12/7/2011

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Titus Beu 2011

Titus BeuUniversity “Babes-Bolyai”Department of Theoretical and Computational PhysicsCluj-Napoca, Romania

12. Non12. Non--orthogonal tightorthogonal tight--binding MDbinding MD

Titus Beu 2011

Bibliography

Basic formulation

TB parametrization

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Rapaport, D.C., The Art of Molecular Dynamics Simulation. Second Edition (Cambridge University Press, 2004).

W.A. Harrison, Solid State Theory, 1970.

W.A. Harrison, Electronic Structure and the Properties of Solids, 1980.

J.C. Slater, G.F. Koster, Phys. Rev. 94, 1498 (1954).

D.A. Papaconstantopoulos, M.J. Mehl, S.C. Erwin, and M.R. Pederson, Math. Res. Soc. Symp. Proc. 491, 221 (1998).

D.A. Papaconstantopoulos et al., Phys. Rev. B 62, 4477 (2000).

Titus Beu 2011

TB total-energy models – simplified two-center-oriented ab initio methods

Electronic structure – parameterized representation of the Kohn-Sham equation

TB parameterizations consider atoms to be permanently surrounded by the tightly bound valence electrons

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One-electron energies εk – eigenvalues of characteristic equation:

H and S - representation of nonorthogonal atom-centered orbitals

Total energy – sum of one-electron energies + repulsive contributions:

Vrep = 0 for certain parametrizations

Interatomic forces:

( ) 0k k

ε− =H S C

{ }( ) { }( ) ( )occ

tot rep

1

0

n

I k k I J K

k J K

E n Vε= <

= + −∑ ∑∑R R R R

���������

{ }( ) occtot

1

nI k

I k k k

kI k k I I

E nε+

+=

∂ ∂ ∂= − = − −

∂ ∂ ∂ ∑

R H SF C C

R C SC R R

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Details:

Multiplying on the left with Ck+ and noting that Ck

+(H − εkS) = 0:

Hence

( ) 0k k

I

ε∂

− =∂

H S CR

1kk k k

I k k I I

εε+

+

∂ ∂ ∂= − −

∂ ∂ ∂

H SC C

R C SC R R

( ) ( ) 0k k k k

I I

ε ε ∂ ∂

− + − = ∂ ∂

H S C H S CR R

0kk k k k k

I I I

εε+ + ∂∂ ∂

− − = ∂ ∂ ∂

H SC C C SC

R R R

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Bond orbitals are constructed from sp3 hybrids

Fullerenes – nominal sp2 bonding occurs on a curved surface → sp3 bonding

Natom atoms → 4Natom valence electrons → nocc = n/2

S has a similar form, with diagonal elements equal to 1.

atom atom

0 0 0

0 0 0

0 0 0

0 0 0

I I I I J J J J

x y z x y z

I IJ IJ IJ IJ I

s ss sx sy sz

I IJ IJ IJ IJ I

p sx xx xy xz x

I IJ IJ IJ IJ

p sy xy yy yz

I IJ IJ IJ IJ

p sz xz yz zz

I J

s p p p s p p p

h H H H H s

h H H H H p

h H H H H

h H H H H

−

= −

−

H

������� �����������

�

�

atom I

y

I

z

Ip

p

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Non-diagonal elements of the Hamiltonian matrix HSlater-Koster form (Phys. Rev. 94, 1498 (1954)):

Hssσ, Hspσ, Hppσ, Hppπ – two-center hopping parameters

γxIJ, γx

IJ, γxIJ – direction cosines for atoms i and j

Identical expressions for the overlap matrix S

On-site elements hl and hopping parameters must be parameterized

( )

IJss ss

IJ IJsx x sp

IJ IJ IJxy x y pp pp pp xy

H H

H H

H H H H

σ

σ

σ π σ

γ

γ γ δ

= = = − +

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Relative position vector – in terms of direction cosines:

Auxiliary derivatives:

( )IJ IJ IJ

IJ J I IJ x y zR γ γ γ= − = + +R R R i j k

( ) ( ) ( )2 2 2

IJ J I J I J IR x x y y z z= − + − + −

, ,IJ IJ IJJ I J I J Ix y z

IJ IJ IJ

x x y y z z

R R Rγ γ γ

− − −= = =

IJIJ IJx

I J

IJIJ IJy

I J

IJIJ IJz

I J

R R

x x

R R

y y

R R

z z

γ

γ

γ

∂ ∂= − = −

∂ ∂∂ ∂

= − = −∂ ∂

∂ ∂= − = −

∂ ∂

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Auxiliary derivatives: Force derivatives:

( )

( )

( )

