Accurate and Efficient Parallel Implementation of an Effective Linear-Scaling Direct Random Phase Approximation Method

Beyond Electrons: Correlation and Self-Energy in Multicomponent Density Functional Theory

Post-Kohn-Sham methods are used to evaluate the ground-state correlation energy and the orbital self-energy of systems consisting of multiple flavors of different fermions. Starting from multicomponent density functional theory, suitable ways to arrive at the corresponding multicomponent random-phase approximation and the multicomponent Green's functionG W ${GW}$ approximation, including relativistic effects, are outlined. Given the importance of both of this methods in the development of modern Kohn-Sham density functional approximations, this work will provide a foundation to design advanced multicomponent density functional approximations. Additionally, theG W ${GW}$ quasiparticle energies are needed to study light-matter interactions with the Bethe-Salpeter equation.

Basis Set Requirements of σ-Functionals for Gaussian- and Slater-Type Basis Functions and Comparison with Range-Separated Hybrid and Double Hybrid Functionals

σ-Functionals belong to the class of Kohn-Sham (KS) correlation functionals based on the adiabatic-connection fluctuation-dissipation theorem and are technically closely related to the random phase approximation (RPA). They have the same computational demand as the latter, with the computational effort of an energy evaluation for both methods being lower than that of a preceding hybrid DFT calculation for typical systems but yield much higher accuracy, reaching chemical accuracy of 1 kcal/mol for quantities such as reactions and transition energies in main group chemistry. In previous work on σ-functionals, rather large Gaussian basis sets have been used. Here, we investigate the actual basis set requirements of σ-functionals and present three setups that employ smaller Gaussian basis sets ranging from quadruple-ζ (QZ) to triple-ζ (TZ) quality and represent a good compromise between accuracy and computational efficiency. Furthermore, we introduce an implementation of σ-functionals based on Slater-type basis sets and present two setups of QZ and TZ quality for this implementation. We test the accuracy of these setups on a large database of various physical properties and types of reactions, as well as equilibrium geometries and vibrational frequencies. As expected, the accuracy of σ-functional calculations becomes somewhat lower with a decreasing basis set size. However, for all setups considered here, calculations with σ-functionals are clearly more accurate than those within the RPA and even more so than those of the conventional KS methods. For the smallest setup using Gaussian-type basis functions and Slater-type basis functions, we introduce a reparametrization that reduces the loss in accuracy due to the basis set error to some extent. A comparison with the range-separated hybrid ωB97X-V and the double hybrid DSD-BLYP-D3 shows that σ functionals outperform in accuracy both of these accurate and, for their class, representative functionals.

Translational eigenstates of He@C60 from four-dimensional ab initio potential energy surfaces interpolated using Gaussian process regression

We investigate the endofullerene system 3He@C60 with a four-dimensional potential energy surface (PES) to include the three He translational degrees of freedom and C60 cage radius. We compare second order Møller-Plesset perturbation theory (MP2), spin component scaled-MP2, scaled opposite spin-MP2, random phase approximation (RPA)@Perdew, Burke, and Ernzerhof (PBE), and corrected Hartree-Fock-RPA to calibrate and gain confidence in the choice of electronic structure method. Due to the high cost of these calculations, the PES is interpolated using Gaussian Process Regression (GPR), owing to its effectiveness with sparse training data. The PES is split into a two-dimensional radial surface, to which corrections are applied to achieve an overall four-dimensional surface. The nuclear Hamiltonian is diagonalized to generate the in-cage translational/vibrational eigenstates. The degeneracy of the three-dimensional harmonic oscillator energies with principal quantum number n is lifted due to the anharmonicity in the radial potential. The (2l + 1)-fold degeneracy of the angular momentum states is also weakly lifted, due to the angular dependence in the potential. We calculate the fundamental frequency to range between 96 and 110 cm-1 depending on the electronic structure method used. Error bars of the eigenstate energies were calculated from the GPR and are on the order of ∼±1.5 cm-1. Wavefunctions are also compared by considering their overlap and Hellinger distance to the one-dimensional empirical potential. As with the energies, the two ab initio methods MP2 and RPA@PBE show the best agreement. While MP2 has better agreement than RPA@PBE, due to its higher computational efficiency and comparable performance, we recommend RPA as an alternative electronic structure method of choice to MP2 for these systems.

