Multireference Coupled Cluster Method Research Articles

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.

Vibrational multi-reference coupled cluster theory in bosonic representation

The vibrational multi-reference coupled cluster method is developed to calculate the vibrational excitation energies of polyatomic molecules. The method is implemented on ozone and formaldehyde molecules and the results are compared with full vibrational configuration interaction (FVCI) method. A good agreement is found between the vibrational multi-reference coupled cluster method and converged FVCI method for lower lying vibrational states.

Linear-response theory for Mukherjee's multireference coupled-cluster method: Excitation energies

The recently presented linear-response function for Mukherjee's multireference coupled-cluster method (Mk-MRCC) [T.-C. Jagau and J. Gauss, J. Chem. Phys. 137, 044115 (2012)] is employed to determine vertical excitation energies within the singles and doubles approximation (Mk-MRCCSD-LR) for ozone as well as for o-benzyne, m-benzyne, and p-benzyne, which display increasing multireference character in their ground states. In order to assess the impact of a multireference ground-state wavefunction on excitation energies, we compare all our results to those obtained at the single-reference coupled-cluster level of theory within the singles and doubles as well as within the singles, doubles, and triples approximation. Special attention is paid to the artificial splitting of certain excited states which arises from the redundancy intrinsic to Mk-MRCC theory and hinders the straightforward application of the Mk-MRCC-LR method.

Linear-response theory for Mukherjee's multireference coupled-cluster method: Static and dynamic polarizabilities

The formalism of response theory is applied to derive expressions for static and dynamic polarizabilities within the state-specific multireference coupled-cluster theory suggested by Mukherjee and co-workers (Mk-MRCC) [J. Chem. Phys. 110, 6171 (1998)]. We show that the redundancy problem inherent to Mk-MRCC theory gives rise to spurious poles in the Mk-MRCC response functions, which hampers the reliable calculation of dynamic polarizabilities. Furthermore, we demonstrate that in the case of a symmetry-breaking perturbation a working response theory is obtained only if certain internal excitations are included in the responses of the cluster amplitudes. Exemplary calculations within the singles and doubles approximation (Mk-MRCCSD) are carried out on aryne compounds to illustrate the impact of a multireference ansatz on the polarizability.

Electronic Structure and Bonding of Cobalt Monoxide, CoO, and Its Ions CoO+ and CoO–: An Ab Initio Study

We present a systematic and high-level ab initio study of CoO and its ions, CoO(+) and CoO(-). Employing variational multireference (MRCI) and single-reference coupled-cluster methods combined with basis sets of quintuple quality, we have calculated 50, 31, and 7 bound states for CoO, CoO(+), and CoO(-), respectively. For all these states, complete potential energy curves have been constructed at the MRCI level of theory, whereas for a few low-lying states core subvalence and scalar relativistic effects have been taken into account. We report energetics, spectroscopic parameters, dipole moments, and spin-orbit coupling constants. The ground states of CoO, CoO(+), and CoO(-) are X(4)Δ, X(5)Δ, and X(5)Δ, respectively, the latter established for the first time. The CoO is quite ionic with a Co to O Mulliken charge transfer of ~0.6 electrons and a dipole moment μ(X(4)Δ) = 4.5 ± 0.1 D. The overall agreement between theory and experiment is good, but there are also important deviations. Despite the seeming simplicity of these diatomic species, reliable results can only be obtained at a high level of theory.

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.

A sequential transformation approach to the internally contracted multireference coupled cluster method

The internally contracted multireference coupled cluster (ic-MRCC) approach is formulated using a new wave function ansatz based on a sequential transformation of the reference function (sqic-MRCC). This alternative wave function simplifies the formulation of computationally viable methods while preserving the accuracy of the ic-MRCC approach. The structure of the sqic-MRCC wave function allows folding the effect of the single excitations into a similarity-transformed Hamiltonian whose particle rank is equal to the one of the Hamiltonian. Consequently, we formulate an approximation to the sqic-MRCC method with singles and doubles (included respectively up to fourfold and twofold commutators, sqic-MRCCSD[2]) that contains all terms present in the corresponding single-reference coupled cluster scheme. Computations of the potential energy curves for the dissociation of BeH(2) show that the untruncated sqic-MRCCSD scheme yields results that are almost indistinguishable from the ordinary ic-MRCCSD method. The energy obtained from the computationally less expensive sqic-MRCCSD[2] approximation is found to deviate from the full ic-MRCCSD method by less than 0.2 mE(h) for BeH(2), while, in the case of water, the harmonic vibrational frequencies of ozone, the singlet-triplet splitting of p-benzyne, and the dissociation curve of N(2), sqic-MRCCSD[2] faithfully reproduces the results obtained via the ic-MRCCSD scheme truncated to two commutators. A formal proof is given of the equivalence of the ic-MRCC and sqic-MRCC methods with the internally contracted and full configuration interaction approaches.

