Auto-Ionizing States in Helium

The validity of the method of diagonalizing a truncated, finite-dimensional submatrix of the Hamiltonian for the calculation of some properties of some autoionizing states is demonstrated, using projection operators. The truncation procedure is shown to be an example of the Feshbach formalism. It differs from other approaches employing the Feshbach formalism mainly in point of view and in the particular choice of the projection operators. Detailed calculations using the diagonalization method are carried out for the lowest autoionizing states of helium with symmetry ${}_{}{}^1{}_{}{}^{}S_{}^e{}_{}{}^{}$, ${}_{}{}^1{}_{}{}^{}P_{}^o{}_{}{}^{}$, or ${}_{}{}^3{}_{}{}^{}P_{}^o{}_{}{}^{}$. The basis used for the singlet $S$ states is of dimension 42, including all properly antisymmetrized hydrogenic functions with $Z=2\mathrm{}$ for all angular configurations up to $g-g$. For the $P$ states all angular configurations up to $f-g$ are included, resulting in a 55-dimensional basis set. The eigenvalues are compared in detail with several different calculations of resonances in $e$-${\mathrm{He}}^{+}$ elastic scattering, and with the observation of these states by optical absorption in He and by inelastic scattering of electrons, protons and ${\mathrm{H}}_2^{+}$ on He. The eigenvectors are considered to represent the doubly excited states after the excitation has occurred and before they decay, so that they can be used to calculate absorption cross sections and lifetimes. The detailed characteristics of the eigenvectors corroborate the classification scheme of Cooper, Fano, and Prats as far as they have gone. Good agreement is also found between the calculated level structure and the experimental results.