Multireference Coupled-cluster Research Articles

An Intruder-Free Fock Space Coupled-Cluster Study of the Potential Energy Curves of LiMg+ within the (2,0) Sector.

The potential energy curves (PECs) and spectroscopic constants of the ground and excited states of a LiMg+ molecular cation were investigated. We obtained accurate results for the fifteen lowest-lying states of the LiMg+ cation using the Intermediate Hamiltonian Fock Space Multireference Coupled Cluster (IH-FS-CC) method applied to the (2,0) sector. Relativistic corrections were accounted for using the third-order Douglas-Kroll method. In each instance, smooth PECs were successfully computed across the entire range of interatomic distances from equilibrium to the dissociation limit. The results are in good accordance with previous studies of this molecular cation. Notably, this study marks the first application of IH-FS-CC in investigating a mixed alkali and alkaline earth molecular cation, proving its usability in determining accurate PECs of such diatomics and their spectroscopic constants.

State-Specific Coupled-Cluster Methods for Excited States.

We reexamine ΔCCSD, a state-specific coupled-cluster (CC) with single and double excitations (CCSD) approach that targets excited states through the utilization of non-Aufbau determinants. This methodology is particularly efficient when dealing with doubly excited states, a domain in which the standard equation-of-motion CCSD (EOM-CCSD) formalism falls short. Our goal here to evaluate the effectiveness of ΔCCSD when applied to other types of excited states, comparing its consistency and accuracy with EOM-CCSD. To this end, we report a benchmark on excitation energies computed with the ΔCCSD and EOM-CCSD methods for a set of molecular excited-state energies that encompasses not only doubly excited states but also doublet-doublet transitions and (singlet and triplet) singly excited states of closed-shell systems. In the latter case, we rely on a minimalist version of multireference CC known as the two-determinant CCSD method to compute the excited states. Our data set, consisting of 276 excited states stemming from the quest database [Véril et al., WIREs Comput. Mol. Sci. 2021, 11, e1517], provides a significant base to draw general conclusions concerning the accuracy of ΔCCSD. Except for the doubly excited states, we found that ΔCCSD underperforms EOM-CCSD. For doublet-doublet transitions, the difference between the mean absolute errors (MAEs) of the two methodologies (of 0.10 and 0.07 eV) is less pronounced than that obtained for singly excited states of closed-shell systems (MAEs of 0.15 and 0.08 eV). This discrepancy is largely attributed to a greater number of excited states in the latter set exhibiting multiconfigurational characters, which are more challenging for ΔCCSD. We also found typically small improvements by employing state-specific optimized orbitals.

Rapidly convergent quantum Monte Carlo using a Chebyshev projector.

The multireference coupled-cluster Monte Carlo (MR-CCMC) algorithm is a determinant-based quantum Monte Carlo (QMC) algorithm that is conceptually similar to Full Configuration Interaction QMC (FCIQMC). It has been shown to offer a balanced treatment of both static and dynamic correlation while retaining polynomial scaling, although application to large systems with significant strong correlation remained impractical. In this paper, we document recent algorithmic advances that enable rapid convergence and a more black-box approach to the multireference problem. These include a logarithmically scaling metric-tree-based excitation acceptance algorithm to search for determinants connected to the reference space at the desired excitation level and a symmetry-screening procedure for the reference space. We show that, for moderately sized reference spaces, the new search algorithm brings about an approximately 8-fold acceleration of one MR-CCMC iteration, while the symmetry screening procedure reduces the number of active reference space determinants with essentially no loss of accuracy. We also introduce a stochastic implementation of an approximate wall projector, which is the infinite imaginary time limit of the exponential projector, using a truncated expansion of the wall function in Chebyshev polynomials. Notably, this wall-Chebyshev projector can be used to accelerate any projector-based QMC algorithm. We show that it requires significantly fewer applications of the Hamiltonian to achieve the same statistical convergence. We benchmark these acceleration methods on the beryllium and carbon dimers, using initiator FCIQMC and MR-CCMC with basis sets up to cc-pVQZ quality.