Communication: Spin-orbit splittings in degenerate open-shell states via Mukherjee's multireference coupled-cluster theory: A measure for the coupling contribution

We propose a generally applicable scheme for the computation of spin-orbit (SO) splittings in degenerate open-shell systems using multireference coupled-cluster (MRCC) theory. As a specific method, Mukherjee's version of MRCC (Mk-MRCC) in conjunction with an effective mean-field SO operator is adapted for this purpose. An expression for the SO splittings is derived and implemented using Mk-MRCC analytic derivative techniques. The computed SO splittings are found to be in satisfactory agreement with experimental data. Due to the symmetry properties of the SO operator, SO splittings can be considered a quality measure for the coupling between reference determinants in Jeziorski-Monkhorst based MRCC methods. We thus provide numerical insights into the coupling problem of Mk-MRCC theory.

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.

Parallel Implementation of Multireference Coupled-Cluster Theories Based on the Reference-Level Parallelism.

A novel algorithm for implementing a general type of multireference coupled-cluster (MRCC) theory based on the Jeziorski-Monkhorst exponential ansatz [Jeziorski, B.; Monkhorst, H. J. Phys. Rev. A1981, 24, 1668] is introduced. The proposed algorithm utilizes processor groups to calculate the equations for the MRCC amplitudes. In the basic formulation, each processor group constructs the equations related to a specific subset of references. By flexible choice of processor groups and subset of reference-specific sufficiency conditions designated to a given group, one can ensure optimum utilization of available computing resources. The performance of this algorithm is illustrated on the examples of the Brillouin-Wigner and Mukherjee MRCC methods with singles and doubles (BW-MRCCSD and Mk-MRCCSD). A significant improvement in scalability and in reduction of time to solution is reported with respect to recently reported parallel implementation of the BW-MRCCSD formalism [Brabec, J.; van Dam, H. J. J.; Kowalski, K.; Pittner, J. Chem. Phys. Lett.2011, 514, 347].

An explicitly correlated Mukherjee's state specific coupled cluster method: development and pilot applications

This paper reports development of the explicitly correlated variant of Mukherjee's state specific multireference coupled cluster method (MkCC-F12). The current implementation is restricted to conventional single and double excitations and to pseudo-double excitations related to the Slater Type Geminal (STG) correlation factor using the SP ansatz. The performance of the MkCCSD-F12 was tested on calculations of singlet methylene, dissociation curve of the fluorine molecule, and the BeH(2) insertion pathway. As expected, the results of the newly developed method reconfirm the significantly faster convergence with respect to the basis set limit compared to the traditional expansion in Slater determinants. Results prove that treating the correlation factor separately for each reference is appropriate.

Fock-space multi-reference coupled-cluster response with the effect of triples on dipole moment of ClO and SF radicals#

Dipole moment calculations of SF and ClO radicals have been carried out using the recently developed partial triples correction to Fock-space multi-reference coupled cluster method. Theoretical calculation of the doublet SF and ClO radicals is useful due to their importance in atmospheric chemistry. The dipole moments of these radicals are extremely sensitive to correlation effects. A brief insight to the way the triples correction has been implemented is presented. We compare the results obtained from our analytic response treatment with that of restricted open Hartree-Fock (ROHF) calculations. Results are presented for both relaxed and non-relaxed approach in the ROHF method. Results suggest the importance of triples corrections. The effects of orbital relaxation are also analysed from the results.

Wavefunction quality of single- and multi-reference coupled-cluster and configuration interaction methods: The P4 and H8 model systems

Projective and expectation value energies, variance and full CI overlap calculations for single- and multi-reference coupled cluster and configuration interaction wavefunctions are carried out. The variance proves to give an accurate measure for the quality of the wavefunction and the error of the correlation energy with respect to full CI. Special attention to the consistency case (two non-interacting H2 and four non-interacting H2 systems) is drawn. Due to the lack of valence extensivity [M. Hanrath, Theor. Chem. Acc. 121 (2008) 187] the multi-reference exponential wavefunction approach [M. Hanrath, J. Chem. Phys. 123 (2005) 84102] lacks exact separability for delocalized orbitals. The single reference formalism based multi-reference coupled cluster approach [O. Oliphant, L. Adamowicz, J. Chem. Phys. 94 (1991) 1229; P. Piecuch, O. Oliphant, L. Adamowicz, J. Chem. Phys. 99 (1993) 1875] proves to be size consistent for localized and delocalized orbitals but does not treat all references on the same footing (lack of Fermi vacuum invariance) causing symmetry issues and discontinuities when calculating potential energy surfaces. Externally restricted exponential approaches are shown to remove extensivity and consistency properties of the coupled cluster approaches.

