Density functional (B3LPY) method has been utilized to optimize the possible structures of PuN, PuO, NO and PuNO molecules using the contracted valence basis set (LANL2 DZ) for Pu atom, and the AUG-cc-pVTZ basis set for N and O atoms. It is shown that the ground state of the PuNO molecules has Cv (Pu-N-O) symmetry and the ground electronic state is 6-. The equilibrium nuclear distances for Pu-N and N-O bonds in the PuNO molecules are RPuN=0.22951 nm and RNO=0.12257 nm, and the dissociation energy is De=8.10537 eV. Furthermore, the other two metastable states of the PuNO molecules are also obtained, and the electronic states of the two configurations are 6- and A with Cv (Pu-O-N) and Cs (O-Pu-N) symmetry, respectively. Then the Murrell-Sorbie potential energy functions of the PuN, PuO and NO molecules have been simulated and the analytical potential energy function of the PuNO molecules has been derived using the many-body expansion theory. The contours of the potential energy functions reproduce exactly the most stable equilibrium structures, the two metastable state structures as well as the dissociation energy of the PuNO molecules. The molecular static reaction pathway, based on the potential energy function, is also discussed.