Real-space multigrid methods for DFT and TDDFT: - TDDFT.org

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)

R
∇ 2 V =
=
f
− ∇
f
2
V
∇ 2 ( V
fine
+ Vcoarse)
= f
⇒ ∇ 2 V =
Multigrid methods
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
coarse
R fine
The idea can be applied recursively (Vcycle).
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 MGpreconditioning
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).

• Discretized Schrödinger equation
• With search vector d vary α to minimize
the Rayleigh quotient
Rayleigh quotient multigrid method
J. Mandel and S. F. Cormick, J. Comput. Phys. 80, 442 (1989).
M. Heiskanen et al., Phys. Rev. B 63, 245106, (2001).
R(
α)
=
u + αd
H u + αd
u + αd
B u + αd
• Coordinate relaxation: choose a coordinate vector d=e.
f
• RQMG – method : on coarse grids minimize the fine grid RQ with: d = Ic
e
• The fine grid Rayleigh quotient can be evaluated entirely on the coarse grid :
R(
α)
=
u
f
u
f
H
B
f
f
u
u
f
f
+ 2α
I
+ 2α
I
c
f
c
f
H
B
f
f
u
u
f
f
e
e
c
c
2
+ α
2
+ α
• If eigenpairs other than the lowest one are required, add a penalty functional to take
care of the orthogonality requirement:
R[
u
u
Hu = εBu
Hu
k + 1 k + 1
k + 1] = +
uk
+ 1
Buk
+ 1
∑
e
e
c
c
H
c
c
B e
e
2
k ui
uk
+ 1
qi
i= 1 ui
Bui
uk
+ 1
uk
+ 1
c
c

Rayleigh quotient multigrid method (continued)
• Galerkin conditions should hold :
f c T
c f
I
c
= ( I
f
) , H
c
= I
f
H
f
Ic
, Bc
=
I
c
f
B
f
I
f
c
• In the original implementation, approximated by discretization coarse grid
approximation (DCA). In MIKA/rspace 1.0 also the Galerkin case implemented
• 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)
R(
α)
=
Hu
−εBu Hu
u u
−εBu

u
( k )
Response iteration method: full response
J. Auer and E. Krotscheck, Comp. Phys. Comm. 151 (2003), 265-271
• Newton-Raphson method for the equation
k
0 = F[
ρ
(
( r)
= ∆ρ
2
∑
p,
h
k )
(
( r)
−δρ
φ ( r)
φ ( r)
p
p
h
ε −ε
h
k )
∫
( r)
=
2
F[
ρ](
r)
= ∑ φh[
ρ](
r)
− ρ(
r)
= 0
• Full response equation (needs unoccupied states) (solve with CG or GMRES)
where
(
]( r)
= F[
ρ
(
+ δρ
(
]( r)
= F[
ρ
dr'
dr''
φ ( r')
φ ( r')
V
p
h
δF[
ρ](
r)
( k )
(
]( r)
+ ∫ dr'
δρ ( r')
+ O((
δρ
δρ(
r')
( + 1)
k ) k )
k )
k )
h
p−h
(
( r',
r'')
δρ
k )
)
( r'')
2
)
V
p−h
( r,
r')
=
r
1
− r'
δVxc(
r)
+
δρ(
r')

Response iteration method : collective approximation
J. Auer and E. Krotscheck, Comp. Phys. Comm. 151 (2003), 265-271
• requires only occupied states
• implemented in MIKA/cyl2 and MIKA/RS2Dot
write
u(
r)
=
[ H
p,
h
0
where
+ 2S
p,
h
φ ( r)
φ ( r)
u
~
∗V
p,
h
with the assumption that u
=
∫
∑
p,
h
drφ
( r)
φ ( r)
ω(
r)
After some manipulation one arrives at
S
u
F
F
p
p
p−h
∗ S
] ∗
~ ω = 2S
1 1
= δ ( r − r')
−
υ ρ(
r)
ρ(
r')
h
h
F
are matrix elements of
F
~
∗V
∑
h
p−h
∆ρ
∗
ρ
φ ( r)
φ ( r')
h
h
2
a local function :

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
Left: SDFT density of 24-electron QD at 5T showing 14 vortice
Right: CSDFT density and currents at the edge of the QD.
• 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

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
2
2e
2 ⎡π
⎤
G = sin ⎢
( N e
− No
)
h ⎣ 2 ⎥⎦
• 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 nanowires
E. 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)
T. Torsti et al., Phys. Rev. B 66, 235420 (2002)
• 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

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 methods
M. 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 2DEG
Saarikoski, 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 currentspin
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