EXCITATION OF POLYENES AND PORPHYRINS 



77 



transition should become weaker and be polarized on a line from one tip 

 of the molecule to the other. The second, instead of being forbidden, 

 may become stronger than the first and will be polarized along the axis 

 of the hairpin. 



This behavior may be observed in gojng from the spectrum of butadiene 

 to that of cyclohexadiene, where the conjugated system is doubled back 

 on itself in this w&y (MuUiken, 1939). It was also seen in the cis-trans 

 isomerism of longer chains by Zechmeister and Polgar (1943), Sandoval 

 and Zechmeister (1947), and Pinckard et at. (1948). These authors called 

 the transition designated as ^C in Figs. 2-1 and 2 the "cis band" because 

 of its strength in this molecular configuration. The increase of intensity 

 of a transition in the cis form is then an excellent criterion for the assign- 

 ment of this transition to the forbidden class in the trans form; the loss of 

 intensity, for its assignment as allowed. Of course, two cis bonds in a 

 molecule, if properly placed, may restore its center of symmetry and make 

 the ^C transition forbidden again, as 

 Zechmeister found in several cases. 

 It is illuminating to compare his 

 fine classical interpretation of the 

 spectra (1944) with its quantum-me- 

 chanical counterpart as given here. 



Alternation of Electron Densities. 

 We may use the free-electron model 

 to determine the electron densities 

 in a polyene. For any wave func- 

 tion i/', the density is given b}' i'"- if 

 4/ is real, as it is here. In Fig. 2-4 

 the density is shown in the lower 

 curves for the different i/'„'s of hexatriene. The total 7r-electron density 

 per unit length of the trough for A^ electrons in the ground state is 



N/2 NJ2 



2 . „ rnrx 



r; ^/^^^N^^^^v v^^V 



t 



^2 ^«^^^^^^ ^.<<^^^^^ 



Fk;. 2-4. Electron density in hexatriene 

 on the free-electron model. 



C = 2 J ^2 = 2 J 



Sin- 



(2-13) 



The sum can be transformed into 



D = 



2 sin ■Kx/2d 



cos 



d L sin -wx/L 

 = at X = 0, .r = L; 



= -^ at a: = 2d, -id, . . 



(N - 2)d; 



(2-14) 

 (2-14a) 

 (2-146) 



^ + J:, at X- = d, M, 5d, . . . , (N - l)d. (2-14c) 

 a J\a 



This curve is shown at the top of Fig. 2-4. 



