Density Fitting Research Articles

Basis set limit MP2 energies for extended molecules via a reduced-cost explicitly correlated approach

A reduced-cost explicitly correlated second-order Møller–Plesset (MP2-F12) approach is presented. Based on our recent reduced-cost explicitly correlated coupled-cluster method [J. Chem. Theory Comput., 19, 174 (2023)], the frozen natural orbital, the natural auxiliary basis, and the natural auxiliary function approximations are utilised to decrease the size of the molecular orbital basis and the auxiliary basis sets required for the density fitting and for the expansion of the explicitly correlated geminals. The necessary modifications of the above approximations for MP2-F12 are discussed. The performance of the new method is tested for atomisation and reaction energies. Our results demonstrate that speedups of 1.5–4 can be achieved in the calculation of MP2-F12 correlation energies with a moderate loss of accuracy.

Unravelling material properties of halide perovskites by combined microwave photoconductivity and time-resolved photoluminescence spectroscopy

Simultaneous measurements of trPL and microwave photoconductivity facilitate a global fit of charge carrier density in perovskite materials. This enables the extraction of fundamental rate constants and the mobility ratio of electrons and holes.

Efficient Hartree–Fock exchange algorithm with Coulomb range separation and long-range density fitting

Separating the Coulomb potential into short-range and long-range components enables the use of different electron repulsion integral algorithms for each component. The short-range part can be efficiently computed using the analytical algorithm due to the locality in both the Gaussian-type orbital basis and the short-range Coulomb potentials. The integrals for the long-range Coulomb potential can be approximated with the density fitting method. A very small auxiliary basis is sufficient for the density fitting method to accurately approximate the long-range integrals. This feature significantly reduces the computational efforts associated with the N4 scaling in density fitting algorithms. For large molecules, the range separation and long-range density fitting method outperforms the conventional analytical integral evaluation scheme employed in Hartree–Fock calculations and provides more than twice the overall performance. In addition, this method offers a higher accuracy compared to conventional density fitting methods. The error in the Hartree–Fock energy can be easily reduced to 0.1 μEh per atom or smaller.

Can lifelong endurance exercise improve ageing through beneficial effects on circadian timing function, muscular performance and health status in men? Protocol for a comparative cross-sectional study.

A well-synchronized circadian system is a manifestation of an individual's health. A gradual weakening of the circadian timing function characterizes aging. Regular exercise has been suggested as a modality to improve many detrimental changes associated with aging. Therefore, we aim to examine the benefits and risks of lifelong endurance exercise on age-dependent changes in the circadian time-keeping function, the performance of the muscular system and health status. The study protocol has a comparative cross-sectional design, including groups of senior (65 to 75 years old, n=16) and young (20-30 years old, n=16) endurance runners and triathletes. Age-matched groups of young and elderly sedentary men are included as controls. The circadian function is evaluated mainly by measurement of urinary 6-sulphatoxymelatonin, a metabolite of the hormone melatonin shown to participate in the modulation of sleep cycles. The 6-sulphatoxymelatonin will be assessed in urine samples collected upon awakening in the morning and in the late evening, as a marker of melatonin production. In addition, sleep/activity rhythms and sleep quality will be measured by wrist actigraphy. Performance of the muscular system will be assessed by examination of muscular strength and quantifying of gene expression in the skeletal muscle tissue samples. Health status and age-induced reduction in immune function are to be analysed via the balance of pro- and anti-inflammatory immune markers in the plasma and skeletal muscle, body composition, bone density and physical fitness.

Nuclear Quantum Effects Made Accessible: Local Density Fitting in Multicomponent Methods.

