MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 15: The Stretching Potential of a Chemical Bond
The stretching potential of a chemical bond is approximately described by the function
V (s) = De(1 - e-s
)2
where De is the energy required to break the bond,  is a parameter, and s is the coordinate that gives the
displacement from the equilibrium bond length.
Problems:
1. Find a power series expansion of V (s) about the equilibrium, s = 0, in terms of s through the s8
term.
2. Compare this expansion with the potential for stretching a simple ("harmonic") spring
Vspring(s) =
1
2
ks2
.
3. The restoring force of a spring is
V
s
, and the force constant is
2
V
s2
|s=0. What are the restoring forces
and the force constants obtained from V (s) and Vspring(s)? How and what does this tell you about ?
4. If the power series were truncated (as an approximation to V (s)), how far would you have to go to be
sure that the asymmetry of V (s) (plot it!) was present for small displacements about s = 0?
[A particular V (s) is said to be symmetric about s = 0 if, for every choice of s, V (s) = V (-s). It is
asymmetric if this is not satisfied. For chemical bonds, V (s) may be nearly symmetric for small displacements
away from the equilibrium, but it has to be asymmetric overall because breaking a bond, i.e. s > 0, is different
than compressing it, i.e. s < 0.]
Contributor: Dr. Clifford Dykstra
Department of Chemistry, IUPUI