Treecode-accelerated Green iteration for Kohn-Sham density functional theory

Highlights: • Integral equation method for Kohn-Sham Density Functional Theory using Green's functions. • A fixed-point problem for the integral operator is solved with Green Iteration. • High accuracy is achieved with adaptive mesh refinement, singularity subtraction, and Fejér quadrature • A GPU-accelerated barycentric Lagrange treecode computes fast approximations of discrete convolution sums. • This method is demonstrated for ground-state energy calculations of atoms and small molecules. We present a real-space computational method called treecode-accelerated Green Iteration (TAGI) for all-electron Kohn-Sham Density Functional Theory. TAGI is based on a reformulation of the Kohn-Sham equations in which the eigenvalue problem in differential form is converted into a fixed-point problem in integral form by convolution with the modified Helmholtz Green's function. In each self-consistent field (SCF) iteration, the fixed-points are computed by Green Iteration, where the discrete convolution sums are efficiently evaluated by a GPU-accelerated barycentric Lagrange treecode. Other techniques used in TAGI include a-priori adaptive mesh refinement, Fejér quadrature, singularity subtraction, gradient-free eigenvalue update, and Anderson mixing to accelerate convergence of the SCF and Green Iterations. Ground state energy computations of several atoms (Li, Be, O) and small molecules (H₂, CO, C₆H₆) demonstrate TAGI's ability to efficiently achieve chemical accuracy.