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´ ısica de Materiales Universidad del Pa´ ıs Vasco UPV/EHU May 21, 2013

Localized atomic orbitals in GPAW Real-space mode, mode=’fd’ ◮ Very accurate ◮ Excellent parallelization ( k -points, domains, bands) ◮ Can be a bit expensive at times LCAO mode, mode=’lcao’ ◮ Very efficient, particularly for large systems ◮ Good parallelization ( k -points, domains, bands/orbitals) ◮ Not all that accurate Planewave mode, mode=’pw’ ◮ Very accurate, planewaves are very neat ◮ But less parallel than ’fd’ ( k -points, bands only)

Localized atomic orbitals in GPAW Localized basis sets ◮ Expand pseudowavefunctions in fixed orbitals: | ˜ � ψ n � = | Φ µ � c µn µ ◮ Now c µn are the variational variables, and | Φ µ � are fixed localized functions ◮ Derive new Kohn–Sham equations and solve: � � H µν c νn = S µν c νn ǫ n ν ν Thanks to Marco Vanin for the collaboration back in the days

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

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 ()

Localized atomic orbitals in GPAW Choosing basis functions — atomic orbitals 4s Basis functions of Fe 4s conf. 3d 3d conf. 4 0 2 6 8 10 Radius [Bohr] ◮ 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)

Localized atomic orbitals in GPAW Choosing basis functions 4s Basis functions of Fe 3d 4s-dz 3d-dz p polarization 0 2 4 6 8 10 Radius [Bohr] ◮ 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

Localized atomic orbitals in GPAW Basis set quality 11 Reasonably optimized H 2 O total energy [eV] szp Not optimized at all!! 12 dz tz qz 13 dzdp dzp tzdp qzdp tztp qztp tzp qzp 14 fd 15 Basis set ◮ 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)

Localized atomic orbitals in GPAW Calculation procedure ∇ 2 ˜ Solve v H ( r ) = − 4 π ˜ ρ ( r ) , ρ ( r ) = ˜ ˜ n ( r ) + atoms ... O ( N ) � Calculate V µν = Φ ∗ µ ( r )[˜ v H ( r ) + ˜ v xc ( r ) + ¯ v ( r )]Φ ν ( r ) d r O ( N ) � P a ∗ iµ ∆ H a ij P a Calculate H µν = T µν + V µν + O ( N ) jν aij T µν = � Φ µ | ˆ P a p a with: T | Φ ν � , iµ = � ˜ i | Φ µ � O ( N ) � � O ( N 3 ) Solve H µν c νn = S µν c νn ǫ n ν ν � O ( N 3 ) Calculate ρ νµ = c νn f n c ∗ µn n � Calculate ˜ n ( r ) = Φ ∗ µ ( r )Φ ν ( r ) ρ νµ + ˜ n core ( r ) O ( N ) µν (Repeat as necessary)

Localized atomic orbitals in GPAW Operation Parallelization Complexity Multigrid Poisson O ( N ) r Density ˜ n ( r ) r , σ O ( N ) XC ˜ v xc ( r ) r , σ O ( N ) Potential V µν ν , r , σ , k O ( N ) O ( N 3 ) Diagonalize H µν µ , ν , σ , k O ( N 3 ) Density matrix ρ µν µ , ν , σ , k 1 . 5 log10(time/scf step/seconds) 1 . 0 0 . 5 0 . 0 [1.26] V µν , ˜ n ( r ) − 0 . 5 [0.71] xc − 1 . 0 [0.92] poisson [2.80] serial diag − 1 . 5 [2.56] parallel diag − 2 . 0 [2.75] ρ µν − 2 . 5 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 log10(Number of atoms)

Localized atomic orbitals in GPAW Parallelization of O ( N 3 ) operations 0 2 4 6 0 2 4 6 1 3 5 7 1 3 5 7 0 2 4 6 0 2 4 6 0 2 4 6 1 3 5 7 1 3 5 7 0 2 4 6 0 2 4 6 1 3 5 7 1 3 5 7 ◮ Left: 2D block cyclic layout for diagonalization H µν , c µn , ρ µν ◮ Right: 1D columnated layout � Φ ∗ µ ( r )Φ ν ( r ) ρ µν → ˜ n ( r ) µν ◮ Must transfer back and forth between two layouts

Localized atomic orbitals in GPAW Parallelization modes ◮ Parallelize over k -points/spins first ( N K cores) ◮ Parallelize over domains next ( N D = N x × N y × N z ) ◮ Parallelize over bands to use less memory ( N B ) ◮ Total number of CPUs should be N K × N D × N B ◮ CPUs for ScaLAPACK are taken within groups of N D × N B 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

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 ()

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

Localized atomic orbitals in GPAW Application: Size effects in transition metal clusters s 1 d 10 s 2 d 10 s 1 d 7 s 1 d 8 s 1 d 9 2.5 Fe Co Ni Cu Zn 3.0 O adsorption energy [eV] 3.5 Ru Rh Pd Ag Cd 4.0 Os Ir Pt Au Hg 4.5 ◮ Smooth variations 5.0 can be attributed to Pd Au 5.5 Rh Ag “geometric effects” Ru 6.0 20 40 60 80 100 120 140 160 180 200 ◮ Noble metals show Number of atoms strong oscillations Figure: Adsorption energies of oxygen on from electronic shell different transition metal clusters structure

Localized atomic orbitals in GPAW Conclusions ◮ LCAO calculations are quite fast ◮ Do mind the parallelization options ◮ We should work on improving basis sets ◮ Maybe some day we should implement sparse / iterative eigensolver ◮ Thank you for listening!