EXCITATION OF POLYENES AND PORPHYRINS 75 



A polymethine ionic dye like those of Fig. 2-2, with A'' odd but with 

 an even number of electrons, has the first (A" + l)/2 orbitals filled, and 



AE = 153,000 ^ St^ ^ - (2-66) 



\"alues predicted by this formula are shown by the slanted line in Fig. 2-2 

 and fit the observations extremely well. [Dewar (1950) has computed 

 the first absorption frequencies of a number of such dyes by the LCAO 

 method. He has also predicted (1952) the polyene frequencies by an 

 ingenious approximation based on the simple zero-energy molecular 

 orbitals.] 



The frequency v of the quantum-mechanical absorption is given by 



VQu = AE/h. (2-7) 



The corresponding classical electromagnetic absorption frequency of a dipole 

 antenna of the same length as the molecule is given by 



vci = c/2L, (2-8) 



where c is the velocity of light. This frequency is lower than the frequency deter- 

 mined from Eqs. (2-6a) and (2-7) by a factor h/imcd, or about H so- This simply 

 means that the velocities of the electrons that carry the quantum-mechanical 

 oscillation are smaller b}' this factor than the electromagnetic-wave veloci};y, the 

 speed of light. The wave length absorbed is therefore not 2L, as it would be 

 classically, but about 



X = 500L. (2-9) 



This linear relation between wave length and molecular length is far from exact 

 in the polyenes, but it helps us understand the approximately constant wave- 

 length shifts introduced by adding additional units to a conjugated chain. 



The second absorption frequency of a polyene will be determined by 

 the energy jump that is shown by a dashed line in Fig. 2-3. This corre- 

 sponds to the absorption regions marked 'C in Figs. 2-1 and 2. 



Allowed and Forbidden Transitions. The lowest absorption frequency 

 is "allowed," but the second lowest frequency is "forbidden " if the conju- 

 gated system is truly a straight line or even if it has a "center of sym- 

 metry," as it would have in a polyene in the zigzag trans, trans, . . . , 

 trans form. 



These terms have the following significance: For any molecule with a 

 center of symmetry, there are two classes of wave functions, "even" 

 and "odd." Even functions are given by odd values of the integer n in 

 Eqs. (2-1) to (2-4). Such a function xp is exactly equal to itself when 

 reflected in the center of the molecule. The odd functions are given by 

 even n in Eqs. (2-1) to (2-4). In them x// is changed into its negative on 

 reflection in the center; it therefore has a node at the center and vanishes 



