Linear-scaling atomic orbital-based second-order Møller–Plesset perturbation theory by rigorous integral screening criteria

Condensed Matter Systems Exposed to Radiation: Multiscale Theory, Simulations, and Experiment

This roadmap reviews the new, highly interdisciplinary research field studying the behavior of condensed matter systems exposed to radiation. The Review highlights several recent advances in the field and provides a roadmap for the development of the field over the next decade. Condensed matter systems exposed to radiation can be inorganic, organic, or biological, finite or infinite, composed of different molecular species or materials, exist in different phases, and operate under different thermodynamic conditions. Many of the key phenomena related to the behavior of irradiated systems are very similar and can be understood based on the same fundamental theoretical principles and computational approaches. The multiscale nature of such phenomena requires the quantitative description of the radiation-induced effects occurring at different spatial and temporal scales, ranging from the atomic to the macroscopic, and the interlinks between such descriptions. The multiscale nature of the effects and the similarity of their manifestation in systems of different origins necessarily bring together different disciplines, such as physics, chemistry, biology, materials science, nanoscience, and biomedical research, demonstrating the numerous interlinks and commonalities between them. This research field is highly relevant to many novel and emerging technologies and medical applications.

High-Performance Multi-GPU Analytic RI-MP2 Energy Gradients

This article presents a novel algorithm for the calculation of analytic energy gradients from second-order Møller-Plesset perturbation theory within the Resolution-of-the-Identity approximation (RI-MP2), which is designed to achieve high performance on clusters with multiple graphical processing units (GPUs). The algorithm uses GPUs for all major steps of the calculation, including integral generation, formation of all required intermediate tensors, solution of the Z-vector equation and gradient accumulation. The implementation in the EXtreme Scale Electronic Structure System (EXESS) software package includes a tailored, highly efficient, multistream scheduling system to hide CPU-GPU data transfer latencies and allows nodes with 8 A100 GPUs to operate at over 80% of theoretical peak floating-point performance. Comparative performance analysis shows a significant reduction in computational time relative to traditional multicore CPU-based methods, with our approach achieving up to a 95-fold speedup over the single-node performance of established software such as Q-Chem and ORCA. Additionally, we demonstrate that pairing our implementation with the molecular fragmentation framework in EXESS can drastically lower the computational scaling of RI-MP2 gradient calculations from quintic to subquadratic, enabling further substantial savings in runtime while retaining high numerical accuracy in the resulting gradients.

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.

Cluster-in-Molecule Local Correlation Method for Dispersion Interactions in Large Systems and Periodic Systems

ConspectusThe noncovalent interactions, including dispersion interactions, control the structures and stabilities of complex chemical systems, including host-guest complexes and the adsorption process of molecules on the solid surfaces. The density functional theory (DFT) with empirical dispersion correction is now the working horse in many areas of applications. Post-Hartree-Fock (post-HF) methods have been well recognized to provide more accurate descriptions in a systematic way. However, traditional post-HF methods are mainly limited to small- or medium-sized systems, and their applications to periodic condensed phase systems are still very limited due to their expensive computational costs.To extend post-HF calculations to large molecules, the cluster-in-molecule (CIM) local correlation approach has been established, allowing highly accurate electron correlation calculations that are routinely available for very large systems. In the CIM approach, the electron correlation energy of a large molecule could be obtained from electron correlation calculations on a series of clusters, each of which contains a subset of occupied and virtual localized molecular orbitals. The CIM method could be massively and efficiently parallelized on general computer clusters. The CIM method has been implemented at various electron correlation levels, including second-order Mo̷ller-Plesset perturbation theory (MP2), coupled cluster singles and doubles (CCSD), CCSD with perturbative triples correction [CCSD(T)], etc. The CIM-MP2 energy gradient algorithm was developed and applied to the geometry optimizations of large systems. The CIM method has also been extended to condensed-phase systems under periodic boundary conditions (PBC-CIM). For periodic systems, the correlation energy per unit cell could be evaluated with correlation energy contributions from a series of clusters that are built with localized Wannier functions.CIM-based electron correlation calculations have been employed to investigate a number of chemical problems in which the dispersion interaction is important. CIM-based post-HF methods including CIM domain-based local pair natural orbital (DLPNO) CCSD(T) are applied to compute the relative or binding energies of biological systems or supramolecular complexes, the reaction barrier in a relatively complex chemical reaction. The CIM-MP2 method is used to obtain the optimized geometry of large systems. CIM-based post-HF calculations have also been used to compute the cohesive energies of molecular crystals and adsorption energies of molecules on the solid surfaces. The CIM and its PBC variant are expected to become a powerful theoretical tool for accurate calculations of the energies and structures for a broad range of large systems and condensed-phase systems with significant dispersion interactions.

