Q-Next: A Fast, Parallel, and Diagonalization-Free Alternative to Direct Inversion of the Iterative Subspace

Abstract

As computer systems dedicated to scientific calculations become massively parallel, the poor parallel performance of the Fock matrix diagonalization becomes a major impediment to achieving larger molecular sizes in self-consistent field (SCF) calculations. In this Article, a novel, highly parallel, and diagonalization-free algorithm for the accelerated convergence of the SCF procedure is presented. The algorithm, called Q-Next, draws on the second-order SCF, quadratically convergent SCF, and direct inversion of the iterative subspace (DIIS) approaches to enable fast convergence while replacing the Fock matrix diagonalization SCF bottleneck with higher parallel efficiency matrix multiplications. Performance results on both parallel multicore CPU and GPU hardware for a variety of test molecules and basis sets are presented, showing that Q-Next achieves a convergence rate comparable to the DIIS method while being, on average, one order of magnitude faster.

Economical quasi-Newton unitary optimization of electronic orbitals

“…Although some approaches evaluate the exponential approximately and return to the target manifold by additional orthogonalization, 9,19 accurate evaluation is important to be able to maintain the fidelity of the relationship between parameters and the objective function values in the context of extrapolation methods like BFGS. Evaluation of the matrix exponential could be improved further, e.g., by leveraging the block-sparse antihermitian structure of r. 34 It is important to express both the gradients and parameter updates for each iteration in the same coordinate frame (basis). In the context of the DIIS methods this is usually done by working in the AO basis (e.g., as noted by Pulay in ref.…”

“…Level-shift s of the L-BFGS Hessian is updated iteratively until eqn (34) is satisfied to the desired precision controlled by parameters T 1,2 (see Table 2 and Algorithm 2). The TR solver in QUOTR is based on the solver described by Burdakov et al 56 that leverages the low-rank structure of the L-BFGS Hessian.…”

“…Despite the long history of innovation, 2–29 development of improved orbital optimizers continues to this day. 30–35 Although the relevant functionals of the orbitals are nonconvex, and global and local nonconvex optimization is NP-hard, 36,37 it is known that many practical orbital optimization problems are easily solved using existing heuristics. For the crucial HF/KS SCF use case, the most popular solvers in the molecular context are based on the Roothaan–Hall (RH) iterative diagonalization of the Fock matrix 2,38 augmented by convergence accelerators such as Anderson mixing 39 or the closely related direct inversion in the iterative subspace (DIIS) method, 10,13,40,41 as well as others.…”

A Story of Three Levels of Sophistication in SCF/KS-DFT Orbital Optimization Procedures

“…where H ̂is the Hamiltonian operator. For the purpose of facilitating the orbital optimization, the energy is expressed as a Taylor expansion 27 (here explicitly expressed up to second order)…”

Quantum Mechanics/Molecular Mechanics Simulations on NVIDIA and AMD Graphics Processing Units

“…Typically, computing QM forces takes more than 95% of the total QM/MM time. In the recent past, various efforts have been undertaken to develop computationally affordable novel QM methods or reimplement traditional QM methods to harness the power of massively parallel central processing unit (CPU) and graphics processing unit (GPU) hardware platforms. − Most notably, a number of leading quantum chemistry software packages have been empowered with GPU acceleration allowing users to achieve unprecedented simulation speeds and model larger molecular systems efficiently. For instance, our own GPU-accelerated QUICK ab initio quantum chemistry and density functional theory package is highly efficient on NVIDIA hardware. , QM/MM simulations with QUICK/AMBER have displayed respectable speedups of up to 53 times for a single GPU with respect to a CPU core for a moderate-sized QM region size that was benchmarked at the time .…”