Single-reference Coupled-cluster Methods Research Articles

Reasons Why Most Single-Reference Coupled Cluster Methods Fail to Provide the Correct Adiabatic Potentials of a Diatomic Carbon Molecule: UCCSDecCCSD Potential Study.

Many single-reference coupled cluster (CC) methods offer adiabatically incorrect potentials when calculating the diatomic carbon molecule, so this problem has been studied extensively. Analysis of the full configuration interaction (FCI) wave function indicates that the main cause of the adiabatic collapse of potentials calculated by the CC method with singles, doubles, and triples (CCSDT) is the strongly increasing bonding character of the T4FCI cluster contribution. In turn, comparative analysis of the CCSDTQ adiabats X1Σg+ and B'1Σg+ demonstrates that the gap between them near the avoided crossing geometry is significantly reduced by quantitatively differentiating the character of the T4 and cluster contributions. These observations clearly indicate the need to take into account the T4 cluster contribution in the standard CC wave function to obtain the correct adiabatic potential. Further analysis of this issue shows that the T4 contribution must be additionally bonding to ensure the adiabatic correctness of the potential. What also seems very interesting is that when the UCCSDecCCSD method [Toboła, Chem. Phys. Lett. 2014, 614, 82-88] is used for the potential calculation, the shape of the potential is entirely determined by a subset of unrestricted Hartree-Fock (UHF) configurations, which are structurally identical to the ground-state configuration in the UHF-based CCSD wave function.

S-Diagnostic─An a Posteriori Error Assessment for Single-Reference Coupled-Cluster Methods.

We propose a novel a posteriori error assessment for the single-reference coupled-cluster (SRCC) method called the S-diagnostic. We provide a derivation of the S-diagnostic that is rooted in the mathematical analysis of different SRCC variants. We numerically scrutinized the S-diagnostic, testing its performance for (1) geometry optimizations, (2) electronic correlation simulations of systems with varying numerical difficulty, and (3) the square-planar copper complexes [CuCl4]2-, [Cu(NH3)4]2+, and [Cu(H2O)4]2+. Throughout the numerical investigations, the S-diagnostic is compared to other SRCC diagnostic procedures, that is, the T1, D1, max T2, and D2 diagnostics as well as different indices of multideterminantal and multireference character in coupled-cluster theory. Our numerical investigations show that the S-diagnostic outperforms the T1, D1, max T2 and D2 diagnostics and is comparable to the indices of multideterminantal and multireference character in coupled-cluster theory in their individual fields of applicability. The experiments investigating the performance of the S-diagnostic for geometry optimizations using SRCC reveal that the S-diagnostic correlates well with different error measures at a high level of statistical relevance. The experiments investigating the performance of the S-diagnostic for electronic correlation simulations show that the S-diagnostic correctly predicts strong multireference regimes. The S-diagnostic, moreover, correctly detects the successful SRCC computations for [CuCl4]2-, [Cu(NH3)4]2+, and [Cu(H2O)4]2+, which have been known to be misdiagnosed by T1 and D1 diagnostics in the past. This shows that the S-diagnostic is a promising candidate for an a posteriori diagnostic for SRCC calculations.

Analysis of the single reference coupled cluster method for electronic structure calculations: the full-coupled cluster equations

The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian—a semi-unbounded, self-adjoint operator acting on an L^2-type Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high-accuracy quantum chemical simulations. The existing numerical analysis of coupled cluster methods relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.

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.

Coupled-Cluster theory revisited

In a series of two articles, we propose a comprehensive mathematical framework for Coupled-Cluster-type methods. In this second part, we analyze the nonlinear equations of the single-reference Coupled-Cluster method using topological degree theory. We establish existence results and qualitative information about the solutions of these equations that also sheds light of the numerically observed behavior. In particular, we compute the topological index of the zeros of the single-reference Coupled-Cluster mapping. For the truncated Coupled-Cluster method, we derive an energy error bound for approximate eigenstates of the Schrödinger equation.

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.

Ground-state potential energy functions and vibration-rotation energy levels of beryllium lithium and its cation.

