General Model Space Research Articles

Multi-reference state-universal coupled-cluster approaches to electronically excited states

The multi-reference (MR), general model space (GMS), state-universal (SU), coupled-cluster (CC) method with singles and doubles (GMS-SU-CCSD), as well as its triple-corrected versions GMS-SU-CCSD(T), are employed to assess their ability to describe low-lying excited states of various molecules, with an emphasis on a simultaneous handling of several states of the same symmetry species. A special attention is given to the role of the so-called C-conditions that account for non-vanishing internal cluster amplitudes when relying on an incomplete GMS, as well as to the choice of suitable model spaces and a perturbative account of secondary triples. The ambiguities arising when using large basis sets are also pointed out. To achieve a general assessment of the potential of the GMS-type SU-CC approaches, the vertical excitation energies of several species, including the challenging BN diatomic as well as larger systems, namely formaldehyde, trans-butadiene, formamide, and benzene are considered. These results are compared with those provided by the equation-of-motion EOM-CCSD method and, whenever available, the density functional theory results and experimental data. These comparisons clearly demonstrate the usefulness of GMS-type MR-CC approaches.

Multireference coupled-cluster study of the symmetry breaking in the $\rm {C_{2}B}$C2B radical

The potential energy surfaces (PESs) for both the ground and the excited electronic states of the C(2)B radical are investigated using various multireference (MR) coupled-cluster (CC) approaches. In the ground state case we employ the reduced MR (RMR) CC approach with singles (S) and doubles (D), the RMR CCSD method, as well as its RMR CCSD(T) version corrected for secondary triples, relying on various model spaces and basis sets. The reliability of this approach is also tested against the benchmark full configuration interaction results obtained for a small Dunning-Hay (DH) basis set. The results imply a clear preference for a cyclic structure which, however, breaks the C(2v) symmetry. This symmetry breaking manifests itself strongly at the level of the independent particle model, as represented by the restricted open-shell Hartree-Fock approximation, but the tendency toward symmetry breaking diminishes with the increasing size of the basis set employed as well as with the enhanced account of the correlation effects. It is likely to disappear in the complete basis set limit. The general model space CCSD method is then used to compute vertical excitation energies for a number of excited states as well as the cuts of the PES as the boron atom moves around the C(2) fragment. These results also explain why no symmetry breaking is found when relying on a spin contaminated unrestricted Hartree-Fock reference, as in the UMP2 method.

Multireference general-model-space state-universal and state-specific coupled-cluster approaches to excited states

The concept of C-conditions, originally introduced in the framework of the multireference (MR), general-model-space (GMS), state-universal (SU), coupled-cluster (CC) approach with singles and doubles (GMS-SU-CCSD) to account for the internal amplitudes that vanish in the case of a complete model space, is applied to a state-selective or state-specific Mukherjee MR-CC method (MkCCSD). In contrast to the existing applications, the emphasis is on the description of excited states, particularly those belonging to the same symmetry species. The applicability of the C-conditions in all MR-SU-CC approaches is emphasized. Convergence problems encountered in the MkCCSD method when handling higher-lying states are pointed out. The performance of the GMS-SU-CCSD and MkCCSD methods is illustrated by considering low-lying vertical excitation energies of the ethylene molecule and para-benzyne diradical. A comparison with the equation-of-motion CCSD results, as well as with the available experimental data and recent multireference configuration interaction theoretical results, is also provided.

A Multireference Coupled-Cluster Study of Electronic Excitations in Furan and Pyrrole†

A multireference (MR), general-model-space (GMS), state-universal (SU), coupled-cluster (CC) method that considers singly (S) and doubly (D) excited cluster amplitudes relative to the reference configurations spanning the model space (GMS SU CCSD) is employed to investigate a number of low-lying excited states of two five-membered, six pi -electron aromatic ring molecules, namely, furan and pyrrole. An extended basis set that includes diffuse functions and molecule-centered Rydberg functions is employed. Computed vertical excitation energies are compared with experimental values as well as with other theoretical results, namely, those obtained by the equation-of-motion (EOM) CCSD method or, equivalently, the symmetry adapted cluster configuration interaction (SAC-CI) method, by the MR CI method, by MR perturbation theory, and by other methods whenever available. For excited states dominated by single-excitations the GMS SU CCSD energies represent an improvement relative to the most closely related EOM CCSD results by about 0.1 eV.

