Truncation of the Exponential Ansatz

Recall that the exponentiated operator may be expanded in a power series 

Inserting this into the energy expression Eq. [44] we obtain 

Note that f is at most a two-particle operator and that T is at least a one-particle excitation operator. Then, assuming that the reference wavefunction is a single determinant constructed from a set of one-electron functions. Slater s rules state that matrix elements of the Hamiltonian between determinants that differ by more than two orbitals are zero. Thus, the fourth term on the left-hand side of Eq. [48] contains, at the least, threefold excitations, and, as a result, that matrix element (and all higher order elements) necessarily vanish. The energy equation then simplifies to 

This is the natural truncation of the coupled cluster energy equation an analogous phenomenon occurs for the amplitude equation (Eq. [45]). This truncation depends only on the form of f and not on that of T or on the number of electrons. Equation [49] is correct even if T is truncated to a particular excitation level. 

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