Real-space multigrid methods for DFT and TDDFT:

Tuomas Torsti

CSC – The finnish IT center for Science

Laboratory of Physics, Helsinki University of Technology

http://www.csc.fi/physics/mika

Acknowledgements

• For Funding– CSC – The finnish IT center for Science

– COMP, Helsinki University of Technology

• For advice– Martti Puska (COMP)

– Risto Nieminen (COMP)

– Janne Ignatius (CSC)

• For collaboration using MIKA/cyl2– Bo Hellsing (Chalmers)

– Vanja Lindberg (Växjö, Chalmers)

– Nerea Zabala (San Sebastian)

– Eduardo Ogando (Bilbao)

– Paula Havu (COMP)

– Tero Hakala (COMP)

• For development of RQMG– Mika Heiskanen (then COMP)

• For collaboration in development of MIKA/rspace

– Sampsa Riikonen (now San Sebastian)

– Ville Lehtola (COMP)

– Kaarle Ritvanen (COMP)• For work done with MIKA/RS2Dot

– Henri Saarikoski

– Esa Räsänen

• For response iterations– Eckhardt Krotscheck (Linz)

– Michael Aichinger (Linz)

• For work done with MIKA/doppler– Ilja Makkonen

Motivation for using real-space grids

• With uniform grids the control of the ”basis set” is simple : Only one parameter (the grid spacing h)

• Flexible choice of boundary conditions : cluster, wire, surface, bulk.• cluster• wire • surface• bulk• ...

• Parallelization using domain decomposition

• It is possible to use nonuniform grids to refine the mesh close to atomic nuclei or ”hard” pseudopotential, and/or to push the vacuum boundary far away in cluster calculations : • adaptive grids• composite grids • finite elements

• Multigrid techniques can be used to obtain optimal scaling for PDE’s

• Natural framework for Order-N (localized orbitals required)

Multigrid methods

fV 2

VfR 2

fVV coarsefine )(2

A. Brandt. Math. Comput. 31, 333 (1977)., T. L. Beck. Rev. Mod. Phys. 72, 1041 (2000).

W. L. Briggs et al., A Multigrid Tutorial, Second Edition. (SIAM 2000).

As a simple example, take the Poisson equation

Simple relaxation schemes (e.g. the Gauss-Seidel method) efficiently remove the short wavelength components of the residual

(they are good smoothers), while critical slowing down occurs for the long wavelength components. Solution: treat long wavelength components of V on a coarse grid

finecoarse RV 2

The idea can be applied recursively (V-cycle). Linear scaling with problem size can be acchieved with the full-multigrid method.

Classification of MG-methods for the eigenproblem

• Steepest descent (or CG or RMM-DIIS) with MG-preconditioning e.g. Bernholc et al., Phys. Rev. B 54 14362 (1996)

• Full approximation storage A. Brandt et al. SIAM J. Sci. Comput. 4, 244 (1983) J. Wang and T. L. Beck , J. Chem. Phys. 112, 9223 (2000)

• Rayleigh Quotient Multigrid method (RQMG) J. Mandel and S. F. Cormick, J. Comput. Phys. 80, 442 (1989). M. Heiskanen et al., Phys. Rev. B 63, 245106, (2001).

Rayleigh quotient multigrid method

• With search vector d vary α to minimize the Rayleigh quotient

duBdu

duHduR

)(

J. Mandel and S. F. Cormick, J. Comput. Phys. 80, 442 (1989).

M. Heiskanen et al., Phys. Rev. B 63, 245106, (2001).

• Coordinate relaxation: choose a coordinate vector d=e.

• RQMG – method : on coarse grids minimize the fine grid RQ with: eId fc

• The fine grid Rayleigh quotient can be evaluated entirely on the coarse grid :

ccccffcffff

ccccffcffff

eBeeuBIuBu

eHeeuHIuHuR

2

2

2

2)(

• If eigenpairs other than the lowest one are required, add a penalty functional to take care of the orthogonality requirement:

k

i kkii

ki

ikk

kkk

uuBuu

uuq

Buu

HuuuR

1 11

2

1

11

111][

• Discretized Schrödinger equation BuHu

Rayleigh quotient multigrid method (continued)

fcf

cfc

fcf

cfc

Tcf

fc IBIBIHIHII ,,)(

• Can we get rid of the penalty functional by minimizing the residual norm instead of the Rayleigh Quotient (In analogy with the familiar RMM-DIIS method) ?

