Editing Rotational transition (section)

=== Nuclear wave function ===
Quantum theoretical analysis of a molecule is simplified by use of [[Born–Oppenheimer approximation]]. Typically, rotational energies of molecules are smaller than [[molecular electronic transition|electronic transition]] energies by a factor of {{math|''m''/''M''}} ≈ 10<sup>−3</sup>–10<sup>−5</sup>, where {{math|''m''}} is electronic mass and {{math|''M''}} is typical nuclear mass.<ref>Chapter 10, ''Physics of Atoms and Molecules'', B.H. Bransden and C.J. Jochain, Pearson education, 2nd edition.</ref> From [[uncertainty principle]], period of motion is of the order of the [[Planck constant]] {{math|''h''}} divided by its energy. Hence nuclear rotational periods are much longer than the electronic periods. So electronic and nuclear motions can be treated separately. In the simple case of a diatomic molecule, the radial part of the [[Schrödinger Equation]] for a nuclear wave function {{math|''F''<sub>''s''</sub>('''R''')}}, in an electronic state {{math|''s''}}, is written as (neglecting spin interactions)
<math display="block">\left[- \frac{\hbar^2}{2\mu R^2} \frac{\partial}{\partial R} \left(R^2 \frac{\partial}{\partial R}\right)+ \frac{\langle \Phi_s|N^2|\Phi_s \rangle}{2\mu R^2}+ E_s(R)-E\right]F_s(\mathbf R) = 0 </math>
where {{math|''μ''}} is [[reduced mass]] of two nuclei, {{math|'''R'''}} is vector joining the two nuclei, {{math|''E''<sub>''s''</sub>(''R'')}} is energy [[eigenvalue]] of electronic wave function {{math|Φ<sub>''s''</sub>}} representing electronic state {{math|''s''}} and {{math|''N''}} is orbital [[momentum operator]] for the relative motion of the two nuclei given by
<math display="block"> N^2 = -\hbar^2 \left[ \frac{1}{\sin\Theta} \frac{\partial}{\partial \Theta}\left(\sin \Theta \frac{\partial}{\partial \Theta}\right)+ \frac{1}{\sin^2\Theta} \frac{\partial^2}{\partial \Phi^2} \right] </math>
The total [[wave function]] for the molecule is
<math display="block"> \Psi_s = F_s(\mathbf R)\Phi_s(\mathbf R,\mathbf r_1, \mathbf r_2, \dots, \mathbf r_N)</math>
where {{math|'''r'''<sub>''i''</sub>}} are position vectors from center of mass of molecule to {{math|''i''}}th electron.
As a consequence of the Born-Oppenheimer approximation, the electronic wave functions {{math|Φ<sub>s</sub>}} is considered to vary very slowly with {{math|'''R'''}}. Thus the Schrödinger equation for an electronic wave function is first solved to obtain {{math|''E''<sub>''s''</sub>(''R'')}} for different values of {{math|''R''}}. {{math|''E''<sub>''s''</sub>}} then plays role of a [[potential well]] in analysis of nuclear wave functions {{math|''F''<sub>''s''</sub>('''R''')}}.

[[File:Angular Momentum of a Diatomic Molecule.png|thumb|Vector addition triangle for orbital angular momentum of a diatomic molecule with components of orbital angular momentum of nuclei and orbital angular momentum of electrons, neglecting coupling between electron and nuclear orbital motion and spin-dependent coupling.Since angular momentum {{math|'''N'''}} of nuclei is perpendicular to internuclear vector {{math|'''R'''}}, components of electronic angular momentum {{math|'''L'''}} and total angular momentum {{math|'''J'''}} along {{math|'''R'''}} are equal.]]