Classical versus Quantum Mechanical Interpretation of Absorption

What once was, and perhaps still is, the standard treatise on atomic spectra, by Condon and Shortley, fills 432 pages of text. Herzberg's treatises fill 581 pages for diatomic molecules, 538 pages for polyatomic molecules, and 670 pages for the electronic spectra of polyatomic molecules. Townes and Schawlow devote 648 pages to microwave spectroscopy. And the physical strain of just lifting these nearly 3000 pages is as nothing compared to the mental strain of absorbing them. Add to these tomes the dozens if not hundreds of books on quantum mechanics and modern physics. Spectroscopy is the science of details. Indeed, this is the source of the almost religious awe in which quantum mechanics is held by physicists. A theory capable of ordering - if not explaining - so many details is akin to magic.

This somewhat depressing inventory of details, systematized by a theory that, as Richard Feynman has so often been quoted as saying, "nobody understands", signals that what follows can only scratch the surface of a vast subject. Given the few pages we can devote to it, our aim must be the modest one of conveying the bare bones of what students of atmospheric science should know about the infrared spectra of molecules that inhabit the atmosphere.

According to classical mechanics, the mechanics of Newton, a mechanical system can have a continuous set of energies (kinetic plus potential). This seems so self-evidently true that it is rarely stated explicitly and even more rarely questioned. But in reality, energy is like dollar bills: you can have one bill in your wallet or two or three, but never 1.523. Dollar bills are quantized. And so is energy, but this is not evident until we consider mechanical systems on a scale not directly accessible to our senses. We do not live at this microscopic (or atomic) scale, so we have no right to expect that the physics of ordinary macroscopic objects is valid there. This is much like moving from one country to another. In one society, certain rules of behavior and customs are taken for granted. But when you enter a different society, you sometimes discover that many of the familiar rules no longer apply. In India, Pakistan, and Sri Lanka, people eat with their fingers. To do otherwise would seem unnatural to the inhabitants of these countries. But in Western countries, eating with your fingers is considered to be extremely impolite. Children who do this are told in no uncertain terms to desist.

The differences between the rules of behavior of macroscopic and microscopic objects are considerably greater than the differences between those of the inhabitants of New York penthouses and the inhabitants of huts in the Amazon jungle.

If the laws of quantum mechanics must be taken on faith, consider that so must the laws of classical mechanics. Why F = ma? You learn this first as an axiom, and so what follows it may seem strange. But if you had learned quantum mechanics first, F = ma might seem strange. Whatever you learn first sets the standard for what is normal.

Consider first a harmonic oscillator with natural frequency w0. According to classical mechanics this oscillator can have a continuous range of energies depending on its initial amplitude. But according to quantum mechanics the energies of this harmonic oscillator are quantized, having only the discrete set of energies where n = 0,1,2,... . This result seems to contradict common sense, but it really doesn't because common sense is based on our experience with macroscopic objects. The difference in energy between physically realizable (allowed) adjacent energy states of the harmonic oscillator is

The oscillators we encounter at the macroscopic level might have natural frequencies as high as 100 Hz. This corresponds to an energy difference of 6.63 x 10~32 J. To get a feeling for how much energy this is, consider how much energy the eyebrow of a flea has when it falls from a flea on the back of a dog. A flea, which can be seen without a microscope, has dimensions of a millimeter or so, and hence the total volume of the flea is of order 10~9 m3. The characteristic linear dimension of a flea's eyebrow is at least 100 times smaller than the

overall dimensions of the flea. Thus the volume of the flea's eyebrow is of order 10-15 m3. We estimate the density of flea flesh to be that of water, 1000 kgm-3, which gives a mass of 10-12 kg for the flea's eyebrow. Suppose that the flea resides on the back of a Great Dane, about 1 m high at the shoulders. The potential energy of the flea's eyebrow, relative to that at the ground, is about 10-11 J. As small as this potential energy is, it is still 1021 times greater than the difference between adjacent energy levels of a harmonic oscillator with natural frequency 100 Hz. Although the energies of macroscopic oscillators are quantized in principle, the spacing of energy levels, as they are called, is so small macroscopically that in practice the levels are continuous. We are forced to come to grips with the discreteness of energy levels only when we consider systems with very high natural frequencies. All else being equal, natural frequencies increase with decreasing mass [Eq. (2.76)], and hence the consequences of discrete energies are not negligible at the atomic scale.

