In search of a rational dressing of intermediate effective Hamiltonians

The intermediate effective Hamiltonians are designed to provide M exact energies and the components of the corresponding eigenvectors in the N-dimensional model space, with N > M. The effective Hamiltonian is not entirely defined by these N × M conditions, and several dressings of the Hamiltonian matrix in the model space are possible. Some of them lead to unreliable N - M roots associated with the intermediate model space. This defect appears dramatically when one refers to the weak separability property, namely, the fact that in a noninteracting A···B problem where the model space only involves excitations on A, the consideration of the excitations on B should not affect the spectrum of A. We suggest variants that should maintain the physical meaning of the intermediate roots. Numerical comparisons illustrate the relevance of this proposal.