Normal coordinates for linear molecules

For an unsymmetrical linear molecule there are N—1 non-degenerate vibrational normal coordinates belonging to the symmetry species 2+ (see Table 7-8.1) and they are of the longitudinal type and N—2 pairs of vibrational normal coordinates (doubly degenerate), each pair belonging to the II symmetry species and they are of the transverse type. 

Classifying the vibrational energy levels means finding out to which irreducible representation of the molecular point group the vibrational wavefunction(s) associated with a given level belong. 

First let us consider the vibrational ground state. The corresponding wavefunction is (see eqn (9-3.9)) 

Under the symmetry operation R the normal coordinate Q,u is transformed to Q p(k) which is a linear combination of Qptnp) and 

Consequently generates the identical representation T1 of the point group and we say that the ground state vibrational wavefunction is totally symmetric. 

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The classification cf normal coordinates Purthcr examples of normal coordinate clsasification Normal coordinates for linear molecules Classification of the vibrational levels 9-10. Infra-red spectra 0-11. Bamaa spectea... 

In Figure 1 are shown potential energy-normal coordinate curves for the vaig mode for site A as Fe111 in the reactants (R) and Fe11 in the products (P). The coordinate is a normal coordinate of the molecule composed of an equally weighted linear combination of the displacement coordinates (q) for the six Fe—O local coordinates, Q = l/V6[q(Fe—0)i + g(Fe—0)2 +. .. + g(Fe—0)6] = V6[q(Fe—O)]. It has no connection with an electron transfer distance nor with the intersite separation between reactants. 

According to classical theory the vibrational motion of a polyatomic molecule can be represented as a superposition of 3N-6 harmonic modes in each of which the atoms move synchronously (i.e. in phase) with a definite frequency v. These normal modes are characterized by time-dependent normal coordinates which indicate, on a mass-weighted scale, the relative displacement of the atoms from their equilibrium positions (Wilson et al., 1955). Figure 2 shows the general shape of the normal coordinates for a non-linear symmetric molecule AB2. The... 

For the normal mode analysis of translational and rotational coordinates of linear molecules the potential energy is expanded to second order around the configuration Ro with R= (ri. <1], ipi, , ca , d/v, d /v. r, is the centre-of-mass position, 0, and (p, arc the orientational variables of molecule i. In terms of the 5.V-dimensional... 

A normal coordinate for a polyatomic molecule can be expressed as a linear combination of the internal coordinates. The vibrational behavior of the atoms can be represented by attaching arrows to show their direction of motion. The lengths of the arrows are in proportion to the maximum amplitudes of each atom s normal coordinate excursion. The normal vibrations of water (H2O) are shown in Figure 9. 

Vibrational frequencies may be extracted from the PES by performing a normal mode analysis. This analysis of the normal vibrations of the molecular configurations is a difficult topic and can be pursued efficiently only with the aid of group theory and advanced matrix algebra. In essence, the 3 translational, 3 rotational and 3N-6 vibrational modes (2 rotational and 3N-5 vibrational modes for linear molecules) may be determined by a coordinate transformation such that all the vibrations separate and become independent normal modes, each performing oscillatory motion at a well defined vibrational frequency. As a more concrete illustration, assume harmonic vibrations and separable rotations. The PES can thus be approximated by a quadratic form in the coordinates... 

In either case it is necessary to obtain the kinetic and potential energies as quadratic forms in the velocities Si and the coordinates Si, respectively, using only the constant part of the coefficients as before. Iffie secular (Miuation will then have the same form as previously, (8), except that it will consist of only BN — 6 rows and columns BN — 5 for linear molecules). It is this reduction in the size of the secular equation which makes the use of internal coordinates useful, inasmuch as in most applica-1ions of the method of normal coordinates one of the most troublesome slops is the solution of the secular equation, a difficulty which increases rapidly with the degree of the equation. 

Vibrations are considered in terms of the classical expressions governing motion of nuclei vibrating about their equilibrium positions with a simple harmonic motion (40). The potential and kinetic potential energies of molecules are defined in terms of the coordinates most appropriate to the molecular structures. All relative motions of atoms about the center of mass (vibrations) are linear combinations of a set of coordinates, known as normal coordinates. For every normal mode of vibration, all coordinates vary periodically with the same frequency and pass through equilibrium simultaneously. 

There are many ways to describe the possible motions of the atoms in a molecule. However, a mathematical treatment of vibrations shows that there will always be a way to assign the changes in coordinates such that all of the possible motions of the atoms can be broken down into 3N — 5 (for linear molecules) or 3N — 6 (for nonlinear molecules) independent motions where for each motion the frequency of every atom s vibration is exactly the same. Such coordinate changes are called normal modes of vibration, or just the normal modes. 

