Energy of Small Fullerene IsomersZacharias G. Fthenakis
Institute of Electronic Structure and Laser - FORTHand
T.E.I. of CreteHeraklion - Crete - Greece
Energy of Small Fullerene Isomers – p. 1
Fullerenes
C60 Buckminsterfullerene
H.W.Kroto, J.R.Heath,
S.C.O’Brien, R.F.Curl
and R.E.Smalley,
Nature,318, 162, (1985)
Energy of Small Fullerene Isomers – p. 2
. . . but not only C60
E.A.Rohlfing,
D.M.Cox and
A.Kaldor,
J.Chem.Phys.81,
3322, (1984)
Energy of Small Fullerene Isomers – p. 3
Fullerenes constructionDefinition: Closed cages with 5- and 6-member rings
Energy of Small Fullerene Isomers – p. 4
Fullerenes constructionDefinition: Closed cages with 5- and 6-member rings
How many pentagons and hexagons are they ?
Energy of Small Fullerene Isomers – p. 4
Fullerenes constructionDefinition: Closed cages with 5- and 6-member rings
How many pentagons and hexagons are they ?
Euler’s theorem for the geometrical solids
⇓• 12 pentagons
• N2 −10 hexagons
N = 20, 24, 26, . . . , 2n, n ∈ N
Energy of Small Fullerene Isomers – p. 4
Fullerenes constructionDefinition: Closed cages with 5- and 6-member rings
How many pentagons and hexagons are they ?
Euler’s theorem for the geometrical solids
⇓• 12 pentagons
• N2 −10 hexagons
N = 20, 24, 26, . . . , 2n, n ∈ N
Different arrangement of the pentagons and thehexagons gives different structures (isomers)
Energy of Small Fullerene Isomers – p. 4
How many isomers are they ?
N Isomers N Isomers N Isomers
20 1 42 45 62 2385
24 1 44 89 64 3465
26 1 46 116 66 4487
28 2 48 199 68 6332
30 3 50 271 70 8149
32 6 52 437 72 11190
34 6 54 580 74 14246
36 15 56 924 76 19151
38 17 58 1205 78 24109
40 40 60 1812 80 31924
P. W. Fowler and
D. E. Manolopoulos,
An atlas of fullerenes,
(Oxford: Clarendon Press),
1995
Energy of Small Fullerene Isomers – p. 5
The question:Can we find a prediction equation, which could
predict the cohesive energy of any fullerene isomer ?
The method:Tight Binding Molecular Dynamics
Energy Calculations for:• All CN isomers,20 ≤ N ≤ 42 (139 structures)
• All IPR-CN isomers,60 ≤ N ≤ 80 (16 structures)
Totaly: 155 structures
Energy of Small Fullerene Isomers – p. 6
The Energy
U = Uatr + Urep + Ubond
Uatr =occ∑
i
εi, Urep =N∑
i<j
Φ0e−4α(rij−d)
Ubond = N
[
A(nb
N
)2
+ B(nb
N
)
+ C
]
Φ0 = 4.0156eV , α = 0.7775Å−1 (adjustable parameters)
d = 1.54Å, nb/N = 1.5
Energy of Small Fullerene Isomers – p. 7
Tight Binding Hamiltonian H
HΨi = εiΨi
base:|l, k〉, l=s, px, py, pz (orbitals),k = 1, 2, . . . , N (atoms)
Hk′,kl′,l = 〈l′, k′|H|l, k〉 = εlδl′lδk′k +Hl′,l(1− δk′,k)
Hl′l =
min{µ′,µ}∑
m=0
f(l̄, m̄, n̄)Vl′lm(r)
(example:Hxx = l̄2Vppσ + (1− l̄2)Vppπ)Scaling:Vll′m(r) = Vll′m(d)e
−α(r−d),Harrison’s scheme:Vll′m(d) = ηl′lm
~2
md2 ,ηssσ = −1.40, ηspσ = 1.84, ηppσ = 3.24 andηppπ = −0.81,~2/m = 7.62eV ·Å2
Energy of Small Fullerene Isomers – p. 8
Molecular Dynamics
Fq = mqd2rqdt2
= −∇rqU, q = 1, 2, · · ·N
∇rqUatr =occ∑
i
〈Ψi|∇rqH|Ψi〉
Damping molecular dynamics
vq(t) −→ vq(t) = (1− a)vq(t)
