Multireference Coupled Cluster Approach Research Articles

Investigating the Heaviest Halogen: Lessons Learned from Modeling the Electronic Structure of Astatine’s Small Molecules

We present a systematic study of electron-correlation and relativistic effects in diatomic molecular species of the heaviest halogen astatine (At) within relativistic single- and multireference coupled-cluster approaches and relativistic density functional theory. We establish revised reference ab initio data for the ground states of At2, HAt, AtAu, and AtO+ using a highly accurate relativistic effective core potential model and in-house basis sets developed for accurate modeling of molecules with large spin-orbit effects. Spin-dependent relativistic effects on chemical bonding in the ground state are comparable to the binding energy or even exceed it in At2. Electron-correlation effects near the equilibrium internuclear separation are mostly dynamical and can be adequately captured using single-reference CCSD(T). However, bond elongation in At2 and, especially, AtO+ results in rapid manifestation of its multireference character. While useful for evaluating the spin-orbit effects on the ground-state bonding and properties, the two-component density functional theory lacks predictive power, especially in combination with popular empirically adjusted exchange-correlation functionals. This drawback supports the necessity to develop new functionals for reliable quantum-chemical models of heavy-element compounds with strong relativistic effects.

Extension of the Fock-space coupled-cluster method with singles and doubles to the three-valence sector.

The single-reference coupled-cluster method has proven very effective in the ab initio description of atomic and molecular systems, but its successful application is limited to states dominated by a single Slater determinant, which is used as the reference. In cases where several determinants are important in the wave function expansion, i.e., we have to deal with nondynamic correlation effects, a multi-reference version of the coupled-cluster method is required. The multi-reference coupled-cluster approaches are based on the effective Hamiltonian formulation providing a two-step procedure, in which dynamic correlation effects can be efficiently evaluated by the wave operator, while nondynamic correlation contributions are given by diagonalization of the effective Hamiltonian in the final step. There are two classical multi-reference coupled-cluster formulations. In this paper, the focus is on the so-called Fock-space coupled-cluster method in its basic version with one- and two-particle operators in the exponent. Computational schemes using this truncation of the cluster operator have been successfully applied in calculations in one- and two-valence sectors of the Fock space. In this paper, we show that the approach can be easily extended and effectively employed in the three-valence sector calculations.

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.

A second-order multi-reference quasiparticle-based perturbation theory

The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141:134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented.

A quasiparticle-based multi-reference coupled-cluster method.

The purpose of this paper is to introduce a quasiparticle-based multi-reference coupled-cluster (MRCC) approach. The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference (MR) basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. On the basis of these quasiparticles, a generalization of the normal-ordered operator products for the MR case can be introduced as an alternative to the approach of Mukherjee and Kutzelnigg [Recent Prog. Many-Body Theor. 4, 127 (1995); Mukherjee and Kutzelnigg, J. Chem. Phys. 107, 432 (1997)]. Based on the new normal ordering any quasiparticle-based theory can be formulated using the well-known diagram techniques. Beyond the general quasiparticle framework we also present a possible realization of the unitary transformation. The suggested transformation has an exponential form where the parameters, holding exclusively active indices, are defined in a form similar to the wave operator of the unitary coupled-cluster approach. The definition of our quasiparticle-based MRCC approach strictly follows the form of the single-reference coupled-cluster method and retains several of its beneficial properties. Test results for small systems are presented using a pilot implementation of the new approach and compared to those obtained by other MR methods.

Intermediate Hamiltonian Fock Space Multireference Coupled Cluster Approach to Core Excitation Spectra.

The Fock space multireference coupled cluster (FSMRCC) method provides an efficient approach for the direct calculation of excitation energies. In intermediate Hamiltonian (IH-FSMRCC) formulation, the method is free from intruder state problems and associated convergence difficulties, even with a large model space. In this paper, we demonstrate that the IH-FSMRCC method with suitably chosen model space can be used for the accurate description of core excitation spectra of molecules, and our results are in excellent agreement with the experimental values. We have investigated the effect of choice of model space on the computed results. Unlike the equation-of-motion (EOM)-based method, the IH-FSMRCC does not require any special technique for convergence and in singles and doubles approximation gives a performance comparable to that of the standard EOMEE-CCSD method, even better in some of the cases.

