Origami: Self Organizing Polyhexagonal Carbon Structures for Formation of Fullerenes, Nanotubes and Other Carbon Structures - Exotic Properties of Carbon Nanomatter - page 21

Here the atom-centred basis function φ ν ((r-R ν )) is centred around the atom with the

position vector R ν and m equals to the number of basis functions centred on the
atoms in the positions R k with 1

≤ k ≤ N where N is the number of atoms and R ν
equals to one of the vectors R k .P k is the momentum of the atom k with the mass
M k . The eigenvalues and the eigenvectors are determined by the equations

m

C ν i (H μν − ε i S μν ) = 0 (2.4)

ν = 1

For each μ and i, where

H μν = φ μ |

H | φ ν (2.5)

and

S μν = φ μ |φ ν

(2.6)

is the overlap integral. The H μν and S μν off diagonal matrix elements are given with
the help of the function H sp σ ( R ), H pp σ ( R ), H ss σ ( R ), H pp π ( R ), S sp σ ( R ), S pp σ ( R ),
S ssσ ( R ), S pp π ( R ). Namely we were using only carbon atoms and each of them was
supplied by one sorbital and three p orbital. That is was s, p x , p y or p z orbital.

The corresponding functions are given in ref. (Porezag et al. 1995 ) with the help
of Chebyshev polynomials. The matrix element were given with the help of the
Slater-Koster relations (Slater and Koster 1954 ).

φ

H ss ( R j − R i ) = H ss σ (R ij ) (2.7)

H sx ( R j − R i ) = cos ( α x )H sp σ (R ij ) (2.8)

cos 2 ( α x )H pp σ (R ij ) cos 2 ( α x ))H pp π (R ij ) H xx ( R j − R i ) = + (1 − (2.9)

H xy ( R j − R i ) = cos ( α x ) cos ( α y )H pp σ (R ij ) − cos ( α x ) cos ( α y )H pp π (R ij ) (2.10)

H xz ( R j − R i ) = cos ( α x ) cos ( α z )H pp σ (R ij ) − cos ( α x ) cos ( α z )H pp π (R ij ) (2.11)

Here we have supposed that the atomic orbital φ μ was centred at R i and the atomic

orbital R i =

0 are cos ( α x ), cos ( α y ) and cos ( α z ). R ij is the distance between the two atoms. If
the atomic orbitals

φ ν was centered on the atom at R j . The direction cosines of the vector R j −

φ μ and φ ν are centred on the same atom, H ss =− 13 . 573 eV and

H xx = H yy = H zz =− 5 . 3715 eV for μ = ν and H μν = 0 for μ = ν . The other

matrix elements can be generated with the help of symmetry.
Similar relations are valid for the S μν overlap matrix elements as well. That is,

S ss ( R j − R i ) = S ss σ (R ij ) (2.12)

S sx ( R j − R i ) = cos ( α x )S sp σ (R ij ) (2.13)

cos 2 ( α x )S pp σ (R ij ) cos 2 ( α x ))S pp π (R ij )

S xx ( R j − R i ) = + (1 − (2.14)