Local Second-Order Møller-Plesset Theory with a Single Threshold Using Orthogonal Virtual Orbitals: Theory, Implementation, and Assessment

It has long been clear that electron correlation methods exhibit unphysical compute scalings with molecular size, which has motivated the development of local correlation methods to discard effectively zero contributions in a controlled way to yield an approximate correlation energy. The ideal local correlation method should have a single numerical threshold that controls the dropping of terms with the ability to have that threshold set small enough so that the correlation energy is reproduced to enough significant figures such that the result is chemically identical. This work reports such a method for the second-order Møller-Plesset (MP2) theory. The theory, implementation, and testing of this local MP2 theory are reported. Thresholds ranging from 10-5 to 10-8 and basis sets ranging from split valence plus polarization through to quadruple-ζ are assessed for local MP2 calculations on a range of molecules, including linear chains and molecules with two- and three-dimensional character. The implementation is shared memory parallel via OpenMP and yields roughly 50% parallel efficiency with 16 cores for a large job. Considerable efforts were made to minimize memory demands, which increased as thresholds were tightened. A variety of relative energy calculations are presented as a function of threshold to provide some guidance to users on how to obtain adequate precision at a low compute cost. It is particularly clear that derivative properties require tighter thresholds in order to achieve an adequate precision.

Further investigations into a Laplace MP2 method using range separated Coulomb potential and orbital selective virtuals: Multipole correction, OSV extrapolation, and critical assessment

We report further investigations to aid the development of a Laplace MP2 (second-order Møller Plesset) method with a range separated Coulomb potential partitioned into short- and long-range parts. The implementation of the method extensively uses sparse matrix algebra, density fitting techniques for the short-range part, and a Fourier transformation in spherical coordinates for the long-range part of the potential. Localized molecular orbitals are employed for the occupied space, whereas virtual space is described by orbital specific virtual orbitals (OSVs) associated with localized molecular orbitals. The Fourier transform is deficient for very large distances between localized occupied orbitals, and a multipole expansion for widely separated pairs is introduced for the direct MP2 contribution, which is applicable also to non-Coulombic potentials that do not satisfy the Laplace equation. For the exchange contribution, an efficient screening of contributing localized occupied pairs is employed, which is discussed more completely here. To mitigate errors due to the truncation of OSVs, a simple and efficient extrapolation procedure is used to obtain results close to MP2 for the full basis set of atomic orbitals Using a suitable set of default parameters, the accuracy of the approach is demonstrated. The current implementation of the approach is not very efficient, and the aim of this paper is to introduce and critically discuss ideas that can have more general applicability beyond MP2 calculations for large molecules.

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.

Analytic gradients for local density fitting Hartree-Fock and Kohn-Sham methods

We present analytic gradients for local density fitting Hartree-Fock (HF) and hybrid Kohn-Sham (KS) density functional methods. Due to the non-variational nature of the local fitting algorithm, the method of Lagrange multipliers is used to avoid the solution of the coupled perturbed HF and KS equations. We propose efficient algorithms for the solution of the arising Z-vector equations and the gradient calculation that preserve the third-order scaling and low memory requirement of the original local fitting algorithm. In order to demonstrate the speed and accuracy of our implementation, gradient calculations and geometry optimizations are presented for various molecular systems. Our results show that significant speedups can be achieved compared to conventional density fitting calculations without sacrificing accuracy.

Efficient implementation of time-dependent auxiliary density functional theory

The random phase approximation of time-dependent auxiliary density functional theory (TDADFT) is rederived from auxiliary density perturbation theory. Our exhaustive validation of TDADFT reveals an upshift of the excitation energies by ∼0.1 eV with respect to standard time-dependent density functional theory. For the computationally efficient implementation of TDADFT, floating point operation optimized three-center electron repulsion integral recurrence relations and their double asymptotic expansions are implemented into the Davidson solver. The computational efficiency of TDADFT is benchmarked with four sets of molecules comprising alkanes, fullerenes, DNA fragments, and zeolites. The results show that TDADFT has a computational scaling between 1.3 and 1.9 with respect to the number of basis functions, which is lower than the scaling of standard time-dependent density functional theory. Due to its computational simplifications, TDADFT is particularly well suited for Born-Oppenheimer molecular dynamics simulations. As illustrative examples, we present the temperature effects on the gas-phase absorption spectra of benzene, naphthalene, and anthracene.

Multinode Multi-GPU Two-Electron Integrals: Code Generation Using the Regent Language

The computation of two-electron repulsion integrals (ERIs) is often the most expensive step of integral-direct self-consistent field methods. Formally it scales as O(N⁴), where N is the number of Gaussian basis functions used to represent the molecular wave function. In practice, this scaling can be reduced to O(N²) or less by neglecting small integrals with screening methods. The contributions of the ERIs to the Fock matrix are of Coulomb (J) and exchange (K) type and require separate algorithms to compute matrix elements efficiently. We previously implemented highly efficient GPU-accelerated J-matrix and K-matrix algorithms in the electronic structure code TeraChem. Although these implementations supported the use of multiple GPUs on a node, they did not support the use of multiple nodes. This presents a key bottleneck to cutting-edge ab initio simulations of large systems, e.g., excited state dynamics of photoactive proteins. We present our implementation of multinode multi-GPU J- and K-matrix algorithms in TeraChem using the Regent programming language. Regent directly supports distributed computation in a task-based model and can generate code for a variety of architectures, including NVIDIA GPUs. We demonstrate multinode scaling up to 45 GPUs (3 nodes) and benchmark against hand-coded TeraChem integral code. We also outline our metaprogrammed Regent implementation, which enables flexible code generation for integrals of different angular momenta.

