Vibration-Rotation Spectra: P, Q, R Branches

High-resolution rovibrational spectroscopy of exactly this P/R(/Q) type underlies atmospheric remote sensing: satellite and ground-based FTIR instruments measure CO₂, CH₄, N₂O, and CO absorption bands, and the individual rotational line intensities and positions (not just the band envelope) are what allow retrieval of gas concentration, temperature, and pressure altitude profiles used in climate monitoring (the HITRAN spectroscopic database is essentially an enormous, continuously updated tabulation of exactly the $\nu_0$, $B_v$, $\alpha_e$, $D_v$ constants derived by the methods above, for thousands of transitions across dozens of molecules). Combustion diagnostics use rovibrational CO and OH spectra, resolved into P/R structure, to extract flame temperature from the relative intensities of different $J$ lines (a Boltzmann population analysis riding on top of the branch assignment). Interstellar and cometary spectroscopy identifies molecules such as HCl, CO, and CH⁺ in space by matching observed rotational-line spacing patterns to laboratory $B_v$ constants extracted exactly this way. In the pharmaceutical and materials industries, gas-phase IR of small volatile fragments (HCl evolved in acid-catalyzed reactions, CO in metal-carbonyl decomposition) is used analytically via rotational fine structure to confirm species identity even when the "vibrational" wavenumber alone would be ambiguous. Finally, the entire P/R/Q branch machinery reappears, essentially unchanged in its selection-rule logic, wrapped around the vibrational progressions of Electronic Transitions and Franck-Condon band systems, where the much larger change in equilibrium geometry between electronic states (rather than the small change between vibrational levels of the same electronic state) produces dramatically pronounced band heads — the diatomic-fundamental "no band head" result derived above is precisely the boundary case that electronic spectroscopy routinely violates.

Ask the tutor: How would the P/R branch formulas change for a symmetric top or a molecule with two different rotational constants like a bent triatomic?

A diatomic molecule's infrared spectrum is a ladder of vibrational levels, each vibrational level itself split into a rotational ladder. A vibrational transition — say $v=0 \to v=1$ — does not connect single states; it connects an entire rotational manifold in $v=0$ to an entire rotational manifold in $v=1$. Because the rotational constant depends on vibrational state, those two manifolds are not identical copies of each other, and because rotational selection rules restrict which $J$-to-$J'$ jumps are dipole-allowed, only a discrete set of lines survives. For a $^1\Sigma$ diatomic (no electronic angular momentum along the bond, e.g. HCl, CO, or ¹²C¹⁶O), the rotational selection rule accompanying $\Delta v = +1$ is strictly $\Delta J = \pm 1$: the photon carries one unit of angular momentum, and the molecule must absorb or emit it into or out of rotation. Lines with $J' = J+1$ (final rotational quantum number exceeds initial) form the R branch; lines with $J' = J - 1$ form the P branch. There is no $\Delta J = 0$ line for such molecules because the vibrational transition dipole moment has no component that can conserve angular momentum without a rotational change — this is why the very center of the band, at the vibrational origin $\nu_0$, is a gap, not a line. Molecules with net electronic angular momentum along the axis (e.g. NO, with unpaired electron spin/orbital angular momentum giving $\Lambda \ne 0$ analogs, or more simply linear molecules undergoing a bending vibration such as the $\nu_2$ mode of CO$_2$) can have $\Delta J = 0$ allowed, producing a Q branch exactly at (or very near) $\nu_0$. In CO and HCl the Q branch is absent; in the bending mode of CO$_2$ or in many polyatomic parallel/perpendicular bands it is prominent.

The key qualitative feature — visible in that strip at the bottom of the figure — is that the R-branch "sticks" are drawn closer together as you move away from the gap, while the P-branch sticks spread farther apart. This asymmetry is the direct fingerprint of $B_1 < B_0$, i.e. of bond lengthening upon vibrational excitation, encoded through the anharmonic and rotation–vibration coupled term-value expression developed below.

The combined vibration–rotation term value, to the order needed for quantitative branch analysis, is

where:

• $\omega_e$ is the harmonic vibrational wavenumber (cm⁻¹),
• $\omega_e x_e$ is the anharmonicity constant,
• $v = 0,1,2,\dots$ is the vibrational quantum number,
• $B_v$ is the rotational constant in vibrational state $v$ (cm⁻¹),
• $D_v$ is the centrifugal distortion constant in state $v$ (cm⁻¹, $D_v \ll B_v$),
• $J = 0,1,2,\dots$ is the rotational quantum number in that vibrational state.

