Summation bands

As the intensity of summation bands do not depend on Boltzmann factors, most of the extra frequencies mentioned in the last two paragraphs will persist at least to some degree at low temperatures, i.e. they will contribute to a temperature independent residual band width. 

Fermi resonance of the vXH vibration with neighbouring overtone and summation frequencies—It has been explained above that Fermi resonance can occur between an anharmonic fundamental vibration such as rXH and other combination (summation) frequencies provided that the latter are of similar frequency to the fundamental and of the same symmetry class. In addition to the frequencies rXH j- nvXH Y that have already been discussed, other interacting summation frequencies might, for example, involve overtones of the SX.H vibration, or combinations of this with rXH Y. Most of the H-bonded systems that can conveniently be studied are part of complex molecules so that many other types of summation bands can often occur in the appropriate region. 

The enhancement of intensity of the overtone and summation bands occurs more strongly the smaller the frequency difference... 

An interaction, through anharmonicity, of the rXH vibration with other overtone and summation bands of similar frequency by Fermi resonance. This factor is more important with complex molecules. 

With regard to the intensities of the overtone and summation bands brought up by Fermi resonance with rXH, my own opinion is that these are more prominent than usual because the rXH fundamental has several components that cover a range of frequencies as envisaged in Stepanov s energy level scheme. However, it should be said that in both infrared and Raman spectra even the fairly harmonic rCH vibrations are often associated with strong extra bands caused by Fermi resonance with overtones, etc., of <5CH vibrations. 

Fig. 3.18 Schematic representation of summation bands. Data reproduced from C. W. Young, R. B. DuVall and N. Wright (1951). Analyt. Chent., 23,709.

It is good to remember too that we are dealing with combination tones. For example (0, 0)- (l, 1) or (vj + v3) is a binary combination (summation) band. Should (0,0)—+(1,1) be more intense than (0, 0)- (l, 0) that means that the combination is more intense than the fundamental. This is quite possible but it requires a large X13 coupling constant or/and a large amount of electrical anharmonicity. Finally, whether or not (0,0)->(l, 0) is the strongest band among the subbands due to combinations of Vj and v3 with various quantum numbers depends on several... 

These bands are partly hot and, of course many other such combinations could be imagined if for the lower frequency we used vp instead of v . The calculations of Robertson17) which take account of both the Franck-Condon and Boltzmann factors substantiate these conditions. For the HC1 complex at 300 °K the (0,1)- (1,0) difference band, the (0,0)->(l, 0) fundamental and the (Q, 0)-+(1,1) summation band which have relative intensities 0.39, 0.80 and 1,00 in this order receive contibutions from (other) hot bands of 0.16, 0.135 and 0.54 respectively. This means that the summation band can be about as sensitive to temperature as the difference band. 

One more comment should be made at this point. It concerns the anharmonic coupling constant between the v3 and va modes. In the case of the HC1 complex the great diffuseness of the 2670 cm-1 subband and the argument about the assignments made its evaluation unsafe. For the HF complex the situation is somewhat more favorable. As has been said in Section 2.1 the separation between vt and the difference band (vt — v ) is just v3 — v0 but the separation between vt and the summation band (vt — v ) is equal to + v + X1o where X1o is the coupling constant. On the basis of the above assignments its value is... 

For the methanol-HF complex the associated HF band vx has its renter at 3530 cm-1. Unfortunately the (vx -1- v ) summation band is overlapped by the methanol OH tend. The difference band is at 3340 cm-1 so that v can be estimated to be 190 cm-1. With DF vx moves to 2602 cm-1. All this is in line with the observation reported on other HF complexes in Section 3.2 and does not supply essentially new information on the nature of H-bonding. 

A binary combination band is due to a transition in which two vibrations change by one quantum. In a summation band each vibration gains one quantum V3u = vj + where Vj and are the combining frequencies. If the initial state of the molecule is not the vibrationless ground state (e.g. vj is singly excited in the initial state) the two vibrations can combine to yield a difference band, v irr = provided... 

Vfc > vj. Normally difference bands are much weaker than summation bands owing to the Boltzmann factor jjj general com-... 

For two fundamental frequencies, a and b, first overtones will occur near 2a and 2b, second overtones near 3a and 3b, etc., and combination bands can appear at a + b and a — hem . Summation bands are commonly observed in the near-infrared spectra of many molecules, but combination bands arising from difference tones are improbable in the near-infrared region at room temperature (Kaye, 1954). 

The strong, poorly resolved summation band c is superimposed on the very broad band b. Except for an expected relative intensity variation due to the differing number of ring methylene groups in the bile acids, this band is common to all six compounds. 

Whiffen [64] has recently been able to show that all the bands in this region can be assigned with reasonably high precision to summation bands of the CH out-of-plane fundamentals which occur between 1000 and 700 cm" This accounts for the variations in characteristic pattern with alterations in the positions of substitution and explains why the general shapes are more important in this case than the absolute frequencies. This also explains the simplification of the bands with more symmetrical structures in which a smaller number of out-of-plane CH fundamentals occur. 

Fig 8.8 Characteristic patterns of summation bands for various substituted benzene rings. The double letters (ee) are the binary combinations of modes seen in Fig 8.6... 

The out-of-plane aromatic C—H bending vibrations in the 1000-700 cm region give rise to relatively prominent summation bands in the infrared 2000-1650 cm" region, seen in Fig. 8.8. Since all the CH out-of-plane... 

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