# D. Bonded Interaction Energy

The mathematical form of the energy terms varies from force-field to force-field. The more common forms will be described.

Stretching Energy: \(\mathrm{E_{stretch} = \sum_{bonds}k_b (r - r_o)^2}\)

The stretching energy equation is based on Hook's law. The kb parameter defines the stiffness of the bond spring. R_{0} is the equilibrium distance between the two atoms. It should make sense that deviations from the equilibrium length would be associated with higher energy. The E vs r curves is hence a parabola:

Obviously only small changes in r are allowed as to large an r value would lead to bond breaking.

Bending Energy: \(\mathrm{E_{bending} = \sum_{angles} k_Θ (Θ - Θ_o)^2 }\)

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The bending energy equation is also based on Hook's law. The kΘ parameter controls the stiffness of the angle spring, while the Θ_{0} is the equilibrium angle. As above, the graph of E vs theta is expected to be a parabola.

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Torsion Energy \(\mathrm{E_{torsion} = \sum_{torsions} A [1 + \cos ( ntau - Θ) ] }\)

The torsion energy is modeled by a periodic function, much as you have seen with energy plots associated with Newman projections sighting down C-C bonds fro butane, for example.

Wolfram Mathematica CDF Player - Torsional Energy (free plugin required)

The parameters (determined for different 4 bonded atoms small molecules using curve fitting) for these are:

- amplitude A
- periodicity n (ethane, sighting along the C-C axis in a Newman projects displays a periodicity of 120 degrees)
- phase shift Phi: shifts curve along rotation (tau) axis. parameter controls its periodicity, and phi shifts the entire curve along the rotation angle axis (tau).