Vibrational frequencies normal modes

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. 

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... 

The method of vibrational analysis presented here can work for any polyatomic molecule. One knows the mass-weighted Hessian and then computes the non-zero eigenvalues which then provide the squares of the normal mode vibrational frequencies. Point group symmetry can be used to block diagonalize this Hessian and to label the vibrational modes according to symmetry. 

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

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. 

Calculated Frequencies. Table II contains the normal-mode vibrational frequencies vu of the light isotopic species, and the frequency shifts A Vi = vii — V2i upon isotopic substitution, calculated with the force fields listed in Table I. The force field for NOa" reproduces the observed frequencies and frequency shifts very well, whereas the calculated frequencies and shifts for N02 differ somewhat from those observed. However, we consider the general quadratic potential used in the calculation the best fit to the observed frequencies. The discrepancy is caused by a disagreement of the observed (2) frequencies with the Teller-Redlich product rule, which is, of course, assumed in the calculations. 

Thus, the /, in Eq. (2.46) are one of the 3Arows of L while the l i in Eq. (2.35) are one of the 3N columns. For a nonlinear molecule there are six zero eigenvalues in A, which correspond to translation and external rotation motions. The remaining 3N — 6 nonzero eigenvalues equal where the v, s are the normal mode vibrational frequencies. 

The force constants associated with a molecule s potential energy minimum are the harmonic values, which can be found from harmonic normal mode vibrational frequencies. For small polyatomic molecules it is possible (Duncan et al., 1973) to extract harmonic normal mode vibrational frequencies from the experimental anhar-monic n = 0 — 1 normal mode transition frequencies (the harmonic frequencies are usually approximately 5% larger than the anharmonic 0 - 1 transition frequencies). Using a normal mode analysis as described in chapter 2, internal coordinate force constants (e.g., table 2.4) may be determined for the molecule by fitting the harmonic frequencies. 

For his calculations, Burton chose the simplest possible material—a cluster of atoms interacting with nearest-neighbor harmonic forces and with the atoms packed onto lattice positions of a close-packed cubic material. He then calculated the partition functions in Eq. (43) and ultimately the cluster concentrations and nucleation rates. The major problem in this calculation was the internal partition function Zmt> which was calculated by diagonalizing the 3i X 3i dynamical matrices of i atom clusters to obtain normal mode vibrational frequencies and ultimately harmonic oscillator partition functions. This calculation was very expensive and could not be done for i larger than about 100. 

Other cross-coupling terms include bond-torsion and bend-bend-torsion. Cross-coupling terms are important for accurate modeling of normal mode vibrational frequencies and to better model the potential at large deformation (i.e., positions far from the potential minimum). 

V = normal-mode vibrational frequency kfi = Boltzmann constant h = Planck constant T = absolute temperature a = symmetry number... 

Normal mode Vibration frequencies Tetrahedral Multiplicity methyl in cm" Planar methyl... 

The parameter vo,with dimensions of velocity, is determined by the saddle point procedure discnssed in Problem F. It is found to be a nearly linear function of the vibrational frequency, as expected from Eq. (9.8). Similarly, the collision number ZvT for V—T transfer in polyatomics is found to be an approximate exponential function of the lowest normal mode vibrational frequency... 

Normal-mode vibrational frequencies (see Normal Modes) are then obtained in order to confirm that the equilibrium structures are minima on the relevant molecular potential surfaces. Local minima have no imaginary frequencies. The theoretical harmonic vibrational frequencies are also used to determine zero-point vibrational energies (ZPEs) and to correct calculated thermochemical data to 298 K (see below). Since... 

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