Tensor-Structured Techniques for Electronic Structure Calculations
Lin, Ian
2022
Abstract
Electronic structure calculations have been one of the most successful scientific fields in the past 50 years. Density functional theory (DFT) with its nice balance between accuracy and efficiency has now become the standard technique for many materials research. On the other hand, the size of systems that wavefunction-based methods can handle has also been improved thanks to the recent developments in both computation power and more advanced methodologies. However, these methods are still suffering from the curse of dimensionality. The cubic scaling of a typical DFT calculation restricts the handleable system size of DFT to only a few thousand electrons. To this end, tensor-structured techniques provide a route to constructing a reduced-order algorithm and hence are promising to improve the computational efficiency. In this dissertation work, we aim at exploring the applications of the tensor-structured techniques on different aspects of electronic structure calculations. In the first part of this work, we present a tensor-structured algorithm for efficient large-scale DFT calculations by constructing a Tucker tensor basis that is adapted to the Kohn–Sham Hamiltonian and localized in real-space. The proposed approach uses an additive separable approximation to the Kohn–Sham Hamiltonian and an L1 localization technique to generate the 1-D localized functions that constitute the Tucker tensor basis. Numerical results show that the resulting Tucker tensor basis exhibits exponential convergence in the ground-state energy with increasing Tucker rank. Further, the proposed tensor-structured algorithm demonstrated sub-quadratic scaling with system-size for both systems with and without a gap, and involving many thousands of atoms. This reduced-order scaling has also resulted in the proposed approach outperforming plane-wave DFT implementation for systems beyond 2000 electrons. In the second part of this work, we present an acceleration scheme for the proposed tensor-structured algorithm. In the proposed algorithm, the discrete Kohn-Sham problem is solved using the Chebyshev filtering subspace iteration method that relies on matrix-matrix multiplications of a sparse symmetric Hamiltonian matrix and a dense wavefunction matrix, expressed in the localized Tucker tensor basis. These matrix-matrix multiplication operations, which constitute the most computationally intensive step of the solution procedure, are GPU accelerated providing ~8-fold GPU-CPU speedup for these operations on the largest systems studied. The computational performance of the GPU accelerated code is presented using benchmark studies on aluminum nano-particles and silicon quantum dots with system sizes ranging up to ~7,000 atoms. Finally, we present a computation kernel using tensor-structured techniques to evaluate the one-electron and two-electron integrals, which are the central quantities to construct the Hamiltonian matrix of a full configuration interaction (FCI) calculation, for any molecular orbitals projected on a tensor-structured finite-element mesh. The proposed computation kernel is used to compute the integrals using Hartree-Fock molecular orbitals, which is subsequently used to perform an FCI calculation. In our numerical study, the FCI energy using integrals from the proposed computation kernel is confirmed to be consistent with a standard FCI calculation using the same Hartree-Fock orbitals. The proposed computation kernel provides a useful tool to investigate the behavior of Kohn-Sham orbitals as a basis for FCI calculations in the future, which could be a route to construct a reduced-order basis for an FCI calculation to reduce the computational cost and enable FCI calculation for the previously computationally inaccessible systems.Deep Blue DOI
Subjects
Density functional theory Tensor-structured algorithm L-1 localization Two-electron integrals
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