12. Non12. Non--orthogonal tight orthogonal tight--binding ...phys.ubbcluj.ro/~tbeu/MD/C12.pdf ·...
Transcript of 12. Non12. Non--orthogonal tight orthogonal tight--binding ...phys.ubbcluj.ro/~tbeu/MD/C12.pdf ·...
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).
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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 = −
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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
= + + −
= + + + −
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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