Vibrational motion polyatomics

Treating the full internal nuclear-motion dynamics of a polyatomic molecule is complicated. It is conventional to examine the rotational movement of a hypothetical "rigid" molecule as well as the vibrational motion of a non-rotating molecule, and to then treat the rotation-vibration couplings using perturbation theory. 

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). 

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

The harmonic oscillator energies and wavefunctions comprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomic molecule are often characterized in terms of individual bond-stretching and angle-bending motions each of which is, in turn, approximated harmonically. This results in a total vibrational wavefunction that is written as a product of functions one for each of the vibrational coordinates. 

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... 

The vibration-rotation interaction term makes the Hamiltonian for nuclear motion of a polyatomic molecule difficult to deal with. Frequently, this term is small compared to the other terms. We shall make the initial approximation of omitting Tvib rot. The rotational kinetic energy TTOt involves the moments of inertia of the molecule, which in turn depend on the instantaneous nuclear configuration. However, the vibrational motions are much faster than the rotational motions, so that we can make the approximation of calculating the moments of inertia averaged over the vibrational motions. 

To consider the quantum mechanics of rotation of a polyatomic molecule, we first need the classical-mechanical expression for the rotational energy. We are considering the molecule to be a rigid rotor, with dimensions obtained by averaging over the vibrational motions. The classical mechanics of rotation of a rigid body in three dimensions is involved, and we shall simply summarize the results.2... 

We have considered the case of vibrational motion of the photofragments accompanied by slow relative motion. We have developed the adiabatic approach to evaluate the nuclear wave-function (Jp and obtained eqs. 74 and 96. Note, that instead of a system of electrons and nuclei (Born-Oppenheimer approximation), we considered here only nuclear motion of a polyatomic system with several degrees of freedom, one of which is "fast" relative to the others. 

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... 

The vibrational motion of polyatomic molecules encompasses all nuclei in the molecule and, as long as the displacement from the equilibrium configuration is sufficiently small, it can be broken down into the so-called normal-mode vibrations (see Appendix E). In special cases these vibrations take a particular simple form. Consider, e.g., a partially deuterated water molecule HOD. In this molecule, the H OD and HO-D stretching motions are largely independent and the normal modes are, essentially, equivalent to the local bond-stretching modes. To that end, consider the following reaction that has been studied experimentally [6,7] as well as theoretically [8] ... 

A polyatomic molecule contains more than one atom17 and can undergo rotational and vibrational motions in addition to translational. Overall, rotational motion is unhindered and therefore consists only of a kinetic energy term of the... 

The harmonic approximation reduces to assuming the PES to be a hyperparaboloid in the vicinity of each of the local minima of the molecular potential energy. Under this assumption the thermodynamical quantities (and some other properties) can be obtained in the close form. Indeed, for the ideal gas of polyatomic molecules the partition function Q is a product of the partition functions corresponding to the translational, rotational, and vibrational motions of the nuclei and to that describing electronic degrees of freedom of an individual molecule ... 

The vibrations of polyatomic molecules are more complicated, for the number of possible interactions rises sharply as the number of atoms increases. However, it is possible to handle the equations of motion governing even the most complicated vibrations by lineal combination (p. 48) of the equations of motions of rather simple vibrations. For example, all vibrations of the C02 molecule are said to be derived from superposition of the four modes of vibration indicated in Figure 25-4(a), whereas all vibrations of the SO2 molecule may be likewise broken down into combinations of one or more of the vibrations indicated in 25-4(b). 

The energy levels of the vibrational modes can be predicted with a reasonable accuracy on the basis of the standard Wilson vibrational analysis (241,244) (called GF analysis). The vibrational motion of atoms in the polyatomic system is approximated by harmonic oscillations in a quadratic force field. Computations of the force constants are the subject of quantum chemistry. 

Almost all infrared work makes use of absorption techniques in which radiation from a source emitting all infrared frequencies is passed through a sample of the material to be studied. When the frequency of this radiation is the same as a vibrational frequency of the molecule, the molecule may be vibrationally excited this results in loss of energy from the radiation and gives rise to an absorption band. The spectrum of a polyatomic molecule generally consists of several such bands arising from different vibrational motions of the molecule. This experiment involves diatomic molecules, which have only one vibrational mode. 

The analysis given thus far applies to diatomic molecules. For simple polyatomic molecules, several characteristic vibrational motions, called normal modes, are possible. As an example, the normal modes of vibration for the SOi molecule are shown in Fig. 14.60. 

The application of this result to the determination of bond stretching force constants in molecules encounters two difficulties. First, real molecules are not exactly harmonic oscillators. Secondly, although the only mode of vibration possible in diatomic molecules is a bond stretching motion, the vibrations of polyatomic molecules are much more complicated, and cannot be expressed as consisting only of a combination of bond stretching motions. We discuss these two problems in turn in the next two sections. 

For a diatomic or polyatomic ideal gas, Cy is greater than j R, because energy can be stored in rotational and vibrational motions of the molecules a greater amount of heat must be transferred to achieve a given temperature change. Even so, it is still true that... 

In a polyatomic molecule, several types of vibrational motion are possible (Fig. 20.9). Each has a different frequency, and each gives rise to a series of allowed quantum vibrational states. Infrared absorption spectra provide useful information about vibrational frequencies and force constants in molecules. For... 

Whereas monatomic molecules can only possess translational thermal energy, two additional kinds of motions become possible in polyatomic molecules. A linear molecule has an axis that defines two perpendicular directions in which rotations can occur each represents an additional degree of freedom, so the two together contribute a total of 1/2 R to the heat capacity. For a non-linear molecule, rotations are possible along all three directions of space, so these molecules have a rotational heat capacity of 3/2 R. Finally, the individual atoms within a molecule can move relative to each other, producing a vibrational motion. A molecule consisting of N atoms can vibrate in 3N-6 different ways or modes1. For mechanical reasons that we cannot go into here, each vibrational mode contributes R (rather than 1/2 R) to the total heat capacity. 

The straightforward way to treat the rotational and vibrational motion of a polyatomic molecule would be to set up the wave equation for (Eq. 34-4), introducing for [/ ( ) an expres-... 

The vibrational motion of polyatomic molecules is usually treated with an accuracy equivalent to that of the simple discussion of diatomic molecules given in Section 35c, that is, with the assumption of Hooke s-law forces between the atoms. When greater accuracy is needed, perturbation methods are employed. 

Progress in the quantum description of vibrational motion of polyatomic molecules... 

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