Complete Model Space Research Articles

Application of the complete-model-space MR-CCSD theory to the 2s 21S and 2p 21S states of Be

A complete-model-space (CMS) valence-universal coupled-cluster (VU-CC) method is applied to the problem of calculating the energies and wavefunctions for the 2s 2 1S and 2p 2 1S quasidegenerate states of Be. The difficulties encountered with this problem have supported the viewpoint that CMS formulations of CC theories are inapplicable in the presence of intruder states. The present approach is based on Lindgren's normal ordered Ansatz for the wave operator. The cluster operator is approximated by its one-and two-electron components represented in terms of configurational excitations, which results in a drastic reduction of cluster amplitudes compared with the spin—orbital representation. The results obtained for a 9s7p5d STO basis set are in good agreement with their accurate counterparts. The multi-reference CC formulation improved the ground state correlation enery by 1.3% in comparison with the single-reference approach.

E2 and M1 Transition Probabilities in Ions of the Nitrogen Isoelectronic Sequence Calculated using MBPT

Forbidden electric quadrupole (E2) and magnetic dipole (M1) transitions are of extreme importance in astrophysics. Up to now the most extensive calculations for the nitrogen isoelectronic sequence have been done using the method proposed by C.J. Zeippen [1] or in MCHF approximation [2]. To account for electron correlations both these methods use a large list of configurations. We have chosen the stationary many-body perturbation theory (MBPT) [3] for the inclusion of the electron correlations. The calculations have been perfomed in the second order in the complete model space 1s22s2p3+ 1s22p5. Relativistic corrections have been accounted for in the Breit-Pauli approximation. In the Table we present probabilities for electric quadrupole W(E2) and magnetic dipole W(M1) transitions (in s−1), wavelengths λ (in A). The comparision of the results shows that our second order calculation data in the most cases are closer to term-energy corrected ones from [1].

The multi-reference Hilbert space coupled-cluster study of the Li2 molecule. Application in a complete model space

The ground and three low-lying excited states of the Li2 molecule are studied by our new multi-reference fully quadratic Hilbert space coupled-cluster method, using a complete model space of four reference functions. DZ and DZP results are reported and compared with full-Cl. The deviations are usually within the range of a few tenths of microhenry. Questions of convergence, including the modification of the one-electron potential and its critical influence on the computational performance of the present method are discussed.

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.

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.

Comparison of complete model space quasidegenerate many-body perturbation theory for LiH with multireference coupled cluster method

The relative efficacy of using low order trucations with large reference space vs high order methods with small reference space is tested by comparing quasidegenerate many-body perturbation theory (QDMBPT) calculations of potential curves for the five lowest electronic states of LiH with the multireference coupled cluster calculations of Ben-Shlomo and Kaldor [J. Chem. Phys. 89, 956 (1988)]. The infinite order coupled cluster calculations use two configurational reference spaces involving the 2σ, 3σ, and 1π orbitals, while the QDMBPT computations are truncated at either second or third orders and employ the full active reference space formed either from the 2σ, 3σ, and 1π or from the 2σ, 3σ, 4σ, and 1π orbitals. This gives us the opportunity of testing the dependence of QDMBPT computations on the size of reference space, the available freedom in choosing valence orbitals and orbital energies, and the order of truncation. Second order, four valence orbital space QDMBPT calculations provide good agreement with the repulsive portion of the coupled cluster potentials, but yield a separated atom limit that is too high and that therefore distorts the remainder of the potential. Third order improves the separated atom limit considerably, providing good agreement with the coupled cluster calculations. The ‘‘full chemical’’ five orbital reference space, on the other hand, yields very good agreement with coupled cluster potentials when using only the simpler second order QDMBPT calculations, and third order corrections in this case are very small but generally improve agreement with coupled cluster potentials. The five orbital reference space calculations are quite insensitive to a wide range of different choices of valence orbitals and orbital energies, demonstrating a robustness to the QDMBPT formalism used.

