A model interaction is introduced for quantum many-body simulations of Coulomb systems using periodic boundary conditions. The interaction gives much smaller finite size effects than the standard Ewald interaction and is also much faster to compute. Variational quantum Monte Carlo simulations of diamond-structure silicon with up to 1000 electrons demonstrate the effectiveness of our method.
A model interaction is introduced for quantum many-body simulations of Coulomb systems using periodic boundary conditions. The interaction gives much smaller finite size effects than the standard Ewald interaction and is also much faster to compute. Variational quantum Monte Carlo simulations of diamond-structure silicon with up to 1000 electrons demonstrate the effectiveness of our method.
Quantum Monte Carlo (QMC) calculations are only possible in finite systems and so solids and liquids must be modeled using small simulation cells subject to periodic boundary conditions. The resulting finite-size errors are often corrected using data from local-density functional or Hartree-Fock calculations, but systematic errors remain after these corrections have been applied. The results of our jellium QMC calculations for simulation cells containing more than 600 electrons confirm that the residual errors are significant and decay very slowly as the system size increases. We show that they are sensitive to the form of the model Coulomb interaction used in the simulation cell Hamiltonian and that the usual choice, exemplified by the Ewald summation technique, is not the best. The finite-size errors can be greatly reduced and the speed of the calculations increased by a factor of 20 if a better choice is made. Finite-size effects plague most methods used for extended Coulomb systems and many of the ideas in this paper are quite general: they may be applied to any type of quantum or classical Monte Carlo simulation, to other many-body approaches such as the GW method, and to Hartree-Fock and density-functional calculations.
Further developments are introduced in the theory of finite-size errors in quantum many-body simulations of extended systems using periodic boundary conditions. We show that our recently introduced model periodic Coulomb interaction [A. J. Williamson et al., Phys. Rev. B 55, R4851 (1997)] can be applied consistently to all Coulomb interactions in the system. The model periodic Coulomb interaction greatly reduces the finite-size errors in quantum many-body simulations. We illustrate the practical application of our techniques with Hartree-Fock and variational and diffusion quantum Monte Carlo calculations for ground- and excited-state calculations. We demonstrate that the finite-size effects in electron promotion and electron addition/subtraction excitation energy calculations are very similar.
We analyze the problem of eliminating finite-size errors from quantum Monte Carlo (QMC) energy data. We demonstrate that both (i) adding a recently proposed [S. Chiesa et al., Phys. Rev. Lett. 97, 076404 (2006)] finite-size correction to the Ewald energy and (ii) using the model periodic Coulomb (MPC) interaction [L. M. Fraser et al., Phys. Rev. B 53, 1814 (1996); P. R. C. Kent et al., Phys. Rev. B 59, 1917 (1999); A. J. Williamson et al., Phys. Rev. B 55, R4851 (1997)] are good solutions to the problem of removing finite-size effects from the interaction energy in cubic systems provided the exchange-correlation (XC) hole has converged with respect to system size. However, we find that the MPC interaction distorts the XC hole in finite systems, implying that the Ewald interaction should be used to generate the configuration distribution. The finite-size correction of Chiesa et al. [Phys. Rev. Lett. 97, 076404 (2006)] is shown to be incomplete in systems of low symmetry. Beyond-leading-order corrections to the kinetic energy are found to be necessary at intermediate and high densities; we investigate the effect of adding such corrections to QMC data for the homogeneous electron gas. We analyze finite-size errors in two-dimensional systems and show that the leading-order behavior differs from that which has hitherto been supposed. We compare the efficiencies of different twist-averaging methods for reducing single-particle finite-size errors and we examine the performance of various finite-size extrapolation formulas. Finally, we investigate the system-size scaling of biases in diffusion QMC.
Ab initio quantum Monte Carlo methods, in principle, allow for the calculation of exact properties of correlated many-electron systems but are, in general, limited to the simulation of a finite number of electrons N under periodic boundary conditions. Therefore, an accurate theory of finite-size effects is indispensable to bridge the gap to realistic applications in the thermodynamic limit. In this work, we revisit the uniform electron gas at finite temperature, as it is relevant to contemporary research, e.g., in the field of warm dense matter. In particular, we present a new scheme to eliminate finite-size effects both in the static structure factor S(q) and in the interaction energy v, which is based on the density response formalism. We demonstrate that this method often allows us to obtain v in the thermodynamic limit within a relative accuracy of ∼0.2% from as few as N = 4 electrons without any empirical choices or knowledge of results for other values of N. Finally, we evaluate the applicability of our method upon increasing the density parameter rs and decreasing the temperature T.
The interaction and exchange-correlation contributions to the ground-state energy of an arbitrary many-electron system can be obtained from a spherical average of the wave-vector-dependent diagonal structure factor (SF). We model the continuous-k spherically averaged SF using quantum Monte Carlo calculations in finite simulation cells. We thus derive a method that allows us to substantially reduce the troublesome Coulomb finite-size errors that are usually present in ground-state energy calculations. To demonstrate this, we perform variational Monte Carlo calculations of the interaction energy of the homogeneous electron gas. The method is, however, equally applicable to arbitrary inhomogeneous systems.
Simulations of extended quantum systems are typically performed by extrapolating results of a sequence of finite-system-size simulations to the thermodynamic limit. In the quantum Monte Carlo community, twist-averaging was pioneered as an efficient strategy to eliminate one-body finite size effects. In the dynamical mean field community, cluster generalizations of the dynamical mean field theory were formulated to study systems with nonlocal correlations. In this work, we put the twist-averaging and the dynamical cluster approximation variant of the dynamical mean field theory onto equal footing, discuss commonalities and differences, and compare results from both techniques to the standard periodic boundary technique. At the example of Hubbard-type models with local, short-range and Yukawa-like longer range interactions we show that all methods converge to the same limit, but that the convergence speed differs in practice. We show that embedding theories are an effective tool for managing both one-body and two-body finite size effects, in particular if interactions are averaged over twist angles.