Electric fields cause mixing of opposite-parity atomic states, and consequent energy shifts. If all relevant atomic states are included in the system, the mixing and splitting can be found by finding the eigenvalues and eigenvectors of the Hamiltonian.
Here we analyze a J=2 state mixed with a J=1 state by a z-directed electric field.
The eigenvectors are the eigenstates in the presence of the electric field, written as a linear combination of the original sublevels. However, they are not listed in the order corresponding to the original sublevels. Here we reorder the eigenvalues so that they match the unperturbed quantities when the electric field approaches zero.