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Localized atomic orbitals in GPAW
- r as we like to call it:
Localized atomic orbitals in GPAW
Localized atomic orbitals in GPAW or as we like to call it: LCAO - - PowerPoint PPT Presentation
Localized atomic orbitals in GPAW or as we like to call it: LCAO - - PowerPoint PPT Presentation
Localized atomic orbitals in GPAW or as we like to call it: LCAO mode Ask Hjorth Larsen Nano-bio Spectroscopy Group Departamento de F sica de Materiales Universidad del Pa s Vasco UPV/EHU May 21, 2013 Localized atomic
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Localized atomic orbitals in GPAW
Localized basis sets
◮ Expand pseudowavefunctions in fixed orbitals:
| ˜ ψn =
- µ
|Φµ cµn
◮ Now cµn are the variational variables, and |Φµ are fixed
localized functions
◮ Derive new Kohn–Sham equations and solve:
- ν
Hµνcνn =
- ν
Sµνcνnǫn Thanks to Marco Vanin for the collaboration back in the days
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Localized atomic orbitals in GPAW
Advantages
◮ More locality → better scaling of many operations ◮ Smaller dimension of Hamiltonian → we can use direct solver ◮ Direct solver → easier to converge (fewer steps) ◮ Small basis allows e.g. Green’s function based transport
calculations
Disadvantages
◮ Basis set is much less “complete” than real-space/planewaves ◮ Binding energies not so accurate ◮ (Alternatively: Spend ages choosing good basis functions) ◮ No simple way to crank up precision
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Localized atomic orbitals in GPAW
Using LCAO mode
from ase.structure import molecule from gpaw import GPAW system = molecule(’H2O’) system.center(vacuum =5.0) calc = GPAW(mode=’lcao ’, # important basis=’dzp’, # also important h=0.18 , # (the usual stuff) xc=’PBE’) system.set_calculator (calc)
- system. get_potential_energy ()
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Localized atomic orbitals in GPAW
Choosing basis functions — atomic orbitals
2 4 6 8 10 Radius [Bohr] Basis functions of Fe
4s 4s conf. 3d 3d conf.
◮ Solve spherical Kohn–Sham equations for isolated atom ◮ Use external potential to confine wavefunctions within some
cutoff
◮ Cutoff defined by requiring that each orbital increases in
energy a bit (e.g. 0.1 eV)
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Localized atomic orbitals in GPAW
Choosing basis functions
2 4 6 8 10 Radius [Bohr] Basis functions of Fe
4s 3d 4s-dz 3d-dz p polarization
◮ Obtain one atomic orbital from each valence state l, n by
solving radial atomic equation
◮ Add more functions with different radial parts ◮ Add more functions with different angular momentum
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Localized atomic orbitals in GPAW
Basis set quality
Basis set 15 14 13 12 11 H2 O total energy [eV] szp dz dzp tz qz tzp qzp dzdp tzdp qzdp tztp qztp fd
Reasonably optimized Not optimized at all!!
◮ Convergence of total energies with basis ◮ Basis sets are optimized only up to dzp! ◮ LCAO is better suited for structures ◮ (Note: Energy differences converge faster)
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Localized atomic orbitals in GPAW
Calculation procedure
Solve ∇2˜ vH(r) = −4π˜ ρ(r), ˜ ρ(r) = ˜ n(r) + atoms... O(N) Calculate Vµν =
- Φ∗
µ(r)[˜
vH(r) + ˜ vxc(r) + ¯ v(r)]Φν(r) dr O(N) Calculate Hµν = Tµν + Vµν +
- aij
P a∗
iµ ∆Ha ijP a jν
O(N) with: Tµν = Φµ| ˆ T|Φν , P a
iµ = ˜
pa
i |Φµ
O(N) Solve
- ν
Hµνcνn =
- ν
Sµνcνnǫn O(N3) Calculate ρνµ =
- n
cνnfnc∗
µn
O(N3) Calculate ˜ n(r) =
- µν
Φ∗
µ(r)Φν(r)ρνµ + ˜
ncore(r) O(N) (Repeat as necessary)
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Localized atomic orbitals in GPAW
Operation Parallelization Complexity Multigrid Poisson r O(N) Density ˜ n(r) r, σ O(N) XC ˜ vxc(r) r, σ O(N) Potential Vµν ν, r, σ, k O(N) Diagonalize Hµν µ, ν, σ, k O(N 3) Density matrix ρµν µ, ν, σ, k O(N 3)
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 log10(Number of atoms) −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 log10(time/scf step/seconds)
[1.26] Vµν, ˜ n(r) [0.71] xc [0.92] poisson [2.80] serial diag [2.56] parallel diag [2.75] ρµν
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Localized atomic orbitals in GPAW
Parallelization of O(N 3) operations
2 4 6 2 4 6 2 4 6 1 3 5 7 1 3 5 7 2 4 6 2 4 6 1 3 5 7 1 3 5 7 2 4 6 2 4 6 1 3 5 7 1 3 5 7
◮ Left: 2D block cyclic layout for diagonalization
Hµν, cµn, ρµν
◮ Right: 1D columnated layout
- µν
Φ∗
µ(r)Φν(r)ρµν
→ ˜ n(r)
◮ Must transfer back and forth between two layouts
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Localized atomic orbitals in GPAW
Parallelization modes
◮ Parallelize over k-points/spins first (NK cores) ◮ Parallelize over domains next (ND = Nx × Ny × Nz) ◮ Parallelize over bands to use less memory (NB) ◮ Total number of CPUs should be NK × ND × NB ◮ CPUs for ScaLAPACK are taken within groups of ND × NB
Rules of thumb for LCAO calculations
◮ ScaLAPACK might help beyond 30–60 atoms ◮ ScaLAPACK operations require high bandwidth (run within
same node/infiniband)
◮ Parallelize a lot more over domains than bands ◮ Band parallelization helps reduce memory usage ◮ Band parallelization is useless without ScaLAPACK
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Localized atomic orbitals in GPAW
Parallelization
from ase.io import read from gpaw import GPAW system = read(’hundredsofatoms .traj ’) # Assume we have 32 cores calc = GPAW(mode=’lcao ’, basis=’dzp’, parallel=dict(domain =(2, 2, 4), band=2, sl_default =(4, 2, 64)), nbands = len(system) * 6, xc=’PBE’) system.set_calculator (calc)
- system. get_potential_energy ()
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Localized atomic orbitals in GPAW
Example application
◮ Large numbers of similar calculations to investigate trends ◮ Study of reactivity of metal clusters of 20–200 atoms
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Localized atomic orbitals in GPAW
Application: Size effects in transition metal clusters
20 40 60 80 100 120 140 160 180 200 Number of atoms 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 O adsorption energy [eV]
Pd Rh Ru Au Ag
Figure: Adsorption energies of oxygen on different transition metal clusters
s1d7 s1d8 s1d9 s1d10 s2d10 Fe Co Ni Cu Zn Ru Rh Pd Ag Cd Os Ir Pt Au Hg
◮ Smooth variations
can be attributed to “geometric effects”
◮ Noble metals show
strong oscillations from electronic shell structure
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Localized atomic orbitals in GPAW