Convergence of many-body wavefunction expansions using a plane wave basis: from the homogeneous electron gas to the solid state

TL;DR

This paper studies how many-body wavefunction expansions using plane wave bases converge in the homogeneous electron gas and solid state, proposing new basis set truncation methods to improve convergence and applying these to various quantum chemistry methods.

Contribution

It introduces new basis set truncation schemes based on momentum transfer vectors that enhance convergence in plane wave expansions for many-electron systems.

Findings

Basis set incompleteness error decays as 1/M, enabling straightforward CBS extrapolation.

Proposed truncation schemes significantly improve convergence rates.

Finite basis energies closely match exact benchmarks across multiple methods.

Abstract

Using the finite simulation-cell homogeneous electron gas (HEG) as a model, we investigate the convergence of the correlation energy to the complete basis set (CBS) limit in methods utilising plane-wave wavefunction expansions. Simple analytic and numerical results from second-order M{\o}ller-Plesset theory (MP2) suggest a 1/M decay of the basis-set incompleteness error where M is the number of plane waves used in the calculation, allowing for straightforward extrapolation to the CBS limit. As we shall show, the choice of basis set truncation when constructing many-electron wavefunctions is far from obvious, and here we propose several alternatives based on the momentum transfer vector, which greatly improve the rate of convergence. This is demonstrated for a variety of wavefunction methods, from MP2 to coupled-cluster doubles theory (CCD) and the random-phase approximation plus second-order screened exchange (RPA+SOSEX). Finite basis-set energies are presented for these methods and compared with exact benchmarks. A transformation can map the orbitals of a general solid state system onto the HEG plane wave basis and thereby allow application of these methods to more realistic physical problems.