Diatomic_molecule - Wikiparency

# Diatomic molecule  ## Microsims — three.js ### Diatomic molecule (three.js) <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Diatomic_molecule.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Diatomic molecule — three.js microsim"></iframe> </div> **Open it full-screen:** [Diatomic_molecule.html](https://wikitube-3d-microsims.netlify.app/Diatomic_molecule.html) · library `threejs` · route `microsim/threejs/` ### Related microsims Live sims on neighbouring articles: - [[Hydrogen_atom]] - [[Ammonia]] - [[Metallic_hydrogen]] - [[Hydrogen_line]] *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*  ## Overview A diatomic molecule is two nuclei held together by a shared electron cloud. Take away the translation of the molecule as a whole and exactly two internal motions remain, and no others: the bond stretches along its own axis, and the whole thing tumbles about its centre of mass. Both are quantised, and their quanta sit about two orders of magnitude apart -- a vibrational quantum of order 1000 cm-1 against a rotational one of order 10 cm-1. Every vibrational level therefore carries a whole ladder of rotational rungs on top of it, and reading that two-tier ladder is most of what molecular spectroscopy does. H2 -- two protons, two electrons -- is the simplest neutral molecule there is, and the calibration point for every other covalent bond. ## The physics A potential energy curve V(r) is not given for free. Born and Oppenheimer supplied it in 1927: a proton is 1836 times heavier than an electron, so on the timescale of nuclear motion the electrons relax essentially instantly to whatever positions the nuclei hold. Freeze the nuclei at separation r, solve the electronic problem there, and you get back one number, E_el(r). Sweep r and you have a curve. Bond length, force constant, vibrational ladder and dissociation energy are all properties of it; without the separation, none of them are defined. Near the bottom that curve is nearly parabolic and a harmonic oscillator works. It fails at high v, where the real curve flattens towards dissociation while a parabola climbs forever, giving evenly spaced levels and no dissociation at all. The Morse potential, V(r) = D_e [1 - exp(-a(r - r_e))]^2, flattens correctly and supports a finite ladder that converges onto the top of the well. Levels are written as term values: E(v,J) = omega_e(v+1/2) - omega_e x_e (v+1/2)^2 + B_v J(J+1) - D_J[J(J+1)]^2, with B_v = B_e - alpha_e(v+1/2). The omega_e x_e term is the anharmonicity that makes the rungs converge; alpha_e is vibration-rotation interaction, since a vibrationally excited molecule sits at larger mean r and so has a smaller rotational constant; D_J is centrifugal distortion, angular momentum stretching the bond so the rotational levels sag below a rigid rotor's. Both motions run on the reduced mass, mu = m1 m2 / (m1 + m2), and this is where isotopes enter. V(r) is built from charges and electrons, so within Born-Oppenheimer it cannot tell a proton from a deuteron: H2, HD and D2 share one curve and one well depth. Only mu changes, hence omega_e as 1/sqrt(mu), hence the zero-point energy -- about 2175 cm-1 for H2 against roughly 1540 for D2. Since D_0 = D_e - ZPE, D2 is bound some 630 cm-1 (0.08 eV) more tightly out of an identical well. Kinetic isotope effects are that difference and nothing else. For H2, r_e = 1.4011 a_0 = 0.741 43 angstrom = 74.143 pm. The two dissociation energies must be kept apart. D_e, the well depth measured from the minimum of the Born-Oppenheimer curve, is 38 293.0 cm-1 = 4.7477 eV = 458.1 kJ/mol -- a property of a fictitious vibrationless molecule, and what quantum-chemistry codes compute. D_0, measured from the actual lowest level (v = 0, J = 0) to two free atoms, is 36 118.069 45(53) cm-1 = 4.478 07 eV = 432.068 kJ/mol. The difference is the zero-point energy, about 5.7 per cent of the well depth, because the proton is so light. D_0 is the bond energy. Two rival numbers circulate. 35 999.582 834 cm-1 is correct but is D_0 from the N = 1 ortho level, not J = 0; add its 118.487 cm-1 of rotation and it becomes 36 118.070. And 435.996 kJ/mol is the bond dissociation enthalpy at 298.15 K, a thermally corrected quantity. The two protons are identical fermions, so the total wavefunction must change sign under exchange. The electronic ground state and the vibrational factor are both symmetric, forcing the nuclear-spin and rotational parts to have opposite exchange symmetry: ortho-hydrogen takes the spin triplet (weight 3) and odd J, para-hydrogen the singlet (weight 1) and even J. The 3:1 ortho:para ratio of "normal" hydrogen is that weight ratio, but it reaches the rotational populations only because the Pauli principle ties spin symmetry to J parity. At 20 K equilibrium is about 99.8 per cent para -- J = 1 lies 118.49 cm-1 (about 170.5 K) above J = 0 -- and conversion takes hours to days. That slowness is expensive. Liquefy normal hydrogen without a catalyst and you bottle 75 per cent ortho, which converts in the tank and releases about 525 kJ/kg, more than the latent heat of vaporization. The liquid boils itself away, order half the contents in a week. Industrial liquefiers therefore pass the gas over a paramagnetic catalyst during cooling, delivering 95 per cent para or better. A homonuclear molecule is symmetric end for end, so its dipole moment is zero at every bond length and stays zero as it stretches and turns: no allowed electric-dipole rotational spectrum, no allowed infrared band. H2 is not spectrum-free -- far-ultraviolet Lyman and Werner bands, weak electric-quadrupole lines -- but the lowest quadrupole line, 0-0 S(0) at 28.221 micrometres, has an upper level 510 K above ground, and S(1) at 17.035 micrometres needs 1015 K. In a 10-20 K cloud those levels are empty, so the most abundant molecule in the universe is invisible where it is coldest. Astronomers count CO instead: 10^-4 as abundant, a small but real dipole of 0.11 D, and a J = 1 to 0 line at 115.271 GHz whose upper level lies only 5.5 K up. The conversion back to H2 column density uses X_CO = 2 x 10^20 cm-2 (K km/s)-1 for the Milky Way disc, with 30 per cent uncertainty that grows to factors of several in low-metallicity dwarfs and galaxy centres -- an empirical calibration, not a constant of nature. Neutral *atomic* hydrogen is the opposite case: it is seen directly, by a magnetic-dipole hyperfine transition, at [[Hydrogen_line]]. ## Controls -> what each maps to | Control | Maps to | Range / values | Physical meaning | |---|---|---|---| | Molecule | the spectroscopic constant set omega_e, omega_e x_e, B_e, alpha_e, D_J, r_e and the two masses | H2 / HD / D2 / HCl / CO | H2-HD-D2 is the isotope series: one potential curve, three reduced masses. HCl and CO are the heavy contrasts | | v | vibrational quantum number | 0 to 17 | Raises the level on the Morse ladder; the rungs converge and the classical turning points fly apart | | J | rotational quantum number | 0 to 20 | Speeds up the tumble, lifts the level by B_v J(J+1), and stretches the bond centrifugally | | Speed | wall-clock rate | 1 to 40 vibrational periods per second | Vibration always looks the same speed, so the rotation rate carries the comparison | | Morse panel | V(r) with the drawn level ladder and turning points | on / off | Where D_e, r_e and the level convergence are legible | | Spectrum | P and R branch strip of the fundamental band | on / off | Struck out in red for homonuclear H2 and D2: no allowed dipole lines | | c.o.m. + arms | centre of mass and the two lever arms | on / off | Shows the mass-weighted split of the amplitude; in HD the deuteron barely moves | | Running | run / pause the integrator | on / off | Pauses the trajectory without changing the state | | Reset | -- | button | Returns the molecule to its outer turning point | The HUD prints mu, the force constant k, the ratio omega_e/2B_e, the split of E(v,J) into its vibrational G(v) and rotational F(J) parts, B_v against B_e, the live bond length r against r_e and the centrifugally stretched r_J, the count of bound levels, the vibrational and rotational periods, and the integrator's energy drift dE/E. ## Learning objective After playing, a learner can name the two internal motions of a diatomic molecule and the quantum number that counts each, read a rovibrational term value off the ladder, explain why H2, HD and D2 share a well depth but not a dissociation energy, and say why cold H2 cannot be seen while CO can. ## Limits and connections The Morse curve is a two-parameter fit, honest only near the bottom of the well. Its implied well depth, D_e(Morse) = omega_e^2 / (4 omega_e x_e), is not the measured one -- within a few per cent for H2 and CO, about 14 per cent high for HCl -- and the HUD prints both so the error stays visible. Its exponential tail decays too fast where the real long-range interaction falls off as a power law in r, so the topmost levels and the approach to dissociation are the least trustworthy part of the picture. The animation integrates a classical trajectory at the quantum energy E(v,J): a legible orbit, not a wavepacket, with no tunnelling modelled. One nucleus instead of two, solved exactly, is [[Hydrogen_atom]]; the transition that makes neutral atomic hydrogen visible where H2 is not is [[Hydrogen_line]]. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Diatomic_molecule) : [Wikitube](https://en.wikitube.io/wiki/Diatomic_molecule) ## Previous hub tags Tree parent: [[Hydrogen]]. Legacy hubs: `HYDROGEN`. --- *Created 2026-08-05 - append-only - portal-microsim-pass to WIKI_REPOPULATION_PROTOCOL v1.0 section 5 - 0 deletions*