Siesta I - Psi-k

Linear-scaling DFT
and the SIESTA method
Emilio Artacho
Department of Earth Sciences
University of Cambridge

DFT: successful but heavy
• Computationally much more expensive than
empirical atomic simulations
• Several thousand atoms in massively parallel
supercomputers
!
• Computational load ~ N 3
" n
" m
= " n
*
# ( r )" m
( r )d 3 r = $ n,m
An order of magnitude increase in computer
power allows a mere doubling of system size

LINEAR SCALING
CPU
load
~ N 3
Early
90’s
~ N
~ 100
N (# atoms)
G. Galli and M. Parrinello, Phys. Rev Lett. 69, 3547 (1992)

Linear-scaling KS-DFT methods
• LCAO: - Gaussian based + QC machinery
G. Scuseria (GAUSSIAN),
M. Head-Gordon (Q-CHEM)
M. Challacombe
- Numerical atomic orbitals (NAO)
SIESTA
S. Kenny &. A Horsfield (PLATO)
- Gaussian with hybrid machinery
J.Vandevondele, J.Hutter, M.Parrinello (CP2K)
• Bessel functions in ovelapping spheres
P. Haynes & M. Payne
• B-splines in 3D grid (finite-elements)
D. Bowler, E. Hernandez & M. Gillan (CONQUEST)
• Grid-based (plane-wave like) methods
J. Bernholc (finite differences, “nearly order-N”), Fattebert
A.Mostofi, C.Skylaris, P.Haynes, M.Payne (FFT-box, ONETEP)

Orbital-free linear-scaling DFT
• Directly propose an approximate energy functional of
the density
• Difficulty: Kinetic energy functional. No ψ
KS
n for
(and local pseudopotentials)
• Linear scaling & extremely efficient
(as empirical potentials)
T = " h2
2
occ
%
n
# 2 $ n
• Simple metals
!
Paul Madden, Emily Carter & collaborators

TWO STEPS IN KS-DFT
two different problems for linear scaling
• Given a Hamiltonian, H ,
obtain Ground State properties
92-94 fever
tight binding
• Obtain H (and S) matrices
(selfconsistently)
- Long range interactions
(electrostatics)
- The rest (T,V xc , S…)
Since 95
DFT, HF, etc

HARTREE (electrostatics)
• SIESTA (and others): real-space grids
FFT, Multigrid (Numerical Recipes)
• Quantum Chemistry: Fast multipoles
Head-Gordon
Scuseria
Multigrid allows 0D, 1D, 2D & 3D bound cond
and implicit solvers, Poisson-Boltzmann etc.

KEY: LOCALITY
Large system
x 2
“Divide and Conquer”
W. Yang, Phys. Rev. Lett. 66, 1438 (1992)

SIESTA method
Linear-scaling DFT based on
NAOs (Numerical Atomic Orbitals)
P. Ordejon, E. Artacho & J. M. Soler , Phys. Rev. B 53, R10441 (1996)
•Born-Oppenheimer (relaxations, mol. dynamics)
•DFT
(LDA, GGA: BLYP, PBE, RPBE, revPBE)
•Pseudopotentials (norm conserving, factorised)
•Numerical atomic orbitals as basis (finite range)
•Numerical evaluation of matrix elements (3D grid)
Implemented in the SIESTA program (and others)
J. M. Soler, E. Artacho, J. D. Gale, A. Garcia, J. Junquera, P. Ordejon &
D. Sanchez-Portal, J. Phys.: Condens. Matter 14, 2745 (2002)
http://www.uam.es/siesta

Finite-support atomic orbitals as basis
s
p
d
f
Strictly localised
(zero beyond cut-off radius)

Hard confining potentials
A single parameter
Energy shift
E. Artacho et al. Phys. Stat. Solidi (b) 215, 809 (1999)
Fireballs
O. F. Sankey & D. J. Niklewski,
Phys. Rev. B 40, 3979 (1989)
Convergence vs Energy shift of
Bond lengths Bond energies
BUT:
A different cut-off radius for
each orbital

Soft confining potentials
1 3 5 7
r (a.u.)
1 3 5 7
r (a.u.)
• Better basis, variationally, & other results
• Removes the discontinuity in the derivative
J. Junquera, O. Paz, D. Sanchez-Portal & E. Artacho, Phys. Rev. B, 64, 235111 (2001)
E. Anglada, J. M. Soler, J. Junquera & E. Artacho, Phys. Rev. B 66, 205101 (2002)

Multiple-zeta
E. Artacho et al. , Phys. Stat. Solidi (b) 215, 809 (1999).