Ground and excited state geometries via Mukherjee’s multireference coupled-cluster method

A comprehensive study of molecular equilibrium structures is conducted to benchmark the multireference coupled-cluster (CC) method suggested by Mukherjee and coworkers (Mk-MRCC). We determine equilibrium structures and adiabatic excitation energies by applying the Mk-MRCC method within the singles and doubles (SD) approximation to ground and excited states of various small and medium-sized molecules. The results are compared to those obtained using other multireference or single-reference CC methods. For most molecules with a multireference ground state, it is found that equilibrium structures and excitation energies computed at the Mk-MRCCSD, equation-of-motion CCSD, multireference averaged-quadratic CC, and CCSD(T) levels of theory qualitatively agree, while the single-reference CCSD method often leads to distinctly different results.

Fock-space multireference coupled-cluster calculations of the hyperfine structure of isoelectronic33S−and35,37Cl

Due to its flexibility and possible systematic improvement, the Fock-space (FS) multireference coupled-cluster (MRCC) method remains a very important tool for the computation of energy differences of spectroscopic interest. In the present work, the FS MRCC method for the electron detachment process has been applied to determine the magnetic hyperfine constant ${A}_{J}$ and nuclear quadrupole moments $Q$ (related to electric hyperfine constant ${B}_{J}$) for the lowest multiplets of ${}^{33}$S${}^{\ensuremath{-}}$, ${}^{35}$Cl, and ${}^{37}$Cl with Dirac-Fock orbitals. In addition, we also report ${}^{2}$${P}_{3/2}$([Ne]$3{s}^{2}3{p}^{5}$) $\ensuremath{\rightarrow}$ ${}^{2}$${P}_{1/2}$([Ne]$3{s}^{2}3{p}^{5}$) magnetic dipole transition matrix element and electron affinity of ${}^{35}\mathrm{Cl}$ (i.e., ionization energy of Cl${}^{\ensuremath{-}}$). Calculated properties are in very good agreement with the available new standard or reference values.

Multi-reference Fock space coupled-cluster method in the intermediate Hamiltonian formulation for potential energy surfaces

The effective and intermediate Hamiltonian multi-reference coupled-cluster (CC) method with singles and doubles for the doubly ionized (0,2) sector of Fock space (FS) is formulated and implemented. The intermediate Hamiltonian realization of the (0,2) FS problem provides a robust computational scheme for solving the FS-CC equations free from the intruder state problem. By introducing an efficient factorization strategy, we obtain a very efficient tool that can be used for computing double ionization potentials but more significantly to describe multi-reference problems in CC theory, illustrated by twisted ethylene and the potential energy curve for F(2). The latter separates smoothly to two F atoms, while the former avoids the cusp behavior at the 90° dihedral. We also explore the double ionization potentials for several small molecules, H(2)O, CO, C(2)H(2), and C(2)H(4).

A universal state-selective approach to multireference coupled-cluster non-iterative corrections

A new form of the asymmetric energy functional for multireference coupled cluster (MRCC) theories is discussed from the point of view of an energy expansion in a quasidegenerate situation. The resulting expansion for the exact electronic energy can be used to define the non-iterative corrections to approximate MRCC approaches. In particular, we show that in the proposed framework the essential part of dynamic correlation can be encapsulated in the so-called correlation Hamiltonian, which in analogy to the effective Hamiltonian, is defined in the model space (M(0)). The proper parametrization of the exact/trial wavefunctions leads to the cancellation of the overlap-type numerators and to a connected form of the correlation Hamiltonian and size-extensive energies. Within this parametrization, when the trial wavefunctions are determined without invoking a specific form of the MRCC sufficiency conditions, the ensuing correction can be universally applied to any type of the approximate MRCC method. The analogies with other MRCC triples corrections to MRCC theories with singles and doubles (MRCCSD) are outlined. In particular, we discuss the approach, which in analogy to the Λ-Mk-MRCCSD(T) method [F. A. Evangelista, E. Prochnow, J. Gauss, H. F. Schaefer III, J. Chem. Phys. 132, 074107 (2010)], introduces an approximate form of the triply-excited clusters into the effective and correlation Hamiltonians. Since the discussed corrections can be calculated as a sum of independent reference-related contributions, possible parallel algorithms are also outlined.

Potential Energy Curve for Ring-Opening Reactions: Comparison Between Broken-Symmetry and Multireference Coupled Cluster Methods

The spin-unrestricted Hartree-Fock (UHF)-based coupled cluster singles and doubles (UHF-CCSD) and Mukherjee's state-specific multireference CCSD (MkCCSD) methods are applied to four ring-opening reactions. The spin-restricted Hartree-Fock (RHF)-based CCSD (RHF-CCSD) calculations are also performed for comparison. In the case of the UHF-CCSD method, an approximate spin-projection (AP) method is applied to the broken-symmetry (BS) singlet solution to remove the spin contamination effect. For potential energy curves (PECs) of all reactions presented in this study, the results of RHF-CCSD and UHF-CCSD are substantially different from those of MkCCSD, while the results after the AP method (AP-UCCSD) reproduce the MkCCSD results well. It strongly suggests that the spin contamination effect should be removed by the AP correction even at the UHF-CCSD level to predict reliable energetics of these reactions.

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.