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Using the finite simulation-cell homogeneous electron gas (HEG) as a model,we investigate the convergence of the correlation energy to the complete basisset (CBS) limit in methods utilising plane-wave wavefunction expansions. Simpleanalytic 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 numberof plane waves used in the calculation, allowing for straightforwardextrapolation to the CBS limit. As we shall show, the choice of basis settruncation 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 avariety 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 andcompared with exact benchmarks. A transformation can map the orbitals of ageneral solid state system onto the HEG plane wave basis and thereby allowapplication of these methods to more realistic physical problems.

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