Scaled Hartree-Fock orbitals for perturbation treatment of ground and excited electronic states.

Perturbation theory is a standard method used to calculate electronic correlation effects in atomic and molecular systems. It is able to provide a size-consistent description of electronic energies. Perturbation theory for excited states is often only applicable in a restricted way due to convergence problems (intruder-state problems). The starting point for a perturbation treatment is the decomposition of the total Hamiltonian H into ${H}_{0}$+V. The present study is devoted to proper definitions of ${\mathrm{H}}_{0}$ for a convergent treatment of ground and excited states. We use the scaled Hartree-Fock operator as an unperturbed Hamiltonian. It contains scaling factors to modify the Coulomb and exchange terms. These factors can be chosen in order to improve the convergence properties for the treatment of ground and excited electronic states. As a simple illustrative example, second-order perturbation calculations are reported in complete model spaces for $^{1}\mathrm{\ensuremath{\Sigma}}_{\mathrm{g}}^{+}$ states of the hydrogen molecule. The total energies obtained show an agreement of better than 0.23 eV with the exact results.

Multireference many-body perturbation theory

The multireference many-body perturbation theory (MBPT) method is presented pedagogically. The distinctions between complete and incomplete model spaces are discussed. All second- and third-order diagrams for the diagonal and off-diagonal elements of the effective Hamiltonian matrix are presented in a compact form for complete and incomplete model spaces, including a new treatment of the renormalization diagrams. Illustrative numerical results are presented for the excitation energies of C2H4 using an incomplete single excitation model space, and for a study of FH potential energy surfaces with several different sizes of reference space.

Connected-diagram expansions of effective Hamiltonians in incomplete model spaces. I. Quasicomplete and isolated incomplete model spaces

In this and the following paper, we formulate a Fock space theory for incomplete model spaces (IMS) that applies both to coupled-cluster expansions and to perturbation theory. We stress in this paper that the concept of the ‘‘connected’’ nature of extensive quantities like an effective Hamiltonian Heff is more fundamental than the ‘‘linkedness’’ that is conventionally used in many-body perturbation theory. The ‘‘connectedness’’ of Heff follows when the wave operator W is multiplicatively separable into noninteracting subsystems. This is ensured by writing W as an exponential Fock space operator with the exponent connected. It is demonstrated in particular that the connectedness of the exponent in W requires that the normalization condition of W be separable as well. Unlike the situation in a complete model space, the definition of ‘‘diagonal’’ or ‘‘nondiagonal’’ operators depends generally on the particular m-valence IMS. There are, however, special categories of IMS, the ‘‘quasicomplete’’ and the ‘‘isolated’’ model spaces, for which these definitions are possible without reference to the particular IMS. The formal properties of these IMS are discussed. It is shown that for the quasicomplete model space, the intermediate normalization is not separable, while it is so for the isolated model space.

A note on the direct calculation of excitation energies by quasi-degenerate MBPT and coupled-cluster theory

For excitation energy calculations using quasi-degenerate MBPT or coupled-cluster (CC) theory, the hitherto chosen strategy has been to have particle-hole (p-h) determinants forming the model space and to use standard formalisms originally developed for complete model spaces. In view of our recent analysis, the p-h determinants constitute an incomplete model space for which intermediate normalization (IN) of the wave-operator Ω is not appropriate if a linked formulation is desired. The theoretical status of earlier applications which had used formulae appropriate to IN, yet ignored disconnected diagrams, is analyzed. It is shown that if only excitation energies, rather than the full Ω, are desired, then no theoretical error is made. The situation is, however, fortuitous in that for mp- mh model spaces with m > 1, a similar procedure using IN would necessarily generate disconnected diagrams.