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.

Third-Order Many-Body Expansion of OSV-MP2 Wave Function for Low-Order Scaling Analytical Gradient Computation

We present a many-body expansion (MBE) formulation and implementation for efficient computation of analytical energy gradients from the orbital-specific-virtual second-order Møllet-Plesset perturbation theory (OSV-MP2) based on our earlier work (Zhou et al. J. Chem. Theory Comput. 2020, 16, 196-210). The third-order MBE(3) expansion of OSV-MP2 amplitudes and density matrices was developed to adopt the orbital-specific clustering and long-range termination schemes, which avoids term-by-term differentiations of the MBE energy bodies. We achieve better efficiency by exploiting the algorithmic sparsity that allows us to prune out insignificant fitting integrals and OSV relaxations. With these approximations, the present implementation is benchmarked on a range of molecules that show an economic scaling in the linear and quadratic regimes for computing MBE(3)-OSV-MP2 amplitude and gradient equations, respectively, and yields normal accuracy comparable to the original OSV-MP2 results. The MPI-3-based parallelism through shared memory one-sided communication is further developed for improving parallel scalability and memory accessibility by sorting the MBE(3) orbital clusters into independent tasks that are distributed on multiple processes across many nodes, supporting both global and local data locations in which selected MBE(3)-OSV-MP2 intermediates of different sizes are distinguished and accordingly placed. The accuracy and efficiency level of our MBE(3)-OSV-MP2 analytical gradient implementation is finally illustrated in two applications: we show that the subtle coordination structure differences of mechanically interlocked Cu-catenane complexes can be distinguished when tuning ligand lengths; and the porphycene molecular dynamics reveals the emergence of the vibrational signature arising from softened N-H stretching associated with hydrogen transfer, using an MP2 level of electron correlation and classical nuclei for the first time.

Toward Laplace MP2 method using range separated Coulomb potential and orbital selective virtuals

We report the development of a new Laplace MP2 (second-order Møller-Plesset) implementation using a range separated Coulomb potential, partitioned into short- and long-range parts. The implementation heavily relies on the use of sparse matrix algebra, density fitting techniques for the short-range Coulomb interactions, while a Fourier transformation in spherical coordinates is used for the long-range part of the potential. Localized molecular orbitals are employed for the occupied space, whereas orbital specific virtual orbitals associated with localized molecular orbitals are obtained from the exchange matrix associated with specific localized occupied orbitals. The range separated potential is crucial to achieve efficient treatment of the direct term in the MP2, while extensive screening is employed to reduce the expense of the exchange contribution in MP2. The focus of this paper is on controllable accuracy and linear scaling of the data entering the algorithm.

Tight distance-dependent estimators for screening two-center and three-center short-range Coulomb integrals over Gaussian basis functions

We derive distance-dependent estimators for two-center and three-center electron repulsion integrals over a short-range Coulomb potential, erfc(ωr₁₂)/r₁₂. These estimators are much tighter than the ones based on the Schwarz inequality and can be viewed as a complement to the distance-dependent estimators for four-center short-range Coulomb integrals and for two-center and three-center full Coulomb integrals previously reported. Because the short-range Coulomb potential is commonly used in solid-state calculations, including those with the Heyd-Scuseria-Ernzerhof functional and with our recently introduced range-separated periodic Gaussian density fitting, we test our estimators on a diverse set of periodic systems using a wide range of the range-separation parameter ω. These tests demonstrate the robust tightness of our estimators, which are then used with integral screening to calculate periodic three-center short-range Coulomb integrals with linear scaling in system size.

Low-Order Scaling Quasiparticle Self-Consistent GW for Molecules

Low-order scaling GW implementations for molecules are usually restricted to approximations with diagonal self-energy. Here, we present an all-electron implementation of quasiparticle self-consistent GW for molecular systems. We use an efficient algorithm for the evaluation of the self-energy in imaginary time, from which a static non-local exchange-correlation potential is calculated via analytical continuation. By using a direct inversion of iterative subspace method, fast and stable convergence is achieved for almost all molecules in the GW100 database. Exceptions are systems which are associated with a breakdown of the single quasiparticle picture in the valence region. The implementation is proven to be starting point independent and good agreement of QP energies with other codes is observed. We demonstrate the computational efficiency of the new implementation by calculating the quasiparticle spectrum of a DNA oligomer with 1,220 electrons using a basis of 6,300 atomic orbitals in less than 4 days on a single compute node with 16 cores. We use then our implementation to study the dependence of quasiparticle energies of DNA oligomers consisting of adenine-thymine pairs on the oligomer size. The first ionization potential in vacuum decreases by nearly 1 electron volt and the electron affinity increases by 0.4 eV going from the smallest to the largest considered oligomer. This shows that the DNA environment stabilizes the hole/electron resulting from photoexcitation/photoattachment. Upon inclusion of the aqueous environment via a polarizable continuum model, the differences between the ionization potentials reduce to 130 meV, demonstrating that the solvent effectively compensates for the stabilizing effect of the DNA environment. The electron affinities of the different oligomers are almost identical in the aqueous environment.

