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Journal of Chemical Theory and Computation

Low Scaling Algorithms for the Random Phase Approximation: Imaginary Time and Laplace Transformations

Merzuk Kaltak, Jiří Klimeš, Georg Kresse

Published Apr 23, 2014• 154 citations
Publisher

Abstract

In this paper, we determine efficient imaginary frequency and imaginary time grids for second-order Møller-Plesset (MP) perturbation theory. The least-squares and Minimax quadratures are compared for periodic systems, finding that the Minimax quadrature performs slightly better for the considered materials. We show that the imaginary frequency grids developed for second order also perform well for the correlation energy in the direct random phase approximation. Furthermore, we show that the polarizabilities on the imaginary time axis can be Fourier-transformed to the imaginary frequency domain, since the time and frequency Minimax grids are dual to each other. The same duality is observed for the least-squares grids. The transformation from imaginary time to imaginary frequency allows one to reduce the time complexity to cubic (in system size), so that random phase approximation (RPA) correlation energies become accessible for large systems.

DOI
10.1021/ct5001268

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In this paper, we determine efficient imaginary frequency and imaginary time grids for second-order Møller-Plesset (MP) perturbation theory. The least-squares and Minimax quadratures are compared for periodic systems, finding that the Minimax quadrature performs slightly better for the considered materials. We show that the imaginary frequency grids developed for second order also perform well for the correlation energy in the direct random phase approximation. Furthermore, we show that the polarizabilities on the imaginary time axis can be Fourier-transformed to the imaginary frequency domain, since the time and frequency Minimax grids are dual to each other. The same duality is observed for the least-squares grids. The transformation from imaginary time to imaginary frequency allows one to reduce the time complexity to cubic (in system size), so that random phase approximation (RPA) correlation energies become accessible for large systems.
Electronic structure calculations based on many-body perturbation theory (e.g. GW or the random-phase approximation (RPA)) require function evaluations in the complex time and frequency domain, for example inhomogeneous Fourier transforms or analytic continuation from the imaginary axis to the real axis. For inhomogeneous Fourier transforms, the time-frequency component of the GreenX library provides time-frequency grids that can be utilized in low-scaling RPA and GW implementations. In addition, the adoption of the compact frequency grids provided by our library also reduces the computational overhead in RPA implementations with conventional scaling. In this work, we present low-scaling GW and conventional RPA benchmark calculations using the GreenX grids with different codes (FHI-aims, CP2K and ABINIT) for molecules, two-dimensional materials and solids. Very small integration errors are observed when using 30 time-frequency points for our test cases, namely <10−8<10^{-8}<10−8 eV/electron for the RPA correlation energies, and 10 meV for the GW quasiparticle energies.
Electronic structure calculations based on many-body perturbation theory [e.g., GW or the random-phase approximation (RPA)] require function evaluations in the complex time and frequency domain, for example, inhomogeneous Fourier transforms or analytic continuation from the imaginary axis to the real axis. For inhomogeneous Fourier transforms, the time-frequency component of the GreenX library provides time-frequency grids that can be utilized in low-scaling RPA and GW implementations. In addition, the adoption of the compact frequency grids provided by our library also reduces the computational overhead in RPA implementations with conventional scaling. In this paper, we present low-scaling GW and conventional RPA benchmark calculations using the GreenX grids with different codes (FHI-aims, CP2K, and ABINIT) for molecules, two-dimensional materials and solids. Very small integration errors are observed when using 30 time-frequency points for our test cases, namely $<{10}^{\ensuremath{-}8}$ eV/electron for the RPA correlation energies, and $\ensuremath{\le}10$ meV for the GWGWGW quasiparticle energies.
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.
First-principles methods are a valuable tool to study the electronic structure of systems and explore the quantum-mechanical mechanisms driving them on the microscopic scale. Semi-local Density-Functional Approximations (DFAs) are still very popular due to the good balance between accuracy and computational efficiency. However, the instantaneous electronic interactions are described with an averaged potential in these methods, which is often insufficient to describe charge transfers or subtle many-body correlations. A promising way to improve the description of such interactions is the usage of advanced correlation methods which explicitly treat the electronic interactions in a many-body picture. Well known methods form this class are e.g. the Random- Phase Approximation (RPA), 2nd Order