With this preamble, consider absorption of electromagnetic energy by a single isolated oscillator from the classical and quantum-mechanical points of view. According to the classical analysis in Section 2.6, the rate at which power is absorbed by an oscillator from a time-harmonic electromagnetic wave of given amplitude depends on its frequency w. Absorption is sharply peaked in a narrow range of frequencies, called an absorption line (or band), centered on the natural frequency of the oscillator. The width of the line is a consequence of damping of the oscillator.

Now consider the same process from a quantum-mechanical point of view. The incident monochromatic electromagnetic wave is considered to be a stream of photons, each with energy frw. Absorption of electromagnetic energy is a consequence of absorption of photons. If the oscillator absorbs a photon, the energy of the oscillator must increase. But this increase can be only one of a set of discrete values. Unless the energy of the photon is equal to the difference between two energy levels of the oscillator, it cannot absorb the photon. This accounts for the narrowness of absorption lines.

2.7.1 Molecular Energy Levels

An electrically neutral atom consists of a nucleus composed of uncharged neutrons and positively charged protons bound together and surrounded by negatively charged electrons equal in number to the protons. Almost the entire mass of an atom resides in its nucleus, which is smaller by a factor of about 1000 than the atom as a whole. That is, the electrons, on average, are at distances from the nucleus large compared with its size.

A molecule is a collection of two or more atoms bound together. They can vibrate about their equilibrium positions and the molecule as a whole can rotate. A vibrating object has vibrational kinetic and potential energy; a rotating object has rotational kinetic energy. The center of mass of the molecule can move and it can interact with neighboring molecules. And electrons have potential and kinetic energies. These various modes of motion of a molecule can be decomposed approximately into translational (position and velocity of the center of mass of the molecule), rotational, vibrational, and electronic modes, and hence the total energy of the molecule is

That this decomposition is only approximate is evident from the rotations and vibrations of two mass points connected by a spring. The rotational kinetic energy of this system depends on its moment of inertia, which in turn depends on the separation between the mass points. But if they vibrate, their separation changes, and hence so does the moment of inertia. Thus the two types of motion, rotation and vibration, are not completely decoupled. Although Eq. (2.93) often is a good approximation, don't be surprised when it fails to explain what is observed.

Each of the separate contributions to the total energy E in Eq. (2.93) is quantized except the translational kinetic energy of the center of mass of a molecule. A truly isolated molecule is an idealization. We are almost always faced with an ensemble of many molecules. Even in a low density gas, interactions between molecules are not completely negligible. Strictly speaking, translational energy is also quantized but the level separation is so small that it cannot be observed. To the extent that Eq. (2.93) is a good approximation we can therefore speak of the electronic energy levels of a molecule, its vibrational energy levels, and so on. The lowest allowed energy of any kind is called the ground state. All higher energy levels, or states, are called excited states. When a molecule absorbs a photon it is said to be excited into a higher-energy state or to undergo a transition from one energy state to another of higher energy. A molecule in an excited state can then spontaneously drop to a lower energy state accompanied by the emission of a photon equal in energy to the difference in energy levels, which underscores our previous assertions about absorption and emission being inverse processes. The Lorentz line shape [Eq. (2.71)] is just as valid quantum-mechanically as it is classically, but the terms in it are interpreted quite differently. Classically, w0 is the natural frequency of an oscillator and 7 is a factor in a viscous damping term; quantum-mechanically, hw0 is the difference between two energy levels and 1/7 is the lifetime of the transition between them (i.e., the average time the molecule exists in the higher energy state).

The energy of a photon is hc

Except for the factor hc, a universal constant, the energy of a photon is inversely proportional to the wavelength of the associated wave. Thus we may express photon energies as inverse wavelengths, which are called wavenumbers, usually with units cm-1. Why wavenumber? Because it is the number of waves in unit length. Spectroscopists are careless in the use of the symbol v: sometimes it denotes frequency, sometimes wavenumber (inverse wavelength). And to make matters worse, the units of wavenumber are usually cm-1, and so if hc is in SI units and v in cm-1, photon energy is 100hcv. We try to be consistent and use v for frequency, v for wavenumber. A word of caution: 2n/A, often written as k or q, is sometimes called wavenumber.

A wavelength of 10 pm corresponds to a wavenumber of 1000 cm-1. Wavelengths in the middle of the visible spectrum correspond to wavenumbers of around 20,000 cm-1. Sometimes you will encounter statements about a photon energy being so many wavenumbers, 2000, say. Unless stated otherwise, this probably means 2000 cm-1.