Performance of multireference and equation-of-motion coupled-cluster methods for potential energy surfaces of low-lying excited states: Symmetric and asymmetric dissociation of water

Multireference (MR), general-model-space (GMS), state-universal (SU) coupled-cluster (CC) method that considers singly (S) and doubly (D) excited cluster amplitudes relative to the reference configurations spanning the model space (GMS SU CCSD), as well as its externally corrected (ec) version (N,M)-CCSD that employs N-reference MR CISD as an external source of higher-than-pair cluster amplitudes in a M-reference GMS CCSD, are employed to investigate low-lying states of the water molecule. The emphasis is on a generation of several low lying states belonging to the same symmetry species. Cuts of the potential energy surface (PES) corresponding to the breaking of a single OH bond and leading to the OH+H fragments, as well as the simultaneous breaking of both bonds into the O+2H are considered. Relying on a simple ab initio model that enables a comparison with the exact full configuration interaction energies, the performance of the GMS-based methods is assessed in the whole relevant range of internuclear separations. It is shown that the ec (N,M)-CCSD version provides best results for both the singlet and the triplet states considered. The same cuts of the PES are then explored using a realistic aug-cc-pVTZ basis set. For triplets, the use of high-spin (M(S)=1) references is to be preferred.

Model space incompleteness in multireference state-universal and state-selective coupled-cluster theories

The role of the so-called C-conditions, originally introduced in the general model space (GMS), state-universal (SU), coupled-cluster (CC) theory with singles and doubles (GMS SU CCSD) to account for the internal cluster amplitudes that vanish in the case of a complete model space, is explored in the context of the state-selective Mukherjee MR-CC method (MkCCSD). Using three examples that involve a considerable quasidegeneracy we show that the C-conditions can be usefully employed also in the MkCCSD method once an incomplete (or general) model space is used.

Multireference coupled‐cluster methods for ground and low‐lying excited states. A benchmark illustration on CH + potentials

AbstractMultireference (MR), general‐model‐space (GMS), state‐universal (SU) coupled‐cluster (CC) method that considers singly (S) and doubly (D) excited cluster amplitudes relative to the reference configurations spanning the model space (GMS SU CCSD), as well as its externally corrected (ec) version (N,M)‐CCSD that uses N‐reference MR CISD as an external source of higher‐than‐pair cluster amplitudes in a M‐reference GMS CCSD, are used to investigate low‐lying states of the CH+ ion. Relying on a simple ab initio model that enables a comparison with the exact full configuration interaction energies, the performance of the GMS‐based methods is assessed in the whole relevant range of internuclear separations. It is shown that the ec (N,M)‐CCSD version provides best results for both the singlet and the triplet states considered. For triplets, the use of high‐spin (MS = 1) references is to be preferred. The GMS‐based MR SU CC results for the ten low‐lying states of CH+ clearly indicate the usefulness and reliability of these approaches. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Method of moments for the continuous transition between the Brillouin–Wigner-type and Rayleigh–Schrödinger-type multireference coupled cluster theories†

We apply the method of moments to the multireference (MR) coupled cluster (CC) formalism representing the continuous transition between the Brillouin–Wigner-type and Rayleigh–Schrödinger-type theories based on the Jeziorski–Monkhorst wave function ansatz and derive the formula for the noniterative energy corrections to the corresponding MRCC energies that recover the exact, full configuration interaction energies in the general model space case, including complete and incomplete model spaces. We also extend the relationship between the generalised moments of the state-universal (SU) MRCC equations within the Jeziorski–Monkhorst and Kucharski–Bartlett formulations of the SUMRCC theory to the general model space case. Finally, we argue that in the complete model space case, the relationship between moments of the SUMRCC equations corresponding to the Jeziorski–Monkhorst and Kucharski–Bartlett formulations of the SUMRCC theory, derived in this work, implies an equivalence of these two formulations of the SUMRCC approach, provided that the disconnected linked terms are included in the Kucharski–Bartlett formulation, and verify this statement numerically.