Speeding up Hartree-Fock and Kohn-Sham calculations with first-order corrections

Several approaches are presented to improve the efficiency of Hartree-Fock and Kohn-Sham self-consistent field (SCF) calculations relying on a simple first-order energy correction reminiscent of the scheme used in dual-basis SCF methods. The basic idea is to perform an initial SCF calculation computing approximate Fock-matrices and, in the final iteration step, to use a more complete Fock-matrix builder together with the energy correction to diminish the error. The approximation is tested for conventional and local density fitting (DF) SCF approaches combining various auxiliary basis sets, fitting metrics, and Fock-matrix construction algorithms in the initial and final iterations as well as for seminumerical SCF methods combining integration grids of different qualities. We also report the implementation of the occupied orbital resolution of identity exchange construction algorithm with local DF approximations. Benchmark calculations are presented for total energies, reaction energies, and molecular geometries. Our results show that speedups of up to 80% can be expected utilizing the new approaches without significant loss of accuracy.

Linear-Scaling Open-Shell MP2 Approach: Algorithm, Benchmarks, and Large-Scale Applications

A linear-scaling local second-order Møller–Plesset (MP2) method is presented for high-spin open-shell molecules based on restricted open-shell (RO) reference functions. The open-shell local MP2 (LMP2) approach inherits the iteration- and redundancy-free formulation and the completely integral-direct, OpenMP-parallel, and memory and disk use economic algorithms of our closed-shell LMP2 implementation. By utilizing restricted local molecular orbitals for the demanding integral transformation step and by introducing a novel long-range spin-polarization approximation, the computational cost of RO-LMP2 approaches that of closed-shell LMP2. Extensive benchmarks were performed for reactions of radicals, ionization potentials, as well as spin-state splittings of carbenes and transition-metal complexes. Compared to the conventional MP2 reference for systems of up to 175 atoms, local errors of at most 0.1 kcal/mol were found, which are well below the intrinsic accuracy of MP2. RO-LMP2 computations are presented for challenging protein models of up to 601 atoms and 11 000 basis functions, which involve either spin states of a complexed iron ion or a highly delocalized singly occupied orbital. The corresponding runtimes of 9–15 h obtained with a single, many-core CPU demonstrate that MP2, as well as spin-scaled MP2 and double-hybrid density functional methods, become widely accessible for open-shell systems of unprecedented size and complexity.

A scaled explicitly correlated F12 correction to second-order Møller-Plesset perturbation theory

An empirically scaled version of the explicitly correlated F12 correction to second-order Møller-Plesset perturbation theory (MP2-F12) is introduced. The scaling eliminates the need for many of the most costly terms of the F12 correction while reproducing the unscaled explicitly correlated F12 interaction energy correction to a high degree of accuracy. The method requires a single, basis set dependent scaling factor that is determined by fitting to a set of test molecules. We present factors for the cc-pVXZ-F12 (X = D, T, Q) basis set family obtained by minimizing interaction energies of the S66 set of small- to medium-sized molecular complexes and show that our new method can be applied to accurately describe a wide range of systems. Remarkably good explicitly correlated corrections to the interaction energy are obtained for the S22 and L7 test sets, with mean percentage errors for the double-zeta basis of 0.60% for the F12 correction to the interaction energy, 0.05% for the total electron correlation interaction energy, and 0.03% for the total interaction energy, respectively. Additionally, mean interaction energy errors introduced by our new approach are below 0.01 kcal mol^-1 for each test set and are thus negligible for second-order perturbation theory based methods. The efficiency of the new method compared to the unscaled F12 correction is shown for all considered systems, with distinct speedups for medium- to large-sized structures.

Scalable Electron Correlation Methods. 8. Explicitly Correlated Open-Shell Coupled-Cluster with Pair Natural Orbitals PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12

