Polyatomic molecules normal mode

Even for molecules in the ground electronic state, our knowledge about cross sections is largely limited to the room-temperature condition, in which vibrational and rotational states are populated in a thermal distribution. Then, for a diatomic molecule, the ground vibrational state is predominantly populated. However, for a polyatomic molecule, normal modes with small quanta must be appreciably excited. For the full understanding of kinetics in plasma chemistry, it is important to assess the role of the internal energy of reactant molecules. 

Vibrational Motion in Polyatomic Molecules. Normal-mode analysis of vibrational motion in polyatomic molecules is the method of choice when there are several vibrational degrees of freedom. The actual vibrations of a polyatomic molecule are completely disordered, or aperiodic. However, these complicated vibrations can be simplified by expressing them as linear combinations of a set of vibrations (i.e., normal modes) in which all atoms move periodically in straight lines and in phase. In other words, all atoms pass through their equilibrium positions at the same time. Each normal mode can be modeled as a harmonic oscillator. The following rules are useful to determine the number of normal modes of vibration that a molecule possesses ... 

The infrared spectra of polyatomic molecules contain one fundamental band for each normal mode whose motion modulates the dipole moment of the molecule. Normal modes that do not modulate the dipole moment of the molecule are not seen in the infrared spectrum. Overtone bands occur as with diatomic molecules, along with combination bands, which are produced when two normal modes make simultaneous transitions. 

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... 

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

In the case of H2O it is easy to see from the form of the normal modes, shown in Figure 4.15, that all the vibrations Vj, V2 and V3 involve a change of dipole moment and are infrared active, that is w=l-0 transitions in each vibration are allowed. The transitions may be labelled Ig, 2q and 3q according to a useful, but not universal, convention for polyatomic molecules in which N, refers to a transition with lower and upper state vibrational quantum numbers v" and v, respectively, in vibration N. 

For a polyatomic molecule, the complex vibrational motion of the atoms can be resolved into a set of fundamental vibrations. Each fundamental vibration, called a normal mode, describes how the atoms move relative to each other. Every normal mode has its own set of energy levels that can be represented by equation (10.11). A linear molecule has (hr) - 5) such fundamental vibrations, where r) is the number of atoms in the molecule. For a nonlinear molecule, the number of fundamental vibrations is (3-q — 6). 

The results obtained above for a diatomic molecule can be generalized for polyatomic molecules. Each of the 3N-6 normal modes of vibration (or 3N-5 for linear molecules) will contribute an energy given by an expression analogous to Eq. (63), namely,... 

For polyatomic molecules, the stretching force constant for a particular bond cannot in general be obtained in an unambiguous manner because any given vibrational mode generally involves movements of more than two of the atoms, which prevent the expression of the observed frequency in terms of the force constant for just one bond. The vibrational modes of a polyatomic molecule can be analyzed by a method known a normal coordinate analysis to... 

The Section on Molecular Rotation and Vibration provides an introduction to how vibrational and rotational energy levels and wavefunctions are expressed for diatomic, linear polyatomic, and non-linear polyatomic molecules whose electronic energies are described by a single potential energy surface. Rotations of "rigid" molecules and harmonic vibrations of uncoupled normal modes constitute the starting point of such treatments. 

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. 

Vibrational states can be described in terms of the normal mode (NM) [50, 51] or the local mode (LM) [37, 52, 53] model. In the former, vibrations in polyatomic molecules are treated as infinitesimal displacements of the nuclei in a harmonic potential, a picture that naturally includes the coupling among the bonds in a molecule. The general formula for the energies of the vibrational levels in a polyatomic molecule is given by [54]... 

The representation as a two-dimensional potential energy diagram is simple for diatomic molecules. But for polyatomic molecules, vibrational motion is more complex. If the vibrations are assumed to be simple harmonic, the net vibrational motion of TV-atomic molecule can be resolved into 3TV-6 components termed normal modes of ibrations (3TV-5 for... 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

The factor in (6.67) that multiplies the integral (6.73) contains the derivatives of the dipole-moment components with respect to the normal coordinate Qk, evaluated at the equilibrium configuration. We conclude that a radiative infrared transition in which the vibrational quantum number of the A th normal mode changes by one is forbidden unless the Acth mode has a change in dipole moment associated with it. The value of the equilibrium dipole moment de is irrelevant for infrared transitions of a polyatomic molecule. 

Up to this point, the molecule has been considered to be a rigid rotor, but the work in Chapter 4 on diatomics shows that we must add corrections for rotation-vibration interaction and centrifugal distortion. For a polyatomic molecule, there are several normal modes of vibration, each with its own vibrational quantum number (see Chapter 6). By analogy to (4.75), we write for polyatomic molecules... 

Raman spectroscopy (Section 4.10) aids the study of the vibrations of polyatomic molecules. For a vibration to be Raman active, it must give a change in the molecular polarizability. For many molecules with some symmetry, one or more of the normal modes correspond to no change in... 

Figure 2.6 Potential energy of a diatomic molecule as a function of internuclear separation r. The equilibrium separation is re. A normal mode in a polyatomic molecule would have a similar potential curve, with a parameter characterizing the phase of the motion replacing r.

The potential energy of such an oscillator can be plotted as a function of the separation r, or, for a normal mode in a polyatomic molecule, as a function of a parameter characterizing the phase of the oscillation. For a simple harmonic oscillator, the potential energy function is parabolic, but for a molecule its shape is that indicated in Figure 2.6. The true curve is close to a parabola at the bottom, and it is for this reason that the assumption of simple harmonic motion is justified for vibrations of low amplitude. 

For a polyatomic molecule there will be a potential energy curve like that of Figure 2.6 for each of the 3N — 6 vibrational modes. The potential energy is therefore characterized by a surface in 3N — 6 + 1-dimensional space. To plot such a surface is clearly impossible we must be content with slices through it along the coordinates of the various normal modes, each of which will resemble Figure 2.6. 

Note that to first order this is simply the sum of the fundamental frequencies, after allowing for anharmonicity. This is an oversimplification, because, in fact, combination bands consist of transitions involving simultaneous excitation of two or more normal modes of a polyatomic molecule, and therefore mixing of vibrational states occurs and... 

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... 

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