Machine Learning K-Means Clustering of Interpolative Separable Density Fitting Algorithm for Accurate and Efficient Cubic-Scaling Exact Exchange Plus Random Phase Approximation within Plane Waves

The exact-exchange plus random-phase approximation (EXX+RPA) method has emerged as a crucial tool for precisely characterizing electronic structures in molecular and solid systems. We present an accurate and efficient implementation of EXX+RPA calculations that scale cubically and are conducted within plane waves. Our approach incorporates the interpolative separable density fitting (ISDF) algorithm, effectively mitigating the computational challenges associated with the plane wave basis set. To overcome the constraints of the conventional ISDF algorithm, characterized by the exceptionally high prefactor in QR factorization for interpolation point selection, we introduce an enhanced machine learning K-means method. This method incorporates a novel empirical weight function called "SSM+" for more precise interpolation point selection, capturing physical information more accurately across diverse systems. Our machine learning approach offers a quasiquadratic scaling alternative, effectively replacing the computationally demanding cubic-scaling QRCP algorithm in plane-wave-based EXX+RPA calculations. Furthermore, we enhance the method's capabilities by optimizing GPU acceleration using MATLAB's integrated GPU toolkit. In particular, our approach reduces the computational scaling of χ0 from 3.80 to 2.13 and the overall computational scaling of EXX from 2.74 to 2.10. We achieve a remarkable GPU acceleration speedup of up to 35×. Regarding CPU computation time, the standard quartic-scaling method requires 22 h to compute Si128, while QRCP completes the calculation in only around 1 h, achieving a speedup up to 20×. However, the utilization of the K-means algorithm reduces the time to 800 s, a substantial improvement of 100× compared to the standard algorithm. By employing the K-means algorithm, the computational time for interpolative point calculation using QRCP decreases from 1 h to 1 min, resulting in a 55× speed increase. With this improved algorithm, we successfully computed the dissociation curve of H2 and the equilibrium polyynic geometry of C18 molecules.

Benchmarking the accuracy of the separable resolution of the identity approach for correlated methods in the numeric atom-centered orbitals framework

Four-center two-electron Coulomb integrals routinely appear in electronic structure algorithms. The resolution-of-the-identity (RI) is a popular technique to reduce the computational cost for the numerical evaluation of these integrals in localized basis-sets codes. Recently, Duchemin and Blase proposed a separable RI scheme [J. Chem. Phys. 150, 174120 (2019)], which preserves the accuracy of the standard global RI method with the Coulomb metric and permits the formulation of cubic-scaling random phase approximation (RPA) and GW approaches. Here, we present the implementation of a separable RI scheme within an all-electron numeric atom-centered orbital framework. We present comprehensive benchmark results using the Thiel and the GW100 test set. Our benchmarks include atomization energies from Hartree-Fock, second-order Møller-Plesset (MP2), coupled-cluster singles and doubles, RPA, and renormalized second-order perturbation theory, as well as quasiparticle energies from GW. We found that the separable RI approach reproduces RI-free HF calculations within 9 meV and MP2 calculations within 1 meV. We have confirmed that the separable RI error is independent of the system size by including disordered carbon clusters up to 116 atoms in our benchmarks.

Deterministic/Fragmented-Stochastic Exchange for Large-Scale Hybrid DFT Calculations

We develop an efficient approach to evaluate range-separated exact exchange for grid- or plane-wave-based representations within the generalized Kohn-Sham-density functional theory (GKS-DFT) framework. The Coulomb kernel is fragmented in reciprocal space, and we employ a mixed deterministic-stochastic representation, retaining long-wavelength (low-k) contributions deterministically and using a sparse ("fragmented") stochastic basis for the high-k part. Coupled with a projection of the Hamiltonian onto a subspace of valence and conduction states from a prior local-DFT calculation, this method allows for the calculation of the long-range exchange of large molecular systems with hundreds and potentially thousands of coupled valence states delocalized over millions of grid points. We find that even a small number of valence and conduction states is sufficient for converging the HOMO and LUMO energies of the GKS-DFT. Excellent tuning of long-range separated hybrids (RSH) is easily obtained in the method for very large systems, as exemplified here for the chlorophyll hexamer of Photosystem II with 1320 electrons.