Energy of Small Fullerene Isomers – p. 9
Optimum isomers
C20 C24 C26
Energy of Small Fullerene Isomers – p. 10
Fullerene stability (I)• The Isolated Pentagon Rule (IPR) The most
stable fullerene is the one with its pentagonsisolated
• The rule can be applied onCn, n ≥ 60
• The smallest possible IPR-fullerene: C60
T. G. Schmalz, W. A. Seitz, D. J. Klein and G. E. Hite, J. Am. Chem. Soc.
110, 1113 (1988); ibid Chem. Phys. Lett.130, 203 (1986)
Energy of Small Fullerene Isomers – p. 11
Fullerene stability (II)The Pentagon Adjacency Penalty Rule (PAPR): The energiesrise linearly with the numberNp of the pentagon adjacencies (forn ≤ 70)
For C40 E. Albertazzi, C. Domene, P. W. Fowler, T. Heine, G. Seifert,C.
Van Alsenoy and F. Zerbetto, Phys. Chem. Chem. Phys.1,2913 (1999)
Energy of Small Fullerene Isomers – p. 12
Cohesive Energy vs Np (20 ≤ N ≤ 32)
10 15 20 25 30Pentagon Adjacencies N
p
0.2
0.3
0.4
0.5
0.6
Coh
esiv
e en
ergy
diff
eren
ce (
eV)
C20
C24
C26
C28
C30
C32
(α)
Energy of Small Fullerene Isomers – p. 13
Cohesive Energy vs Np (34 ≤ N ≤ 42)
8 10 12 14 16 18 20Pentagon Adjacencies N
p
0.15
0.2
0.25
0.3
0.35
0.4
Coh
esiv
e en
ergy
diff
eren
ce (
eV)
C34
C36
C38
C40
C42
Energy of Small Fullerene Isomers – p. 14
Cohesive Energy vs Np (for all)
0 5 10 15 20 25 30Pentagon Adjacencies N
p
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Coh
esiv
e en
ergy
diff
eren
ce (
eV)
C20
C24
C26
C28
C30
C32
C34
C36
C38
C40
C42
C60
C80
(γ)
Energy of Small Fullerene Isomers – p. 15
Cohesive Energy (I)Ecoh = aNp + b
N a(eV ) b(eV ) r2 ∆Emax(eV )
28 0.00037 0.435895 1.000 0.000
30 0.0110793 0.188846 0.980 0.004
32 0.0219506 -0.014340 0.960 0.014
34 0.0225614 -0.022061 0.928 0.013
36 0.0186716 0.027488 0.947 0.022
38 0.0144732 0.076364 0.936 0.021
40 0.0147164 0.063971 0.965 0.020
42 0.0145274 0.055595 0.918 0.029
ForN = 40, a = 56.7kjoule/mole
Albertazi et al values fora: 24.4− 99.5kjoule/moleEnergy of Small Fullerene Isomers – p. 16
Cohesive Energy vs Np (20 ≤ N ≤ 32)
10 15 20 25 30Pentagon Adjacencies N
p
0.2
0.3
0.4
0.5
0.6
Coh
esiv
e en
ergy
diff
eren
ce (
eV)
C20
C24
C26
C28
C30
C32
(α)
Energy of Small Fullerene Isomers – p. 17
Cohesive Energy vs Np (20 ≤ N ≤ 32)
10 15 20 25 30Pentagon Adjacencies N
p
0.2
0.3
0.4
0.5
0.6
Coh
esiv
e en
ergy
diff
eren
ce (
eV)
C20
C24
C26
C28
C30
C32
(α)
Energy of Small Fullerene Isomers – p. 18
Cohesive Energy vs Np (34 ≤ N ≤ 42)
8 10 12 14 16 18 20Pentagon Adjacencies N
p
0.15
0.2
0.25
