The paper discusses the non-adiabatic crossing of energy levels in a molecular system. When a parameter (such as internuclear distance R) is varied slowly, the system remains in its initial state. However, if the parameter is varied rapidly, the system may transition between states, leading to a non-adiabatic transition. The transition probability is calculated under certain conditions, such as the smallness of the coupling between states and the linear behavior of energy differences during the transition. The analysis uses linear combinations of adiabatic eigenfunctions to simplify the wave equation. This leads to a set of differential equations for the coefficients of the wave function. Solving these equations involves transforming them into a form resembling the Weber equation, which is then solved using special functions. The solution shows that the transition probability depends on the coupling strength and the rate of change of the energy difference. The probability is found to be exponentially small, with the exponent depending on the square of the coupling strength divided by the rate of change of the energy difference. The results are applied to two cases: one where the energy difference is constant and the coupling varies, and another where the energy difference varies linearly and the coupling is constant. In both cases, the transition probability is found to depend on the relative velocity of the atoms. The paper concludes that the transition probability can be rigorously calculated under certain conditions, and the results are consistent with previous theoretical predictions. The findings have implications for understanding non-adiabatic transitions in molecular systems.The paper discusses the non-adiabatic crossing of energy levels in a molecular system. When a parameter (such as internuclear distance R) is varied slowly, the system remains in its initial state. However, if the parameter is varied rapidly, the system may transition between states, leading to a non-adiabatic transition. The transition probability is calculated under certain conditions, such as the smallness of the coupling between states and the linear behavior of energy differences during the transition. The analysis uses linear combinations of adiabatic eigenfunctions to simplify the wave equation. This leads to a set of differential equations for the coefficients of the wave function. Solving these equations involves transforming them into a form resembling the Weber equation, which is then solved using special functions. The solution shows that the transition probability depends on the coupling strength and the rate of change of the energy difference. The probability is found to be exponentially small, with the exponent depending on the square of the coupling strength divided by the rate of change of the energy difference. The results are applied to two cases: one where the energy difference is constant and the coupling varies, and another where the energy difference varies linearly and the coupling is constant. In both cases, the transition probability is found to depend on the relative velocity of the atoms. The paper concludes that the transition probability can be rigorously calculated under certain conditions, and the results are consistent with previous theoretical predictions. The findings have implications for understanding non-adiabatic transitions in molecular systems.