# 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?

• Oct 8 '16 at 21:05

## 1 Answer

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 to the extent of the accuracy of the potential. From the "Limits" section:

The term Van der Waals (vdW) forces roughly includes the following intermolecular forces:

• the permanent dipole-permanent dipole force (Keesom force), which is an electrostatic term,
• the permanent dipole-induced dipole force (Debye force), which is an induction or polarization term,
• the induced dipole-induced dipole force (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, 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 is a nice set of notes that describes the interaction between electric multipoles to different orders.

• Just keep in mind that the potentials for Keesom force and Debye force here are all averaged for ALL angles. The exact dependence on the inter electronic distance w/o being averaged are here: chem1.com/acad/webtext/states/interact.html Jan 26 '17 at 16:26