Electronic structures and molecular dynamics of gadolinium-doped formamidinium lead iodine (FAPbI₃) perovskite crystal were studied for improving the photovoltaic performance with stability. The band structure, partial density of state, electron density distribution, enthalpy, kinetic energy, molecular dynamics were expected by first-principles calculation. The narrow band dispersion, delocalization of 5d, 4f orbital of gadolinium, 6p orbital of lead ion, 5p orbital of iodine ion near valence and conduction band states promoted the charge transfer between 5d orbital of gadolinium ion and 6p orbital of lead ion, expecting increase of carrier mobility related with short circuit current density as the photovoltaic performance. Addition of gadolinium ion into the crystal caused the slight distortion in the crystal field, stabilizing the crystal while suppression of the decomposition with desorption and diffusion of the ions in the crystal. The stability of the performance would be explained by the thermodynamic characteristics The enthalpy and kinetic energy based on the molecular dynamics indicates stabilization of the gadolinium-doped perovskite crystal with slight distortion of coordination structure, as compared with that of the FAPbI₃ perovskite crystal. Addition of gadolinium ion into the perovskite crystal supported the stability of the photovoltaic performance. The calculation prediction expect that the gadolinium-doped FAPbI₃ perovskite crystal have great potential to apply for the industry product of the photovoltaic devices with the photovoltaic performance.