6: Plane waves, unit cells, k- points and all that
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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 Group with support from the ESF ψ 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 Nuts and Bolts 2001 Lecture 6: Plane waves etc. 2
3 The Starting Point The object is to find the single particle solutions to the Kohn-Sham equation. ( ) 2 + V ψ = ε ψ eff i i i The single particle orbitals (bands) can be represented in any complete basis set. Nuts and Bolts 2001 Lecture 6: Plane waves etc. 3
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 Nuts and Bolts 2001 Lecture 6: Plane waves etc. 4
5 Plane Waves Represent the orbital in Fourier space ψ = ~ g.r 3 ( r) ψ ( g) e i d g For a periodic system (Bloch s Theorem) ψ ( = k r ) c e i ( ). k, G G Where the Gs are reciprocal lattice vectors and k is a symmetry label in the 1 st Brillouin zone. G + k r Nuts and Bolts 2001 Lecture 6: Plane waves etc. 5
6 Reciprocal Space Nuts and Bolts 2001 Lecture 6: Plane waves etc. 6
7 The Cut-off Energy Limit the number of plane wave components to those such that ( G + k ) 2 This defines a length scale 2 E cut λ = π E cut Nuts and Bolts 2001 Lecture 6: Plane waves etc. 7
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 E cut increases Converge required property with respect to cut-off energy Nuts and Bolts 2001 Lecture 6: Plane waves etc. 8
9 Convergence with E cut (Si8) Total Energy (ev) Cut-off Energy (ev) Nuts and Bolts 2001 Lecture 6: Plane waves etc. 9
10 The Reciprocal Space Sphere Nuts and Bolts 2001 Lecture 6: Plane waves etc. 10
11 The FFT Grid grid_scale *2k cut 2k cut 4k cut Nuts and Bolts 2001 Lecture 6: Plane waves etc. 11
12 The Charge Density Grid 2* fine_gmax Nuts and Bolts 2001 Lecture 6: Plane waves etc. 12
13 Periodic Systems SiC beta 8-atom unit cell Nuts and Bolts 2001 Lecture 6: Plane waves etc. 13
14 SiC beta Crystal Nuts and Bolts 2001 Lecture 6: Plane waves etc. 14
15 Defects H defect in Si Bond Centre Site (64 Si cell) 001 direction 111 direction Nuts and Bolts 2001 Lecture 6: Plane waves etc. 15
16 Defects H defect in Si Tetrahedral Site (64 Si cell) 001 direction 111 direction Nuts and Bolts 2001 Lecture 6: Plane waves etc. 16
17 Surfaces SiC (110) Nuts and Bolts 2001 Lecture 6: Plane waves etc. 17
18 SiC (110) Surface Slabs Nuts and Bolts 2001 Lecture 6: Plane waves etc. 18
19 Reducing interactions H termination Nuts and Bolts 2001 Lecture 6: Plane waves etc. 19
20 Molecules (Aflatoxin B1) Nuts and Bolts 2001 Lecture 6: Plane waves etc. 20
21 Aflatoxin B1 Crystal Nuts and Bolts 2001 Lecture 6: Plane waves etc. 21
22 Convergence with Supercell Size (NH 3 ) N-H Bond Length (A) Supercell Side Length (A) Nuts and Bolts 2001 Lecture 6: Plane waves etc. 22
23 Now, Where Were We? ψ ( = k r ) c e i ( ). k, G G G + k r For all ( G + k ) 2 2 E cut And k within the first Brillouin zone. Nuts and Bolts 2001 Lecture 6: Plane waves etc. 23
24 Integrating over the 1 st Brillouin Zone Observables are calculated as an integral over all k-points within the 1 st Brillouin zone. For example: E tot = V 1 BZ 1stBZ E ( k ) d 3 k n ( r ) = n ( r) d k V 1 BZ 1stBZ k 3 Nuts and Bolts 2001 Lecture 6: Plane waves etc. 24
25 Example Integration Nuts and Bolts 2001 Lecture 6: Plane waves etc. 25
26 K-points and Metals Nuts and Bolts 2001 Lecture 6: Plane waves etc. 26
27 Defining the k-point Grid Standard method is the Monkhorst-Pack grid. A regular grid in k-space. Monkhorst, Pack. Phys. Rev. B 13 p (1997) 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. Moreno, Soler. Phys. Rev. B 45 p (1992) Probert. Phys. Rev. B (in press) Nuts and Bolts 2001 Lecture 6: Plane waves etc. 27
28 Monkhorst-Pack Grids for SC Cell q Number of Points (full grid) Number of Points (Symmetrised) R 2 Offset Number of Points (Symmetrised Offset) R 2 (Offset) /4,1/4,1/ /8,1/8,1/ /4,0,1/ /16,1/16,1/ Nuts and Bolts 2001 Lecture 6: Plane waves etc. 28
29 Monkhorst-Pack Grids for BCC Cell q Number of Points (full grid) Number of Points (Symmetrised) R 2 Offset Number of Points (Symmetrised Offset) R 2 (Offset) ,1/4,1/ /4,1/4,1/ /2,1/2,1/ /8,1/8,1/ Nuts and Bolts 2001 Lecture 6: Plane waves etc. 29
30 Monkhorst-Pack Grids for FCC Cell q Number of Points (full grid) Number of Points (Symmetrised) R 2 Offset Number of Points (Symmetrised Offset) R 2 (Offset) ,1/2,1/ ,0, /2,1/2,1/ ,0, Nuts and Bolts 2001 Lecture 6: Plane waves etc. 30
31 Why Plane Waves? Systematic convergence with respect to single parameter E cut 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 N PW lnn PW The obvious choice for periodic and works well for aperiodic systems Nuts and Bolts 2001 Lecture 6: Plane waves etc. 31
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 Nuts and Bolts 2001 Lecture 6: Plane waves etc. 32
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 1 st Brillouin zone Nuts and Bolts 2001 Lecture 6: Plane waves etc. 33
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