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Accurate and efficient algorithm for Bader charge integration

Min Yu, Dallas R. Trinkle

arXiv:1010.4916v1cond-mat.mtrl-sciphysics.chem-ph

TL;DR

Discrete-grid Bader integration must accurately assign basin volumes despite difficult zero-flux boundaries and grid-dependent errors. The paper derives a trajectory-flux weight for fractional grid-volume assignments, achieving efficient linear scaling and quadratic convergence, with reported atomic integration errors at least an order of magnitude smaller than the near-grid method and a 0.003 e result.

  • Problem

    Accurate integration over Bader basins is difficult on discrete grids because zero-flux surfaces and grid-based assignments introduce integration errors.

  • Method

    The weight method computes each grid volume’s fractional contribution to a basin from the fraction of trajectories flowing to that basin.

  • Results

    The method has quadratic convergence and atomic integration errors at least one order of magnitude smaller than the near-grid method, including a reported 0.003 e result.

  • Takeaways & Limitations

    The weight method provides an efficient and accurate way to identify attraction basins and integrate functions over them on discrete grids.

Abstract

from arXiv · show

We propose an efficient, accurate method to integrate the basins of attraction of a smooth function defined on a general discrete grid, and apply it to the Bader charge partitioning for the electron charge density. Starting with the evolution of trajectories in space following the gradient of charge density, we derive an expression for the fraction of space neighboring each grid point that flows to its neighbors. This serves as the basis to compute the fraction of each grid volume that belongs to a basin (Bader volume), and as a weight for the discrete integration of functions over the Bader volume. Compared with other grid-based algorithms, our approach is robust, more computationally efficient with linear computational effort, accurate, and has quadratic convergence. Moreover, it is straightforward to extend to non-uniform grids, such as from a mesh-refinement approach, and can be used to both identify basins of attraction of fixed points and integrate functions over the basins.

I. INTRODUCTION

Bader partitioning assigns space to basins whose gradient trajectories reach the same electron-density maximum, but discrete-grid methods face boundary and integration errors. The proposed weight method uses fractional grid-volume assignments to improve accuracy while retaining efficient scaling and extending to nonuniform grids.

  • Bader partitioning: Bader volumes consist of points whose density-gradient trajectories reach the same unique electron-density maximum, separated by zero-flux surfaces.The dividing surface satisfies ∇ρ · n̂ = 0.
  • Problem: Accurate zero-flux-surface determination is challenging when charge density is available only on a discrete real-space grid.This difficulty motivates grid-based approximations and fractional volume assignments.
  • Prior methods: Existing approaches have important limitations, including huge computational cost for complicated topologies and failure on surfaces with sharp cusps or points.These limitations are reported for octal-tree and elastic-sheet approaches, respectively.
  • Prior methods: The near-grid method removes lattice bias and scales linearly with grid size, but still requires self-consistent volume-assignment iteration and has integration error linear in grid spacing.Consequently, accurate calculations for large systems may require very fine grids.
  • Weight method: The proposed weight function represents the fraction of each grid volume belonging to a Bader basin, improving integration accuracy for charge density and kinetic energy.The weight is computed from the integrated flux of trajectories from a grid volume to neighboring volumes.
  • Contributions: The algorithm combines linear computational effort with quadratic integration-error scaling and is straightforward to extend to nonuniform adaptive mesh-refinement grids.It is also presented as a simple, extendable method for basin identification and integration over Bader volumes.

II. THE WEIGHT METHOD

The weight method approximates basin membership on a discrete grid by tracking gradient-flow fractions through Voronoi volumes, producing weights for integration over attraction basins. Its formulation yields quadratic convergence and supports efficient, non-uniform-grid computation.

  • Bader basins are sets of points whose charge-density trajectories flow to the same local maximum, with boundaries satisfying zero flux.
  • The method assigns each grid point a weight wA(X) equal to the fraction of its Voronoi volume whose trajectories end in basin A.
  • The resulting discrete integral converges quadratically because linear-order weight errors produce quadratic-order integration errors.
  • Gradient-flow probability evolves through neighboring Voronoi volumes, and fluxes are approximated across their shared facets.
  • Weights are solved after sorting grid points by decreasing density, with local maxima initiating basins and boundary points receiving fractional assignments.
  • The algorithm avoids self-consistency, has linear computational effort, and extends to non-uniform grids by computing Voronoi volumes and facets.

III. EVALUATION OF NUMERICAL CONVERGENCE

The evaluation estimates Bader-volume integration error by integrating the Laplacian of charge density within each basin. It compares the weight and near-grid methods across analytic and real charge densities.