We present explicitly correlated open-shell pair natural orbital local coupled-cluster methods, PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12. The methods are extensions of our previously reported PNO-R/UCCSD methods (J. Chem. Theory Comput., 2020, 16, 3135-3151, https://pubs.acs.org/doi/10.1021/acs.jctc.0c00192) with additions of explicit correlation and perturbative triples corrections. The explicit correlation treatment follows the spin-orbital CCSD-F12b theory using Ansatz 3*A, which is found to yield comparable or better basis set convergence than the more rigorous Ansatz 3C in computed ionization potentials and reaction energies using double- to quaduple-ζ basis sets. The perturbative triples correction is adapted from the spin-orbital (T) theory to use triples natural orbitals (TNOs). To address the coupling due to off-diagonal Fock matrix elements, the local triples amplitudes are iteratively solved using small domains of TNOs, and a semicanonical (T0) domain correction with larger domains is applied to reduce the domain errors. The performance of the methods is demonstrated through benchmark calculations on ionization potentials, radical stabilization energies, reaction energies of fragmentations and rearrangements in radical cations, and spin-state energy differences of iron complexes. For a few test sets where canonical calculations are feasible, PNO-RCCSD(T)-F12 results agree with the canonical ones to within 0.4 kcal mol^-1, and this maximum error is reduced to below 0.2 kcal mol^-1 when large local domains are used. For larger systems, results using different thresholds for the local approximations are compared to demonstrate that 1 kcal mol^-1 level of accuracy can be achieved using our default settings. For a couple of difficult cases, it is demonstrated that the errors from individual approximations are only a fraction of 1 kcal mol^-1, and the overall accuracy of the method does not rely on error compensations. In contrast to canonical calculations, the use of spin-orbitals does not lead to a significant increase of computational time and memory usage in the most expensive steps of PNO-R/UCCSD(T)-F12 calculations. The only exception is the iterative solution of the (T) amplitudes, which can be avoided without significant errors by using a perturbative treatment of the off-diagonal coupling, known as (T1) approximation. For most systems, even the semicanonical approximation (T0) leads only to small errors in relative energies. Our program is well parallelized and capable of computing accurate correlation energies for molecules with 100-200 atoms using augmented triple-ζ basis sets in less than a day of elapsed time on a small computer cluster.

Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Møller-Plesset Perturbation Theory

We employ various reduced scaling techniques to accelerate the recently developed least-squares tensor hypercontraction (LS-THC) approximation [Parrish, R. M., Hohenstein, E. G., Martínez, T. J., Sherrill, C. D. J. Chem. Phys. 137, 224106 (2012)] for electron repulsion integrals (ERIs) and apply it to second-order Møller-Plesset perturbation theory (MP2). The grid-projected ERI tensors are efficiently constructed using a localized Cholesky molecular orbital basis from density-fitted integrals with an attenuated Coulomb metric. Additionally, rigorous integral screening and the natural blocking matrix format are applied to reduce the complexity of this step. By recasting the equations to form the quantized representation of the 1/r operator Z into the form of a system of linear equations, the bottleneck of inverting the grid metric via pseudoinversion is removed. This leads to a reduced scaling THC algorithm and application to MP2 yields the (sub-)quadratically scaling THC-ω-RI-CDD-SOS-MP2 method. The efficiency of this method is assessed for various systems including DNA fragments with over 8000 basis functions and the subquadratic scaling is illustrated.

Quantum Information and Algorithms for Correlated Quantum Matter

Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.

Basis set extrapolation in pair natural orbital theories

We present the results of a benchmark study of the effect of Pair Natural Orbital (PNO) truncation errors on the performance of basis set extrapolation. We find that reliable conclusions from the application of Helgaker's extrapolation method are only obtained when using tight PNO thresholds of at least 10^-7. The use of looser thresholds introduces a significant risk of observing a false basis set convergence and underestimating the residual basis set errors. We propose an alternative extrapolation approach based on the PNO truncation level that only requires a single basis set and show that it is a viable alternative to hierarchical basis set extrapolation methods.

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.

Double hybrid DFT calculations with Slater type orbitals

On a comprehensive database with 1,644 datapoints, covering several aspects of main-group as well as of transition metal chemistry, we assess the performance of 60 density functional approximations (DFA), among them 36 double hybrids (DH). All calculations are performed using a Slater type orbital (STO) basis set of triple-ζ (TZ) quality and the highly efficient pair atomic resolution of the identity approach for the exchange- and Coulomb-term of the KS matrix (PARI-K and PARI-J, respectively) and for the evaluation of the MP2 energy correction (PARI-MP2). Employing the quadratic scaling SOS-AO-PARI-MP2 algorithm, DHs based on the spin-opposite-scaled (SOS) MP2 approximation are benchmarked against a database of large molecules. We evaluate the accuracy of STO/PARI calculations for B3LYP as well as for the DH B2GP-PLYP and show that the combined basis set and PARI-error is comparable to the one obtained using the well-known def2-TZVPP Gaussian-type basis set in conjunction with global density fitting. While quadruple-ζ (QZ) calculations are currently not feasible for PARI-MP2 due to numerical issues, we show that, on the TZ level, Jacob's ladder for classifying DFAs is reproduced. However, while the best DHs are more accurate than the best hybrids, the improvements are less pronounced than the ones commonly found on the QZ level. For conformers of organic molecules and noncovalent interactions where very high accuracy is required for qualitatively correct results, DHs provide only small improvements over hybrids, while they still excel in thermochemistry, kinetics, transition metal chemistry and the description of strained organic systems.