Hilbert space multireference coupled cluster tailored by matrix product states

In the past decade, the quantum chemical version of the density matrix renormalization group method has established itself as the method of choice for strongly correlated molecular systems. However, despite its favorable scaling, in practice, it is not suitable for computations of dynamic correlation. Several approaches to include that in post-DMRG methods exist; in our group, we focused on the tailored coupled cluster (TCC) approach. This method works well in many situations; however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias toward the reference determinant representing the Fermi vacuum. Although sometimes it is possible to use a compensation scheme to avoid this bias for energy differences, it is certainly a drawback. In order to overcome this bias of the TCC method, we have developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We have implemented and compared the performance of three Hilbert-space multireference coupled cluster (MRCC) variants—the state universal one and the Brillouin–Wigner and Mukherjee’s state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethyleneethane molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. We have also investigated the sensitivity of the results on the orbital rotation of the highest occupied and lowest unoccupied molecular orbital (HOMO–LUMO) pair, as it is well known that Hilbert-space MRCC methods are not invariant to such transformations.

Performance Tests of the Second-Order Approximate Internally Contracted Multireference Coupled-Cluster Singles and Doubles Method icMRCC2.

Benchmark results are presented for the second-order approximation of the internally contracted multireference coupled-cluster method with single and double excitations, icMRCC2 [Köhn, Bargholz, J. Chem. Phys. 2019, 151, 041106], which was designed as a multireference analogue of the single-reference second-order approximate coupled-cluster method CC2 [Christiansen, Koch, Jørgensen, Chem. Phys. Lett. 1995, 243, 409-418]. Vertical excitation energies of various small to medium-sized organic molecules are investigated based on established test sets from the literature. Additionally, the spectroscopic constants of ground and excited states of diatomics and the geometric parameters of excited triatomic molecules were determined and compared to the experimental data. The results show that the method clearly extends the applicability of single-reference CC2, including doubly excited states, and also artifacts of CC2 like too low Rydberg excitations and too weak multiple bonds are eliminated. The method is computationally more demanding than standard multireference second-order perturbation theories but improves significantly in accuracy, as shown by the benchmark results. In addition, it is demonstrated that small active spaces are often sufficient to obtain accurate energies with icMRCC2. Example applications like the automerization of cyclobutadiene, the deactivation pathway of ethylene, and the excited states of an iron complex with a noninnocent nitrosyl ligand demonstrate the potential of icMRCC2 in cases with strong multireference character.

A systematic improvement to UGA-SSMRCCSD equations and its implication for potential energy curves.

The Unitary Group Adaptation (UGA) offers a very compact and efficient spin adaptation strategy for any spin-free Hamiltonian in a many body framework. Our use of UGA in the context of state-specific (SS) Jeziorski-Monkhorst Ansatz based multireference coupled cluster (MRCC) theory obviates the non-commutativity between the spin-free cluster operators via a normal ordered exponential parametrization in the wave operator. A previous formulation of UGA-SSMRCC by us [R. Maitra, D. Sinha, and D. Mukherjee, J. Chem. Phys. 137, 024105 (2012)], using the same ansatz, employed certain sufficiency conditions to reach the final working equations, which cannot be improved systematically. In this article, we will present a more rigorous formulation that follows from an exact factorization of the unlinked terms of the Bloch equation, resulting in equations on which a hierarchy of approximations can be systematically performed on the emergent additional terms. This derivation was shown in our recent article [D. Chakravarti, S. Sen, and D. Mukherjee, Mol. Phys. 119, e1979676 (2021)] in the context of a single open shell CC formalism and was applied to spectroscopic energy differences where the contribution of the new terms was found to be of the order of ∼0.001eV for ionization potential, electron affinity, and excitation energy. In the current work, we will present a comparison between the earlier and current formulations via both a theoretical analysis and a numerical demonstration of the dramatic effect of the additional terms brought in by the factorization on potential energy curves. The contribution of such terms was found to gain importance with an increase in the number of singly occupied active orbitals in the model space functions.

