Defining subsystems and related basic quantities

Now we pass to the formal derivations of a hybrid method. We assume that the orbitals forming the basis for the entire molecular system may be ascribed either to the chemically active part of the molecular system (reactive or R-states) or to the chemically inactive rest of the system (medium or M-states). In the present context, the orbitals are not necessarily the basis AO, but any set of their orthonormal linear combinations thought to be distributed between the subsystems. The numbers of electrons in the R-system (chemically active subsystem) Nr and in the M-system (chemically inactive subsystem) NM = Ne — Nr, respectively, are good quantum numbers at least in the low energy range. We also assume that the orbital basis in both the systems is formed by the strictly local orbitals proposed in [59]. The strictly local orbitals are orthonormalized linear combinations of the AOs centered on a single atom. In that sense they are the classical hybrid orbitals (HO)  

The electronic Hamiltonian for the whole system is now a sum of subsystem Hamiltonians and of their interaction which is taken to comprise the terms of two types - the Coulomb Wc and the resonance (electron transfer) Wr interactions  

The Hamiltonian for the M-system is a sum of the free M-system Hamiltonian II h and of the attraction of electrons in the M-system to the cores of the R-system VR. Analogous subdividing is true for the R-system. On the other hand the interaction terms further subdivide to  

The exact wave function of the system is represented by a generalized group function (GGF) where numbers of electrons in subsystems are not fixed  

The electron variables related to the R- and M-systems entering the wave functions of the complex system in eq. (1.227) are entangled. Separating them is reached by projecting the wave function of the eq. (1.227) of the entire system on the subspace of the GF of the eq. (1.216). This basically repeats the moves done in the previous section in a narrower context here we are going to pay more attention to the resolvent term Wrr and to setting conditions on the subsystem wave functions. 

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