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SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005. Introduction to the SIESTA method. Some technicalities. Born-Oppenheimer DFT Pseudopotentials Numerical atomic orbitals Numerical evaluation of matrix elements.

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introduction to the siesta method

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

Introduction to the SIESTA method

Some technicalities

linear scaling dft based on numerical atomic orbitals naos
Born-Oppenheimer

DFT

Pseudopotentials

Numerical atomic orbitals

Numerical evaluation of matrix elements

relaxations, MD, phonons.

LDA, GGA

norm conserving, factorised.

finite range

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

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

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)

matrix equations

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

unknown

Def.

and

HCn=ɛnSCn

~ - ~-

Matrix equations:

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

slide4

Initial guess

Mixing

Total energy Charge density Forces

Self-consistent iterations

basis set atomic orbitals

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

s

p

d

f

Basis Set: Atomic Orbitals

Strictly localised

(zero beyond a cut-off radius)

long range potentials

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

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

Long range

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

dVH(r) = VH[rSCF(r)] - VH[ratoms(r)]

Two-center

integrals

Grid integrals

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

Long-range potentials
sparsity

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

5

4

1

2

3

Non-overlap interactions

Basis orbitals

KB pseudopotential projector

Sparsity

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andHare sparse

is not strictly sparse but only a sparse subset is needed

two center integrals

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

Two-center integrals

Convolution theorem

grid work

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

FFT

Grid work
egg box effect

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

Grid points

Orbital/atom

E

x

Egg-box effect
  • Affects more to forces than to energy
  • Grid-cell sampling
grid smoothness energy cutoff

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

x  kc=/x  Ecut=h2kc2/2me

Grid smoothness: energy cutoff
grid smoothness convergence

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

Grid smoothness: convergence
boundary conditions

SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE

University of Illinois at Urbana-Champaign, June 13-23, 2005

Boundary conditions
  • Isolated object (atom, molecule, cluster):
    • Open boundary conditions (defined at infinity)
  • 3D Periodic object (crystal):
    • Periodic Boundary Conditions
  • Mixed:
    • 1D periodic (chains)
    • 2D periodic (slabs)

Unit cell