Møller-Plesset Perturbation Theory (MP2) and Coupled-Cluster Theory (CC). Unfortunately, the time and memory requirements for all these methods suffer from a steep scaling with system size in their canonical formulations. Several lower-scaling reformulations exist for these methods, but the at least cubic memory requirements and the increased algorithmic complexity prevented any efficient parallelization of these methods so far. In my thesis I developed a new lower-scaling technique for advanced correlation methods to address this challenge. The key components of this technique are the Laplace-Transformation (LT) and a new local Resolution of Identity (RI) strategy, named RI-LVL. Together, they significantly reduce the memory requirement without sacrificing accuracy and allow for an efficient parallelization by means of à priori known sparsity patterns. In combination with established integral screening techniques, the computational scaling can also be reduced considerably. The potential of this new approach is demonstrated at the example of an in-memory LT-MP2 with a MPI/OpenMP hybrid parallelization. It features an at worst cubic computational scaling, a quadratic memory scaling and a high parallel efficiency. Based on the Numeric Atom-Centered Orbital (NAO) framework used in the Fritz Haber Institute ab initio molecular simulations (FHI-aims) code, this implementation can handle both molecular and periodic systems with the same numerical footing. I will also discuss in detail how the combination of the LT and RI- LVL can be applied to RPA calculations. The accuracy and performance of my LT- MP2 is demonstrated in detail for both molecular and periodic systems. In addition, the convergence of the MP2 level adsorption energy of water on a rutile titanium dioxide (TiO2) surface in the low coverage limit is explored. This new LT-MP2 recovers more than 99.9% of the canonical MP2 correlation energy and can handle TiO2 surfaces with up to 270 atoms in the unit cell.
New methods to efficiently calculate energetics and first order-properties for mean-field and correlated electronic structure theories are presented. In linear-scaling short-range hybrid calculations new integral screening criteria exploiting the locality of the attenuated Coulomb operator are introduced that allow to significantly increase the performance of these density functionals. This enables short-range hybrid calculations at a similar cost as pure semi-local density functional theory (DFT) calculations with increased accuracy due to the admixture of exchange. At the level of correlated electronic structure theory, the realm of systems which can be calculated with a linear-scaling random-phase-approximation (RPA) method is extended with a novel multi-node parallel algorithm. Furthermore, more efficient quadrature schemes are used based on the equivalence of the employed integral transforms with the Fourier transform of the non-interacting polarizability. In combination with a new way to introduce Cholesky orbitals this results in a more effient and numerically more accurate linear-scaling RPA correlation energy method with improved memory requirements. Combining these techniques further with an approach to calculate analytical first order properties using quantities from many body perturbation theory, allows to present a low-scaling method that gives access to RPA gradients for molecular systems with several hundred atoms. Moreover linear- and low-scaling methods to calculate different beyond RPA correlation energies are devised, exploiting the locality of exchange type contractions. This enables to apply these more accurate RPA methods to significantly larger systems, which allows to demonstrate the gain in accuracy for large, dispersion dominated systems. The newly developed methods extend the scope of RPA and RPA with exchange correlation energies, and first order RPA properties significantly, while the accuracy is under full numerical control. Furthermore, there is no overhead to the respective canonical algorithm, making the presented methods competitive also for small system sizes. Finally, a new beyond RPA scheme is presented that combines the benefits of plain RPA and RPA with second-order screened exchange, leading to a more balanced, highly accurate post Kohn-Sham method.
The Matsubara Green's function that is used to describe temperature-dependent behavior is expressed on a numerical grid. While such a grid usually has a couple of hundred points for low-energy model systems, for realistic systems with large basis sets the size of an accurate grid can be tens of thousands of points, constituting a severe computational and memory bottleneck. In this paper, we determine efficient imaginary time grids for the temperature-dependent Matsubara Green's function formalism that can be used for calculations on realistic systems. We show that, because of the use of an orthogonal polynomial transform, we can restrict the imaginary time grid to a few hundred points and reach micro-Hartree accuracy in the electronic energy evaluation. Moreover, we show that only a limited number of orthogonal polynomial expansion coefficients are necessary to preserve accuracy when working with a dual representation of the Green's function or self-energy and transforming between the imaginary time and frequency domain.