Intensive development of plasma physics, astrophysics, thermo-nuclear fusion, laser physics and especially accelerator physics in the past several decades led to an incresing interest towards physics of atomic collision processes.

Knowledge of certain fundamental features of collisions among elementary atomic particles, such as electrons, nuclei, atoms, ions, moleculs, photons, etc., is of great importance for understanding a large number of phenomena and effects of a macroscopic order in the mentioned areas of modern physics. Among a considerable variaty of atomic collisions, the most important are those inelastic scattering

Here, the total Schroedinger wave-function (stationary or time-dependent ) for the whole projectile-target system is expanded in terms of atomic or molecular basis set functions. The choice of this basis is governed by the concrete problem under study. One usually selects the hydrogenic or Slater-type orbitals (STO), pseudostates, Sturmians, Gaussians, etc. The general functional form of these orbitals is given by the product of spherical harmonics with the radial function R(r) representing the corresponding radial counterpart (Laguerre, Gegenbauer or Hermite polynomials). The present exercise is aimed to give a brief introduction to these two models. For definitness, we choose to illustrate the mentioned techniques in the case of one-electron processes. phenomena, that lead to alteration of the initial structure of the colliding particles.

Such occurrance is due to transitions of one or more electrons light particles, e.g. muons) from one to the other colliding partners. These processes change the basic characteristcs (e.g. identity, charge, mass, etc.) of scattering constituents. This, in turn, alters the very structure and properties (e.g. electrical, optical, transport) of the gaseous enviroment, in which these collision processes customarily take place. Therefore, experimental and theoretical investigation of such atomic scattering processes is very important, not only for numerous applications, but also for understanding the mechanisms of electronic transitions, as well as for illuminating various aspects of the nature of interaction among atomic particles.

Inelastic atomic collisions are studied experimentally through the measurements of various observables (cross sections, rate coefficients, etc) for e.g. excitation, ionization, electron transfer (also know as charge exchange or electron capture), etc. In addition to application in neighbooring branches in physics and in other sciences (chemistry, biology, medicine), better understanding of the underlying physics in these atomic collision processes is also important as a guidance for development of theoretical general purpose methods with a possible usage within a larger context of the time-dependent quantum mechanics.

Among these general techniques, a long-term experiance resulted in emergence of two close-coupling procedures known as the Atomic Orbital (AO) and Molecular Orbital (MO) expansion methods.

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