The derivatives to any order of the confluent hypergeometric (Kummer) function F=F11(a,b,z) with respect to the parameter a or b are investigated and expressed in terms of generalizations of multivariable Kampé de Fériet functions. Various properties (reduction formulas, recurrence relations, particular cases, and series and integral representations) of the defined hypergeometric functions are given. Finally, an application to the two-body Coulomb problem is presented: the derivatives of F with respect to a are used to write the scattering wave function as a power series of the Sommerfeld parameter.

Simple and accurate wavefunctions for the He atom and He-like isoelectronic ions are presented. These functions-the product of hydrogenic one-electron solutions and a fully correlated part-satisfy all the coalescence cusp conditions at the Coulomb singularities. Functions with different numbers of parameters and different degrees of accuracy are discussed. Simple analytic expressions for the wavefunction and the energy, valid for a wide range of nuclear charges, are presented. The wavefunctions are tested, in the case of helium, through the calculations of various cross sections which probe different regions of the configuration space, mostly those close to the two-particle coalescence points.

In this letter, a set of ground state wavefunctions for the He atom is given. The functions are constructed in terms of exponential and power series as similar as possible to the Hylleraas functions of Chandrasekhar and Herzberg (1955 Phys. Rev. 98 1050). The accuracy of the calculated energies is found to be about 10 −4 au and all the cusp conditions at the Coulomb singularities are satisfied. The nine-parameter functions proposed here are found to have better local energy than those given by the 6 and 14 terms Hylleraas functions of Chandrasekhar. The mean value of various functions evaluated with the different proposals shows their good quality. These properties highly qualify the function to be used as an alternative to the Chandrasekhar functions in collisional problems. The whole set of functions given here can be considered as an alternative to the proposals of Chandrasekhar (1955

The two-body Coulomb Schrödinger equation with different types of nonhomogeneities are studied. The particular solution of these nonhomogeneous equations is expressed in closed form in terms of a two-variable hypergeometric function. A particular representation of the latter allows one to study efficiently the solution in the asymptotic limit of large values of the coordinate and hence the associated physics. Simple sources are first considered, and a complete analysis of scattering and bound states is performed. The solutions corresponding to more general (arbitrary) sources are then provided and written in terms of more general hypergeometric functions.

No abstract