Electron correlation is crucial for the description of a wide range of phenomena in physics, chemistry and biology - it plays a vital role in the description of chemical transformations, accounts for defect stability in the solid state, and is a key factor in the structure and dynamics of water. Electron correlation can be straightforwardly and systematically accounted for in wavefunction theory. However, accurate wavefunction-based methods exhibit steep scaling and often sluggish convergence with respect to the basis set used. Plane waves (PWs) would be ideally suited for the calculation of basis-set limit correlation energies because their use allows for simple extrapolation of the correlation energy in particular in solid-state and condensed-phase setups. However, the routine use of correlated wavefunction approaches in a PW basis is hampered by a prohibitive scaling as the number of virtual continuum states quickly reach the tens of thousands. Among the correlated wavefunction methods, second-order Møller-Plesset perturbation theory (MP2) is the most expedient approach and exhibits a comparably flat formal scaling of O(N5). Still, MP2 calculations in PWs have not been feasible for all but the smallest systems. By extension, the same restrictions apply to double hybrid (DH) exchange-correlation functionals of Density Functional Theory. Here, we introduce a method that is able to overcome the current limitations of MP2 theory (and DH functionals) in PWs. We introduce a stochastic sampling based on Monte-Carlo summation over continuum orbitals, which enables speedups of up to a factor of 1000 with respect to standard calculations. At the same time, stochastic sampling reduces the formal scaling from O(N5) to an unprecedented O(N2). Absolute correlation energies obtained with this approach are accurate to &lt; 0.1 kcal/mol or &lt; 0.1 meV/electron, respectively, when compared to conventional PW-calculations. Calculations were performed for isolated and periodic systems containing up to several hundreds of electrons. We will show how extrapolated, condensed-phase basis-set limit MP2 energies for a hydronium ion solvated in 32 water molecules can be obtained in less than a day on conventional computational infrastructure. Estimators for the stochastic error will also be discussed. Finally, we will demonstrate that stochastic summation also be applied to the Random Phase Approximation (RPA), which has recently regained popularity in particular in solid state physics.

Monte-Carlo summation over continnum states: Basis-set limit correlation energies for gaseous, condensed-phase and solid-state systems with unprecedented scaling

ACS Fall 2021

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