Scalable Electron Correlation Methods. 7. Local Open-Shell Coupled-Cluster Methods Using Pair Natural Orbitals: PNO-RCCSD and PNO-UCCSD

We present well-parallelized local implementations of high-spin open-shell coupled cluster methods with single and double excitations (CCSD) using pair natural orbitals (PNOs). The methods are based on the spin-orbital coupled cluster theory using restricted open-shell Hartree-Fock (ROHF) reference functions. Two variants, namely, PNO-UCCSD and PNO-RCCSD are implemented and compared. In PNO-UCCSD, the coupled cluster amplitudes are spin-unrestricted, while in PNO-RCCSD the linear terms are spin-adapted by a spin-projection approach as described in J. Chem. Phys. 1993, 99, 5219-5227. Near linear scaling of the computational cost with the number of correlated electrons is achieved by applying domain and pair approximations. The PNOs are spin-independent and obtained using a semicanonical spin-restricted MP2 approximation with large domains of projected atomic orbitals (PAOs). The pair approximations of our previously described closed-shell PNO-LCCSD method are carefully revised so that they are compatible to the UCCSD theory, and PNO-UCCSD or PNO-RCCSD calculations for closed-shell molecules yield exactly the same results as corresponding spin-free closed-shell PNO-LCCSD calculations. The convergence of the results with respect to the thresholds and options that control the domain and pair approximations is demonstrated. It is found that large domains are required for the single excitations in open-shell calculations in order to obtain converged results. In general, the errors of relative energies caused by the local approximations can be reduced to below 1 kcal mol^-1, even for difficult cases. Presently, PNO-RCCSD and PNO-UCCSD calculations for molecules with 100-200 atoms and augmented triple-ζ basis sets can be carried out in a few hours of elapsed time using ∼100 CPU cores. In addition, the program is also capable of performing distinguishable cluster (PNO-RDCSD and PNO-UDCSC) calculations. The present work is a critical step in developing fully local open-shell PNO-RCCSD(T)-F12 methods.

A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals

We report a production level implementation of pair atomic resolution of the identity (PARI) based second-order Møller-Plesset perturbation theory (MP2) in the Slater type orbital (STO) based Amsterdam Density Functional (ADF) code. As demonstrated by systematic benchmarks, dimerization and isomerization energies obtained with our code using STO basis sets of triple-ζ-quality show mean absolute deviations from Gaussian type orbital, canonical, basis set limit extrapolated, global density fitting (DF)-MP2 results of less than 1 kcal/mol. Furthermore, we introduce a quadratic scaling atomic orbital based spin-opposite-scaled (SOS)-MP2 approach with a very small prefactor. Due to a worst-case scaling of [Formula: see text], our implementation is very fast already for small systems and shows an exceptionally early crossover to canonical SOS-PARI-MP2. We report computational wall time results for linear as well as for realistic three-dimensional molecules and show that triple-ζ quality calculations on molecules of several hundreds of atoms are only a matter of a few hours on a single compute node, the bottleneck of the computations being the SCF rather than the post-SCF energy correction.

Wave Function Perspective and Efficient Truncation of Renormalized Second-Order Perturbation Theory

We present an approach to a renormalized second-order Green's function perturbation theory (GF2), which avoids all dependency on continuous variables, grids, or explicit Green's functions and is instead formulated entirely in terms of static quantities and wave functions. Correlation effects from MP2 diagrams are iteratively incorporated to modify the underlying spectrum of excitations by coupling the physical system to fictitious auxiliary degrees of freedom, allowing for single-particle orbitals to delocalize into this additional space. The overall approach is shown to be rigorously O[N⁵], after an appropriate compression of this auxiliary space. This is achieved via a novel scheme, which ensures that a desired number of moments of the underlying occupied and virtual spectra are conserved in the compression, allowing a rapid and systematically improvable convergence to the limit of the effective dynamical resolution. The approach is found to then allow for the qualitative description of stronger correlation effects, avoiding the divergences of MP2, as well as its orbital-optimized version. On application to the G1 test set, we find that modification up to only the third spectral moment of the underlying spectrum from which the double excitations are built are required for accurate energetics, even in strongly correlated regimes. This is beyond simple self-consistent changes to the density matrix of the system but far from requiring a description of the full dynamics of the frequency-dependent self-energy.

Domain-based local pair natural orbital methods within the correlation consistent composite approach

Ab initio composite approaches have been utilized to model and predict main group thermochemistry within 1 kcal mol^-1 , on average, from well-established reliable experiments, primarily for molecules with less than 30 atoms. For molecules of increasing size and complexity, such as biomolecular complexes, composite methodologies have been limited in their application. Therefore, the domain-based local pair natural orbital (DLPNO) methods have been implemented within the correlation consistent composite approach (ccCA) framework, namely DLPNO-ccCA, to reduce the computational cost (disk space, CPU (central processing unit) time, memory) and predict energetic properties such as enthalpies of formation, noncovalent interactions, and conformation energies for organic biomolecular complexes including one of the largest molecules examined via composite strategies, within 1 kcal mol^-1 , after calibration with 119 molecules and a set of linear alkanes. © 2019 Wiley Periodicals, Inc.

