Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm

Abstract

Construction of the exact exchange matrix, K, is typically the rate-determining step in hybrid density functional theory, and therefore, new approaches with increased efficiency are highly desirable. We present a framework with potential for greatly improved efficiency by computing a compressed exchange matrix that yields the exact exchange energy, gradient, and direct inversion of the iterative subspace (DIIS) error vector. The compressed exchange matrix is constructed with one index in the compact molecular orbital basis and the other index in the full atomic orbital basis. To illustrate the advantages, we present a practical algorithm that uses this framework in conjunction with the resolution of the identity (RI) approximation. We demonstrate that convergence using this method, referred to hereafter as occupied orbital RI-K (occ-RI-K), in combination with the DIIS algorithm is well-behaved, that the accuracy of computed energetics is excellent (identical to conventional RI-K), and that significant speedups can be obtained over existing integral-direct and RI-K methods. For a 4400 basis function C 68 H 22 hydrogen-terminated graphene fragment, our algorithm yields a 14× speedup over the conventional algorithm and a speedup of 3.3× over RI-K. C 2015 AIP Publishing LLC. [http://dx

Occupied-Orbital Fast Multipole Method for Efficient Exact Exchange Evaluation

“…Total timings for an SCF iteration and timings for the far-field exchange evaluation were averaged over the first 5 iterations. The cost of the far-field exchange evaluation for C 96 H 24 , which was about 10% of the total cost per SCF iteration, was 188 sec with 1 compute node, and reduced to 99, 52, 29, 15, 10, and 9 sec using 2, 4, 8, 16, 32, and 64 compute nodes. The timings for C 150 H 30 were 216, 111, 62, 37, 21, and 16 sec using 2, 4, 8, 16, 32, and 64 compute nodes.…”

“…24 As shown below, this trick is essential for utilizing a hierarchy of boxes with upward and downward translation of multipoles in the FMM algorithm. We report an efficient, parallel implementation of the algorithm, which is publicly available as part of the bagel package.…”

“…D tu are the density matrix elements, which become diagonal in the canonical molecular orbital (MO) representation. There have been extensive studies to optimize the exchange evaluation: for instance, the LinK method, 5,6 multipole accelerated algorithms, 7,8 rigorous integral screening, [9][10][11][12] density fitting with local domains, 13,14 truncated or short-range exchange kernels with and without the use of the resolution-ofthe-identity (RI) approximation, [15][16][17][18] the chain-of-sphere exchange method based on quadrature, 19 the auxiliary density matrix method, 20 the pair-atomic RI approximation, [21][22][23][24] and the low-rank decomposition of the exchange operator. 25,26 In this work, we report an efficient algorithm, termed occupied-orbital fast multipole method for exchange (occ-FMM-K), for computing the exchange contributions based on the fast multipole method (FMM).…”

Advanced Potential Energy Surfaces for Molecular Simulation

“…Indeed, DFT based ab initio molecular dynamics calculations of pure water and aqueous solutions using fully converged discrete variable representation basis sets show that both structural and dynamical properties are sensitive to the size of the basis set [131][132][133][134][135][136][137] . Examples of existing approaches that help defray the computational cost are the use of resolution of the identity (RI) approximations for the Coulomb and exchange interactions [138][139][140][141] , which are nowadays standard, as well as the dual-basis SCF methods. [142][143] In order to approach the basis set limit with potentially lower computational effort, we have been working on a new SCF scheme using a minimal adaptive basis (MAB) consisting of functions that are variationally optimized on each atomic site.…”

Communication: Almost error-free resolution-of-the-identity correlation methods by null space removal of the particle-hole interactions

“…Several attempts and discussions have therefore revolved around improving the RI approach. [14][15][16][17] While a lot of effort has been put into advancing upon the approximation formula itself [18][19][20][21][22][23] as well as the auxiliary basis sets, [24][25][26] the null space structure of the physical models remains more or less opaque.…”