Inactive normal vibration

We encounter here for the first time the occurrence of a normal vibration which is completely inactive as a fundamental. This phenomenon is not commonplace but is encountered occasionally in relatively symmetrical molecules. 

The infrared spectrum of physically adsorbed methane had a band at 2899 cm.-1 which is not present in the equivalent path-length of either the gas or the liquid. This corresponds to the symmetrical C—H stretching vibration which produces a Raman band at 2916 cm.-1. The spectrum of adsorbed ethylene shows an extra weak shoulder at 3010 cm.-1 which was assumed to be the normally infrared inactive v vibration in which all four hydrogens are vibrating in phase and which produces a Raman band at 3019 cm.-1. 

Here, fix, ny and /rz are the x, y and z components of the dipole moment at the electronic ground state, respectively, i/v and l/v, are vibrational wavefunctions where v and v" are the vibrational quantum numbers before and after the transition, respectively. Qa is the normal coordinate of the normal vibration, a. If one of these integrals is nonzero, this vibration is infrared-active. If all three integrals are zero, it is infrared-inactive. 

Christe and co-workers (3) obtained the XeFj ion as the tetra-methylammonium salt at — 86°C, and determined its structure by x-ray diffraction as well as vibrational spectroscopy. This anion takes a highly unusual D5h structure, shown in Fig. 4-1, which can be derived from that of a pentagonal bipyramidal IF7 in which the two axial fluorine ligands are replaced by two sterically active free valence electron pairs. The XeFj anion has 12 (3x6 — 6) normal vibrations that are classified into 1/1, (R) + 2 ,, (IR) + 2E 2(R) + E2 (inactive) under Dsh symmetry. 

The normal vibrations of H3 are shown in Fig. 1. The totally symmetric Vj mode is infrared inactive, and the doubly degenerate mode is infrared active. The doubly degenerate mode has unit vibrational angular momentum (fj = — 1) as initially shown by Teller. The vibrational states of H3 are specified by two vibrational quantum numbers and and the vibrational angular momentum quantum number 1. The vibrational energy structure of H3 relevant for astronomical observations is shown in Fig. 2. The transitions observed in the laboratory are shown by upward-pointing arrows, while the emissions observed in astronomical objects are shown by bold downward-pointing arrows. The numbers in parentheses are 111. 

Fig. 1 Normal vibrational modes for Hj. The totally symmetric v, mode (3178.3 cm" ) is Raman active and infrared inactive. The degenerate mode (2521.3 cm" ) is infrared active.

For simple molecules with n atoms, the vibrational spectra may be analyzed in terms of the 3n - 6 normal vibrations. According to the relevant symmetry point group, the number of IR- and Raman-active as well as the inactive... 

If one of these integrals is not zero, the normal vibration associated with is Raman-active. If all the integrals are zero, the vibration is Raman-inactive. As shown below, it is possible to determine whether the integrals of Eqs. 1.102 and 1.104 are zero or nonzero from a consideration of symmetry ... 

Tetrahedral X4-type molecules of Tj symmetry exhibit three normal vibrations as shown in Fig. 2.14, and Vi(Ai) and V2(E) are Raman-active whereas V3(F2) is IR as well as Raman-active. Table 2.5a lists observed frequencies of tetrahedral X4 molecules. Comparison of vibrational frequencies of the X (X = Si,Ge,Sn) series and the corresponding isoelectronic Y4 (Y = P,As,Sb) series shows that the ratio, v(X4 )/v (Y4) is approximately 0.77. On the basis of this criterion, Kliche et al. [770] predicted that the vibrational frequencies of the Pb4 ion are 115 (Ai), 69 ( ), and 93 (F2) cm On the other hand, the X4-type cation such as assumes a square-planar ring structure of D4/, symmetry, and its 6 (3 x 4-6) vibrations are classified into Aig (R)+ Big (R)+ B2g (R)+ 2M(inactive)+ (IR). Assignments of these vibrations have... 

