Multireference Fock-space coupled-cluster and equation-of-motion coupled-cluster theories: The detailed interconnections

Equation-of-motion orbital-optimized coupled-cluster doubles method with the density-fitting approximation: An efficient implementation

Orbital-optimized coupled-cluster methods are very helpful for theoretical predictions of the molecular properties of challenging chemical systems, such as excited states. In this research, an efficient implementation of the equation-of-motion orbital-optimized coupled-cluster doubles method with the density-fitting (DF) approach, denoted by DF-EOM-OCCD, is presented. The computational cost of the DF-EOM-OCCD method for excitation energies is compared with that of the conventional EOM-OCCD method. Our results demonstrate that DF-EOM-OCCD excitation energies are dramatically accelerated compared to EOM-OCCD. There are almost 17-fold reductions for theC 5 H 12 molecule in an aug-cc-pVTZ basis set with the RHF reference. This dramatic performance improvement comes from the reduced cost of integral transformation with the DF approach and the efficient evaluation of the particle-particle ladder (PPL) term, which is the most expensive term to evaluate. Further, our results show that the DF-EOM-OCCD approach is very helpful for the computation of excitation energies in open-shell molecular systems. Overall, we conclude that our new DF-EOM-OCCD implementation is very promising for the study of excited states in large-sized challenging chemical systems.

"Best" Iterative Coupled-Cluster Triples Model? More Evidence for 3CC

To follow up on the unexpectedly good performance of several coupled-cluster models with approximate inclusion of 3-body clusters [Rishi, V.; Valeev, E. F. J. Chem. Phys. 2019, 151, 064102.] we performed a more complete assessment of the 3CC method [Feller, D. . J. Chem. Phys. 2008, 129, 204105.] for accurate computational thermochemistry in the standard HEAT framework. New spin-integrated implementation of the 3CC method applicable to closed- and open-shell systems utilizes a new automated toolchain for derivation, optimization, and evaluation of operator algebra in many-body electronic structure. We found that with a double-ζ basis set the 3CC correlation energies and their atomization energy contributions are almost always more accurate (with respect to the CCSDTQ reference) than the CCSDT model as well as the standard CCSD(T) model. The mean absolute errors in cc-pVDZ {3CC, CCSDT, and CCSD(T)} electronic (per valence electron) and atomization energies relative to the CCSDTQ reference for the HEAT data set [Tajti, A. . J. Chem. Phys. 2004, 121, 11599-11613.], were {24, 70, 122} μEh/e and {0.46, 2.00, 2.58} kJ/mol, respectively. The mean absolute errors in the complete-basis-set limit {3CC, CCSDT, and CCSD(T)} atomization energies relative to the HEAT model reference, were {0.52, 2.00, and 1.07} kJ/mol, The significant and systematic reduction of the error by the 3CC method and its lower cost than CCSDT suggests it as a viable candidate for post-CCSD(T) thermochemistry applications, as well as the preferred alternative to CCSDT in general.

Multireference Fock Space Coupled-Cluster Method for the (3,0) Sector

This work reports an implementation of a novel realization of the multireference coupled cluster theory formulated in Fock space. Extending the previous formulation carried out in the (1,1) [M. Musial, R. J. Bartlett, J. Chem. Phys. 129, 044101 (2008)], (0,2) [M. Musial, R. J. Bartlett, J. Chem. Phys. 135, 044121 (2011)], and (2,0) [M. Musial, J. Chem. Phys. 136, 134111 (2012)] sectors to the (3,0) sector, we are able to treat structures with three valence electrons. The (3,0) sector describes systems with three electrons added to the reference, which means that in order to perform correlated calculations for the neutral AB molecule, we have to adopt as the reference a triply ionized structure AB3+. A desirable situation occurs when such an ion has a closed-shell structure and also dissociates into closed shell fragments. This feature makes it possible to apply the restricted Hartree-Fock scheme for the whole range of interatomic distances. Examples of molecules of this type are the diatomics formed by the atoms of alkali metals and alkaline earth metals. An analogous structure is also exhibited by alkali metal monocarbides. In the current work, we have calculated the potential energy curves and spectroscopic constants for the LiBe, LiC, and NaC molecules.