Reading: the first two terms are the anharmonic vibrational ladder alone (Morse-oscillator term values); the last two terms are the rigid-rotor-plus-centrifugal-distortion ladder, but crucially evaluated with a rotational constant $B_v$ that is not the same number in every vibrational state. The vibration-dependence of $B_v$ is captured, to first order, by

where:

• $B_e = \dfrac{h}{8\pi^2 c \mu r_e^2}$ is the rotational constant at the (hypothetical) equilibrium bond length $r_e$,
• $\alpha_e$ is the vibration–rotation coupling constant (cm⁻¹), positive for essentially all diatomics,
• $\mu$ is the reduced mass, $c$ the speed of light.

Reading: as $v$ increases, the molecule spends more time at larger bond extensions (anharmonicity means the outer turning point moves out faster than the inner one moves in), so the effective average $1/r^2$ shrinks, and $B_v$ decreases with $v$. This is why $B_1 < B_0$ always, and why the two branches curve oppositely. The transition wavenumber for absorption $v''=0 \to v'=1$ (double-prime = lower/initial state, single-prime = upper/final state, standard spectroscopic convention) is

where:

• $\nu_0 = \omega_e - 2\omega_e x_e$ is the fundamental band origin (the $v=0\to1$ vibrational transition wavenumber for $J=0\to0$, even though that specific line is dipole-forbidden),
• $J'$ and $J''$ are related by the selection rule for each branch.

Reading: every observed line is a difference of two term values, one from each vibrational manifold; the branch label tells you exactly how $J'$ and $J''$ are related, collapsing this general formula into three closed expressions.

The three branch formulas

Applying $J' = J''+1$ (R branch, $J''=0,1,2,\dots$), $J' = J''-1$ (P branch, $J''=1,2,3,\dots$; must start at $J''=1$ since $J'\ge0$), and defining $m$ as a single running index:

where here $J \equiv J''$, the rotational quantum number of the lower (initially populated) state — this is the universal spectroscopic convention: label every line by its lower-state $J$.

Reading: both formulas are quadratic in $J$, and because $B_1 - B_0 = -\alpha_e < 0$, the $R$-branch quadratic term is negative (lines crowd together, eventually turning over into a "band head" in molecules with large $\alpha_e$) while the $P$-branch quadratic term is positive (lines spread apart monotonically). Neither branch is a rigid ladder; both curve, in opposite senses, and by exactly the same physical constant $\alpha_e$. A single unified Fortrat-type formula covering both branches at once, using signed running index $m$ ($m = J+1 = 1,2,3,\dots$ for R; $m = -J = -1,-2,-3,\dots$ for P), is

where:

• $m>0$ labels R-branch lines ($m = J+1$),
• $m<0$ labels P-branch lines ($m=-J$),
• $m=0$ would be the missing Q line at exactly $\nu_0$.

Reading: this is the celebrated Fortrat diagram relation — plotting observed $\tilde\nu$ against integer $m$ traces out a single parabola, and a quadratic least-squares fit to all the lines in a spectrum, both branches together, yields $\nu_0$, $B_0+B_1$, and $B_0-B_1$ in one shot, which is exactly how real spectra are fit.

Example 1 — Foundational: building the first few HCl lines from tabulated constants. For H³⁵Cl, spectroscopic tables give (these are the real, non-round literature values):

First compute the band origin:

Next the state-dependent rotational constants, using $B_v=B_e-\alpha_e(v+\tfrac12)$:

Now compute R(2) — the line from $J''=2\to J'=3$:

And P(3) — the line from $J''=3\to J'=2$:

Both values land within a fraction of a wavenumber of tabulated HCl absorption lines, confirming the formula's internal consistency. Note the asymmetry already: R(2) is only $\approx 64.4$ cm⁻¹ above $\nu_0$, while P(3) is $\approx 64.4$ cm⁻¹ below — nearly symmetric at low $J$ because $\alpha_e \ll B_e$, but the small quadratic terms ($\pm 2.76$ cm⁻¹) are exactly what breaks that symmetry and is measurable at high resolution.

Limiting check: if $\alpha_e \to 0$ (no vibration-rotation coupling, $B_1=B_0=B$), both formulas collapse to $\tilde\nu_R(J)=\nu_0+2B(J+1)$ and $\tilde\nu_P(J)=\nu_0-2BJ$ — a perfectly symmetric, evenly spaced double comb with spacing exactly $2B$ on either side, and no band-head curvature at all. This is the "rigid rotor + separate harmonic oscillator" limit, recovering the naive picture before coupling is included.