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 Linked-Diagram and Coupled-Cluster Expansions for Complete and Incomplete Model Spaces

Without abstract; see full text

Linked-Diagram and Coupled-Cluster Expansions for Multi-Configurational, Complete and Incomplete Model Spaces

The coupled-cluster or exp S formalism, where the wave operator is expressed in exponential form, is treated for a general, multi-Configurational model space. It is shown that the cluster operator, S, is rigorously connected when the model space is complete or "quasi-complete" in the sense that it contains all configurations that can be formed by distributing the valence electrons within certain groups of valence orbitals with given occupation number in each group. For this class of model spaces also the linked-diagram theorem is valid in the sense that the diagrams of the wave operator and the effective Hamiltonian do not contain any separate, closed part. The diagrams of the effective Hamiltonian are connected, while those of the wave operator may contain disconnected, open parts.For a more general, multi-configurational (incomplete) model space a formal expansion of coupled-cluster type is still possible, but it is found that the cluster operator is no longer necessarily connected, which leads to unlinked diagrams in the expansion of the wave operator and the effective Hamiltonian. A general procedure for generating the cluster operator in this case is described and applied particularly in the pair approximation.

General-model-space many-body perturbation theory: The(2s3p)P1,3states in the Be isoelectronic sequence

The energies of the $(2s3p)^{1,3}P$ states of the Be sequence ions N iv-Ne vii are calculated, with use of a general-model-space many-body perturbation theory (MBPT) to third order. The triplet states are described accurately by a one-configuration model ($P$) space including only $2s3p$, but the singlets are not; the singlet-triplet separation has the right sign (singlet below triplet) but is only 30% of experiment. Incorporation of the $2p3s$ configuration in the model space greatly improves the results. This incomplete, two-configuration model space yields virtually the same energies as the larger, complete model space required by other quasidegenerate MBPT methods, at one-fifth the computational cost.

General-model-space perturbation theory: Excitation and ionization of N2

Vertical excitation and ionization energies are calculated by diagrammatic many-body perturbation theory (MBPT) for 20 states of N2 and N+2 at R=2.068 bohr. Nondegenerate or degenerate MBPT with complete or incomplete model spaces is employed, depending on the molecular state. A variety of zero-order Hamiltonians (H0) is tried. It is found that [2/1] Padé approximants depend little on the choice of H0, making them superior to straightforward summation to third order. They also provide good approximations to the CI limit for the basis (average error for all states 0.35 eV).

Size consistency in perturbation theories

Several versions of Rayleigh–Schrödinger (RS) and Brillouin–Wigner (BW) perturbation theories are tested for size consistency using a composite system consisting of N identical noninteracting subsystems. None of the BW versions and only two implementations of RS perturbation theory pass the test. The nondegenerate Brueckner–Goldstone expansion and the quasidegenerate theory of Brandow which employs a complete model space are size consistent. The incomplete model space version of Hose and Kaldor is not generally size consistent, except under special circumstances.

A General-Model-Space Diagrammatic Perturbation Theory

A diagrammatic many-body perturbation theory applicable to arbitrary model spaces is presented. The necessity of having a complete model space (all possible occupancies of the partially-filled shells) is avoided. This requirement may be troublesome for systems with several well-spaced open shells, such as most atomic and molecular excited states, as a complete model space spans a very broad energy range and leaves out states within that range, leading to poor or no convergence of the perturbation series. The method presented here would be particularly useful for such states. The solution of a model problem (He2 excited Σg+ states) is demonstrated.

Diagrammatic many-body perturbation theory for general model spaces

Previous formulations, due to Brandow (1977) and subsequent authors, are limited to complete model spaces, including all possible occupations of open-shell orbitals. This definition of the model space may be troublesome for systems with several open shells, such as most molecular excited states, leading to intruder states and slowing or even destroying the convergence of the perturbation series. The formalism presented here applies to any model space, whether complete or incomplete, degenerate, quasi-degenerate or non-degenerate. Certain types of unlinked diagrams may appear for incomplete, quasi-degenerate model spaces. The orbitals are partitioned into two classes (holes and particles) for the purpose of diagram summation instead of Brandow's three (core, valence and particles), with the partitioning for a particular element of the effective interaction matrix determined by the ket state of that element.