Efficient Treatment of Correlation Energies at the Basis-Set Limit by Monte Carlo Summation of Continuum States

Abstract

The calculation of electron correlation is vital for the description of atomistic phenomena in physics, chemistry, and biology. However, accurate wavefunction-based methods exhibit steep scaling and often sluggish convergence with respect to the basis set at hand. Because of their delocalization and ease of extrapolation to the basis-set limit, plane waves would be ideally suited for the calculation of basis-set limit correlation energies. However, the routine use of correlated wavefunction approaches in a plane-wave basis set is hampered by prohibitive scaling due to a large number of virtual continuum states and has not been feasible for all but the smallest systems, even if substantial computational resources are available and methods with comparably beneficial scaling, such as the Møller–Plesset perturbation theory to second order (MP2), are used. Here, we introduce a stochastic sampling of the MP2 integrand based on Monte Carlo summation over continuum orbitals, which allows for speedups of up to a factor of 1000. Given a fixed number of sampling points, the resulting algorithm is dominated by a flat scaling of . Absolute correlation energies are accurate to <0.1 kcal/mol with respect to conventional calculations for several hundreds of electrons. This allows for the calculation of unbiased basis-set limit correlation energies for systems containing hundreds of electrons with unprecedented efficiency gains based on a straightforward treatment of continuum contributions.

Plane waves versus correlation-consistent basis sets: A comparison of MP2 non-covalent interaction energies in the complete basis set limit

“…The latter are directly linked to the delocalized nature of the virtual states that contribute to the correlation energy, and therefore necessitate a balanced and complete space coverage. In that sense, plane waves are capable of capturing high-lying (continuum-like) states as well as localized occupied states, at the cost of a sufficiently large cutoff energy 47 . When compared to plane waves, we noticed that the ability of the correlation-consistent bases to cope with the BSIE decreases as the number of electrons increases, therefore questioning the capability of localized basis sets to recover most of the correlation energy as the system size increases.…”

“…Assuming that a sufficiently high energy cutoff may be chosen to faithfully describe pair densities over a wide range of systems, and that n max increases less than linearly with N (as we have observed 47 ) , the PW basis set then becomes more favorable in the limit of large systems 2 , provided that Ω does not increase significantly for the correct convergence of Coulomb interactions in a periodic setup (Section II B 2).…”

“…G values of the pair densities (Eq.8) for calculating the integrals over N ρ ia G ∼ 10 4 -10 5 integrands (Eq.7) as well as the contribution of ∼ 10 9 -10 11 ijab sums (Eq.9), so that only a parallel approach can handle such intense RAM and CPU requirements in a reasonable time. The PW/pseudopotential MP2 method used herein has been developed and implemented in the CPMD program 35,47 for which we give the pseudocode of the parallel implementation in the Supplementary Material (Algorithm 1).…”

Plane Waves Versus Correlation-Consistent Basis Sets: A Comparison of MP2 Non-Covalent Interaction Energies in the Complete Basis Set Limit

“…2). Assuming that a sufficiently high energy cutoff may be chosen to faithfully describe pair densities over a wide range of systems and that n max increases less than linearly with N (as we have observed 47 ), the PW basis set then becomes more favorable in the limit of large systems, 2 provided that Ω does not increase significantly for the correct convergence of Coulomb interactions in a periodic setup (Section 2.2.2). Although the correlation-consistent basis sets provide gradual and monotonic progress, their associated computational cost grows faster than the rate of convergence.…”

Focal-point approach with pair-specific cusp correction for coupled-cluster theory

“…The decomposition of the CCSD correlation energy into contributions that dominate the BSIE (MP2 and PPL) and the rest can be very useful in ab initio calculations for the following two reasons. Firstly, the MP2 correlation energy in the com-plete basis set limit can be computed using algorithms with low computational complexity, removing the corresponding BSIE almost exactly [38][39][40][41]. Secondly, constructing approximations to the BSIE in the PPL term is a much simpler challenge than correcting the BSIE in all possible diagrammatic contributions to the CCSD correlation energy.…”