MICHAEL KASHA 



41 



Si// = Si/^ 



E(7r) = Ktt) 



CgCTT) = -1 (tt) 



O-^ ( TT ) = 1 ( TT ) 



c-;(7r) = -Ktt) 



E(n) = 1(n) 

 C2(n) = -1 (n) 

 o-i (n) = -1 (n) 

 o-^' ( n ) = 1 { n ) 

 .-. x\ '^ bg 



Fig. 6. Eigenvalue equations for symmetry operators of the Coy point group 

 acting on the Tp-orbital and hjghest n-orbital of the formaldehyde molecule. 



Fig. 6 and states that when the operator 5 acts on the wave function i/-, 

 the result should be uniquely a product of a constant s and the wave- 

 function xj/. Applying the four operators E, Co, cr/, cr/' to the TT-orbital 

 of formaldehyde (Figs. 3 and 5) , it is evident that the values of the 

 constants s are in turn -|-1, — 1, +1, — 1. In other words, the Tr-orbital 

 is symmetric under the operator E and (j^', but antisymmetric under 

 C2 and cTv'. A different set of eigenvalues (or constants) s is obtained 

 for the same set of four operators applied to the n-orbital of formal- 

 dehyde (Figs. 3 and 5, where the 2py-lone pair is taken as the highest 

 7i-orbital) . 



Comparing this set of eigenvalues s with the sets (horizontal rows) 

 corresponding to the syinmetry species A^, A 2, B^, B2 of the Cg^ 

 Character Table (Fig. 7, left) allows a unique symmetry classification. 

 Thus, the 7r-orbital of formaldehyde is said to "transform as the sym- 

 metry species Bi" or to "belong to the representation B^." The con- 

 vention is used, that for orbitals, lower case Schoenflies species symbols 

 are used, in this case b^ (Fig. 6, bottom); while the upper case Schoen- 



T 



X 



y 



■2v 



A, 



A2 



B, 

 B2 



E C2 cr^(xz) CTvCyz) 



1 



-1 

 -I 



! 



Fig. 7. Character tables for the C>„ and Cj^ point groups, with 

 transformation properties of translations. 



