** Ab initio methods**
These methods are used to describe ion-atom and ion-molecule collisions. The electronic degrees of freedom are
treated quantaly, while the relative motion between target and projectile could be described classically (eikonal approximation) or quantally (common reaction coordinate formalism).
The vibrational degrees of freedom of the target can be described using several approximations, including Frank-Condon, vibrational sudden, and close-coupling.

The electronic wave function is expanded in terms of the clumped-nuclei electronic Hamiltonian, and they are calculated using quantum-chemistry packages, usually MELD or MOLPRO.

**Effective hamiltonians methods**
The study of complex systems (many particles) is carried out using effective hamiltonians methods in which the electron interaction with a core ion is
represented by model potentials. In this treatment, many-electron cross sections are obtained by using different statistical interpretations (independent particle model, independent event model, etc.)

**Classical mechanics methods**
These methods start with the statistical description of the initial state of the system and perform the classical evolution according to the corresponding Hamilton equations. The projectile-target collisions are described withing the eikonal approximation.
To represent complex systems, one can use model potentials. Many-electron cross sections are obtained using the statistical interpretations. These methods
provide very good description of the ionization mechanism.