Triple Electron Attachments with a New Intermediate-Hamiltonian Fock-Space Coupled-Cluster Method

The implementation of a new intermediate-Hamiltonian Fock-space coupled-cluster (IHFSCC) scheme for the (3,0) sector of the Fock space is reported. In this IHFSCC approach, the three-body contributions in the cluster operator S(3,0) corresponding to the (3,0) sector of the Fock space are considered, while S(1,0) and S(2,0) at the (1,0) and (2,0) level only include one- and two-body operators. By introducing a suitable partition of the wave operator, the intermediate Hamiltonian, which only depends on the two-body operator of S(1,0), is obtained. S(2,0) and S(3,0) are not required within this new IHFSCC scheme, and a large reference space can be possibly employed. The performance of this (3,0) IHFSCC method in calculating triple ionization potentials and excitation energies for atoms and cations with a ground p3 configuration as well as adiabatic excitation energies for some molecules is investigated. The effect of the number of active virtual orbitals and three different types of orbitals, i.e., reference orbitals, restricted open-shell Hartree-Fock orbitals (ROHF) of the target systems, and canonicalized ROHF orbitals, on IHFSCC results, is also studied. Our calculations indicate that reasonable results can be achieved with this (3,0) IHFSCC method when a minimal reference space is employed. Further increasing the number of active orbitals does not necessarily improve the results. In addition, the IHFSCC results using canonicalized ROHF orbitals generally agree well with reference values, and they are not very sensitive to the number of active orbitals compared with results using the reference orbitals. The new (3,0) IHFSCC method can be applied to open-shell systems with three unpaired electrons with reasonable accuracy at a relatively low computational cost.

Fock Space Coupled-Cluster Method for the Ground and Excited States of the NaMg+ Molecular Cation

This study presents a theoretical investigation of the potential energy curves (PECs) and spectroscopic constants of the NaMg+ molecular cation using the intermediate Hamiltonian Fock space coupled-cluster method with singles and doubles within the (2,0) sector [IH-FS-CCSD(2,0)]. The NaMg+ cation has gained scientific interest due to its potential applications in ultracold chemistry, yet it remains experimentally unexplored. We computed 20 lowest-lying PECs and assigned them to 10 dissociation limits. Our results demonstrated high accuracy, and computed curves are smooth and continuous over the entire range of interatomic distances. The study validates the effectiveness of the IH-FS-CCSD(2,0) method in describing PECs of diatomic molecular cations composed of alkali and alkaline earth metals and provides a solid theoretical foundation for further studies of NaMg+ and similar systems.

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.

Benchmark Study of the Electronic States of the LiRb Molecule: Ab Initio Calculations with the Fock Space Coupled Cluster Approach

Accurate potential energy curves (PECs) are determined for the twenty-two electronic states of LiRb. In contrast to previous studies, the applied approach relies on the first principle calculations involving correlation among all electrons. The current methodology is founded on the multireference coupled cluster (CC) scheme constructed within the Fock space (FS) formalism, specifically for the (2,0) sector. The FS methodology is established within the framework of the intermediate Hamiltonian formalism and offers an intruder-free, efficient computational scheme. This method has a distinctive feature that, when applied to the doubly ionized system, provides the characteristics of the neutral case. This proves especially beneficial when investigating PECs in situations where a closed-shell molecule dissociates into open-shell fragments, yet its double positive ion forms closed-shell species. In every instance, we successfully computed continuous PECs spanning the entire range of interatomic distances, from the equilibrium to the dissociation limit. Moreover, the spectroscopic characteristic of various electronic states is presented, including relativistic effects. Relativistic corrections included at the third-order Douglas-Kroll level have a non-negligible effect on the accuracy of the determined spectroscopic constants.

Intermediate Hamiltonian Fock-space coupled-cluster theory for excitation energies, double ionization potentials, and double electron attachments with spin–orbit coupling

The intermediate Hamiltonian Fock-space coupled-cluster methods at the singles and doubles level (IHFSCCSD) for excitation energies in the (1p, 1h) sector, double ionization potentials in the (0p, 2h) sector, and double electron attachments in the (2p, 0h) sector of the Fock space are implemented based on the CCSD method with spin–orbit coupling (SOC) included in the post-Hartree–Fock treatment using a closed-shell reference in this work. The active space is chosen to contain those orbitals that have the largest contribution to principal ionized or electron-attached states obtained from the equation-of-motion coupled-cluster calculations. Both time-reversal symmetry and spatial symmetry are exploited in the implementation. Our results show that the accuracy of IHFSCCSD results is closely related to the active space, and the sufficiency of the active space can be assessed from the percentage of transitions within the active space. In addition, unreasonable results may be encountered when the ionized or electron-attached states with a somewhat larger contribution from double excitations are included to determine the active space and cluster operators in the (0p, 1h) or (1p, 0h) sector of the Fock space. A larger active space may be required to describe SO splitting reliably than that in the scalar-relativistic calculations in some cases. The IHFSCCSD method with SOC developed in this work can provide reliable results for heavy-element systems when a sufficient active space built upon the principal ionization potential/electron affinity states is adopted.

