74 RADIATION BIOLOGY 



where L is the length of the trough and x is the distance measured from 

 one end. The normalization factor that multiplies the sine function is 

 chosen so that the integral of the electron density (the square of the wave 



function) over the length of the trough, / yp'^ dx, will be unity. 



The lowest wave function has a wave length X = 2L, with no nodes in 

 the trough; the next has X = 2L/2, with one node; the nth has X = 2L/n, 

 with n — \ nodes; and so on, like the oscillations of a vibrating string. 



Orbital Energies. The energies of these orbitals are determined by the 

 de Broglie relation between electron momentum mvn and wave length, 



mvn = h/\n, (2-2) 



where m is the electron mass and h is Planck's constant. This relation 

 fixes the electron velocity v„; and the energy E,, is then 



£/„ = - mvl = 7^^ = ^-r^ (2-3) 



2 2m X^ 8mL- 



The energy varies quadratically with n. 



If we assume that the length L for a conjugated chain of N atoms is 

 Nd, where d is the average interatomic distance, then 



The second expression gives the energy in wave numbers if d is 1.40 A — 

 the average C — C distance in a conjugated system. 



The Pauli principle specifies that only two electrons, of opposite spin, 

 may occupy each orbital. For a polyene where there are N x-electrons 

 in the lowest possible energy state, they then fill up the lowest N/2 

 orbitals (A^ is even). The energy required to lift an electron from the 

 highest filled orbital to the lowest unfilled orbital becomes 



_ {N/2 + 1)W {N/2rh' 

 iiw2)+i - A^/2 ^^^^^^_ 8mrfW2 ^^"^^ 



or 



This should represent approximately the energy of the first spectroscopic, 

 transition. For long chains it should vary as 1/A^. Note the absence 

 of adjustable parameters. Values predicted by this formula are given 

 by the slanted "theoretical" line in Fig. 2-1 for comparison with the 

 observed first transitions (marked ^B). The predicted wave lengths are 

 about right for butadiene. The predictions get poorer with increasing 

 wave length, varying about twice as fast with N as observed. 