Use of a convenient size-extensive normalization in multi-reference coupled cluster (MRCC) theory with incomplete model space: A novel valence universal MRCC formulation

We present in this paper a size-extensive formulation of a valence universal multi-reference coupled cluster (VU-MRCC) theory which uses a general incomplete model space (IMS). The earlier formulations by Mukherjee [D. Mukherjee, Chem. Phys. Lett. 125 (1986) 207] led to size-extensive Heff which was both connected and ‘closed’, thereby leading to size-extensive energies. However, this necessitated abandoning the intermediate normalization (IN) for the valence universal wave-operator Ω when represented as a normal ordered exponential cluster Ansatz Ω≡{exp(S)} with S as the cluster operator. The lack of IN stemmed from the excitation operator Sq-op which leads to excitations into the complementary model space by their action on at least one model function. The powers of Sq-op can in general bring a model function ϕi back to another model function ϕj, and this is the reason why Ω does not respect IN. Sq-op are all labelled by active orbitals only. To achieve connectivity of Heff, it must be a ‘closed’ operator. A closed operator is one which always produces a model function by its action on another model function. Since the decoupling conditions Lq-op=0, and Lop=0 for the transformed operator L=Ω-1HΩ would be in conflict with Ωq-op=1q-op, the model space projection of Ω, PΩP=P cannot be maintained for the normal ordered Ansatz. This leads to a somewhat awkward expression for Heff. Bera et al. [N. Bera, S. Ghosh, D. Mukherjee, S. Chattopadhyay, J. Phys. Chem. A 109 (2005) 11462] recently tried to simplify the expression for Heff, and accomplished this by introducing suitable counter-terms Xcl in Ω to enforce Ωcl=1cl. We show in this paper that Heff in this formulation leads to a disconnected Heff, though it is equivalent by a similarity transformation to a connected effective hamiltonian H∼eff. Guided by the insight gleaned from this demonstration, we have proposed in this paper a new form of the wave-operator which never generates any powers of Sq-op, which is closed. This ‘externally projected’ wave-operator does not need counter-terms Xcl and automatically ensures Ωcl=1cl, thereby yielding directly a closed connected H∼eff. The desirable features of the traditional normal ordered Ansatz, such as the valence universality, subsystem embedding conditions hierarchical decoupling of the VU-MRCC equations for decreasing valence ranks are all satisfied by this new Ansatz for the wave-operator.

Diagonal perturbative triple corrections to the general‐model‐space state‐universal coupled‐cluster method: Are they warranted and useful?

The recently developed general‐model‐space (GMS) state‐universal (SU) coupled‐cluster (CC) approach, together with its version corrected for triples via a single‐reference (SR) CCSD(T)‐type correction of the diagonal elements of the effective Hamiltonian, is applied to several molecular electronic structure problems in order to assess their performance and the role of triples. These results are compared with an alternative handling of higher‐than‐pair clusters via the externally corrected SU CCSD method, denoted (M,N)‐CCSD, which employs N wave functions of the M‐reference CISD as an external source for N‐reference SU CCSD. These methods are applied to the problem of bond breaking in the ground and excited states of the F2 and HF molecules, where the high‐spin triplet component is also handled via the SR CCSD method. We further examine the vertical excitation energies of water and the basic spectroscopic constants (equilibrium geometries, harmonic frequencies, and excitation energies) for several low‐lying states of oxygen. The results are encouraging and are discussed from the viewpoint of the applicability and usefulness of perturbative‐type triple corrections.

General-model-space state-universal coupled-cluster method: excitation energies of water

A recently developed multireference (MR), general-model-space (GMS), state-universal (SU) coupled-cluster (CC) method [J. Chem. Phys. 119, 5320 (2003)] in its extended form that considers singly (S) and doubly (D) excited cluster amplitudes relative to the reference configurations spanning the model space—which can differ by up to and including quadruple excitations—is employed to investigate a number of low-lying excited states of the water molecule. The role played by different atomic orbital basis sets is examined by employing both the aug-cc-pVTZ and cc-pVTZ + diff bases, the latter extending the cc-pVTZ basis by an s diffuse function on hydrogen and two sets of s and p diffuse functions on oxygen. The computed vertical excitation energies are compared with those obtained earlier with a cc-pVDZ basis set, as well as with other—both theoretical and experimental—results, when available. For a number of relevant states the equilibrium geometry, and the implied T 0 excitation energies, are also given. It is shown that two sets of s and p diffuse functions on oxygen are essential for an accurate rendering of the experimental spectra and thus play a much more important role than the higher-angular-momentum diffuse functions (i.e. d and f on oxygen and p and d on hydrogen) characterizing the aug-cc-pVTZ basis set.

