Electrostatics in periodic boundary conditions and real-space corrections

We address periodic-image errors arising from the use of periodic boundary conditions to describe systems that do not exhibit full three-dimensional periodicity. The difference between the periodic potential, as straightforwardly obtained from a Fourier transform, and the potential satisfying any other boundary conditions can be characterized analytically. In light of this observation, we present an efficient real-space method to correct periodic-image errors, based on a multigrid solver for the potential difference, and demonstrate that excellent convergence of the energy with respect to cell size can be achieved in practical calculations. Additionally, we derive rapidly convergent expansions for determining the Madelung constants of point-charge assemblies in one, two, and three dimensions.

Figure 1

(Color online) (a) Open-boundary electrostatic potential $v$, (b) periodic electrostatic potential $v^{\prime }$, and (c) electrostatic-potential correction $v^{\text{corr}}=v-v^{\prime }$ for a pyridazine cation in a cubic cell of length $L=15\text{bohr}$. The potentials are plotted in three orthogonal planes $\left(Oxy\right)$, $\left(Oxz\right)$, and $\left(Oyz\right)$ passing through the center of the cell.

Figure 2

(Color online) Open-boundary electrostatic potential $v$, periodic potential $v^{\prime }$, and electrostatic-potential correction $v^{\text{corr}}$ averaged in the $xy$ plane parallel to the surface for (a) carbon monoxide adsorbed on a neutral platinum slab and (b) carbon monoxide adsorbed on a charged platinum slab.

Figure 3

(Color online) Corrective potential $v_0^{\text{corr}}$ for a cubic lattice of point charges and its parabolic approximation in the vicinity of the origin.

Figure 4

(Color online) PCC (e.g., Makov–Payne), GCC (e.g., Blöchl and LMCC), and DCC corrective potentials for a pyridazine cation ${\text{C}}_4{\text{H}}_5{\text{N}}_2^{+}$ in a cubic cell of length $L=15\text{bohr}$. The corrective potentials are plotted along the $z$ axis perpendicular to the plane of the molecule, as defined in Fig. 1. The PCC and GCC corrections are calculated up to dipole order. The spread of the Gaussian countercharges is $\sigma =0.5\text{bohr}$.

Figure 5

(Color online) Electrostatic energy of two Gaussians of unit charge and unit spread calculated via (a) real-space integration, (b) reciprocal-space integration with the PCC energy correction given by Eq. (21), and (c) reciprocal-space integration with the energy correction given by Eq. (15) in Ref. 5. The Gaussian charges are positioned at ${\mathbf{r}}_0=\left(-5,-5,-5\right)$ and ${\mathbf{r}}_1=\left(5,5,5\right)$ (corresponding to a quadrupole moment $Q$ of 153 a.u.).

Figure 6

(Color online) DCC total energy as a function of cell size for a pyridazine cation varying the coarse-grid cutoff $E_{\text{cut}}^{\text{corr}}$ from 10 Ry $\left(M=13\times 13\times 13\right)$ to 250 Ry $\left(M=N=73\times 73\times 73\right)$. Also depicted is the corrective potential $v^{\text{corr}}$ in the plane of the molecule as a function of the coarse-grid resolution at a cell size of 15 bohr.

Figure 7

(Color online) Total energy as a function of cell size for a neutral pyridazine molecule without correction and corrected using the PCC, GCC, and DCC schemes. The PCC and GCC corrections are calculated up to quadrupole order. The inset shows a pyridazine molecule in a cell of size $L=15\text{bohr}$.

Figure 8

(Color online) Total energy as a function of cell size for a pyridazine cation without correction and corrected using the PCC, GCC, and DCC schemes. The PCC and GCC corrections are calculated up to quadrupole order. The inset of the top graph shows a pyridazine cation in a cell of size $L=15\text{bohr}$.

Figure 9

(Color online) Accuracy of the total energy as a function of computational time without correction and using the PCC, GCC, and DCC schemes for cell sizes in the range of 12–19 bohr. For each scheme, the corrective potential is updated every five SCF iterations.

Figure 10

(Color online) Total energy as a function of transverse cell size for a PVDF chain without correction and using the GCC and DCC schemes.

Figure 11

(Color online) Convergence of the opposite Fermi energy $-{ϵ}_F$ as a function of transverse cell size for a Pt(100) slab without correction and using the GCC and DCC schemes.

Figure 12

(Color online) Fourier-decomposition calculation of the electrostatic potential $v\left({\mathbf{r}}_{\perp },z\right)={\sum }_{g_z}v\left({\mathbf{r}}_{\perp };g_z\right)e^{i{g}_{z}z}$ for an infinite PVDF chain. (1) The longitudinal Fourier transform of the charge density is calculated to obtain the contributions from each axial wave vector $g_z$. (2) The electrostatic potential generated by each Fourier component of the charge density is calculated using the Green’s functions. (3) The electrostatic potential is then transformed back to real space.

Figure 13

(Color online) Total energy as a function of transverse cell size for a $–{\left[{\text{CH}}_2{\text{CF}}_2\right]}_3–{\left[{\text{CF}}_2{\text{CH}}_2\right]}_3–$- polymer chain without correction, corrected using the density-countercharge scheme with full Fourier decomposition (DCC) and by limiting the density-countercharge decomposition to the linear-average $\mathbf{g}=0$ component (DCC/LA).

Figure 14

(Color online) Force on one of the fluorine atoms along a transverse lattice direction as a function of transverse cell size for a $–{\left[{\text{CH}}_2{\text{CF}}_2\right]}_3–{\left[{\text{CF}}_2{\text{CH}}_2\right]}_3–$ polymer chain without correction, corrected using the density-countercharge scheme with full Fourier decomposing (DCC) and by limiting the density-countercharge decomposition to the linear-average $\mathbf{g}=0$ component (DCC/LA).

Figure 15

(Color online) Charge-density linear response function $\chi \left(g_{\parallel }\right)$ as a function of transverse cell size for a graphene sheet without correction and corrected using the density-countercharge potential corresponding to $g_{\parallel }$ (DCC).

Figure 16

(Color online) Convergence of the Madelung constant as a function of the geometric parameter $L/\sigma$ for a cubic unit cell using the approximation given by Eq. (A6). [The black curve is Eq. (A6) without the $\pi {\sigma }^2/L^2$ and the complementary-error-function terms, the red curve is Eq. (A6) without the complementary-error-function term, and the blue curve is Eq. (A6)]. Note the negligible contribution of the complementary-error-function term beyond $L/\sigma =3$ and the improvement in convergence brought about by the term $\pi {\sigma }^2/L^2$.

Figure 17

(Color online) Accuracy of the total energy of a pyridazine cation as a function of computational time using the PCC, GCC, and MCC schemes for cell sizes in the range of 12–19 bohr. The labels $D$ (dipole) and $Q$ (quadrupole) indicate the order of the multipole expansion. For each scheme, the corrective potential is updated every five SCF iterations.