Higher-Order Overtones

The thirty-two silent modes of Coo have been studied by various techniques [7], the most fruitful being higher-order Raman and infra-red spectroscopy. Because of the molecular nature of solid Cqq, the higher-order spectra are relatively sharp. Thus overtone and combination modes can be resolved, and with the help of a force constant model for the vibrational modes, various observed molecular frequencies can be identified with specific vibrational modes. Using this strategy, the 32 silent intramolecular modes of Ceo have been determined [101, 102]. 

It has been pointed out that the central ( 2, transition does not experience any first-order quadrupole interaction. The absence of first-order broadening effects is a general property of symmetric (m, - m) transitions. There are cases where this can be a distinct advantage, the most direct instance being for integer spin nuclei (e.g. D and both 1=1) where there is no ( /2, — /2) transition. The main problem is to excite and detect such higher-order transitions, for which there are two separate approaches. The sample may either be irradiated and detected at the multiple quantum frequency (called overtone spectroscopy) or the MQ transition can be excited and a 2D sequence used to detect the effect on the observable magnetisation. 

Figure 1. Difference in energy between approximate and exact results for the (m,0) overtones of linear HDO. In local modes, m is the number of quanta in the O-D bond. The approximate results are uncoupled harmonic oscillators in normal modes (HO), uncoupled normal modes including all higher-order diagonal anharmonidties (H0-normal), and SCF in normal modes. From Ref. 17.

Only fundamental bands can have an intensity different from zero in the double harmonic approximation. Including higher order terms in the expansion allows the calculation of intensities of overtone bands, as well as adding contributions to the fundamental bands. 

Fig. 3.14 Level diagram for the generation of higher-order Stokes sidebands, which differ from the vibrational overtone frequencies...

The benzene preresonance Raman spectra show also fairly strong activity of the first overtone of mode Vi4(b2 ). This mode is dipole forbidden and thus cannot directly obtain its intensity through coupling of B2 with the ground state as discussed in Section IX,D. However, it is possible to formulate several higher-order vibronic coupling schemes that yield such overtone activity, depending on which state is the resonant state. The Raman intensity probably derives from the dipole allowed 2 transition... 

Most Raman-active substances show only one or two Stokes lines at the frequencies cos = cd] — coy in stimulated emission. At higher pump intensities, however, lines at the frequencies 0 = 001 —ncoy (n = l,2, 3), which do not correspond to overtones of vibrational frequencies, have been observed besides these Stokes lines. Because of the anharmonicity of molecular vibrations, the spontaneous Raman lines from vibrational overtones are shifted against n>L by Acu = ncoy — n - -n)xk, where Xk represent the anharmonicity constants. In Fig. 8.12 is illustrated that these higher-order Stokes lines are generated by consecutive Raman processes, induced by the pump wave, the Stokes wave, etc. 

Under normal conditions, the fundamental frequency oscillates in the quartz resonator. However, the characteristics of the external network can be utilized to promote oscillation in a higher order overtone mode. The exact overtone frequency is typically not an exact harmonic of the fundamental frequency. However, the overtone is normally close to a harmonic value. The external circuit is tuned to a frequency near the desired overtone frequency. 

The selection rules on Av can be extracted by applying the second quantization formalism to Eq. 6.98. In particular, the first-order terms proportional to Qi permit transitions with Avi = 1, the second-order terms in QiQj are responsible for the overtone and combination bands with A Vi + Vj) = 0 or +2, and so on. The symmetry selection rule must simultaneously be satisfied. It is well known to students in organic chemistry that overtone and combination bands are frequently prominent in infrared spectra, and so the second- and higher order terms in Eq. 6.98 are not negligible. 

The low correlation coefficient of 0.747 for the pH is caused by the small pH range of about 0.6 pH unit. On the other hand, determination of pH in cheese by NIRS is the result of many overlapping overtones that are also very weak. Consequently, it seems impossible to make a high precision calibration for the measurement of pH by NIRS. Mathematical manipulation of the raw NIR data of the cheese samples shows a high correlation or intercorrelation between the parameters — water sol. N/tot. N, TCA sol. N/tot. N, and water-soluble primary amines — and the total protein content. This is to be expected because during cheese ripening proteins are broken down to peptides and amino acids, mainly by enzymatic processes. Because the measurement of protein by NIRS is based on the absorption of casein molecules as well as a variety of peptides and amino acids, it is very difficult to resolve the protein absorption bands in a lot of smaller bands and to correlate these bands to the constituents from which they arise. The only possibility probably is to use higher order mathematical data transformations. 

Since the characters of the second overtone of an a2 fundamental are the same as those of the forbidden fundamental itself, this overtone is not allowed in the infrared. A simple repetition of the procedure used to obtain the characters of the second overtone will lead to the characters of overtones of higher order. 

For convenience, the NIR region is often divided into three sections. As summarized in Table 18.2, bands due to different origins are observed in the respective regions and the intensities of bands also vary with the regions. The intensities of bands in the shorter-wavelength region are usually much weaker, as they are due to higher-order overtones and combinations. 

Two other types of absorption band may also be observed overtone and combination bands. Overtone bands are observed at approximately twice the frequency of strong fundamental absorption bands (overtones of higher order having too low an intensity to be observed). Combination bands result from the combination (addition or subtraction) of two fundamental frequencies. 

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