Another quantity with dimensions of energy that keeps cropping up in all kinds of problems is kBT, which we can divide by hc so as to express it in the same units as wavenumbers. At typical terrestrial temperatures (300 K say), kBT/hc is around 200 cm-1.

Consider now a gas of molecules in thermal equilibrium at temperature T. These molecules can collide and exchange energy. Indeed, a molecular collision is defined as an interaction between two or more molecules in which the energy of each molecule after the interaction is different from that before (total energy being conserved). The most probable kinetic energy of a gas molecule is kBT/2 (see Sec. 1.2), so this is how much energy can be exchanged in a typical collision. Some molecules have greater energies than the most probable, some less, the probability of energy E being proportional to exp(—E/kBT). (We omit the other factor dependent on energy because it does not vary so strongly). This quantity, called the Boltzmann factor, crops up in all kinds of problems. Indeed, it would not be an exaggeration to say that it, not love, is what makes the world go round. Rates of chemical reactions are determined by the Boltzmann factor. Our bodies are complicated, finely-tuned engines in which many chemical reactions are continuously taking place, their rates determined by the Boltzmann factor. A small change in temperature changes not only the absolute rates of these reactions but their relative rates as well. This is why our bodies constantly attempt to keep our deep core temperature at around 37 °C. An increase in this temperature of only 1-2 °C is sufficient to send us to bed wracked with pain and fever. An increase of a few more degrees might be fatal. When you cook food you are exploiting the Boltzmann factor. An egg will rot before it hardens at room temperature, but immerse it in water at 100 ° C and it is hard-boiled in minutes. A rough rule of thumb used by chemists is that chemical reaction rates double with every 10 °C increase in temperature. A temperature increase from 20 °C to 100 °C corresponds to 8 doubling times, a factor of 256.

If a gas is in thermal equilibrium its molecules are distributed in their energy states. The ratio, on average, of the number Nj of molecules having energy Ej and the number of molecules Ni having energy Ei is approximately

Typically, the separation between electronic energy levels is around 10,000 cm-1. Suppose that Ei corresponds to the ground electronic state and that Ej corresponds to the first excited electronic state. Because the difference AE between these two energy states is around 10,000 cm-1, the ratio of the number of molecules with electrons in the first excited state to the number in the ground electronic state at 300 K is of order exp( —10,000/200) = exp(-50) « 10-25. At typical terrestrial temperatures almost all molecules in a gas are in their ground electronic state. To populate excited electronic states would require a ten-fold or more increase in absolute temperature. Typical separations between adjacent vibrational energy levels are about 1000 cm-1. Again, this is large compared with kB T for ordinary temperatures, and hence most gas molecules at these temperatures are in their vibrational ground states. Typical separations between rotational energy states are 10-100 cm-1, which are comparable with kB T. Thus at ordinary temperatures many molecules are in excited rotational states.

Merely because molecules vibrate and rotate does not necessarily mean that they radiate (emit). Absorption, as we have seen, is the inverse of emission. It is perhaps easier to discuss spontaneous emission because it occurs in the absence of an external exciting field. One of the results of classical electromagnetic theory is that accelerated charges radiate electromagnetic waves. A corollary of this is that for an electrically neutral charge distribution (e.g., molecule) to radiate, its dipole moment must change with time, either in magnitude or direction or both. A molecule with a permanent dipole moment is said to be polar. A common example is the water molecule (H2O), which owes its permanent dipole moment to its asymmetry: it is composed of two atoms of one kind (hydrogen) and one atom of another (oxygen), which all do not lie on a line. As a consequence of its permanent dipole moment, a rotating water molecule radiates (emits).

The two most abundant molecules in the atmosphere are the diatomic molecules nitrogen (N2) and oxygen (O2). Both of these molecules are homonuclear, which is just a fancy way of saying that they are composed of identical atoms. Such molecules cannot have a permanent dipole moment. To have such a moment would require the center of positive charge to be associated with one atom and the center of negative charge to be associated with the other. But symmetry rules this out: both atoms are identical, and hence one cannot be positively and the other negatively charged. Although nitrogen and oxygen molecules can rotate, in so doing they do not radiate (much). These molecules also can vibrate. But again symmetry rules out the possibility of a changing dipole moment during this vibration. The two atoms must enter into the vibration in a symmetric way, and hence the dipole moment associated with one atom is equal and opposite to that associated with the other.

We have seen that the motions of a molecule can be expressed as a sum of normal modes, each with a characteristic frequency. These frequencies lie in the infrared and a radiating mode is called infrared active. A mode that does not radiate is called infrared inactive. The terms infrared active and inactive, which are familiar to infrared spectroscopists, are preferable to the popular but misleading term "greenhouse gas." Water vapor is infrared active; nitrogen and oxygen, for the most part, are infrared inactive. Greenhouse gases are produced by resident cats with digestive problems.

