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The Journal of Chemical Physics

Regularized second-order correlation methods for extended systems

Elisabeth Keller, Theodoros Tsatsoulis, Karsten Reuter, Johannes T. Margraf

Published Dec 22, 2021• 17 citations
Publisher

Abstract

Second-order Møller-Plesset perturbation theory (MP2) constitutes the simplest form of many-body wavefunction theory and often provides a good compromise between efficiency and accuracy. There are, however, well-known limitations to this approach. In particular, MP2 is known to fail or diverge for some prototypical condensed matter systems like the homogeneous electron gas (HEG) and to overestimate dispersion-driven interactions in strongly polarizable systems. In this paper, we explore how the issues of MP2 for metallic, polarizable, and strongly correlated periodic systems can be ameliorated through regularization. To this end, two regularized second-order methods (including a new, size-extensive Brillouin-Wigner approach) are applied to the HEG, the one-dimensional Hubbard model, and the graphene-water interaction. We find that regularization consistently leads to improvements over the MP2 baseline and that different regularizers are appropriate for the various systems.

DOI
10.1063/5.0078119

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Second-order Møller-Plesset perturbation theory (MP2) constitutes the simplest form of many-body wavefunction theory and often provides a good compromise between efficiency and accuracy. There are, however, well-known limitations to this approach. In particular, MP2 is known to fail or diverge for some prototypical condensed matter systems like the homogeneous electron gas (HEG) and to overestimate dispersion-driven interactions in strongly polarizable systems. In this paper, we explore how the issues of MP2 for metallic, polarizable, and strongly correlated periodic systems can be ameliorated through regularization. To this end, two regularized second-order methods (including a new, size-extensive Brillouin-Wigner approach) are applied to the HEG, the one-dimensional Hubbard model, and the graphene-water interaction. We find that regularization consistently leads to improvements over the MP2 baseline and that different regularizers are appropriate for the various systems.
We present a scalable single-particle framework to treat electronic correlation in molecules and materials motivated by Green's function theory. We derive a size-extensive Brillouin-Wigner perturbation theory from the single-particle Green's function by introducing the Goldstone self-energy. This new ground state correlation energy, referred to as Quasi-Particle MP2 theory (QPMP2), avoids the characteristic divergences present in both second-order Møller-Plesset perturbation theory and Coupled Cluster Singles and Doubles within the strongly correlated regime. We show that the exact ground state energy and properties of the Hubbard dimer are reproduced by QPMP2 and demonstrate the advantages of the approach for larger Hubbard models where the metal-to-insulator transition is qualitatively reproduced, contrasting with the complete failure of traditional methods. We apply this formalism to characteristic strongly correlated molecular systems and show that QPMP2 provides an efficient, size-consistent regularization of MP2.
Second-order Møller-Plesset perturbation theory is well-known as a computationally inexpensive approach to the electron correlation problem that is size-consistent with a size-consistent reference but fails to be regular. On the other hand, the less well-known many-body version of Brillouin-Wigner perturbation theory has the reverse properties: it is regular but fails to be size-consistent when used with the standard MP partitioning. Consequently, its widespread use remains limited. In this work, we analyze the ways in which it is possible to use alternative non-MP partitions of the Hamiltonian to yield variants of BW2 that are size-consistent as well as regular. We show that there is a vast space of such BW2 theories and also show that it is possible to define a repartitioned BW2 theory from the ground state density alone, which regenerates the exact correlation energy. We also provide a general recipe for deriving regular, size-consistent, and size-extensive partitions from physically meaningful components, and we apply the result to small model systems. The scope of these results appears to further set the stage for a revival of BW2 in quantum chemistry.
We propose progressive downsampling of wave vectors in the Brillouin zone integrations occurring in the second-order many-body or Møller-Plesset perturbation (MP2) method for extended systems of one-dimensional periodicity. Higher-lying unoccupied and lower-lying occupied Bloch orbitals are subject to downsampling by an exponentially increasing factor (with base n), making the total number of Bloch orbitals included in the MP2 lattice sums grow only logarithmically with respect to the number of basis functions per unit cell. Unlike the mod n downsampling scheme proposed earlier, this log n scheme reduces the scaling of the computational cost and thus achieves a greater speedup as the unit cell size increases. Correct band indexing is essential for accuracy. Two-electron integrals entering the MP2 energy and quasiparticle energy expressions must be multiplied by quadrature weights that are a function of the energy bands involved, and an algorithm to compute the weights is proposed. A combined use of the log n and mod n schemes can speedup the MP2/6-31G** calculation of polyacetylene typically by a factor of 20 with an error in the correlation energy within a few percent relative to the conventional calculation. Similar combinations can reproduce the MP2 quasiparticle energy bands accurately at a fraction of the usual computational cost.
Abstract The multireference, state‐specific, second‐order, Brillouin–Wigner perturbation theory is presented as a robust approach to the electron correlation problem for systems demanding the use of a multireference function. A posteriori modifications are made which, in the case of a single reference function, recover the well known formula of second‐order many‐body perturbation theory (MBPT2), i.e. Møller–Plesset (MP2) theory, and in the multireference case can be shown to be equivalent to state‐specific multireference Rayleigh–Schrödinger‐like perturbation theory. It is shown that multireference many‐body Brillouin–Wigner perturbation theory when restricted to second‐order is a true many‐body theory, that is, it is a theory, which has the property of extensivity. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Møller-Plesset (MP2) perturbation theory. In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, O(N4), with respect to the system size N and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 formulation using plane wave basis sets and fast Fourier transforms. Analogously, this approach could allow us to calculate second order screened exchange as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies.
The work herein is concerned with developing computational models to understand molecules. The underlying theme of this research is the reassessment of zeroth-order approximations for higher-level methods. For second-order Moller-Plesset theory (MP2), qualitative failures of the Hartree-Fock orbitals in the form of spin contamination can lead to catastrophic errors in the second order energies. By working with orbitals optimized in the presence of correlations, orbital-optimized MP2 can fix the spin contamination problem that plague radicals, aromatics, and transition metal complexes. In path integral Monte Carlo for vibrational energies, the zeroth-order propagator is typically chosen to be the most general possible, the free particle propagator; we chose to be informed by the molecular structure we have already attained and apply a propagator based on the harmonic modes of the molecule, improving sampling efficiency and our Trotter approximation.