Analytic energy gradients for orbital-optimized MP3 and MP2.5 with the density-fitting approximation: An efficient implementation

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.

State-Of-The-Art Computations of Vertical Electron Affinities with the Extended Koopmans' Theorem Integrated with the CCSD(T) Method

Accurate computation of electron affinities (EAs), within 0.10 eV, is one of the most challenging problems in modern computational quantum chemistry. The extended Koopmans' theorem (EKT) enables direct computations of electron affinities (EAs) from any level of the theory. In this research, the EKT approach based on the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method is applied to computations of EAs for the first time. For efficiency, the density-fitting (DF) technique is used for two-electron integrals. Further, the EKT-CCSD(T) method is applied to three test sets of atoms and closed- and open-shell molecules, denoted A16, C10, and O33, respectively, for comparison with the experimental electron affinities. For the A16, C10, and O33 sets, the EKT-CCSD(T) approach, along with the aug-cc-pV5Z basis set, provide mean absolute errors (MAEs) of 0.05, 0.08, and 0.09 eV, respectively. Hence, our results demonstrate that high-accuracy computations of EAs can be achieved with the EKT-CCSD(T) approach. Further, when the EKT-CCSD(T) approach is not computationally affordable, the EKT-MP2.5, EKT-LCCD, and EKT-CCSD methods can be considered, and their results are also reasonably accurate. The huge advantage of the EKT method for the computation of IPs is that it comes for free in an analytic gradient computation. Hence, one needs neither separate computations for neutral and ionic species, as in the case of common approaches, nor additional efforts to obtain IPs, as in the case of equation-of-motion approaches. Overall, we believe that the present research may open new avenues in EA computations.

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

Efficient implementations of the symmetric and asymmetric triple excitation corrections for the orbital-optimized coupled-cluster doubles (OCCD) method with the density-fitting approach, denoted by DF-OCCD(T) and DF-OCCD(T)_Λ, are presented. The computational cost of the DF-OCCD(T) method is compared with that of the conventional OCCD(T). In the conventional OCCD(T) and OCCD(T)_Λ methods, one needs to perform four-index integral transformations at each coupled-cluster doubles iterations, which limits its applications to large chemical systems. Our results demonstrate that DF-OCCD(T) provides dramatically lower computational costs compared to OCCD(T), and there are more than 68-fold reductions in the computational time for the C₅H₁₂ molecule with the cc-pVTZ basis set. Our results show that the DF-OCCD(T) and DF-OCCD(T)_Λ methods are very helpful for the study of single bond-breaking problems. Performances of the DF-OCCD(T) and DF-OCCD(T)_Λ methods are noticeably better than that of the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method for the potential energy surfaces of the molecules considered. Specifically, the DF-OCCD(T)_Λ method provides dramatic improvements upon CCSD(T), and there are 8-14-fold reductions in nonparallelity errors. Overall, we conclude that the DF-OCCD(T)_Λ method is very promising for the study of challenging chemical systems, where the CCSD(T) fails.

Resolution of the identity approximation applied to PNOF correlation calculations

In this work, the required algebra to employ the resolution of the identity approximation within the Piris Natural Orbital Functional (PNOF) is developed, leading to an implementation named DoNOF-RI. The arithmetic scaling is reduced from fifth-order to fourth-order, and the memory scaling is reduced from fourth-order to third-order, allowing significant computational time savings. After the DoNOF-RI calculation has fully converged, a restart with four-center electron repulsion integrals can be performed to remove the effect of the auxiliary basis set incompleteness, quickly converging to the exact result. The proposed approach has been tested on cycloalkanes and other molecules of general interest to study the numerical results, as well as the speed-ups achieved by PNOF7-RI when compared with PNOF7.

Energy and analytic gradients for the orbital-optimized coupled-cluster doubles method with the density-fitting approximation: An efficient implementation

Efficient implementations of the orbital-optimized coupled-cluster doubles (or simply "optimized CCD," OCCD, for short) method and its analytic energy gradients with the density-fitting (DF) approach, denoted by DF-OCCD, are presented. In addition to the DF approach, the Cholesky-decomposed variant (CD-OCCD) is also implemented for energy computations. The computational cost of the DF-OCCD method (available in a plugin version of the DFOCC module of PSI4) is compared with that of the conventional OCCD (from the Q-CHEM package). The OCCD computations were performed with the Q-CHEM package in which OCCD are denoted by OD. In the conventional OCCD method, one needs to perform four-index integral transformations at each of the CCD iterations, which limits its applications to large chemical systems. Our results demonstrate that DF-OCCD provides dramatically lower computational costs compared to OCCD, and there are almost eightfold reductions in the computational time for the C₆H₁₄ molecule with the cc-pVTZ basis set. For open-shell geometries, interaction energies, and hydrogen transfer reactions, DF-OCCD provides significant improvements upon DF-CCD. Furthermore, the performance of the DF-OCCD method is substantially better for harmonic vibrational frequencies in the case of symmetry-breaking problems. Moreover, several factors make DF-OCCD more attractive compared to CCSD: (1) for DF-OCCD, there is no need for orbital relaxation contributions in analytic gradient computations; (2) active spaces can readily be incorporated into DF-OCCD; (3) DF-OCCD provides accurate vibrational frequencies when symmetry-breaking problems are observed; (4) in its response function, DF-OCCD avoids artificial poles; hence, excited-state molecular properties can be computed via linear response theory; and (5) symmetric and asymmetric triples corrections based on DF-OCCD [DF-OCCD(T)] have a significantly better performance in near degeneracy regions.

Psi4 1.4: Open-source software for high-throughput quantum chemistry

PSI4 is a free and open-source ab initio electronic structure program providing implementations of Hartree-Fock, density functional theory, many-body perturbation theory, configuration interaction, density cumulant theory, symmetry-adapted perturbation theory, and coupled-cluster theory. Most of the methods are quite efficient, thanks to density fitting and multi-core parallelism. The program is a hybrid of C++ and Python, and calculations may be run with very simple text files or using the Python API, facilitating post-processing and complex workflows; method developers also have access to most of PSI4's core functionalities via Python. Job specification may be passed using The Molecular Sciences Software Institute (MolSSI) QCSCHEMA data format, facilitating interoperability. A rewrite of our top-level computation driver, and concomitant adoption of the MolSSI QCARCHIVE INFRASTRUCTURE project, makes the latest version of PSI4 well suited to distributed computation of large numbers of independent tasks. The project has fostered the development of independent software components that may be reused in other quantum chemistry programs.

Third-Order Møller-Plesset Perturbation Theory Made Useful? Choice of Orbitals and Scaling Greatly Improves Accuracy for Thermochemistry, Kinetics, and Intermolecular Interactions

We develop and test methods that include second- and third-order perturbation theory (MP3) using orbitals obtained from regularized orbital-optimized second-order perturbation theory, κ-OOMP2, denoted as MP3:κ-OOMP2. Testing MP3:κ-OOMP2 shows RMS errors that are 1.7-5 times smaller than those of MP3 across 7 data sets. To do still better, empirical training of the scaling factors for the second- and third-order correlation energies and the regularization parameter on one of those data sets led to an unregularized scaled (c₂ = 1.0; c₃ = 0.8) denoted as MP2.8:κ-OOMP2. MP2.8:κ-OOMP2 yields significant additional improvement over MP3:κ-OOMP2 in 4 of 6 test data sets on thermochemistry, kinetics, and noncovalent interactions. Remarkably, these two methods outperform coupled cluster with singles and doubles in 5 of the 7 data sets considered, at greatly reduced cost (no O(N6) iterations).