecture 6: Plane waves etc.

  3. The Starting Point • The object is to find the single particle solutions to the Kohn-Sham equation. • The single particle orbitals (bands) can be represented in any complete basis set. Lecture 6: Plane waves etc.

  4. Basis Sets – Some Choices • Linear combination of atomic orbitals (LCAO) • STO-xG • 6-31G • 6-311G* • 6-311G+* • Real space grid • Wavelets • Plane Waves Lecture 6: Plane waves etc.

  5. Plane Waves Represent the orbital in Fourier space For a periodic system (Bloch’s Theorem) Where the Gs are reciprocal lattice vectors and k is a symmetry label in the 1st Brillouin zone. Lecture 6: Plane waves etc.

  6. Reciprocal Space Lecture 6: Plane waves etc.

  7. The Cut-off Energy Limit the number of plane wave components to those such that This defines a length scale Lecture 6: Plane waves etc.

  8. How to Choose a Cut-off Energy • The minimum length scale depends on the elements in the system • Variational principle  energy monotonically decreases to ground state energy as Ecutincreases • Converge required property with respect to cut-off energy Lecture 6: Plane waves etc.

  9. Convergence with Ecut(Si8) Lecture 6: Plane waves etc.

  10. The Reciprocal Space Sphere Lecture 6: Plane waves etc.

  11. The FFT Grid grid_scale *2kcut  4kcut 2kcut Lecture 6: Plane waves etc.

  12. The Charge Density Grid 2* fine_gmax Lecture 6: Plane waves etc.

  13. Periodic Systems SiC beta 8-atom unit cell Lecture 6: Plane waves etc.

  14. SiC beta Crystal Lecture 6: Plane waves etc.

  15. Defects – H defect in SiBond Centre Site (64 Si cell) 001 direction 111 direction Lecture 6: Plane waves etc.

  16. Defects – H defect in SiTetrahedral Site (64 Si cell) 001 direction 111 direction Lecture 6: Plane waves etc.

  17. Surfaces – SiC (110) Lecture 6: Plane waves etc.

  18. SiC (110) Surface Slabs Lecture 6: Plane waves etc.

  19. Reducing interactions – H termination Lecture 6: Plane waves etc.

  20. Molecules (Aflatoxin B1) Lecture 6: Plane waves etc.

  21. Aflatoxin B1 “Crystal” Lecture 6: Plane waves etc.

  22. Convergence with Supercell Size (NH3) Lecture 6: Plane waves etc.

  23. Now, Where Were We…? For all And k within the first Brillouin zone. Lecture 6: Plane waves etc.

  24. Integrating over the 1st Brillouin Zone Observables are calculated as an integral over all k-points within the 1st Brillouin zone. For example: Lecture 6: Plane waves etc.

  25. Example Integration Lecture 6: Plane waves etc.

  26. K-points and Metals Lecture 6: Plane waves etc.

  27. Defining the k-point Grid • Standard method is the Monkhorst-Pack grid. A regular grid in k-space. • The symmetry of the cell may be used to reduce the number of k-points which are needed. • Shifting the origin of the grid may improve convergence with k-points. Monkhorst, Pack. Phys. Rev. B 13 p. 5188 (1997) Moreno, Soler. Phys. Rev. B 45 p. 13891 (1992) Probert. Phys. Rev. B (in press) Lecture 6: Plane waves etc.

  28. Monkhorst-Pack Grids for SC Cell Lecture 6: Plane waves etc.

  29. Monkhorst-Pack Grids for BCC Cell Lecture 6: Plane waves etc.

  30. Monkhorst-Pack Grids for FCC Cell Lecture 6: Plane waves etc.

  31. Why Plane Waves? • Systematic convergence with respect to single parameter Ecut • Non-local. Cover all space equally • Cheap forces (No Pulay term) • No basis-set superposition error • Numerically efficient • Use FFTs to transform between real and reciprocal space • Calculation scales as NPWlnNPW • The obvious choice for periodic and works well for aperiodic systems Lecture 6: Plane waves etc.

  32. Disadvantages of Plane Waves • Need lots of basis functions/atom • ‘Waste’ basis functions in vacuum regions • But, rapidly becomes more efficient than localised basis set due to better scaling • Need pseudopotentials for tractability • To represent core features would require huge cut-off energy • But, Core features can be reconstructed • Does not encode ‘local’ properties • But, can overcome this using projection analysis Lecture 6: Plane waves etc.

  33. Summary • Bands – single particle solutions to Kohn-Sham equation • Plane wave basis set – bands represented on reciprocal space grid within cut-off • Supercells – approximating aperiodic system with a periodic one • K-points – integration grid in 1st Brillouin zone Lecture 6: Plane waves etc.

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