Efficient implementations of the symmetric and asymmetric triple excitation corrections for the orbital-optimized coupled-cluster doubles method with the density-fitting approximation

Linear-Scaling Systematic Molecular Fragmentation Approach for Perturbation Theory and Coupled-Cluster Methods

The coupled-cluster (CC) singles and doubles with perturbative triples [CCSD(T)] method is frequently referred to as the “gold standard” of modern computational chemistry. However, the high computational cost of CCSD(T) [O(N⁷)], where N is the number of basis functions, limits its applications to small-sized chemical systems. To address this problem, efficient implementations of linear-scaling coupled-cluster methods, which employ the systematic molecular fragmentation (SMF) approach, are reported. In this study, we aim to do the following: (1) To achieve exact linear scaling and to obtain a pure ab initio approach, we revise the handling of nonbonded interactions in the SMF approach, denoted by LSSMF. (2) A new fragmentation algorithm, which yields smaller-sized fragments, that better fits high-level CC methods is introduced. (3) A modified nonbonded fragmentation scheme is proposed to enhance the existent algorithm. Performances of the LSSMF-CC approaches, such as LSSMF-CCSD(T), are compared with their canonical versions for a set of alkane molecules, C_nH_2n+2 (n = 6–10), which includes 142 molecules. Our results demonstrate that the LSSMF approach introduces negligible errors compared with the canonical methods; mean absolute errors (MAEs) are between 0.20 and 0.59 kcal mol^–1 for LSSMF(3,1)-CCSD(T). For a larger alkanes set (L12), C_nH_2n+2 (n = 50–70), the performance of LSSMF for the second-order perturbation theory (MP2) is investigated. For the L12 set, various bonded and nonbonded levels are considered. Our results demonstrate that the combination of bonded level 6 with nonbonded level 2, LSSMF(6,2), provides very accurate results for the MP2 method with a MAE value of 0.32 kcal mol^–1. The LSSMF(6,2) approach yields more than a 26-fold reduction in errors compared with LSSMF(3,1). Hence, we obtain substantial improvements over the original SMF approach. To illustrate the efficiency and applicability of the LSSMF-CCSD(T) approach, we consider an alkane molecule with 10,004 atoms. For this molecule, the LSSMF(3,1)-CCSD(T)/cc-pVTZ energy computation, on a Linux cluster with 100 nodes, 4 cores, and 5 GB of memory provided to each node, is performed just in ∼24 h. As a second test, we consider a biomolecular complex (PDB code: 1GLA), which includes 10,488 atoms, to assess the efficiency of the LSSMF approach. The LSSMF(3,1)-FNO–CCSD(T)/cc-pVTZ energy computation is completed in ∼7 days for the biomolecular complex. Hence, our results demonstrate that the LSSMF-CC approaches are very efficient. Overall, we conclude the following: (1) The LSSMF(m, n)-CCSD(T) methods can be reliably used for large-scale chemical systems, where the canonical methods are not computationally affordable. (2) The accuracy of bonded level 3 is not satisfactory for large chemical systems. (3) For high-accuracy studies, bonded level 5 (or higher) and nonbonded level 2 should be employed.

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.

Cumulants as the variables of density cumulant theory: A path to Hermitian triples

We study the combination of orbital-optimized density cumulant theory and a new parameterization of reduced density matrices in which the variables are the particle-hole cumulant elements. We call this combination OλDCT. We find that this new Ansatz solves problems identified in the previous unitary coupled cluster Ansatz for density cumulant theory: the theory is now free of near-zero denominators between occupied and virtual blocks, can correctly describe the dissociation of H₂, and is rigorously size-extensive. In addition, the new Ansatz has fewer terms than the previous unitary Ansatz, and the optimal orbitals delivered by the exact theory are the natural orbitals. Numerical studies on systems amenable to full configuration interaction show that the amplitudes from the previous ODC-12 method approximate the exact amplitudes predicted by this Ansatz. Studies on equilibrium properties of diatomic molecules show that even with the new Ansatz, it is necessary to include triples to improve the accuracy of the method compared to orbital-optimized linearized coupled cluster doubles. With a simple iterative triples correction, OλDCT outperforms other orbital-optimized methods truncated at comparable levels in the amplitudes, as well as coupled cluster single and doubles with perturbative triples [CCSD(T)]. By adding four more terms to the cumulant parameterization, OλDCT outperforms CCSDT while having the same O(V⁵O³) scaling.