= Linear rigid rotor
{title2=$E_J=\hbar^2J(J+1)/(2I)$}
For a <diatomic molecule>, let $m_1,m_2$ be the two <masses> and $r$ their fixed separation. The distances from the <centre of mass> are $r_1=m_2r/(m_1+m_2)$ and $r_2=m_1r/(m_1+m_2)$. A perpendicular axis through the <centre of mass> therefore has <moment of inertia>
$$
I=m_1r_1^2+m_2r_2^2=\mu r^2,\qquad \mu=\frac{m_1m_2}{m_1+m_2}.
$$
The <reduced mass> $\mu$ here is not the <electric dipole moment>, for which we use $\mathbf d$. For a longer <linear molecule>, replace $\mu r^2$ by $\sum_i m_i\rho_i^2$, where $\rho_i$ is each distance along the axis from the <centre of mass>. In the point-mass idealization, the axial <moment of inertia> is zero; turning about that axis changes no nuclear positions, so no independent axial rotational degree of freedom is included.
For the field-free <linear rigid rotor>, assume a fixed internal <quantum state> with no electronic <angular momentum> or electronic <spin>, and omit couplings to nuclear <spin>. The rotational <Hamiltonian> is $\widehat H_{\rm rot}=\widehat{\mathbf L}^{\,2}/(2I)$. Its orientation <wavefunctions> are <spherical harmonics> $Y_{JM}$, and its <angular momentum eigenstates> satisfy
$$
\widehat{\mathbf L}^{\,2}Y_{JM}=\hbar^2J(J+1)Y_{JM},\qquad
\widehat L_zY_{JM}=\hbar M Y_{JM},\qquad
E_J=\frac{\hbar^2}{2I}J(J+1)=hB J(J+1),
$$
where $J=0,1,2,\ldots$ and $M=-J,-J+1,\ldots,J$. Each $E_J$ belongs to $2J+1$ <angular momentum eigenstates> distinguished by the <magnetic quantum number> $M$, an instance of <degenerate energy levels>. The <Planck constant> is $h=2\pi\hbar$; $B$ is the frequency-valued <rotational constant>. Notice that the <energy eigenvalues> scale as $J(J+1)$, not $J^2$.
\Image[/linear-rigid-rotor-levels-and-transitions.png]
{title=<Linear rigid rotor>: increasing gaps between <energy eigenvalues> and equally spaced transition <frequencies>}
{id=image-linear-rigid-rotor-levels-and-transitions}
{description=Original dimensionless model: $E_J/(hB)=J(J+1)$. Upward arrows show <photon absorption> with $\Delta J=1$; the sticks show positions only, not measured or predicted intensities. The $2J+1$ states at each <energy eigenvalue> are represented by one horizontal line.}
Subtracting consecutive <energy eigenvalues> gives
$$
\nu_{J\to J+1}=\frac{E_{J+1}-E_J}{h}=2B(J+1).
$$
Thus increasing level gaps produce a regularly spaced series of lines at $2B,4B,6B,\ldots$, with spacing $2B$. Allowed positions do not guarantee that every line is visible: <electric-dipole selection rules>, occupation of the initial <quantum state>, and instrument response also matter.
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