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Abstract Abstract The interaction of Nucleon-Nucleon (NN) has certain physical characteristics, which indicated by nucleon, meson and quark degrees of freedom. The main purpose of this work is calculating the ground state energies of 21 H and 42 He nuclei through the two-body system with the exchange of two (; !), three (; ; !), and four (; ; !; ) mesons that mediated between two nucleons, each one of these calculations is considered beside the quark-quark interaction in the constructed hybrid model. The calculations are divided into two parts: The rst part deals with the meson exchange between two nucleons through the approach of one-boson exchange potential (OBEP) based on the quasirelativistic decoupled Dirac equation and self-consistent Hartree-Fock formulation, where each nucleon acts as a source of pseudoscalar , scalar , and vector elds !; . The OBEP is analytically derived with two static functions of meson, the single particle energy dependent (SPED) and generalized Yukawa (GY) functions. The obtained theoretical results are compared with other theoretical models and their corresponding experimental data, it is found that the SPED function gives a satised agreement more than the GY function in case of the considered nuclei. The contribution of -meson is more clear than -meson since, the considered nuclei are light and the repulsive eect of -meson demands more massive nuclei. The ground state energies are well reproduced by using all possible degrees of freedom according to our interest |