State-specific Multireference Coupled-cluster Research Articles

Alternative definition of excitation amplitudes in multi-reference state-specific coupled cluster.

A central difficulty of state-specific Multi-Reference Coupled Cluster (MR-CC) in the multi-exponential Jeziorski-Monkhorst formalism concerns the definition of the amplitudes of the single and double excitation operators appearing in the exponential wave operators. If the reference space is a complete active space (CAS), the number of these amplitudes is larger than the number of singly and doubly excited determinants on which one may project the eigenequation, and one must impose additional conditions. The present work first defines a state-specific reference-independent operator T∼^m which acting on the CAS component of the wave function |Ψ0m⟩ maximizes the overlap between (1+T∼^m)|Ψ0m⟩ and the eigenvector of the CAS-SD (Singles and Doubles) Configuration Interaction (CI) matrix |ΨCAS-SDm⟩. This operator may be used to generate approximate coefficients of the triples and quadruples, and a dressing of the CAS-SD CI matrix, according to the intermediate Hamiltonian formalism. The process may be iterated to convergence. As a refinement towards a strict coupled cluster formalism, one may exploit reference-independent amplitudes provided by (1+T∼^m)|Ψ0m⟩ to define a reference-dependent operator T^m by fitting the eigenvector of the (dressed) CAS-SD CI matrix. The two variants, which are internally uncontracted, give rather similar results. The new MR-CC version has been tested on the ground state potential energy curves of 6 molecules (up to triple-bond breaking) and two excited states. The non-parallelism error with respect to the full-CI curves is of the order of 1 mEh.

A state-specific multi-reference coupled-cluster approach with a cost-effective treatment of connected triples: implementation to geometry optimisation

ABSTRACTRecently, we have suggested an approximate state-specific multi-reference coupled-cluster (SS-MRCC) singles, doubles and triples method based on the CCSDT-1a+d approximation applied to the single-reference CC approach, in which the contribution of the connected triple excitations is iteratively treated. The method, abbreviated as SS-MRCCSDT-1a+d is intruder-free and fully size-extensive. It has been employed for geometry optimisations of various systems possessing quasi-degeneracy of varying degrees (like N2H2 and O3) by invoking numerical gradient scheme. The method is also applied to CH2 and square cyclobutadiene in their excited states. For all systems under study, the computed values are in good accordance with state-of-the-art theoretical estimates indicating that the method might be a promising candidate for an accurate treatment of geometrical parameters of states plagued by electronic degeneracy in a computationally tractable manner.

Revisiting the ‘cis-effect’ in 1,2-difluoro derivatives of ethylene and diazene using ab initio multireference methods

The relative stabilities of cis- and trans- isomers of 1,2-difluoroethylene and 1,2-difluorodiazene have been studied via the state-specific multireference coupled cluster (SS-MRCC) method and its perturbative counterpart through the computation of the optimised structures and corresponding energies. Despite the existence of several cis-destabilising mechanisms, present calculations reveal the energetic preference of the cis- isomer (the cis-effect) for the systems considered here. Differences in structural parameters and vibrational frequencies among cis- and trans- isomers have been discussed. Very good agreement of our estimates has been found with the benchmark theoretical and experimental results. The SS-MRCC methods produce a smooth and consistent behaviour across the entire torsional surface for the cis–trans isomerisation indicating that the method has sufficient flexibility to model large changes in electronic structure that accompany chemical changes.

Proposal of multi-root multi-reference coupled cluster formalisms

This work first returns on the intrinsic difficulties of multi-reference coupled cluster (MR-CC) formalisms. They may be formulated either in an effective Hamiltonian frame or in an intermediate effective Hamiltonian (IEH) one. In the former case complete model space approach being intractable, the incomplete model space approach is re-examined, and is formulated in terms of an IEH, despite the fact that the model space dimension is equal to the number of desired roots. Some of its drawbacks are illustrated on the magnetic systems problem. Then one proposes a multi-root complete active space (CAS)-based CC-SD, which only handles single and double excitation operators, generalising a previously proposed State Specific MR-CC formalism. The method proceeds through an iterative dressing of the matrix elements between the singles and doubles and the CAS determinants.

