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Parallel Self-Consistent-Field Calculations via Chebyshev-Filtered Subspace Acceleration

Yunkai Zhou, Yousef Saad, Murilo L. Tiago, James R. Chelikowsky

TL;DR

The work tackles the high cost of solving the nonlinear Kohn–Sham eigenproblem in DFT SCF, especially for large systems. It introduces CheFSI, a nonlinear subspace iteration method that replaces repeated inner diagonalizations with Chebyshev-filtered subspace updates, using a single initial diagonalization via Chebyshev–Davidson. Implemented in the real-space, MPI-parallel PARSEC framework, CheFSI achieves significant speedups over traditional eigensolvers and enables DFT calculations on clusters with thousands of atoms that were previously infeasible. The approach yields self-consistent solutions with physical observables (e.g., IP, EA, DOS) consistent with expectations and demonstrates strong scalability for large-scale materials simulations. Overall, CheFSI broadens the practical reach of high-accuracy DFT in large-scale, real-space computations.

Abstract

Solving the Kohn-Sham eigenvalue problem constitutes the most computationally expensive part in self-consistent density functional theory (DFT) calculations. In a previous paper, we have proposed a nonlinear Chebyshev-filtered subspace iteration method, which avoids computing explicit eigenvectors except at the first SCF iteration. The method may be viewed as an approach to solve the original nonlinear Kohn-Sham equation by a nonlinear subspace iteration technique, without emphasizing the intermediate linearized Kohn-Sham eigenvalue problem. It reaches self-consistency within a similar number of SCF iterations as eigensolver-based approaches. However, replacing the standard diagonalization at each SCF iteration by a Chebyshev subspace filtering step results in a significant speedup over methods based on standard diagonalization. Here, we discuss an approach for implementing this method in multi-processor, parallel environment. Numerical results are presented to show that the method enables to perform a class of highly challenging DFT calculations that were not feasible before.

Parallel Self-Consistent-Field Calculations via Chebyshev-Filtered Subspace Acceleration

TL;DR

The work tackles the high cost of solving the nonlinear Kohn–Sham eigenproblem in DFT SCF, especially for large systems. It introduces CheFSI, a nonlinear subspace iteration method that replaces repeated inner diagonalizations with Chebyshev-filtered subspace updates, using a single initial diagonalization via Chebyshev–Davidson. Implemented in the real-space, MPI-parallel PARSEC framework, CheFSI achieves significant speedups over traditional eigensolvers and enables DFT calculations on clusters with thousands of atoms that were previously infeasible. The approach yields self-consistent solutions with physical observables (e.g., IP, EA, DOS) consistent with expectations and demonstrates strong scalability for large-scale materials simulations. Overall, CheFSI broadens the practical reach of high-accuracy DFT in large-scale, real-space computations.

Abstract

Solving the Kohn-Sham eigenvalue problem constitutes the most computationally expensive part in self-consistent density functional theory (DFT) calculations. In a previous paper, we have proposed a nonlinear Chebyshev-filtered subspace iteration method, which avoids computing explicit eigenvectors except at the first SCF iteration. The method may be viewed as an approach to solve the original nonlinear Kohn-Sham equation by a nonlinear subspace iteration technique, without emphasizing the intermediate linearized Kohn-Sham eigenvalue problem. It reaches self-consistency within a similar number of SCF iterations as eigensolver-based approaches. However, replacing the standard diagonalization at each SCF iteration by a Chebyshev subspace filtering step results in a significant speedup over methods based on standard diagonalization. Here, we discuss an approach for implementing this method in multi-processor, parallel environment. Numerical results are presented to show that the method enables to perform a class of highly challenging DFT calculations that were not feasible before.

Paper Structure

This paper contains 7 sections, 6 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Eigenvalue problems in DFT SCF calculations
  3. The parallel environment in PARSEC
  4. The nonlinear Chebyshev-filtered subspace iteration
  5. Chebyshev-Davidson algorithm for the first SCF iteration
  6. Numerical Results
  7. Concluding Remarks

Figures (3)

  • Figure 1: Sample decomposition of a physical domain in PARSEC.
  • Figure 2: Ionization potential (IP, crosses) and electron affinity (EA, "plus" signs) for various clusters with diameters ranging from 0 nm ($SiH_{4}$) to 7 nm ($Si_{9041}H_{1860}$). Squares denote the negative of the highest occupied eigenvalue energy ($-E_{HOMO}$) of the neutral cluster. Diamonds denote the negative of the lowest unoccupied eigenvalue energy ($-E_{LUMO}$).
  • Figure 3: Density of states (DOS) of the cluster $Si_{9041}H_{1860}$ (upper panel) compared with periodic crystalline silicon (lower panel). As a consequence of the large size, the DOS of the $Si_{9041}H_{1860}$ cluster is very close to that of bulk silicon (the infinite-size limit).