Polarization
E. Artacho et al. , Phys. Stat. Solidi (b) 215, 809 (1999).

Linear-scaling matrix-element
calculations
FINITE SUPPORT BASIS FUNCTIONS
S µ"
,T µ"
,V µ"
PS,nl
Two-centre integrals
1D Numerical (tabulated)
!
!
V xc µ"
(#),V Hartree NA
µ"
($#),V µ"
atoms
3D integral in finite real-space grid
!
atoms
&
n
"# $ # % # n
atomic
V NA " #{
V PS,local $V Hartree (% atomic n
)}
n

!
Grid integrations
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . .
"
. . . . . . . . . .
. . . . . µ
( r )
. . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . ". (. r r ) #
. . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
Finite-range orbitals => Lists of points
!
Sparse matrices

Eggbox
50 Ry
4.4 Bohr
Common to
many
methods!
Breaking of translational invariace with 3D grid:
- Grid-cell sampling (JM Soler et al JPCM 2002)
- Filtering (E Anglada & JM Soler, PRB 2006)

Solving H (diagonalising)
Nearsightedness principle
Implies localisation:
W. Kohn, Phys. Rev. Lett. 76, 3168 (1996)
"( r r , r # r )
"( r r # r
R )
' '' % 0
r $ r #%&
& && $ 0
r # R r
$%
Wannier-like unitary transformations of eigenvectors:
!
localised basis of occupied space
!

Linear-scaling solution
!
!
• Search directly for localised quantities:
-
E BS
"
occ
$ # n
= E %( r
n
[ r , r &)]
- by minimising E
E BS
= E
[{ }]
" ( r r ) i
• Make them strictly local (finite support)
(the approximation is here)

Functional of LWFs
Where H and S are the overlap and Hamiltonian matrices in
the basis of the (unknown) localised wave functions ϕ i .
E. B. Stechel, A. R. Williams & P. J. Feibelman, Phys. Rev. B 49, 10088 (1994)
!
E BS
J.-L. Fattebert & J. Bernholc, Phys. Rev. B 62 1713 (2000)
The inverse of the overlap matrix can been avoided:
E BS
[{" }] i
= 2 Tr{ S #1 H}
[ ] = 2 Tr 2 # S
{" } i
{ }
( )H
P. Ordejon et al. Phys. Rev. B 48, 14646 (1993)
F. Mauri & G. Galli, Phys. Rev. B 50, 4316 (1994)
!
The functionals are different but equal at the minimum

Localised solutions
R c

Problem: local minima
Excess charge
on C atoms
Slab of 16 layers of diamond
Kim, Mauri and Galli, PRB 52, 1640 (1995)

Local minima: a way to see it
Consider a square lattice with
• 4 electrons per atom
• 4 bonds surrounding each atom
• Each bond described by a LWF
• => 2 LWF per atom
• Distribute: each atom “donates” 2
bonds and “receives” 2

Local minima: a way to see it
Consider a square lattice with
• 4 electrons per atom
• 4 bonds surrounding each atom
• Each bond described by a LWF
• => 2 LWF per atom
• Distribute: each atom “donates” 2
bonds and “receives” 2

Kim, Mauri & Galli’s functional
Minimise the following varying φ i (LWF’s)
It removes problem of local minima but
requires predetermination of Fermi level
Kim, Mauri and Galli, PRB 52, 1640 (1995)

KIM’s eta & ill-conditioning
Effect of eta.
Need to predefine it
Ill-conditioning: Minimisation algorithm (CG) prop to
Condition Number: Max curvature / Min curvature
(problem common to many of the schemes)