Communication: Application of state-specific multireference coupled cluster methods to core-level excitations

The concept of the model space underlying multireference coupled-cluster (MRCC) formulations is a powerful tool to deal with complex correlation effects for various electronic states. Here, we demonstrate that iterative state-specific MRCC methods (SS-MRCC) based on properly defined model spaces can be used to describe core-level excited states even when Hartree-Fock orbitals are utilized. We show that the SS-MRCC models with single and double excitations are comparable in accuracy to high-level single reference equation-of-motion coupled cluster (EOMCC) formalism.

Implementation of the multireference Brillouin-Wigner and Mukherjee's coupled cluster methods with non-iterative triple excitations utilizing reference-level parallelism

In this paper we discuss the performance of the non-iterative state-specific multireference coupled cluster (SS-MRCC) methods accounting for the effect of triply excited cluster amplitudes. The corrections to the Brillouin-Wigner and Mukherjee's MRCC models based on the manifold of singly and doubly excited cluster amplitudes (BW-MRCCSD and Mk-MRCCSD, respectively) are tested and compared with exact full configuration interaction results for small systems (H(2)O, N(2), and Be(3)). For the larger systems (naphthyne isomers) the BW-MRCC and Mk-MRCC methods with iterative singles, doubles, and non-iterative triples (BW-MRCCSD(T) and Mk-MRCCSD(T)) are compared against the results obtained with single reference coupled cluster methods. We also report on the parallel performance of the non-iterative implementations based on the use of processor groups.

Towards large-scale calculations with State-Specific Multireference Coupled Cluster methods: Studies on dodecane, naphthynes, and polycarbenes

Multireference Coupled Cluster (MRCC) methods are essential for accurate descriptions of molecular systems with degenerate states. However, their use has been hampered by their high computational cost, which can be alleviated with scalable MRCC codes. Exploiting large computational resources our recent parallelization work has demonstrated MRCC on chemically relevant molecules within practical per core resource constraints.We report on parallel implementations of Mukherjee’s MRCC (MR MkCCSD) and Brillouin-Wigner MRCC (MR BWCC) models. Especially abstracting a methyl-group from dodecane, and singlet-high-spin splittings in naphthyne isomers and polycarbenes demonstrate systems with significant multireference character and size that currently can be handled.

Diagnosis of the performance of the state-specific multireference coupled-cluster method with different truncation schemes

We have tested the linked version of a iterative (partial) triples correction for the Jeziorski-Monkhorst ansatz based state-specific multireference coupled cluster (SS-MRCC) approach with singles and doubles (SD) excitations [abbreviated as SS-MRCCSDT-1a and SS-MRCCSDT-1a+d]. The assessments of SS-MRCCSDT-1a and SS-MRCCSDT-1a+d schemes have been performed on the ground potential energy surface (PES) of P4, Li(2),Be(2) systems which demand the MR description, and on study of the excitation energy between the ground and first excited state for P4 system. Illustrations in the isomerization of cyclobutadiene also show the power of the schemes. One of the designed features of the SS-MRCCSDT-n methods introduced here is that they do not require storage of the triples amplitudes. In the entire range of geometries, we found a definite improvement provided by SS-MRCC with SDT-1a and SDT-1a+d schemes over the standard SD one. In the nondegenerate regions of PES, the closeness of the performance of the single-reference CC to the SS-MRCC methods increases after inclusion of even partial triple excitations. Generally, the performance of the SS-MRCCSDT-1a+d approach is closer to the corresponding full configuration interaction (FCI) one than to the SS-MRCCSDT-1a specially in the degenerate geometries (as is evident from nonparallelism error). The deviation from FCI for the first excited state of the P4 model using various SS-MRCC theories with different truncation schemes obtained by converging on the second root of the effective Hamiltonian has also been reported. We also compare our results with the current generation state-of-the-art single and multireference CC calculations to envisage the usefulness of the present approach. Initial implementation indicates that the SS-MRCCSDT-n formalism can provide not only reliable excitation energies and barrier height even when used in a relatively small model space, but also offers a considerable promise in generating the entire energy surface with low nonparallelity error.

