Hyperspherical Method

The hyperspherical adiabatic approach is used to obtain the highly excited series 1sns 1 S e and 1s(n + 1)p 1 P o of the helium atom. The introduction of appropriate asymptotic conditions at large values of the hyperspherical radius results in a stable algorithm that allows the calculation of the full atomic spectrum with precision of a few parts per million. Comparison with the variational calculations available in the literature shows that the accuracy of the results improves with increasing principal quantum number. We present the energies up to n = 31 which is the typical value used in multiphoton excitation experiments.

We present a nonadiabatic calculation, within the hyperspherical adiabatic approach, for the ground state energy of the alkali-metal negative ions. An application to the sodium negative ion (Na Ϫ ) is considered. This system is treated as a two-electron problem in which a model potential is used for the interaction between the Na ϩ core and the valence electrons. Potential curves and nonadiabatic couplings are obtained by a direct numerical calculation, as well as the channel functions. An analysis of convergence is made and comparisons of the electron affinity with results of prior work of other authors are given.

Using the hyperspherical adiabatic approach in a coupled-channel calculation, we present precise binding energies of excitons trapped by impurity donors in semiconductors within the effective-mass approximation. Energies for such three-body systems are presented as a function of the relative electron-hole mass in the range 1р1/р6, where the Born-Oppenheimer approach is not efficiently applicable. The hyperspherical approach leads to precise energies using the intuitive picture of potential curves and nonadiabatic couplings in an ab initio procedure. We also present an estimation for a critical value of (crit) for which no bound state can be found. Comparisons are given with results of prior work by other authors.

We present a nonadiabatic calculation, within the hyperspherical adiabatic approach, for the ground state energy of the alkali-metal negative ions. An application to the sodium negative ion (Na Ϫ ) is considered. This system is treated as a two-electron problem in which a model potential is used for the interaction between the Na ϩ core and the valence electrons. Potential curves and nonadiabatic couplings are obtained by a direct numerical calculation, as well as the channel functions. An analysis of convergence is made and comparisons of the electron affinity with results of prior work of other authors are given.

Energies and wavefunctions are calculated for the bound states of the helium atom in the hyperspherical adiabatic approach by the full inclusion of nonadiabatic couplings. We show that the use of appropriate asymptotic radial boundary conditions not only allows the efficient calculation of energies accurate up to a few ppm for the ground state but also gives increasingly precise results for high-lying excited states with a unique set of equations. The accuracy of the wavefunctions is demonstrated by the calculation of oscillator strengths in the length form for transitions between states n 1 S e and (n + 1) 1 P o up to n = 29, in agreement with variational calculations.

We present angular basis functions for the Schrödinger equation of two-electron systems in hyperspherical coordinates. By using the hyperspherical adiabatic approach, the wave functions of two-electron systems are expanded in analytical functions, which generalizes the Jacobi polynomials. We show that these functions, obtained by selecting the diagonal terms of the angular equation, allow efficient diagonalization of the Hamiltonian for all values of the hyperspherical radius. The method is applied to the determination of the 1 S e energy levels of the Li ϩ and we show that the precision can be improved in a systematic and controllable way.