The accurate ground-state potential energy functions of the beryllium lithium dimer, BeLi, and its cation, BeLi+ , have been determined from ab initio calculations using the multi-reference averaged coupled-pair functional (MR-ACPF) method in conjunction with the correlation-consistent core-valence basis sets up to septuple-zeta quality. The importance of the electron correlation beyond the MR-ACPF level of approximation, scalar relativistic, and adiabatic effects was discussed. For BeLi+ , the similar calculation was performed using the single-reference coupled-cluster method, up to the CCSDTQ level of approximation. The vibration-rotation energy levels of the two species were predicted to near the "spectroscopic" accuracy. Strangely enough, the predicted vibrational term values do not compare well with the recent experimental data for either BeLi or BeLi+ , with discrepancies reaching as much as 20-60 cm-1 for highly excited vibrational levels.

Variational coupled cluster for ground and excited states

In single-reference coupled-cluster (CC) methods, one has to solve a set of non-linear polynomial equations in order to determine the so-called amplitudes that are then used to compute the energy and other properties. Although it is of common practice to converge to the (lowest-energy) ground-state solution, it is also possible, thanks to tailored algorithms, to access higher-energy roots of these equations that may or may not correspond to genuine excited states. Here, we explore the structure of the energy landscape of variational CC and we compare it with its (projected) traditional version in the case where the excitation operator is restricted to paired double excitations (pCCD). By investigating two model systems (the symmetric stretching of the linear H4 molecule and the continuous deformation of the square H4 molecule into a rectangular arrangement) in the presence of weak and strong correlations, the performance of variational pCCD (VpCCD) and traditional pCCD is gauged against their configuration interaction (CI) equivalent, known as doubly occupied CI, for reference Slater determinants made of ground- or excited-state Hartree-Fock orbitals or state-specific orbitals optimized directly at the VpCCD level. The influence of spatial symmetry breaking is also investigated.

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.

Initial excited-state relaxation of locked retinal protonated schiff base chromophore. An insight from coupled cluster and multireference perturbation theory calculations.

The initial S1 excited-state relaxation of retinal protonated Schiff base (RPSB) analog with central C11C12 double bond locked by eight-membered ring (locked-11.8) was investigated by means of multireference perturbation theory methods (XMCQDPT2, XMS-CASPT2, MS-CASPT2) as well as single-reference coupled-cluster CC2 method. The analysis of XMCQDPT2-based geometries reveals rather weak coupling between in-plane and out-of-plane structural evolution and minor energetical relaxation of three locked-11.8 conformers. Therefore, a strong coupling between bonds length inversion and backbone out-of-plane deformation resulting in a very steep S1 energy profile predicted by CASSCF/CASPT2 calculations is in clear contradiction with the reference XMCQDPT2 results. Even though CC2 method predicts good quality ground-state structures, the excited-state structures display more advanced torsional deformation leading to ca. 0.2 eV exaggerated energy relaxation and significantly red shifted (0.4-0.7 eV) emission maxima. According to our findings, the initial photoisomerization process in locked-11.8, and possibly in other RPSB analogs, studied fully (both geometries and energies) by multireference perturbation theory may be somewhat slower than predicted by CASSCF/CASPT2 or CC2 methods. © 2018 Wiley Periodicals, Inc.

New ab initio adiabatic potential energy surfaces and bound state calculations for the singlet ground X̃(1)A1 and excited C̃(1)B2(2(1)A(')) states of SO2.

We report new and more accurate adiabatic potential energy surfaces (PESs) for the ground X̃(1)A1 and electronically excited C̃(1)B2(2(1)A(')) states of the SO2 molecule. Ab initio points are calculated using the explicitly correlated internally contracted multi-reference configuration interaction (icMRCI-F12) method. A second less accurate PES for the ground X̃ state is also calculated using an explicitly correlated single-reference coupled-cluster method with single, double, and non-iterative triple excitations [CCSD(T)-F12]. With these new three-dimensional PESs, we determine energies of the vibrational bound states and compare these values to existing literature data and experiment.

Benchmark Applications of Variations of Multireference Equation of Motion Coupled-Cluster Theory.

