The conventional Rayleigh-Ritz variational method in which one uses pure Slater-type orbitals and correlated factors r""* in the basis functions has been applied to obtain the eigenvalues of the five lowest lying states with symmetries 4 P°, 4 j? e , and 4 5 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 He"~. For the other members of the isoelectronic sequence up to Z=10, 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° state is found to be bound in He~ with a binding energy > 0.033 eV. No sign of binding is indicated for the lowest 4 S« state, but the lowest i 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«(1)-4 P°(1) and 4 P«(1)-4 P°(1) are responsible for the two observed multiplets present at 2934 and 3714 A, 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.
The present investigation should provide some insight into the convergence properties of some particular upper roots of the full Hamiltonian. In this connection detailed calculations using the scaling-variation orthogonalization procedure with different subsets of the basis up to order 54 have been carried through both for He and H−.
Subsets including basic functions up to g—g are used, and a complete mapping of the ten lowest roots of the energy matrix by a continuous variation of the scale factor has been carried out within a definite range. The investigation clearly demonstrates the stabilizing ability of particular upper roots throughout the extension of the basis if the orthogonalization is properly achieved. These roots in the second-step solution may be associated with certain autoionizing states (quasi doubly excited, discrete states). The eigenvalues, obtained by optimizing the scale factor for these roots, are compared with those obtained for resonance positions in He+ and H elastic-scattering calculations by using other approximate methods, and by experiments.
The spherically symmetric component of the ground-state wave functions of the two-electron series (H~, He, Li + , Be 2+ , C 4+ , 0 6+ , and Ne 8+ ) has been investigated by using the orthonormal complete set of (2q-\-2) -order associated Laguerre functions as radial orbitals in the method of superposition of configurations. The results for the total energies and the corresponding expansion coefficients demonstrate the excellent convergence properties of these functions. They are also useful practically since only one single-orbital exponent is used. Applied to excited quantum states, they indicate a more slowly convergent process.
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