The divide-expand-consolidate family of coupled cluster methods: Numerical illustrations using second order Møller-Plesset perturbation theory

Abstract

Previously, we have introduced the linear scaling coupled cluster (CC) divide-expand-consolidate (DEC) method, using an occupied space partitioning of the standard correlation energy. In this article, we show that the correlation energy may alternatively be expressed using a virtual space partitioning, and that the Lagrangian correlation energy may be partitioned using elements from both the occupied and virtual partitioning schemes. The partitionings of the correlation energy leads to atomic site and pair interaction energies which are term-wise invariant with respect to an orthogonal transformation among the occupied or the virtual orbitals. Evaluating the atomic site and pair interaction energies using local orbitals leads to a linear scaling algorithm and a distinction between Coulomb hole and dispersion energy contributions to the correlation energy. Further, a detailed error analysis is performed illustrating the error control imposed on all components of the energy by the chosen energy threshold. This error control is ultimately used to show how to reduce the computational cost for evaluating dispersion energy contributions in DEC.

Linear-Scaling Coupled Cluster with Perturbative Triple Excitations: The Divide–Expand–Consolidate CCSD(T) Model

“…In the DEC-CC framework, [19][20][21][22][23] we assign local HF orbitals to atomic sites P, Q, R, S, . .…”

“…These advances are made possible through a reformulation of the (T) correction to the CCSD energy such that its expression in terms of localized orbitals aligns with our recently developed divide-expand-consolidate (DEC) coupled cluster framework. [19][20][21][22][23] In a DEC-CC calculation, the inherent locality of the electron correlation problem is efficiently exploited in order to express the correlated wave function calculation on the full molecular system in terms of numerous small and independent fragment calculations that each uses a subset of the total orbital space. Importantly, the local orbital spaces used in the individual fragment calculations are determined in a black-box manner during the calculation to ensure that the calculated final energy is determined to within a predefined precision compared to a conventional calculation.…”

Communication: Improved pair approximations in local coupled-cluster methods

“…Furthermore, it is not sufficient to use LMP2 weak pair amplitudes to avoid this error. This should also affect the accuracy of fragmentation approaches [27][28][29][30][31][32][33][34][35][36][37] and implies that the fragments must be very large in order to obtain converged results.…”

“…Pair approximations are also implicit in fragmentation approaches, [27][28][29][30][31][32][33][34][35][36][37] in which (overlapping) groups of orbitals are correlated independently. These methods rely on the assumption that the contribution of amplitudes that are not included in a fragment has a negligible effect on the computed pair energies.…”

Calculation of excitation energies from the CC2 linear response theory using Cholesky decomposition

“…Considerable efforts are now oriented in reducing the scaling of expensive methods like coupled cluster and a lot of progress has been made. Some groups develop local approximations, [1][2][3][4] others focus on the approximation of the two-electron integrals [5][6][7] and totally new algorithm are developed to be well adapted to new technologies (massive parallelization, 8,9 graphics processing units, 10,11 etc.). In particular, the Cholesky decomposition arises as an efficient tool for the reduction of computational efforts as well as storage requirements.…”