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5 Shape Analysis of Carbon Nanotubes, Nanotori and Nanotube Junctions

5 Shape Analysis of Carbon Nanotubes, Nanotori and Nanotube Junctions

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106



I. La´szlo´ et al.



Fig. 6.5 Drawing of nanotube using three bi-lobal eigenvectors of the adjacency matrix (a) and of

the Laplacian (b)



Fig. 6.6 Top and side view of nanotube junctions obtained by three bi-lobal eigenvectors of the

adjacency matrix (a) and of the Laplacian (b)



graphs describing nanotubes nanotori and nanotube junctions. In each cases we

can note some kind of flattening. The nanotube ends are spiked and they are

turning back in the case of adjacency matrices. This phenomenon shows that for

non-regular graphs the eigenvectors of the Laplacian matrix give better drawings

than those of the adjacency matrix.

Let us suppose that we have calculated the Descartes coordinates ðxi ; yi ; zi Þ of the

atoms in a non-spherical structure of n atoms. Thus X, Y and Z are n-dimensional

vectors containing the x, y and z coordinates of the atoms in order. Let us suppose

further that the centre of mass of the molecule is in the origin of the coordinate

system and the eigenvectors of its tensor of inertia are showing to the direction of

the x, y and z axis.



6 Graph Drawing with Eigenvectors



107



With the help of the following scalar products

aXk ¼ Xck ; aYk ¼ Yck and aZk ¼ Zck



(6.34)



the atomic coordinates can be written as





n

X



aXk ck ; Y ¼



k¼1



n

X



aYk ck and Z ¼



k¼1



n

X



aZk ck :



(6.35)



k¼1



Here ck is the eigenvector of the Laplacian L and the corresponding eigenvalues lk

are ordered in increasing order. We say that the weights of the eigenvector ck in X,

Y and Z are in order jaXk j2 , jaYk j2 and jaZk j2 .

The measure of the convergence using only m n terms in the summation can

be described with the following notations:

mị



Xi







m

X

kẳ1



mị



aXk cki ; Yi







m

X



mị



aYk cki and Zi







kẳ1



m

X



aZk cki



(6.36)



kẳ1



and





R ẳ X; Y; Zị; Rmị ẳ Xmị ; Ymị ; Zmị :



(6.37)



The convergence of the structure is quantified as follows:

n 

2

X





ðmÞ

X À XðmÞ  ẳ 1

X i Xi

n iẳ1



!12



n 

2

X





mị

Y Ymị  ẳ 1

Yi Yi

n iẳ1

n 

2

X





mị

Z Zmị  ¼ 1

Zi À Zi

n i¼1



(6.38)

!12

(6.39)



!12



n 

2 

2 

2 12

X





ðmÞ

ðmÞ

ðmÞ

R À Rmị  ẳ 1

Xi X i

ỵ Y Yi

ỵ Z À Zi

n i¼1



(6.40)



(6.41)



In Graovac et al. (2008b) we obtained that only few of the coefficients jaXk j2 , jaYk j2

and jaZk j2 are significantly greater than the others. Thus we made such kind of

summations similar to the Eq. 6.36 which contained only the significant jaXk j2 ,

jaYk j2 and jaZk j2 terms. In these summations we have found that three bi-lobal



108



I. La´szlo´ et al.



Fig. 6.7 Drawing of

nanotorus using ten

eigenvectors of the Laplacian.

Top view and side view



eigenvectors of the Laplacian reproduce well the nanotubes without any spiky

phenomenon at the ends (Fig. 6.5b). For nanotori we had to use 8–10 eigenvectors

of the Laplacian and most of them were bi-lobal. One representative structure is

in the Fig. 6.7.

The details of our nanotube junction analysis can be found in Graovac et al.

(2008a).

We have found that the greatest absolute values of aX2 , aY3 and aZ7 are attributed

to the three bi-lobal eigenvectors . If the number of eigenfunctions (m) in Eq. 6.36 is

smaller than 7 the picture of the structure can be described as a planar or a curved

two dimensional surface (Fig. 6.8). The eigenvectors c4 , c5 and c6 are 4, 4 and

5 –lobal but they have relatively small weight in Z. Although the eigenvectors

c8 , c9 , c10 , c11 and c12 have relatively small weight in Z and their lobality is from 3

to 5 they are important in eliminating the spiky features at the tube ends (Fig. 6.8).

We can obtain no flattened structure only if m is greater than the serial number of

any bi-lobal eigenfunction. The weight of the eigenvectors c13 and c14 is also

significant in the vector Z. If m is greater than 16, practically there are no

appreciated changing in the picture of the structure.