Highly accurate σ- and τ-functionals for beyond-RPA methods with approximate exchange kernels

σ-Functionals are promising new developments for the Kohn-Sham correlation energy based upon the direct Random Phase Approximation (dRPA) within the adiabatic connection formalism, providing impressive improvements over dRPA for a broad range of benchmarks. However, σ-functionals exhibit a high amount of self-interaction inherited from the approximations made within dRPA. Inclusion of an exchange kernel in deriving the coupling-strength-dependent density-density response function leads to so-called τ-functionals, which - apart from a fourth-order Taylor series expansion - have only been realized in an approximate fashion so far to the best of our knowledge, most notably in the form of scaled σ-functionals. In this work, we derive, optimize, and benchmark three types of σ- and τ-functionals including approximate exchange effects in the form of an antisymmetrized Hartree kernel. These functionals, based on a second-order screened exchange type contribution in the adiabatic connection formalism, the electron-hole time-dependent Hartree-Fock kernel (eh-TDHF) otherwise known as RPA with exchange (RPAx), and an approximation thereof known as approximate exchange kernel (AXK), are optimized on the ASCDB database using two new parametrizations named A1 and A2. In addition, we report a first full evaluation of σ- and τ-functionals on the GMTKN55 database, revealing our exchange-including functionals to considerably outperform existing σ-functionals while being highly competitive with some of the best double-hybrid functionals of the original GMTKN55 publication. In particular, the σ-functionals based on AXK and τ-functionals based on RPAx with PBE0 reference stand out as highly accurate approaches for a wide variety of chemically relevant problems.

Analytical Second-Order Properties for the Random Phase Approximation: Nuclear Magnetic Resonance Shieldings

A method for the analytical computation of nuclear magnetic resonance (NMR) shieldings within the direct random phase approximation (RPA) is presented. As a starting point, we use the RPA ground-state energy expression within the resolution-of-the-identity approximation in the atomic-orbital formalism. As has been shown in a recent benchmark study using numerical second derivatives [Glasbrenner, M. J. Chem. Theory Comput. 2022, 18, 192], RPA based on a Hartree-Fock reference shows accuracies comparable to coupled cluster singles and doubles (CCSD) for NMR chemical shieldings. Together with the much lower computational cost of RPA, it has emerged as an accurate method for the computation of NMR shieldings. Therefore, we aim to extend the applicability of RPA NMR to larger systems by introducing analytical second-order derivatives, making it a viable method for the accurate and efficient computation of NMR chemical shieldings.

Corrected density functional theory and the random phase approximation: Improved accuracy at little extra cost

We recently introduced an efficient methodology to perform density-corrected Hartree-Fock density functional theory [DC(HF)-DFT] calculations and an extension to it we called "corrected" HF DFT [C(HF)-DFT] [Graf and Thom, J. Chem. Theory Comput. 19 5427-5438 (2023)]. In this work, we take a further step and combine C(HF)-DFT, augmented with a straightforward orbital energy correction, with the random phase approximation (RPA). We refer to the resulting methodology as corrected HF RPA [C(HF)-RPA]. We evaluate the proposed methodology across various RPA methods: direct RPA (dRPA), RPA with an approximate exchange kernel, and RPA with second-order screened exchange. C(HF)-dRPA demonstrates very promising performance; for RPA with exchange methods, on the other hand, we often find over-corrections.

Robust relativistic many-body Green's function based approaches for assessing core ionized and excited states

A two-component contour deformation (CD) based GW method that employs frequency sampling to drastically reduce the computational effort when assessing quasiparticle states far away from the Fermi level is outlined. Compared to the canonical CD-GW method, computational scaling is reduced by an order of magnitude without sacrificing accuracy. This allows for an efficient calculation of core ionization energies. The improved computational efficiency is used to provide benchmarks for core ionized states, comparing the performance of 15 density functional approximations as Kohn-Sham starting points for GW calculations on a set of 65 core ionization energies of 32 small molecules. Contrary to valence states, GW calculations on core states prefer functionals with only a moderate amount of Hartree-Fock exchange. Moreover, modern ab initio local hybrid functionals are also shown to provide excellent generalized Kohn-Sham references for core GW calculations. Furthermore, the core-valence separated Bethe-Salpeter equation (CVS-BSE) is outlined. CVS-BSE is a convenient tool to probe core excited states. The latter is tested on a set of 40 core excitations of eight small inorganic molecules. Results from the CVS-BSE method for excitation energies and the corresponding absorption cross sections are found to be in excellent agreement with those of reference damped response BSE calculations.

