Finite-Size Error in Many-Body Simulations with Long-Range Interactions

We discuss the origin of the finite-size error of the energy in many-body simulation of systems of charged particles and we propose a correction based on the random-phase approximation at long wavelengths. The correction is determined mainly by the collective charge oscillations of the interacting system. Finite-size corrections, both on kinetic and potential energy, can be calculated within a single simulation. Results are presented for the electron gas and silicon.

Figure 1

Lower panel: static structure factor for the electron gas at $r_s=10$. Upper panel: $\Delta S_N=S_N(k)-S_{66}(k)$. The difference is computed using a spline function interpolation of $S_{66}$.

Figure 2

Electron gas variational energies per particle at $r_s=10$ using periodic boundary condition (PBC) and twist-averaged boundary condition (TABC). ${\Delta }_N\equiv \Delta T_N+\Delta V_N=\hslash {\omega }_p/2N$ (see text for the definition of $\Delta T_N$ and $\Delta V_N$). Error bars are smaller than symbol size.

Figure 3

Structure factor (left) and Jastrow potential (right) for diamond silicon at ambient pressure. The continuous lines are fit to the data (see text). The Jastrow potential shows a $k^{-2}$ divergence at small $k$ that was not explicitly imposed but obtained through energy variance minimization using the casino code. The smallest cell is the conventional fcc cubic cell of diamond (32 electrons). The two intermediate ones are, respectively, $2\times 2\times 2$ and $3\times 3\times 3$ supercells of the primitive cell (8 electrons). The largest one is a $2\times 2\times 2$ supercell of the conventional cubic cell.

Figure 4

Diffusion Monte Carlo energies per electron in diamond silicon at $r_s=2.0$ using periodic boundary condition (PBC) and the model periodic Coulomb interaction (MPC). “A” corrects for momentum quantization effects (see text). ${\Delta }_N\equiv \Delta T_N+\Delta V_N$ (see text). Energies and the $S_N(\mathbf{k})$ and $u_N(\mathbf{k})$ used to compute ${\Delta }_N$ are all obtained in simulations with the same number of particles. Error bar on the energies are smaller than the size of the symbols. See caption of Fig. 3 for the description of the cells.