• Galerkin conditions should hold :

• In the original implementation, approximated by discretization coarse grid approximation (DCA). In MIKA/rspace 1.0 also the Galerkin case implemented

uu

BuHuBuHuR

)(

Response iteration method: full response

h

h rrrF 0)()]([)]([2

)''()'','()'()'(''')()(

2

)()()(

)(

,

)()()(

rrrVrrdrdrrr

rrru

khphp

hp hp

hp

kkk

)'(

)(

'

1)',(

r

rV

rrrrV xc

hp

J. Auer and E. Krotscheck, Comp. Phys. Comm. 151 (2003), 265-271

• Newton-Raphson method for the equation

• Full response equation (needs unoccupied states) (solve with CG or GMRES)

where

))(()'()'(

)]([')]([)]([)]([0 2)()()()()()1( kkkkkk Or

r

rFdrrFrFrF

Response iteration method : collective approximation

J. Auer and E. Krotscheck, Comp. Phys. Comm. 151 (2003), 265-271

2

0

,

,

,,

)'()()'()(

11)'(

where

~2~]

~2[

at arrives oneon manipulati someAfter

)()()(

:functiona of elementsmatrix are that assumption with the

)()()(

write

hhhF

hpFFhpF

hphp

hp

hphphp

rrrr

rrS

VSSVSH

rrrdru

local u

urrru

• requires only occupied states

• implemented in MIKA/cyl2 and MIKA/RS2Dot

MIKA/rspace 1.0

• Parallelized over k-points and real-space domains• Periodic and cluster boundary-conditions implemented• Norm-concerving nonlocal pseudopotentials of the Kleynman-Bylander form (usually

Troullier-Martins pseudopotentials are used), double-grid method• Hellman-Feynman Forces• Structural optimization with the BFGS-method (two implementations)• Mixing schemes:

– Pulay – Broyden – GR-Pulay (D. R Bowler and M. J. Gillan. Chem. Phys. Lett. 325, 473 (2000) ),

– ”screened Coulomb interaction” (M. Manninen et al., Phys. Rev. B 12, 4012 (1975). )

– Pulay-Kerker (Note: rough Fourier components obtained using a MG-technique)

– Pulay-Kerker with metric (motivated by Kresse and Furthmuller, PRB 54, 11169).

MIKA/rspace (future)

• Mixed boundary conditions for surface computations

• Spin-dependent version of the code

• Alternative MG-solver (e.g. RMM-DIIS with MG-preconditioning)

• PBE (Perdew, Burke, Ernzerhof) GGA correction – already implemented, and will be included in the next release

• Response iterations (already implemented in other MIKA-codes, 3D subroutines from prof. Krotscheck available)

• Build an interface to Octopus for time-dependent calculations

Double grid method for nonlocal pseudopotentials

T. Ono and K. Hirose, PRL 82, 5016 (1999)

• Replaces the fourier filtering of pseudopotentials of Briggs et al. • The idea should be understood as a general scheme to transfer a function from a fine

grid to a coarse grid, and is in fact equivalent to the MG restriction operation.• Should be applied also to the local part, and compensating gaussian charges (all

functions that are transferred from a radial grid to the computational grid) • Thanks to J. J. Mortensen (CAMP, DTU) who implemented this in grid-based PAW.

All-electron finite-element calculations with ELMER

• These are outside the scope of the MIKA-project, but demonstrated the

capabilities of CSC’s ELMER package.

Vortex clusters in quantum dots

• Saarikoski et al. Phys. Rev. Lett (2004) (cond-mat/0402514)• Exact diagonalization and DFT (both CSDFT and SDFT) give corresponding results –

limitations and differences of the methods discussed.• Finding the vortex solution in DFT requires high numerical accuracy. Our real-space

implementation is superior to existing plane-wave schemes in describing the vanishing density at the vortex core

Left: SDFT density of 24-electron QD at 5T showing 14 vortice

Right: CSDFT density and currents at the edge of the QD.

Conductance oscillations in metallic nanocontacts

P. Havu et al., Phys. Rev. B, 66, 075401 (2002).

• We model a chain of N Na atoms between two conical stabilized jellium leads

• Since only one channel contributes to the conductance, and because of the mirror symmetry, the Friedel sum rule can be applied for the conductance

oe NN

h

eG

2sin

2 22

• We observe the even-odd behaviour of the conductance as the function of N

• In addition, the important role of the leads is manifested as an additional oscillation as a function of the cone opening angle

Ultimate jellium model for breaking nanowiresE. Ogando et al., Phys. Rev. B 67, 075417 (2003).

• Ultimate jellium is a locally neutral model, the compensating background charge density equals the electron density at every point.

• The shape of the system in the central part is free to vary to minimize the total energy.

• The shape of the leads is frozen to the uniform wire solution.

• In the beginning of the elongation, classical catenoid shape is observed

• Quantum mechanical shell structure in cylindrical symmetry -> cylinders with magic radii.

• Quantum mechanical shell structure in sperical symmetry -> Cluster derived structures (CDS)

• Oscillation of elongation force

Model study of adsorbed metallic quantum dots: Na on Cu(111)

• Roughly hexagonal islands are observed to form on the second monolayer of Na grown on Cu(111)

• Bandgap at Fermi level in Cu for electrons moving in the (111) direction –> quantum well states

• We developed a two-jellium model to fit the bottoms of two surface state bands

• The infinite monolayer is described with as a hexagonal lattice of circles, the k-space is sampled with two points.

• In the largest system studied, 2400 states are solved – the code is parallelized over the 65*2 different values of (m,k). This is also a demanding test for the charge density (or potential) mixing.

