General Model Space Research Articles

A general model-space coupled-cluster method using a Hilbert-space approach

An extension of the recently reported multireference coupled-cluster method [L. Meissner, S. A. Kucharski, and R. J. Bartlett, J. Chem. Phys. 91, 6187 (1989)] to the general incomplete model space is presented. Connectedness of the cluster operators and the effective Hamiltonian is achieved by the admission of a special kind of internal excitation in the standard definition of the cluster operators and by setting corresponding matrix elements of the effective Hamiltonian equal to zero. This special type of similarity transformation leads to a unique partition of the excitations from the model space into those generated by the cluster operators and those generated by the effective Hamiltonian. The new form of the effective Hamiltonian preserves the size extensivity of the method. A discussion of the application of the method to the problem of the simultaneous calculation of energies of the ground closed-shell state and excited states is considered.

The general model space effective Hamiltonian in order-for-order expansion

The order-by-order perturbation expansion of the effective Hamiltonian for a multireference space is presented. The concept of the general model space (complete or incomplete) of Hose and Kaldor is used as well as their concept of a graphical representation of the perturbation expansion. The obtained result differs, however, from that of Hose and Kaldor. In the Hose and Kaldor derivation many types of disconnected diagrams of the effective Hamiltonian are declared reducible and omitted by in fact they do not completely cancel and these give an irreducible contribution to the effective Hamiltonian maxtrix causing the method to not be fully extensive. To ensure that all terms are included, we present modified diagram rules for the effective Hamiltonian.

The open-shell coupled-cluster method in general model space: Five states of LiH

The open-shell coupled-cluster method is used to calculate the potential functions of the five lowest states of LiH, at internuclear separation of 2.0≤R≤10.0 bohr, with a 18σ9π2δ basis. The usual complete-model-space calculation diverges, and a general, incomplete model space, comprising the physically important 2σ 2, 2σ3σ, and 2σ1π configurations (but excluding the high-energy determinants 3σ 2 and 1π 2 from the P space) is used. Good agreement with experimental molecular constants is obtained.

Connected-diagram expansions of effective Hamiltonians in incomplete model spaces. II. The general incomplete model space

We generalize here the formalism of the preceeding paper to encompass the case of the general incomplete model space. The classification of operators as diagonal or nondiagonal depends in this case upon the specific m-valence model space. It is stressed that even then one has to work in Fock space in order to get connected-diagram expansions, since connectedness is a Fock space property. Two choices of separable normalization of the wave operator W leading to a connected Heff are discussed. It is shown that the intermediate normalization is not separable in general and hence not compatible with a connected-diagram expansion. We also discuss how to generate ‘‘subduced’’ incomplete model spaces of lower particle rank such that Heff remains a valid effective Hamiltonian for these subduced model spaces as well. We discuss the nature of the various disconnected diagrams encountered in many-body formalisms and point out which of these are really worth worrying about. We finally comment on the question of the separability of the wave function into proper fragments.

On the connectivity criteria in the open-shell coupled-cluster theory for general model spaces

In this paper we study the following aspects of the open-shell coupled-cluster (CC) theories: (a) we examine the current theoretical status regarding the existence or non-existence of a linked-cluster theorem, ensuring the connectedness of the cluster amplitudes and the effective Hamiltonian; (b) we lay down the necessary and sufficient conditions for connectivity for a general (incomplete) model space, involving valence particles as well as valence holes; and (c) critically re-assess the earlier theoretical works from a comprehensive and unifying view point.

Aspects of linked cluster expansion in general model space many-body perturbation and coupled-cluster theory

In this paper, a method of generating separable forms of the wave-operator for incomplete model spaces is discussed. With a time-dependent access to the many-body perturbation and coupled-cluster theories, it is shown how one can extract the regular part of the wave-operator which consists of linked cluster-operators only in the adiabatic limit. The procedure naturally suggests a hierarchy of lower valence model spaces P (k) . once a particular m-valence incomplete model space P (m) is specified. The wave-operator Ω and the effective Hamiltonian H eff are linked in this development and are valence-universal in the sense of being valid for all P (k)' s. 0 k m. We have derived two distinct forms for Ω: (i) Ω = {exp(S)}, with { } as normal order with respect to suitable vacuum, where S are open operators inducing transitions from P (m) to outside it; (ii) Ω N = {exp(S + X)}, where X are additional closed operators which are introduced to maintain isometry of Ω N : P (k) Ω N + Ω N P (k) = P (k) . In neither of these choices do we have intermediate normalization. It is also possible to develop an alternative strategy with the complete model spaces, such that an effective valence-universal operator H may be found which generates roots, only a subset of which are equal to the eigenvalues of H. These subsets are the ones that H eff would have furnished. This may thus be viewed as a Fock-space realization of the intermediate Hamiltonian approach.

A note on the convergence of multiconfigurational many‐body perturbation theory

AbstractThe convergence of multiconfigurational many‐body perturbation theory (MC MBPT) is discussed in connection with the intruder state. Its convergence properties are first examined with a fictitious three‐level system employing a Hermitian version of MC MBPT, which permits a general model space. It is then applied to the H2—H2 and N2 systems. The results suggest that a more extensive model space is likely to embrace new intruder states and the space extension be executed with due caution.

A hermitian open-shell many-body perturbation theory for treating intruder states

In this paper we have formulated an open-shell many-body perturbation theory (mbpt) that applies to an incomplete model space. The effective HamiltonianH eff generated in our theory is hermitian. We follow the resolvent-operator based time-dependent formulation ofmbpt of Banerjeeet al, and show quite generally that, by classifying the various determinants spanning the Hilbert space as model valence spaceP, virtual valence spaceR and virtual spaceQ, a diagrammaticmbpt satisfying size consistency can be developed. The chief new features of the theory are (i) manifest hermiticity ofH eff; (ii) presence of disconnected diagrams no part of which is a legitimate diagram (irreducible disconnected diagrams); (iii) presence of folded diagrams with components that may be irreducible disconnected diagrams; (iv) a consistent treatment of the variousn-valence sectors of the Hilbert space that may be interacting throughH eft if valence-holes are present—in particular a modified treatment of the core-valence separation. The generalisation afforded by the theory offers useful conceptual as well as computational advantages because the convergence difficulty encountered in a complete valence space formulation may be bypassed here by keeping the offending valence space determinants out of the model space. A brief critique of the developmentvis-a-vis the only other general model space development leading to a nonhermitianH eff is also given.

Applications of a non-perturbative many-body formalism to general open-shell atomic and molecular problems: calculation of the ground and the lowest π-π* singlet and triplet energies and the first ionization potential of trans-butadiene

In this paper we explore the feasibility of widening the scope of the non-perturbative open-shell many-body formalism recently developed by us [1], which utilizes an Ursell type of cluster expansion about certain starting wavefunctions spanning a model space. We show that, by generalizing the definition of the cluster expansion operator, we can incorporate into the model space (a) determinants differing widely in energy and (b) determinants differing in their number of electrons. This flexibility is useful for the calculation of difference energies of interest, like transition energies and ionization potentials of atomic and molecular systems. The generalized scheme has been tested on the 4π-electron problem trans-butadiene for which, by choosing a very general model space, we have calculated the energies of the ground, the lowest π-π* singlet and triplet and the first ionization potential by choosing a single composite cluster expansion operator for all states. Results for some more restricted choice of ...