BASIC CONCEPTS AND AN OVERVIEW - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 131

FIGURE 1.6 Geometry of a simple chain molecule, illustrating the definition of
interatomic distance r 23 , bend angle, and torsion angle .

1
2

( ) 2 ∑

r

U = kr r −

(107)

intramolecular ij ij eq

bonds

1

( ) 2

∑

0

+ k 00 −

(108)

ijk ijk eq

2

bonds
angles

1

( ) 2 ∑∑

m
ijkl

φ ,

+ k 1 + cos( m φ −g

(109)

ijkl m

2

torsion

m

angles

The geometry is illustrated in Fig. 1.6. The “bonds” will typically involve the
separation between adjacent pairs of atoms in a molecular framework, and we
assume inEq. (107), a harmonic form with specified equilibrium separation, al-
though this is not the only possibility. The “bend angles”are between successive
bond vectors such as and , and therefore involve three atom coordinates:

( ) ( ) ( )

− 1/ 2 − 1/ 2

(110)

ˆˆ

cos q = rr . = rr . r r . r r .

ijk ij jk ij ij jk jk jk jk

where Usually this bending term is taken to be quadratic in the angular displace-
ment from the equilibrium value, as in Eq. (110), although periodic functions are
also used. The “torsion angles”ijkl are defined in terms of three connected bonds,
hence four atomic coordinates:

(111)

cos φ =− n ˆ . n ˆ , where n = × r r , n = × r r

ijkl ijk jkl ijk ij jk jkl jk kl