Finite-Size Correction in Many-Body Electronic Structure Calculations

Finite-size (FS) effects are a major source of error in many-body (MB) electronic structure calculations of extended systems. A method is presented to correct for such errors. We show that MB FS effects can be effectively included in a modified local density approximation calculation. A parametrization for the FS exchange-correlation functional is obtained. The method is simple and gives post-processing corrections that can be applied to any MB results. Applications to a model insulator ($P_2$ in a supercell), to semiconducting Si, and to metallic Na show that the method delivers greatly improved FS corrections.

Figure 1

Calculated and parameterized jellium exchange and correlation energies per electron vs $r_s$, for a range of supercell sizes $L$ (in Bohr). Top panel: exchange energies, with solid lines given by the fit in Eq. (3), and open symbols by $\mathbf{k}$-point averaged calculations. Bottom panel: correlation energies, with solid lines given by the fit in Eq. (6), open symbols by AF QMC calculations, and small filled symbols by the large-$N$ asymptotic expression in Eq. (5).

Figure 2

$P_2$ PBC total energy convergence vs supercell size. Standard ${\mathrm{DFT}}^{\infty }$ FS effect is different (too small) from that of MB AF QMC calculations. ${\mathrm{DFT}}^{\mathrm{FS}}$ parallels the MB calculation and leads to much more rapid convergence. The inset focuses on larger $L$ and shows the raw and corrected AF QMC results plotted as a function of $1/L^3$.

Figure 3

Total energy per atom of bulk Si in fcc supercells of size $n\times n\times n$ ($2n^3$ atoms). DMC energies in the main graph are from Kent et al. [1] (shifted here relative to the extrapolated infinite-size limit). FS corrections are shown from both standard DFT and the present method. The inset shows the calculated two-body correction as a function of inverse the number of atoms, for fcc and cubic supercells.

Figure 4

Top: Equation of state for bcc Na. The AF QMC energy is shown vs lattice constant $a$ for 16- and 54-atom supercells, with one-body (1B) and full FS corrections. With full correction (solid lines), 16- and 54-atom results are almost indistinguishable. QMC statistical errors include that of $\mathbf{k}$-point averaging. The vertical arrows indicate the calculated equilibrium $a$. Bottom: persistence of the two-body FS error, with finite slopes (number of atoms indicated).