An overlap fitted chain of spheres exchange method

Abstract

The "chain of spheres" (COS) algorithm, as part of the RIJCOSX SCF procedure, approximates the exchange term by performing analytic integration with respect to the coordinates of only one of the two electrons, whereas for the remaining coordinates, integration is carried out numerically. In the present work, we attempt to enhance the efficiency of the method by minimizing numerical errors in the COS procedure. The main idea is based on the work of Friesner and consists of finding a fitting matrix, Q, which leads the numerical and analytically evaluated overlap matrices to coincide. Using Q, the evaluation of exchange integrals can indeed be improved. Improved results and timings are obtained with the present default grid setup for both single point calculations and geometry optimizations. The fitting procedure results in a reduction of grid sizes necessary for achieving chemical accuracy. We demonstrate this by testing a number of grids and comparing results to the fully analytic and the earlier COS approximations. This turns out to be favourable for total and reaction energies, for which chemical accuracy can now be reached with a corresponding ~30% speedup over the original RIJCOSX procedure for single point energies. Results are slightly less favourable for the accuracy of geometry optimizations, but the procedure is still shown to yield geometries with errors well below the method inherent errors of the employed theoretical framework.

Strengths, Weaknesses, Opportunities and Threats: Computational Studies of Mn- and Fe-Catalyzed Epoxidations

“…This strategy effectively saves computational effort when calculating the Coulombic repulsion term in pure DFT functionals [141]. Hybrid DFT methods can also use a variant of the RI algorithm to achieve near-linear scaling in the calculation of the exact Hartree-Fock exchange term, in which the exchange integral is computed from fitting an analytical expression for the electron density instead of the usual numerical integration scheme [142]. Such methods have the advantage of being easily implemented into existing quantum chemistry software codes.…”

QM/MM Study of the Nitrogenase MoFe Protein Resting State: Broken-Symmetry States, Protonation States, and QM Region Convergence in the FeMoco Active Site

“…A triple-zeta ZORA-def2-TZVP basis set was used for all Mo, Fe and S atoms (and interstitial carbide of FeMoco) while a double-zeta ZORA-def2-SVP basis set was used for all other atoms. The RIJCOSX approximation 68,69 was used to speed up Coulomb and Exchange integrals. An example ORCA inputfile used in the QM/MM and QM calculations is shown in the supporting information.…”

The INV24 test set: how well do quantum-chemical methods describe inversion and racemization barriers?

“…109 Most DFT calculations were carried out with ORCA3.0.2 with the self-consistent-field (SCF) convergence option "TightSCF" and a numerical integration grid "grid5". All calculations also made use of the resolution of the identity (RI) approximation for the evaluation of Coulomb integrals 110,111 and the chain-of-spheres approximation 112 for the evaluation of exchange integrals in hybrid and double-hybrid functionals. The second-order perturbation correction in double hybrids was also assessed with the RI approximation.…”