Functions and the Exponential Ansatz

The complicated notation of Eq. [12] can be drastically reduced by using a simple analytic form for the cluster functions. Note again that each determinant involving a cluster function is actually a linear combination of determinants each of which differs from the reference, C o by specific number of orbitals. For example, the 27th term on the right-hand side of in Eq. [12] expands to become 

We will define a creation operator by its action on a Slater determinant  

Note also that an annihilation operator acting on the vacuum state gives a zero result, 

Pairwise permutations of the operators introduce changes in the sign of the resulting determinant, for example. 

Therefore, the anticommutation relation for a pair of creation operators is simply 

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