In this work, several variations of the multireference equation of motion (MR-EOM) methodology are investigated for the calculation of excitation spectra. These variants of MR-EOM are characterized by the following aspects: (1) the operators included in the sequence of similarity transformations of the molecular electronic Hamiltonian, (2) whether permutational symmetries (i.e., hermitization, vertex symmetry) are imposed on the final elements of the similarity-transformed Hamiltonian, (3) the size of the manifold over which the similarity-transformed Hamiltonian is diagonalized, (4) whether the two-body cumulant is included in the expressions defining the amplitudes and the elements of the transformed Hamiltonian. The MR-EOM methods are benchmarked for the calculation of the excitation energies of a test set of organic molecules. With the availability of reliable benchmark data for this test set, it is possible to gauge the relative accuracy of these approaches. We also further examine a subset of the MR-EOM methods for the calculation of the excitation energies of some transition-metal complexes. These systems prove to be particularly difficult for single-reference coupled-cluster methods.

Many interacting fermions in a one-dimensional harmonic trap: a quantum-chemical treatment

We employ ab initio methods of quantum chemistry to investigate spin-1/2 fermions interacting via a two-body contact potential in a one-dimensional harmonic trap. The convergence of the total energy with the size of the one-particle basis set is analytically investigated for the two-body problem and the same form of the convergence formula is numerically confirmed to be valid for the many-body case. Benchmark calculations for two to six fermions with the full configuration interaction method equivalent to the exact diagonalization approach, and the coupled cluster (CC) method including single, double, triple, and quadruple excitations are presented. The convergence of the correlation energy with the level of excitations included in the CC model is analyzed. The range of the interaction strength for which single-reference CC methods work is examined. Next, the CC method restricted to single, double, and noniterative triple excitations, CCSD(T), is employed to study a two-component Fermi gas composed of 6–80 atoms in a one-dimensional harmonic trap. The density profiles of trapped atomic clouds are also reported. Finally, a comparison with experimental results for few-fermion systems is presented. Upcoming possible applications and extensions of the presented approach are discussed.

Profiling the binding motif between Be and Mg in the ground state via a single-reference coupled cluster method

We present a study on the performance of our iterative triples correction for the coupled cluster singles and doubles excitations (CCSDT-1a+d) method for computation of potential energy surface (PES), spectroscopic constants, and vibrational spectrum for the ground state (X1Σ+) BeMg, where the ostensible inadequacy of the CCSD and CCSD(T) methods is quite expected. We compare our results with those obtained using state-of-the-art multireference configuration interaction (MRCI) investigations reported earlier by Kerkines and Nicolaides. Our estimated dissociation energy (417.37 cm−1), equilibrium distance (3.285 Å), and vibrational frequency (82.32 cm−1) are in good agreement with recent results of advanced MRCI calculations for X1Σ+ BeMg PES, which exhibits a shallow well of 469.4 cm−1 with a minimum at 3.241 Å and a harmonic vibrational frequency of 85.7 cm−1. Very weakly bound nature of X1Σ+ BeMg is clearly reflected from these values. In accord with MRCI studies, a comparison of BeMg with iso-valence weakly bound ground-state species, Be2 and Mg2, suggests that its characteristics do not exhibit any resemblance to Be2 rather, it shows a close kinship to Mg2. The agreement of our derived vibrational levels with those obtained via the high-level MRCI calculations is very encouraging reflecting the potential of the suitably modified single-reference coupled cluster (SRCC) method, CCSDT-1a+d as a tool for the study of multireference van der Waals systems.

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.

An ab initio study of the electronic structure of the boron oxide neutral (BO), cationic (BO⁺), and anionic (BO⁻) species.

The BO neutral, cationic, and anionic molecular species have been painstakingly studied through multireference configuration interaction and single reference coupled cluster methods employing basis sets of quintuple cardinality. Potential energy curves have been constructed for 38 (BO), 37 (BO(+)), and 12 (BO(-)) states and the usual molecular parameters have been extracted most of which are in very good agreement with the scarce experimental data. Numerous avoided crossings appear on more or less all of the studied states of the neutral and cationic species challenging the validity of the Born Oppenheimer approximation. Finally, all excited states of the anionic system lie above the ground state of the neutral BO system and are therefore resonances.

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.

