State-specific Multireference Coupled Cluster Method Research Articles

Reaction Profiles and Kinetics for Radical-Radical Hydrogen Abstraction via Multireference Coupled Cluster Theory.

Radical-radical abstractions in hydrocarbon oxidation chemistry are disproportionation reactions that are generally exothermic with little or no barrier yet are underappreciated and poorly studied. Such challenging multireference electronic structure problems are tackled here using the recently developed state-specific multireference coupled cluster methods Mk-MRCCSD and Mk-MRCCSD(T), as well as the companion perturbation theory Mk-MRPT2 and the established MRCISD, MRCISD+Q, and CASPT2 approaches. Reaction paths are investigated for five prototypes involving radical-radical hydrogen abstraction: H + BeH → H2+ Be, H + NH2 → H2 + NH, CH3 + C2H5 → CH4 + C2H4, H + C2H5 → H2 + C2H4, and H + HCO → H2 + CO. Full configuration interaction (FCI) benchmark computations for the H + BeH, H + NH2, and H + HCO reactions prove that Mk-MRCCSD(T) provides superior accuracy for the interaction energies in the entrance channel, with mean absolute errors less than 0.3 kcal mol-1 and percentage deviations less than 10% over the fragment separations of relevance to kinetics. To facilitate combustion studies, energetics for the CH3 + C2H5, H + C2H5, and H + HCO reactions were computed at each level of theory with correlation-consistent basis sets (cc-pVXZ, X = T, Q, 5) and extrapolated to the complete basis set (CBS) limit. These CBS energies were coupled with CASPT2 projected vibrational frequencies along a minimum energy path to obtain rate constants for these three reactions. The rigorous Mk-MRCCSD(T)/CBS results demonstrate unequivocally that these three reactions proceed with no barrier in the entrance channel, contrary to some earlier predictions. Mk-MRCCSD(T) also reveals that the economical CASPT2 method performs well for large interfragment separations but may deteriorate substantially at shorter distances.

Perturbative triples correction to domain-based local pair natural orbital variants of Mukherjee's state specific coupled cluster method.

In this article we report an implementation of the perturbative triples correction to Mukherjee's state-specific multireference coupled cluster method based on the domain-based pair natural orbital approach (DLPNO-MkCC). We tested the performance of DLPNO-MkCCSD(T) in calculations involving tetramethyleneethane and isomers of naphthynes. These tests show that more than 97% of triples energy was recovered with respect to the canonical MkCCSD(T) method, which together with the DLPNO-MkCCSD part accounts for about 99.70-99.85% of the total correlation energy. The applicability of the method was demonstrated on calculations of singlet-triplet gaps for several large systems: triangulene, dynemicin A, and a beryllium complex.

Domain-Based Local Pair Natural Orbital Version of Mukherjee's State-Specific Coupled Cluster Method.

This article reports development of a local variant of Mukherjee's state-specific multireference coupled cluster method based on the domain-based pair natural orbital approach (DLPNO-MkCC). The current implementation is restricted to connected single and double excitations and model space with up to biexcited references. The performance of the DLPNO-MkCCSD was tested on calculations of tetramethyleneethane. The results show that above 99.9% of the correlation energy was recovered, with respect to the conventional MkCC method. To demonstrate the applicability of the method to large systems, singlet-triplet gaps of triangulene and bis(1-(2,6-diisopropylphenyl)-3,3,5,5-tetramethylpyrrolidine-2-ylidene)beryllium complex were studied. For the last system (105 atoms), we were able to perform a calculation in cc-pVTZ with 2158 basis functions on a single CPU in less than 9 days.

Perturbative universal state-selective correction for state-specific multi-reference coupled cluster methods.

In this work, we report an extension of our previous development of the universal state-selective (USS) multireference coupled-cluster (MRCC) formalism. It was shown [Brabec et al., J. Chem. Phys. 136, 124102 (2012)] and [Banik et al., J. Chem. Phys. 142, 114106 (2015)] that the USS(2) approach significantly improves the accuracy of Brillouin-Wigner and Mukherjee MRCC formulations, however, the numerical and storage costs associated with calculating highly excited intermediates pose a significant challenge, which can restrict the applicability of the USS(2) method. Therefore, we introduce a perturbative variant of the USS(2) approach (USS(pt)), which substantially reduces numerical overhead of the full USS(2) correction while preserving its accuracy. Since the new USS(pt) implementation calculates the triple and quadruple projections in on-the-fly manner, the memory bottleneck associated with the need of storing expensive recursive intermediates is entirely eliminated. On the example of several benchmark systems, we demonstrate accuracies of USS(pt) and USS(2) approaches and their efficiency in describing quasidegenerate electronic states. It is also shown that the USS(pt) method significantly alleviates problems associated with the lack of invariance of MRCC theories upon the rotation of active orbitals.

A Local Pair Natural Orbital-Based Multireference Mukherjee's Coupled Cluster Method.

