Landau–Zener transitions for eigenvalue avoided crossings in the adiabatic and Born–Oppenheimer approximations

In the Born–Oppenheimer approximation context, we study the propagation of Gaussian wave packets through the simplest type of eigenvalue avoided crossings of an electronic Hamiltonian 𝒞⁴ in the nuclear position variable. It yields a two‐parameter problem: the mass ratio ε⁴ between electrons and nuclei and the minimum gap δ between the two eigenvalues. We prove that, up to first order, the Landau–Zener formula correctly predicts the transition probability from a level to another when the wave packet propagates through the avoided crossing in the two different regimes: δ being asymptotically either smaller or greater than ε when both go to 0.