Analytic gradients for the single-reference driven similarity renormalization group second-order perturbation theory

We derive and implement analytic energy gradients for the single-reference driven similarity renormalization group second-order perturbation theory (DSRG-PT2). The resulting equations possess an asymptotic scaling that is identical to that of the second-order Møller-Plesset perturbation theory (MP2), indicating that the exponential regularizer in the DSRG equations does not introduce formal difficulties in the gradient theory. We apply the DSRG-PT2 method to optimizing the geometries of 15 small molecules. The equilibrium bond lengths computed with DSRG-PT2 are found similar to those of MP2, yielding a mean absolute error of 0.0033 Å and a standard deviation of 0.0045 Å when compared with coupled cluster with singles, doubles, and perturbative triples.

Scalable Electron Correlation Methods. 6. Local Spin-Restricted Open-Shell Second-Order Møller-Plesset Perturbation Theory Using Pair Natural Orbitals: PNO-RMP2

We present a (near) linear scaling implementation of high-spin open-shell Møller-Plesset perturbation theory using pair natural orbitals (PNO-RMP2). The theory is based on a new variant of open-shell MP2 which is fully spin-adapted and uses a single set of spin-free amplitudes, as in closed-shell MP2. This method, denoted SROMP2, is invariant to unitary orbital transformations within the closed, open, and virtual orbital subspaces. Accordingly, only a single set of PNOs per spatial orbital pair is needed, and the efficiency is similar to closed-shell calculations. The PNOs are obtained using a semicanonical approximation with large domains of projected atomic orbitals (PAOs). Linear scaling is achieved provided that the open-shell orbitals are local, and distant pairs are treated by multipole approximations. The method is efficiently parallelized. The convergence of ionization and reaction energies as a function of the PAO and PNO domain sizes is demonstrated and found to be very similar as for closed-shell calculations. The suitability of the PNOs for explicitly correlated PNO-RCCSD-F12 calculations is also tested. So far, this method is only simulated using a conventional program with appropriate projections to the PAO and PNO subspaces. It is demonstrated for radical stabilization energies as well as ionization potentials that the errors caused by the local domain approximations with our default thresholds are negligible.

Assessment of DFT for endohedral complexes' dipole moment: PNO-LCCSD-F12 as a reference method

We present a systematic evaluation of the performance of a wide range of exchange-correlation functionals and related dispersion correction schemes for the computation of dipole moments of endohedral complexes, formed through the encapsulation of an AB molecule (AB = LiF, HCl) inside carbon nanotubes (CNTs) of different diameter. The consistency and accuracy of (i) generalized gradient approximation, (ii) meta GGA, (iii) global hybrid, and (iv) range-separated hybrid density functionals are assessed. In total, 37 density functionals are tested. The results obtained using the highly accurate pair natural orbitals based explicitly correlated local coupled cluster singles doubles (PNO-LCCSD-F12) method of Werner and co-workers [Schwilk et al., J. Chem. Theory Comput., 2017, 13, 3650; Ma et al., J. Chem. Theory Comput., 2017, 13, 4871] with the aug-cc-pVTZ basis set serve as a reference. The static electric dipole moment is computed via the finite field response or, when possible, as the expectation value of the dipole operator. Among others, it is shown that functionals belonging to the class of range-separated hybrids, provide results closest to the coupled cluster reference data. In particular, the ωB97X as well as the M11 functional may be considered as a promising choice for computing electric properties of noncovalent endohedral complexes. On the other hand, the worst performance was found for the functionals which do not include the Hartree-Fock exchange. The analysis of both the coupled cluster and the DFT results indicates a strong coupling of dispersion and polarization that may also explain why lower level DFT methods, as well as Hartree-Fock and MP2, cannot yield dipole moments beyond a qualitative agreement with the higher order reference data. Interestingly, the much smaller and less systematically constructed basis sets of Pople of moderate size provide results of accuracy at least comparable with the extended Dunning's aug-cc-pVTZ basis set.

Lowering of the complexity of quantum chemistry methods by choice of representation

The complexity of the standard hierarchy of quantum chemistry methods is not invariant to the choice of representation. This work explores how the scaling of common quantum chemistry methods can be reduced using real-space, momentum-space, and time-dependent intermediate representations without introducing approximations. We find the scalings of exact Gaussian basis Hartree-Fock theory, second-order Møller-Plesset perturbation theory, and coupled cluster theory (specifically, linearized coupled cluster doubles and the distinguishable cluster approximation with doubles) to be O(N³), O(N³), and O(N⁵), respectively, where N denotes the system size. These scalings are not asymptotic and hold over all ranges of N.

Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis

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.

Quantum Chemistry in Dataflow: Density-Fitting MP2

We demonstrate the use of dataflow technology in the computation of the correlation energy in molecules at the Møller-Plesset perturbation theory (MP2) level. Specifically, we benchmark density fitting (DF)-MP2 for as many as 168 atoms (in valinomycin) and show that speed-ups between 3 and 3.8 times can be achieved when compared to the MOLPRO package run on a single CPU. Acceleration is achieved by offloading the matrix multiplications steps in DF-MP2 to Dataflow Engines (DFEs). We project that the acceleration factor could be as much as 24 with the next generation of DFEs.

