Computational Chemistry

Calculations on staggered butane gave for the C–C–C angle

aeqðCCCÞ¼ 112 : 5  (3.16)

kbendðCCCÞ¼ 0 :110 KJ mol$^1 deg$^2 (3.17)

Parameterizing the Torsional Term For the ethane case (Fig.3.4), the equation
for energy as a function of dihedral angle can be deduced fairly simply by adjusting
the basic equationE¼cosyto giveE¼1/2Emax[1 + cos3(y+ 60)].
For butane (Fig.3.5), using Eq.3.4and experimenting with a curve-fitting
program shows that a reasonably accurate torsional potential energy function can
be created with five parameters,k 0 andk 1 – k 4 :

EtorsionðCH 3 CH 2 $CH 2 CH 3 Þ¼k 0 þ

X^4

r¼ 1

kr½ 1 þcosðryފ (3.18)

The values of the parametersk 0 – k 5 are given in Table3.2. The calculated curve
can be made to match the experimental one as closely as desired by using more
terms (Fourier analysis).
Parameterizing the Nonbonded Interactions Term To parameterize Eq.3.5we
might perform ab initio calculations in which the separation of two atoms or groups
in different molecules (to avoid the complication of concomitant changes in bond
lengths and angles) is varied, and fit Eq.3.5to the energy vs. distance results. For
nonpolar groups this would require quite high-level calculations (Chapter 5 ), as van
der Waals or dispersion forces are involved. We shall approximate the nonbonded
interactions of methyl groups by the interactions of methane molecules, using
experimental values ofknbands, derived from studies of the viscosity or the
compressibility of methane. The two methods give slightly different values [ 7 b],
but we can use the values

knb¼ 4 :7 kJ mol$^1 (3.19)

Table 3.2The experimental potential energy values for rotation about the central C–C
bond of CH 3 CH 2 –CH 2 CH 3 can be approximated by EtorsionðCH 3 CH 2 $CH 2 CH 3 Þ¼k 0 þ
P^4

r¼ 1

kr½ 1 þcosðryފwithk 0 ¼20.1,k 1 ¼$4.7,k 2 ¼1.91,k 3 ¼$7.75,k 4 ¼0.58. Experimental

energy values at 30, 90, and 150were interpolated from those at 0, 60, 120, and 180;
energies are in kJ mol$^1
y(deg) E(calculated) E(experimental)
0 0.15 0
30 6.7 7.0
60 14 14
90 8.8 9.0
120 3.5 3.3
150 15 15
180 25 25

3.2 The Basic Principles of Molecular Mechanics 55