Superior performance of Mukherjee’s state-specific multi-reference coupled-cluster theory at the singles and doubles truncation scheme with localized active orbitals

The state-specific multi-reference coupled-cluster (SS-MRCC) theory of Mukherjee et al., in its singles and doubles truncation scheme (SS-MRCCSD), misses important couplings between the virtual functions reached by single and double excitations from different model functions. Since the SS-MRCC theory is not invariant with respect to the transformations among the active orbitals, the results are dependent on the active orbitals chosen. We demonstrate in this paper with results for potential energy curves for several example molecules involving single and multiple bond dissociation that the performance of SS-MRCCSD is significantly improved if localized active orbitals are used. The improvement is remarkable both in terms of the non-parallelity error and the magnitude of correlation energy recovered vis-a-vis the full configuration interaction results with the same basis set. The results bolster our claim that SS-MRCCSD with localized orbitals is an accurate general theory for potential energy surfaces.

Ab initio wavefunctions in bioinorganic chemistry: More than a succès d’estime?

Ab initio wavefunctions in bioinorganic chemistry: More than a succès d’estime?

Application of the uncoupled state-specific multireference coupled cluster method to a weakly bonded system: exploring the ground state Be2

Recently, Mukherjee and co-workers (2008 Chem. Phys. 349, 115) have proposed the uncoupled state-specific multireference coupled cluster approach (UC-SSMRCC), attuned particularly for the multireference systems to provide a relaxed description of the nondynamical correlation (stemming from quasidegeneracy) in the presence of dynamical correlation. By invoking an analogue of the anonymous parentage approximation in the coupling terms of the state-specific multireference coupled cluster (SS-MRCC) theory one obtains the UC-SSMRCC method without significantly sacrificing the accuracy and simultaneously retains all the key features of the parent theory. For this very reason the UC-SSMRCC calculations using large basis sets are computationally more feasible even with the inclusion of connected triples than the parent one. As far as our knowledge is concerned, no application of either the parent SS-MRCC or UC-SSMRCC has been reported in weakly bound systems to date. Hence, we have performed a UC-SSMRCC study of a weakly bound dimer (say X1Σ+ Be2) by the singles-, doubles-, triples-1a (SDT-1a) level of approximation. Because of the strong near-degeneracy of 2s and 2p atomic orbitals, X1Σ+ Be2 is a very suitable benchmark problem for quantum chemistry and a critical test for new theoretical methods and procedures. This study reveals that the quality of the description of X1Σ+ Be2 critically depends on the level of truncation of the cluster operators (and also on the choice of basis set) and on a delicately poised treatment of dynamic and nondynamic correlation effects. Our computed spectroscopic constants are in acceptably good agreement with other previously reported current generation correlation recovery theoretical estimates, indicating that the UC-SSMRCC method with SDT-1a is capable of achieving a qualitatively correct description of the X1Σ+ Be2 energy surface even with the smallest reference space. The UC-SSMRCC scheme is also capable of providing estimates with almost no deterioration of the quality vis-á-vis the parent SS-MRCC. Accordingly, it is hoped that the realm of the application of the UC-SSMRCC method will be increased.

