Electronic Structure Methods based onLocal Orbitals

• LCAO: Linear Combination of Atomic Orbitals. Matrix Elements.

• Tight-Binding Method (TB):

• basic understanding of band structures.

• Slater-Koster two-center approximation: empirical TB.

• Total Energy and Forces in TB: TB Molecular Dynamics.

• Fully self-consistent solution of the Kohn-Sham equations in

localized basis

Atomic-like (or pseudoatomic) Orbitals

DFT: Kohn-Sham Eqns & Total Energy

Linear Combination of Atomic Orbitals

Diatomic Molecule

Diatomic Homonuclear Molecule

Diatomic Heteronuclear Molecule

Evolution of eigenvalues with distance

LCAO with Traslational Symmetry (I)

LCAO with Traslational Symmetry (II)

Interatomic Matrix Elements

Two-center Interatomic

Matrix Elements

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Two-center matrix elements

Slater-Koster Two-center Approximation

Slater-Koster Approximation: d orbitals

Slater-Koster Interatomic Matrix Elements (I)

l, m, n = direction

cosines of the vector

from the left state to

the right state

l = cos θ

Es,x= l Vspσ = Vspσ cos θ

Slater-Koster Interatomic Matrix Elements (II)

Zinc-Blende (ZB) Structure: Hybrid orbitals

LCAO for ZB: Homopolar solids (Si)

LCAO for ZB: Heteropolar solids (GaAs)

LCAO Hamiltonian for ZB 8 different Bloch states build from (s,p) anion & cation orbitals

LCAO Hamiltonian for ZB (II)

Slater-Koster LCAO Hamiltonian for ZB Slater-Koster: Two-center

approximation to the

interatomic matrix elements:

(Empirical) Tight-BindingTwo-center interatomic matrix elements fitted to reproduce

states at high-symmetry points in BZ: 6 (8) parameters for

homopolar (heteropolar) semiconductors

Tight-Binding Bands for Ge

(using the parameters in previous slide)

LCAO Bands for ZB

Fully-selfconsistent solution of the Kohn-Sham equations in localized bases

KS eqns have the same form as the Tight-Binding eqns BUT:

• The matrix elements must be computed explicitly.

• The potential must be derived self-consistently.

• Selection of orbitals: minimal basis vs overcompleteness

• Analytic basis functions: Gaussians, Slater-type

• Numerical orbitals: “confined” pseudoatomic orbitals.

FIREBALL, SIESTA, OPENMX …

Numerical Orbitals: Construction“compressed” short-range orbitals generated with confining potentials

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• confined orbitals are more realistic in a solid.

• fewer matrix elements need to be included.

Total Energy & Forces in TB Models

Transferability inTB: non-orthogonalityand environment dependence

Integrals involving numerical orbitals