  We present here the mathematical background of our approach, presented in
Phys. Rev. B 86, 134526 (2012) regarding the gap function and symmetry for the
noncentrosymmetric (NCS) superconductor $Li_2Pt_3B$. As revealed by the
experiment, this NCS superconductor gives rise to line nodes in the
superconducting order parameter, which is responsible for many of its
experimental behaviors. Owing to the enhanced d-character of the relevant bands
that cross the Fermi level,the system gets weakly correlated. The nature and
symmetry of this nodal behavior is explained from a microscopic viewpoint. In
this article starting with an Hubbard model relevant for this NCS system by
considering the effect of the onsite Coulomb repulsion on the pairing potential
perturbatively, we extract the superconducting gap equation. Further analysis
of this equation predicts a $s_\pm$ wave gap function with line nodes as the
most promising candidate in the superconducting state.
