Selection rule
The selection rules may differ according to the technique used to observe the transition.
In quantum mechanics the basis for a spectroscopic selection rule is the value of the transition moment integral [1] where
If the transition moment function is symmetric over all of the totally symmetric representation of the point group to which the atom or molecule belongs, then the integral's value is (in general) not zero and the transition is allowed.
The symmetry of the transition moment function is the direct product of the parities of its three components.
The symmetry characteristics of each component can be obtained from standard character tables.
Rules for obtaining the symmetries of a direct product can be found in texts on character tables.
[2] The Laporte rule is a selection rule formally stated as follows: In a centrosymmetric environment, transitions between like atomic orbitals such as s-s, p-p, d-d, or f-f, transitions are forbidden.
The Laporte rule (law) applies to electric dipole transitions, so the operator has u symmetry (meaning ungerade, odd).
In formal terms, only states with the same total spin quantum number are "spin-allowed".
Both can be observed, in spite of the Laporte rule, because the actual transitions are coupled to vibrations that are anti-symmetric and have the same symmetry as the dipole moment operator.
It is, therefore, a basis for the totally symmetric representation in the point group of the molecule.
[7] In infrared spectroscopy, the transition moment operator transforms as either x and/or y and/or z.
The excited state wave function must also transform as at least one of these vectors.
In Raman spectroscopy, the operator transforms as one of the second-order terms in the right-most column of the character table, below.
[9] In the harmonic approximation, it can be shown that overtones are forbidden in both infrared and Raman spectra.
[11] Displacements from the ideal structure can result in relaxation of the selection rules and appearance of these unexpected phonon modes in the spectra.
Therefore, the appearance of new modes in the spectra can be a useful indicator of symmetry breakdown.
[15] In rovibronic transitions, the excited states involve three wave functions.
This is typical of the infrared spectra of heteronuclear diatomic molecules.
[16] Resonance Raman spectroscopy involves a kind of vibronic coupling.
[17] In spite of appearances, the selection rules are the same as in Raman spectroscopy.
[18] In general, electric (charge) radiation or magnetic (current, magnetic moment) radiation can be classified into multipoles Eλ (electric) or Mλ (magnetic) of order 2λ, e.g., E1 for electric dipole, E2 for quadrupole, or E3 for octupole.
In transitions where the change in angular momentum between the initial and final states makes several multipole radiations possible, usually the lowest-order multipoles are overwhelmingly more likely, and dominate the transition.
The corresponding quantum numbers λ and μ (z-axis angular momentum) must satisfy and Parity is also preserved.
These considerations generate different sets of transitions rules depending on the multipole order and type.
[20] Semi-forbidden transitions (resulting in so-called intercombination lines) are electric dipole (E1) transitions for which the selection rule that the spin does not change is violated.
is the secondary total angular momentum quantum number.
In hyperfine structure, the total angular momentum of the atom is
The dipole moment of the molecule and the image charges perpendicular to the surface reinforce each other.
In contrast, the dipole moments of the molecule and the image charges parallel to the surface cancel out.
