Molecular-Orbital Calculation of the Shape Resonance in N 2 −

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