The Mirror-Image Law

The fluorescence emission spectrum of a molecule usually is approximately a mirror image of the absorption spectmm, as illustrated in Fig. 5.3. Several factors contribute to this synuneby. First, if the Bom-Oppenheimer approximation holds, and if the vibrational modes are harmonic and have the same frequencies in the ground and excited electronic states (all significant approximations), then the 

You can see this by examining the products of the harmonic-osciUator wavefunctions plotted in Fig. 5.4. Since XbO- XafO) becomes superimposable on Xa(2 b(P) if it is inverted with respect to the vibrational axis, the results of integrating the two products from oo to oo over this axis must be identical. The product becomes superimposable onXaO)Xb(0) if it is inverted with respect 

Corresponding upward and downward transitions with identical Franck-Condon factors thus occur at frequencies displaced equally on either side of Vgg. But the 

We have assumed that any coherence in the temporal parts of the individual vibrational wavefunctions is lost very rapidly relative to the lifetime of the excited 

so that we can treat the fluorescence from each vibrational level independently. This condition is assured if the vibrational levels have reached thermal equilibrium. We return to vibrational coherence in Chap. 11. 

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If the mirror-image law (Eq. 5.13) holds, an alternative approach is to recast Eq. (5.19) to give the rate constant for fluorescence at frequency Va — S as a function of the absorption coefficient at frequency v a + S, and then to integrate over the absorption spectmm instead of emission. Letting the fluorescence frequency be v = 2vqo — V, this gives ... 

The mirror-image law and the fluorescence and absorption methods for obtaining all assume that thermal equilibration of the excited molecule with the surroundings occurs rapidly relative to the lifetime of the excited state. If this assumption is valid, the ratio of fluorescence to absorbance at a given frequency should have a predictable dependence on temperature. Returning to Eq. (5.19) and collecting the terms that depend on v, we see that... 

About 50 years ago, physicists were amazed to discover that the universe, which had previously been regarded as completely symmetrical, had a certain preference for left-handedness. It had been considered impossible that basic natural laws would distinguish between left and right. This assumption formed the basis for the physical law of the conservation of parity according to this, the sum of the parities before and after each physical process must be equal. In other words the mirror image of each physical phenomenon is also a real phenomenon (Ball, 1994). 

These laws are extremely helpful in luminescence analysis, especially in interpreting the spectra and in establishing the nature of energy levels of molecules. If either the mirror image or the universal relationship is strictly satisfied, then the shape of one of the spectra (absorption or fluorescene) can be determined from the shape of the other. The mirror image shows that the vibrational systems of the electron levels of S0- and Srstates have identical structures and can be used to evaluate their relative populations and relative probabilities of absorptive and radiative transitions as well as to determine the frequency of pure electron transitions. 

The third law of crystallography states that all crystals of the same compound possess the same elements of symmetry. There are three types of symmetry a plane of symmetry, a line of symmetry, and a center of symmetry (14). A plane of symmetry passes through the center of the crystal and divides it into two equal portions, each of which is the mirror image of the other. If it is possible to draw an imaginary line through the center of the crystal and then revolve the crystal about this line in such a way as to cause the crystal to appear unchanged two, three, four, or six times in 360° of revolution, then the crystal... 

As may be recalled from our discussion of the law of definite proportion, isomers are molecules that have been built from the same number and type of atoms but arranged in a different order. We cited, as examples, fulminic acid, cyanic acid, and isocyanic acid HONC, HOCN, and HCNO, respectively. We saw that this simple rearrangement of elements made the first explosive, the second a poison, and the third a pacific participant in several, more constructive, organic syntheses. Isomers that differ only by being mirror images of each other are termed chiral isomers (pronounced kiral, with a hard c sound, the way chemist is pronounced kemisf). 

The effective diffusivity becomes essentially constant and equal to the limiting value for the Henry s law region [i) ss > /(l - )h J and Eq. (6.19) reduces to the form of Eq. (6.16). Under these conditions adsorption and desorption curves are mirror images and the uptake rate is independent of step size but for larger concentration steps the rate becomes dependent on step size and adsorption is much faster than desorption. 

Euclid, about 300 B.C., treated mathematically the size relations of the object and the image for plane and spherical mirrors. He located the focal point of concave mirrors and laid a foundation for the study of convergence and divergence of light beams reflected from mirrors. About 100 B.C., Hero presumably laid the basis of the law concerning the equality of the angles of... 

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