The simplest description of vibrational degrees of lieedom of a molecule with N atoms is in terms of 3N — 6 or 3N — 5 (for linear molecules) normal vibrational modes. Vibrational analysis concerns the study of these normal vibrational modes. It is possible to define mass-weighted normal mode coordinates which provide an equivalent description of the molecular vibrations. Normal mode coordinate Qk a given normal mode k (k = 1, 3N — 6) corresponds to a specific vibrational pattern (displacements from equilibrium) on the molecule, for which all atoms oscillate at the same frequency k-... 

The description of the vibrations of polyatomic molecules only becomes mathematically tractable by treating the system as a set of coupled harmonic oscillators. Thus a set of 3N - 6 (3N - 5 for linear molecules) normal modes of vibrations can be described in which aU the nuclei in the molecule move in phase in a simple harmonic motion with the same frequency, normal-mode frequencies are solved, the normal coordinates for the vibrations can be determined, and how the nuclei move in each of the normal modes of vibration can be shown. There are two important points that follow from this. First, each normal mode can be classified in terms of the irreducible representations of the point group describing the overall symmetry of the molecule [7, 8]. This symmetry classification of the... 

The number of degrees of freedom equals the number of normal modes of vibration. The normal modes, also called fundamental modes, are a set of harmonic motions, each independent of the others and each having a distinct frequency. It is possible for two or more of the frequencies to be identical, and the corresponding modes are said to be degenerate. However, the total number of modes in the individual degenerate states are counted separately and still total 3A — 6 for nonlinear and 3 A — 5 for linear molecules. A set of coordinates can be defined, each of which gives the displacement in one of the normal modes of vibration. The normal coordinates can be expressed as combinations of the x-, y-, and z-coordinates of the individual nuclei. 

For molecules with an even number of electrons, the spin function has only single-valued representations just as the spatial wave function. For these molecules, any degenerate spin-orbit state is unstable in the symmetric conformation since there is always a nontotally symmetric normal coordinate along which the potential energy depends linearly. For example, for an - state of a C3 molecule, the spin function has species da and E that upon... 

The potential energy curve in Figure 6.4 is a two-dimensional plot, one dimension for the potential energy V and a second for the vibrational coordinate r. For a polyatomic molecule, with 3N — 6 (non-linear) or 3iV — 5 (linear) normal vibrations, it requires a [(3N — 6) - - 1]-or [(3A 5) -F 1]-dimensional surface to illustrate the variation of V with all the normal coordinates. Such a surface is known as a hypersurface and clearly cannot be illustrated in diagrammatic form. What we can do is take a section of the surface in two dimensions, corresponding to V and each of the normal coordinates in turn, thereby producing a potential energy curve for each normal coordinate. 

If we use a contour map to represent a three-dimensional surface, with each contour line representing constant potential energy, two vibrational coordinates can be illustrated. Figure 6.35 shows such a map for the linear molecule CO2. The coordinates used here are not normal coordinates but the two CO bond lengths rj and r2 shown in Figure 6.36(a). It is assumed that the molecule does not bend. 

Each normal mode of vibration can be described by a normal coordinate Qi which is a linear combination of nuclear displacement coordinates of the molecule. For the symmetric stretching vibration vi of C02, the normal coordinate is of the form... 

Work has also been conducted that involved the investigation, via infrared spectroscopy, of matrix-isolated, plutonium oxides (40), with the appropriate precautions being taken because of the toxicity of plutonium and its compounds. A sputtering technique was used to vaporize the metal. The IR spectra of PuO and PUO2 in both Ar and Kr matrices were identified, with the observed frequencies for the latter (794.25 and 786.80 cm", respectively) assigned to the stretchingmode of Pu 02. Normal-coordinate analysis of the PUO2 isotopomers, Pu 02, Pu 02, and Pu 0 0 in Ar showed that the molecule is linear. The PuO molecule was observed in multiple sites in Ar matrices, but not in Kr, with Pu 0 at 822.28 cm" in the most stable, Ar site, and at 817.27 cm" in Kr. No evidence for PuOa was observed. 

With the aid of a normal coordinate analysis involving different isotopomers a linear structure of the Pd-Si-0 molecule is deduced. The results of ab initio MP2 calculations (Tab. 4) confirm the experimentally obtained IR spectra and their interpretation. The Pd-C bond in PdCO is similar to the Pd-Si bond in PdSiO which means, that the donor bond is strengthened by x acceptor components. This conclusion is in line with the high value of the Pd-Si force constant (exp. f(PdSi) = 2.69, f(SiO) = 8.92 mdyn/A) as well as with the energy of PdSiO (Pd + SiO —> PdSiO + 182 kJ/mol for comparison Pd + CO —> PdCO + 162 kJ/mol, MP2 level of theory). 

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