## AMBER force field If we add potentials together of all particles, or say nuclei, we may get a description of potential energy vs. $r$. This is the force field, we may write it as $V(r)=E_{bonded}+E_{nonbonded}$ Or so-called AMBER force field, it can be expressed as $V(r) = \sum_{bonds} k_b(b-b_0)^2 + \sum_{angles}k_\theta (\theta-\theta_0)^2 + \sum_{dihedral}\frac{V_n}{2}(1+\cos n\phi -\delta)+\sum_{nonbonded, ij} \frac{A_{ij}}{r_{ij}^{12}}-\frac{B_{ij}}{r_{ij}^6} +\frac{q_iq_j}{r_{ij}}$ >[!Note] >AMBER is *assisted model building with energy refinement*. From left to right we have bounding energy (transitional vibration), bonding angle, dihedral, VdW term ([[Details on LJ|LJ form]]) and Coulombic term. >[!Notice] > We cannot use force field to describe bond creation or breaking process, also electronic properties since there is no electron. Above expressions get parameters, which can be obtained from solving Schrödinger equation or experiments. (AMBER is a parameterized potential, one should first fit for parameters to use) Now let's see each of the terms. #### bond energy: Typically use a quadratic expansion is enough. $V= k(r-r_{eq})^2$ #### bond angle, i.e., angular term: Similarly, use the harmonic term to show the deviation from equilibrium $E = k(\theta-\theta_{eq})^2$ #### dihedral terms: This gives the angular repair density of the structure, i.e., ![[Drawing 2023-09-24 19.26.01.excalidraw.svg]] #### Columbic interactions $E=\sum_i\sum_{j>i}C \frac{q_iq_j}{r_ij}$ #### VdW term We use the LJ for VdW, which has [[Details on LJ#^3e2485|the form]] $V = 4\epsilon \left[ \left( \frac{\sigma}{r_{ij}} \right)^{12}-\left( \frac{\sigma}{r_{ij}} \right)^6 \right]$![[Drawing 2023-09-24 19.33.43.excalidraw.svg]] >[!Note] >$\epsilon$ is called dispersion energy, may be understood as depth of potential well. $\sigma$ is spacing when $U=0$. > More on [[Details on LJ|next chapter]].