Computational Chemistry

The potential energy curve for two nonpolar nonbonded atoms has the general
form shown in Fig.3.6. A simple way to approximate this is by the so-called
Lennard-Jones 12–6 potential [ 7 ]:

Enonbond¼knb

s

r

 12

$

s

r

 6

(*3.5)

r¼the distance between the centers of the nonbonded atoms or groups.
The function reproduces the small attractive dip in the curve (represented by the
negative term) as the atoms or groups approach one another, then the very steep rise
in potential energy (represented by the positive, repulsive term raised to a large
power) as they are pushed together closer than their van der Waals radii. SettingdE/
dr¼0, we find that for the energy minimum in the curve the corresponding value of
r isrmin¼ 21 =^6 s,

i.e: s¼ 2 $^1 =^6 rmin (3.6)

If we assume that this minimum corresponds to van der Waals contact of the
nonbonded groups, thenrmin¼(RA+RB), the sum of the van der Waals radii of the
groups A and B. So

21 =^6 s¼ðRAþRBÞ

MeMe

Me

Me

Me

Me

Me

Me

H 3 C CH 3

C C

H

H H

H

CCCC dihedral, degrees

energy

kJ mol–1

0 60 120 180

10

20

25 kJ mol–1

3 kJ mol–1

14 kJ mol–1

Fig. 3.5Variation of the energy of butane with dihedral angle. The curve can be represented by a
sum of cosine functions

3.2 The Basic Principles of Molecular Mechanics 51