Incomplete Model Spaces Research Articles

Explication and revision of generalized Van Vleck perturbation theory for molecular electronic structure

A revision of second-order Generalized Van Vleck Perturbation Theory (GVVPT2) for the description of dynamic electron correlation in molecules is presented. It is shown that the basic formulas of the suggested method are well-defined approximations to the theoretically carefully constructed self-consistent quasidegenerate perturbation theory. Furthermore, it is shown that nonlinear responses to the perturbations can be obtained by explicit formulas. The revised GVVPT2 makes active use of the recently introduced concept of macroconfigurations, whereby vast numbers of null Hamiltonian matrix elements are prescreened with minimal computational cost and the remainders are organized for facile computation by Table-CI-like methodology. Moreover, use of macroconfigurations allows the efficient use of incomplete model spaces, which extends drastically the applicability of the method. Representative calculations on model systems studied previously with the original formulation show close agreement and on additional model systems show the wide applicability of the revised formulation.

Open-shell coupled-cluster theory for a general incomplete model space using eigenvalue-independent partitioning

Multireference coupled-cluster (MRCC) theory using an eigenvalue-independent partitioning (EIP) technique is formulated for a general incomplete model space, taking a valence universal normal ordered wave operator …ifΩ = srmexp(S)s that acts upon the model space P. For the size-extensivity requirement the intermediate normalization is abandoned (i.e. PΩP ≠ P) and hence the closed part of Ω, Ω cl , is not unity. This leads to an implicit equation for the H eff. The EIP structure of the MRCC equation is capable of producing the main as well as the satellite states together.

Ground and excited states of K2 and K2+ by the open-shell coupled cluster method

The Fock space open-shell coupled cluster method is used to calculate the potential functions of the five lowest states of K2 at internuclear separation of 2≤R≤7 Å and the three lowest states of K2+ at 2≤R≤15 Å, with a 112-function contracted Gaussian basis. The usual complete-model-space calculation diverges and a general, incomplete model space is used. Core and core–valence correlation is found to be important, and very good agreement with experimentally known molecular spectroscopic constants is obtained when it is included: equilibrium internuclear separations (except for the very flat x 3Σu+ state of K2) are accurate to 0.07 Å, dissociation and excitation energies to 0.09 eV, and vibrational constants to 6 cm−1.

Hilbert space coupled-cluster method in an incomplete model space

The Hilbert space multireference coupled-cluster method with single and double excitations (HS-MRCCSD) is implemented in complete and special classes of incomplete model spaces. Vertical excitation energies of the water molecule are calculated. Good agreement (within 0.1 eV) is obtained with experimental values, and also with results of Fock space calculations which include effects of triple excitations in lowest order (FS-MRCCSD+T). The HS-MRCCSD functions are not pure spin states, and small energy differences occur between sublevels which should be degenerate.

Coupled-cluster method for an incomplete model space

The coupled-cluster method with multidimensional reference space is studied in the case of the incomplete active space (IAS ). The latter was chosen as a subspace of the Hilbert space corresponding to a fixed number of valence particles. Two different approaches for the normalization condition are analyzed. When not imposing intermediate normalization, the cancellation of disconnected terms is proven, ensuring that extensive energies are obtained. © 1992 John Wiley & Sons, Inc.

The multireference coupled-cluster method in Hilbert space: An incomplete model space application to the LiH molecule

The first results from a Hilbert space, multireference coupled-cluster (CC) method in an incomplete model (active) space are reported for the five lowest states of LiH. The active space is spanned by several configurations at the level of single and double excitations, where the configuration(s) causing intruder state problems are excluded from the complete Hilbert reference space. Full inclusion of single- and double-excitation operators is considered in the expansion for the cluster operator, with all quadratic terms in the renormalization part. The multireference CC results for the ground (X 1Σ+) and four low-lying excited states (a 3Σ+, A 1Σ+, a 3Π, A 1Π ) of LiH are compared with the corresponding full configuration-interaction (FCI) energies. The agreement between FCI and CC values within a few hundredths of mH for the Π states proves the feasibility of the present method to describe, quantitatively, the quasicomplete reference space problem. Deviations of the incomplete multireference results from the exact FCI energies for for excited Σ states vary with respect to the quasidegeneracies involved from a few tenths of mH for the nondegenerate (near-equilibrium) geometries to several mH in the dissociation region.

