Approximation canonical transformation theory

Up to this point our discussion of canonical transformations has been exact. We now proceed to the specific approximations that characterize our formulation of CT theory and discuss their relationship with approximations commonly made in other theories involving canonical (i.e., unitary) transformations. 

Solving this partial differential equation, which is required to construct the canonical transformation, is in general as difficult as solving Eq. (1). However, Eq. (5) is very useful in obtaining approximate solutions using, for example, perturbation theory. 

Since both initial and final states satisfy the same equations of motion, the transformation of Eq. (6) is a natural canonical transformation. This kind of canonical transformation is a basic tool in the so-called Lie canonical perturbation theory for obtaining approximate constants of the motion. 

Recently Sokolov and Chan published a paper where another quasiparticle-based framework was outlined [40], This approach has the enviable feature of Fermi-vacnnm independence. In that paper the concept of a non-particle-number-conserving canonical transformation was introduced and, as a low-order approximation, the application of a Bogoliubov transformation in a second-order perturbation theory was investigated to describe MR situations. The presented results, as well as the intruder problem on the PES in the BeH2 model, indicate that the applied approximation needs further improvements. 

The localized many-body perturbation theory (LMBPT) applies localized HF orbitals which are unitary transforms of the canonical ones in the diagrammatic many-body perturbation theory. The method was elaborated on models of cyclic polyenes in the Pariser-Parr-Pople (PPP) approximation. These systems are considered as not well localized so they are suitable to study the importance of non local effects. The description of LMBPT follows the main points as it was first published in 1984 (Kapuy etal, 1983). 

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