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.

Further investigations into a Laplace MP2 method using range separated Coulomb potential and orbital selective virtuals: Multipole correction, OSV extrapolation, and critical assessment.

We report further investigations to aid the development of a Laplace MP2 (second-order Møller Plesset) method with a range separated Coulomb potential partitioned into short- and long-range parts. The implementation of the method extensively uses sparse matrix algebra, density fitting techniques for the short-range part, and a Fourier transformation in spherical coordinates for the long-range part of the potential. Localized molecular orbitals are employed for the occupied space, whereas virtual space is described by orbital specific virtual orbitals (OSVs) associated with localized molecular orbitals. The Fourier transform is deficient for very large distances between localized occupied orbitals, and a multipole expansion for widely separated pairs is introduced for the direct MP2 contribution, which is applicable also to non-Coulombic potentials that do not satisfy the Laplace equation. For the exchange contribution, an efficient screening of contributing localized occupied pairs is employed, which is discussed more completely here. To mitigate errors due to the truncation of OSVs, a simple and efficient extrapolation procedure is used to obtain results close to MP2 for the full basis set of atomic orbitals Using a suitable set of default parameters, the accuracy of the approach is demonstrated. The current implementation of the approach is not very efficient, and the aim of this paper is to introduce and critically discuss ideas that can have more general applicability beyond MP2 calculations for large molecules.

Toward Laplace MP2 method using range separated Coulomb potential and orbital selective virtuals.

We report the development of a new Laplace MP2 (second-order Møller-Plesset) implementation using a range separated Coulomb potential, partitioned into short- and long-range parts. The implementation heavily relies on the use of sparse matrix algebra, density fitting techniques for the short-range Coulomb interactions, while a Fourier transformation in spherical coordinates is used for the long-range part of the potential. Localized molecular orbitals are employed for the occupied space, whereas orbital specific virtual orbitals associated with localized molecular orbitals are obtained from the exchange matrix associated with specific localized occupied orbitals. The range separated potential is crucial to achieve efficient treatment of the direct term in the MP2, while extensive screening is employed to reduce the expense of the exchange contribution in MP2. The focus of this paper is on controllable accuracy and linear scaling of the data entering the algorithm.

Toward the efficient local tailored coupled cluster approximation and the peculiar case of oxo-Mn(Salen).

We introduce a new implementation of the coupled cluster method with single and double excitations tailored by the matrix product state wave functions (DMRG-TCCSD), which employs the local pair natural orbital (LPNO) approach. By exploiting locality in the coupled cluster stage of the calculation, we were able to remove some of the limitations that hindered the application of the canonical version of the method to larger systems and/or with larger basis sets. We assessed the accuracy of the approximation using two systems: tetramethyleneethane (TME) and oxo-Mn(Salen). Using the default cut-off parameters, we were able to recover over 99.7% and 99.8% of the canonical correlation energy for the triplet and singlet state of TME, respectively. In the case of oxo-Mn(Salen), we found that the amount of retrieved canonical correlation energy depends on the size of the complete active space (CAS)-we retrieved over 99.6% for the larger 27 orbital CAS and over 99.8% for the smaller 22 orbital CAS. The use of LPNO-TCCSD allowed us to perform these calculations up to quadruple-ζ basis set, amounting to 1178 basis functions. Moreover, we examined dependence of the ground state of oxo-Mn(Salen) on the CAS composition. We found that the inclusion of 4dxy orbital plays an important role in stabilizing the singlet state at the DMRG-CASSCF level via double-shell effect. However, by including dynamic correlation, the ground state was found to be triplet regardless of the size of the basis set or the composition of CAS, which is in agreement with previous findings by canonical DMRG-TCCSD in smaller basis.

Perturbative triples correction to domain-based local pair natural orbital variants of Mukherjee's state specific coupled cluster method.

In this article we report an implementation of the perturbative triples correction to Mukherjee's state-specific multireference coupled cluster method based on the domain-based pair natural orbital approach (DLPNO-MkCC). We tested the performance of DLPNO-MkCCSD(T) in calculations involving tetramethyleneethane and isomers of naphthynes. These tests show that more than 97% of triples energy was recovered with respect to the canonical MkCCSD(T) method, which together with the DLPNO-MkCCSD part accounts for about 99.70-99.85% of the total correlation energy. The applicability of the method was demonstrated on calculations of singlet-triplet gaps for several large systems: triangulene, dynemicin A, and a beryllium complex.

Domain-Based Local Pair Natural Orbital Version of Mukherjee's State-Specific Coupled Cluster Method.