Hermitian formulation of the coupled-cluster approach

A Hermitian formulation of the coupled-cluster approach (CCA) is developed, based on the Jorgensen condition, P Omega Dagger Omega P=P, Omega being the wave operator and P the projection operator for the model space. This leads to a formalism where the exact as well as the model functions are orthonormal, and the effective Hamiltonian has the manifestly Hermitian form Heff=P Omega Dagger H Omega P. It is shown that the Jorgensen condition is compatible with the connectivity criteria (connected cluster operator and effective Hamiltonian) for a general, incomplete model space. Even with an effective Hamiltonian of this form, however, nonHermiticity may be introduced when the cluster expansion is truncated. This can be remedied by a reformulation of the coupled-cluster equations, where additional terms, which cancel in the complete expansion, preserve Hermiticity at each truncation. The new equations also lead to additional terms in the cluster operator itself, which make it possible, for instance, to include important effects in the pair approach that otherwise would require the evaluation of three- and four-body clusters.

The Fock space coupled cluster method: theory and application

The Fock space coupled cluster method and its application to atomic and molecular systems are described. The importance of conserving size extensivity is demonstrated by the electron affinities of the alkali atoms. Two types of intruder states are discussed, one attributable to the orbital energy spectrum and the other caused by two-electron interactions. They are illustrated by the excited states of Li2 and by1S states of Be, respectively. It is shown how both problems may be solved using incomplete model spaces. The selection of the model space in a moderately dense spectrum is discussed in connection with N2 excited states.

Na2 Ground and Excited States by the Open‐Shell Coupled‐Cluster Method

AbstractThe open‐shell coupled‐cluster method is used to calculate the potential functions of the five lowest states of Na2, at internuclear separation of 3.5 bohr ≤ R ≤ 10 bohr, with an 80—function contracted Gaussian basis. The usual complete model space calculation diverges, and a general, incomplete model space is used. Core (2s, 2p) correlation plays an important role. Very good agreement with experimentally known molecular constants is obtained when the 2s, 2p, and 3s electrons are correlated.

Multireference coupled cluster theory in Fock space

A Fock space multireference coupled cluster method based on incomplete model spaces is described. Some of the essential computational aspects of the theory are discussed with the aid of the diagrammatic representation of the equations. An application to the calculation of ionization potentials and excitation energies ofs-tetrazine is presented along with comparisons with conventionalab initio calculations and experimental results.

Excitation energies with multireference many-body perturbation theory

Excitation energy calculations with multireference many-body perturbation theory (MRMBPT) are theoretically and numerically studied. An extension of the Hose–Kaldor (HK) scheme is presented, which removes disconnected terms and, hence, the size-extensivity error for higher-order MRMBPT approximations. The excitation problem requires the use of an incomplete model space for which connectivity of the effective Hamiltonian Heff, is incompatible with intermediate normalization. In our formulation a proper choice of the model space as an introduction of a ‘‘quasiintermediate’’ normalization leads to the connected structure of Heff. This guarantees size extensivity of the method which generally could not be achieved with the earlier Hose–Kaldor (HK) framework based upon intermediate normalization. Special attention is paid to the case when the Hartree–Fock (HF) approximation is used in the zeroth-order step. In this specific case the HK formalism applied to a subspace of the model space spanned by singly excited determinants gives the same result through third order which means that in the HF case disconnected contributions to Heff disappear and size extensivity is preserved to that order. MRMBPT(3) results for N2 and CO are presented to offer illustrative comparisons with the recently proposed EOM-CCSD and Fock space MRCCSD results.

Li 2 ground and excited states by the open-shell coupled-cluster method

The open-shell coupled-cluster method is used to calculate the potential functions of the nine lowest states of Li 2, at internuclear separation of 4⩽ R⩽11 bohr, with a 74-function contracted-Gaussian basis. The usual complete-model-space calculation diverges, and a general, incomplete model space is used. Excellent agreement is obtained with experimentally known molecular constants: equilibrium internuclear separations are accurate to 0.01 Å, dissociation and excitation energies to 0.02 eV, and vibrational constants to 3 cm −1. Possible difficulties in model space structure are discussed.

