3.9

Proof of equation ()

 

In eq. (3.8), consider the special case of :
 
Let be
 
and f a function that is completely symmetric in the arguments . The restricted sum of eq. (B1) can be written as the unrestricted sum
 
The matrix elements vanish unless the all appear in . Consider a term that satisfies this condition. There are ways in which the indices r that agree with one of can be distributed over the postions . Therefore, imposing the requirement that the first indices should agree with (up to a permutation) one obtains

 
Here, the restriction means that none of the indices ,, is allowed to coincide with any one of the indices , . The short hand notation means the same. Eq. (B4) is obviously the same as
 
which is easily generalized to eq. (3.9).

  
Table i: Two-body interaction in the exciton picture.

 
Table i: (Continued) Two-body interaction in the exciton picture.