  • Atomic integration error is estimated from the nonzero integral of the charge-density Laplacian within each Bader volume.
  • The comparison uses zero-flux surfaces and includes analytic charge densities in orthogonal and non-orthogonal cells.
  • The evaluation also tests real systems, including an ionic compound, a semiconductor, and the Na atom in crystalline NaCl.

A. Gaussian densities

For Gaussian charge densities, the near-grid method misassigns points near dividing surfaces, whereas the weight method reduces integration error and approaches quadratic convergence.

  • The Gaussian test constructs a three-dimensional model density from three Gaussian functions and varies grid resolution from N=20 to 100.
  • Near-grid basin assignments are wrong for grid points close to dividing surfaces where charge-density gradients are nearly parallel to those surfaces.
  • 130 reduction in error from the near-grid method is reported for the weight method in the FCC-cell comparison.
  • 0.71 is the fitted convergence rate for the weight method, compared with 0.45 for the near-grid method.
  • The weight method has lower absolute error and faster convergence, with no large-grid-spacing crossover favoring near-grid.

B. Titania bulk

In rutile TiO2, the weight method systematically lowers maximal atomic integration error relative to the near-grid method and converges faster across charge-density grids.

  • The TiO2 test uses DFT charge densities for rutile’s six-atom tetragonal unit cell across grids from 45×45×30 to 150×150×100 points.
  • The weight method corrects grid points misassigned by the near-grid method, with gradient directions used to verify basin assignments.
  • The weight method gives maximal atomic integration error one order of magnitude lower than the near-grid method systematically.
  • 1.0eV is exceeded by the near-grid atomic integration error at the minimal 45×45×30 grid.
  • ∼2/3 convergence for the weight method corresponds to quadratic convergence, versus ∼1/3 and linear convergence for near-grid.
  • Improved error and faster convergence allow more accurate density integration with fewer grid points than near-grid.

C. NaCl crystal

In NaCl, the weight method yields lower atomic-integration errors and faster convergence than the near-grid method, while both produce smooth, monotonic Bader-charge convergence.

  • The weight method’s maximal atomic integration error is at least one order of magnitude lower than the near-grid method across tested NaCl grids.
  • The weight method error scales approximately as the 2/3 power of grid points, compared with approximately the 1/3 power for the near-grid method.
  • Both methods show monotonic, smooth convergence for Na Bader charge, with converged values of 0.878e and 0.881e for the near-grid and weight methods, respectively.
  • The near-grid method’s systematic misassignment may not improve with increasing grid-point density, potentially complicating continually refined-grid approaches.
  • At 60^3 grid points, the near-grid method underestimates Na Bader charge by 0.01e, while the weight method underestimates it by 0.005e.

IV. COMPUTATIONAL EFFORT

The weight method analyzes charge-density grids with linear scaling and a smaller computational prefactor than the near-grid method, including for large systems.

  • The weight method requires overall computational effort that scales linearly with the number of grid points.
  • Only surface grid points require multiple atomic-weight calculations, while interior points require one calculation each.
  • For grids approaching 10^7–10^8 points and systems with hundreds of atoms, the authors find the weight method more accurate and efficient than the near-grid method.
  • Both methods scale linearly with grid points, but the weight method has a smaller computer-time prefactor.

V. CONCLUSIONS

The paper presents a weight method that assigns Voronoi-volume fractions to attraction basins, supports uniform and non-uniform grids, and improves integration accuracy over the near-grid method.

  • The weight method integrates functions over basins of attraction by assigning grid-point Voronoi-volume fractions to surrounding basins.
  • Weights are propagated from higher-density neighbors after grid points are sorted in descending density order.
  • The flow formulation across neighboring-cell dividing surfaces applies to both uniform and non-uniform grids.
  • Atomic integration error decreases inversely with the 2/3 power of grid points for the weight method, versus the 1/3 power for the near-grid method.
  • In TiO2 and NaCl tests, the weight method’s maximal atomic integration error is at least one order of magnitude smaller than the near-grid method’s.

Appendix A: Quadratic error in one-dimension

The appendix shows that the weight integration reproduces the one-dimensional basin contribution with an error of order h2, yielding quadratic convergence for smooth functions.

  • The one-dimensional analysis isolates boundary grid points as the possible source of errors linear in h.
  • Interior-grid integration contributes a total error quadratic in h.
  • For a boundary point, the weight method’s integration contribution differs from the true contribution by an error of order h2 rather than h.
  • The Taylor-expansion derivation agrees with the weight integration through an error of order h2.
  • The weight wA(X) represents the fraction of a grid volume whose trajectory ends in basin A, and corresponds to the basin’s Voronoi-volume fraction to first order in h.
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