Operator manifolds partitioning

Partitioning the operator manifold can lead to efficient strategies for finding poles and residues that are based on solutions of one-electron equations with energy-dependent effective operators [16]. In equation 15, only the upper left block of the inverse matrix is relevant. After a few elementary matrix manipulations, a convenient form of the inverse-propagator matrix emerges, where... 

As shown in Section II, we wish to calculate the poles and residues of P Q . However, even using moderately large operator manifolds, the inverse matrix becomes so large that we cannot evaluate all elements of it equally well. We therefore wish to treat one part of it better than the rest. Which part we choose will be directed by the physics of the problem. In order to do so it is convenient to partition (Lowdin, 1963) the inverse matrix, for instance in the following way (Nielsen et al, 1980), letting hf, = h —... 

Because of the high research activity level on how to use an MCSCF reference in the GFs (EP and PP), it is not presently clear how to optimally choose truncated sets of 1V operators. It is likely that many workers will carry out test calculations involving many choices of the pertinent operator manifolds before this situation is improved. Moreover, questions concerning when and how to partition the resulting (1V / T/) matrix so as to reduce the dimension of the matrix whose poles are to be found remain unanswered... 

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

If the projection manifold spans the entire field operator space, there is no approximation. However, that is not feasible and one proceeds by partitioning the projection manifold so that h = (ql q) f. Provided f is orthogonalized against the particle-hole and the hole-particle space, i.e.,... 

---