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A pendular state is a field-dressed rotational quantum state in which a molecule's orientation is concentrated near a preferred direction, resembling small angular librations of a pendulum. It arises when the orienting or aligning field interaction competes strongly with rotational kinetic energy.

A permanent dipole in a static electric field has an interaction proportional to $-\cos \theta$, which distinguishes opposite directions. An anisotropic induced-polarization interaction often depends on $-{\cos }^2\theta$, aligning an axis without necessarily orienting its two ends. The field strength and rotational constant determine whether a weak-field perturbation treatment is adequate.

The quadratic rotational Stark effect is a small-field expansion, rather than a response law valid at every field. For a polar linear rigid rotor with rotational constant $B$ in energy units and permanent electric dipole moment $p$, a static field $\mathcal{E}$ gives the Hamiltonian operatorThe dimensionless ratio $p\mathcal{E}/B$ measures competition between rotation and orientation. Once it is not small, the field mixes many field-free angular momentum eigenstates, so diagonalizing the fixed-magnetic quantum number matrix replaces a truncated second-order correction. Increasing the basis cutoff tests numerical convergence.

For a differentiable nondegenerate eigenvalue $E_{\alpha }(\mathcal{E})$, the Hellmann–Feynman theorem gives the mean dipole component $-d{E}_{\alpha }/d\mathcal{E}=p⟨\cos \theta ⟩$. Its magnitude cannot exceed $p$. Continuing the weak-field linear induced-dipole formula indefinitely would violate this bound. Strongly oriented low-lying pendular states instead describe restricted angular motion about the field direction.^[ref]