Practical Post-Kohn-Sham Methods for Time-Reversal Symmetry Breaking References

The applicability of reduced scaling algorithms based on auxiliary subspace methods for the correlation energy from the random phase approximation (RPA) as well as the correlation part of the self-energy obtained from the GW method is outlined for time-reversal symmetry breaking Kohn-Sham (KS) references. The updated algorithms allow for an efficient evaluation of RPA energies and GW quasiparticle energies for molecular systems with KS references that break time-reversal symmetry. The latter occur, for example, in magnetic fields. Furthermore, KS references for relativistic open-shell molecules also break time-reversal symmetry due to the single determinant ansatz used. Errors of the updated reduced-scaling algorithms are shown to be negligible compared to reference implementations, while the overall computational scaling is reduced by 2 orders of magnitude. Ionization energies obtained from the GW approximation are shown to be robust even for the electronically complicated group of trivalent lanthanoid ions. Starting from GW quasiparticle energies, it is subsequently shown that light-matter interactions of these systems can be calculated using the Bethe-Salpeter equation (BSE). Using the combined GW-BSE method, the absorption and emission spectra of a molecular europium(III) complex can be obtained including spin-orbit coupling.

Toward Pair Atomic Density Fitting for Correlation Energies with Benchmark Accuracy

Pair atomic density fitting (PADF) has been identified as a promising strategy to reduce the scaling with system size of quantum chemical methods for the calculation of the correlation energy like the direct random-phase approximation (RPA) or second-order Møller-Plesset perturbation theory (MP2). PADF can however introduce large errors in correlation energies as the two-electron interaction energy is not guaranteed to be bounded from below. This issue can be partially alleviated by using very large fit sets, but this comes at the price of reduced efficiency and having to deal with near-linear dependencies in the fit set. One posibility is to use global density fitting (DF), but in this work, we introduce an alternative methodology to overcome this problem that preserves the intrinsically favorable scaling of PADF. We first regularize the Fock matrix by projecting out parts of the basis set which gives rise to orbital products that are hard to describe by PADF. After having thus obtained a reliable self-consistent field solution, we then also apply this projector to the orbital coefficient matrix to improve the precision of PADF-MP2 and PADF-RPA. We systematically assess the accuracy of this new approach in a numerical atomic orbital framework using Slater type orbitals (STO) and correlation consistent Gaussian type basis sets up to quintuple-ζ quality for systems with more than 200 atoms. For the small and medium systems in the S66 database we show the maximum deviation of PADF-MP2 and PADF-RPA relative correlation energies to DF-MP2 and DF-RPA reference results to be 0.07 and 0.14 kcal/mol, respectively. When the new projector method is used, the errors only slightly increase for large molecules and also when moderately sized fit sets are used the resulting errors are well under control. Finally, we demonstrate the computational efficiency of our algorithm by calculating the interaction energies of large, non-covalently bound complexes with more than 1000 atoms and 20000 atomic orbitals at the RPA@PBE/CC-pVTZ level of theory.

Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation

A method for the evaluation of analytical frozen-core gradients within the random phase approximation is presented. We outline an efficient way to evaluate the response of the density of active electrons arising only when introducing the frozen-core approximation and constituting the main difficulty, together with the response of the standard Kohn-Sham density. The general framework allows to extend the outlined procedure to related electron correlation methods in the atomic orbital basis that require the evaluation of density responses, such as second-order Møller-Plesset perturbation theory or coupled cluster variants. By using Cholesky decomposed densities─which reintroduce the occupied index in the time-determining steps─we are able to achieve speedups of 20-30% (depending on the size of the basis set) by using the frozen-core approximation, which is of similar magnitude as for molecular orbital formulations. We further show that the errors introduced by the frozen-core approximation are practically insignificant for molecular geometries.

Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies

With increasing interelectronic distance, the screening of the electron–electron interaction by the presence of other electrons becomes the dominant source of electron correlation. This effect is described by the random phase approximation (RPA) which is therefore a promising method for the calculation of weak interactions. The success of the RPA relies on the cancellation of errors, which can be traced back to the violation of the crossing symmetry of the 4-point vertex, leading to strongly overestimated total correlation energies. By the addition of second-order screened exchange (SOSEX) to the correlation energy, this issue is substantially reduced. In the adiabatic connection (AC) SOSEX formalism, one of the two electron–electron interaction lines in the second-order exchange term is dynamically screened (SOSEX(W, v_c)). A related SOSEX expression in which both electron–electron interaction lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, v_c), the evaluation of this correlation energy expression does not require an expensive numerical frequency integration and is therefore advantageous from a computational perspective. We compare the accuracy of the statically screened variant to RPA and RPA+SOSEX(W, v_c) for a wide range of chemical reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, v_c) for charged excitations and noncovalent interactions where they lead to major improvements over RPA.

Tension between predicting accurate ground state correlation energies and excitation energies from adiabatic approximations in TDDFT

The connection between the adiabatic excitation energy of time-dependent density functional theory and the ground state correlation energy from the adiabatic connection fluctuation–dissipation theorem (ACFDT) is explored in the limiting case of one excited state. An exact expression is derived for any adiabatic Hartree-exchange–correlation kernel that connects the excitation energy and the potential contribution to correlation. The resulting formula is applied to the asymmetric Hubbard dimer, a system where this limit is exact. Results from a hierarchy of approximations to the kernel, including the random phase approximation (RPA) with and without exchange and the adiabatically exact (AE) approximation, are compared to the exact ones. At full coupling, the numerical results indicate a tension between predicting an accurate excitation energy and an accurate potential contribution to correlation. The AE approximation is capable of making accurate predictions of both quantities, but only in parts of the parameter space that classify as weakly correlated, while RPA tends to be unable to accurately predict these properties simultaneously anywhere. For a strongly correlated dimer, the AE approximation greatly overestimates the excitation energy yet continues to yield an accurate ground state correlation energy due to its accurate prediction of the adiabatic connection integrand. If similar trends hold for real systems, the development of correlation kernels will be important for applications of the ACFDT in systems with large potential contributions to correlation.

Benchmarking the Accuracy of the Direct Random Phase Approximation and σ-Functionals for NMR Shieldings

A method for computing NMR shieldings with the direct random phase approximation (RPA) and the closely related σ-functionals [Trushin, E.; Thierbach, A.; Görling, A. Toward chemical accuracy at low computational cost: density functional theory with σ-functionals for the correlation energy. J. Chem. Phys. 2021, 154, 014104] is presented, which is based on a finite-difference approach. The accuracy is evaluated in benchmark calculations using high-quality coupled cluster values as a reference. Our results show that the accuracy of the computed NMR shieldings using direct RPA is strongly dependent on the density functional theory reference orbitals and improves with increasing amounts of exact Hartree-Fock exchange in the functional. NMR shieldings computed with direct RPA using a Hartree-Fock reference are significantly more accurate than MP2 shieldings and comparable to CCSD shieldings. Also, the basis set convergence is analyzed and it is shown that at least triple-zeta basis sets are required for reliable results.

Chemical accuracy with σ-functionals for the Kohn-Sham correlation energy optimized for different input orbitals and eigenvalues

Recently, a new type of orbital-dependent functional for the Kohn-Sham (KS) correlation energy, σ-functionals, was introduced. Technically, σ-functionals are closely related to the well-known direct random phase approximation (dRPA). Within the dRPA, a function of the eigenvalues σ of the frequency-dependent KS response function is integrated over purely imaginary frequencies. In σ-functionals, this function is replaced by one that is optimized with respect to reference sets of atomization, reaction, transition state, and non-covalent interaction energies. The previously introduced σ-functional uses input orbitals and eigenvalues from KS calculations with the generalized gradient approximation (GGA) exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE). Here, σ-functionals using input orbitals and eigenvalues from the meta-GGA TPSS and the hybrid-functionals PBE0 and B3LYP are presented and tested. The number of reference sets taken into account in the optimization of the σ-functionals is larger than in the first PBE based σ-functional and includes sets with 3d-transition metal compounds. Therefore, also a reparameterized PBE based σ-functional is introduced. The σ-functionals based on PBE0 and B3LYP orbitals and eigenvalues reach chemical accuracy for main group chemistry. For the 10 966 reactions from the highly accurate W4-11RE reference set, the B3LYP based σ-functional exhibits a mean average deviation of 1.03 kcal/mol compared to 1.08 kcal/mol for the coupled cluster singles doubles perturbative triples method if the same valence quadruple zeta basis set is used. For 3d-transition metal chemistry, accuracies of about 2 kcal/mol are reached. The computational effort for the post-self-consistent evaluation of the σ-functional is lower than that of a preceding PBE0 or B3LYP calculation for typical systems.

