The most of the experimental measurable quantities in many-body electron system

such as the optical absorption

luminescent spectra and exciton effect

are related to the excitation states description. The density functional theory with local density approximation (DFT-LDA) as a ground state theory

i.e.

the solutions based on the Kohn-Sham equation is a powerful tool for studying the ground state properties in many-particle systems. However

the first principles computation of excited states is more complexity than ground-state calculations. A key problem is that the exchange-correlation interaction in excited states is differ from the ground-states. In recent years

nevertheless

several electronic-excitation theories have been developed. The most important theoretical and computational method include the many-body perturbation theory which is based on the quasi-particle concept and the Green function equation

the time-dependent DFT and the Bethe-Salpeter equation for describing the electron-hole interaction. Among them the central ingredient is an electron’s self-energy operator Σ that describes the exchange correlation effect beyond Hartree approximation. The implementations of the above theories are unavoidable to introduce some of approximations. A good approximation for Σ is the GW approach by Hedin. It is shown from computation simulations for many real condensed matter systems that the GW approach is a successful method for electronic-excitation problems. Another effective method for excited-states is so-called screen-exchange local-density approximation (sX-LDA). It is a combination of LDA and Hartree-Fock (HF)theory

in which the intrinsic local screen-exchange interaction has been replaced by a non-local one and a generalized Kohn-Sham equation (GKS) is given. Based on a PWscf package

the sX-LDA-PWscf method has been tested and indicates that an available approach for treating band-gap problem in semiconductors. In this paper

the many-body theories for the excitation process

its development and signification are reviewed. The relation and differences consist in different theoretical methods are discussed. On the based of above discussions

the applications of many-body theory to the band-band transition (band-gap underestimation problem) and exciton effect in semiconductors are presented.