In Search of a Rational Dressing of Intermediate Effective Hamiltonians Barthe ́ le ́ my Pradines, Nicolas Suaud, and Jean-Paul Malrieu* Laboratoire de Chimie et Physique Quantiques, IRSAMC, Universite ́ de Toulouse 3 Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France Laboratoire de Chimie et Physique Quantiques UMR 5626, CNRS, 118 route de Narbonne, 31062 Toulouse Cedex, France ABSTRACT: 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. 1. INTRODUCTION Single reference perturbation theory is a major tool for the study of closed shell molecules, providing an order-by-order generation of the wave function, of the energy, and of the other properties. It may be a computational tool, 1-5 but one of its major merits is conceptual, since it generates a diagrammatic representation of the wave function and of the energy, leads to the fundamental linked cluster theorem, 6 and has established the basis of the most rational methods for the study of the electron correlation problem, in particular the coupled cluster method. 7,7-10 When one considers the states that are intrinsically of multireference character, as are most of the excited states, the classical counterpart becomes the quasi degenerate perturbation theory (QDPT) 11-13 This theory defines an N-dimensional model space, a projector P ̂ 0 onto this space, and provides an order-by-order generation of an effective Hamiltonian, built in this model space. At convergence, the diagonalization of this Hamiltonian provides N exact eigenenergies of the exact Hamiltonian H ̂ , and the projections of the corresponding eigenvectors in the model space. These eigenvectors define the target space, of projector P ̂ , and this theory establishes the wave operator Ω ̂ , which sends from P ̂ 0 to P ̂ , as will be recalled later. This powerful tool faces major difficulties. On one hand it has been used to establish the generalization of the linked cluster theorem 12 to multireference situations, as well as the corresponding diagrams, but on the other hand the conditions to derive it were severe: the model space must be complete (full configuration interaction of a given number of electrons and orbitals), and the zeroth-order Hamiltonian was monoelec- tronic. This second condition is not strictly necessary to achieve the strict separability, provided that localized orbitals are used, but under the first of these conditions the perturbation theory will diverge in most of the atomic and molecular problems, due to the near degeneracy or paradoxical energy ordering between the model space determinants of higher energy and the outer space determinants of lowest energy. Nevertheless, the effective Hamiltonian theory offers the most rigorous derivation of model Hamiltonians, as soon as the target space eigenvectors are known or reliably approached by nonperturbative means. Two reasons have prompted us to propose, 30 years ago, the concept of intermediate effective Hamiltonians. 14 The inter- mediate effective Hamiltonians are less demanding then the effective Hamiltonians, they work in a N-dimensional model space but only ask for M < N exact roots and the corresponding eigenvectors in the full model space. One may frequently define an M-dimensional main model space and the complementary intermediate model space of dimension N - M. The main model space vectors are well separated in energy from the outer space vectors, which enables the perturbation theory to converge. This is a practical advantage. The other advantage is that one may eventually establish a model Hamiltonian in cases where the target space is not accessible or even not identifiable. The best example of these situations concerns the research of valence model Hamiltonians, which work in a basis involving both neutral valence bond (VB) determinants, of low energy, and ionic VB determinants, of high energy. The identification of the exact eigenvectors having the largest projections on the model space is neither easy nor always Special Issue: Jacopo Tomasi Festschrift Received: September 30, 2014 Revised: December 3, 2014 Article pubs.acs.org/JPCA © XXXX American Chemical Society A dx.doi.org/10.1021/jp509893r | J. Phys. Chem. A XXXX, XXX, XXX-XXX