Electronic structure and spectrum of R-system

Wave function of electrons in quantum R-system Ap satisfies the Schrodinger equation with the effective Hamiltonian iTff eq. (1.246), which is obtained by averaging the interaction operators in eq. (1.232) over the ground state of the M-system, i.e. over Ap, and acts on the quantum numbers (variables) of electrons in the R-system. 

In the frame of the hybrid methods it must be computed by a QM method. The Schrodinger equation with the effective Hamiltonian Hjf has multiple solutions, which describe excited states of the R-system provided the M-system is frozen in its ground state. Electronic energy of the system in the state expressed by the wave function eq. (1.231), has the form [29,30]  

The description of the electronic structure of the complex molecular system given by the system eq. (1.246) is perfectly sufficient when it goes about the hybrid QM/QM methods, when both the parts of the complex system are described by some QM methods. In the case of the hybrid methods in a narrow sense i.e. of the QM/MM methods, further refinements are necessary. The problem is that the description provided by eq. (1.246) suffers from the need to calculate the expectation values in these expressions over the wave function i.e. over the solution of the self-consistency equations eq. (1.246) in the presence of the R-system. This result does not seem to be particularly attractive since the functions Y are not known and are not supposed to be calculated in the frame of the MM procedure. Thus the theory must be reformulated in a spirit of the theory of intermolecular interactions [67] and to express necessary quantities in terms of the observable characteristics of free parts of the complex system. 

The reformulation of the theory of interaction between the R- and M-systems in terms of observables pertinent to the M-system assumes certain procedure for evaluating either the wave functions 4 ,)/ or directly the necessary expectation values taken over it. To do so, we notice that according to eq. (1.235) the effective Hamiltonian H fj for the M-system in the presence of R-system, defining is close to the Hamiltonian Hfor the free M-system. The assumption that the M-system is inert implies that its characteristic excitation energies are large, thus the reduced interac- 

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