0.3
0.35
0.4
Coh
esiv
e en
ergy
diff
eren
ce (
eV)
C34
C36
C38
C40
C42
Energy of Small Fullerene Isomers – p. 19
Cohesive Energy vs Np (34 ≤ N ≤ 42)
8 10 12 14 16 18 20Pentagon Adjacencies N
p
0.15
0.2
0.25
0.3
0.35
0.4
Coh
esiv
e en
ergy
diff
eren
ce (
eV)
C34
C36
C38
C40
C42
(β)
Energy of Small Fullerene Isomers – p. 20
Cohesive Energy vs Np (for all)
0 5 10 15 20 25 30Pentagon Adjacencies N
p
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Coh
esiv
e en
ergy
diff
eren
ce (
eV)
C20
C24
C26
C28
C30
C32
C34
C36
C38
C40
C42
C60
C80
(γ)
Energy of Small Fullerene Isomers – p. 21
Cohesive Energy vs Np (for all)
0 5 10 15 20 25 30Pentagon Adjacencies N
p
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Coh
esiv
e en
ergy
diff
eren
ce (
eV)
C20
C24
C26
C28
C30
C32
C34
C36
C38
C40
C42
C60
C80
(γ)
Energy of Small Fullerene Isomers – p. 22
Cohesive Energy (II)Value of interceptb, if the slope is fixed to the valuea = 0.0145 for all N
N b (eV ) N b (eV ) N b (eV )
20 0.16997 34 0.10020 70 -0.0026724 0.19275 36 0.08728 72 -0.0148226 0.18743 38 0.07600 74 -0.0283728 0.16743 40 0.06714 76 -0.0283530 0.12605 42 0.05592 78 -0.0339432 0.11232 60 0.00000 80 -0.03453
Energy of Small Fullerene Isomers – p. 23
Intercept b vs N
20 30 40 50 60 70 80Number of atoms (n)
0
0.05
0.1
0.15
0.2
Coh
esiv
e E
nerg
y di
ffere
nce
(eV
) (α)
Energy of Small Fullerene Isomers – p. 24
Intercept b vs N−1
0 0.01 0.02 0.03 0.04 0.051/n
-0.05
0
0.05
0.1
0.15
0.2
0.25
Coh
esiv
e E
nerg
y di
ffere
nce
(eV
)
y = 8.1038 n-1
- 0.1364 (R=0.9972)
(β)
WithoutN = 20 case: b =8.1038
N−0.1364, (R = 0.9972)
Energy of Small Fullerene Isomers – p. 25
Intercept b vs N−1
0 0.01 0.02 0.03 0.04 0.051/n
-0.05
0
0.05
0.1
0.15
0.2
0.25
Coh
esiv
e E
nerg
y di
ffere
nce
(eV
)
y = 8.1038 n-1
- 0.1364 (R=0.9972)
(β)
WithoutN = 20 case: b =8.1038
N−0.1364, (R = 0.9972)
Conclusion:b is a function ofNEnergy of Small Fullerene Isomers – p. 25
Cohesive Energy vs NFor large IPR-fullerenes:
• G. B. Adams, O. F. Sankey, M. O’Keffe, J. B. Page and D. A. Drabold,
Science256, 1792 (1992)
Ecoh =a
N+
b
N 2
• J. Tersoff, Phys. Rev. B46, 15546 (1992)
Ecoh =a
N+ b
lnN
N• A. Maiti, C. J. Brabec and J. Bernholc, Phys. Rev. Lett.70, 3023
(1993)
ECoh = Egraphene + 4.6164N−(0.561+0.0022 lnN)
Energy of Small Fullerene Isomers – p. 26
Cohesive Energy vs N• T. A. Witten and H. Li, Europhys. Lett23, 51 (1993)
Ecoh = AN−5/6
• J. L. Morán-López, K. H. Bennemann, M. Cabrera-Trujillo and J.
