Let the unperturbed atom or molecule be in a g-fold degenerate state with orthonormal zeroth-order state functions . (Non-degeneracy is the special case g = 1). According to perturbation theory the first-order energies are the eigenvalues of the g × g  matrix with general element 
If g = 1 (as is often the case for electronic states of molecules)  the first-order energy becomes proportional to the expectation (average) value of the dipole operator ,
Because the electric dipole moment is a vector (tensor of the first rank), the diagonal elements of the perturbation matrix V vanish between states with a certain parity. Atoms and molecules possessing inversion symmetry do not have a (permanent) dipole moment and hence do not show a linear Stark effect.
In order to obtain a non-zero matrix V for systems with an inversion center it is necessary that some of the unperturbed functions  have opposite parity (obtain plus and minus under inversion), because only functions of opposite parity give non-vanishing matrix elements. Degenerate zeroth-order states of opposite parity occur for excited hydrogen-like (one-electron) atoms or Rydberg states. Neglecting fine-structure effects, such a state with the principal quantum number n is n-fold degenerate and
where  is the azimuthal (angular momentum) quantum number. For instance, the excited n = 4 state contains the following  states,
The one-electron states with even  are even under parity, while those with odd  are odd under parity. Hence hydrogen-like atoms with n>1 show first-order Stark effect.
The first-order Stark effect occurs in rotational transitions of symmetric top molecules (but not for linear and asymmetric molecules). In first approximation a molecule may be seen as a rigid rotor. A symmetric top rigid rotor has the unperturbed eigenstates
with 2(2J+1)-fold degenerate energy for |K| > 0 and (2J+1)-fold degenerate energy for K=0.
Here D is an element of the Wigner D-matrix. The first-order perturbation matrix  on basis of the unperturbed rigid rotor function is non-zero and can be diagonalized. This gives shifts and splittings
in  the rotational spectrum. Quantitative analysis of these Stark shift yields the permanent electric dipole moment of the symmetric top molecule.