On the calculation of radial wave functions corresponding to energies in the continuum part of the helium spectrum

Summary  For the purpose of recalculating the helium primary specific ionization values in an attempt to bridge the gap existing between theory and experiment in this field, this paper deals with the derivation of new wave functions corresponding to energies from the continuum part of the He spectrum, which are more accurate than those resulting from the commonly adopted approximations. Sect.1 is devoted to an appropriate series expansion of the wave function for the two helium planetary electrons and to the corresponding transformation of the Schrödinger equation into an infinite system of simultaneous differential equations which the expansion entails. Further, it is shown how a suitable truncation leads to the basic, one-electron, Schrödinger-type differential equation for the approximate wave function describing the positive energy electron in the electric field of the helium nucleus and the bound electron. The physical reasons underlying the truncation procedure are discussed in detail. In Sect.2, it is shown that the correction due to the motion of the nucleus vanishes identically to within the present lowest-order approximation scheme. Sect.3 deals with the radial part of the desired wave function, its reduction to atomic units, the ordinary second order differential equation which it satisfies, its Maclaurin series expansion and its general characteristics, especially its asymptotic behavior. Sect.4 describes the numerical integration of the mentioned radial wave equation for 27 pairs of energy and orbital angular momentum quantum number values, using an IBM 650 ordinator. All wave functions have been obtained in an interval of at least 8 atomic units. Finally, the normalization of the resulting curves to unit amplitude at infinity is examined in Sect.5. Two very elegant methods are developed and studied in detail. In the first procedure, the amplitude at infinity and the phase angle are rigorously given by certain integrals which are directly calculable. In the second and actually adopted method, the basic idea consists in introducing the Madelung transformation, leading to a non-linear differential equation for the local amplitude of the radial wave function. Calculating an asymptotic series expansion for this local amplitude, it becomes possible, in principle, to compute the normalized wave function starting at infinity down to a certain minimum abscissa dictated by the desired limit of precision. The corresponding curve to be normalized could then be fitted to the normalized one. However, it is shown that the entire procedure is enormously simplified by carrying out the matching operation at a zero of the oscillating radial wave function.