= Spin isomers of hydrogen
{wiki}
= Hydrogen spin isomers
{synonym}
<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 $\widehat{\mathbf r}$, and the <spherical harmonic> rotational factor satisfies
$$
Y_{JM}(-\widehat{\mathbf r})=(-1)^JY_{JM}(\widehat{\mathbf r}).
$$
The 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>.
\Image[/hydrogen-spin-symmetry-and-rotational-levels.png]
{title=<Proton> exchange selects even-$J$ <parahydrogen> and odd-$J$ <orthohydrogen>}
{id=image-hydrogen-spin-symmetry-and-rotational-levels}
{description=Original <linear rigid rotor> diagram under the electronic and vibrational <ground state> assumptions above. Each line represents $g=(2I_{\rm n}+1)(2J+1)$ <quantum states>, including both <nuclear spin> and rotational <degenerate energy levels>. The <energy> separation between the two lowest allowed levels is rotational. No measured spectrum or spin-dependent splitting is shown.}
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