= Centrifugal distortion
{title2=$E_J/h=BJ(J+1)-D[J(J+1)]^2+\cdots$}
Real <chemical bonds> can stretch as <molecular rotation> increases. The resulting larger <moment of inertia> lowers the <energy eigenvalues> relative to a <rigid rotor>. Within one vibrational <quantum state>, a leading effective correction is
$$
\frac{E_J}{h}=B_vJ(J+1)-D_v[J(J+1)]^2+\cdots,\qquad
\nu_{J\to J+1}=2B_v(J+1)-4D_v(J+1)^3+\cdots.
$$
Both $B_v$ and $D_v$ are in <frequency> units here; $v$ labels the vibrational <quantum state>. For positive $D_v$, the adjacent-line spacing becomes
$$
\nu_{J+1\to J+2}-\nu_{J\to J+1}=2B_v-4D_v(3J^2+9J+7),
$$
so the departures grow with $J$. <Molecular vibration> also changes the effective <rotational constant>: even the lowest vibrational <quantum state> does not have motionless <atomic nuclei>.<Diatomic Spectral Database by National Institute of Standards and Technology/1Σ-Ground State Molecules>[§2a, equations (7)–(8): distortion and vibration-dependent constants]{ref}
The quartic correction is a limited expansion, not a model to extrapolate to arbitrary $J$. It is small relative to the rigid contribution only while $(D_v/B_v)J(J+1)\ll1$; eventually higher terms, changed internal motion, or <chemical bond> breaking matter. In the synthetic plot, the largest upper-state label is $J=5$, giving a stipulated correction ratio of $0.03$. No universal maximum valid $J$ follows without a particular <molecule> and precision target.
The <quantum numbers> also have limits of meaning. Here $J$ describes spin-free rotation. In a <molecule> with electronic <spin>, rotation alone is commonly labelled $N$, while $J$ includes electronic <angular momentum>; including nuclear <spin> introduces additional total-<angular momentum> labels and resolved splittings. An applied <electric field> can mix different field-free $J$ states through the <Stark effect>. For identical <atomic nuclei>, the <Pauli exclusion principle> can restrict combinations of rotational and nuclear-<spin> states. The simple $2J+1$ degeneracy and every-integer-$J$ ladder therefore describe the stated idealization, not all observed sublevels.<Diatomic Spectral Database by National Institute of Standards and Technology/1Σ-Ground State Molecules>[§2a: hyperfine discussion; §2b: distinct definitions of J, N and F]{ref}
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