The simulation of nuclear quantum effects (NQEs) is crucial for an accurate description of systems and processes involving light nuclei, such as hydrogen atoms. Within the last years, the importance of those effects has been highlighted for a vast range of systems with tremendous implications in chemistry, biology, physics, and materials sciences. However, while electronic structure theory methods have become routine tools for quantum chemical investigations, there is still a lack of approaches to address NQEs that are computationally accessible and straightforward to use. To address this, we present the first combination of the nuclear-electronic orbital Hartree-Fock approach with both local and density fitting approximations (LDF-NEO-HF). This results in a low-order scaling approach that enables the inclusion of NQEs for large systems within a fraction of a day and for small to medium size systems in minutes. Moreover, we demonstrate the qualitative accuracy and robustness of our approach to retrieve NQEs for three real-use cases motivated by chemical, biological, and materials science applications.

The versatility of the Cholesky decomposition in electronic structure theory

AbstractThe resolution‐of‐the‐identity (RI) or density fitting (DF) approximation for the electron repulsion integrals (ERIs) has become a standard component of accelerated and reduced‐scaling implementations of first‐principles Gaussian‐type orbital electronic‐structure methods. The Cholesky decomposition (CD) of the ERIs has also become increasingly deployed across quantum chemistry packages in the last decade, even though its early applications were mostly limited to high‐accuracy methods such as coupled‐cluster theory and multiconfigurational approaches. Starting with a summary of the basic theory underpinning both the CD and RI/DF approximations, thus underlining the extremely close relation of the CD and RI/DF techniques, we provide a brief and largely chronological review of the evolution of the CD approach from its birth in 1977 to its current state. In addition to being a purely numerical procedure for handling ERIs, thus providing robust and computationally efficient approximations to the exact ERIs that have been found increasingly useful on modern computer platforms, CD also offers highly accurate approaches for generating auxiliary basis sets for the RI/DF approximation on the fly due to the deep mathematical connection between the two approaches. In this review, we aim to provide a concise reference of the main techniques employed in various CD approaches in electronic structure theory, to exemplify the connection between the CD and RI/DF approaches, and to clarify the state of the art to guide new implementations of CD approaches across electronic structure programs.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry Quantum Computing > Theory Development

Accelerating Coupled-Cluster Calculations with GPUs: An Implementation of the Density-Fitted CCSD(T) Approach for Heterogeneous Computing Architectures Using OpenMP Directives.

An algorithm is presented for the coupled-cluster singles, doubles, and perturbative triples correction [CCSD(T)] method based on the density fitting or the resolution-of-the-identity (RI) approximation for performing calculations on heterogeneous computing platforms composed of multicore CPUs and graphics processing units (GPUs). The directive-based approach to GPU offloading offered by the OpenMP application programming interface has been employed to adapt the most compute-intensive terms in the RI-CCSD amplitude equations with computational costs scaling as , , and (where NO and NV denote the numbers of correlated occupied and virtual orbitals, respectively) and the perturbative triples correction to execute on GPU architectures. The pertinent tensor contractions are performed using an accelerated math library such as cuBLAS or hipBLAS. Optimal strategies are discussed for splitting large data arrays into tiles to fit them into the relatively small memory space of the GPUs, while also minimizing the low-bandwidth CPU-GPU data transfers. The performance of the hybrid CPU-GPU RI-CCSD(T) code is demonstrated on pre-exascale supercomputers composed of heterogeneous nodes equipped with NVIDIA Tesla V100 and A100 GPUs and on the world's first exascale supercomputer named "Frontier", the nodes of which consist of AMD MI250X GPUs. Speedups within the range 4-8× relative to the recently reported CPU-only algorithm are obtained for the GPU-offloaded terms in the RI-CCSD amplitude equations. Applications to polycyclic aromatic hydrocarbons containing 16-66 carbon atoms demonstrate that the acceleration of the hybrid CPU-GPU code for the perturbative triples correction relative to the CPU-only code increases with the molecule size, attaining a speedup of 5.7× for the largest circumovalene molecule (C66H20). The GPU-offloaded code enables the computation of the perturbative triples correction for the C60 molecule using the cc-pVDZ/aug-cc-pVTZ-RI basis sets in 7 min on Frontier when using 12,288 AMD GPUs with a parallel efficiency of 83.1%.

