Page 187 - Advanced Organic Chemistry Part A - Structure and Mechanisms, 5th ed (2007) - Carey _ Sundberg

mechanics computations indicate that this is indeed the minimum energy conformation 167
for cyclododecane. 80 86
As the ring size increases, the number of possible conformations increases further SECTION 2.3
so that many alternative diamond lattice conformations are available. 87 Molecular Mechanics

2.3. Molecular Mechanics

The analysis of molecular conformation can be systematically and quantitatively
approached through molecular mechanics. 88 A molecule adopts the geometry that
minimizes its total strain energy. The minimum energy geometry is strained (destabi-
lized) to the extent that its structural parameters deviate from their ideal values. The
energy for a particular kind of distortion is a function of the amount of distortion and
the opposing force. The total strain energy is the sum of several contributions:

E strain = E r +E
+E +E d (2.9)

where E r is the energy associated with stretching or compression of bonds, E
is the energy of bond angle distortion, E is the torsional strain, and E d are the
energy increments that result from nonbonded interactions between atoms.
Molecular mechanics calculations involve summation of the force fields for each
type of strain. The original mathematical expressions for the force fields were derived
from classical mechanical potential energy functions. The energy required to stretch a
bond or to bend a bond angle increases as the square of the distortion:
Bond stretching E r = 0 5k r −r 2 (2.10)
r
0
where k , is the stretching force constant, r the bond length, and r the normal bond
r 0
length.
Bond angle bending E
= 0 5k
2 (2.11)

where k is the bending force constant and
is the deviation of the bond angle from

its normal value. The torsional strain is a sinusoidal function of the torsion angle.
Torsional strain results from the barrier to rotation about single bonds, as described
for ethane on p. 142–143. For molecules with a threefold barrier such as ethane, the
form of the torsional barrier is:
E = 0 5V 1+cos3 (2.12)
0
where V is the rotational energy barrier and is the torsional angle. For hydrocarbons,
0
V can be taken as being equal to the ethane barrier (2.9 kcal/mol).
0
Nonbonded interaction energies, which may be attractive or repulsive, are the
most difficult contributions to evaluate. When two uncharged atoms approach each
other, the interaction between them is very small at large distances, becomes slightly
86
M. Saunders, J. Comput. Chem., 12, 645 (1991).
87 M. Saunders, J. Am. Chem. Soc., 109, 3150 (1987); V. L. Shannon, H. L. Strauss, R. G. Snyder,
C. A. Elliger, and W. L. Mattice, J. Am. Chem. Soc., 111, 1947 (1989); M. Saunders, K. N. Houk,
Y. D. Wu, W. C. Still, M. Lipton, G. Chang, and W. C. Guida. J. Am. Chem. Soc., 112, 1419 (1990).
88
For general reviews see: W. Gans, A. Amann, and J. C. A. Boeyens, Fundamental Principles of
Molecular Modeling, Plenum Press, New York. 1996; A. K. Rappe and C. J. Casewitt, Molecular
Mechanics Across Chemistry, University Science Books, Sausalito, CA, 1997; J. C. A. Boeyens and
P. Comba, Coordn. Chem. Rev., 212, 3 (2001).