Thermodynamics of Tetravalent Thorium and Uranium Complexes from First-Principles Calculations

Enthalpies of formation for the ThX4 and UX4 (X = F, Cl, OH) species have been investigated with density functional theory and coupled-cluster methods. ThX4 molecules are all confirmed as tetrahedral, while all UX4 molecules are predicted to adopt D2d symmetry using density functional theory. Multireference coupled cluster approaches confirm the D2d symmetry for UF4. The bonding is mostly ionic, and predicted formation energies for the halogen species show good agreement with experiment. Our calculated hydration energy of UF4 (-54.0 kcal/mol) is in very good agreement with the experimental data (-54.8 kcal/mol). We predict CCSD(T) formation energies of ΔfG[U(OH)4(g)] = -286.3 kcal/mol and ΔfG[U(OH)4(aq)] = -318.7 kcal/mol. ΔfG[U(OH)4(aq)] is 21 kcal/mol less stable than the established experimental thermodynamic data.

Ab initio calculation of correlation effects for the O 1s core electron binding energy in MgO

Wavefunction based ab initio embedded cluster calculations are employed to calculate the O 1s core electron binding energies (CEBEs) of bulk MgO and the MgO(001) surface. A quantum cluster consisting of 61 atoms in five layers and embedded in a large point charge field is used for bulk MgO, the cluster for the MgO(001) surface is chosen accordingly. The O 1s CEBEs are calculated at the Koopmans’ theorem (KT) and ΔSCF levels and with inclusion of correlation effects by means of the MC-CEPA method (multi-configuration coupled electron pair approximation), which is an approximate multi-reference coupled cluster approach. The correlation contributions to the O 1s CEBE of the central O atom due to the Mg atoms in the first and the O atoms in the second coordination shell turned out to be additive to a large extent. Therefore, they could be evaluated in an incremental fashion by a series of smaller calculations, where only a few atoms are included in the correlation treatment rather than all atoms of the first coordination shells or of the full quantum cluster. This makes the calculations feasible, even if large basis sets are used. The final results for the O 1s CEBEs are 533.47 and 533.50 eV for bulk MgO and the MgO(001) surface, to which electron correlation contributes 0.77 and 0.70 eV, respectively.

Universal state-selective corrections to multi-reference coupled-cluster theories with single and double excitations

The recently proposed universal state-selective (USS) corrections [K. Kowalski, J. Chem. Phys. 134, 194107 (2011)] to approximate multi-reference coupled-cluster (MRCC) energies can be commonly applied to any type of MRCC theory based on the Jeziorski-Monkhorst [B. Jeziorski and H. J. Monkhorst, Phys. Rev. A 24, 1668 (1981)] exponential ansatz. In this paper we report on the performance of a simple USS correction to the Brillouin-Wigner and Mukherjee's MRCC approaches employing single and double excitations (USS-BW-MRCCSD and USS-Mk-MRCCSD). It is shown that the USS-BW-MRCCSD correction, which employs the manifold of single and double excitations, can be related to a posteriori corrections utilized in routine BW-MRCCSD calculations. In several benchmark calculations we compare the USS-BW-MRCCSD and USS-Mk-MRCCSD results with the results obtained with the full configuration interaction method.

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.

On the approximation of the similarity-transformed Hamiltonian in single-reference and multireference coupled cluster theory

We consider the recursive single commutator (RSC) approximation of the Baker–Campbell–Hausdorff expansion introduced by Yanai and Chan [T. Yanai, G.K.-L. Chan, J. Chem. Phys. 124 (2006) 194106] and apply it in order to approximate the similarity transformation of the Hamiltonian in both traditional and unitary coupled cluster theory. The equilibrium bond distance, harmonic vibrational frequency, and anharmonic constant of H2, HF, N2, CuH, and Cu2 were computed using the coupled cluster approach with single and double excitations (CCSD) and CCSD with the RSC approximation of the similarity-transformed Hamiltonian (CCSD-RSC). Our results demonstrate that the RSC approximation introduces substantial errors in the estimates of molecular properties. The leading pejorative effects of the RSC approximation can be traced back to the imbalanced description of diagrams arising from the term 12H^,T^2,T^2. Following this analysis we consider a modified RSC scheme correct to fourth-order in the energy, which is found to reproduce CCSD results more closely. The RSC scheme is also applied in conjunction with the state-specific multireference coupled cluster approach of Mukherjee and co-workers [U.S. Mahapatra, B. Datta, D. Mukherjee, J. Chem. Phys. 110 (1999) 6171] to compute the potential energy curve of the BeH2 model, the vibrational frequencies of ozone, and the singlet–triplet splitting of p-benzyne. These examples show that the deterioration of the results caused by the RSC scheme is analogous to the one observed in the single-reference case. Implications for the formulation of approximate internally contracted multireference theories are discussed.

