Scalable Electron Correlation Methods I.: PNO-LMP2 with Linear Scaling in the Molecular Size and Near-Inverse-Linear Scaling in the Number of Processors

Abstract

We propose to construct electron correlation methods that are scalable in both molecule size and aggregated parallel computational power, in the sense that the total elapsed time of a calculation becomes nearly independent of the molecular size when the number of processors grows linearly with the molecular size. This is shown to be possible by exploiting a combination of local approximations and parallel algorithms. The concept is demonstrated with a linear scaling pair natural orbital local second-order Møller-Plesset perturbation theory (PNO-LMP2) method. In this method, both the wave function manifold and the integrals are transformed incrementally from projected atomic orbitals (PAOs) first to orbital-specific virtuals (OSVs) and finally to pair natural orbitals (PNOs), which allow for minimum domain sizes and fine-grained accuracy control using very few parameters. A parallel algorithm design is discussed, which is efficient for both small and large molecules, and numbers of processors, although true inverse-linear scaling with compute power is not yet reached in all cases. Initial applications to reactions involving large molecules reveal surprisingly large effects of dispersion energy contributions as well as large intramolecular basis set superposition errors in canonical MP2 calculations. In order to account for the dispersion effects, the usual selection of PNOs on the basis of natural occupation numbers turns out to be insufficient, and a new energy-based criterion is proposed. If explicitly correlated (F12) terms are included, fast convergence to the MP2 complete basis set (CBS) limit is achieved. For the studied reactions, the PNO-LMP2-F12 results deviate from the canonical MP2/CBS and MP2-F12 values by <1 kJ mol(-1), using triple-ζ (VTZ-F12) basis sets.

Perspective: Explicitly correlated electronic structure theory for complex systems

“…Readers can find more comprehensive publication lists elsewhere. 135,136 In addition, we will explain other solutions for large molecules using massively parallel computing techniques.…”

“…Although PNOs have large linear dependencies amongst pairs, their use leads to only dense operations over pairs and appears to be suitable for parallel implementations in distributed memory. 136,147 Other explicitly correlated methods utilizing localities involve explicitly correlated CCSD(T) 148,149 in the incremental scheme of Stoll 150 and Divide-Expand-Consolidate MP2-F12 (DEC-RIMP2-F12) 151 with simpler structures for F12 implementations to complement the domain-based approaches.…”

“…One major technical challenge is to avoid different many-electron integrals in explicitly correlated theories, as these are expensive, increasing the computational cost compared with their parent methods for electron correlation. Many promising approaches to overcome these problems have already been proposed for large or periodic systems, including Monte Carlo techniques, 115,116 local explicitly correlated methods, 136,145,146 and plane wave basis set techniques. 44,121 These methods exhibit a trade-off between computational cost and accuracy that is strongly dependent on the system size, dimensionality, and possibly the band gap.…”

Communication: Improved pair approximations in local coupled-cluster methods

“…The errors caused by weak pair approximations were assumed to be small since the pair correlation energies decay as r −6 i j with the distance r i j between the charge centroids of the local orbitals φ i and φ j . However, this situation has changed, since with modern pair natural orbital (PNO) approaches, [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] the domain error can be systematically controlled and made very small (<1 kJ mol −1 for relative energies), in particular when explicitly correlated terms are included. 5,12,25,26 On the other hand, we demonstrate in the current work that pair approximations based on local second-order Møller-Plesset theory (LMP2) can lead to very large errors (up to ≈ 40 kJ mol −1 in the present examples) for reaction and activation energies of complex chemical reactions.…”

“…Pairs with E i j ≥ T close are strong, those with T close > E i j ≥ T weak are close, those with T weak > E i j ≥ T dist are weak, and the remaining ones distant. The latter are treated using a non-iterative dipole-dipole approximation, 13,25,40 which is also used to select the pairs in this class. Throughout this work, we will use T dist = 1 µH.…”