Efficient Implementation of Equation-of-Motion Coupled-Cluster Singles and Doubles Method with the Density-Fitting Approximation: An Enhanced Algorithm for the Particle-Particle Ladder Term

An efficient implementation of the density-fitted equation-of-motion coupled-cluster singles and doubles (DF-EOM-CCSD) method is presented with an enhanced algorithm for the particle-particle ladder (PPL) term, which is the most expensive part of EOM-CCSD computations. To further improve the evaluation of the PPL term, a hybrid density-fitting/Cholesky decomposition (DF/CD) algorithm is also introduced. In the hybrid DF/CD approach, four virtual index integrals are constructed on-the-fly from the DF factors; then, their partial Cholesky decomposition is simultaneously performed. The computational cost of the DF-EOM-CCSD method for excitation energies is compared with that of the resolution of the identity EOM-CCSD (RI-EOM-CCSD) (from the Q-chem 5.3 package). Our results demonstrate that DF-EOM-CCSD excitation energies are significantly accelerated compared to RI-EOM-CCSD. There is more than a 2-fold reduction for the C₈H₁₈ molecule in the cc-pVTZ basis set with the restricted Hartree-Fock (RHF) reference. This cost savings results from the efficient evaluation of the PPL term. In the RHF based DF-EOM-CCSD method, the number of flops (NOF) is 1/4O²V⁴, while that of RI-EOM-CCSD was reported (Epifanovsky et al. J. Chem. Phys. 2013, 139, 134105) to be 5/8O²V⁴ for the PPL contraction term. Further, the NOF of VVVV-type integral transformation is 1/2V⁴N_aux in our case, while it appears to be V⁴N_aux for RI-EOM-CCSD. Hence, our implementation is 2.5 and 2.0 times more efficient compared to RI-EOM-CCSD for these expensive terms. For the unrestricted Hartree-Fock (UHF) reference, our implementation maintains its enhanced performance and provides a 1.8-fold reduction in the computational time compared to RI-EOM-CCSD for the C₇H₁₆ molecule. Our results indicate that our DF-EOM-CCSD implementation is 1.7 and 1.4 times more efficient compared with RI-EOM-CCSD for average computational cost per EOM-CCSD iteration. Moreover, our results show that the new hybrid DF/CD approach improves upon the DF algorithm, especially for large molecular systems. Overall, we conclude that the new hybrid DF/CD PPL algorithm is very promising for large-sized chemical systems.

An efficient Fock space multi-reference coupled cluster method based on natural orbitals: Theory, implementation, and benchmark

We present a natural orbital-based implementation of the intermediate Hamiltonian Fock space coupled-cluster method for the (1, 1) sector of Fock space. The use of natural orbitals significantly reduces the computational cost and can automatically choose an appropriate set of active orbitals. The new method retains the charge transfer separability of the original intermediate Hamiltonian Fock space coupled-cluster method and gives excellent performance for valence, Rydberg, and charge-transfer excited states. It offers significant computational advantages over the popular equation of motion coupled cluster method for excited states dominated by single excitations.

A Fock space coupled cluster based probing of the single- and double-ionization profiles for the poly-cyclic aromatic hydrocarbons and conjugated polyenes

Sequential formation of a poly-cyclic aromatic hydrocarbon (PAH) dication in the H I regions of the interstellar medium (ISM) is proposed to be a function of internal energy of the doubly ionized PAHs, which, in turn, is dependent on the single- and double-ionization potentials of the system. This sets a limit on the single- and double-ionization energies of the system(s) that can further undergo sequential absorption of two photons, leading to a dication (PAH⁺²). Here, we report the single-ionization (I⁺¹) and double-ionization (I⁺²) energies and the I⁺²/I⁺¹ ratio for some selected PAHs and conjugated polyenes obtained using the Fock space coupled cluster technique, enabling simultaneous consideration of several electronic states of different characters. The I⁺² to I⁺¹ ratio bears a constant ratio, giving allowance to determine I⁺² from the knowledge of single-ionization (I⁺¹) and vice versa. Our observations are in good agreement with the established literature findings, confirming the reliability of our estimates. The measured single- and double-ionization energies further demonstrate that the sequential formation and fragmentation of a PAH dication in the H I regions of the ISM for systems such as benzene and conjugated polyenes such as ethylene and butadiene are quite unlikely because I⁺²-I⁺¹ for such system(s) is higher than the available photon energy in the H I regions of the ISM. Present findings may be useful to understand the formation and underlying decay mechanisms of multiply charged ions from PAHs and related compounds that may accentuate the exploration of the phenomenon of high-temperature superconductivity.