• The local density of states is calculated at a realistic STM-tip distance (15 a.u.) above the surface and compared with measured differential conductance

T. Torsti et al., Phys. Rev. B 66, 235420 (2002)

Quantum corrals (Tero Hakala, M.Sc. project)

• We use a pseudopotential (E. Ogando et al. submitted to PRB, cond-mat/0310533) for the Cu(111) surface

• A ring of 45 Pb atoms on both sides of a Cu(111) slab with 5 atomic layers and radius 60 bohr : a localized surface state observed within the corral

• The total system size was 3272 electrons and required about 2000 SCF-iterations to converge (about 1 day with 8 processor in the IBM server cluster of CSC).

Quantum corrals (continued)

• Charge transfer in a corral with 8 Pb-atoms on both sides of a Cu(111)-slab with15 atomic layers. This transfer is due to the equilibration of chemical potentials between Pb and Cu.

• It has been observed also in 1D-calculations of Pb on top of Cu(111) by Ogando et al.

Partial list of publications related to MIKA

Numerical methodsM. Heiskanen, T. Torsti, M.J. Puska, and R.M. Nieminen, Multigrid method for electronic structure calculations, Phys. Rev. B 63, 245106 (2001).T. Torsti, M. Heiskanen, M.J. Puska, and R.M. Nieminen, MIKA: a multigrid-based program package for electronic structure calculations, Int. J. Quantum Chem. 91, 171-176 (2003). T. Torsti, Real-Space Electronic Structure Calculations for Nanoscale Systems, CSC Research Reports R01/03 (Ph. D. -thesis).

Applications to axially symmetric model systems P. Havu, T. Torsti, M.J. Puska, and R.M. Nieminen, Conductance oscillations in metallic nanocontacts, Phys. Rev. B 66, 075401 (2002). T. Torsti, V. Lindberg, M. J. Puska, and B. Hellsing Model study of adsorbed metallic quantum dots: Na on Cu(111) Physical Review B 66, 235420 .E. Ogando, T. Torsti, N. Zabala, and M. J. Puska, ”Electronic resonance states in metallic nanowires ... simulated with the ultimate jellium model”, Phys. Rev. B. 67, 075417 T. Torsti, Real-Space Electronic Structure Calculations for Nanoscale Systems, CSC Research Reports R01/03 (Ph. D. -thesis)

Applications to quantum dots in 2DEGSaarikoski, H. , Harju, A. , Puska, M. J., Nieminen, R. M., Vortex Clusters in Quantum Dots, Submitted to Physical Review Letters on 19.2.2004 Harju, A., Räsänen, E., Saarikoski, H., Puska, M.J., Nieminen, R.M., and Niemelä, K., Broken symmetry in density-functional theory: Analysis and cure, Submitted to Physical Review B on 3.2.2004 Räsänen, E., Harju, A., Puska, M. J., and Nieminen, R. M., Rectangular quantum dots in high magnetic fields, Submitted to Physical Review B on 27.11.2003.Räsänen, E., Puska, M.J., and Nieminen, R.M., Maximum-density-droplet formation in hard-wall quantum dots, Submitted to Physica E on 9.6.2003. Räsänen, E., Saarikoski, H., Stavrou, V. N., Harju, A., Puska, M.J., and Nieminen, R.M., Electronic structure of rectangular quantum dots, Physical Review B 67, 235307 (2003) .Saarikoski, H., Räsänen, E.,Siljamäki, S., Harju, A., Puska, M.J., Nieminen, R.M., Testing of two-dimensional local approximations in the current-spin and spin-density-functional theories, Physical Review B 67, 205327 (2003) . Räsänen, E., Saarikoski, H., Puska, M. J., and Nieminen, R. M., Wigner molecules in polygonal quantum dots: A density-functional study, Physical Review B 67 , 035326 (2003) . Saarikoski, H., Räsänen, E., Siljamäki S., Harju A., Puska, M.J., and Nieminen, R.M., Electronic properties of model quantum-dot structures in zero and finite magnetic fields, European Physical Journal B 26 , 241-252 (2002) .

Applications of the RQMG method to one-dimensional problems Engström, K., Kinaret, J., Puska, M.J., and Saarikoski, H., Influence of Electron-Electron Interactions on Supercurrent in SNS structures, Low Temperature Physics 29, 546 (2003). Ogando,E. Zabala,N., Chulkov,E.V., Puska,M.J., Quantum size effects in Pb islands on Cu(111): Electronic-structure calculations, Submitted to Phys. Rev. B on 22.10.2003

Summary

• MIKA (Multigrid Instead of the K-spAce) is a collection of programs that solve the Kohn-Sham equations of DFT in one, two and three dimensional cartesian coordinate systems or in axial symmetry

• The core numerical method is the Rayleigh quotient multigrid method for the eigenproblem

• No TDDFT yet, but this has a high priority as a future development.

• MIKA / rspace 1.0 was released on 2.9.2004. Along with the other codes, it is licensed with the GPL, and available from http://www.csc.fi/physics/mika