The XeFs anion has 12 normal vibrations, which are classified into A ](R)+A2(IR)+2 j(IR)+2 2(R)+ 2 (inactive) under D5/, symmetry. Thus, only three vibrations (A 2 and 2Il ) are IR-active and only three vibrations (A +2 2) are Raman-active. In agreement with this prediction, the observed Raman spectmm of (CH3)4N[XeF5] exhibits three bands at 502 (symmetric stretch, A j), 423 (asymmetric stretch, E 2), and 377 cm (in-plane bending, As discussed in Sec. 1.11, the trigonal-bipyramidal and tetragonal-pyramidal structures can be ruled out since many more IR- and Raman-active bands are expected for these structures. 

The normal vibrational modes of ethylene, numbered according to the convention of Herzberg ( Infra-red and Raman spectra , van Nostrand, 1945) are shown in table 1. Since the molecule has a centre of symmetry no mode symmetrical with respect to this centre can be active in the infra-red and no antisymmetrical mode can be active in the Raman spectrum. The twisting mode number 4 is inactive in both. The most recent assignment (Crawford, Lancaster, and Inskeq>, J. Chem. Phys. 1953, 21, 678) of wave-numbers for the remaining eleven modes are given. 

If one of these integrals is not zero, the normal vibration associated with is infrared active. If all the integrals are zero, the vibration is infrared inactive. 

Besides the molecules just discussed, ozone, O3, should be mentioned in this connection. It is of special chemical interest since it usually has been supposed to have an equilateral triangular form. If this is so, the molecule should have a doubly degenerate and a non-degenerate normal vibration, of which the former should be active in absorption or emission and inactive in the Raman effect, while for the latter the converse would be true. The experimental investigations, however, appear to show that there are three active vibrations in absorption. This can only be accounted for by the assumption that the nuclear equilibrium configuration is triangular but not equilateral. Further work will be required to clear up this question entirely. 

Methane, CH4, is the most important representative of the regular tetrahedral structure. That it has a centre of symmetry is supported by its vanishing dipole moment and the absence of a rotational spectrum. Just as for linear molecules with a centre of symmetry we have for any other molecules with a centre of symmetry, that the normal vibrations symmetrical with respect to this centre are inactive in emission or absorption. The very strong frequency 2915 cm which only appears in the Raman spectrum must hence be identified with one of the normal vibrations v(s), 8 (s). More particularly it can be proved to be the vibration v (s) in which the outside atoms move in a radial direction with respect to the central atoms. The reason for this identification is the absence of rotational structure in the Raman line, in harmony with the circumstance that the rotation of the molecule vibrating in the way described will not affect its polarisability. Another frequency appearing both in the Raman and infra-red spectrum has the value 3022 cm and must be identified with v a). 

Summarizing, in the crystal there are 36 Raman active internal modes (symmetry species Ug, hig, 2g> and 26 infrared active internal modes (biw b2w hsu) as well as 12 Raman active and 7 infrared active external vibrations (librations and translations). Vibrations of the type are inactive because there appears no dipole moment along the normal coordinates in these vibrations of the crystal. 

Vibrational frequencies for various normal modes must be estimated and active as well as inactive energies should be decided. Numerical methods may be used to calculate rate constant k at various concentrations obtained by RRKM theory. The rate constant has been found to be same as given by conventional transition state theory, i.e. 

The analysis of vibration spectra proceeds by the use of normal modes. For instance, the vibration of a nonlinear water molecule has three degrees of freedom, which can be represented as three normal modes. The first mode is a symmetric stretch at 3586 cm , where the O atom moves up and the two H atoms move away from the O atom the second is an asymmetric stretch at 3725 cm where one H atom draws closer to the O atom but the other H atom pulls away and the third is a bending moment at 1595 cm , where the O atom moves down and the two H atoms move up and away diagonally. The linear CO2 molecule has four normal modes of vibration. The first is a symmetric stretch, which is inactive in the infrared, where the two O atoms move away from the central C atom the second is an asymmetric stretch at 2335 cm where both O atoms move right while the C atom moves left and the third and fourth together constitute a doubly degenerate bending motion at 663 cm where both O atoms move forward and the C atom moves backward, or both O atoms move upward and the C atom moves downward. 

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