Laplace-transformed multi-reference second-order perturbation theories in the atomic and active molecular orbital basis

In the present article, we show how to formulate the partially contracted n-electron valence second-order perturbation theory (NEVPT2) energies in the atomic and active molecular orbital basis by employing the Laplace transformation of orbital-energy denominators (OEDs). As atomic-orbital (AO) basis functions are inherently localized and the number of active orbitals is comparatively small, our formulation is particularly suited for a linearly scaling NEVPT2 implementation. In our formulation, there are two kinds of NEVPT2 energy contributions, which differ in the number of active orbitals in the two-electron integrals involved. Those involving integrals with either no or a single active orbital can be formulated completely in the AO basis as single-reference second-order Møller-Plesset perturbation theory and benefit from sparse active pseudo-density matrices-particularly if the active molecular orbitals are localized only in parts of a molecule. Conversely, energy contributions involving integrals with either two or three active orbitals can be obtained from Coulomb and exchange matrices generalized for pairs of active orbitals. Moreover, we demonstrate that Laplace-transformed partially contracted NEVPT2 is nothing less than time-dependent NEVPT2 [A. Y. Sokolov and G. K.-L. Chan, J. Chem. Phys. 144, 064102 (2016)] iff the all-active intermediates are computed with the internal-contraction approximation. Furthermore, we show that for multi-reference perturbation theories it is particularly challenging to find optimal parameters of the numerical Laplace transformation as the fit range may vary among the 8 different OEDs by many orders of magnitude. Selecting the number of quadrature points for each OED separately according to an accuracy-based criterion allows us to control the errors in the NEVPT2 energies reliably.

Optimization of the linear-scaling local natural orbital CCSD(T) method: Redundancy-free triples correction using Laplace transform

An improved algorithm is presented for the evaluation of the (T) correction as a part of our local natural orbital (LNO) coupled-cluster singles and doubles with perturbative triples [LNO-CCSD(T)] scheme [Z. Rolik et al., J. Chem. Phys. 139, 094105 (2013)]. The new algorithm is an order of magnitude faster than our previous one and removes the bottleneck related to the calculation of the (T) contribution. First, a numerical Laplace transformed expression for the (T) fragment energy is introduced, which requires on average 3 to 4 times fewer floating point operations with negligible compromise in accuracy eliminating the redundancy among the evaluated triples amplitudes. Second, an additional speedup factor of 3 is achieved by the optimization of our canonical (T) algorithm, which is also executed in the local case. These developments can also be integrated into canonical as well as alternative fragmentation-based local CCSD(T) approaches with minor modifications. As it is demonstrated by our benchmark calculations, the evaluation of the new Laplace transformed (T) correction can always be performed if the preceding CCSD iterations are feasible, and the new scheme enables the computation of LNO-CCSD(T) correlation energies with at least triple-zeta quality basis sets for realistic three-dimensional molecules with more than 600 atoms and 12 000 basis functions in a matter of days on a single processor.

Efficient evaluation of three-center Coulomb integrals

In this study we pursue the most efficient paths for the evaluation of three-center electron repulsion integrals (ERIs) over solid harmonic Gaussian functions of various angular momenta. First, the adaptation of the well-established techniques developed for four-center ERIs, such as the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys quadrature schemes, and the combinations thereof for three-center ERIs is discussed. Several algorithmic aspects, such as the order of the various operations and primitive loops as well as prescreening strategies, are analyzed. Second, the number of floating point operations (FLOPs) is estimated for the various algorithms derived, and based on these results the most promising ones are selected. We report the efficient implementation of the latter algorithms invoking automated programming techniques and also evaluate their practical performance. We conclude that the simplified Obara-Saika scheme of Ahlrichs is the most cost-effective one in the majority of cases, but the modified Gill-Head-Gordon-Pople and Rys algorithms proposed herein are preferred for particular shell triplets. Our numerical experiments also show that even though the solid harmonic transformation and the horizontal recurrence require significantly fewer FLOPs if performed at the contracted level, this approach does not improve the efficiency in practical cases. Instead, it is more advantageous to carry out these operations at the primitive level, which allows for more efficient integral prescreening and memory layout.

The Laplace transformed divide-expand-consolidate resolution of the identity second-order Møller-Plesset perturbation (DEC-LT-RIMP2) theory method

The divide-expand-consolidate resolution of the identity second-order Møller-Plesset perturbation (DEC-RI-MP2) theory method introduced in Baudin et al. [J. Chem. Phys. 144, 054102 (2016)] is significantly improved by introducing the Laplace transform of the orbital energy denominator in order to construct the double amplitudes directly in the local basis. Furthermore, this paper introduces the auxiliary reduction procedure, which reduces the set of the auxiliary functions employed in the individual fragments. The resulting Laplace transformed divide-expand-consolidate resolution of the identity second-order Møller-Plesset perturbation method is applied to the insulin molecule where we obtain a factor 9.5 speedup compared to the DEC-RI-MP2 method.