1. Molecular Energy Levels and Absorption and Emission Spectroscopy
We shall deal first with the simplest system, that of diatomic molecules, and then extrapolate to polyatomic systems.
Atoms in molecules undergo a variety of motions relative to each other. As illustrated in Fig. 3.1 for the water molecule, these can be separated into vibrational motions involving the various chemical bonds in the molecule, rotation of the molecule as a whole, and translational motion of the molecule, i.e., movement in the three coordinates x, y, and z. However, since the interaction of light with the molecule only directly changes the vibrational and rotational motions, we shall not consider translation further here. Of course, once the molecule has absorbed the light initially, the energy may also be converted into translational energy, at least in part, leading to an increase in temperature.
In addition to inducing changes in the positions of atoms within the molecule, the absorption of light can lead to changes in electron distribution. In contrast to vibrational and rotational changes, such electronic transitions typically require sufficient energy (in the ultraviolet and visible regions) that they can lead to breaking of chemical bonds; i.e., photochemistry can occur. It is this latter process that, in general, is of most interest in atmospheric chemistry.
a. Diatomic Molecules
(1) Vibrational energy and transitions As seen in Fig. 3.2a, the bond between the two atoms in a diatomic molecule can be viewed as a vibrating spring in which, as the internuclear distance changes from the equilibrium value rc, the atoms experience a force that tends to restore them to the equilibrium position. The ideal, or harmonic, oscillator is defined as one that obeys Hooke's law; that is, the restoring force F on the atoms in a diatomic molecule is proportional to their displacement from the equilibrium position.
Substitution of the potential energy for this harmonic oscillator into the Schrodinger wave equation gives the allowed vibrational energy levels, which are quantified and have energies Ev given by
where vw-h is a constant characteristic of the molecule and is related to the strength of the bond and the
3. SPECTROSCOPY AND PHOTOCHEMISTRY: FUNDAMENTALS
Symmetric Stretch v., 3652 cm"1
Symmetric Stretch v., 3652 cm"1
FIGURE 3.1 (a) Internal vibrations of the bonds in the water molecule, (b) rotational motion of water, and (c) translation of the water molecule.
reduced mass of the molecule. The vibrational quantum number v can have the integral values 0, 1,2,... . Thus the vibrational energy levels of this ideal oscillator are equally spaced.
However, as seen in Fig. 3.2, this idealized harmonic oscillator (Fig. 3.2b) is satisfactory only for low vibrational energy levels. For real molecules, the potential energy rises sharply at small values of r, when the atoms approach each other closely and experience significant charge repulsion; furthermore, as the atoms move apart to large values of r, the bond stretches until it ultimately breaks and dissociation occurs (Fig. 3.2c).
To obtain the allowed energy levels, Ev, for a real diatomic molecule, known as an anharmonic oscillator, one substitutes the potential energy function describing the curve in Fig. 3.2c into the Schrodinger equation; the allowed energy levels are
Once again v is the vibrational quantum number with allowed values of 0, 1, 2,..., and xc and yc are anharmonicity constants characteristic of the molecule.
Equation (B) is often expressed in wavenumbers, wvjb; the allowed energy states, Ev, in units of wavenumbers (cm 1) become
Note that throughout this book we use a bar over a parameter (e.g., E) if it is expressed in units of wavenumbers. Values for cjc, x0, and yc for a number of diatomic molecules are found in Herzberg's classic Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules (1950) and in Huber and Herzberg (1979).
An important consequence of using the potential energy for a real molecule in the Schrodinger equation is that the vibrational energy levels become more closely spaced with increasing quantum number v (Fig. 3.2c versus 3.2b).
When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are At' = +1; that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to At' = ± 2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and
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