# Can someone help me understand the motivation behind and visualize the dihedral potential and the Urey-Bradley potential?

I am studying statistical mechanics and force-fields, and I see a lot of this formula being thrown around with no explanation: $$U_{\mathrm{dihedral}} = \begin{cases} k(1+\cos (n\psi + \phi)),& n>0 \\ k(\psi - \phi)^2, & n=0 \\ \end{cases}$$ The only explanation I read is: this is a 4-body torsion angle (known as the dihedral angle) potential, which embodies the potential due to the angular spring between the planes formed by the first 3 atoms, and the other 3 atoms. $$\psi$$ is the angle between the planes, and $$\phi$$ is the phase shift angle, while $$k$$ is a multiplicative constant.

Could anyone please provide a visualization for this? and why should there be a $$\cos$$ term in the potential? Why is there an $$n$$ in the mix? Looks an awful lot like an eigenvalue of sorts.

Same applies for the Urey-Bradley potential. But I want to understand why such a potential was introduced in the first place. The 3-body potential described angular vibrational motion: $$U_{UB} = k(\theta - \theta_0)^2 + k_{ub}(r_{ik} - r_{ub})^2$$ where $$\theta$$ is the angle between $$r_{ij}$$ and $$r_{jk}$$, and the second term is for the covalent spring bond between atoms $$i$$ and $$k$$. I don't understand - if these atoms are not bonded, why cook up a new bond?

• – Karsten Theis Feb 26 at 4:08
• this is great, thank you @KarstenTheis – megamence Feb 26 at 4:20
• It turns out you cannot complete an internal coordinate system for a molecule without including parameters that include four atoms, in many cases. – Bertram Feb 26 at 16:22