Inclusion of selected higher excitations involving active orbitals in the state-specific multireference coupled-cluster theory

The parent state-specific multireference coupled-cluster (SS-MRCC) theory proposed by Mukherjee et al. [J. Chem. Phys. 110, 6171 (1999)], though rigorously size-extensive and also size-consistent with localized orbitals, has some deficiencies in the minimal truncation scheme, viz. at the singles and doubles (SD) level (SS-MRCCSD). SS-MRCCSD does not involve the direct coupling of all the model functions with a given virtual function belonging to the uncontracted multiconfiguration CISD space. It also does not involve, even in the linear power of a cluster operator T(μ), the direct coupling of the virtual functions χ(l(μ)), which are up to doubly excited with respect to a model function φ(μ) to the other virtual functions of the MRCISD space which can be generated by triple and quadruple excitations from φ(μ). We argue that inclusion of a selection of triples and quadruples involving at most two inactive orbital excitations from every φ(μ) would ameliorate the shortcoming of the incomplete coupling of the triply and quadruply excited virtual functions which can couple with the singly and doubly excited ones. This extended ansatz for our SS-MRCC theory, to be called SS-MRCCSDtq by us, would still miss the direct coupling of the manifold of the model functions {φ(λ),λ ≠ μ} to singly and doubly excited virtual functions. However, this effect is expected to be less significant than the lack of the more complete virtual space couplings, these functions being many more numerous, suggesting the new methods to be significantly improved schemes. Excellent results on the potential energy surfaces of small molecules involving single, double, and triple bond dissociation bear out our expectations fully.

MkMRCC, APUCC and APUBD approaches to 1,n-didehydropolyene diradicals: the nature of through-bond exchange interactions

Mukherjee-type (Mk) state specific (SS) multi-reference (MR) coupled-cluster (CC) calculations of 1,n-didehydropolyene diradicals were carried out to elucidate singlet-triplet energy gaps via through-bond coupling between terminal radicals. Spin-unrestricted Hartree–Fock (UHF) based coupled-cluster (CC) computations of these diradicals were also performed. Comparison between symmetry-adapted MkMRCC and broken-symmetry (BS) UHF-CC computational results indicated that spin-contamination error of UHF-CC solutions was left at the SD level, although it had been thought that this error was negligible for the CC scheme in general. In order to eliminate the spin contamination error, approximate spin-projection (AP) scheme was applied for UCC, and the AP procedure indeed eliminated the error to yield good agreement with MRCC in energy. The CCD with spin-unrestricted Brueckner's orbital (UB) was also employed for these polyene diradicals, showing that large spin-contamination errors at UHF solutions are dramatically improved, and therefore AP scheme for UBD removed easily the rest of spin-contaminations. Pure- and hybrid-density functional theory (DFT) calculations of the species were also performed. Three different computational schemes for total spin angular momentums were examined for the AP correction of the hybrid DFT. The AP DFT calculations yielded the singlet-triplet energy gaps that were in good agreement with those of MRCC, AP UHF-CC and AP UB-CC. Chemical indices such as the diradical character were calculated with all these methods. Implications of the present computational results are discussed in relation to previous RMRCC calculations of diradical species and BS calculations of large exchange coupled systems.

Potential energy surface studies via a single root multireference coupled cluster theory

