Rotational-Vibrational Spectrum: P and R Branches (2D)

Energy levels

The vibration of a diatomic molecule is an anharmonic oscillator, G(v) = w_e (v + 1/2) - w_e x_e (v + 1/2)^2 + w_e y_e (v + 1/2)^3, and the end-over-end rotation is a rotor, F_v(J) = B_v J (J + 1) - D J^2 (J + 1)^2, with B_v = B_e - a_e (v + 1/2). Everything is in wavenumbers (cm^-1), so a level energy is E / hc = G(v) + F_v(J). The rotational constant is B = h / (8 pi^2 c mu r^2), which is why a rotational spectrum gives the bond length: r = sqrt(16.8576 / (mu B)) in angstrom for mu in u and B in cm^-1.

Selection rules and the two branches

A molecule absorbs a photon only if the vibration changes its dipole moment (so N₂ and O₂ are silent in the infrared) and only if delta J = +1 or -1, because the photon carries one unit of angular momentum. Jumps with J to J+1 form the R branch on the high-wavenumber side, jumps with J to J-1 the P branch on the low side. There is no delta J = 0 line, so the band origin nu_0 = G(v') - G(0) shows as a gap between P(1) and R(0) of about 4B.

Line positions

With B' = B_1 for the upper and B'' = B_0 for the lower vibrational level: R(J) = nu_0 + (B' + B'')(J + 1) + (B' - B'')(J + 1)^2 and P(J) = nu_0 - (B' + B'') J + (B' - B'') J^2. A rigid rotor (B' = B'') gives equally spaced lines nu_0 + 2B m. Because the bond is longer in the upper state, B' < B'': R lines crowd together (and in extreme cases pile up in a band head, which you can create with the coupling slider) while P lines spread out. Differences between lines remove one of the two states: R(J-1) - P(J+1) = 4 B'' (J + 1/2) depends only on the lower state, R(J) - P(J) = 4 B' (J + 1/2) only on the upper one, and the last two graphs plot exactly these.

Intensities: Boltzmann populations

The fraction of molecules in level J of v = 0 is n_J = (2J + 1) exp(-hc F_0(J) / kT) / Q, which rises (more orientations) then falls (less thermal energy). It peaks at J_max = sqrt(kT / 2hcB) - 1/2 and Q = kT / hcB + 1/3 + hcB / 15kT. The strength of a line is the population times a Honl-London factor, (J + 1) / (2J + 1) for R and J / (2J + 1) for P, times nu (1 - exp(-hc nu / kT)). A hotter gas spreads the comb out and lowers the peak; a light molecule such as HF has few, widely spaced lines.

Isotope shift

Replacing an atom changes the reduced mass mu but not the electronic potential, so with rho = sqrt(mu / mu') the constants scale as w_e ~ rho, B_e ~ rho^2, a_e ~ rho^3, D ~ rho^4. H-37Cl is heavier than H-35Cl, so its lines sit about 2.1 cm^-1 lower, giving the famous doublet in every HCl spectrum with the 3 : 1 intensity ratio of the natural chlorine isotopes.

Line shapes

Each line is drawn as a pseudo-Voigt profile: a Doppler Gaussian (sigma = nu sqrt(kT / m c^2)), pressure (Lorentz) broadening of about 0.07 cm^-1 per atmosphere, and the instrument resolution, which at the default setting is much wider than both. The absorbance follows Beer-Lambert, A = -log10(T), proportional to pressure times path length over temperature.

Simplifications

Only the ground vibrational level is populated (a hot band from v = 1 matters above roughly 1000 K for CO and is left out); band strengths are the same for all four molecules and overtones use an illustrative strength ratio rather than a computed one; the Herman-Wallis correction, nuclear-spin statistics, Lambda-doubling and the spin of open-shell molecules such as NO are ignored (NO is therefore not offered); the spin and vibration of the 3D and 2D models are slowed by roughly 10¹¹ so you can see them. The constants are standard literature values for the most abundant isotopologue, and the first check compares four lines with published positions.

1. Start with HCl at room temperature. Count the lines on both sides of the gap in the middle: the gap between P(1) and R(0) is the missing delta J = 0 line, about 4B wide.
2. Click Next line and watch the molecule and the ladder: an R line lifts the molecule from J to J+1, a P line from J to J-1.
3. Raise the temperature. The most populated J (graph 2) moves up like sqrt(T), the comb broadens and the strongest line flattens.
4. Choose HCl isotope doublets. Every line is a pair, 2 cm-1 apart: H-35Cl and H-37Cl. Set the heavy fraction to 0 and the partners vanish; push it to 100 % and the whole band moves.
5. Switch to Ideal rigid rotor and harmonic oscillator: the spacing graph turns flat. Then increase the coupling slider step by step to watch R lines close up and P lines spread out.
6. Try Strong coupling to create a band head, and CO to see a much denser comb (a heavier molecule has a smaller B).
7. Use the first overtone (v = 0 to 2): it lies near twice the fundamental, a bit lower because of anharmonicity, and is far weaker, so raise pressure and path length to see it.