Canonical transformation theory reference function

Consequently, with the simplihcations above, all the working equations of the canonical transformation theory can be evaluated entirely in terms of a limited number of reduced density matrices (e.g., one- and two-particle density matrices) and no explicit manipulation of the complicated reference function is required. 

A further point is of interest in the formal discussion of the canonical transformation theory. So far we have assumed that the reference function is fixed and have considered only solving for the amplitudes in the excitation operator. We may also consider optimization of the reference function itself in the presence of the excitation operator A. This consideration is useful in understanding the nature of the cumulant decomposition in the canonical transformation theory. 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

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