Rovibrational selection rules

For weak light in the electric-dipole approximation, the transition amplitude involves the polarization projection of the molecular dipole between initial and final states. Separate electronic, vibrational, and rotational factors where the Born–Oppenheimer and rotation-vibration approximations allow it. Within one electronic state, expand the body-fixed dipole as a function of stretch displacement q about equilibrium. A constant dipole term gives no transition between different orthogonal vibrational states. The linear derivative term is the leading infrared contribution.

For harmonic vibrational wavefunctions, q connects levels differing by one quantum. The leading rule is therefore $\Delta v=\pm1$ when the dipole derivative is nonzero. Anharmonic wavefunctions and higher derivatives of the dipole allow overtones and modify intensities, so the harmonic rule is not an absolute prohibition in every molecular Hamiltonian. A permanent dipole is sufficient for elementary pure rotational electric-dipole activity, but vibrational activity specifically asks how that dipole changes with the normal coordinate.

For a polar diatomic in a closed-shell $^1\Sigma$ electronic state, the elementary electric-dipole rovibrational transitions within that electronic state have $\Delta J=\pm1$. The negative change forms the P branch and the positive change the R branch. A Q branch with zero change is absent for this specified parallel Sigma-to-Sigma case. Other electronic angular momenta or polyatomic band types can admit Q branches, so the restriction must not be promoted into a universal rule for every vibration.

If upper and lower vibrational states share one approximate rotational constant, subtracting their rotor terms gives particularly simple branch positions about the vibrational band origin. P lines lie below the origin and R lines above it. There is no P(0) because angular momentum cannot become negative. Unequal rotational constants produce quadratic J-dependent shifts and possible branch heads. Nuclear-spin restrictions and population factors further determine which permitted lines appear and with what strength.

Raman scattering couples through the polarizability tensor, because the incident field induces a dipole. A vibrational mode can be Raman active when polarizability changes along that coordinate even if the permanent dipole remains zero. A homonuclear diatomic stretch is the standard example: inversion forbids its isolated electric-dipole vibrational transition within the same electronic state, while its polarizability can vary. For centrosymmetric molecules the familiar infrared/Raman mutual exclusion refers to the symmetry of fundamental modes within the appropriate approximations.

Raman shifts measure the energy gained or lost by the molecule, rather than a requirement that the incident photon itself match an infrared absorption frequency. Stokes light leaves the molecule more excited, while anti-Stokes light removes an initial excitation. Rotational Raman rules arise from scalar and rank-two polarizability components and differ from dipole rules. These statements identify symmetry and coupling mechanisms; actual line intensities still require populations, matrix elements, polarization geometry, and the optical frequency factors appropriate to the experiment.