Dorantes-Dávila, Solid State Commun.89, 977 (1994)
∆Ecoh =A
N+
B√N
+ C
• S. Itoh, P. Ordejón, D. A. Drabold and R. M. Martin,Phys. Rev. B53,
2132 (1996)
Ecoh = AN−B
Energy of Small Fullerene Isomers – p. 27
Testing the models (I)Error for all (155) isomers:
0 40 80 120Isomer’s number
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Coh
esiv
e E
nerg
y D
iffer
ence
s (e
V)
Adams et alTersoff
ANp+ Bn
-1+ C
Moran-Lopez et alWitten & LiItoch et al
Energy of Small Fullerene Isomers – p. 28
Testing the models (II)Error for all isomers withoutN = 20 case (154):
0 40 80 120Isomer’s Number
-0.02
-0.01
0
0.01
0.02
0.03
Coh
esiv
e E
nerg
y D
iffer
ence
s (e
V)
Adams et al.Tersoff
ANp+ BN
-1+ C
Moran-Lopez et alWitten & LiItoch et al
Energy of Small Fullerene Isomers – p. 29
Testing the models (III)Fitting for 30 ≤ N ≤ 42Error from extrapolation for IPR60 ≤ N ≤ 80
0 40 80 120Isomer’s number
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05C
ohes
ive
Ene
rgy
Diff
eren
ces
(eV
)Adams et alTersoff
ANp+ Bn
-1+ C
Moran-Lopez et alWitten & LiItoch et al
n = 60
Energy of Small Fullerene Isomers – p. 30
Testing the models (IV)Fitting for 60 ≤ N ≤ 80Error from extrapolation for30 ≤ N ≤ 42
20 30 40 50 60 70 80Number of atoms (n)
-0.05
0
0.05
0.1
0.15
0.2
Ene
rgy
(eV
)
Ecoh
- 0.01542Np
13.89973/n - 126.39465/n2- 0.21401
5.88468/n + 84.52596/n2- 0.12178
(a)
20 30 40 50 60 70 80Number of atoms (n)
0
0.05
0.1
0.15
0.2
Ene
rgy
(eV
)
Ecoh
- 0.01542Np
-11.41535/n + 7.07094ln(n)/n - 0.31219 15.93761/n - 2.34858ln(n)/n - 0.10559
(b)
20 30 40 50 60 70 80Number of atoms (n)
-0.05
0
0.05
0.1
0.15
0.2
Ene
rgy
(eV
)
Ecoh
- 0.01541Np
6.96332/n - 0.119728.31507/n - 0.13903
(c)
20 30 40 50 60 70 80Number of atoms (n)
0
0.05
0.1
0.15
0.2
Ene
rgy
(eV
)
Ecoh
- 0.01542Np
-7.3172/n + 4.71907n-1/2
- 0.50868
12.9426/n - 1.10686n-1/2
- 0.07305
(d)
20 30 40 50 60 70 80Number of atoms (n)
-0.05
0
0.05
0.1
0.15
0.2
Ene
rgy
(eV
)
Ecoh
- 0.01541Np
4.5891n-5/6
- 0.1577
4.9162n-5/6
- 0.1627
(e)
20 30 40 50 60 70 80Number of atoms (n)
-0.05
0
0.05
0.1
0.15
0.2
Ene
rgy
(eV
)
Ecoh
- 0.01541Np
1.911n-0.22609
- 0.7761
21.862n-1.2872
- 0.1126
(f)
(a) Adams et al, (b) Tersoff, (c)n−1, (d) Morán-López et al, (e) Witten and Li
and (f) Itoh et al Energy of Small Fullerene Isomers – p. 31
Results• Whitten and Lee
Ecoh = 0.01477Np+4.9528N−5/6−0.16585, R2 = 0.9927.
• N−1
Ecoh = 0.01504Np +7.5657
N− 0.1299, R2 = 0.9929
Energy of Small Fullerene Isomers – p. 32
Results• Whitten and Lee
Ecoh = 0.01477Np+4.9528N−5/6−0.16585, R2 = 0.9927.
• N−1
Ecoh = 0.01504Np +7.5657
N− 0.1299, R2 = 0.9929
• If N → ∞ then:Ecoh = ANp + BIf ∆Np = 1 then∆Ecoh ≈ 0.015eVjust for a single defect !!!
