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Nature

Towards an exact description of electronic wavefunctions in real solids

George H. Booth, Andreas Grüneis, Georg Kresse, Ali Alavi

Published Dec 18, 2012• 567 citations
Publisher
DOI
10.1038/nature11770

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The dimension of the Hilbert space needed for the description of an interacting electron system increases exponentially with electron number N. As pointed out by Kohn this exponential wall problem (EWP) limits the concept of many-electron wavefunctions based on solutions of Schrödinger’s equation to [Formula: see text], where [Formula: see text] [Rev. Mod. Phys. 71, 1253 (1999)]. This limitation does not hold when the electronic interactions are neglected or treated in a mean-field approximation. The EWP has directed electronic structure calculations for solids to methods like density-functional theory which avoid dealing with many-electron wavefunctions. We show that the highly unsatisfactory limitation of many-electron wavefunctions to [Formula: see text] can be overcome by generalizing their definition. We show for the ground state of a large system that it is the logarithm of the solution of Schrödinger’s equation which should be used to characterize this system. In this case the wavefunctions of independent subsystems add up rather than multiply. This feature is realized by the action function. It provides a simple physical picture for the resolution of the EWP.
Electronic structure calculations for solids based on many-electron wavefunctions have been hampered by the argument that for large electron numbers wavefunctions are not a legitimate scientific concept, because they face an exponential wall problem. We show that this problem can be avoided by using cumulant techniques in formulating wavefunctions. Therefore calculations for solids based on many-electron functions are possible and useful. This includes also systems with strongly correlated electrons.
While Diffusion Monte Carlo (DMC) is in principle an exact stochastic method for \textit{ab initio} electronic structure calculations, in practice the fermionic sign problem necessitates the use of the fixed-node approximation and trial wavefunctions with approximate nodes (or zeros) must be used. This approximation introduces a variational error in the energy that potentially can be tested and systematically improved. Here, we present a computational method that produces trial wavefunctions with systematically improvable nodes for DMC calculations of periodic solids. These trial wavefunctions are efficiently generated with the configuration interaction using a perturbative selection made iteratively (CIPSI) method. A simple protocol in which both exact and approximate results for finite supercells are used to extrapolate to the thermodynamic limit is introduced.