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.

Size‐extensivity in multireference many‐body perturbation theories: A direct comparison between single‐reference and multireference perturbation theories in the nondegenerate case

AbstractLinkage properties of the diagrammatic representation of the energies obtained in the multireference many‐body perturbation calculations with respect to the incompleteness or completeness of the model space are discussed. The case of not completely degenerate model space is considered for which a comparison with the standard single‐reference many‐body perturbation expansion is possible. The Hose–Kaldor type of graphical representation of the perturbation expansion for the effective Hamiltonian is used in this comparison. It is shown that for an incomplete model space the perturbation expansion is not size‐extensive. In this case, for a truncated expansion of the effective Hamiltonian, the energies obtained by diagonalization of the effective Hamiltonian matrix are represented by both linked and unlinked irreducible contributions. The unlinked ones do not appear when the complete model space is used.

A multireference coupled-cluster method for special classes of incomplete model spaces

A multireference coupled-cluster (MRCC) method for special but general classes of incomplete model spaces is developed within a Hilbert space framework. The formulation avoids the valence universality requirement and related Fock space considerations that require hierarchical solutions for different number of electrons. Consequently, the Hilbert space approach has fewer amplitudes to determine. It is shown that diagonalization of the effective Hamiltonian leads to purely extensive energies, providing the formal basis for a general MR-CC methods for potential energy surfaces.

Size-extensive effective hamiltonian formalisms using quasi-Hilbert and quasi-Fock space strategies with incomplete model spaces

An open-shell coupled cluster (CC) theory is developed using an incomplete model space (IMS) that works entirely within one particular n-valence Hilbert space sector. It is proved that the theory is size-extensive since both H eff and the energies obtained therefrom are connected. The method may be viewed as the Fock space CC theory for an IMS projected onto a given Hilbert space, and is denoted here as a “quasi-Hilbert” space theory. A “quasi-Fock“ formalism is also outlined where additional information on the lower valence model spaces are deliberately retained.

On the extensivity of the roots of effective Hamiltonians in many-body formalisms employing incomplete model spaces

We analyze the current controversies regarding the extensivity of energies computed from an effective Hamiltonian defined over an incomplete model space (IMS). We show that the recently developed formalism in Fock space, using a size-extensive normalization for a valence universal operator Ω, generates both a connected H eff and size-extensive energies. In contrast, the corresponding Hilbert space formalisms, with intermediate normalization for Ω, produce size-inextensive energies. It is emphasized that the extensivity of energies for the Fock space theory stems not just from the connectivity of H eff but also due to the existence of certain special null matrix-elements in the matrix of H eff demanded by the decoupling conditions defined in Fock space.

The eigenvalue-independent partitioning technique in Fock space: An alternative route to open-shell coupled-cluster theory for incomplete model spaces

Our recent open-shell coupled-cluster (CC) theory for incomplete model spaces, having valence holes and valence particles, is cast in an alternative form having computational advantages. An eigenvalue-independent partitioning technique in Fock space converts the CC equations for each m-hole, n-particle model space of dimension N (m,n) to a non-Hermitian eigenproblem of larger dimension, whose N (m,n) roots are the energies of interest. This alternative strategy works even in the presence of intruders, as is demonstrated by applying it to the main and satellite peaks of the IP and Auger spectra of HF and to the excitation energies of N 2.

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.

Atomic and molecular applications of open-shell cluster expansion techniques with incomplete model spaces

Pilot numerical applications are reported of two recently formulated cluster expansion formalisms that generate connected effective Hamiltonians even for incomplete model spaces. Results are presented for prototype systems such as (a) the He atom, (b) the H 2 molecule and (c) the He 2 2+ ion. For He and H 2 we give results for typical examples of incomplete model spaces, and for He 2 2+ we present adiabatic potential energy curves the first four 1Σ g + and the first two 3Σ g +states, calculated with an incomplete model space.