12.12.2 Supplementing XPol with Empirical Potentials

In order to obtain physical results, one must either supplement the XPol energy expression with either empirical intermolecular potentials or else with an ab initio treatment of intermolecular interactions. The latter approach is described in Section 12.14. Here, we describe how to add Lennard-Jones or Buckingham potentials to the XPol energy, using the $xpol_mm and $xpol_params sections described below.

The Lennard-Jones potential is

$$V_{\mathrm{LJ}}(R_{ij})=4{ϵ}_{ij}\left[{\left(\frac{{\sigma }_{ij}}{R_{ij}}\right)}^{12}-{\left(\frac{{\sigma }_{ij}}{R_{ij}}\right)}^6\right],$$

(12.44)

where $R_{ij}$ represents the distance between atoms $i$ and $j$. This potential is characterized by two parameters, a well depth ${ϵ}_{ij}$ and a length scale ${\sigma }_{ij}$. Although quite common, the $R^{-12}$ repulsion is unrealistically steep. The Buckingham potential replaces this with an exponential function,

$$V_{\mathrm{Buck}}(R_{ij})={ϵ}_{ij}\left[A{e}^{-B\frac{R_{ij}}{{\sigma }_{ij}}}-C{\left(\frac{{\sigma }_{ij}}{R_{ij}}\right)}^6\right],$$

(12.45)

Here, $A$, $B$, and $C$ are additional (dimensionless) constants, independent of atom type. In both Eq. (12.44) and Eq. (12.45), the parameters ${ϵ}_{ij}$ and ${\sigma }_{ij}$ are determined using the geometric mean of atomic well-depth and length-scale parameters. For example,

$${\sigma }_{ij}=\sqrt{{\sigma }_i{\sigma }_j}.$$

(12.46)

The atomic parameters ${\sigma }_i$ and ${ϵ}_i$ must be specified using a $xpol_mm section in the Q-Chem input file. The format is a molecular mechanics-like specification of atom types and connectivities. All atoms specified in the $molecule section must also be specified in the $xpol_mm section. Each line must contain an atom number, atomic symbol, Cartesian coordinates, integer atom type, and any connectivity data. The $xpol_params section specifies, for each atom type, a value for $ϵ$ in kcal/mol and a value for $\sigma$ in Ångstroms. A Lennard-Jones potential is used by default; if a Buckingham potential is desired, then the first line of the $xpol_params section should contain the string BUCKINGHAM followed by values for the $A$, $B$, and $C$ parameters.