Exact solutions of the rigid rotor in the electric field

We first present a new constraint condition on the confluent Heun function H_C(α, β, γ, δ, η;z) (β, γ ≥ 0, z ∈ [0, 1]) and then illustrate how to solve the rigid rotor in the electric field. We find its exact solutions unsolved previously through solving the Wronskian determinant. The present results compared with those by the perturbation methods are found to have a big difference for a large parameter a. We also present 2D and 3D probability density distributions by choosing different angular momentum quantum numbers l. We observe that the original eigenvalues with degeneracy (2 l + 1) are split into the (l + 1) state with approximate eigenvalues l(l + 1) for small a but large l.