Element orbitals for Kohn-Sham density functional theory

We present a method to discretize the Kohn-Sham Hamiltonian matrix in the pseudopotential framework by a small set of basis functions automatically contracted from a uniform basis set such as plane waves. Each basis function is localized around an element, which is a small part of the global domain containing multiple atoms. We demonstrate that the resulting basis set achieves meV accuracy for three-dimensional densely packed systems with a small number of basis functions per atom. The procedure is applicable to both insulating and metallic systems.

Figure 1

Sketch for the construction of adaptive local basis functions and element orbitals. Each adaptive local basis function is supported in an element. Each element orbital is supported in an extended element.

Figure 2

(a) Convergence of adaptive local basis functions (ALB) for a 3D bulk Na system with 432 atoms. (b) Convergence of element orbitals (EOs) for the same Na system with fixed number of ALBs. (c) Convergence in terms of the localization radius for the same Na system with fixed number of ALBs and fixed number of EOs.

Figure 3

The isosurface of the first nine element orbitals belonging to the same extended element, for a 3D disordered bulk Na system in a supercell with 432 atoms. The 27 Na atoms nearest to the element orbitals within a sphere of radius $6.0$ a.u. are plotted as gold balls. The positive and negative part of the element orbitals are represented by red and blue color, respectively.

Figure 4

(a) The value of the element orbitals ${\phi }_1$ (blue solid line), ${\phi }_4$ (red dashed line), and ${\phi }_7$ (black dot dashed line) along one $\left[100\right]$ direction of a 3D disordered bulk Na system with 432 atoms. The two red circles indicate the boundary of the extended element. (b) Zoom-in of (a) to the region near the boundary of the extended element. The same set of element orbitals ${\phi }_1$ (blue solid line with circles), ${\phi }_4$ (red dashed line with triangles), and ${\phi }_7$ (black dot dashed line with diamonds) are shown, with the symbols indicating the position of the numerical grids. The red circle indicates the boundary of the extended element.

Figure 5

The total free energy per atom for 3D bulk Na system with 432 atoms (a) and 3D bulk Si system with 512 atoms (b), with different lattice constants calculated from abinit (red line with diamonds) and from element orbitals (blue dashed line with circles).

Figure 6

Graphene sheet consisting of 32 C atoms (cyan balls) with 1 C atom substituted by a Si atom (gold ball). Each black box represents an element. (a) The first element orbital ${\phi }_1$ (green) for the upper element with 2 C atoms, and the first element orbital ${\phi }_1$ (red) for the lower element with 1 C atom and 1 Si atom. (b) The second element orbital ${\phi }_2$ (green for the positive part and black for the negative part) for the upper element with 2 C atoms, and the second element orbital ${\phi }_2$ (red for the positive part and blue for the negative part) for the lower element with 1 C atom and 1 Si atom.

Figure 7

(a) The atomic configuration of a graphene sheet consisting of 512 C atoms (cyan balls), with 128 C atoms randomly selected and substituted by Si atoms (gold balls). (b) The electron density across $y-z$ plane.

Figure 8

The total computational time per SCF iteration (red solid line with upward-pointing triangles) for 3D bulk Na systems ranging from 128 to 4394 atoms. The breakdown of the total computational time includes the time for using lobpcg to generate adaptive local basis functions (blue dashed line with diamonds), the time for constructing the element orbitals from adaptive local basis functions (black dot dashed line with circles), the time for solving the generalized eigenvalue problem using the dense scalapack solver (green solid line with left-pointing triangles), the overhead time for solving the DG problem (magenta dashed line with right-pointing triangles), and the rest of the time in a SCF iteration (cyan dot dashed line with stars).

Figure 9

The memory cost per processor (a) and the communication percentage (b) for 3D bulk Na systems ranging from 128 to 4394 atoms.