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