Exact Coulomb cutoff technique for supercell calculations

We present a reciprocal space analytical method to cut off the long range interactions in supercell calculations for systems that are infinite and periodic in one or two dimensions, generalizing previous work to treat finite systems. The proposed cutoffs are functions in Fourier space, that are used as a multiplicative factor to screen the bare Coulomb interaction. The functions are analytic everywhere except in a subdomain of the Fourier space that depends on the periodic dimensionality. We show that the divergences that lead to the nonanalytical behavior can be exactly canceled when both the ionic and the Hartree potential are properly screened. This technique is exact, fast, and very easy to implement in already existing supercell codes. To illustrate the performance of the scheme, we apply it to the case of the Coulomb interaction in systems with reduced periodicity (as one-dimensional chains and layers). For these test cases, we address the impact of the cutoff on different relevant quantities for ground and excited state properties, namely: the convergence of the ground state properties, the static polarizability of the system, the quasiparticle corrections in the GW scheme, and the binding energy of the excitonic states in the Bethe-Salpeter equation. The results are very promising and easy to implement in all available first-principles codes.

Figure 1

Schematic description for the supercell construction in a 2D system. The upper sketch corresponds to the 2D-periodic case (i.e., a 2D bulk crystal). The middle sketch corresponds to a 0D-periodic system (i.e., a finite 2D system), and the bottom one to a 1D-periodic (i.e., an isolated chain). In the 0D-periodic case the electrons belonging to different cells do not interact, while in the 1D-periodic a chain does not interact with another, but all the electrons of the chain do interact with each other (see text for details).

Figure 2

Calculated total and ionic and Hartree potentials for a 3D-periodic (top) and 1D-periodic (bottom) Na chain.

Figure 3

Effect of the cutoff in a Na linear chain in a supercell size of $7.5\times 19\times 19\mathrm{a.u.}$ The bands obtained with an ordinary supercell calculation with no cutoff (dashed lines) are compared to the bands obtained applying the 1D cylindrical cutoff (solid line). As is explained in the text, only the unoccupied levels are affected by the cutoff.

Figure 4

Band structure of a planar sheet of Na atoms in a supercell size of $7.5\times 7.5\times 19\mathrm{a.u.}$ The bands with and without cutoff are identical for all the occupied states. The unoccupied states are influenced by different boundary conditions.

Figure 5

Top: Polarizability per unit cell of an $H_2$ chain in RPA as a function of the inverse supercell volume. The solid line extrapolates the values obtained with the cutoff potential, while the dashed lines extrapolates the values obtained with the bare Coulomb potential. The cutoff radius is $8.0\mathrm{a.u.}$ $\left(4.2\mathrm{\AA }\right)$. The interchain distance is indicated in the top axis. Bottom: Polarizability of the $H_2$ chain calculated with the finite cutoff potential of Ref. 35. In abscissa different values of the cutoff length along the chain axis. The dashed straight line indicates the value obtained with the cutoff of Eq. (13). In the inset we show the convergence of the polarizability with respect to the $k$ points sampling along the chain axis obtained with the cutoff of Eq. (13). In the upper axis it is indicated the maximum allowed length $h$ for each $k$ point sampling used in the calculation of the $G_x=0$ components by Eq. (29).

Figure 6

Convergence of the GW quasiparticle gap for the $H_2$ chain as a function of the inverse of the cell size, using the bare Coulomb potential (dashed line) and the cutoff potential (solid line). In the inset the behavior of the GW quasiparticle gap as a function of the value of the cutoff radius for a supercell with inter-chain distance of $32\mathrm{a.u.}$ $\left(17\mathrm{\AA }\right)$ is shown. The plateau obtained around a radius of $8\mathrm{a.u.}$ (i.e., one fourth of the supercell size) corresponds to the situation in which the radial images of chains no longer mutually interact, and the calculation is converged. Increasing the radius above approximately $12\mathrm{a.u.}$ the interaction is back and produces oscillations in the value of the gap.

Figure 7

Top: Photo absorption cross section of a linear chain of $H_2$ for different supercell volumes. In the legend the interchain distances corresponding to each volume are indicated. The intensity has been normalized to the volume of the supercell. The noninteracting absorption spectra and the spectra obtained with the cutoff potential are also included. Bottom: exciton binding energy vs supercell volume calculated using the cut off potential (solid line), and the bare Coulomb potential (dashed line).

Figure 8

Values of ${ϵ}_{00}^{-1}(q_x,q_y)$ in the $q_z=0$ plane for the $H_2$ chain with an interchain distance of $25\mathrm{a.u.}$, using the bare Coulomb (top) and the cutoff potential (bottom). The axis of the chain is along the $x$ direction.