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?
