Real-space electronic-structure calculations with a time-saving double-grid technique

We present a set of efficient techniques in first-principles electronic-structure calculations utilizing the real-space finite-difference method. These techniques greatly reduce the overhead for performing integrals that involve norm-conserving pseudopotentials, solving Poisson equations, and treating models which have specific periodicities, while keeping a high degree of accuracy. Since real-space methods are inherently local, they have a lot of advantages in applicability and flexibility compared with the conventional plane-wave approach and promise to be well suited for large and accurate ab initio calculations. In order to demonstrate the potential power of these techniques, we present several applications for electronic structure calculations of atoms, molecules, a helical nanotube, and a fullerene bulk.

Figure 1

Double grid adopted in the text. The “×” and “•” correspond to coarse- and dense-grid points, respectively. The circle shows the core region of an atom that is taken to be sufficiently large to contain the cutoff region of nonlocal pseudopotentials.

Figure 2

Functions on coarse- and dense-grid points in the one-dimensional case. $X_J\left(x_j\right)$ represents a coarse (dense-) grid point with $j=nJ+t(0⩽t<n)$, and so $X_J=x_{nJ}$. (a) The inner product between the wave function $\psi \left(x\right)$ and the nonlocal part of pseudopotential $v\left(x\right)$ evaluated over coarse-grid points. (b) The inner product evaluated over dense-grid points.

Figure 3

Convergence of the total energy for the hydrogen atom as a function of the coarse-grid cutoff energy $E_c^{\mathrm{coars}}$. The atom is located at the center between adjacent coarse-grid points.

Figure 4

Variation of the total energy as a function of the displacement of the flourine atom relative to coarse-grid points along a coordinate axis. The coarse-grid spacing $h_{\mu }$ is 0.33 a.u.

Figure 5

Relative positions with respect to the grids for ${\mathrm{F}}_2$ and ${\mathrm{Cu}}_2$ molecules and adiabatic potential curves. (a) The center of gravity of the molecule is placed at the center of neighboring grids with respect to the $x$, $y$, and $z$ directions. In the figure, $\times$ indicates the coarse grid and • represents nuclei. (b) The center of gravity of the molecule is placed on the grid plane which orthogonally intersects the coupling axis. For the grid plane parallel to the coupling axis, the center of gravity is placed at the center of neighboring grids. (c) Adiabatic potential curves of ${\mathrm{F}}_2$ molecules. The grid spacing is 0.33 a.u. (d) Adiabatic potential curve for ${\mathrm{Cu}}_2$ molecules. The grid spacing is 0.27 a.u. The calculated points are fit to spline functions as a guide to the eye.

Figure 6

Relative errors between the Hartree potential obtained using boundary values computed by each multipole expansion method and that evaluated by the analytical solution for the cross section at $z=0.075\mathrm{a}.\mathrm{u}.$ Results using boundary values (a) obtained by the FCD-MPE method and (b) obtained by the multipole expansion around one point. Spheres show the locations of the nuclei.

Figure 7

(Color online) (a) Front and (b) axial views of the (5, 5) carbon nanotube. $r$, $\phi$, and $L_z$ are the radius, the helical twist angle, and the translational shift of the nanotube, respectively.

Figure 8

(a) Electronic band structure of the (5, 5) carbon nanotube using the supercell with length $L_z$, (b) pseudoband structure, and (c) electronic band structure using the conventional supercell with the length $2L_z$. The zero of energy is chosen to be the Fermi level.

Figure 9

Schematic image of uneven periodic boundary conditions. The shaded area represents a supercell, and $dx$, $dy$, and $dz$ are grid spacings in $x$, $y$, and $z$ direction, respectively.

Figure 10

(Color online) Computational model of the orthorhombic phase of bulk ${\mathrm{C}}_{20}$. The rectangular solid represents the supercell, and the red and blue balls are carbon atoms in the supercell. The bond between C1 and C2 is a double bond, and others are single bonds.

Figure 11

Variation of high-frequency dielectric constant with respect to the direction of applied external electric field.

Figure 12

(Color online) Induced charge on $y\text{−}z$ planes including (a) atoms C1 and C2 and (b) atoms C3 and C4 due to applied external electric field. Contour interval is $5\times {10}^{-4}$ electrons per unit cell. The large and small spheres indicate the atomic positions on and below the plane, respectively. The atoms labeled by C1, C2, C3, and C4 correspond to those in Fig. 10. Red (blue) regions represent positive (negative) contours.