The normal modes of vibration of the water molecule are shown schematically in Fig. 2.17. There are three modes: the O-H (symmetric) stretching mode, the H-O-H bending mode, and the O-H (asymmetric) stretching mode. The corresponding normal frequencies are v1 = 3657.1cm"1 (2.73 pm), ¿>2 = 1594.8 cm"1 (6.27 pm), and ¿>3 = 3755.9 cm"1 (2.66 pm). The O-H stretching modes are so named because the vibration occurs approximately along the O-H bond.

Keep in mind that these normal frequencies are for the isolated water molecule, water in the gaseous phase. When water vapor condenses to form liquid water, the positions of the absorption bands shift. And when liquid water freezes to form ice, there is yet another shift in the absorption bands. Nevertheless, the normal frequencies of the isolated water molecule are good guides to the approximate positions of absorption bands in the condensed phases. This is evident in Fig. 2.2. Note the large dips in the absorption length around 3 pm and 6 pm. These are the consequence, even in the condensed phases, of vibrations of the isolated water molecule, which does not lose its identity completely when it condenses.

According to quantum mechanics, the vibrational energy levels of the water molecule are quantized:

where vi is any one of the three normal mode frequencies of the water molecule. On the basis of this equation alone we would expect absorption bands not only at the fundamental frequency vi but at overtones as well, integral multiples of the fundamental frequency: 2vi, 3vi,... . But this expectation is not quite borne out by experience. Equation (2.96) does not tell the entire story. We have to account for selection rules, rules that tell us which energy transitions are allowed. The different energy states described by Eq. (2.96) are also states of different angular momentum. The photon carries quantized angular momentum. Thus when a photon is absorbed by a molecule, and it undergoes a transition to a higher energy level, the transition must be such that angular momentum is conserved. Thus if one unit of angular momentum is annihilated (so to speak) when a photon is absorbed, the molecule must increase its angular momentum, and hence energy state, by the same amount. And similarly for emission: when a photon is emitted, the one unit of angular momentum created upon the birth of the photon must be compensated for by the same decrease in the angular momentum of the molecule. Remember that angular momentum is a vector, and hence the angular momentum of the emitted or absorbed photon and the change in angular momentum of the molecule must add vectorially to zero. This requirement of angular momentum conservation leads to the selection rule An = ±1. That is, transitions are allowed only for integral changes in the quantum number n in Eq. (2.96).

This selection rule is not absolute because it is based on the harmonic approximation (Sec. 2.6.1). Because the forces between atoms in the water molecule are not exactly harmonic, the previous section rule can be violated. That is, transitions corresponding to overtones of the fundamental frequencies and even combinations of different frequencies (e.g., v1 + 2v2, v1 - v2, etc.) are possible. These transitions are very weak, but not so weak as to be unobservable.

Although absorption drops precipitously from infrared to visible, it does not reach zero. Moreover, absorption over the visible spectrum is least in the blue-green and rises toward the red in both the liquid and solid phases. (This is evident in Fig. 2.2 but even more so in Fig. 5.12.) Water is a weak blue dye, and this is intrinsic, not the result of some vague impurities. This intrinsic selective absorption by water leads to observable consequences: the blue of the sea, of crevasses in glaciers, ice caves, and frozen waterfalls (see Sec. 5.3.1). What is the molecular mechanism for this blueness?

Overtones of fundamental vibration frequencies in the infrared and combinations of these frequencies make their presence felt, weakly, but observably in the visible. The experimental proof of the vibrational origin of the visible absorption spectrum of water was obtained several years ago by Chuck Braun and Sergei Smirnov, who published their results in a delightful paper, "Why is water blue?", one of those papers you must read before going to your grave. Braun and Smirnov measured the visible and near-visible absorption spectra of ordinary water (H2O) and heavy water (D2O). The vibration spectrum of a molecule depends on the masses of its constituent atoms [see Eq. (2.76)]. Because the nuclear mass of heavy water is greater than that of ordinary water we expect a shift (isotope shift) toward lower frequencies in the vibration spectrum of D2O relative to that of H2O. And this shift was indeed observed. In ordinary water there is a peak at about 750 nm, which shifts to about 1000 nm for heavy water. Braun and Smirnov did not calculate the magnitude of the isotope shift because of the greater mass of deuterium relative to that of hydrogen, so we did using Eq. (2.76) and found good agreement with the measured shift.

