Fundamental transition selection rules

Since the fundamental transitions generally give rise to IR absorption bands and Raman lines which are more intense by at least an order of magnitude than any other kinds of transition, they are of the greatest interest and we shall deal only with the fundamentals here. Selection rules for other types of transition can also be obtained by arguments of the type we shall use, but the reader is cautioned that where degenerate modes are concerned subtle complications often arise.t... 

These signals correspond precisely to the characteristic vibrations of the molecule. If v = 0 they are called fundamental bands. As levels with v > 0 are normally not significantly populated, these bands account for most of the intensity in an absorption spectrum. The IR absorption spectrum and the Stokes Raman scattering spectrum both correspond to v —> v + 1 transitions, so the fundamental vibrational selection rule is normally stated as Av = +1. We should remember that Av = — 1 is also permitted, but this corresponds to the emission of radiation in the IR or to the anti-Stokes lines in the Raman spectrum, which will be weak, as the population of molecules with v > 0 will be small. 

For a fundamental transition to occur by absorption of infrared dipole radiation, it is necessary that one or more of these integrals (and consequently the intensity) be nonzero. It follows from the selection rule given above that in order that a transition be infrared active p must have the same symmetry properties as at least one of x, y, or z. 

In a case where the transition of an energy state is from 0 to 1 in any one of the vibrational states (vi,v2,v3,. ..), the transition is considered as fundamental and is allowed by selection rules. When a transition is from the ground state to v — 2,3,. .., and all others are zero, it is known as an overtone. Transitions from the ground state to a state for which Vj = 1 and vj = 1 simultaneously are known as combination bands. Other combinations, such as v — 1, Vj = 1, v = 1, or v, — 2, v7 — 1, etc., are also possible. In the strictest form, overtones and combinations are not allowed, however they do appear (weaker than fundamentals) due to anharmonicity or Fermi resonance. 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

Anharmonicity leads to deviations of two kinds. At higher quantum numbers, AE becomes smaller, and the selection rule is not rigorously followed as a result, transitions of A 2 or 3 are observed. Such transformations are responsible for the appearance of overtone lines at frequencies approximately two or three times that of the fundamental line the intensity of overtone absorption is frequently low, and the peaks may not be observed. Vibrational spectra are further comphcated by the fact that two different vibrations in a molecule can interact to give absorption peaks with frequencies that are approximately the sums or differences of their fundamental frequencies. Again, the intensities of combination and difference peaks are generally low. 

The infra-red and Raman spectra of molecules are dominated by transitions between the ground state and the fundamental levels but, in practice, the number of fundamental frequencies observed does not reach 3JV—6 since (a) some of the Xt are identical (leading to degenerate fundamental levels) and (b) selection rules forbid certain transitions. Both (a) and (b) are determined by the symmetry of the molecule. 

A complex 7THG can result from one-, two-, or three-photon resonances. One-photon resonance occurs when the fundamental frequency co is close to an allowed electronic transition. Two-photon resonance occurs when 2co is close to a two-photon allowed electronic transition. For centrosymmetric molecules the two-photon selection rule couples states of like inversion symmetry, e.g. g <- g. For acentric molecules one-photon transitions can also be two-photon allowed. Three-photon resonance occurs when 3co is close to the energy of an electronic transition the same symmetry rules apply as for one-photon transitions. 

Allowed transitions in the harmonic approximation are those for which the vibrational quantum number changes by one unit. Overtones - that is, the absorption of light at a whole number times the fundamental frequency - would not be possible. A general selection rule for the absorption of a photon is that the dipole... 

Step 4. For a vibrational mode to be infrared (IR) active, it must bring about a change in the molecule s dipole moment. Since the symmetry species of the dipole moment s components are the same as rx, ry, and 1, a normal mode having the same symmetry as Ix, Ey, or 1 will be infrared active. The argument employed here is very similar to that used in the derivation of the selection rules for electric dipole transitions (Section 7.1.3). So, of the six vibrations of NH3, all are infrared active, and they comprise four normal modes with distinct fundamental frequencies. 

Avk=(vk+jic)= Avj=(vj+jj)=0, j k. A photon-induced transition via an external field occurs only for a given fundamental frequency, for w = between adjacent vibrational states, Vk and (vk l). Non-vanishing terms, to higher order in Eq. (28), mechanical anharmonicity, and non-linear terms in the dipole-moment function, Eq. (29), lead to breakdown in these strict selection rules, and frequencies other than the set, a n l>u2> u... . . to, become observable in the dipole-allowed spectrum. 

Although we have been able to see on inspection which vibrational fundamentals of water and acetylene are infrared active, in general this is not the case. It is also not the case for vibrational overtone and combination tone transitions. To be able to obtain selection rules for all infrared vibrational transitions in any polyatomic molecule we must resort to symmetry arguments. 

According to quantum mechanics, only those transitions involving Ad = 1 are allowed for a harmonic oscillator. If the vibration is anhar-monic, however, transitions involving Au = 2, 3,. .. (overtones) are also weakly allowed by selection rules. Among many Au = 1 transitions, that of u = 0 <-> 1 (fundamental) appears most strongly both in IR and Raman spectra. This is expected from the Maxwell-Boltzmann distribution law, which states that the population ratio of the u = 1 and u = 0 states is given by... 

Raman Selection Rules. For polyatomic molecules a number of Stokes Raman bands are observed, each corresponding to an allowed transition between two vibrational energy levels of the molecule. (An allowed transition is one for which the intensity is not uniquely zero owing to symmetry.) As in the case of infrared spectroscopy (see Exp. 38), only the fundamental transitions (corresponding to frequencies v, V2, v, ...) are usually intense enough to be observed, although weak overtone and combination Raman bands are sometimes detected. For molecules with appreciable symmetry, some fundamental transitions may be absent in the Raman and/or infrared spectra. The essential requirement is that the transition moment F (whose square determines the intensity) be nonzero i.e.. 

Selection Rules. The harmonic-oscillator, rigid-rotor selection mles are Av = 1 and AJ = 1 that is, infrared emission or absorption can occur only when these allowed transitions take place. For an anharmonic diatomic molecule, the A7 = 1 selection mle is still valid, but weak transitions corresponding to An = 2, 3, etc. (overtones) can now be observed. Since we are interested in the most intense absorption band (the fundamental ), we are concerned with transitions from various J levels of the vibrational ground... 

The second type of fundamental vibrations involves an oscillating dipole moment, oriented perpendicularly to the unique molecular axis. The corresponding infrared selection rules for the rotational transitions are given by... 

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