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Spin isomers of hydrogen are two forms of molecular hydrogen, ${}^1{\mathrm{H}}_2$, distinguished by the combined nuclear spin of their two protons. They have the same chemical formula and electronic ground state, but different allowed molecular rotation states. Parahydrogen has total nuclear spin zero and even rotational quantum number $J$; orthohydrogen has total nuclear spin one and odd $J$. The difference makes heat capacity a probe of fermionic exchange symmetry, and makes cooling history matter for a tank of molecular hydrogen.

Assume the electronic ground state $X{}^1{\Sigma }_g^{+}$ and the vibrational ground state of a free molecule. The electronic factor is symmetric under exchanging the two protons, and the vibrational factor depends only on their separation, so it is also symmetric. These assumptions concern the exchange of protons; the electronic wavefunction separately obeys fermionic exchange symmetry under exchange of electrons. Neglect electronic excitation, vibrational excitation, external fields, and small interactions involving nuclear spin when calculating rotational energy eigenvalues.

The two protons are identical particles with spin $1/2$. Their complete wavefunction must change sign under their particle exchange operator. Interchanging the two ends reverses the internuclear direction $\hat{\mathbf{r}}$, and the spherical harmonic rotational factor satisfiesThe antisymmetric spin-one-half singlet state therefore requires symmetric, even-$J$ rotation; the symmetric spin-one-half triplet state requires antisymmetric, odd-$J$ rotation. This is exchange symmetry of two identical spin-one-half fermions applied inside a diatomic molecule, not an extra rotational force. A linear rigid rotor may omit spin-dependent splittings while still requiring this restriction on its allowed quantum states.

Take an invented diatomic molecule with a Morse potential, setting the separated-atom energy to zero:Here $a$ is an inverse length, $r_e$ the equilibrium bond length, and ${\omega }_e=a\sqrt{2D_e/\mu }$, with reduced mass $\mu$. These stipulated parameters describe a simple model, not a measured chemical substance. For the one-dimensional Morse oscillator, energy eigenvalues above the minimum areConsequently,The harmonic estimate $A/2=0.1000\mathrm{eV}$ is slightly larger than this model's exact zero-point energy. The physical bound-level restriction matters: the quadratic expression is not an infinite ladder of vibrational quantum states.

For a separate thermal illustration using this same stipulated $D_0$, assume an ideal gas, fully active molecular rotation of the parent, negligible thermal molecular vibration, and negligible electronic excitation for all species. Each monatomic product contributes $\delta h=5RT/2$; the diatomic molecule contributes $7RT/2$. Thus the net increment is $3RT/2$, with molar gas constant $R$. At $T=300\mathrm{K}$,This last result follows from the stated thermal model, not a universal correction for every chemical bond. Frozen rotations, accessible excited states, different fragment composition, or a condensed phase change the thermal increments.

The University of Florida's Rotation Vibration Spectrum of the HCl Molecule, University of Florida laboratory protocol, printed pp. 2–3 and 10–11, uses a scanning Michelson interferometer for Fourier transform spectroscopy. An air-filled silicon-window cell supplies the background. Supervised preparation adds concentrated hydrogen chloride liquid; the cleaned cell subsequently receives DCl liquid. Acquisition settings are identical.

The readout is transmission or absorbance against spectroscopic wavenumber, resolving molecular rotation alongside stretching. Background addresses instrument and cell contributions; cleaning addresses carryover. Excessive absorption impairs analysis. Assigning isotopes and rotational branches precedes fitting band parameters to test isotope scaling of diatomic vibration. These are prescribed measurements, not numerical observations supplied here.^[ref]

For a transparent ideal model, add the linear rigid rotor to the harmonic vibrational energy levels and assume the same frequency-valued rotational constant $B$ in both vibrational states:For a spin-free diatomic molecule with no electronic angular momentum, electric-dipole selection rules give $\Delta J=\pm 1$ for this vibrational absorption. Subtracting initial from final energy eigenvalues yieldsThese are two rotational branches about the vibrational band origin, not a single stretch line. For stipulated dimensionless values ${\tilde{\nu }}_{\mathrm{harm}}=100$ and $\tilde{B}=1$, the first three R-branch positions are $102,104,106$, and the first three P-branch positions are $98,96,94$. There is no $J=0\to -1$ transition. The numbers illustrate level subtraction; they are not HCl measurements. Real upper and lower rotational constants differ, and centrifugal distortion shifts lines further.

In the independent harmonic approximation, each diatomic molecule has one bond-stretching quantum harmonic oscillator with energies $(n+1/2)\hslash \omega$. Write $x=\hslash \omega /(k_{B}T)$. Its canonical partition function is $z=e^{-x/2}/(1-e^{-x})$, and differentiating its logarithm givesFor $T\ll \hslash \omega /k_B$, excitation is exponentially suppressed and the heat capacity tends to zero. At high temperature it tends to $N{k}_B$, one $k_B/2$ contribution from each quadratic kinetic and potential term. The zero-point energy does not contribute to the derivative with temperature. For a dilute gas whose rotations are already active, adding translation and rotation gives molar $C_V\simeq 5R/2$ before vibration is activated and $7R/2$ afterwards, provided dissociation and electronic excitations are negligible.

In the ideal electric-dipole interaction, a vibrational transition needs a nonzero matrix element of the electric dipole moment. For a diatomic bond stretch, expand the body-axis dipole component as $d(q)=d_e+d_e^{\prime }q+\cdots$. A changing electric dipole moment, rather than a permanent value $d_e$ alone, permits the harmonic fundamental. Keeping only the linear term and harmonic wavefunctions gives $\Delta v=\pm 1$.^[ref]

The derivation is sharper than a picture of a shaking molecule. Orthogonality removes the constant term for $v^{\prime }\ne v$, and creation and annihilation operators giveHere $\delta$ is the Kronecker delta. Thus absorption raises $v$ by one when the dipole derivative and angular factor are nonzero, and its photon satisfiesA nonzero vibrational matrix element is only one factor: rotational electric-dipole selection rules, initial populations, and spectral resolution also constrain visibility.

For an isolated diatomic molecule with identical atomic nuclei, such as molecular hydrogen in its symmetric electronic ground state, stretching preserves zero electric dipole moment. By symmetry, this homonuclear stretch is infrared inactive in the ideal dipole model, although it still vibrates. Other interactions or environments can permit detection.^[ref]