Three Lagrangians for the complete-active space coupled-cluster method.

Three fully variational formulations of the complete-active space coupled-cluster method are derived. The formulations include the ability to approximate the model vectors by smooth manifolds, thereby opening up the possibility for overcoming the exponential wall of scaling for model spaces of complete-active space type. In particular, model vectors of matrix-product states are considered, and it is argued that the present variational formulation allows not only favorably scaling multireference coupled-cluster calculations but also systematic correction of tailored coupled-cluster calculations and of quantum chemical density-matrix renormalization group methods, which are fast and polynomial scaling but lack the ability to properly resolve dynamical correlation at chemical accuracy. The extension of the variational formulations to the time domain is also discussed, with derivations of abstract evolution equations.

A hybrid stochastic configuration interaction-coupled cluster approach for multireference systems.

The development of multireference coupled cluster (MRCC) techniques has remained an open area of study in electronic structure theory for decades due to the inherent complexity of expressing a multiconfigurational wavefunction in the fundamentally single-reference coupled cluster framework. The recently developed multireference-coupled cluster Monte Carlo (mrCCMC) technique uses the formal simplicity of the Monte Carlo approach to Hilbert space quantum chemistry to avoid some of the complexities of conventional MRCC, but there is room for improvement in terms of accuracy and, particularly, computational cost. In this paper, we explore the potential of incorporating ideas from conventional MRCC-namely, the treatment of the strongly correlated space in a configuration interaction formalism-to the mrCCMC framework, leading to a series of methods with increasing relaxation of the reference space in the presence of external amplitudes. These techniques offer new balances of stability and cost against accuracy, as well as a means to better explore and better understand the structure of solutions to the mrCCMC equations.

Toward an efficient implementation of internally contracted coupled-cluster methods.

A new implementation of the internally contracted multireference coupled-cluster with singles and doubles (icMRCCSD) method is presented. The new code employs an efficient tensor contraction kernel and can also avoid full four-external integral transformations, which significantly extends the scope of the applicability of icMRCCSD. The new implementation is currently restricted to the simple case of two active electrons in two orbitals and also supports the computation of spin-adapted doublet and triplet coupled-cluster wavefunctions. This contribution describes the basic approach for the automated derivation of working equations and benchmarks the current code against efficient implementations of standard methods, such as single-reference coupled-cluster singles and doubles (CCSD) and internally contracted multireference configuration interaction (icMRCI). Run times for linearized variants of icMRCCSD are only twice as long as comparable CCSD runs and similar to those of the icMRCI implementation, while non-linear terms of more complete variants of icMRCCSD lead to an order of magnitude longer computation times. Nevertheless, the new code allows for computations at larger scales than it was possible previously, with less demands on memory and disk-space resources. This is exemplified by numerical structure optimizations and harmonic force field determinations of NC2H5 isomers and the singlet and triplet states of m-benzyne. In addition, the exchange coupling of a dinuclear copper complex is determined. This work also defines a new commutator approximation for icMRCCSD, which includes all terms that are also present in the single-reference CCSD method, thus yielding a consistent pair of single-reference and multireference coupled-cluster methods.

A new intermediate Hamiltonian Fock-space coupled-cluster formalism for the three-valence sector

The single-reference coupled-cluster (CC) method is one of the most efficient approaches for describing electron correlation effects in atoms and molecules but its successful application is limited to states dominated by a single Slater determinant. In cases where several determinants dominate the wave function expansion, the multi-reference (MR) CC formulations are required. They usually use the effective Hamiltonian formalism which allows dealing with two types of electron correlation appearing in the MR cases: nondynamic and dynamic. There are two MR-CC formulations of this type: the Hilbert space CC method and the Fock-space (FS) one. In the paper, a version of the FS-CC method designated for the description of systems with three-valence electrons is being considered. The idea of the intermediate Hamiltonian (IH) reformulation of the standard FS-CC method, which simplifies the calculations and avoids the intruder state problem, is discussed in this context. A new IH-FS-CC approach is presented which takes advantage of the redefinition of the normal ordering of the FS cluster expansion introduced by Lindgren. The redefinition significantly simplifies the IH-FS-CC formalism and lowers the computational cost of the calculations. The advantages of the new IH formulation are discussed in detail for the case of the FS-CC method in which the lowest-order cluster operators are included in all sectors of the Fock-space, including the three-valence one.

