Efficient Integral-Direct Methods for Self-Consistent Reduced Density Matrix Functional Theory Calculations on...

Reduced density matrix functional theory (RDMFT), a promising direction in the problem of describing strongly correlated systems, is currently limited by its explicit dependence on natural orbitals and, by extension, the costly need to construct two-electron integrals in the molecular orbital basis. While a resolution-of-the-identity approach can reduce the asymptotic scaling behavior from O(N⁵) to O(N⁴), this is still prohibitively expensive for large systems, especially considering the usually slow convergence and the resulting high number of orbital optimization steps. In this work, efficient integral-direct methods are derived and benchmarked for various approximate functionals. Furthermore, we show how these integral-direct methods can be integrated into existing self-consistent energy minimization frameworks in an efficient manner, including improved methods for calculating diagonal elements of the two-electron integral tensor as required in self-interaction-corrected functionals and second derivatives of the energy with respect to the occupation numbers. In combination, these methods provide speedups of up to several orders of magnitude while greatly diminishing memory requirements, enabling the application of RDMFT to large molecular systems of general chemical interest, such as the challenging triplet-quintet gap of the iron(II) porphyrin complex.