This paper reports the development of a local variant of Mukherjee's state-specific multireference coupled cluster method based on the pair natural orbital approach (LPNO-MkCC). The current implementation is restricted to single and double excitations. The performance of the LPNO-MkCCSD method was tested on calculations of naphthyne isomers, tetramethyleneethane, and β-carotene molecules. The results show that 99.7-99.8% of correlation energy was recovered with respect to the MkCC method based on canonical orbitals. Moreover, the errors of relative energies between different isomers or along a potential energy curve (with respect to the canonical method) are below 0.4 kcal/mol, safely within the chemical accuracy. The computational efficiency of our implementation of LPNO-MkCCSD in the ORCA program allows calculation of the β-carotene molecule (96 atoms and 1984 basis functions) on a single CPU core.

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.

Note: Excited state studies of ozone using state-specific multireference coupled cluster methods

Vertical excitation energies obtained with state-specific multi-reference coupled cluster (MRCC) methods are reported for the low-lying singlet and triplet excited of the ozone molecule. The MRCC results are also compared with those obtained with high-order equation-of-motion coupled cluster methods.

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.

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.

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.

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.

Analytic gradients for the state-specific multireference coupled cluster singles and doubles model

The general theory of analytic energy gradients is presented for the state-specific multireference coupled cluster method introduced by Mukherjee and co-workers [Mol. Phys. 94, 157 (1998)], together with an implementation within the singles and doubles approximation, restricted to two closed-shell determinants and Hartree-Fock orbitals. Expressions for the energy gradient are derived based on a Lagrangian formalism and cast in a density-matrix notation suitable for implementation in standard quantum-chemical program packages. In the present implementation, we exploit a decomposition of the multireference coupled cluster gradient expressions, i.e., lambda equations and the corresponding density matrices, into a so-called single-reference part for each reference determinant and a coupling term. Our implementation exhibits the proper scaling, i.e., O(dN6) with d as the number of reference determinants and N as the number of orbitals, and it is thus suitable for large-scale applications. The applicability of our multireference coupled cluster gradients is illustrated by computations for the equilibrium geometry of the 2,6-isomers of pyridyne and the pyridynium cation. The results are compared to those from single-reference coupled cluster calculations and are discussed with respect to the future perspectives of multireference coupled cluster theory.

A companion perturbation theory for state-specific multireference coupled cluster methods

A partitioning scheme is applied to the state-specific Mukherjee multireference coupled cluster method to derive a companion perturbation theory (Mk-MRPT2). A production-level implementation of Mk-MRPT2 is reported. The effectiveness of the Mk-MRPT2 method is demonstrated by application to the classic F(2) dissociation problem and the lowest-lying electronic states of meta-benzyne, including computations with up to 766 atomic orbitals. We show that Mk-MRPT2 theory is particularly useful in multireference focal point extrapolations to determine ab initio limits.

Applications of size-consistent state-specific multi-reference coupled cluster (SS-MRCC)theory to study the potential energy curves of some interesting molecular systems

In this paper we apply the state-specific multi-reference coupled cluster method(SS-MRCC) (Mahapatra et al 1999 J. Chem. Phys. 110 6171) to compute the potential energycurves (PECs) of some interesting diatomic molecular systems. The theory is built oncomplete active space (CAS) reference functions. This formulation relies upon asmall CAS, but is capable of producing energies to a high degree of accuracy. In the SS-MRCC method, depending upon the method of iteration, we maydetermine cluster amplitudes and consequently the effective Hamiltonian either with the frozen (unrelaxed scheme) or with the converged coefficients (relaxedscheme) which are the model space coefficients in the final wavefunction. Thus the energycan be obtained either as an expectation value with respect to the unrelaxed function or bydiagonalizing within the CAS. The SS-MRCC method is rigorously size-extensive and size-consistent.The method bypasses the intruders by focusing on a single state of interest in anatural manner. The method could thus serve as a reliable choice for PEC studiesover a wide range of geometries. We will also compare the relative performancesof the description of ‘unrelaxed’ versus ‘relaxed’ coefficients. We also evaluatecertain spectroscopic constants and compare them with experimental results.

Development of a linear response theory based on a state-specific multireference coupled cluster formalism

We present in this paper a linear response theory based on our recently developed state-specific multireference coupled cluster (SS-MRCC) method to compute excited state energies for systems whose ground state has a pronounced multireference character. The SS-MRCC method is built on complete active space reference functions, and is designed to treat quasidegeneracy of varying degrees while bypassing the intruder problem. The linear response theory based on such a function [multireference coupled cluster based linear response theory (MR-CCLRT)] offers a very convenient access to computation of excited states and, in particular, to generation of potential energy surfaces (PES) for excited states where a traditional response formulation based on a single reference theory will fail due to the quasidegeneracy at some regions of the PES and the effective Hamiltonian-based multireference response methods would be plagued by intruders. An attractive feature of the MR-CCLRT is that the computed excitation energies are size intensive in the sense that they become asymptotically equal to the sum of fragment excitation energies in the limit of noninteracting fragments. Illustrative numerical results are presented for the excited state PES of the rectangular H4 (P4) model, the trapezoidal H4 (H4) model, for Li2, and for some sample points on the excited states PES of the BeH2 complex. The ground states of all the three examples possess quasidegeneracy at some point on the PES, and there are potential intruders at some other points in the PES, and hence are appropriate to test the efficacy of the MR-CCLRT. A comparison with the (CI) full configuration interaction and MR-CCLRT results in the same basis for all the molecules shows very good performance of the theory in general, and indicates the efficacy of the method.