We have employed complete active space based single root multireference coupled cluster method (the resulting method is referred to by the acronym sr-MRCC) to compute the potential energy surfaces (PESs) of some well studied "protypical model" systems for which a highly accurate and reliable database is available for comparison. As that of state-specific theory, the sr-MRCC approach focuses and correlates one state while using a multiconfigurational reference and thus it naturally avoids intruder states. The present method is structurally different from the well known state specific multireference coupled cluster (SS-MRCC) method introduced by Mahapatra et al. [Mol. Phys. 94, 157 (1998)]. As that of the SS-MRCC theory, the present method is also based on the Jeziorski-Monkhorst ansatz where a different exponential cluster operator exp(T(mu)) acts on its corresponding model function phi(mu). The final cluster finding equations contain coupling between the cluster operators for all the mu, which are mainly responsible to prove the extensivity of both the cluster amplitudes and the energy. The present sr-MRCC theory is size-extensive and size-consistent when localized orbitals are used. The systems considered here exhibit varying degrees of degeneracy at different regions of PES. The treatment of these systems via traditional effective Hamiltonian based methods suffers from divergence problems in the iterative solution of the CC equations (the issue termed as "intruder state"). The sr-MRCC results lie closer to the ones obtained by the SS-MRCC method for these systems. To judge the efficacy of the present method, we have compared our results with other previously published theoretical estimations, which clearly indicate that the present method is reliable in studying the dissociation PES of states plagued by electronic degeneracy as well as notorious intruder effects. The highly satisfactory performance of the sr-MRCC method, vis-a-vis the other sophisticated methods, in describing the lowest and the first excited singlet states of BeH(2) at points of high degeneracy is noticeable.

Full implementation and benchmark studies of Mukherjee’s state-specific multireference coupled-cluster ansatz

The state-specific multireference coupled-cluster (SS-MRCC) ansatz developed by Mukherjee and co-workers [J. Chem. Phys. 110, 6171 (1999)] has been implemented by means of string-based techniques. The implementation is general and allows for using arbitrary complete active spaces of any spin multiplicity and arbitrarily high excitations in the cluster operators. Several test calculations have been performed for single- and multiple-bond dissociations of molecular systems. Our experience shows that convergence problems are encountered when solving the working equations of the SS-MRCC in the case the weight of one or more reference functions tends to take on very small values. This is system specific and cannot yet be handled in a black-box fashion. The problem can be obviated by either dropping all the cluster amplitudes from the corresponding model functions with coefficients below a threshold or by a regularization procedure suggested by Tikhonov or a combination of both. In the current formulation the SS-MRCC is not invariant with respect to transformation of active orbitals among themselves. This feature has been extensively explored to test the degree of accuracy of the computed energies with both pseudocanonical and localized active orbitals. The performance of the method is assessed by comparing the results with the corresponding full configuration interaction (CI) values with the same set of orbitals (correlated and frozen). Relative efficacies of CI methods such as MRCI singles and doubles with the same active space and size-extensivity corrected ones such as MR averaged coupled pair functional and MR averaged quadratic CC have also been studied. Allied full-fledged CC methods have also been employed to see their relative performance vis-à-vis the SS-MRCC. These latter methods are the complete-active-space-inspired single-reference (SR) CC based SS theory and the single-root MR Brillouin-Wigner CC. Our benchmark results indicate that the performance of the SS-MRCC is generally quite good for localized active orbitals. The performance with the pseudocanonical orbitals, however, is sometimes not as satisfactory as for the localized orbitals.

Multireference state-specific coupled-cluster methods. State-of-the-art and perspectives

This work reviews the state-specific multireference coupled-cluster (CC) approaches which have been developed as approximate methods for performing high-level quantum mechanical calculations on quasidegenerate ground and excited states of atomic and molecular systems. The term "quasidegenerate" refers to a state that cannot be described even in the first approximation by a single-determinant wavefunction (a Slater determinant), but requires two or more determinants for this purpose. The main challenge with applying the coupled-cluster theory to such states is in describing the electron correlation effects in the wavefunctions representing these states in a manner that is size-extensive, yet accurate and simple enough so the method can be routinely applied to small and medium-size molecular systems. We are describing how this can be accomplished within a theory that focuses on only one state of the system in a single CC calculation (the state-specific theory).

