^{1,a)}, Luigi Genovese

^{2,b)}, Alessandro Mirone

^{1}and Vicente Armando Sole

^{1}

### Abstract

We present an explicit solver of the three-dimensional screened and unscreened Poisson's equation, which combines accuracy, computational efficiency, and versatility. The solver, based on a mixed plane-wave/interpolating scaling function representation, can deal with any kind of periodicity (along one, two, or three spatial axes) as well as with fully isolated boundary conditions. It can seamlessly accommodate a finite screening length, non-orthorhombic lattices, and charged systems. This approach is particularly advantageous because convergence is attained by simply refining the real space grid, namely without any adjustable parameter. At the same time, the numerical method features scaling of the computational cost (*N* being the number of grid points) very much like plane-wave methods. The methodology, validated on model systems, is tailored for leading-edge computer simulations of materials (including *ab initio*electronic structure computations), but it might as well be beneficial for other research domains.

The authors thank Thierry Deutsch for valuable suggestions on the manuscript and Claudio Ferrero for the critical proofreading. A.C. acknowledges the financial support of the French National Research Agency in the frame of the “NEWCASTLE” project.

I. INTRODUCTION

II. FREE BOUNDARY CONDITIONS

III. WIRE-LIKE BOUNDARY CONDITIONS

IV. SURFACE-LIKE BOUNDARY CONDITIONS

V. NUMERICAL RESULTS

VI. CONCLUSION

### Key Topics

- Green's function methods
- 16.0
- Electrostatics
- 14.0
- Poisson's equation
- 12.0
- Carrier density
- 8.0
- Boundary value problems
- 6.0

## Figures

Accuracy of the approximation of the function *e* ^{−x }/*x* with 136 Gaussians used in the solution of the screened Poisson's equation for the case of free BC. The range of the independent variable *x* is [10^{−9}, 33]. We plot both the absolute and the relative error because the latter is a better indicator close to the origin (where the fitted function takes on very large values), while the former is a reliable signature of the goodness of the fit towards the opposite end. Note that at *x* = 33, the function *e* ^{−x }/*x* is already smaller than the machine precision.

Accuracy of the approximation of the function *e* ^{−x }/*x* with 136 Gaussians used in the solution of the screened Poisson's equation for the case of free BC. The range of the independent variable *x* is [10^{−9}, 33]. We plot both the absolute and the relative error because the latter is a better indicator close to the origin (where the fitted function takes on very large values), while the former is a reliable signature of the goodness of the fit towards the opposite end. Note that at *x* = 33, the function *e* ^{−x }/*x* is already smaller than the machine precision.

Accuracy of the approximation of the Green's function with 144 Gaussians as used in the solution of the screened Poisson's equation for the case of wire-like BC. The range of the independent variable *x* is [10^{−9}, 30] for the function and [10^{−9}, 1] for log (*x*).

Accuracy of the approximation of the Green's function with 144 Gaussians as used in the solution of the screened Poisson's equation for the case of wire-like BC. The range of the independent variable *x* is [10^{−9}, 30] for the function and [10^{−9}, 1] for log (*x*).

Accuracy test for the case of free/isolated boundary conditions in the absence of screening (*m* is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of free/isolated boundary conditions in the absence of screening (*m* is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of free/isolated boundary conditions in the presence of screening (*m* = 16 is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of free/isolated boundary conditions in the presence of screening (*m* = 16 is the order of the ISF, *h* the grid spacing).

Influence of the (cubic) simulation box size on the accuracy of the Hartree energy. *L* stands for the box size, whereas *L* _{ref.} stands for the box size for which ρ(*r* = *L*) = 2.21 × 10^{−12} bohr^{−3}.

Influence of the (cubic) simulation box size on the accuracy of the Hartree energy. *L* stands for the box size, whereas *L* _{ref.} stands for the box size for which ρ(*r* = *L*) = 2.21 × 10^{−12} bohr^{−3}.

Accuracy test for the case of surface-like boundary conditions in the absence of screening (*m* is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of surface-like boundary conditions in the absence of screening (*m* is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of surface-like boundary conditions in the presence of screening (*m* = 16 is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of surface-like boundary conditions in the presence of screening (*m* = 16 is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of wire-like boundary conditions in the absence of screening (*m* is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of wire-like boundary conditions in the absence of screening (*m* is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of wire-like boundary conditions in the presence of screening (*m* = 16 is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of wire-like boundary conditions in the presence of screening (*m* = 16 is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of wire-like boundary conditions (WBC) with monopolar charge density distribution (*m* is the order of the ISF, *h* the grid spacing).

Accuracy test for the case of wire-like boundary conditions (WBC) with monopolar charge density distribution (*m* is the order of the ISF, *h* the grid spacing).

Accuracy in the computation of the Hartree linear energy density—see Eq. (37)—in the case of wire-like boundary conditions (WBC) with monopolar charge density distribution (*m* is the order of the ISF, *h* the grid spacing). In our setup, .

Accuracy in the computation of the Hartree linear energy density—see Eq. (37)—in the case of wire-like boundary conditions (WBC) with monopolar charge density distribution (*m* is the order of the ISF, *h* the grid spacing). In our setup, .

Gaussian density charge distribution (red, dashed mesh) as a function of the isolated directions (*x*, *y* in our notation) and the corresponding electrostatic potential (black, solid mesh) evaluated for different values of the screening, namely upon increasing boundary thickness. The charge density is implicitly periodic along *z* (wire-like BC). The amplitude of the plotted density charge distribution is multiplied by a factor 10 with respect to the actual value in order to improve the readability of the picture. Only half of the solution is drawn to highlight its profile.

Gaussian density charge distribution (red, dashed mesh) as a function of the isolated directions (*x*, *y* in our notation) and the corresponding electrostatic potential (black, solid mesh) evaluated for different values of the screening, namely upon increasing boundary thickness. The charge density is implicitly periodic along *z* (wire-like BC). The amplitude of the plotted density charge distribution is multiplied by a factor 10 with respect to the actual value in order to improve the readability of the picture. Only half of the solution is drawn to highlight its profile.

Electrostatic potential generated by a pair of planar charge distributions of opposite sign (positive in red/solid; negative in black/dashed) modelling a planar capacitor unlimited in the periodic directions. The piecewise planar behavior corresponds to the case with no screening (μ_{0} = 0), whereas the other solutions are obtained with upon increasing the boundary thickness. The potential is more and more localized around the capacitor's plates and falls rapidly to zero as μ_{0} is increased. Each curve is normalised to one to improve readability.

Electrostatic potential generated by a pair of planar charge distributions of opposite sign (positive in red/solid; negative in black/dashed) modelling a planar capacitor unlimited in the periodic directions. The piecewise planar behavior corresponds to the case with no screening (μ_{0} = 0), whereas the other solutions are obtained with upon increasing the boundary thickness. The potential is more and more localized around the capacitor's plates and falls rapidly to zero as μ_{0} is increased. Each curve is normalised to one to improve readability.

Electrostatic potential generated by a cylindrical capacitor, periodic in the vertical direction. The different solutions correspond to upon increasing the boundary thickness. Each curve is normalised to one for sake of readability. Only half of the solution is drawn so as to highlight the potential profile.

Electrostatic potential generated by a cylindrical capacitor, periodic in the vertical direction. The different solutions correspond to upon increasing the boundary thickness. Each curve is normalised to one for sake of readability. Only half of the solution is drawn so as to highlight the potential profile.

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