We investigate the quantum dynamics of two interacting bosonic atoms confined in a one-dimensional anharmonic trap. The tunneling rate, an experimentally measurable parameter of the system, is calculated as a function of the effective coupling interatomic constant g from the ground (n=N=0) and first excited atomic states in the trap with respect to relative (n=2,N=0) and center-of-mass (n=0,N=2) atomic motion. This allows us to investigate the initial population and pair correlation, as well as the effective coupling constant g, of the system by comparing the calculated tunneling rate with the experimental one. We observe that the only possible tunneling scenario is a sequential particle tunneling in the cases we consider. We also analyze a rearrangement (0,2)(2,0) of the spectrum in the limit g→±0 of noninteracting atoms. © 2017 American Physical Society.