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Efficient and accurate solver of the three-dimensional screened and unscreened Poisson's equation with generic boundary conditions
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10.1063/1.4755349
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Affiliations:
1 European Synchrotron Radiation Facility, 6 rue Horowitz, BP 220, 38043 Grenoble Cedex 9, France
2 Laboratoire de simulation atomistique (L_Sim), SP2M, UMR-E CEA / UJF-Grenoble 1, INAC, F-38054 Grenoble, France
a) Electronic mail: alessandro.cerioni@esrf.fr.
b) Electronic mail: luigi.genovese@cea.fr.
J. Chem. Phys. 137, 134108 (2012)
/content/aip/journal/jcp/137/13/10.1063/1.4755349
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/13/10.1063/1.4755349
View: Figures

Figures

FIG. 1.

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.

FIG. 2.

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).

FIG. 3.

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).

FIG. 4.

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).

FIG. 5.

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.

FIG. 6.

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).

FIG. 7.

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).

FIG. 8.

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).

FIG. 9.

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).

FIG. 10.

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).

FIG. 11.

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, .

FIG. 12.

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.

FIG. 13.

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 (μ0 = 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 μ0 is increased. Each curve is normalised to one to improve readability.

FIG. 14.

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.

/content/aip/journal/jcp/137/13/10.1063/1.4755349
2012-10-04
2014-04-17

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