Advancing Linear-Scaling Techniques in Computation of Exchange Matrices in Mean-Field and Cluster-in-Molecule Calculations

Abstract

Quantum chemistry faces ongoing challenges in developing methods that combine efficiency with accuracy, especially for large molecular systems. The Cluster-in-Molecule (CiM) technique, integrated with Coupled Cluster theory (CC) methods, offers a promising solution by accurately computing correlated ground state energies through division into computations of small subsystems. These systems utilize a subset of localized natural orbitals (LNO) defined by localized orbital domains [1, 2, 3, 4, 5, 6]. The advantage of CiM-CC approach is that all subsystem calculations can trivially be computed in parallel with a relatively straightforward algorithm. The main challenges are in defining small orbital domains in accurate and efficient ways, and the required integral transformation from the global atomic orbitals (AO) basis to the subset of LNO.

In this work, we enhance the efficiency of calculating two-electron repulsion integral (ERI) through advanced computational techniques that incorporate the Resolution of Identity (RI) metric matrix and a three-index short-range Coulomb potential with Gaussian-Type Geminal (GTG) correction. This aspect of the research, inspired by the thesis work of Dr.Michael J. Lecours in the Nooijen group, focuses on improving the efficiency of calculating the exchange matrices K while maintaining acceptable error margins [7, 8]. Our newly developed algorithms in the Python module for quantum chemistry platform (PySCF) program, especially for calculating Coulomb J and exchange K matrices through the JK-Engine, are shown to achieve a linear correlation between performance and the size of molecular systems. These improvements are not only vital for CiM but also for Hartree-Fock (HF) and (hybrid) Density Functional Theory (DFT) mean-field calculations, with accuracy controlled by a single parameter defining the short-range Coulomb potential’s range.

Utilizing the exchange matrix, we present an efficient orbital domain construction scheme for occupied localized molecular orbitals (LMO) based on the pivoted Cholesky decomposition of the exchange matrix. This method improves the efficiency of the parti- tioning into LMO subspaces, crucial for CiM calculations.

In summary, our advancements in linear-scaling exchange matrix calculations and or- bital domain construction mark significant progress toward more efficient and accurate electronic structure calculations for mean-field and CiM approaches, promising enhanced computational performance for large molecular systems.