Reappraisal of the Role of Size-Extensive Normalization for Multireference Coupled Cluster (MRCC) Theory Using General Model Space: A Valence Universal MRCC Approach

We present a brief description of a valence-universal multireference coupled cluster (VU-MRCC) theory that can handle completely general incomplete model spaces, remaining close to the intermediate normalization (IN) condition for omega as much as possible without violating extensivity and without the use of a post facto correction. In this formalism, the connectedness of the cluster operators as well as effective Hamiltonian and hence the extensivity of the corresponding roots is achieved by invoking appropriate decoupling conditions on the special type of wave operator omega = {exp(S + X(cl))} satisfying the Bloch equations in the Fock-space S in an excitation operator and X is a closed operator (denoted by cl). This special type of wave-operator leads to a unique partition of the excitations from the model space into those generated by the cluster operators (open and quasi-open) and those generated by the effective Hamiltonian (closed). In this formulation, for every X(cl), there is a counterterm from {exp(S)}(cl) canceling each other. This leads to a connected expressions for cluster amplitudes, using the constraint omega(cl) = 1. The new form of the effective Hamiltonian preserves the extensivity of the target energies. Our analysis implies that IN for omega is a valid size-extensive normalization for certain special IMS such as the quasi-complete model space and the isolated incomplete model space.

Multi-reference Brillouin–Wigner coupled-cluster method with a general model space

We describe a multi-reference Brillouin–Wigner coupled-cluster (BWCC) method that employs a general model space (GMS) by relying on the recently introduced C-conditions [X. Li, J. Paldus. J. chem. Phys., 119, 5320 (2003).] This generalization preserves all the essential features of the standard complete model space (CMS) BWCC method. This makes it possible to substantially reduce the dimension of the model space employed by keeping only important configurations and thus to minimize the computational cost. We also present the results of a preliminary numerical test using the cc-pVDZ model of the LiH molecule.

Performance of the general-model-space state-universal coupled-cluster method.

The capabilities of the recently developed multireference, general-model-space (GMS), state-universal (SU) coupled-cluster (CC) method have been extended in order to enable the handling of any excited state that represents a single (S) or a double (D) excitation relative to the ground state. A series of calculations concerning the ground and excited states of the CH(+), HF, F(2), H(2)O, NH(2), and CH(2) molecules were carried out so as to assess the performance of the GMS SU CCSD method. For diatomics we have computed the entire potential energy curves, while for triatomics we have focused on vertical excitation energies. We demonstrate how a systematic enlargement of the model space enables a consideration of a larger and larger number of excited states. A comparison of the CC and full configuration interaction or large-scale CI results enables an assessment of the accuracy and reliability of the GMS SU CCSD method within a given basis set. In all cases very good results have been obtained, including highly excited states and those having a doubly-excited character.

General-Model-Space State–Universal Coupled-Cluster Method: Diagrammatic Approach

The explicit expressions for the recently formulated general-model-space (GMS) version of the multireference state-universal (SU) coupled-cluster (CC) method [X. Li and J. Paldus, J. Chem. Phys. 119 (2003) 5320], truncated at the level of single (S) and double (D) excitations, are re-derived using the spin-orbital form of the diagrammatic technique. The focus is on the so-called coupling coefficients, which represent a new type of quantity that does not appear in the single-reference CC approaches. The role of the connectivity conditions, referred to as the C-conditions, in the elimination of disconnected terms in the GMS SU CC method is analyzed in terms of the connectivity of the resulting diagrams and is illustrated on typical examples.