2

2

2

1

1

1

, , ,

IJIJ IJxx x

I J IJ

IJIJ IJyy y

I J IJ

IJIJ IJzz z

I J IJ

IJ IJIJ IJ

I J IJ

x x R

y y R

z z R

x y zR

α βα α

γγ γ

γγ γ

γγ γ

γ γγ γα β

β β

−∂ ∂ = − = − ∂ ∂ −∂ ∂

= − = − ∂ ∂ −∂ ∂ = − = −

∂ ∂

∂ ∂ = − = ≠ = ∂ ∂

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

IJ IJ IJIJ

x

I J IJ

IJ IJ IJIJ

y

I J IJ

IJ IJ IJIJ

z

I J IJ

F R F R dF R

x x dR

F R F R dF R

y y dR

F R F R dF R

z z dR

γ

γ

γ

∂ ∂= − = −

∂ ∂∂ ∂

= − = −∂ ∂

∂ ∂ = − = −

∂ ∂

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Notation:

Derivatives of non-diagonal Hamiltionian elements:

( )2

IJ IJ IJIJss ss ssx

I I IJ

IJIJspIJ IJsx

x sp

I IJ

IJ

sy IJ IJ IJ

x y sp

I

IJIJ IJ IJszx z sp

I

H H dH

x x dR

HHD

x R

HD

x

HD

x

σ σ

σ

σ

σ

σ

γ

γ

γ γ

γ γ

∂ ∂= = −

∂ ∂

∂= −

∂∂

=∂

∂ =

∂

IJ IJ

sp spIJ

sp

IJ IJ

H dHD

R dR

σ σ

σ = −

Titus Beu 2011

Notations:

Derivatives of non-diagonal Hamiltionian elements:

( )1

2

IJ IJ IJ

pp pp pp

IJ

IJ IJ

pp ppIJ IJ

pp pp

IJ IJ

F H HR

dH dHG F

dR dR

σ π

σ π

= −

= − +

( )

( )

( )

2

2

2

2

IJIJppIJ IJ IJ IJxx

x x pp pp

I IJ

IJ IJ

yy ppIJ IJ IJ

x y pp

I IJ

IJIJppIJ IJ IJzz

x z pp

I IJ

dHHG F

x dR

H dHG

x dR

dHHG

x dR

π

π

π

γ γ

γ γ

γ γ

∂= − −

∂

∂= −

∂

∂ = − ∂

( )

( )

2

2

IJ

xy IJ IJ IJ IJ

y y pp pp

I

IJIJ IJ IJ IJxzz x pp pp

I

IJ

yz IJ IJ IJ IJ

x y z pp

I

HG F

x

HG F

x

HG

x

γ γ

γ γ

γ γ γ

∂ = − ∂∂ = − ∂∂ =

∂

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Papaconstantopoulos et al., Phys. Rev. B 62, 4477 (2000).

Pseudo-atomic density – environment of each atom:

Cutoff function:

On-site terms (diagonal of H) – Birch-like equation (l = s,p):

Slater-Koster hopping parameters (µ = σ,π)

2exp( ) ( )

N

i ij ij

j i

R f rρ λ≠

= −∑

( ){ }1

0 0( ) 1 exp / , 10.5 , 0.5c cf R R R R a a−

= + − ∆ = ∆ =

2/3 4/3 2i

l l l i l i l ih α β ρ γ ρ χ ρ= + + +

2 2

' ' ' ' '

2 3 2

' ' ' ' ' '

( ) ( ) exp( ) ( )

( ) ( ) exp( ) ( )

ll ll ll ll ll

ll ll ll ll ll ll

H R a b R c R d R f R

S R p R q R r R s R f R

µ µ µ µ µ

µ µ µ µ µ µδ

= + + −

= + + + −

Titus Beu 2011

Derivatives of the two-center Slater-Koster terms:

( )( ) ( )( )( )

( )

( )( ) ( )( )( )

( )

′

′

2

2

2 2

2

2 exp ( )

2 3 exp ( )

ll

ll ll ll

ll ll

ll

ll ll ll ll

ll ll

dH Rb c R d R f R

dRf R

d H Rf R

dS Rp q R r R s R f R

dRf R

s S Rf R

µ

µ µ µ

µ µ

µ

µ µ µ µ

µ µ

′′ ′ ′

′ ′

′′ ′ ′ ′

′ ′

= + −

+ −

= + + −

+ −

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C36

Ebind = 9.75 eV

C60

Ebind = 10.02 eV

C70

Ebind = 10.06 eV Ebind = 10.09 eV

C96

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C60

qtot = +10e, Eexc = 50 eV

C60

qtot = +10e, Eexc = 100 eV

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Eexc = 100 eV Eexc = 200 eV

Eexc = 300 eV Eexc = 400 eV