Normal vibration modes

The method of vibrational analysis presented here ean work for any polyatomie moleeule. One knows the mass-weighted Hessian and then eomputes the non-zero eigenvalues whieh then provide the squares of the normal mode vibrational frequeneies. Point group symmetry ean be used to bloek diagonalize this Hessian and to label the vibrational modes aeeording to symmetry. 

The polarlsablllty of a molecule will vary during vibrations which change the Internuclear separations. Thus the vibrations of a molecule sitting In an electrical field will be coupled to the field via the polarlsablllty. This should be particularly noticeable for a molecule adsorbed on an electrode surface where the field strength Is typically In the range 10 -10 V cm, The dipole, perpendicular to the surface, Induced In the molecule by the static electric field will fluctuate In step with the normal mode vibrations of the molecule. 

The number of peaks in an IR spectrum may increase due to overtones. Normally, the vibrational level is allowed to change by +1. If the vibrational energy level changes by 2 or more (a forbidden transition), an overtone results. It is also possible for two normal mode vibrations to combine into a third. 

Figure 4.19 Normal-mode vibrational quantum numbers for a bent triatomic molecule. Contrast the results for water, which is (cf. Table 4.6) near the local-mode limit with that for S02, which is near the normal-mode limit.

How one obtains the three normal mode vibrational frequencies of the water molecule corresponding to the three vibrational degrees of freedom of the water molecule will be the subject of the following section. The H20 molecule has three normal vibrational frequencies which can be determined by vibrational spectroscopy. There are four force constants in the harmonic force field that are not known (see Equation 3.6). The values of four force constants cannot be determined from three observed frequencies. One needs additional information about the potential function in order to determine all four force constants. Here comes one of the first applications of isotope effects. If one has frequencies for both H20 and D20, one knows that these frequencies result from different atomic masses vibrating on the same potential function within the Born-Oppenheimer approximation. Thus, we... 

From Equation 4.79, it is then recognized that the isotope effect is given by a symmetry number factor and terms which depend only on the normal mode vibrational frequencies. There are no terms in the equality that depend explicitly on atomic and molecular masses or on moments of inertia. 

This result indicates that (s2/si)f (compare Equations4.78 and 4.93) is just the quantum effect on the molecular partition functions of the normal mode vibrations. This result has now been derived without the explicit use of the Teller-Redlich product rule. 

One can simplify Equation 4.95 and obtain a very interesting result. We previously obtained the normal mode vibrational frequencies v by diagonalization of the matrix of the harmonic force constants in mass weighted Cartesian coordinates (Chapter 3). These force constants Fy were obtained from the force constants in Cartesian coordinates fq by using... 

The (3N — 6) non-zero eigenvalues A of the matrix F are related to the normal mode vibrational frequencies by... 

TABLE 4. Normal-mode vibrational frequencies (cm ) of CH3CI and CH3MgCl ... 

The functions (6.85) each correspond to a specific pair of the normal-mode vibrational quantum numbers v2a and v2b, whereas the correct zeroth-order functions (6.87) do not. Hence the quantum numbers v2a and v2b are not particularly physically significant. The quantum numbers of physical significance are v2 = v2a -I- v2b, which (together with c, and t>3) specifies which degenerate vibrational level we are dealing with, and the vibrational angular-momentum quantum number. 

One of the earliest models is the quasi-diatomic model (10-13). This model is based on the assumption that the normal modes describing the state(s) of the photofragments are also the normal modes of the precursor molecule. This means, for example, that in the photodissociation of a linear triatomic molecule ABC A + BC (e.g., photodissociation of ICN - I + CN), the diatomic oscillator BC is- assumed to be a normal mode vibration in the description of the initial state of the triatomic molecule ABC. This means that the force constant matrix describing the vibrational motion of the molecule ABC can be written in the form (ignoring the bending motion) ... 

For the predissociative C2N2 (C- TIU) state in the collinear approximation, the nuclear wavefunction is approximated by the product of three harmonic oscillator functions describing the normal modes vibrations. The frequencies and normal coordinates of the three linear stretching vibrations were obtained from ab initio MCHF calculations. The validity of the harmonic approximations is supported from experimental data (8) where absorption spectra of C2N2 is found to give a set of equidistant bands. 

Within the theory presented here, the onset of superconductivity, Tc, is when the condition, v = co, takes place. At this point, the normal conduction through free-electrons becomes zero and therefore is frozen . As stated before, the phonon (antisymmetric normal mode) vibration is written as ... 

Stationary Points and Normal-Mode Vibrations - Zero Point Energy... 

Fig. 2.18 The normal-mode vibrations of water. The arrows indicate the directions in which the atoms move on reaching the maximum amplitude these directions are reversed...

Calculation of the ZPE is more involved it requires calculating the frequencies (i.e. the normal-mode vibrations - Section 2.5) and summing the energies of each... 

The normal-mode vibrational frequencies of a molecule correspond, with qualifications, to the bands seen in the infrared (IR) spectrum of the substance. Discrepancies may arise from overtone and combination bands in the experimental IR, and from problems in accurate calculation of relative intensities (less so, probably, from problems in calculation of frequency positions). Thus the IR spectrum of a substance that has never been made can be calculated to serve as a guide for the experimentalist. Unidentified IR bands observed in an experiment can sometimes be assigned to a particular substance on the basis of the calculated spectrum of a suspect if the spectra of the usual suspects are not available from experiment (they might be extremely reactive, transient species), we can calculate them. 

The vibrational motion of polyatomic molecules encompasses all nuclei in the molecule and, as long as the displacement from the equilibrium configuration is sufficiently small, it can be broken down into the so-called normal-mode vibrations (see Appendix E). In special cases these vibrations take a particular simple form. Consider, e.g., a partially deuterated water molecule HOD. In this molecule, the H OD and HO-D stretching motions are largely independent and the normal modes are, essentially, equivalent to the local bond-stretching modes. To that end, consider the following reaction that has been studied experimentally [6,7] as well as theoretically [8] ... 

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