High-order excitations in state-universal and state-specific multireference coupled cluster theories: Model systems

For the first time high-order excitations (n>2) have been studied in three multireference couple cluster (MRCC) theories built on the wave operator formalism: (1) the state-universal (SU) method of Jeziorski and Monkhorst (JM) (2) the state-specific Brillouin-Wigner (BW) coupled cluster method, and (3) the state-specific MRCC approach of Mukherjee (Mk). For the H4, P4, BeH(2), and H8 models, multireference coupled cluster wave functions, with complete excitations ranging from doubles to hextuples, have been computed with a new arbitrary-order string-based code. Comparison is then made to corresponding single-reference coupled cluster and full configuration interaction (FCI) results. For the ground states the BW and Mk methods are found, in general, to provide more accurate results than the SU approach at all levels of truncation of the cluster operator. The inclusion of connected triple excitations reduces the nonparallelism error in singles and doubles MRCC energies by a factor of 2-10. In the BeH(2) and H8 models, the inclusion of all quadruple excitations yields absolute energies within 1 kcal mol(-1) of the FCI limit. While the MRCC methods are very effective in multireference regions of the potential energy surfaces, they are outperformed by single-reference CC when one electronic configuration dominates.

Potential energy surface of the electron excited states in the state-specific multi-reference coupled cluster theory. Hydrogen fluoride dissociation

The recently proposed state-specific multi-reference coupled cluster (SSMRCC) theory with the complete active space (CAS) reference has been tested in calculations of the potential energy surfaces of electronically-excited states. The algorithm for the method is derived using the computer-based automated approach for generating the coupled cluster diagrams and the energy and amplitude equations. The active space for the calculation of the target state is constructed using natural orbital expansion obtained from the configuration interaction method with single and double excitations (CISD) one-particle density matrix. The spin–orbitals for the CASCC calculation are obtained using the complete active space self consistent field (CASSCF) method. The numerical example shown concerns several electronically excited states with different spatial and spin symmetries of the hydrogen fluoride molecule. The calculations are performed at several internuclear distances and compared with the results obtained using other methods such as the full configuration interaction (FCI) method, the equation-of-motion (EOM) method, and the multi-reference perturbation theory (MRPT). The comparison shows that the CASCC results are better than the EOM and MRPT results.

Multireference State-specific Coupled Cluster Approach with the CAS Reference: Inserting Be into H2

An algorithm for generation of the spin-orbital diagrammatic representation, the corresponding algebraical formulas, and the computer code of the coupled cluster (CC) method with an arbitrary level of the electronic excitations developed earlier in our laboratory have been employed to generate the CAS(2,2)CCSD code. CAS(2,2)CCSD is the state-specific, multireference coupled cluster (SSMRCC) approach with single and double excitations based on the CASSCF(2,2) reference wave function. The CAS(2,2)CCSD was used to describe the model process of inserting the Be atom into the H2 molecule. We show that our method performs better than the “fully-blown” SSMRCC approach of Mukherjee and coworkers (J Chem Phys 110:6171, 1999).

Molecular Applications of a State-Specific Multireference Coupled Electron-Pair Approximation (SS-MRCEPA)-like Method

In this paper, we present a coupled electron-pair (CEPA) type variant of the state-specific multireference coupled cluster (SS-MRCC) method [Mahapatra, U. S., et al. J. Chem. Phys. 1999, 110, 6171]. The method termed as SS-MRCEPA based on complete active space (CAS) can handle quasi-degeneracy of varying degrees over a wide range of potential energy curves (PECs), including regions of real or avoided curve-crossing. The method is size-extensive and avoids the intruder problem in a natural manner. Exploiting a two-dimensional CAS-based SS-MRCEPA method, we consider, in this paper, several demanding molecular systems that benefit from multireference description. The reliability of computational results of the method for PECs of the ground state of P4, H4, H8, perpendicular insertion of Be into H2, Li2, and ground-state energy at the equilibrium point of CH2 will be discussed with respect to the parent SS-MRCC and full CI/large scale CI results. We have also reported the excitation energies corresponding to the ground states of H8 and CH2 systems. The method has also been applied to study the bond breaking in the F2 molecule which is a challenging task for any ab initio method. In all cases, the comparison is also made with the results obtained from other CC- and CEPA-type methods wherever available.