Vibrational anharmonidty

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. K a more elaborate treatment is required, the suimnation for the partition functions must be carried out explicitly. An approximate account for vibrational anharmonidty can be obtained by using the harmonic form for the partition function (and resulting enthalpy and entropy... 

The relationship between the potential function K(R) and the observable spectroscopic parameters is summarized in Figure 2. The harmonic vibration frequencies are obtained as the eigenvalues of a secular determinant involving the quadratic force constants and the atomic masses and molecular geometry (the F and G matrices of Wilson s well-known formalism) by a calculational procedure discussed in detail by Wilson, Decius, and Cross.1 The eigenvectors determine the normal coordinates Q in terms of which the kinetic and quadratic potential energy terms are both diagonal (R = LQ). The various anharmonidty constants and vibration/rotation interaction constants are obtained in terms of the... 

It may seen in Fig. 4 that the calculations and experiments agree well, up to approximately room temperature. Above this, they diverge due to the onset of significant anharmonidty in the vibrations, an effect not incorporated in the calculated values. 

A practical conclusion about force constants is that correlation effeas are relatively unimportant for the anhatmonic constants. Pulay et al. have explained that force constants need to be considered in terms of a valence electronic part and a core-plus-nuclear part. The core contribution is changing more rapidly around the equilibrium than is the valence part, since the core—core potentials are essentially a 1/R repulsion, being that they involve small, positively charged entities. Core—valence attraction will also be changing more rapidly than the valence part, and this may partly offset the core-core repulsion. The relative effect of electron correlation on the core—core potentials should be small, and if they are the leading source of anharmonidty around the equihbrium, then it fits that electron correladon is not important on the anharmonic constants. The practical result is that accurate vibrational analysis is possible by carrying out correlated calculations to obtain the harmonic force constants, but by using only SCF calculations to obtain cubic and quartic force constants. 

The difference between the frequencies of the fundamental and hot modes is 20cm for poly(dioxolane) and 40cm" for polyftetramethylene oxide). It means that anharmonidty of torsional vibrations is much larger for the latter polymer which is consistent with the nuclear magnetic resonance measurements of potential barriers of molecular motion in these polymers [178]. In order to assign unambiguously the lower-ftequou bands considered to be hot transitions modes, temperature measurements are required. 

Apart from the change of the dipole moment D during the vibration, the intensities of overtone absorption bands additionally depend on the mechanical anharmonidty of the osdllator and it has been shown that vibrations with low anharmonidty constants x also have low overtone intensities. X-H stretching vibrations, for example, have large/values and therefore lead to dominant absorption bands in the NIR spectra. Table 1 summarizes the anharmonicity constants of the vibrations of some characteristic chemical functionalities. 

---