46 LIGHT AND LIFE 



77- _> 77-* promotions lead to excited states of the same symmetry, but 

 differing significantly from the former. 



The state symmetry is a compound symmetry which is a product 

 of the orbital symmetries, the latter taken in the product as many 

 times as the orbital is used, i.e., as many times as there are electrons. 

 Group theory offers a simple and powerful tool for evaluating com- 

 pound symmetry in the Direct Product. The direct product of, e.g., 

 ^2 X ^2 in Cov is obtained (Fig. 7) by taking the separate products of 

 the eigenvalues or characters under each symmetry operation for the 

 symmetry species involved, and comparing the new set of characters 

 with the available sets in the table. Thus, ^2 X ^2 leads to the 

 set of products (1) (1), (1) (1), (-1) (-1), (-1) (-1) which yield 

 the set of characters 1, 1, 1, 1; thus A2 X ^2 = ^i- It is evident that 

 here the square of the characters of any symmetry species leads to 

 the characters of the A^, or totally symmetric species (symmetric under 

 all symmetry operations.) Thus, all closed shell electronic configura- 

 tions will be totally symmetric {A^ in Cov', A' in Ci,,) . 



The symmetry classification of an excited state thus can be readily 

 obtained. In formaldehyde, the n -> tt* promotion results in the 

 n-orbital and the 7r*-orbital being singly occupied. The respective 

 orbital symmetries are (cf. Fig. 10) bo and b^; thus Z?2 X ^1 = ^2 

 (where the upper case Schoenflies symbol is used for state symmetry 

 classification, and lower case for orbital symmetry) , and the remain- 

 ing filled orbitals all have a^ symmetry, which has no effect on the 

 state symmetry. In this fashion, all of the state symmetries recorded 

 in column 3 of Fig. 10, for the corresponding electronic states of 

 column 1, may be deduced by consulting Fig. 8 and the electronic 

 configurations of column 2 of Fig. 10, and then obtaining the direct 

 product of the corresponding symmetry species for each state. 



IV. Probability of Electronic Transitions; 

 Optical RoTATOR^ Strength 



A. Transition Moment Integral 



In order to evaluate the probability of an electronic transition in 

 a molecule, and the polarization of the transition with respect to the 

 molecular axes, the appropriate quantum mechanical integral, the 

 "transition moment integral," will be discussed briefly (7) . This 

 integral allows one to gain a physical insight into the nature of the 

 light absorption process, and is given in the top part of Fig. 11. 

 The probability, P, of an electric dipole transition is proportional to 



