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The rotational state populations start at $N_0/N_0=1$. Their $2J+1$ statistical weight factor initially favors higher $J$, while the Boltzmann factor eventually suppresses them, producing one maximum followed by a tail tending to zero. The physically allowed values occur at integer $J$; a continuous curve only guides the eye.

Use the stipulated common Hückel Coulomb parameter $\alpha$, an equal Hückel resonance integral $\beta$ along each cube edge, and zero coupling elsewhere; take an orthonormal site basis. Each lithium vertex has three sodium neighbors. The totally symmetric combinations therefore have the blockFor a coordinate-like $T_2$ combination, the sum of the three neighboring sodium coordinate signs equals the lithium coordinate sign. Thus each of the three equivalent blocks isTheir eigenvalues give the cube nearest-neighbour Hückel model:For the bonding convention $\beta <0$ these are in increasing energy order, with statistical weights $1,3,3,1$. The eight valence electrons occupy $(a_1{)}^2(t_2{)}^6$, filling the four bonding spatial molecular orbitals. Their total one-electron energy is $8\alpha +12\beta$. This is a deliberately simplified model: unequal atomic energies or unequal overlaps require a generalized calculation.

Using mass $7\mathrm{u}$ for the dominant lithium isotope gives a transition wavenumber ${\tilde{\nu }}_0=14925\mathrm{c}{\mathrm{m}}^{-1}$. Inserting the thermal speed into the stated Doppler broadening expression givesThe supplied formula is the Gaussian half width at half maximum; the corresponding full width is about $0.256\mathrm{c}{\mathrm{m}}^{-1}$. Meanwhile exponential population decay with lifetime $\tau$ produces a natural linewidth expressed as a full widthHere $c$ must be in $\mathrm{cm}{\mathrm{s}}^{-1}$ in the second expression. Equivalently the fine-structure separation is about $9.0$ GHz, the Doppler half width about $3.83$ GHz, and the natural linewidth expressed as a full width about $5.89$ MHz. The natural linewidth is therefore negligible compared with thermal broadening; low density also makes collision broadening less important.

The doublet is resolvable in principle, but only marginally against the Doppler width. The separation exceeds one Doppler full width, so two components can survive, although their profiles overlap strongly. The $J=3/2$ and $J=1/2$ upper levels have different statistical weights, so the weaker member need not make an equally conspicuous maximum. A spectral resolving power $R\gtrsim {\tilde{\nu }}_0/0.3\simeq 5\times 10^4$ is the basic instrumental requirement; a width well below $0.1\mathrm{c}{\mathrm{m}}^{-1}$ and adequate signal would make the test more convincing. An arbitrarily fine spectrometer cannot remove the physical Doppler overlap. This conclusion uses the stipulated thermal, optically thin picture; additional photospheric velocity fields or optical-depth effects would require more information.

For a linear rigid rotor with moment of inertia $I$, the energy eigenvalue at angular momentum quantum number $J$ is ${\hslash }^{2}J(J+1)/(2I)$. Each level has statistical weight $2J+1$ before additional symmetry or nuclear-spin restrictions.