A generalized hybrid scheme for multireference methods.

A generalization of the hybrid scheme for multireference methods as recently put forward by Saitow and Yanai [J. Chem. Phys. 152, 114 111 (2020)] is presented. The hybrid methods are constructed by defining internal and external excitation spaces and evaluating these two subsets of excitations at different levels of theory. New hybrids that use the mix of internally contracted multireference coupled-cluster, unshifted multireference coupled electron pair, and multireference perturbation methods are derived and benchmarked. A new separation of the excitation space, which combines all singles and doubles excitations to the virtual orbitals into the external space, is also presented and tested. In general, the hybrid methods improve upon their non-hybrid parent method and offer a good compromise between computational complexity and numerical accuracy.

Dressing of Vertices by Cumulants in Multi-Reference Coupled Cluster.

A new scheme is introduced in Multi-Reference (MR) Coupled Cluster (CC) based on the MR Generalized Normal Ordering (MRGNO) and the corresponding MR Generalized Wick Theorem (MRGWT) of Kutzelnigg and Mukherjee. The key element is the identification of a structure in MRGWT generated terms, facilitated by Goldstone diagram techniques. This allows for bundling the many terms of the MRGWT expansion and introduces a hierarchy in the equations that can be harnessed in devising approximations. One- and two-particle interaction vertices are found to be uniformly substituted for their counterpart dressed by density cumulants. This allows for a straightforward rewriting of the ordinary energy expression of CC with interaction dressed (id) one- and two-particle terms and reveals the presence of three- and higher-rank dressed interaction vertices too. Cumulants appearing out of dressed interaction vertices contribute to the amplitude equations and can be interpreted to have an amplitude dressing role. Dressing of one- and two-particle interaction vertices is the most straightforward to implement and does not hinder computational feasibility, provided that the reference function involves strictly limited active space sizes. The Generalized Valence Bond wave function, underlying pilot numerical tests, fulfills this criterion. Results on multiple bond breaking scenarios point to the need of stepping beyond one- and two-particle id. An extremely simple version of incorporating amplitude dressing in addition to one- and two-particle id is seen to cure the potential energy curves remarkably, stimulating further investigations along this line.

C-H Bond Activation by the Excited Zinc Atom: Gas-Phase Formation of Methylzinc Hydride (HZnCH3) Based on Multireference Second-Order Perturbation Theory and Coupled Cluster Calculations.

The pioneering spectroscopic observations of the methylzinc hydride [HZnCH3(X1A1)] molecule were reported previously by the Ziurys group [J. Am. Chem. Soc.2010, 132, 17186–17192], and the possible formation mechanisms were suggested therein, including those with the participation of excited zinc atoms in reaction with methane. Herein, the ground singlet state and the lowest excited triplet state potential energy surfaces of the Zn + CH4 reaction have been explored using high-level electronic structure calculations with multireference second-order perturbation theory and coupled cluster singles and doubles with perturbative triples (CCSD(T)) methods in conjunction with all-electron basis sets (up to aug-cc-pV5Z) and scalar relativistic effects incorporated via the second-order Douglas–Kroll–Hess (DK) method. Based on the ab initio results, a plausible scenario for the formation of HZnCH3(X1A1) is proposed involving the activation of the C–H bond of methane by the lowest excited 3P state atomic zinc. Calculations also highlight the importance of an agostic-like Zn···H–C interactions in the pre-activation complex and good agreement between the structure of the HZnCH3(X1A1) molecule predicted at the DK-CCSD(T)/aug-cc-pVQZ-DK level of theory and that derived from rotational spectroscopy, as well as the discrepancies between the ab initio and density functional theory predictions.

