User:Cabb99/sandbox

CHARMM22


V(r^N)=\sum_\mbox{bonds} k_b (l-l_0)^2 + \sum_\mbox{angles} k_a (\theta - \theta_0)^2$$ $$+ \sum_\mbox{dihedrals} k_\phi [1+\cos(n \phi- \delta)]$$ $$+\sum_{j=1} ^{N-1} \sum_{i=j+1} ^N \left\{\epsilon_{i,j}\left[\left(\frac{r_{0ij}}{r_{ij}} \right)^{12} - 2\left(\frac{r_{0ij}}{r_{ij}} \right)^{6} \right]+ \frac{q_iq_j}{4\pi \epsilon_0 r_{ij}}\right\} $$

AMBER
The functional form of the AMBER force field is

V(r^N)=\sum_\mbox{bonds} k_b (l-l_0)^2 + \sum_\mbox{angles} k_a (\theta - \theta_0)^2$$ $$+ \sum_\mbox{torsions} \frac{1}{2} V_n [1+\cos(n \omega- \gamma)]$$ $$+\sum_{j=1} ^{N-1} \sum_{i=j+1} ^N \left\{\epsilon_{i,j}\left[\left(\frac{r_{0ij}}{r_{ij}} \right)^{12} - 2\left(\frac{r_{0ij}}{r_{ij}} \right)^{6} \right]+ \frac{q_iq_j}{4\pi \epsilon_0 r_{ij}}\right\} $$ Note that despite the term force field, this equation defines the potential energy of the system; the force is the derivative of this potential with respect to position.

OPLS
The OPLS (Optimized Potentials for Liquid Simulations) force field was developed by Prof. William L. Jorgensen at Purdue University and later at Yale University.

The functional form of the OPLS force field is very similar to that of AMBER:

$$E \left(r^N \right ) = E_\mathrm{bonds} + E_\mathrm{angles} + E_\mathrm{dihedrals} + E_\mathrm{nonbonded}$$

$$E_\mathrm{bonds} = \sum_\mathrm{bonds} K_r (r-r_0)^2 \, $$

$$E_\mathrm{angles} = \sum_\mathrm{angles} k_\theta (\theta-\theta_0)^2 \, $$

$$E_\mathrm{dihedrals} = \frac {V_1} {2} \left [ 1 + \cos (\phi-\phi_0) \right ] + \frac {V_2} {2} \left [ 1 - \cos 2(\phi-\phi_0) \right ] + \frac {V_3} {2} \left [ 1 + \cos 3(\phi-\phi_0) \right ] + \frac {V_4} {2} \left [ 1 - \cos 4(\phi-\phi_0) \right ]$$

$$E_\mathrm{nonbonded} = \sum_{i>j} f_{ij} \left(                   \frac {A_{ij}}{r_{ij}^{12}} - \frac {C_{ij}}{r_{ij}^6}                    + \frac {q_iq_j e^2}{4\pi\epsilon_0 r_{ij}} \right) $$

with the combining rules $$A_{ij} = \sqrt{A_{ii} A_{jj}}$$ and $$C_{ij} = \sqrt{C_{ii}C_{jj}}$$.