Lagrangian-Based Minimal-Overhead Batching Scheme for the Efficient Integral-Direct Evaluation of the RPA Correlation Energy

A highly memory-efficient integral-direct random phase approximation (RPA) method based on our ω-CDGD-RI-RPA method [Graf, D. J. Chem. Theory Comput. 2018, 14, 2505] is presented that completely alleviates the memory bottleneck of storing the multidimensional three-center integral tensor, which severely limited the tractable system sizes. Based on a Lagrangian formulation, we introduce an optimized batching scheme over the auxiliary and basis-function indices, which allows to compute the optimal number of batches for a given amount of system memory, while minimizing the batching overhead. Thus, our optimized batching constitutes the best tradeoff between program runtime and memory demand. Within this batching scheme, the half-transformed three-center integral tensor B_iμ^M is recomputed for each batch of auxiliary and basis functions. This allows the computation of systems that were out of reach before. The largest system within this work consists of a DNA fragment comprising 1052 atoms and 11 230 basis functions calculated on a single node, which emphasizes the new possibilities of our integral-direct RPA method.

Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy

We introduce new functionals for the Kohn-Sham correlation energy that are based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem and are named σ-functionals. Like in the well-established direct random phase approximation (dRPA), σ-functionals require as input exclusively eigenvalues σ of the frequency-dependent KS response function. In the new functionals, functions of σ replace the σ-dependent dRPA expression in the coupling-constant and frequency integrations contained in the ACFD theorem. We optimize σ-functionals with the help of reference sets for atomization, reaction, transition state, and non-covalent interaction energies. The optimized functionals are to be used in a post-self-consistent way using orbitals and eigenvalues from conventional Kohn-Sham calculations employing the exchange-correlation functional of Perdew, Burke, and Ernzerhof. The accuracy of the presented approach is much higher than that of dRPA methods and is comparable to that of high-level wave function methods. Reaction and transition state energies from σ-functionals exhibit accuracies close to 1 kcal/mol and thus approach chemical accuracy. For the 10 966 reactions of the W4-11RE reference set, the mean absolute deviation is 1.25 kcal/mol compared to 3.21 kcal/mol in the dRPA case. Non-covalent binding energies are accurate to a few tenths of a kcal/mol. The presented approach is highly efficient, and the post-self-consistent calculation of the total energy requires less computational time than a density-functional calculation with a hybrid functional and thus can be easily carried out routinely. σ-Functionals can be implemented in any existing dRPA code with negligible programming effort.

A range-separated generalized Kohn-Sham method including a long-range nonlocal random phase approximation correlation potential

Based on our recently published range-separated random phase approximation (RPA) functional [Kreppel et al., "Range-separated density-functional theory in combination with the random phase approximation: An accuracy benchmark," J. Chem. Theory Comput. 16, 2985-2994 (2020)], we introduce self-consistent minimization with respect to the one-particle density matrix. In contrast to the range-separated RPA methods presented so far, the new method includes a long-range nonlocal RPA correlation potential in the orbital optimization process, making it a full-featured variational generalized Kohn-Sham (GKS) method. The new method not only improves upon all other tested RPA schemes including the standard post-GKS range-separated RPA for the investigated test cases covering general main group thermochemistry, kinetics, and noncovalent interactions but also significantly outperforms the popular G₀W₀ method in estimating the ionization potentials and fundamental gaps considered in this work using the eigenvalue spectra obtained from the GKS Hamiltonian.

