= Pure rotational electric-dipole selection rules
{title2=$\Delta J=\pm1$}
For the field-free, spin-free <linear rigid rotor> in one fixed internal <quantum state>, the <electric-dipole interaction> is $-\mathbf d\cdot\mathbf E(t)$. A nonzero permanent <electric dipole moment> along the molecular axis allows pure rotational <photon absorption> with $\Delta J=+1$ and <photon emission> with $\Delta J=-1$. The <matrix element> between initial and final <quantum states> must be nonzero: the existence of an <energy> gap alone is insufficient. Relative to a chosen $z$ axis, <polarization of an electromagnetic wave> along $z$ selects $\Delta M=0$; transverse <polarization of an electromagnetic wave> permits $\Delta M=\pm1$.<Lecture 24 Supplement: Energy levels of a Rigid Rotor by Dudley Herschbach>[printed pp. 2–3: dipole transition integrals, polarization and selection rules]{ref}
<Hydrogen chloride> and <carbon monoxide> are examples with a permanent <electric dipole moment>. By symmetry, <molecular hydrogen> and <carbon dioxide> have no permanent <electric dipole moment> in the ideal symmetric internal state. They still rotate and have rotational <energy eigenvalues>, but the simple permanent-<electric dipole> spectrum described here is absent. This restriction concerns this interaction and these assumptions, not every possible way of detecting <molecular rotation>.
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