EXCITATION OF POLYENES AND PORPHYRINS 99 



refer to the symmetry of the wave functions in standard symmetry 

 notation; in the calculations porphin was assumed to have Dih, or square, 

 symmetry. 



Angular Momentum and Vector Addition. The fundamental difference 

 between porphyrins and the polyenes stems from the fact that the conju- 

 gated system of porphyrins is not linear but has a two-dimensional exten- 

 sion in its own plane. As a result, some orbitals of porphin such as the 

 lowest unfilled one (in the Eg column of Fig. 2-18), which we again call 

 the ^-orbital, are doubly degenerate. That is, they consist of two 

 orbitals, one pointed in the x direction of the square, the other in the y 

 direction, the two components being of equal energy because these direc- 

 tions are physically indistinguishable if and when porphin is scjuare. 



Or we may equally well think of the electron as switching rapidly from 

 one of these components to the other and so traveling clockwise (one 

 component) or counterclockwise (the other component) around the ring. 

 This degeneracy in the ^-orbital makes each of the long-wave-length 

 transitions of porphin also doubly degenerate, with two components polar- 

 ized in mutually perpendicular directions (Fig. 2-19). 



In a two-dimensional conjugated system it also often happens that 

 other pairs of orbitals that are not strictly degenerate, like the £'20, are 

 nevertheless almost degenerate, as, for instance, the highest filled pair, 

 Aiu and A2U, in the y' columns of porphin. This suggests a modification 

 of the LCAO results, as follows: To a certain approximation we may 

 treat such a pair of orbitals in the same way that we treated the Eg 

 orbital, i.e., as though they were components of a doubly degenerate 

 /-orbital, with the electron in one component moving clockwise around 

 the ring, in the other counterclockwise (Piatt, 1949, 1950). 



This combined /-orbital may then be thought of as having an angular 

 momentum, in this case with a value of four atomic units; i.e., the Aiy, 

 and A 2m components out of w^hich we made this orbital are each crossed 

 through the center by four nodal lines. They are both odd functions, 

 which change sign on inversion in the center of symmetry. The (7-orbital, 

 of Eg type, is even and is crossed by five nodal lines through the center, 

 so that it has an effective angular momentum of 5 units. 



On exciting an electron of angular momentum 4 to angular momen- 

 tum 5, the two momenta may be either in the same direction, and add, 

 or in the opposite direction, and subtract. The whole molecule, which 

 began in its ground state ^A with total angular momentum 0, may then 

 change its angular momentum to a value of 1 unit (5 — 4) clockwise 

 or counterclockwise, or of 9 units (5 + 4) clockwise or counterclock- 

 wise. These two values of momentum will give two degenerate singlet 

 states and two degenerate triplet states for the first excited configuration, 

 . . . dh-fg. We call these states 'B°, 'Q° and ^B°, 'Q°, where B refers 

 to 1 unit of angular momentum and Q to 9. Electron interaction will 



