Robust fitting techniques in the chain of spheres approximation to the Fock exchange: The role of the complementary space

Abstract

The chain of spheres seminumerical integration algorithm is reexamined from the point of view of various numerical error cancellation schemes. The two sources of ideas are the robust fitting technique of Dunlap and the dealiasing concept of the pseudospectral methods of Friesner. The most efficient scheme is identified and is found to be a generalization of the overlap fitting procedure discussed earlier. It will be shown that due to the conceptual similarity between complementary auxiliary basis functions of explicitly correlated approaches and the dealiasing functions of pseudospectral methods, the former can be used in a similar manner as the latter to cancel numerical errors. In contrast to the dealiasing functions, the complementary auxiliary basis functions are not specific to the integration grid. The scheme obtained can dramatically improve reaction energies even using small grid setups at a moderate additional cost.

Local hybrid functionals: Theory, implementation, and performance of an emerging new tool in quantum chemistry and beyond

“…We note in passing that efficient semi-numerical integration for exactexchange methods, irrespective of the additional use of an RI technique, is usually combined with an RI method for Coulomb integrals, as otherwise the latter may become the computational bottleneck. 139 One of the most crucial aspects of semi-numerical integration (including COSX and pseudospectral methods), is an adequate numerical grid. While too small grids may deteriorate the numerical accuracy of the integration, too large grids may remove the computational advantages of the method.…”

Efficient Semi-numerical Implementation of Global and Local Hybrid Functionals for Time-Dependent Density Functional Theory

“…A more detailed description can be found in refs. 45,51 For the Coulomb interaction, a pure semi-numerical scheme without de-aliasing functions and other features of the PS method may lead to numerical instabilities. This is due to the more long-ranged character of Coulomb compared to exchange contributions, as has been pointed out by van Wüllen.…”

“…45,51 The resulting combined integration scheme scales as n · N 2 BF for the exchange part and as N aux · N 2 BF (with N aux being the number of auxilliary basis functions) for the Coulomb part. Both the RI-J and the COSX 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 method are well established in TDDFT calculations.…”

Efficient implementation of the analytic second derivatives of Hartree–Fock and hybrid DFT energies: a detailed analysis of different approximations

“…Various Time Dependent Density Functional Theory (TDDFT) variants also benefit enormously from speed-ups offered by the COSX algorithm [32]. Recently, COSX was also introduced for the evaluation of various intermediates of higher level correlation methods: the single Fock term of the localised pair natural orbital coupled-cluster (LPNO-CC) method [33] and the external exchange term of the spin-component-scaled MP3 method [34]. As part of the present work, the COSX approximation is introduced at various stages of the molecular Hessian evaluation procedure, e.g.…”