Rising absorption in the red gives ordinary water its bluish color, whereas absorption by heavy water is flat throughout the visible spectrum and begins to rise only well into the infrared.

When questions about visible absorption by water arise, chemists seem to be divided into two groups: those who adamantly deny that water has a visible absorption spectrum and those who admit that it does but are just as adamant that it is a consequence of hydrogen bonding.

You may think that we are joking about the first group, yet we could tell you many stories. One will suffice. Several years ago one of the authors gave a talk in Italy on colors in nature for an audience mostly of biological scientists. Afterwards, a photochemist accused him of being a criminal and pervert for telling innocent biologists that water has a visible absorption spectrum.

In chemistry, hydrogen bonding is the universal solvent...for ignorance. Whenever a puzzle arises and there is any liquid water in the neighborhood, hydrogen bonding dissolves the puzzle in the same way that Alexander untied the Gordian knot with his sword. Hydrogen bonding plays the same role that friction does in the undergraduate physics laboratory. Liquid water, however, is blue not because of hydrogen bonding but despite it. Hydrogen bonding in liquid water shifts all the infrared absorption bands, including their overtones and combinations, to lower frequencies compared with the gas phase.

Braun and Smirnov are of the opinion that the color of water is the only example of a color in nature that results from vibrational rather than electronic transitions. It is commonly believed that all colors in nature can be traced to electronic excitations. Water provides an example to the contrary.

Carbon dioxide is a linear, symmetric molecule. By linear is meant that its bonds lie on a straight line. By symmetric is meant that it is composed of a carbon atom flanked on either side by oxygen atoms. This symmetry implies that the carbon dioxide molecule does not have a permanent dipole moment, which precludes this molecule from having what is called a rotational absorption band: an absorption band associated with rotation of the molecule unaccompanied by vibrations. But the carbon dioxide molecule can vibrate in such a way that it has a changing dipole moment. This in turn implies that this molecule, when vibrating in one of its vibrational modes, also can rotate to give a changing dipole moment. So carbon dioxide has vibration-rotational bands.

Carbon dioxide has a bending mode with frequency z>2 = 667.4 cm-1 (15 pm) and an asymmetric stretching mode with frequency z>2 = 2349.2 cm-1 (4.25 pm). The symmetric stretching mode is infrared inactive. It is the vibration-rotational band near 15 pm that is the major player in the global warming scenario associated with increased carbon dioxide in the atmosphere.

Rotational energies also are quantized. For example, those of the rigid rotator, a massless rigid rod of length r with one end fixed, the other end attached to a mass m, and free to rotate are given by

where I = mr2 is the moment of inertia about the rotation axis. As with transitions of the harmonic oscillator, those of the rigid rotator are restricted by a selection rule: A J = ±1.

Although molecules can rotate, and hence have rotational energy levels, they are never exactly describable as rigid rotators. Like the harmonic oscillator, the rigid rotator is an idealization, and hence this selection rule is not absolute. When we say with seeming absoluteness that a transition is forbidden, we mean that the transition probability is very small compared with that for an allowed transition but not identically zero. Quantum mechanics is a most permissive theory, allowing improbable events that according to classical mechanics are impossible.

You may wonder why we jumped directly to the quantum-mechanical rigid rotator instead of making a more gradual transition by way of the classical rotator. This is what we did for the harmonic oscillator, beginning with its classical equation of motion and then stating, but not proving, how the discrete energy levels of the quantum-mechanical oscillator are related to the classical natural frequency of the oscillator. Why didn't we follow the same approach for the rigid rotator? And why were we unable to find any authors who do or even bring up the subject?

As it happens, analysis of the classical harmonic oscillator is considerably less difficult than analysis of the quantum-mechanical oscillator. When we turn to the rigid rotator, this happy state of affairs is reversed: the quantum-mechanical rotator is more amenable to analysis. The classical rigid rotator has no natural frequency. It can spin at any angular speed. A simple pendulum does have a natural frequency for small-amplitude oscillations, but this frequency depends on the magnitude of an external force: gravity. Thus even if we could solve the classical equation of motion for a driven rigid rotator, we wouldn't acquire any fruitful analogies or equations we could carry with us into the quantum domain. And similarly for electronic energy levels. But with only a few exceptions, these energy levels are not of great relevance to the atmosphere. One exception is the Chappuis bands of ozone, which play an important role in the color of the zenith twilight sky (see Sec. 8.1.3).

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