Size extensivity of a general‐model‐space state‐universal coupled‐cluster method

AbstractWe show that the recently formulated general‐model‐space (GMS), state‐universal (SU), coupled‐cluster (CC) method, exploiting the so‐called C‐conditions, is size extensive (in the sense that the energy of a composite system consisting of noninteracting subsystems equals the sum of the subsystem energies, while employing the appropriate method in each case). This assumes that the reference or model space employed for the composite system is size consistent, i.e., is given by a tensor product of subsystem model spaces. For pedagogical reasons, the demonstration is carried out for the single reference case before the multireference SU CC case is addressed. The importance of the C‐conditions and a relationship with earlier approaches is discussed, as well as the limiting case of a complete‐model‐space SU CC method. Finally, the size extensivity of a GMS‐based SU CCSD method is illustrated by actual numerical examples. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

The general-model-space state-universal coupled-cluster method exemplified by the LiH molecule

The salient features of the recently introduced general-model-space (GMS) state-universal (SU) coupled-cluster (CC) method are illustrated on the case of the LiH molecule. Describing the breaking of the Li–H bond by relying on an open-shell-type GMS reveals the importance of the connectivity conditions (C conditions), which represent a crucial new ingredient of the GMS SU CC theory. Only when we properly account for these C conditions can we uniquely represent the full configuration interaction (FCI) wave functions in terms of the multireference SU exponential cluster ansatz and recover the FCI energies via the GMS SU CC method, assuming that all the relevant clusters at a given level of the theory are considered. Drawing on various GMSs, we compute the potential energy curves for three Σ+1, two Σ+3, three Π,1 and three Π3 states, using the GMS SU CC method truncated at the singly- and doubly-excited level (GMS SU CCSD), as well as the externally corrected (N,M)-CCSD method that exploits the NR-CISD wave functions as the external source of higher-than-pair clusters in the MR SU CCSD method. In all cases we obtain excellent results: For Σ+ states, the maximum difference between the FCI and various SU CCSD energies is about 0.5 millihartree. These errors are further reduced when we employ the (N,M)-CCSD methods. For the Π states, the deviations of the SU CCSD energies relative to FCI amount to at most a few hundreds of a millihartree. We also report on the size-extensivity tests and the exactness of the formalism for two-electron systems.

N -reference, M-state coupled-cluster method: Merging the state-universal and reduced multireference coupled-cluster theories

We propose a generalization of the reduced multireference coupled-cluster method with singles and doubles (CCSD) to the genuine MR, state-universal (SU) CC approaches. Two key ingredients of this generalization are (i) the algorithm for the cluster analysis of general MR configuration interaction (CI) wave functions that is based on the SU cluster ansatz of Jeziorski and Monkhorst, and (ii) the formulation of the SU CC method employing a general (incomplete) model space. These recent developments enable us to employ modest size MR CISD wave functions that are based on an N-dimensional reference space M1 as a source of higher-than-pair-cluster amplitudes in the externally corrected SU CCSD method that is based on an M-dimensional model space M0, forming a subspace of M1. An appropriate choice of M0 and M1 makes it then possible to avoid the most severe intruder-state problems. The method is illustrated on the often-investigated H4 and H8 model systems.

Theoretical description of doubleβdecay of160Gd

The half-life of the $\ensuremath{\beta}{\ensuremath{\beta}}_{2\ensuremath{\nu}}$ decay of ${}^{160}\mathrm{Gd}$ is estimated by including the pairing interaction within the pseudo-SU(3) model. This process was previously reported as theoretically forbidden in the context of that model, however, the pairing interaction is able to mix different occupations and opens new channels for the decay. Explicit expressions are presented for the mixing induced by the pairing force. Matrix elements for the $\ensuremath{\beta}{\ensuremath{\beta}}_{2\ensuremath{\nu}}$ and $\ensuremath{\beta}{\ensuremath{\beta}}_{0\ensuremath{\nu}}$ decays are now calculated by assuming both a dominant component in the wave function of the ground state of ${}^{160}\mathrm{Gd}$ and a more general model space. The results, of the calculated $\ensuremath{\beta}\ensuremath{\beta}$ half-lives, suggest that the planned experiments would succeed in detecting the $\ensuremath{\beta}{\ensuremath{\beta}}_{2\ensuremath{\nu}}$ decay of ${}^{160}\mathrm{Gd}$ and, eventually, would improve the limits for the zero-neutrino mode.

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.