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Journal of Research of the National Bureau of Standards 41, 379–475 (1948), RP1932. DOI: doi.org/10.6028/jres.041.037

Section III.2 and table 12 give ortho–para equilibrium composition and compare state sums with spectral and thermal evidence. The compilation distinguishes fixed-composition normal hydrogen from equilibrium spin isomers of hydrogen mixtures. It supplies institutional thermodynamic data and theory, rather than answers to a particular examination.

Harold W. Woolley helped compile quantitative properties from thermodynamics of hydrogen, distinguishing its isotopic forms and spin isomers of hydrogen. The compilation compares orthohydrogen and parahydrogen equilibrium calculations with spectral and thermal evidence. It also distinguishes a fixed-composition sample of normal hydrogen from a mixture allowed to reach spin-isomer equilibrium.^[ref]

NBS author of the hydrogen thermodynamic compilation cited in Compilation of Thermal Properties of Hydrogen in Its Various Isotopic and Ortho-Para Modifications by Harold W. Woolley and others. The institutional article byline verifies this author name.

Define ${\theta }_{\mathrm{rot}}={\hslash }^2/(2I{k}_{\mathrm{B}})$. Summing the spatial states counted in part (c),When $T\gg {\theta }_{\mathrm{rot}}$ the closely spaced terms can be replaced by a Gaussian integral:Here labelled means that the two ends of the molecular axis were distinguished in the angular count. In the usual high-temperature molecular convention that removes equivalent orientations and factors out nuclear-spin multiplicity, divide by the rotational symmetry number $s=2$ for homonuclear hydrogen:The full exact spin isomers of hydrogen count is instead the even-$m$ spatial sum plus three times the odd-$m$ sum; its high-temperature limit is $2\sqrt{\pi T/{\theta }_{\mathrm{rot}}}$, including the four nuclear-spin states. These conventions differ only by temperature-independent factors in this limit. All give $U_{\mathrm{rot},\mathrm{m}}=RT/2$ and $C_{V,m}^{\mathrm{rot}}=R/2$, consistent with one quadratic rotational degree of freedom. By comparison, the free linear rigid rotor has $q_{\mathrm{rot}}\propto T$ and $C_{V,m}^{\mathrm{rot}}=R$.

The spin isomers of hydrogen illustrate how nuclear spin statistics constrain the product of electronic, vibrational, rotational and nuclear-spin exchange symmetries. In the ground electronic and vibrational states of these homonuclear molecules, the first two factors are symmetric. Exchanging the nuclei reverses the internuclear direction, so a rotational state has parity $(-1{)}^J$.

For ${}^1{\mathrm{H}}_2$, the protons are spin-one-half fermions and the total nuclear-exchange wavefunction must be antisymmetric. The antisymmetric spin singlet pairs with even $J$ (para-hydrogen, one spin state); the symmetric spin triplet pairs with odd $J$ (ortho-hydrogen, three spin states). For ${}^2{\mathrm{H}}_2$, spin-one deuterons are bosons, so the total must be symmetric. Nuclear spins $I=0,2$ supply six symmetric states paired with even $J$ (ortho-deuterium); $I=1$ supplies three antisymmetric states paired with odd $J$ (para-deuterium). The high-temperature ortho/para statistical ratios are $3:1$ for hydrogen and $2:1$ for deuterium, while low-temperature equilibrium populations also depend on rotational energies and whether spin conversion can occur.

Paul Harteck and Karl Friedrich Bonhoeffer prepared parahydrogen at low temperatures in 1929. The experiment distinguished forms of molecular hydrogen associated with different nuclear-spin states, connecting a measurable sample property with the quantum description of spin isomers of hydrogen.^[ref]

A chemist.^[ref]

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.