Molecules fundamental vibrational frequencies

For a nonlinear molecule composed of N atoms, 3N—6 eigenvalues provide the normal or fundamental vibrational frequencies of the vibration and and the associated eigenvectors, called normal modes give the directions and relative amplitudes of the atomic displacements in each mode. 

In Chapter 10, we will make quantitative calculations of U- U0 and the other thermodynamic properties for a gas, based on the molecular parameters of the molecules such as mass, bond angles, bond lengths, fundamental vibrational frequencies, and electronic energy levels and degeneracies. 

Table 10.2 Fundamental vibrational frequencies of some common molecules.

Tables 10.1, 10.2, and 10.3e summarize moments of inertia (rotational constants), fundamental vibrational frequencies (vibrational constants), and differences in energy between electronic energy levels for a number of common molecules or atoms/The values given in these tables can be used to calculate the rotational, vibrational, and electronic energy levels. They will be useful as we calculate the thermodynamic properties of the ideal gas.

Calculation of Thermodynamic Properties We note that the translational contributions to the thermodynamic properties depend on the mass or molecular weight of the molecule, the rotational contributions on the moments of inertia, the vibrational contributions on the fundamental vibrational frequencies, and the electronic contributions on the energies and statistical weight factors for the electronic states. With the aid of this information, as summarized in Tables 10.1 to 10.3 for a number of molecules, and the thermodynamic relationships summarized in Table 10.4, we can calculate a... 

Vibration The vibrational contribution is calculated from the fundamental vibrational frequencies and the relationship in Table 10.4. CO is a linear molecule with (3r/ - 5) = 4 fundamentals. The values of Jj are obtained from... 

E10.6 For the diatomic molecule Na2, 5 = 230.476 J-K-1-mol" at T= 300 K, and 256.876 J-K-,-mol-1 at T= 600 K. Assume the rigid rotator and harmonic oscillator approximations and calculate u, the fundamental vibrational frequency and r, the interatomic separation between the atoms in the molecule. For a diatomic molecule, the moment of inertia is given by l pr2, where p is the reduced mass given by... 

El0.7 Carbonyl sulfide (OCS) is a linear molecule with a moment of inertia of 137 x 10-40 g em2. The three fundamental vibrational frequencies are 521.50, 859.2, and 2050.5 cm-1, but one is degenerate and needs to be counted twice in calculating the entropy. A Third Law measurement of the entropy of OCS (ideal gas) at the normal boiling point of T = 222.87 K andp = 0.101325 MPa gives a value of 219.9 J-K- -mol"1. Use this result to decide which vibrational frequency should be given double weight. 

Tables A4.2 and A4.3 summarize moments of inertia and fundamental vibrational frequencies of some common molecules, while Table A4.4 gives electronic energy levels for some common molecules or atoms with unpaired electrons.

In the construction of the matrix F of Eq. (63), the symmetrical equivalence of the two O-H bonds was taken into account. Nevertheless, it contains four independent force constants. As the water molecule has but three fundamental vibrational frequencies, at least one interaction constant must be neglected or some other constraint introduced. If all of the off-diagonal elements of F are neglected, the two principal constants, f, and / constitute the valence force field for this molecule. However, to reproduce the three observed vibrational frequencies this force field must be modified to include the interaction constant... 

As presented in the example of ethylene oxide above, it is often beneficial to obtain the IR spectra of isotopomers of the system under study. The isotopomers also were useful in the interpretation of the IR spectra of cyclopropene. In Table 2 the observed and calculated (MP2/6-31G ) isotopic shifts for three of the isotopomers of cyclopropene are given. Comparison of the calculated shifts with those observed indicates that theory reproduces well experimental results. Such calculated shifts can be extremely useful in assigning the origins (symmetries) of the fundamental vibrational frequencies of the parent molecule. 

The photoelectron spectra of the isoelectronic molecules N2 and CO have peaks corresponding to the ionization energies, 7, given in Table 4.3. The peaks corresponding to the ionized molecules have vibrational structure, with vibrational separations in wavenumbers given in Table 4.3. The fundamental vibration frequencies of the neutral molecules N2 and CO are 2345 and 2143 cm1, respectively. 

The quantity Z is the number of collisions per unit time suffered by the oscillator when the gas density is one molecule per unit volume, N is the total concentration of heat bath molecules, and 0 = hv/kT, where h is Planck s constant, k the Boltzmann constant, T the absolute temperature, and v the fundamental vibrational frequency of the oscillators. 

Other systems consisting of molecules other than H2 have similar rotovibrational spectra. However, the various rotational lines cannot usually be resolved, owing to the smallness of the rotational constants B and the typically very diffuse induced lines. One example, the spectrum of compressed oxygen, was shown above, Fig. 1.1. It consists basically of three branches, the Q, S, and O branch. The latter two are fairly well modeled by the envelope of the rotational stick spectra, similar to that shown in Fig. 3.20, but shifted by the fundamental vibration frequency. 

The above discussion has outlined the theoretical approach to the determination of the fundamental vibration frequencies of a molecule. The practical solution of the problem as formulated above presents, however, certain more or less serious difficulties. For example, the completely general potential function of equation (4) is generally not usable even for small molecules, because it contains more independent constants than can be determined from the experimental data. However, by making certain assumptions about the nature of the force field in the molecule, the number of constants can be reduced. One assumption which often works quite well in practice is that of a valence force field [Herzberg 76)]. This assumes that contributions to the potential energy... 

Expressions for the partition function can be obtained for each type of energy level in an atom or molecule. These relationships can then be used to derive equations for calculating the thermodynamic functions of an ideal gas. Table 11.4 or Table A6.1 in Appendix 6 summarize the equations for calculating the translational, rotational, and vibrational contributions to the thermodynamic functions, assuming the molecule is a rigid rotator and harmonic oscillator.yy Moments of inertia and fundamental vibrational frequencies for a number of molecules are given in Tables A6.2 to A6.4 of Appendix 6. From these values, the thermodynamic functions can be calculated with the aid of Table 11.4. 

The more detailed question as to whether multiple loss of vibrational energy quanta is probable is much more difficult to answer decisively. The view is generally accepted that stepwise loss of vibrational energy is usual. The vibrational energy is much more rapidly equilibrated for molecules with low fundamental vibration frequencies such as iodine (co = 215)30 than it is for molecules with high frequencies such as nitrogen (co = 2360)30. 

Calculation of partition functions requires spectroscopic quantities for the rotational and vibrational partition functions. The quantities required are moments of inertia, rotational symmetry numbers and fundamental vibration frequencies for all normal modes of vibration. The translational terms require the mass of the molecule. All terms depend on temperature. Calculation of partition functions is routine for species for which a detailed spectroscopic analysis has been made. 

In a diatomic molecule, the masses mv and m2 vibrate back and forth relative to their centre of mass in opposite directions, as shown in the following figure. The two masses reach the extremes of their respective motions at the same time. The diatomic molecule has only one vibrational degree of freedom, i.e., it has only one frequency, called the fundamental vibrational frequency. During vibrational motion, the bond of the molecule behave like a spring and the molecule exhibits a simple harmonic motion provided the displacement of the nuclei from the equilibrium configuration is not too much. At the two extremes of motion which correspond to extension and compression of the chemical bond between the two atoms, the potential energy is maximum. On... 

Spectroscopic data. This attractive molecule has long been a favourite for spectroscopic studies and only the most comprehensive or recent data are given here. The four fundamental vibrational frequencies have often been measured for the molecule since all four are Raman active but only two (the F2) are IR active we concentrate on the Raman data (Table 24). 

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