Automatic Generation of Auxiliary Basis Sets in Spherical Representation Using the Cholesky Decomposition.

Density fitting techniques that use automatically generated auxiliary basis sets generally rely on the formation of basis function products. Recently, Lehtola [ J. Chem. Theory Comput. 2021, 17, 6886-6900] presented a procedure making use of a purely spherical representation by adding auxiliary basis functions coupled to the required angular momentum quantum numbers for the product of spherical harmonics and then removing linear dependencies by means of a Cholesky decomposition. In this work, we extend this idea by making use of the explicit equations for the product of two spherical harmonics in the angular part of the basis function product. Some of the resulting terms are not directly accessible when popular standard integral libraries are used, which could prevent the widespread use of the exact product form. For these terms, we introduce four approximations of increasing sophistication that require integrals involving only standard Gaussian-type orbitals and thus can be computed with standard libraries. We assess the accuracy of the different schemes in the context of the aCD for the reconstruction of the electron repulsion integral matrix and absolute and relative single point energies and in the framework of optimally tuned range-separated hybrid functionals.

The General Atomic and Molecular Electronic Structure System (GAMESS): Novel Methods on Novel Architectures.

The primary focus of GAMESS over the last 5 years has been the development of new high-performance codes that are able to take effective and efficient advantage of the most advanced computer architectures, both CPU and accelerators. These efforts include employing density fitting and fragmentation methods to reduce the high scaling of well-correlated (e.g., coupled-cluster) methods as well as developing novel codes that can take optimal advantage of graphical processing units and other modern accelerators. Because accurate wave functions can be very complex, an important new functionality in GAMESS is the quasi-atomic orbital analysis, an unbiased approach to the understanding of covalent bonds embedded in the wave function. Best practices for the maintenance and distribution of GAMESS are also discussed.

Automatic Generation of Accurate and Cost-Efficient Auxiliary Basis Sets.

We have recently discussed an algorithm to automatically generate auxiliary basis sets (ABSs) of the standard form for density fitting (DF) or resolution-of-the-identity (RI) calculations in a given atomic orbital basis set (OBS) of any form, such as Gaussian-type orbitals, Slater-type orbitals, or numerical atomic orbitals [J. Chem. Theory Comput. 2021, 17, 6886]. In this work, we study two ways to reduce the cost of such automatically generated ABSs without sacrificing their accuracy. We contract the ABS with a singular value decomposition proposed by Kállay [J. Chem. Phys. 2014, 141, 244113], used here in a somewhat different setting. We also drop the high-angular momentum functions from the ABS, as they are unnecessary for global fitting methods. Studying the effect of these two types of truncations on Hartree-Fock (HF) and second-order Møller-Plesset perturbation theory (MP2) calculations on a chemically diverse set of first- and second-row molecules within the RI/DF approach, we show that accurate total and atomization energies can be achieved by a combination of the two approaches with significant reductions in the size of the ABS. While the original approach yields ABSs whose number of functions NbfABS scales with the number of functions in the OBS, NOBSbf, as NABSbf = γNOBSbf with the prefactor , the reduction schemes of this work afford results of essentially the same quality as the original unpruned and uncontracted ABS with γ ≈ 5-6, while an accuracy that may suffice for routine applications is achievable with a further reduced ABS with γ ≈ 3-4. The observed errors are similar at HF and MP2 levels of theory, suggesting that the generated ABSs are highly transferable and can also be applied to model challenging properties with high-level methods.

Low-Scaling Algorithm for the Random Phase Approximation Using Tensor Hypercontraction with k-point Sampling.

