Interaction Energy between a Helium Atom and a Hydrogen Molecule

The interaction energy between a helium atom and a hydrogen molecule has been calculated from first principles using a simple wave function made up of $1S$ orbitals centered on the three nuclei. All molecular integrals encountered were accurately calculated using an IBM 709 digital computer. If $r$ represents the distance measured along a line drawn from the helium nucleus to the midpoint of the hydrogen molecule bond, and if $\gamma$ is the angle between this line and the axis of the hydrogen molecule, then the interaction energy was computed at 15° increments of $\gamma$ from $r=3.8\mathrm{} \mathrm{to} 5.2$ a.u. With the ${\mathrm{H}}_2$ bond length held constant at 1.406 a.u. it was found possible to represent the computed interaction energy quite accurately by the function $C{e}^{-\kappa r}[1+\delta P_2(cos\gamma \left)\right]$, where $P_2\mathrm{}\left(x\right)\mathrm{}$ is a Legendre polynomial, $C=17.283\mathrm{}$ double Ry, $\kappa =2.027\mathrm{}$ ${\mathrm{}(\mathrm{a}.\mathrm{u}.\mathrm{})\mathrm{}}^{-1\mathrm{}}$, and $\delta =0.375\mathrm{}$. The spherical average of the computed interaction energy agrees quite closely with an interaction energy obtained from gas diffusion measurements. It is shown that it is impossible to represent the calculated interaction energy by means of a dumbbell-type function, i.e., a function of the form $f\left({\rho }_{\mathrm{ac}}\right)+f\left({\rho }_{\mathrm{bc}}\right)$, where $f\left(x\right)\mathrm{}$ is some suitable chosen function and ${\rho }_{\mathrm{ac}}$ and ${\rho }_{\mathrm{bc}}$ represent the distance from the helium nucleus to the two hydrogen nuclei, respectively. Results are also presented for a slightly elongated ${\mathrm{H}}_2$ bond length of 1.486 a.u.