= Molecular structure from rotational constants
Recovering <molecular geometry> from <rotational constants> is an <inverse problem>. In the synthetic assignment, $B=5\times10^9\,\mathrm{Hz}$ gives $I=h/(8\pi^2B)\simeq1.6784\times10^{-45}\,\mathrm{kg\,m^2}$. For a <diatomic molecule> with known <masses>, <rotational constant> explains how to infer an effective <bond length>. For a longer <linear molecule>, one $I$ cannot determine every <bond length>: several different arrangements can have the same <moment of inertia>.
For a fixed three-atom linear model with positions $0,a,a+b$ and positive <masses> $m_1,m_2,m_3$, subtracting the <centre of mass> gives
$$
I=\frac{m_1m_2a^2+m_2m_3b^2+m_1m_3(a+b)^2}{m_1+m_2+m_3}.
$$
This is one equation for two unknown <bond lengths>. Even fixing the order of the <chemical atoms> leaves a family of solutions. Additional <isotope> substitutions change known <masses> and supply independent equations, provided the effective <molecular geometry> can be treated as shared. For nonlinear <molecules>, three principal <moments of inertia> give frequency-valued constants $A=h/(8\pi^2I_a)$, $B=h/(8\pi^2I_b)$, and $C=h/(8\pi^2I_c)$; a single evenly spaced comb is generally insufficient.
Vibrational averaging, unresolved <hyperfine structure>, internal motion, and <centrifugal distortion> limit structural inference. A ground-vibrational fitted <rotational constant> is not automatically the equilibrium constant. Ordinary field-free <rotational spectroscopy> also gives identical spectra for opposite <enantiomers> under the usual parity-symmetric molecular <Hamiltonian>; assigning composition and conformation does not by itself determine handedness. Molecular identification from a rich fingerprint can therefore succeed even when a unique full <molecular geometry> cannot be reconstructed.
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