Scalable Electron Correlation Methods. 3. Efficient and Accurate Parallel Local Coupled Cluster with Pair Natural Orbitals (PNO-LCCSD)

Abstract

A well-parallelized local singles and doubles coupled-cluster (LCCSD) method using pair natural virtual orbitals (PNOs) is presented. The PNOs are constructed using large domains of projected atomic orbitals (PAOs) and orbital specific virtual orbitals (OSVs). We introduce a hierarchy of close, weak, and distant pairs, based on pair energies evaluated by local Møller-Plesset perturbation theory (LMP2). In contrast to most previous implementations of LCCSD methods, the close and weak pairs are not approximated by LMP2 but treated by higher-order methods. This leads to greatly improved accuracy for large systems, in particular when long-range dispersion interactions are important. Close pair amplitudes are optimized using approximate LCCSD equations, in which slowly decaying terms that mutually cancel at long-range are neglected. For weak pairs, the same approximations are used, but in addition, the nonlinear terms are neglected (coupled electron pair approximation). Distant pairs are treated by spin-component scaled (SCS)-LMP2 using multipole approximations. For efficiency reasons, various projection approximations are also introduced. The impact of these approximations on absolute and relative energies depends on the PNO domain sizes. The errors are found to be negligible, provided that sufficiently large PNO domains are used for close and weak pairs. For the selection of these domains the usual natural orbital occupation number criterion is found to be insufficient, and an additional energy criterion is used. For extended one-dimensional systems, the computational effort of the method scales nearly linearly with the number of correlated electrons, but the linear scaling regime is usually not reached in real-life applications for three-dimensional systems. Nevertheless, due to the parallelization that is efficient up to about 100-200 CPU cores (dependent on the molecular size), accurate calculations for three-dimensional molecules with about 100 atoms and augmented triple-ζ basis sets (e.g., cc-pVTZ-F12) can be carried out in 1-3 h of elapsed time (depending on the molecular structure and the number of CPU cores, excluding the time for Hartree-Fock). The convergence of the results with respect to the thresholds and options that control the domain, pair, and projection approximations is extensively tested. Benchmark examples for several difficult and large cases are presented, which demonstrate that the errors of relative energies (e.g., reaction energies, barrier heights) caused by the pair and projection approximations can be reduced to below 1 kJ mol. The remaining errors are mainly caused by the finite PAO domains. The larger these are made, the more intramolecular or intermolecular basis set superposition errors are introduced, and the canonical results are approached only very slowly. This problem is eliminated in the explicitly correlated variant of our method (PNO-LCCSD-F12), which will be described in a separate paper, along with a larger set of benchmark calculations.

Coupled‐cluster singles, doubles and perturbative triples with density fitting approximation for massively parallel heterogeneous platforms

“…Even more crucial has been the recent development of CCSD(T) with reduced complexity, for example, methods based on pair‐natural orbital (PNOs), which allows treatment of systems with hundreds of atoms; systems of this size are far beyond the limits of the conventional‐scaling CCSD(T) on a classical computer. Efficient reduced‐scaling explicitly correlated CCSD(T) has been demonstrated by several groups, including ours.…”

“…[9][10][11][12] Even more crucial has been the recent development of CCSD(T) with reduced complexity, for example, methods based on pair-natural orbital (PNOs), [13][14][15] which allows treatment of systems with hundreds of atoms; systems of this size are far beyond the limits of the conventional-scaling CCSD(T) on a classical computer. Efficient reduced-scaling explicitly correlated CCSD(T) has been demonstrated by several groups, [16][17][18][19][20][21][22] including ours.Although the capability of conventional CCSD(T) has been surpassed by its reduced-scaling variants, the full-scaling (canonical) CCSD(T) is still highly desirable for assessing the impact of numerical approximations responsible for the complexity reduction. With this goal in mind we developed a reference scalable canonical CCSD(T) implementation capable of treating molecules as large as presently feasible.…”

Communication: Multipole approximations of distant pair energies in local correlation methods with pair natural orbitals

“…1 which has been used in previous benchmarks of PNO-LCCSD methods. 12,16 This is an excellent and difficult benchmark case since the correlation effects on the reaction energy are very large (Hartree-Fock, LMP2-F12, and LCCSD-F12 yield 92, 251, and 190 kJ mol −1 , respectively, using the VTZ-F12 basis set 16 ). The results can be directly compared to an experimental gas-phase value 17 of 196.5 ± 11.2 kJ/mol, which has been obtained by subtracting the PW91/cc-pVTZ-pp zero-point correction 17 of −8.2 kJ/mol from the measured value.…”

Validating additive correction schemes against gradient‐based extrapolations

“…With increasing availability of computational power, programming tricks, clever choice of basis sets, and various approximations to canonical coupled cluster (CC) methods (eg, by density fitting [DF], reduction of virtual space, or local correlation with paired orbitals among others), it is now possible to carry out calculations of (nearly) CC through pertubative triplets (CCSD[T]) quality almost routinely. Coupled to correlation consistent basis sets and extrapolation procedures, one can now obtain highly accurate “gold‐standard” results even for rather large systems …”