Three-Body Excitations in Fock-Space Coupled-Cluster: Fourth Order Perturbation Correction to Electron Affinity and Its Relation to Bondonic Formalism.

In this paper, we present a formulation of highly correlated Fock-space multi-reference coupled-cluster (FSMRCC) methods, including approximate triples on top of the FSMRCC with singles and doubles, which correct the electron affinities by at least at third and up to the fourth order in perturbation. We discuss various partial fourth-order schemes, which are reliable and yet computationally more efficient than the full fourth-order triples scheme. The third-order scheme is called MRCCSD+T*(3). We present two approximate fourth-order schemes, MRCCSD+T*−a(4) and MRCCSD+T*(4). The results that are presented allow one to choose an appropriate fourth-order scheme, which is less expensive and right for the problem. All these schemes are based on the effective Hamiltonian scheme, and provide a direct calculation of the vertical electron affinities. We apply these schemes to a prototype Li2 molecule, using four different basis sets, as well as BeO and CH+. We have calculated the vertical electron affinities of Li2 at the geometry of the neutral Li2 molecule. We also present the vertical ionization potentials of the Li2 anion at the geometry of the anion ground state. We have also shown how to calculate adiabatic electron affinity, though in that case we lose the advantages of direct calculation. BeO has been examined in two basis sets. For CH+, four different basis sets have been used. We have presented the partial fourth-order schemes to the EA in all the basis sets. The results are analyzed to illustrate the importance of triples, as well as highlight computationally efficient partial fourth-order schemes. The choice of the basis set on the electron affinity calculation is also emphasized. Comparisons with available experimental and theoretical results are presented. The general fourth-order schemes, which are conceptually equivalent with the Fock-space multi-reference coupled-cluster singles, doubles, and triplets (MRCCSD+T) methods, based on bondonic formalism, are also presented here in a composed way, for quantum electronic affinity.

Ring coupled cluster doubles at the multireference level.

A ring approximation within an internally contracted multireference (MR) Coupled Cluster (CC) framework is worked out and tested. Derivation of equations utilizes MR based, generalized normal ordering and the corresponding generalized Wick-theorem (MR-GWT). Contractions among cluster operators are avoided by adopting a normal ordered exponential ansatz. The original version of the MR ring CC doubles (MR-rCCD) equations [Á. Szabados and Á. Margócsy, Mol. Phys. 115, 2731 (2017)] is rectified in two aspects. On the one hand, over-completeness of double excitations is treated by relying on the concept of frames. On the other hand, restriction on the maximal cumulant rank is lifted from two to four. This is found essential for obtaining reliable correlation corrections to the energy. The MR function underlying the approach is provided by the Generalized Valence Bond (GVB) model. The pair structure of the reference ensures a fragment structure of GVB cumulants. This represents a benefit when evaluating cumulant contractions appearing as a consequence of MR-GWT. In particular, cumulant involving terms remain less expensive than their traditional, pair-contracted counterpart, facilitating an O(N6) eventual scaling of the proposed MR-rCCD method. Pilot applications are presented for covalent bond breaking, deprotonation energies, and torsional potentials.

A multireference coupled cluster theory based on quantum electrodynamics: Physical aspects of open-shell treatment

An electrodynamical coupled cluster (CC) formulation employing standard QED Hamiltonian that is written in Coulomb gauge and uses a multiconfiguration SCF picture of matter fields is discussed here. It employs a radiative cluster, matter clusters and their pair modifications, and considers a number state distribution of photons in thermal equilibrium. QED and electron correlation through CC are treated on the same footing. Thus an averaging over the radiation state is done to generate Lamb, Breit and hyperfine interactions from the radiative cluster. Because of the thermal distribution, it leaves a residual transverse interaction that may modify the static correlation. Dynamical correlation effects are determined next while using the exponential matter cluster in traditional ways of multireference CC. Deexcitations to negative-energy levels, included in matter clusters, generate vacuum polarization effects. The dynamical correlation energy includes relativistic corrections as well as QED contributions, namely, Lamb, Breit, hyperfine and pair energy.

