Variational Calculations for Quartet States of Three-Electron Atomic Systems

The conventional Rayleigh-Ritz variational method in which one uses pure Slater-type orbitals and correlated factors ${r_{\mu \nu }}^2$ in the basis functions has been applied to obtain the eigenvalues of the five lowest lying states with symmetries ${}_{}{}^4{}_{}{}^{}P_{}^0{}_{}{}^{}$, ${}_{}{}^4{}_{}{}^{}P_{}^e{}_{}{}^{}$, and ${}_{}{}^4{}_{}{}^{}S_{}^e{}_{}{}^{}$ for three-electron atomic systems. To find the absolute minimum which is attainable for each eigenvalue, the nonlinear parameters (exponential parameters) have been varied freely in submatrices up to order 30 with 20 noncorrelated and 10 correlated basis functions. This variation has been carried through separately to find the five lowest eigenvalues of each symmetry in Li and for only the lowest one in ${\mathrm{He}}^{-}$. For the other members of the isoelectronic sequence up to $Z=10\mathrm{}$, the absolute minima of the three lowest lying eigenvalues are found approximately by using the correlated subset of order 30 with common fixed exponential parameters for each symmetry and by freely varying the scale parameter. The lowest ${}_{}{}^4{}_{}{}^{}P_{}^0{}_{}{}^{}$ state is found to be bound in ${\mathrm{He}}^{-}$ with a binding energy ≥0.033 eV. No sign of binding is indicated for the lowest ${}_{}{}^4{}_{}{}^{}S_{}^e{}_{}{}^{}$ state, but the lowest ${}_{}{}^4{}_{}{}^{}P_{}^e{}_{}{}^{}$ state is also found to be bound by ≥0.20 eV. The results for Li indicate as certain that the transitions ${}_{}{}^4{}_{}{}^{}S_{}^e{}_{}{}^{}\mathrm{}\left(1\right)\mathrm{}-{}_{}{}^4{}_{}{}^{}P_{}^0{}_{}{}^{}\mathrm{}\left(1\right)\mathrm{}$ and ${}_{}{}^4{}_{}{}^{}P_{}^e{}_{}{}^{}\mathrm{}\left(1\right)\mathrm{}-{}_{}{}^4{}_{}{}^{}P_{}^0{}_{}{}^{}\mathrm{}\left(1\right)\mathrm{}$ are responsible for the two observed multiplets present at 2934 and 3714 Å, respectively in the optical spectrum. These lines cannot be classified in the normal singly excited spectrum of the atom or ion. The results for Li are compared in detail with those obtained by recent electron-impact experiments and by other theoretical calculations.