This article reports development of a local variant of Mukherjee's state-specific multireference coupled cluster method based on the domain-based pair natural orbital approach (DLPNO-MkCC). The current implementation is restricted to connected single and double excitations and model space with up to biexcited references. The performance of the DLPNO-MkCCSD was tested on calculations of tetramethyleneethane. The results show that above 99.9% of the correlation energy was recovered, with respect to the conventional MkCC method. To demonstrate the applicability of the method to large systems, singlet-triplet gaps of triangulene and bis(1-(2,6-diisopropylphenyl)-3,3,5,5-tetramethylpyrrolidine-2-ylidene)beryllium complex were studied. For the last system (105 atoms), we were able to perform a calculation in cc-pVTZ with 2158 basis functions on a single CPU in less than 9 days.

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.

A Local Pair Natural Orbital-Based Multireference Mukherjee's Coupled Cluster Method.

This paper reports the development of a local variant of Mukherjee's state-specific multireference coupled cluster method based on the pair natural orbital approach (LPNO-MkCC). The current implementation is restricted to single and double excitations. The performance of the LPNO-MkCCSD method was tested on calculations of naphthyne isomers, tetramethyleneethane, and ß-carotene molecules. The results show that 99.7-99.8% of correlation energy was recovered with respect to the MkCC method based on canonical orbitals. Moreover, the errors of relative energies between different isomers or along a potential energy curve (with respect to the canonical method) are below 0.4 kcal/mol, safely within the chemical accuracy. The computational efficiency of our implementation of LPNO-MkCCSD in the ORCA program allows calculation of the ß-carotene molecule (96 atoms and 1984 basis functions) on a single CPU core.

Communication: multireference equation of motion coupled cluster: a transform and diagonalize approach to electronic structure.

The novel multireference equation-of-motion coupled-cluster (MREOM-CC) approaches provide versatile and accurate access to a large number of electronic states. The methods proceed by a sequence of many-body similarity transformations and a subsequent diagonalization of the transformed Hamiltonian over a compact subspace. The transformed Hamiltonian is a connected entity and preserves spin- and spatial symmetry properties of the original Hamiltonian, but is no longer Hermitean. The final diagonalization spaces are defined in terms of a complete active space (CAS) and limited excitations (1h, 1p, 2h, …) out of the CAS. The methods are invariant to rotations of orbitals within their respective subspaces (inactive, active, external). Applications to first row transition metal atoms (Cr, Mn, and Fe) are presented yielding results for up to 524 electronic states (for Cr) with an rms error compared to experiment of about 0.05 eV. The accuracy of the MREOM family of methods is closely related to its favorable extensivity properties as illustrated by calculations on the O2-O2 dimer. The computational costs of the transformation steps in MREOM are comparable to those of closed-shell Coupled Cluster Singles and Doubles (CCSD) approach.

Additional global internal contraction in variations of multireference equation of motion coupled cluster theory.

Extensions of multireference equation of motion coupled cluster theory (MR-EOMCC) [D. Datta and M. Nooijen, J. Chem. Phys. 137, 204107 (2012)] are presented that include additional correlation effects into the global, internally contracted similarity transformation, induced by the cluster operators. As a result the final uncontracted diagonalization space can be more compact than in the parent MR-EOMCC approach. A wide range of applications, including transition metal atomic excitation spectra, a large set of valence excited states of organic compounds, and potential energy surfaces of ground and excited states of butadiene, is presented to benchmark the applicability of the parent MR-EOMCC methodology and its new variations.

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

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

Multireference state-specific Mukherjee's coupled cluster method with noniterative triexcitations using uncoupled approximation.

A new version of the multireference Mukherjee's coupled cluster method with perturbative triexcitations has been formulated, which is based on the uncoupled approximation applied to the triples equation. In contrast to the method developed by Evangelista et al. [J. Chem. Phys. 132, 074107 (2010)], the proposed approach does not require to solve the equation for T(3) amplitudes iteratively, yet yields results of essentially the same quality. The method, abbreviated as MR MkCCSD(Tu), has been implemented in the ACES II program package and its assessment has been performed on the BeH(2) model and on the tetramethyleneethane molecule.

Uncoupled multireference state-specific Mukherjee's coupled cluster method with triexcitations.

We have developed the uncoupled version of multireference Mukherjee's coupled cluster method with connected triexcitations. The method has been implemented in ACES II program package. The agreement between the uncoupled and the standard version of Mukherjee's multireference coupled cluster method has been reported previously at the singles and doubles level by Das et al. [J. Mol. Struct.: THEOCHEM 79, 771 (2006); Chem. Phys. 349, 115 (2008)]. The aim of this article is to investigate this method further, in order to establish how its performance changes with the size of the basis set, size of the model space, multireference character of different molecules, and inclusion of connected triple excitations. Assessment of the new method has been performed on the singlet methylene, potential energy curve of fluorine molecule, and third b (1)Σ(g)(+) electronic state of oxygen molecule.

Multireference Mukherjee's coupled cluster method with triexcitations in the linked formulation: Efficient implementation and applications.

