78 



RADIATION BIOLOGY 



The alternating maxima and minima form a free-electron counter- 

 part to the alternating single and double bonds in the simplest classical 

 valence-bond diagram. Kuhn (1950) gives several examples. The aver- 

 age distance between maxima, or between minima, is just 2d, and the 

 maxima are located in the center of the classical double bonds. The 

 alternation becomes less marked with increasing N and is less marked 

 in the center of the polyene than at the ends. The maxima and minima, 

 from the present point of view, depend on the wave character of elec- 

 trons, just as do the maxima and minima of different atomic shells in 

 the radial electron densities for an atom. The relation of these polyene 

 density alternations to the classical bond diagrams is then something like 



the relation between the positions of 

 maximum density in atoms and the 

 Bohr classical orbit radii. 



The effect of high electron density 

 at particular points in the polyenes 

 is to attract the neighboring nuclei, 

 producing a shorter bond length and 

 higher force constant. Low density 

 has the opposite effect, so that an 

 alternation of electron densities im- 

 plies an alternation of bond lengths 

 and force constants. This alterna- 

 tion is well known from the classical 

 valence-bond treatment, as well as 

 experimentally from X-ray diffraction and infrared analysis. 



But from the present point of view it could be said that the alternating 

 bond lengths give direct evidence of the wave character of electrons — 

 almost as direct as the Davisson-Germer experiment on the diffraction of 

 electrons from a crystal. The alternation of bond lengths shows the pres- 

 ence of standing waves produced by the internal interference of electron 

 waves reflected from the ends of the molecule. 



Effect of Alternation on Transition Frequencies. Kuhn (1950) has 

 shown how to correct the errors of the polyene frequency predictions 

 by introducing an alternation of period 2d into the square-well potential, 

 as shown in Fig. 2-5, using deeper minima near the classical double bonds, 

 where the nuclei are closer together. 



The effect of this periodic perturbation on free-electron and LCAO 

 orbital energies in molecules, just as in metals (Seitz, 1940; Brillouin, 

 1946), is to enlarge the energy gap between the orbitals of wave lengths 

 of more than 4c? and those of wave lengths of less than 4d. In the 

 language of metal theory, the periodicity introduces a Brillouin bound- 

 ary between the first and second Brillouin zones. But this enlarged gap 

 is just the energy gap corresponding to the first transition, so that this 



Fig. 2-5. Free-electron potential with 

 Kuhn periodicity from alternating 

 double and single bonds. 



