Introduction to the SIESTA method

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

Some technicalities

• Born-Oppenheimer

• DFT

• Pseudopotentials

• Numerical atomic orbitals

• Numerical evaluation of matrix elements

• relaxations, MD, phonons.

• LDA, GGA

• norm conserving, factorised.

• finite range

P. Ordejon, E. Artacho & J. M. Soler , Phys. Rev. B 53, R10441 (1996)

J. M.Soler et al, J. Phys.: Condens. Matter 14, 2745 (2002)

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

Linear-scaling DFT based on Linear-scaling DFT based on Numerical Atomic Orbitals (NAOs)Numerical Atomic Orbitals (NAOs)

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

)()( rcr nn

)(r

Expand in terms of a finite set of known wave-functions

unknownunknown unknownunknown

)()(ˆ rrh nnn

)()(ˆ rcrhc nnn

rdrhrh 3* )(ˆ)( rdrrS

3* )()( Def. and

nnn cSch

~ - ~-~ - ~-~ - ~-~ - ~-HCHCnn==ɛɛnnSCSCnnHCHCnn==ɛɛnnSCSCnn

Matrix equations:Matrix equations:

Self-consistent iterations

rVrV xcH

,

H

r atom

r Initial guess

out ,

Total energy Charge density

Forces inout

,,

r ,,

Mixing inout ,, ,

Basis Set: Atomic OrbitalsBasis Set: Atomic Orbitals

s

p

d

f

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

Strictly localised (zero beyond a cut-off radius)

Long-range potentialsLong-range potentialsH = T + Vion(r) + Vnl+ VH(r) +

Vxc(r) Long range

H = T + Vnl + Vna(r) + VH (r) + Vxc(r)

Two-centerintegrals Grid integrals

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

VH(r) = VH[SCF(r)] - VH[atoms(r)]

Vna(r) = Vion(r) + VH[atoms(r)] Neutral-atom potential

SparsitySparsity

KB pseudopotential projector

Basis orbitalsNon-overlap interactions

1 23

4

5

1 with 1 and 2

2 with 1,2,3, and 5

3 with 2,3,4, and 5

4 with 3,4 and 5

5 with 2,3,4, and 5

S and H are sparse

is not strictly sparse

but only a sparse subset is needed

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

Two-center integralsTwo-center integrals

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

Convolution theorem

Grid workGrid work

)(r(r)φφρ(r)ψρ(r)

ccρ

(r)φc(r)ψ

νμμν

μνi

2i

iνi

iμμν

μμ

iμi

(r)δVδρ(r)

(r)ρ(r)ρδρ(r)

(r)Vρ(r)

H

atomsSCF

xc

FFT

)(r

)(r

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

)()()()()( rrVrrVrH

Egg-box effectEgg-box effect

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

•Affects more to forces than to energy•Grid-cell sampling

Ex

Grid points

Orbital/atom

Grid smoothness: Grid smoothness: energy cutoffenergy cutoff

x kc=/x Ecut=h2kc2/2me

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

Grid smoothness: Grid smoothness: convergenceconvergence

2)/( xEcut

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005

Isolated object (atom, molecule, cluster):Isolated object (atom, molecule, cluster): Open boundary conditions (defined at infinity)Open boundary conditions (defined at infinity)

3D Periodic object (crystal):3D Periodic object (crystal): Periodic Boundary ConditionsPeriodic Boundary Conditions

Mixed: Mixed: 1D periodic (chains)1D periodic (chains) 2D periodic (slabs)2D periodic (slabs)

Boundary conditionsBoundary conditions

Unit cell

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCEUniversity of Illinois at Urbana-Champaign, June 13-23, 2005