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Let $J$ label the lower rotational state and write the elementary band model asHere $B^{\prime \prime },B^{\prime }$ are lower and upper rotational constants, in inverse centimetres; a constant term associated with upper-state vibrational angular momentum can be included in ${\tilde{\nu }}_{\ast }$. The P branch has $J^{\prime }=J-1$, and the R branch has $J^{\prime }=J+1$.

In this model the six printed labels are assigned asThe low-frequency series steps away from the central origin towards increasing lower $J$. The first two high-frequency R positions are approximately $714.96$ and $717.93{\mathrm{cm}}^{-1}$, so the labels at $720.91$ and $723.90$ correspond to $R(2)$ and $R(3)$; they are not the first two rotational transitions.

There is a physical limitation to taking this schematic model literally. A strict $l=0\to |l|=1$ bending band has $J^{\prime }\ge 1$, so a genuine $P(1)$ transition to $J^{\prime }=0$ is forbidden. The printed $709.04$ feature has the formal $P(1)$ position of the elementary model. The numerical analysis below follows the printed line sequence; it does not infer a physically allowed $J^{\prime }=0$ bending level from that feature.

The resolved lines arise from simultaneous changes of vibration and rotation in hydrogen cyanide. A vibrational fundamental connects many populated lower rotational levels to different upper rotational levels, giving the fine structure of a rovibrational spectrum.

The prominent central Q branch, together with flanking P branch and R branch lines, identifies a perpendicular band. For linear HCN, this is the doubly degenerate bending fundamental, not either stretching fundamental. Bending creates a transition-dipole component perpendicular to the molecular axis; its vibrational angular momentum permits $\Delta J=0$ as well as $\Delta J=\pm 1$.

The two perpendicular bending coordinates share one vibrational frequency. Their degeneracy is different from the series of resolved rotational transitions superimposed on that fundamental. The stretches have their electric dipole transition moment along the molecular axis and would give the usual parallel-band P/R pattern without this strong central Q feature.