Formulas for non-bonded interaction energies

If one were to calculate the non-bonded interaction energy between two atoms, this would equate to the sum of the vdW + electrostatic potential energies:


$$
E\_{\text{non-bond}} = E\_{\text{vdW}} + E\_{\text{electrostatic}}
$$


Could anyone please explain how I could calculate these potentials providing a formula?


Perhaps given an example with zinc ($\ce{Zn^2+}$) and oxygen ($\ce{O}$), where $\sigma$ and $\varepsilon$ values (defined by Charmm27) are given as:


\begin{array}{ccc}
\hline
\text{Species} & \sigma & \varepsilon \\
\hline
\ce{Zn} & 1.942 & 0.25 \\
\ce{O} & 3.029 & 0.12 \\
\hline
\end{array}


Ultimately I just wish to understand how they are calculated.


Simply put, I am trying to calculate non-bonded interaction energy between a ligand and a protein. However, after trying to understand each of these terms I am a little lost.




You've got the Lennard-Jones potential parameters for zinc and oxygen from a certain version of the CHARMM force field. The [commonly-used notation](https://en.wikipedia.org/wiki/Lennard-Jones_potential) is $\varepsilon$ for potential well depth and $\sigma$ is the distance where the pair potential is zero, and $r\_{min}$ is the distance which minimizes the L-J potential energy function. The linked Wikipedia article gives the potential as:


$$E\_{\rm L-J}(r) = 4\varepsilon\left[\left({\sigma\over r}\right)^{12} - \left({\sigma\over r}\right)^{6}\right] = \varepsilon\left[\left({r\_{min}\over r}\right)^{12} - 2\left({r\_{min}\over r}\right)^{6}\right]
$$


At $r = r\_{min}$, $E\_{\rm L-J} = -\varepsilon$ and we can map $r\_{min} = 2^{1/6}\sigma$ to work with either form of the L-J potential function.


[*Accurate Calculation of Hydration Free Energies using Pair-Specific Lennard-Jones Parameters in the CHARMM Drude Polarizable Force Field*. J. Chem. Theory Comput. 6(4) 1181–1198 (2010)](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2853947/) describes (among many other things) how the L-J potential term is calculated for 2 interacting species (as in your case). See equations (1), (2), and (3) in that paper, reproduced below (where I have used $r\_{min}$ instead of $R\_{min}$ for consistency).


In the paper, the Lennard-Jones potential is given as


$$E\_{\rm L-J}(r) = \varepsilon\left[\left({r\_{min}\over r}\right)^{12} - 2\left({r\_{min}\over r}\right)^{6}\right]$$


For a pair of interacting species $i,j$:


$$r\_{min} = {r\_{min,i} + r\_{min,j}\over 2}$$


and


$$\varepsilon = (\varepsilon\_{i}\cdot\varepsilon\_{j})^{1/2}$$


For your system, we compute $r\_{min}$ from the given values of $\sigma$ (using the mean of the two values of $\sigma$, as specified in the manuscript):


$$r\_{min} = 2^{1/6}\sigma = 2^{1/6}\cdot {1.942 + 3.029\over 2} = 2.789\;\unicode{xc5}$$


and


$$\varepsilon = (0.25\cdot 0.12)^{1/2} = 0.1732\;\mathrm{kcal\;mol^{-1}}$$


Now, at some distance $r$, you can compute $E\_{\rm L-J}(r)$.


In the linked publication, the authors also go through the process of how the L-J parameters are derived/fit, which may be of interest to you.


Finally, see [this vintage web page authored by a familiar guy](http://ambermd.org/Questions/vdw.html) which goes through a procedure for taking parameters for computed hydration free energies of cations and deriving L-J parameters for use in the AMBER force field. The procedure is very similar to that used for CHARMM.


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