04/13/16

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

Stretching Energy: E_{stretch} =
Σ_{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: E_{bending} =
Σ_{angles}
k_{Θ}
(Θ - Θ_{o})^{2}

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.

.

Torsion Energy E_{torsion
}=
Σ_{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).

Interactive SageMath Graph: Interactive Graph of 6-12 Lennard Jones Potential

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