76 RADIATION BIOLOGY 



there. (We neglect here the additional node in the plane of the molecule 

 common to all orbitals.) 



Now, the "oscillator strength" of a transition can be found experi- 

 mentally from the integrated absorption intensity, 



/ - 4.32 X 10-9/6, dv, (2-10) 



where ty is the observed molar extinction as a function of the frequency v 

 in wave numbers. This quantity is predicted theoretically by the 

 expression 



/ = 1.085 X lO-^'vQ^ (2-11) 



where Q is the "transition matrix element" for one electron to jump 

 from state n to state m: 



Qx,nm = i^nXxpn, (It (.r-compouent) , (2-12) 



where x is the coordinate measured from the center of the molecule in 

 angstroms and dr is the volume element of integration. 



The components of Q„,„ all vanish when both \pn and ;/'„, are even or 

 when both are odd, and the transition is then "forbidden" (Laporte rule). 

 Actually in a polyatomic molecule there is enough vibrational motion so 

 that the center of symmetry is not preserved. As a result, the transitions 

 that are believed to be of this forbidden type may still be seen, but they 

 are only about one-fifth as strong as their "allowed" counterparts, which 

 are transitions between even and odd orbitals. 



Intensities, Polarizations, and Molecular Configurations. On insertion 

 of the free-electron polyene i/'-functions into the expression for Q, it will 

 be found that Q is largest for \n — m\ = 1, i.e., for the lowest allowed 

 transition, and that the oscillator strength falls off approximately as 

 \/\n — m\ for the higher allowed transitions. 



For polyenes of different lengths, Q for the first allowed transition 

 should be proportional to the length, and / to vQ'^, but since v varies 

 approximately inversely with the length, / is also approximately pro- 

 portional to the length. This was one of the first empirical conclusions 

 from the comparative study of polyenes (Hausser et al., 1935; Smakula, 

 1934). The width of the first polyene absorption region is approximately 

 constant, and emax is therefore also approximately proportional to the 

 length. 



Intensity predictions by the LCAO method (Mulliken and Rieke, 1941) 

 are too large, but their relative values for different molecules are accurate. 

 Predictions using the free-electron model appear to be accurate both abso- 

 lutely and relatively (Bayliss, 1948, 1952; Kuhn, 1948b; Simpson, 1948). 

 The low allowed transitions of 7r-electrons in extended polyenes should 

 be, and apparently are, polarized along the a:-axis; i.e., Qy = Qz — 0. 



If the polyene has one cis link in the middle, so that it is doubled back 

 on itself like a hairpin, these intensity relations are changed. The first 