Alternative single-reference coupled cluster approaches for multireference problems: The simpler, the better

We report a general implementation of alternative formulations of single-reference coupled cluster theory (extended, unitary, and variational) with arbitrary-order truncation of the cluster operator. These methods are applied to compute the energy of Ne and the equilibrium properties of HF and C(2). Potential energy curves for the dissociation of HF and the BeH(2) model computed with the extended, variational, and unitary coupled cluster approaches are compared to those obtained from the multireference coupled cluster approach of Mukherjee et al. [J. Chem. Phys. 110, 6171 (1999)] and the internally contracted multireference coupled cluster approach [F. A. Evangelista and J. Gauss, J. Chem. Phys. 134, 114102 (2011)]. In the case of Ne, HF, and C(2), the alternative coupled cluster approaches yield almost identical bond length, harmonic vibrational frequency, and anharmonic constant, which are more accurate than those from traditional coupled cluster theory. For potential energy curves, the alternative coupled cluster methods are found to be more accurate than traditional coupled cluster theory, but are three to ten times less accurate than multireference coupled cluster approaches. The most challenging benchmark, the BeH(2) model, highlights the strong dependence of the alternative coupled cluster theories on the choice of the Fermi vacuum. When evaluated by the accuracy to cost ratio, the alternative coupled cluster methods are not competitive with respect to traditional CC theory, in other words, the simplest theory is found to be the most effective one.

A state-specific partially internally contracted multireference coupled cluster approach

A state-specific partially internally contracted multireference coupled cluster approach is presented for general complete active spaces with arbitrary number of active electrons. The dominant dynamical correlation is included via an exponential parametrization of internally contracted cluster operators ( ̂T) which excite electrons from a multideterminantal reference function. The remaining dynamical correlation and relaxation effects are included via a diagonalization of the transformed Hamiltonian ̅Ĥ =e(- ̂T)Ĥe( ̂T) in the multireference configuration interaction singles space in an uncontracted fashion. A new set of residual equations for determining the internally contracted cluster amplitudes is proposed. The second quantized matrix elements of ̅Ĥ , expressed using the extended normal ordering of Kutzelnigg and Mukherjee, are used as the residual equations without projection onto the excited configurations. These residual equations, referred to as the many-body residuals, do not have any near-singularity and thus, should allow one to solve all the amplitudes without discarding any. There are some relatively minor remaining convergence issues that may arise from an attempt to solve all the amplitudes and an initial analysis is provided in this paper. Applications to the bond-stretching potential energy surfaces for N(2), CO, and the low-lying electronic states of C(2) indicate clear improvements of the results using the many-body residuals over the conventional projected residual equations.

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.

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.

An orbital-invariant internally contracted multireference coupled cluster approach

We have formulated and implemented an internally contracted multireference coupled cluster (ic-MRCC) approach aimed at solving two of the problems encountered in methods based on the Jeziorski-Monkhorst ansatz: (i) the scaling of the computational and memory costs with respect to the number of references, and (ii) the lack of invariance of the energy with respect to rotations among active orbitals. The ic-MRCC approach is based on a straightforward generalization of the single-reference coupled cluster ansatz in which an exponential operator is applied to a multiconfigurational wave function. The ic-MRCC method truncated to single and double excitations (ic-MRCCSD) yields very accurate potential energy curves in benchmark computations on the Be + H(2) insertion reaction, the dissociation of hydrogen fluoride, and the symmetric double dissociation of water. Approximations of the ic-MRCC theory in which the Baker-Campbell-Hausdorff expansion is truncated up to a given number of commutators are found to converge quickly to the full theory. In our tests, two commutators are sufficient to recover a total energy within 0.5 mE(h) of the full ic-MRCCSD method along the entire potential energy curve. A formal analysis shows that the ic-MRCC method is invariant with respect to rotation among active orbitals, and that the orthogonalization procedure used to produce the set of linearly independent excitation operators plays a crucial role in guaranteeing the invariance properties. The orbital invariance was confirmed in numerical tests. Moreover, approximated versions of the ic-MRCC theory based on a truncated Baker-Campbell-Hausdorff expansion, preserve the orbital invariance properties of the full theory.