In Graovac et al. (2008a) we have investigated altogether 11 junctions and

obtained three-dimensional structures only if all the three bi-lobal eigenfunctions

were included in the summations. We had to take into consideration some other

eigenvectors for eliminating the spiky behaviours at the end of the tubes. We have

found in each case a value for m which gave satisfactory coordinates that can

be used for initial coordinates in molecular mechanics calculations.



6 Graph Drawing with Eigenvectors



109



Fig. 6.8 Shape analysis of a nanotube junction. Top and side view of the structure obtained by

3 (a), 9 (b) and 16 (c) eigenvectors of the Laplacian



6.6

6.6.1



Toward Topological Coordinates of any

Molecular Arrangement

Embedding with Three Eigenvectors into R3



In the previous paragraph we have seen that the applicability of three eigenvectors

is restricted only to spherical structures and nanotubes. Here we shall show that

there exist such kind of matrix or matrices which can reproduce practically exactly

the ðxi ; yi ; zi Þ Descartes-coordinates with the help of three eigenvectors. Let us

suppose that we have the exact ðxi ; yi ; zi Þ coordinates of the atoms. In this case,

as the number of atoms is n, the values of X, Y and Z are n-dimensional vectors

containing the x, y and z coordinates of the atoms in order. We suppose further,

as before that the centre of mass of the molecule is in the origin and the molecule

is directed in such a way that the eigenvectors of its tensor of inertia are showing to

the directions of the x, y and z axis. Thus we have the following relations:

n

X



xi ¼ 0;



i¼1



n

X

i¼1



yi ¼ 0;



n

X



zi ¼ 0



(6.42)



i¼1



and

XY ¼



n

X

i¼1



xi yi ¼ 0; YZ ¼



n

X



yi zi ¼ 0; ZX ¼



i¼1



n

X



z i xi ¼ 0



(6.43)



i¼1



1

If we define the vector U of n dimension with the relations ui ¼ pffiffiffi from

n

Eq. 6.42 follows that

UX ¼ 0; UY ¼ 0; UZ ¼ 0



(6.44)



I. La´szlo´ et al.



110



Equations 6.43 come from the conditions that the off-diagonal matrix elements

of the tensor of inertia are zero because of the special position of the molecule.

Let us suppose that the total energy





Erị ẳ E r12 ; r21 ; :::rij ; rji :::



(6.45)



depends only on the inter-atomic distances

rij ẳ







xi xj



2





2

2 12

ỵ yi yj ỵ zi À zj



(6.46)



Here we suppose further that total energy EðrÞ depends on rij and rji in a

symmetric way. If this is not the case, we substitute rij or rji by the value

rij ẳ rji ẳ rij ỵ rij ị=2.

In the followings we shall use the relations:

drij xi À xj drij yi À yj drij zi À zj

¼

;

¼

;

¼

dxi

rij

dyi

rij

dzi

rij



(6.47)



drij xj À xi drij yj À yi drij zj À zi

¼

;

¼

;

¼

dxj

rij

dyj

rij

dzj

rij



(6.48)



À

Á

As the gradients give the forces Fi ¼ Fxi ; Fyi ; Fzi acting on the i-th atom, we have:

@Erị

@Erị

@Erị

ẳ Fxi ;

ẳ Fyi ;

ẳ ÀFzi

@xi

@yi

@zi



(6.49)



Applying the above mentioned relations we obtain.

n

@E X

¼

wij xj ¼ Fxi

@xi

jẳ1



(6.50)



n

@Erị X



wij yj ẳ Fyi

@yi

jẳ1



(6.51)



n

@E X



wij zj ẳ Fzi

@zi

jẳ1



(6.52)



@Erị @Erị



rij @rij rji @rji



(6.53)



where

wij ¼ À



6 Graph Drawing with Eigenvectors



111



and

wii ¼



n 

X

@Erị

j6ẳi



@Erị



rij @rij rji @rji









n

X



wij



(6.54)



j6ẳi



That is

WX ẳ Fx ; WY ẳ ÀFy WZ ¼ ÀFz



(6.55)



If the atoms are in the equilibrium positions, X, Y and Z can be seen as

eigenvectors of W with zero eigenvalue, that is

WX ¼ 0; WY = 0; WZ = 0



(6.56)



From the relations of Eqs. 6.53 and 6.54 follows that

WU ¼ 0



(6.57)



1

with ui ¼ pffiffiffi .

n

If the centre of mass of the molecule is in the origin and the molecule is directed

in such a way that the eigenvectors of its tensor of inertia are showing to the

directions of the x, y and z axis, from the Eqs. 6.43, 6.44, 6.56, 6.57 follows that X,

Y, Z and U are orthogonal eigenvectors of the matrix W. That is

X ¼ Sx Cx ; Y ¼ Sy Cy ; Z ¼ Sz Cz



(6.58)



where Cx , Cy , Cz and U are orthogonal and normalized eigenvectors of W with zero

eigenvalue and Sx , Sy and Sz are appropriate scaling factors.

