What is the potential to use for non-dispersion interactions?

Although liquid argon’s intermolecular potential can be described using a Lennard-Jones (LJ) potential, which takes into account dispersion forces (induced dipole-induced dipole interactions), what potential do you use when also including the effects of dipole-induced dipole and dipole-dipole interactions?




Because all three of these interactions are of the same order in the interelectronic distance ($1/r^{6}$), they are *all* captured within the [Lennard-Jones potential](https://en.wikipedia.org/wiki/Lennard-Jones_potential) to the extent of the accuracy of the potential. From the "Limits" section:


The term [Van der Waals (vdW) forces](https://en.wikipedia.org/wiki/Van_der_Waals_force) roughly includes the following [intermolecular forces](https://en.wikipedia.org/wiki/Intermolecular_force):



> 
> * the permanent dipole-permanent dipole force ([Keesom force](https://en.wikipedia.org/wiki/Keesom_force)), which is an electrostatic term,
> * the permanent dipole-induced dipole force ([Debye force](https://en.wikipedia.org/wiki/Debye_force)), which is an induction or polarization term,
> * the induced dipole-induced dipole force ([London dispersion force](https://en.wikipedia.org/wiki/London_dispersion_force)), which is the dispersion term.
> 
> 
> 


the forms of which are


$$
\begin{align}
V\_{\text{Keesom}} &= \frac{-m\_{1}^{2}m\_{2}^{2}}{24\pi^{2}\epsilon\_{0}^{2}\epsilon\_{r}^{2}k\_{b}Tr^{6}} \\
V\_{\text{Debye}} &= \frac{-m\_{1}^{2}\alpha\_{2}}{16\pi^{2}\epsilon\_{0}^{2}\epsilon\_{r}^{2}r^{6}} \\
V\_{\text{LJ}} &= 4\epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right] 
\end{align}
$$


Note that the LJ potential contains both a repulsive ($1/r^{12}$) and attractive ($-1/r^{6}$) component. This is not *the* form of the dispersion interaction, but is both a suitable mathematical form that encapsulates a few noncovalent (vdW) interactions together and is easy to evaluate on a computer.


You may also be interested in the form of a more complicated potential, such as the [AMOEBA polarizable force field](http://dx.doi.org/10.1021/jp910674d), which uses a slightly different form for the pairwise vdW interaction ($i,j$ are atoms)


$$
\begin{align}
U\_{\text{vdw}}(ij) &= \epsilon\_{ij} \left( \frac{1.07}{\rho\_{ij} + 0.07} \right)^{7} \left( \frac{1.12}{\rho\_{ij}^{7} + 0.12} - 2 \right) \\
\rho\_{ij} &= \frac{R\_{ij}}{R\_{ij}^{0}} \\
\epsilon\_{ij} &= \frac{4\epsilon\_{ii}\epsilon\_{jj}}{(\epsilon\_{ii}^{1/2}+\epsilon\_{jj}^{1/2})^{2}} \\
R\_{ij}^{0} &= \frac{ (R\_{ii}^{0})^{3} + (R\_{jj}^{0})^{3} }{ (R\_{ii}^{0})^{2} + (R\_{jj}^{0})^{2} }
\end{align}
$$


and also considers up to permanent and induced quadrupole moments


$$
\begin{align}
U\_{\text{elec}}^{\text{perm}}(r\_{ij}) &= M\_{i}^{T} T\_{ij} M\_{j} \\
T\_{ij} &= \begin{bmatrix}
1 & \frac{\partial}{\partial x\_j} & \frac{\partial}{\partial y\_j} & \frac{\partial}{\partial z\_j} & \mathrm{L} \\ 
\frac{\partial}{\partial x\_j} & \frac{\partial^2}{\partial x\_i \partial x\_j} & \frac{\partial^2}{\partial x\_i \partial y\_j} & \frac{\partial^2}{\partial x\_i \partial z\_j} & \mathrm{L} \\ 
\frac{\partial}{\partial y\_j} & \frac{\partial^2}{\partial y\_i \partial x\_j} & \frac{\partial^2}{\partial y\_i \partial y\_j} & \frac{\partial^2}{\partial y\_i \partial z\_j} & \mathrm{L} \\ 
\frac{\partial}{\partial z\_j} & \frac{\partial^2}{\partial z\_i \partial x\_j} & \frac{\partial^2}{\partial z\_i \partial y\_j} & \frac{\partial^2}{\partial z\_i \partial z\_j} & \mathrm{L} \\ 
\mathrm{M} & \mathrm{M} & \mathrm{M} & \mathrm{M} & \mathrm{O}
\end{bmatrix} \left( \frac{1}{r\_{ji}} \right)
\end{align}
$$


along with the other typical bonded terms found in modern force fields.


[Here](https://janheyda.files.wordpress.com/2015/08/electrostatics-multipoles.pdf) is a nice set of notes that describes the interaction between electric multipoles to different orders.


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