Performance of the EOMIP-CCSD(2) Method for Determining the Structure and Properties of Doublet Radicals: A Benchmark Investigation.

We present a benchmark study on the performance of the EOMIP-CCSD(2) method for computation of structure and properties of doublet radicals. The EOMIP-CCSD(2) method is a second-order approximation to the standard EOMIP-CCSD method. By retaining the black box nature of the standard EOMIP-CCSD method and adding favorable N(5) scaling, the EOMIP-CCSD(2) method can become the method of choice for predicting the structure and spectroscopic properties of large doublet radicals. The EOMIP-CCSD(2) method overcomes the typical problems associated with the standard single reference ab initio treatment of doublet radicals. We compare our results for geometries and harmonic vibrational frequencies with those obtained using the standard EOMIP-CCSD method, as well as unrestricted Hartree-Fock (UHF)- and restricted open-shell Hartree-Fock (ROHF)-based single-reference coupled-cluster and second order many-body perturbation theory (MBPT(2)) methods. The effect of the basis set on the quality of the results has been studied using a hierarchy of Dunning's correlation-consistent aug-cc-pVXZ (X = D, T, Q) basis sets. Numerical results show that the EOMIP-CCSD(2) method, despite its N(5) scaling, gives better agreement with experimental results, compared to the UHF- and ROHF-based MBPT(2), as well as the single-reference coupled-cluster methods.

An ab initio study of the electronic structure of BF and BF+

The BF and BF(+) molecular systems have been thoroughly studied by multireference variational and single reference coupled-cluster methods employing basis sets of sextuple cardinality. Potential energy curves have been constructed for 33 (BF) and 8 (BF(+)) states and the usual molecular parameters have been extracted most of which are in excellent agreement with the available experimental data. A characteristic feature of the BF system is the intense valence-Rydberg interaction in most of the studied states. Its X-state is of (1)Σ(+) symmetry featuring a triple bond of roughly 182 kcal∕mol at re = 1.2631 Å. It is interesting indeed that for the last 50 years it was believed that the charge polarity of BF is B(-)F(+). This is the first time that according to the present high level calculations, it is shown that the polarity of BF is the "normal" one B(+)F(-), with a total charge transfer from B to F of about 0.2 electrons. Removing a rather inactive σ electron we get the X(2)Σ(+) BF(+) state sharing similar bonding features as the X(1)Σ(+) one of the parental molecule.

Relative Energies and Geometries of the cis- and trans-HO3 Radicals from the Parametric 2-Electron Density Matrix Method

The parametric 2-electron reduced density matrix (2-RDM) method employing the M functional [Mazziotti, D. A. Phys. Rev. Lett. 2008, 101, 253002], also known as the 2-RDM(M) method, improves on the accuracy of coupled electron-pair theories including coupled cluster with single-double excitations at the computational cost of configuration interaction with single-double excitations. The cis- and trans-HO(3) isomers along with their isomerization transition state were examined using the recent extension of 2-RDM(M) to nonsinglet open-shell states [Schwerdtfeger, C. A.; Mazziotti, D. A. J. Chem. Phys. 2012, 137, 034107] and several coupled cluster methods. We report the calculated energies, geometries, natural-orbital occupation numbers, and reaction barriers for the HO(3) isomers. We find that the 2-RDM(M) method predicts that the trans isomer of HO(3) is lower in energy than the cis isomer by 1.71 kcal/mol in the correlation-consistent polarized valence quadruple-ζ (cc-pVQZ) basis set and 1.84 kcal/mol in the augmented correlation-consistent polarized valence quadruple-ζ (aug-cc-pVQZ) basis set. Results include the harmonic zero-point vibrational energies calculated in the correlation-consistent polarized valence double-ζ basis set. On the basis of the results of a geometry optimization in the augmented correlation consistent polarized valence triple-ζ basis set, the parametric 2-RDM(M) method predicts a central oxygen-oxygen bond of 1.6187 Å. We compare these energies and geometries to those predicted by three single-reference coupled cluster methods and experimental results and find that the inclusion of multireference correlation is important to describe properly the relative energies of the cis- and trans-HO(3) isomers and improve agreement with experimental geometries.