Repulsive Interaction between Two Ground-State Helium Atoms

The repulsion energy between two ground-state helium atoms (${{}_{}{}^1{}_{}{}^{}\Sigma _g^{}{}_{}{}^{}}^{+}$ state of ${\mathrm{He}}_2$) has been investigated in the single-configuration MO (molecular orbital) approximation and then further refined to include the effects of electron correlation by the inclusion of super-position of configurations. In the former case, the wave function is expressed as a single antisymmetrized spin-orbital product (ASOP) of the form $\left|{\sigma }_{g}1s\right(1\left)\mathrm{}\alpha \mathrm{}\right(1\left)\mathrm{}{\sigma }_{g}1s\right(2\left)\mathrm{}\beta \mathrm{}\right(2\left)\mathrm{}{\sigma }_{u}1s^{\prime }\mathrm{}\right(3\left)\mathrm{}\alpha \mathrm{}\right(3\left)\mathrm{}{\sigma }_{u}1s^{\prime }\mathrm{}\right(4\left)\mathrm{}\beta \mathrm{}\right(4\left)\right|$, where the MO's ${\sigma }_{g}1s$ and ${\sigma }_{u}1s^{\prime }$ are approximated as the sum and difference of Slater-type orbitals (STO's) $1s$ and $1s^{\prime }$, respectively, and the energy minimized with respect to the orbital exponents. In addition to the repulsion energy at small internuclear distances the slight polarization of the ${\sigma }_{u}1s^{\prime }$ MO effected by the $\zeta$ variation, permitted the single ASOP wave function to give indications of the van der Waals energy minimum at large distances. The super-position of configurations treatment employed a linear combination of configurations constructed out of a $1s$, $1s^{\prime }$, $2p\sigma$, $2p\pi$ STO basis set, and at five internuclear distances $R$ between 0.5 and 2.0 A, the STO orbital exponents were varied to minimize the energy. Various wave functions including from 10 to 64 electron configurations were tried, which were so chosen that the wave functions would go properly as $R\to 0$ into the ${}_{}{}^1{}_{}{}^{}S$ ground-state function of beryllium and as $R\to \infty$ go into a product of two ${}_{}{}^1{}_{}{}^{}S$ helium atom functions. The computed repulsion energies in the region $0.5\mathrm{A}<~R<~1.0\mathrm{A}$ are 2.8 to 1.2 times higher than values deduced from experimental scattering data obtained several years ago, and since the difference between the lowest computed (64-configuration) energy and the estimated exact energy is much smaller than this, a reinvestigation of the scattering analysis is emphasized.