Example 2 — Applied: extracting B₀, B₁, and $r_e$ for CO from two observed line positions. Suppose (using real literature-quality ¹²C¹⁶O fundamental-band data) two lines are measured at

and additionally the ground-state combination difference at $J=5$ (from R(4) and P(6), which share upper level $J'=5$) is to be used to extract $B_0$ directly, bypassing $\nu_0$ and $B_1$ entirely — exactly the method derived above. Using $\Delta_2F''(J) = \tilde\nu_R(J-1)-\tilde\nu_P(J+1) = 2B_0(2J+1)$ with $J=5$:

This matches the accepted literature value for CO, $B_0=1.9313$ cm⁻¹ scaled appropriately for illustrative purposes here (the numbers above are chosen for pedagogical clarity of the algebra rather than reproducing the exact tabulated CO constant, since real CO has $B_0\approx1.931$ cm⁻¹ and correspondingly larger $J$ spacing between these particular lines; the arithmetic method is identical regardless of which molecule supplies the numbers). Now invert $B_0=h/(8\pi^2c\mu r_0^2)$ for the effective bond length. Using $\mu_{\text{CO}} = \dfrac{(12.000)(15.9949)}{12.000+15.9949}\,u = 6.8562\ u = 1.1385\times10^{-26}\ \text{kg}$:

Plugging in $h=6.6261\times10^{-34}\ \text{J s}$, $c=2.9979\times10^{10}\ \text{cm/s}$, $B_0 = 2.8755\ \text{cm}^{-1}$:

Compute the denominator step by step:

(This is deliberately illustrative arithmetic on constructed data to show the full chain from a line-position difference to a bond length; the true CO bond length, obtained from the actual $B_0=1.9313$ cm⁻¹, is 1.1283 Å — the method, not this particular numerical answer, is the transferable result.)

Example 3 — Exam/competition level: locating the band head and estimating the maximum observable J in the R branch. Some molecules with unusually large $\alpha_e/B_e$ ratios show the R branch literally turning back on itself and forming a band head — a wavelength beyond which no further R lines appear, because $\tilde\nu_R(J)$ reaches a maximum and then decreases. Find the value of $J$ at the band head, given (representative, deliberately non-round, non-tabulated values chosen to test the method) $B_0 = 1.5250$ cm⁻¹, $B_1=1.2870$ cm⁻¹. Treat $\tilde\nu_R(J)$ as a continuous function of $J$ and set its derivative to zero. From

let $x=J+1$ and differentiate with respect to $x$:

Set to zero:

Since $B_1<B_0$ here, $(B_1-B_0)$ is negative, so for this to give a positive $x$ (physically meaningful) we must track signs carefully. Rewrite using $B_1-B_0=-\alpha_e$ with $\alpha_e = B_0-B_1 = 1.5250-1.2870=0.2380$ cm⁻¹:

This is negative as written because the maximum of $\tilde\nu_R(J)$ (a downward parabola in $x$ since the $x^2$ coefficient is $-(B_1-B_0)=+\alpha_e$... Let's redo the sign bookkeeping explicitly, since this is exactly the kind of step where errors creep in. We have

With $B_1-B_0 = -\alpha_e$, the coefficient of $x^2$ is $-(-\alpha_e) = +\alpha_e$, which is positive. A positive coefficient on $x^2$ means the parabola opens upward, so $\tilde\nu_R(x)$ has a minimum, not a maximum, at the vertex — meaning the branch does not turn over for $B_1<B_0$; it simply spreads apart ever more slowly and then, mathematically, would start increasing spacing again only for very distorted (non-physical in this regime) values. The band head in real diatomics ($B_1<B_0$, i.e. $\alpha_e>0$) forms in the R branch when the spacing shrinks to zero and reverses, i.e. when consecutive line spacings $\tilde\nu_R(J)-\tilde\nu_R(J-1)$ change sign. The correct condition is on the discrete first difference, not the continuous derivative extremum with the wrong curvature sign — showing explicitly why band heads occur in the R branch (not P) when $\alpha_e>0$: reconsider the spacing itself,

Using the compact form with $x=J+1$: consecutive R-lines are spaced by

Setting this spacing to zero (the head, where consecutive lines coincide) gives

With $B_1-B_0=-0.2380$ and $B_1+B_0=2.8120$:

which is negative — confirming that for $\alpha_e>0$ (the normal case, $B_1<B_0$) the R branch does not form a band head at any physical positive $J$; instead lines merely crowd together asymptotically without ever reversing. A true band head at positive $J$ requires $B_1>B_0$ (excited state more rotationally rigid than the ground state — rare, but it occurs in a handful of electronically excited states and in some bent-molecule bending overtones). This worked example is deliberately included because it is the single most common conceptual trap: students expect every R branch to show a head, when in fact the ordinary vibrational fundamental of essentially every ground-state diatomic (H₂, HCl, CO, NO) shows converging-but-never-reversing R lines, and genuine band heads belong overwhelmingly to electronic (not vibrational) band systems, foreshadowing exactly the physics of Electronic Transitions and Franck-Condon envelopes where $B'\gg B''$ or $B'\ll B''$ regularly occurs because the excited electronic state can have a drastically different equilibrium geometry.