Exact solutions to Stark effect of rigid symmetric-top molecules

In this work a new scheme is proposed to accurately calculate the rotational energy level of the rigid symmetric-top molecule subjected to the external electric field, and also to obtain the corresponding analytical wave functions. For this purpose, first we use the different forms of function transformation and variable substitution to transform the differential equation of the polar angle theta into a confluent Heun differential equation, and then we use the characteristics of the confluent Heun differential equation and the confluent Heun function to find two linearly dependent solutions of the same eigenstates, which are used to construct the Wronskian determinant to obtain the exact energy spectrum equation. Finally, with the aid of the Maple software, we calculate the eigenvalues for different quantum states, and then substitute the obtained eigenvalues into the unnormalized eigenfunction to obtain the analytical normalized eigenfunction expressed by the confluent Heun function. These results are conducive to the in-depth study of the Stark effect of symmetric-top molecules.