We have formulated the multireference Mukherjee's coupled clusters method with triexcitations (MR MkCCSDT) in the linked version and implemented it in the ACES II program package. The assessment of the new method has been performed on the first three electronic states of the oxygen molecule, on studies of singlet-triplet gap in methylene and twisted ethylene, where a comparison with other multireference CC treatments and with experimental data is available. The MR MkCCSDT results show accuracy comparable to which can be achieved with CCSDT in single reference cases. Comparison of the previously developed MkCCSD(T) method with MkCCSDT as a reference suggests, that MkCCSD(T) might be a promising candidate for an accurate treatment of systems where the static correlation plays an important role, at least for situations where small model spaces are sufficient.

State specific equation of motion coupled cluster method in general active space.

The state specific equation of motion coupled cluster (SS-EOMCC) method is an internally contracted multireference approach, applicable to both ground and excited states. Attractive features of the method are as follows: (1) the SS-EOMCC wave function is qualitatively correct and rigorously spin adapted, (2) both orbitals and dynamical correlation are optimized for the target state, (3) nondynamical correlation and differential orbital relaxation effects are taken care of by a diagonalization of the transformed Hamiltonian in the multireference configuration-interaction singles space, (4) only one- and two-particle density matrices of a complete-active-space self-consistent-field reference state are needed to define equations for the cluster amplitudes, and (5) the method is invariant with respect to orbital rotations in core, active, and virtual subspaces. Prior applications focused on biradical-like systems, in which only one extra orbital is needed to construct the active space, and similarly, single bond breaking processes. In this paper, the applicability of the method is extended to systems of general active spaces. Studies on F(2), H(2)O, CO, and N(2) are carried out to gauge its accuracy. The convergence strategy is discussed in detail.

Multireference state-specific Mukherjee's coupled cluster method with noniterative triexcitations.

We have formulated and implemented the multireference Mukherjee's coupled cluster method with connected singles, doubles, and perturbative triples [MR MkCCSD(T)] in the ACES II program package. Assessment of the new method has been performed on the first three electronic states of the oxygen molecule and on the automerization barrier of cyclobutadiene, where a comparison with other multireference CC treatments and with experimental data where available. The MR MkCCSD(T) method seems to be a promising candidate for an accurate, yet computationally tractable, treatment of systems where the static correlation plays an important role.

Application of double ionization state-specific equation of motion coupled cluster method to organic diradicals.

The state-specific equation of motion coupled cluster method is applied to three systems of diradical character: automerization of cyclobutadiene, singlet-triplet gaps of trimethylmethylene, and Bergman reaction. The aim of the paper is to assess the performance of the method and test numerically the importance of orbital optimization, three-body terms in transformed Hamiltonian, and the choice of cluster equations.

Multireference Brillouin-Wigner coupled cluster method with singles, doubles, and triples: efficient implementation and comparison with approximate approaches.

We have developed an efficient implementation of the multireference Brillouin-Wigner coupled cluster method with full iterative treatment of connected singles, doubles, and triples. Its computational costs are too high for applications to larger molecules; however, it can be used as a useful tool for benchmarking approximate methods. Performance of the method has been tested on the ground and low-lying excited states of the oxygen molecule and on the singlet-triplet gap in methylene; the results are in good agreement with experimental data.

Multireference Brillouin-Wigner coupled clusters method with noniterative perturbative connected triples.

We developed and implemented the state-specific Brillouin-Wigner coupled cluster method with singles, doubles, and noniterative perturbative triples, called MR BWCCSD(T), for a general number of closed- and open-shell reference configurations. To assess the accuracy of the method, we performed calculations of the three lowest electronic states of the oxygen molecule and of the automerization barrier of cyclobutadiene. For the oxygen molecule, the results were in a good agreement in comparison with those of the iterative MR BWCCSDTalpha method. For cyclobutadiene, the effect of connected triples was found to be minor, which is in agreement with the previous study by and Balková and Bartlett [J. Chem. Phys. 101, 8972 (1994)].

Towards the multireference Brillouin-Wigner coupled-clusters method with iterative connected triples: MR BWCCSDT-alpha approximation.

We developed and implemented an approximation of the state-specific Brillouin-Wigner coupled-cluster method with singles, doubles, and triples, called MRBWCCSDT-alpha, for a general number of closed- and open-shell reference configurations. The accuracy of the method is assessed on the calculation of the oxygen molecule in the X3sigma(g-), a1delta(g), and b1sigma(g+) states and the results of this multireference treatment are compared with previous MRBWCCSD results and with those obtained by the doubly ionized similarity transformed equation-of-motion CCSD and multireference configuration interaction methods and with experimental spectroscopic data. Explicit tests of the size-extensivity of the MRBWCCSDT-alpha method with iterative size-extensivity correction are also performed.