Electronic states of NaLi molecule: Benchmark results with Fock space coupled cluster approach

Accurate potential energy curves (PECs) are obtained for 20 lowest lying electronic states of the NaLi molecule. The computational scheme used here is based on the multireference coupled cluster theory formulated in the (2,0) sector of the Fock space. The latter sector provides the description of states obtained by attachment of two electrons to the reference system. This makes it possible to adopt the doubly ionized NaLi⁺² molecule as a Fermi vacuum. The latter has a very concrete advantage in calculations of the PECs since it dissociates into closed shell fragments (NaLi⁺² → Na⁺ + Li⁺); hence, the restricted Hartree-Fock method can be used within the whole range of interatomic distances. Computed PECs and spectroscopic constants stay very close to the experimental values (if the latter are available) with the accuracy exceeding the other theoretical approaches including those based on the effective core polarization potentials. Relativistic corrections included at the infinite-order two-component level have a non-negligible effect on the accuracy of computed excitation and dissociation energies with contributions up to 50 cm^-1.

Resource-Efficient Chemistry on Quantum Computers with the Variational Quantum Eigensolver and the Double Unitary Coupled-Cluster Approach

Applications of quantum simulation algorithms to obtain electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices require careful consideration of resources describing the complex electron correlation effects. In modeling second-quantized problems, the biggest challenge confronted is that the number of qubits scales linearly with the size of the molecular basis. This poses a significant limitation on the size of the basis sets and the number of correlated electrons included in quantum simulations of chemical processes. To address this issue and enable more realistic simulations on NISQ computers, we employ the double unitary coupled-cluster (DUCC) method to effectively downfold correlation effects into the reduced-size orbital space, commonly referred to as the active space. Using downfolding techniques, we demonstrate that properly constructed effective Hamiltonians can capture the effect of the whole orbital space in small-size active spaces. Combining the downfolding preprocessing technique with the variational quantum eigensolver, we solve for the ground-state energy of H₂, Li₂, and BeH₂ in the cc-pVTZ basis using the DUCC-reduced active spaces. We compare these results to full configuration-interaction and high-level coupled-cluster reference calculations.

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 confinement induced spectroscopic study of noble gas atoms using equation of motion architecture: Encapsulation within fullerene's voids

A relativistic study of spectroscopic properties of the endohedral fullerenes Ng@C₆₀^q (where Ng = He, Ne and q=0,±1,±2 are the charges) associated with the C₆₀ molecule has been done using the equation of motion coupled cluster (EOM-CC) methodology. Specific properties estimated are the transition energies, dipole oscillator strengths, and transition probabilities for the low-lying excitations 1s²(¹S₀) → 1snp (¹P₁) (n = 2, 3, 4) for He@C₆₀^q and 1s²2s²2p⁶ (¹S₀) → 1s²2s²2p⁵ns∕nd (¹P₁) (n = 3, 4) for Ne@C₆₀^q, which have been compared with those for the isolated atom to depict the confinement effect of the host molecule on the encapsulated atom. This is accomplished by introducing an effective potential to the atomic Hamiltonian induced by the fullerene moiety and its charge. The EOM-CC results have been compared with those estimated with the random phase approximation (and configuration interaction singles) to understand the effect of electron correlation under such confinement. The systematic and interesting behavior of the properties is highlighted indicating the effect of fullerene cage potential on the redistribution of electron density of the guest atom.

Electron Detachment and Subsequent Structural Changes of Water Clusters

A cost-effective equation of motion coupled cluster method, EOMIP-CCSD(2), is used to investigate vertical and adiabatic ionization potential as well as ionization-induced structural changes of water clusters and compared with CCSD(T), CASPT2, and MP2 methods. The moderate N(5) scaling and low storage requirement yields EOMIP-CCSD(2) calculation feasible even for reasonably large molecules and clusters with accuracy comparable to CCSD(T) method at much cheaper computational cost. Our calculations shed light on the authenticity of EOMIP-CCSD(2) results and establish a reliable method to study of ionization energy of molecular clusters. We have further investigated the performance of several classes of DFT functionals for ionization energies of water clusters to benchmark the results and to get a reliable functionals for the same.

Structure, stability, and properties of the trans peroxo nitrate radical: the importance of nondynamic correlation

We report a comparative single-reference and multireference coupled-cluster investigation on the structure, potential energy surface, and IR spectroscopic properties of the trans peroxo nitrate radical, one of the key intermediates in stratospheric NOX chemistry. The previous single-reference ab initio studies predicted an unbound structure for the trans peroxo nitrate radical. However, our Fock space multireference coupled-cluster calculation confirms a bound structure for the trans peroxo nitrate radical, in accordance with the experimental results reported earlier. Further, the analysis of the potential energy surface in FSMRCC method indicates a well-behaved minima, contrary to the shallow minima predicted by the single-reference coupled-cluster method. The harmonic force field analysis, of various possible isomers of peroxo nitrate also reveals that only the trans structure leads to the experimentally observed IR peak at 1840 cm(-1). The present study highlights the critical importance of nondynamic correlation in predicting the structure and properties of high-energy stratospheric NOx radicals.