Concise Physical Chemistry

c19 JWBS043-Rogers September 13, 2010 11:29 Printer Name: Yet to Come

GEOMETRY AND ENTHALPY: MOLECULAR MECHANICS 309

(Nevins et al. 1996 a, 1996 b) and other groups have devoted four decades to the job,

resulting in rather large MM computer programs that have evolved with time. Among

them is the Allinger series MM1, MM2,..., MM4, with some variants in between.

19.4 GEOMETRY AND ENTHALPY: MOLECULAR MECHANICS

Suppose that we have suitable force parameters for the C H and the C C bonds in

alkanes andr 0 at the minima of the Hooke’s law potential energy wells (Fig. 18.2).

We should be able to construct models of ethane and higher alkanes and find the

structure and energy that best satisfy thekfvalues.

But not quite. Like the members of a large family, the bond–atom combinations in

a molecule do not always agree. The final structure of the molecule is a compromise

in which no bond is exactly at its ideal lengthr 0 but no bond is too far removed from

it. The distance between the idealized bond lengthr 0 and the real oneris assumed

to follow Hooke’s law,F=−kf(r−r 0 ). The restoring forceFresults in a parabolic

potential energy well with the potential energy defined asV=0 at its minimum:

V(r−r 0 )=kf^12 (r−r 0 )^2

Like a large family, the final compromise engenders some strain. The energy of

the real molecule is above the energy of a hypothetical strainless molecule by the

summation of the strain energy experienced by all its members at the compromise

position. The energy of the real molecule is at least

E=Estrain-free+

∑

all bonds

V(r−r 0 )

Bond stretching is not the only energy that enters into the final compromise.

Chemical bonds can also be bent through an angleθ. It is reasonable to assign a

Hooke’s law potential energy for bending. If the compromise structure involves some

bond bending, the potential energy of bending must be included in the sum

E=Estrain-free+

∑

all bonds

Vstretch(r−r 0 )+

∑

all simple angles

Vbend(θ−θ 0 )

Intramolecular interactions result in forces called van der Waals forces, and molecules

may suffer dihedral angular distortion calledtorsional forces. These give rise to

potential energiesVvdWandVtor, respectively. The sum of all potential energies over

a hypothetical strainless enthalpy is

E=Estrain-free+

∑

Vstretch+

∑

Vbend+

∑

VvdW+

∑

Vtors