Molecular-Orbital Calculation of the Shape Resonance inN2−

The 2-eV shape resonance in ${\mathrm{N}}_2$-electron scattering is calculated by a self-consistent-field energy-variational procedure. The resonance state corresponds to the attachment of an incident $d$-wave electron to the $1{\pi }_g$ valence orbital of the metastable ${}_{}{}^2{}_{}{}^{}\Pi _g^{}{}_{}{}^{}$ state of ${\mathrm{N}}_2^{-}$. The resonant behavior is due to the tunnelling of the electron through a $\frac{2(2+1)\mathrm{}}{{\mathcal{r}}^2}$ centrifugal barrier and temporary trapping in an attractive field. This tunnelling is reflected in the bimodal behavior of the calculated $1{\pi }_g$ orbitals; the inner portion of the orbital defines the resonance state. The "potential" curve for ${\mathrm{N}}_2^{-}$ is calculated in the Hartree-Fock approximation; a resonance threshold of 2.5 eV is predicted, with $R_e=2.27\mathrm{}$ a.u. and ${\omega }_e\approx 2000$ ${\mathrm{cm}}^{-1\mathrm{}}$. Expected correlation-energy corrections would improve the agreement with experiment. A local potential for electron scattering is generated by inverting the $1{\pi }_g$ orbital, and resonance widths are calculated. The widths vary from 0.13 eV at the equilibrium distance of ${\mathrm{N}}_2^{-}$ to 0.8 eV at the ${\mathrm{N}}_2$ equilibrium distance.