Insights into the orbital invariance problem in state-specific multireference coupled cluster theory

In this communication we report the results of our studies on the orbital invariance properties of the state-specific multireference coupled cluster approach suggested by Mukherjee and co-workers (Mk-MRCC). In particular, we have gathered numerical evidence to show that even when the linear excitation manifold is modified in order to span the same space for each reference, the resulting method is not orbital invariant. In order to test this conjecture we have proposed a new truncation scheme (Mk-MRCCSDtq) which, in addition to full single and double excitations, contains partial triple and quadruple excitations. For a reference space generated by all possible combinations of two electrons in two orbitals, the linear excitation manifold of Mk-MRCCSDtq spans the same set for each reference determinant. Mk-MRCCSDtq is found to lack energy invariance for rotations among active molecular orbitals but it is less sensitive to orbital rotations than the conventional scheme which includes only singles and doubles (Mk-MRCCSD). Nevertheless, Mk-MRCCSDtq is a very accurate method, superior with respect to multireference configuration interaction approaches, and competitive with the active-space coupled cluster method and the MRexpT ansatz.

Perturbative triples corrections in state-specific multireference coupled cluster theory

We formulated and implemented a perturbative triples correction for the state-specific multireference coupled cluster approach with singles and doubles suggested by Mukherjee and co-workers, Mk-MRCCSD [Mol. Phys. 94, 157 (1998)]. Our derivation of the energy correction [Mk-MRCCSD(T)] is based on a constrained search for stationary points of the Mk-MRCC energy functional together with a perturbative expansion with respect to the appearing triples cluster operator. The Lambda-Mk-MRCCSD(T) approach derived in this way consists in (1) a correction to the off-diagonal matrix elements of the effective Hamiltonian which is unique to coupled cluster methods based on the Jeziorski-Monkhorst ansatz, and (2) an asymmetric energy correction to the diagonal elements of the effective Hamiltonian. The Mk-MRCCSD(T) correction is obtained from the Lambda-Mk-MRCCSD(T) method by approximating the singles and doubles Lagrange multipliers with the corresponding cluster amplitudes. We investigate the performance of the Mk-MRCCSD(T) method by applying it to the potential energy curve of the BeH(2) model and F(2) and the geometry and harmonic vibrational frequencies of ozone. Computation of the energy difference between the mono- and bicyclic forms of the 2,6-pyridyne diradical illustrates the potential of Mk-MRCCSD(T) as a tool for the study of realistic chemical problems requiring multireference zeroth-order wave functions.

Excited and ionized states of the ozone molecule with full triples coupled cluster methods

The role of connected triple excitations in coupled cluster (CC) calculations of vertical excitation energies, ionization potentials, and the electron affinity of the ozone molecule is evaluated. The equation of motion (EOM) and Fock space (FS) multireference CC approaches with full triples have been used in the calculations. The effect of the T(3) and R(3) operators significantly improve the EOM CCSD results for all considered quantities. A similar behavior is observed in the case of the FS-CC calculations. The FS-CC calculations with full triples have been obtained only for the intermediate Hamiltonian realization of the FS approach as the standard formulation diverges. The latter results are rigorously linked, and less expensive since smaller matrices are diagonalized.

Dissociating N2: a multi-reference coupled cluster study on the potential energy surfaces of ground and excited states

This article reports on the calculation of 12 low lying states of the nitrogen molecule along its dissociation using the multi-reference exponential wavefunction ansatz [J. Chem. Phys. 123, 84102 (2005)], the single-reference formalism multi-reference coupled cluster [J. Chem. Phys. 94, 1229 (1991)], and MRCI methods. Energies relative to full CI are given. The results show the multi-reference coupled cluster approaches generally applicable to very demanding problems at high accuracy. In comparison to MRCI both coupled cluster type approaches do not reproduce spatial symmetry exactly. Nevertheless, this error in the degeneracy is much less than the overall error of the MRCI calculations.