The question arises if we have any orthogonal and normalized eigenvectors Ax ,

y

A , Az and U of W with zero eigenvalue are there any appropriate scaling factors

Sx , Sy and Sz for obtaining the Descartes coordinates with a relation,

X ¼ Sx Ax ; Y ¼ Sy Ay ; Z ¼ Sz Az



(6.59)



If the number of eigenvectors with zero eigenvalue is four, the answer is yes, but

in Eq. 6.59 we obtain a rotation of the molecule as the vectors Ax , Ay , Az can be

obtained as linear combination of the vectors Cx , Cy , Cz . If the vectors Cx , Cy , Cz

are mixed with the vector U it means arbitrary translation and a rotation of the

molecule. As the vector U is known it can be easily subtracted from the linear

combinations in the case of mixings.

Usually the first neighbour distances in a molecule do not determine the

positions of the atoms but the full structure can be describe if we know the second

neighbour distances as well. From this follows that if the edges of a molecular graph

G ẳ V; Eịcorrespond to the first and second neighbours of a molecule, the

matrix W can be generated from a total energy EðrÞ which depends only on the



I. La´szlo´ et al.



112

Fig. 6.9 Coordinates of a

nanotube junction of the tubes

(9,6), (8,7) and (10,5)

obtained by three

eigenvectors of the matrix W

(top view and side view)



first and second neighbors of the molecule. If the dimension of the null space of W

is four than this null space contains three eigenvectors which give a proper

embedding of G ẳ V; Eị into R3 .



6.6.2



Examples for Embedding with Three Eigenvectors into R3



We tested our ideas for several structures. Here we present our results obtained for

nanotube junctions nano tori and helical nanotubes. We calculated the interatomic

interactions with the help of the Brenner potential (Brenner 1990) and harmonic

potentials as well. In the Brenner potential there are first neighbour and second

neighbour interactions.

The matrix W has non-zero matrix elements only for the first and the second

neighbours. We calculated the equilibrium position of the carbon atoms in a nanotube junction, in a torus and in a helical structure. In each cases we could reconstruct

the original coordinates with the help of the relations of Eq. 6.59, see Figs. 6.9–6.11.

In each cases we used Sx ¼ Sy ¼ Sz . The final values for the parameters Sx , Sy and

Sz can be obtained from scaling three independent distance to given values.



6 Graph Drawing with Eigenvectors



113



Fig. 6.10 Coordinates of a

torus obtained by three

eigenvectors of the matrix W

(top view and side view)



6.7



Conclusions



We have examined the possibilities of drawing graphs with eigenvectors of the

adjacency matrix A the Laplacian L and the Colin de Verdie`re matrix M. We have

suggested the matrix W for drawing not only spherical graphs with eigenvectors.

For these symmetric matrices the off-diagonal matrix element of ðu; vÞ 2

= E are

equals to zero. The absolute value of other off-diagonal matrix element for

matrices A and L are equals to 1. In matrices L and W the diagonal matrix elements

are calculated in such a way that the sum of matrix elements be equal to zero in each

row. The off-diagonal matrix elements of M and W are determined using special

kind of conditions. We have shown that in each cases where the adjacency matrix

was applicable the Laplacian matrix gave good drawings as well. We have found

also that the Laplacian could be used even for nanotubes where the adjacency matrix

was not applicable. It was demonstrated that three eigenvectors of the matrix W

produced good drawing in each cases under study in this paper. The drawback of W

is that at present there is not a simple algorithm for its construction. In this work we

could generate it using an energy minimalization algorithm with the help of a



114



I. La´szlo´ et al.



Fig. 6.11 Coordinates of a

helix obtained by three

eigenvectors of the matrix W

(top view and side view)



conjugate gradient algorithm. There is a hope, however, that using appropriate

approximations for the matrix elements of W a method can be found for constructing

topological coordinates of complicated non-spherical structures as well.

Acknowledgements I. La´szlo´ thanks for the supports of grants TAMOP-4.2.1/B-09/1/KONV2010-0003, TAMOP-4.2.1/B-09/1/KMR-2010-0002 and for the support obtained in the frame

work of bi-lateral agreement between the Croatian Academy of Science and Art and the Hungarian

Academy of Sciences.



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