Reference spaces for multireference coupled-cluster theory: the challenge of the CoH molecule

In multireference calculations on transition metal compounds, large active spaces including a second set of d orbitals are often used. However, with the increase in the system size and the complexity of the method, such calculations quickly become impractical. In this work, we looked for an inner valence active space for atomic cobalt and its hydride that correctly describes the transition energies among states with distinct configurations. Potential energy curves for the CoH molecule were obtained using a fixed reference space, where the orbitals are optimised only for the isolated atoms, avoiding undesirable sudden variations of the active orbitals. The calculations made use of the multireference configuration interaction (MRCI) and coupled-cluster (MRCC) theories. The accuracy of these calculations is carefully analysed, and we show that MRCC results are accurate, although not all electronic states are quantitatively described at the MRCI level. We conclude that the use of the fixed reference space is a viable approach to reduce the computational cost when large active spaces are prohibitive. Finally, a careful comparison of the present results with experimental values is carried out.

The MRCC program system: Accurate quantum chemistry from water to proteins.

MRCC is a package of ab initio and density functional quantum chemistry programs for accurate electronic structure calculations. The suite has efficient implementations of both low- and high-level correlation methods, such as second-order Møller-Plesset (MP2), random-phase approximation (RPA), second-order algebraic-diagrammatic construction [ADC(2)], coupled-cluster (CC), configuration interaction (CI), and related techniques. It has a state-of-the-art CC singles and doubles with perturbative triples [CCSD(T)] code, and its specialties, the arbitrary-order iterative and perturbative CC methods developed by automated programming tools, enable achieving convergence with regard to the level of correlation. The package also offers a collection of multi-reference CC and CI approaches. Efficient implementations of density functional theory (DFT) and more advanced combined DFT-wave function approaches are also available. Its other special features, the highly competitive linear-scaling local correlation schemes, allow for MP2, RPA, ADC(2), CCSD(T), and higher-order CC calculations for extended systems. Local correlation calculations can be considerably accelerated by multi-level approximations and DFT-embedding techniques, and an interface to molecular dynamics software is provided for quantum mechanics/molecular mechanics calculations. All components of MRCC support shared-memory parallelism, and multi-node parallelization is also available for various methods. For academic purposes, the package is available free of charge.

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.

The intermediate Hamiltonian Fock-space coupled-cluster method with approximate evaluation of the three-body effects.

The exponential parametrization of the wave function used in the coupled-cluster approaches has proven very successful in the ab initio description of atomic and molecular systems. This concerns first of all the single-reference version of the method that is designed for states dominated by a single Slater determinant. Usually, the coupled-cluster methods with one- and two-body excitation operators in the exponent form the basic computational schemes. The inclusion of three-body effects in the cluster operator to increase the accuracy of the results is numerically expensive, so their approximate evaluation is rather used in practice. In the case of the single-reference coupled-cluster approach, the problem of approximate evaluation of three-body effects in the cluster operator has been well studied, and computational schemes of both noniterative and iterative nature have been proposed. The situation is different in the case of multireference coupled-cluster methods which are required to describe open shell and quasidegenerate states. The multireference approaches in their standard effective Hamiltonian formulations are more complicated and less frequently used in routine calculations; however, one of them, the so-called Fock-space coupled-cluster method, becomes very effective if reformulated within the intermediate Hamiltonian framework. Both the basic version of the method with one- and two-body clusters and the extended one that includes up to three-body operators in the exponent are implemented. The latter approach provides more accurate results, but its relatively high numerical cost limits its applicability. For this reason, going beyond the basic scheme with one- and two-body clusters through an approximate evaluation of the impact of three-body clusters is of great interest. In the paper, we investigate different ways of approximate inclusion of the three-body effects in the Fock-space coupled-cluster method designated for excitation energy calculations.