ENERGY EXCHANGE IN PHOTOREACTIONS 



31 



the first several oscillations are shown as they were calculated b}^ Hirsch- 

 felder et al. (1936). The penetration of each valley by the collision path 

 is similar to the collision between two diatomic molecules and could per- 

 haps be treated as an independent collision process involving small parts 

 of the molecule treated as isolated entities. The fact that most of the 

 atoms of a molecule are participants in several independent modes of 

 motion greatly complicates such a calculation. This condition also 

 improves the probability of energy migration. Using the lowest point 

 of the potential well as the origin, the potential energy may be expanded 

 classically in a Taylor's series thus: 



E = 



A/ \dqijc 



2t L^ L^ \dqi dqj/ 



1=1 j=i 





Since the first-order terms are zero 

 and choice of the g's as normal co- 

 ordinates eliminates the second- 

 order cross terms, the energy, which 

 is not strongly dependent on the 

 third-order terms, may be expressed 

 as a sum of independent second- 

 order terms. However, energy mi- 

 gration is entirely due to the third- 

 order cross terms, which do not 

 vanish. In Fig. 1-12 idealized po- 

 tential wells for the cases with and 

 without these so-called "coupling" 

 terms are pictured. Trajectories in 

 the smooth well preserve energy in 

 its original distribution. The dis- 

 tortions of the second well are neces- 

 sary for migration. 



The third-order coupling coeffi- 

 cients are sufficiently large to allow 

 reasonably free exchange of energy 

 among all degrees of freedom, as 

 Franck et al. (1932) emphasized. 

 The rapidity of such energy migra- 

 tion may be explained by the shal- 

 lowness of potential wells for stable 

 molecules. The collision path al- 

 ways moves in the region of suitable 

 gateways. The density of these 



+ 



(1-22) 



(A B) - C DISTANCE 



^AB)-C DISTANCE 



Fig. 1-12. Idealized potential wells for 

 stable diatomic molecules: (a) without 

 coupling term between modes and (f>) 

 with coupling ternl.' 