Energy of Small Fullerene Isomers – p. 32
Error for the Whitten and Lee Model
0 50 100 150Isomer’s number
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Coh
esiv
e E
nerg
y D
Iffer
ence
s (e
V)
Energy of Small Fullerene Isomers – p. 33
Ecoh vs N for the Whitten and Lee Model
20 30 40 50 60 70 80Number of atoms (n)
-0.05
0
0.05
0.1
0.15
0.2
Ene
rgy
(eV
)E
coh - 0.01477N
p
4.9528n-5/6
- 0.16585
Energy of Small Fullerene Isomers – p. 34
Pyramidalization angle
Pyramidalizationangle
θpi = θσπi − 90o
∆Hcf = a
(
1
N
N∑
i=1
θp2i
)
+ b.
∆Hcf = heat of formation per atom
D. Bakowies and W. Thiel, J. Am. Chem. Soc113, 3704 (1991)
Energy of Small Fullerene Isomers – p. 35
Alternatively...• φa = φ1 + φ2 + φ3 (φ(ideal)
a = 360o − 12onpi)
• Defect angle:φd = 360o − φa (φ(ideal)d = 12onpi)
φ1φ3φ2
x
y
z
2
1
3
4
θ2p ∝ φdi
• Descartes theorem:N∑
i=1
φdi = 4π
(T. G. Schmalz, W. A. Seitz, D. J. Klein and G. E. Hite, J. Am. Chem. Soc.
110, 1113 (1988));
P. J. FedericoDescartes on Polyhedra, Springer Verlag, Berlin (1982)Energy of Small Fullerene Isomers – p. 36
Descartes theorem "violation"
The differencen∑
i=1
φdi − 4π vsn
20 30 40 50 60 70 80Number of atoms (N)
0
50
100
150
200
Sum
of d
efec
t ang
les
- 72
0o
Energy of Small Fullerene Isomers – p. 37
Cohesive Energy vs 〈φd〉Ecoh = 0.02824〈φd〉 − 0.30783(eV )
0 10 20 30 40
average distortion angle (o)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Coh
esiv
e E
nerg
y (e
V)
Energy of Small Fullerene Isomers – p. 38
Error
0 40 80 120 160isomer number
-0.04
-0.02
0
0.02
0.04
Coh
esiv
e E
nerg
y D
iffer
ence
(eV
)
Energy of Small Fullerene Isomers – p. 39
Error
0 40 80 120 160isomer number
-0.04
-0.02
0
0.02
0.04
Coh
esiv
e E
nerg
y D
iffer
ence
(eV
)
defect angleWhitten & Lee
Energy of Small Fullerene Isomers – p. 40
Relation between 〈φd〉 and Np
•
Ucoh = ANp +B
N+ C
•Ucoh = a〈φd〉+ b
CONSEQUENTLY:
〈φd〉 = A′Np +B′
N+ C ′
Energy of Small Fullerene Isomers – p. 41
〈φd〉 vs Np
8 10 12 14 16 18 20N
p
17
19
21
23
25
Ave
rage
def
ect a
ngle
(de
g) C32
C34
C36
C38
C40
C42
Energy of Small Fullerene Isomers – p. 42
〈φd〉 prediction
〈φd〉 = 0.3541Np +439.44
N+ 4.2005
10 15 20 25 30 35Defect Angle (deg)
10
15
20
25
30
35
AN
p+ B
/N +
C
Energy of Small Fullerene Isomers – p. 43
Error of the prediction
Error= 〈φd〉 −(
0.3541Np +439.44
N+ 4.2005
)
0 40 80 120 160Isomer Number
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5E
rror
of t
he p
redi
ctio
n (d
eg)
Energy of Small Fullerene Isomers – p. 44
Relative error of the prediction
% Relative Error= 100× 〈φd〉 −(
0.3541Np +439.44
N+ 4.2005
)
〈φd〉
0 40 80 120 160Isomers Number
-2
0
2
4%
Rel
ativ
e E
rror
of p
redi
ctio
n
Energy of Small Fullerene Isomers – p. 45
Conclusion• Ucoh can be predicted vsNp andN
• 〈φd〉 can be predicted vsNp andN• Distortions on planar surfaces of pentagons and
hexagons are related withNp andN• The cohesive energy dependence onNp comes
from the surface distortions of the pentagons andhexagons
Energy of Small Fullerene Isomers – p. 46