Other linear-scaling solvers
• Divide & Conquer (W. Yang)
• Li, Nunes, Vandebilt (LNV)
(and variations)
[ ]
E = E "( r r , r r ')
BS
• Goedecker’s (two) (Projecting onto Occupied space)
!
• Energy renormalisation group (ERG, Baer et al)
METALS

Actual linear scaling
Pentium III 800 MHz
(single processor)
1 Gb of RAM
c-Si supercells, single-ζ
512.000 atoms in 256 nodes
(Julian Gale)

http://www.uam.es/siesta

No DC conduction in λ-DNA
P. J. de Pablo et al. PRL 2000
E. Artacho et al., Mol. Phys. 2003
But possible polaron conduction in dry poly dG-poly dC (A-DNA)
S. Alexandre, E. Artacho, J.M. Soler & H. Chacham, PRL 2003
HOMO band (blue)
LUMO band (red)
Hole polarons:
E binding = 0.3 eV
Very localised
Move along the
HOMO band

Inhibition of cyclin-dependent kinases
enzymes relevant to cell replication (and to cancer)
L. Heady et al., J. Medic. Chem.2006

Geobacter pili’s pilin (geo-bio-nano!)
Soil bacteria feeding from ferrous minerals by electron transfer
Ambient (fluid) conditions
nA, 0.1 G Ohm
G. Reguera et al. Nature 435, 1098 (2005)
G. Feliciano, A. J. R. da Silva, G. Reguera & EA

Candidate material for spintronics
a
20 ì m
B,M
LSMO
[100]
b
CNT
LSMO
c
LSMO
x
CNT
300 nm
Magnetoresistance in device made of LSMO
strips bridged by CNT
L. E. Hueso et al, Nature (in press; cond-mat/0511697)

Magnetoresistance measurements
50
40
5 K
25 mV
a
V (mV)
0 100 200 300 400 500
b
MC (%)
30
20
10
25 mV
5 K
0
-150 -100 -50 0 50 100 150
B (mT)
0 20 40 60 80 100 120
T (K)
Varying external B field, change relative spin orientation,
important change in resistance
L. E. Hueso et al, Nature 2007

Electronic structure at the LSMO/CNT interface
(explicit doping; same supercell plus (6,6) CNT)

Electronic structure at the LSMO/CNT interface
(explicit doping; same supercell plus (6,6) CNT)

GGA: LSMO (001) surface. x = 1/3
• Supercell: Slab 2x4 (x5.5 layers + vacuum) . 224 atoms, 48 Mn. (exposing
MnO 2
layers on both sides)
• Explicit doping: 27 La, 13 Sr (disordered, 1 realisation)

TranSiesta (J. L. Mozos, M. Brandbyge, K. Stokbro, J. Taylor & P. Ordejon)
Utility in Siesta for ballistic electron transport
Non-equilibrium Green’s functions (Keldysh)
Conduction through dithiol
Pulling Au atoms alters conductance
Transmittance vs energy
R. J. C. Batista, P. Ordejon, H. Chacham & E. Artacho, PRB RC 2006
Transport / energetics / geometry on the same footing
But also LDA+U, Pseudo-SIC, Accelerated dynamics, STM, etc

Acknowledgments
Daniel Sanchez-Portal (UPV San Sebastian)
Pablo Ordejon (ICMAB Barcelona)
Jose M. Soler (UAM Madrid)
Julian Gale
(Curtin, Australia)
Alberto Garcia (UPV Bilbao)
Richard Martin (U. Illinois, Urbana)
Javier Junquera (U. Santander)
Oscar Paz
(UAM Madrid)
Eduardo Anglada (UAM Madrid)
SIESTA
gang
Miguel Pruneda, Valeria Ferrari, Nicola Spaldin, Ben Simons,
Peter Littlewood
Luis Hueso, Neil Mathur
Ronaldo Batista, Helio Chacham
Gustavo T. Feliciano, Antonio J. R. da Silva

Summary
• Essentials of linear-scaling DFT around SIESTA
• Exploiting locality explicitly (except for Hartree)
• Still much to do (metals, generality, robustness)
• Larger sizes open up to calculations
• BUT: facing the complexity barrier!