[Paper Review] An orthogonalization-free parallelizable framework for all-electron calculations in density functional theory

Bin Gao, Guanghui Hu+2 more|arXiv (Cornell University)|Jul 28, 2020

Advanced Chemical Physics Studies4 citations

TL;DR

This paper proposes an orthogonalization-free, parallelizable algorithm (PCAL) for all-electron Kohn–Sham density functional theory calculations. By reformulating the total energy minimization problem and avoiding iterative orthogonalization, the method achieves high scalability and efficiency through BLAS-optimized operations, demonstrating superior convergence and speedup in large-scale molecular systems compared to SCF and MOptQR methods.

ABSTRACT

All-electron calculations play an important role in density functional theory, in which improving computational efficiency is one of the most needed and challenging tasks. In the model formulations, both nonlinear eigenvalue problem and total energy minimization problem pursue orthogonal solutions. Most existing algorithms for solving these two models invoke orthogonalization process either explicitly or implicitly in each iteration. Their efficiency suffers from this process in view of its cubic complexity and low parallel scalability in terms of the number of electrons for large scale systems. To break through this bottleneck, we propose an orthogonalization-free algorithm framework based on the total energy minimization problem. It is shown that the desired orthogonality can be gradually achieved without invoking orthogonalization in each iteration. Moreover, this framework fully consists of Basic Linear Algebra Subprograms (BLAS) operations and thus can be naturally parallelized. The global convergence of the proposed algorithm is established. We also present a precondition technique which can dramatically accelerate the convergence of the algorithm. The numerical experiments on all-electron calculations show the efficiency and high scalability of the proposed algorithm.

Motivation & Objective

• Address the computational bottleneck in all-electron Kohn–Sham DFT caused by cubic-complexity orthogonalization procedures.
• Develop a scalable, parallel algorithm that avoids explicit or implicit orthogonalization in each iteration.
• Enable efficient large-scale all-electron calculations by leveraging finite element discretization and BLAS-optimized operations.
• Establish global convergence and improve convergence speed via a novel preconditioner.
• Demonstrate robustness and high parallel scalability on large molecular systems, including C384 with 1152 orbitals.

Proposed method

• Reformulate the Kohn–Sham ground state problem as a total energy minimization with orthogonality constraints, avoiding the need for orthogonalization at each step.
• Propose a novel algorithmic framework (PCAL) that enforces orthogonality implicitly through optimization dynamics, not explicit projection.
• Use finite element discretization with radial mesh adaptation to resolve electron density singularities near nuclei.
• Implement the algorithm using only Basic Linear Algebra Subprograms (BLAS) operations, enabling high parallel scalability.
• Introduce a problem-specific preconditioner to accelerate convergence in the optimization process.
• Apply a line search strategy to ensure global convergence of the iterative scheme.

Experimental results

Research questions

• RQ1 Can the orthogonalization bottleneck in all-electron Kohn–Sham DFT be eliminated without sacrificing convergence or accuracy?
• RQ2 How can a Kohn–Sham total energy minimization framework be designed to achieve high parallel scalability while avoiding orthogonalization?
• RQ3 What is the impact of a novel preconditioner on the convergence rate of the orthogonalization-free algorithm?
• RQ4 How does the proposed method scale on large molecular systems compared to classical SCF and MOptQR solvers?
• RQ5 To what extent is the algorithm robust to initial guesses and algorithmic parameters in practical simulations?

Key findings

• The proposed PCAL algorithm achieves over 99% parallel efficiency when the number of orbitals exceeds 180, indicating strong scalability.
• For the C384 molecule (1152 orbitals), PCAL achieves a speedup factor of 3.76 on 16 cores, approaching ideal scalability.
• PCAL reduces CPU time significantly compared to SCF and MOptQR, with the largest reduction observed in large-scale systems.
• The algorithm shows insensitivity to initial guesses and algorithm parameters, demonstrating robustness in numerical experiments.
• The preconditioner accelerates convergence in all tested molecules (He, LiH, CH4, H2O, C6H6, C12H10N2), improving efficiency without increasing complexity.
• The method maintains global convergence and avoids divergence issues seen in SCF for large systems, unlike SCF which diverged in the C384 test.