Hilbert space matrix approach

In Section IV.A, the adiabatic-to-diabatic transformation matrix as well as the diabatic potentials were derived for the relevant sub-space without running into theoretical conflicts. In other words, the conditions in Eqs. (10) led to a.finite sub-Hilbert space which, for all practical purposes, behaves like a full (infinite) Hilbert space. However it is inconceivable that such strict conditions as presented in Eq. (10) are fulfilled for real molecular systems. Thus the question is to what extent the results of the present approach, namely, the adiabatic-to-diabatic transformation matrix, the curl equation, and first and foremost, the diabatic potentials, are affected if the conditions in Eq. (10) are replaced by more realistic ones This subject will be treated next. 

An abstract approach was taken in Refs. [Gorini 1976 Sudarshan 1992], We assume that the operators act on a Hilbert space H. We list the most general linear transformation on a density matrix, by writing ... 

The ADC(3), NR2, 2ph-TDA, and BD-Tl [24] approximations display this structure in H. In these methods, couplings between simple (h and p) and triple (2hp and 2ph) operators may be evaluated through first or second order in the fluctuation potential. Hilbert space methods of similar computational difficulty, such as IP-CCSD [25], do not have couplings between nh(n-l)p and np(n-l)h sectors. A similar formulation can be achieved by using Kohn-Sham orbitals to define a reference state [26]. Identical expressions for the matrix elements of H can be derived using a diagrammatic approach [27]. 

A further point which should be mentioned concerns the treatment of incomplete model spaces. Meissner and his collaborators [61], have shown that the cancellation of terms corresponding to disconnected diagrams in the equations for amplitudes is, in general, not a sufficient condition for extensivity. In any Hilbert-space MRCC method, extensivity may be destroyed by diagonalization of the effective Hamiltonian matrix. For the complete model spaces employed in the studies reviewed here, the diagonalization has a full Cl-like character and, therefore, extensivity is ensured. Although Meissner and collaborators [61,62] devised an approach which should also be extensive for the case of an incomplete model space, practical implementation appears somewhat complicated. 

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