We investigate a one-dimensional asymmetric square double well potential with an energy barrier to separate two uneven quantum wells. The model study is motivated by gene mutation-induced cancer diseases, chemical kinetics, defects in materials, etc. The solutions of Schrödinger equation show that between two adjacent quantum states, the quantum particle tends to switch the quantum well for eigenstates between the potential of the higher energy quantum well and the energy barrier. In some cases, switching quantum well is disrupted. From the asymptotic behavior of limiting cases, we show that the energy barrier merely perturbs the energy states of both quantum wells. The overall energy levels follow the order of the quantum states from both quantum wells in the absence of tunneling, leading to switching or no switching of quantum wells between two adjacent states. The asymmetry in the potential induces disorder and causes localization similar to Anderson localization. Our work characterizes the preferential positioning of a quantum particle in the two uneven quantum wells. At higher quantum states but below the barrier, intramolecular tunneling splitting of two adjacent quantum states is observed for some cases. While the barrier height approaches infinity, the wavefunctions of these two adjacent states are preferentially located in different quantum wells. As the barrier height is reduced, these wavefunctions display mixing features from both quantum states in contrast to the case when they are separated by an infinite barrier without tunneling. Such mixing states are expected to facilitate transformation from reactants to products via photoexcitation, for example. Here, we develop a two-state model to elucidate intramolecular tunneling splitting. The results provide insights into some non-classical pathways for a chemical process under a one-dimension barrier.