Low-Order Scaling G0W0 by Pair Atomic Density Fitting

We derive a low-scaling G0W0 algorithm for molecules using pair atomic density fitting (PADF) and an imaginary time representation of the Green's function and describe its implementation in the Slater type orbital (STO)-based Amsterdam density functional (ADF) electronic structure code. We demonstrate the scalability of our algorithm on a series of water clusters with up to 432 atoms and 7776 basis functions and observe asymptotic quadratic scaling with realistic threshold qualities controlling distance effects and basis sets of triple-ζ (TZ) plus double polarization quality. Also owing to a very small prefactor, a G0W0 calculation for the largest of these clusters takes only 240 CPU hours with these settings. We assess the accuracy of our algorithm for HOMO and LUMO energies in the GW100 database. With errors of 0.24 eV for HOMO energies on the quadruple-ζ level, our implementation is less accurate than canonical all-electron implementations using the larger def2-QZVP GTO-type basis set. Apart from basis set errors, this is related to the well-known shortcomings of the GW space-time method using analytical continuation techniques as well as to numerical issues of the PADF approach of accurately representing diffuse atomic orbital (AO) products. We speculate that these difficulties might be overcome by using optimized auxiliary fit sets with more diffuse functions of higher angular momenta. Despite these shortcomings, for subsets of medium and large molecules from the GW5000 database, the error of our approach using basis sets of TZ and augmented double-ζ (DZ) quality is decreasing with system size. On the augmented DZ level, we reproduce canonical, complete basis set limit extrapolated reference values with an accuracy of 80 meV on average for a set of 20 large organic molecules. We anticipate our algorithm, in its current form, to be very useful in the study of single-particle properties of large organic systems such as chromophores and acceptor molecules.

Efficient Reduced-Scaling Second-Order Møller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb Metric

We present a novel, highly efficient method for the computation of second-order Møller-Plesset perturbation theory (MP2) correlation energies, which uses the resolution of the identity (RI) approximation and local molecular orbitals obtained from a Cholesky decomposition of pseudodensity matrices (CDD), as in the RI-CDD-MP2 method developed previously in our group [Maurer, S. A.; Clin, L.; Ochsenfeld, C. J. Chem. Phys. 2014, 140, 224112]. In addition, we introduce an attenuated Coulomb metric and subsequently redesign the RI-CDD-MP2 method in order to exploit the resulting sparsity in the three-center integrals. Coulomb and exchange energy contributions are computed separately using specialized algorithms. A simple, yet effective integral screening protocol based on Schwarz estimates is used for the MP2 exchange energy. The Coulomb energy computation and the preceding transformations of the three-center integrals are accelerated using a modified version of the natural blocking approach [Jung, Y.; Head-Gordon, M. Phys. Chem. Chem. Phys. 2006, 8, 2831-2840]. Effective subquadratic scaling for a wide range of molecule sizes is demonstrated in test calculations in conjunction with a low prefactor. The method is shown to enable cost-efficient MP2 calculations on large molecular systems with several thousand basis functions.

Range-Separated Density-Functional Theory in Combination with the Random Phase Approximation: An Accuracy Benchmark

A formulation of range-separated random phase approximation (RPA) based on our efficient ω-CDGD-RI-RPA [J. Chem. Theory Comput. 2018, 14, 2505] method and a large scale benchmark study are presented. By application to the GMTKN55 data set, we obtain a comprehensive picture of the performance of range-separated RPA in general main group thermochemistry, kinetics, and noncovalent interactions. The results show that range-separated RPA performs stably over the broad range of molecular chemistry included in the GMTKN55 set. It improves significantly over semilocal DFT but it is still less accurate than modern dispersion corrected double-hybrid functionals. Furthermore, range-separated RPA shows a faster basis set convergence compared to standard full-range RPA making it a promising applicable approach with only one empirical parameter.

Low-Scaling Self-Consistent Minimization of a Density Matrix Based Random Phase Approximation Method in the Atomic Orbital Space

An efficient minimization of the random phase approximation (RPA) energy with respect to the one-particle density matrix in the atomic orbital space is presented. The problem of imposing full self-consistency on functionals depending on the potential itself is bypassed by approximating the RPA Hamiltonian on the basis of the well-known Hartree-Fock Hamiltonian making our self-consistent RPA method completely parameter-free. It is shown that the new method not only outperforms post-Kohn-Sham RPA in describing noncovalent interactions but also gives accurate dipole moments demonstrating the high quality of the calculated densities. Furthermore, the main drawback of atomic orbital based methods, in increasing the prefactor as compared to their canonical counterparts, is overcome by introducing Cholesky decomposed projectors allowing the use of large basis sets. Exploiting the locality of atomic and/or Cholesky orbitals enables us to present a self-consistent RPA method which shows asymptotically quadratic scaling opening the door for calculations on large molecular systems.