40 



LIGHT AND LIFE 



in Fig. 5. It should be noted that in each case yz is taken to be the 

 molecular plane, and the x-axis is taken to be the out-of-plane axis. 



It will be noted that the three molecules (a) formaldehyde, (b) 

 pyridine, and (c) pyrimidine have identical symmetry properties as 

 far as their molecular skeletons are concerned. Each of these has the 

 identity operation E ('leave-it-alone'-operator) , the two-fold rotation 

 axis C2 (rotation by one half-turn about the z-axis) , the reflection 

 plane cr/ (reflection in xz plane) , and the reflection plane 0-^/' (re- 

 flection in yz plane); both of these are considered to be vertical planes, 

 i.e., containing the Co axis. The unique point group which includes 

 these four (and no other) symmetry operators is labelled the Co^ Point 

 Group (Schoenflies notation) . 



z 



I 

 I 



11/ 

 — --P— 



/ I 



H / 



/ M 



N> 



H' 



H 

 ■N 



X- 



T--f->H 



Fig. 5. Coordinate axes for the (a) formaldehyde, (b) pyridine, 

 (c) pyrimidine, and the (d) purine molecules. 



The purine molecule, Fig. 5 (d) , has a lower symmetry than the 

 previous three molecules. It possesses the identity operation E (in- 

 cluded in all point groups by definition) , which may be considered 

 to be synonymous with a one-fold axis C^ (rotation by a full turn 

 about the .x-axis) . The only other 'covering" symmetry operation 

 (which brings the molecular skeleton into itself) is a reflection plane 

 containing the molecular plane, aj, (reflection in yz plane: called 

 a horizontal plane because it is perpendicular to the highest-fold axis 



-C). 



C. Classification of Orbitals According to Symmeti-y Species 



All molecular orbitals which correspond to proper wavefunctions 



must, under the symmetry operators of the appropriate point group, 



satisfy a simple symmetry eigenvalue equation. This is illustrated in 



Fig. 6 for the Tr-orbital and the highest 7?-orbital of formaldehyde 



(C2V point group). The general equation is given at the top of 