We present a low-scaling algorithm for the random phase approximation (RPA) with k-point sampling in the framework of tensor hypercontraction (THC) for electron repulsion integrals (ERIs). The THC factorization is obtained via a revised interpolative separable density fitting (ISDF) procedure with a momentum-dependent auxiliary basis for generic single-particle Bloch orbitals. Our formulation does not require preoptimized interpolating points or auxiliary bases, and the accuracy is systematically controlled by the number of interpolating points. The resulting RPA algorithm scales linearly with the number of k-points and cubically with the system size without any assumption on sparsity or locality of orbitals. The errors of ERIs and RPA energy show rapid convergence with respect to the size of the THC auxiliary basis, suggesting a promising and robust direction to construct efficient algorithms of higher order many-body perturbation theories for large-scale systems.

Two-Component GW Calculations: Cubic Scaling Implementation and Comparison of Vertex-Corrected and Partially Self-Consistent GW Variants.

We report an all-electron, atomic orbital (AO)-based, two-component (2C) implementation of the GW approximation (GWA) for closed-shell molecules. Our algorithm is based on the space-time formulation of the GWA and uses analytical continuation (AC) of the self-energy, and pair-atomic density fitting (PADF) to switch between AO and auxiliary basis. By calculating the dynamical contribution to the GW self-energy at a quasi-one-component level, our 2C-GW algorithm is only about a factor of 2-3 slower than in the scalar relativistic case. Additionally, we present a 2C implementation of the simplest vertex correction to the self-energy, the statically screened G3W2 correction. Comparison of first ionization potentials (IPs) of a set of 67 molecules with heavy elements (a subset of the SOC81 set) calculated with our implementation against results from the WEST code reveals mean absolute deviations (MAD) of around 70 meV for G0W0@PBE and G0W0@PBE0. We check the accuracy of our AC treatment by comparison to full-frequency GW calculations, which shows that in the absence of multisolution cases, the errors due to AC are only minor. This implies that the main sources of the observed deviations between both implementations are the different single-particle bases and the pseudopotential approximation in the WEST code. Finally, we assess the performance of some (partially self-consistent) variants of the GWA for the calculation of first IPs by comparison to vertical experimental reference values. G0W0@PBE0 (25% exact exchange) and G0W0@BHLYP (50% exact exchange) perform best with mean absolute deviations (MAD) of about 200 meV. Explicit treatment of spin-orbit effects at the 2C level is crucial for systematic agreement with experiment. On the other hand, eigenvalue-only self-consistent GW (evGW) and quasi-particle self-consistent GW (qsGW) significantly overestimate the IPs. Perturbative G3W2 corrections increase the IPs and therefore improve the agreement with experiment in cases where G0W0 alone underestimates the IPs. With a MAD of only 140 meV, 2C-G0W0@PBE0 + G3W2 is in best agreement with the experimental reference values.

Even Faster Exact Exchange for Solids via Tensor Hypercontraction.

Hybrid density functional theory (DFT) remains intractable for large periodic systems due to the demanding computational cost of exact exchange. We apply the tensor hypercontraction (THC) (or interpolative separable density fitting) approximation to periodic hybrid DFT calculations with Gaussian-type orbitals using the Gaussian plane wave approach. This is done to lower the computational scaling with respect to the number of basis functions (N) and k-points (Nk) at a fixed system size. Additionally, we propose an algorithm to fit only occupied orbital products via THC (i.e., a set of points, NISDF) to further reduce computation time and memory usage. This algorithm has linear scaling cost with k-points, no explicit dependence of NISDF on basis set size, and overall cubic scaling with unit cell size. Significant speedups and reduced memory usage may be obtained for moderately sized k-point meshes, with additional gains for large k-point meshes. Adequate accuracy can be obtained using THC-oo-K for self-consistent calculations. We perform illustrative hybrid density function theory calculations on the benzene crystal in the basis set and thermodynamic limits to highlight the utility of this algorithm.

Common predators and factors influencing their abundance in Anopheles funestus aquatic habitats in rural south-eastern Tanzania.

The role of larval predators in regulating the Anopheles funestus population in various malaria-endemic countries remains relatively unknown. This study aimed to investigate the common predators that co-exist with Anopheles funestus group larvae and evaluate factors that influence their abundance in rural south-eastern Tanzania. Mosquito larvae and predators were sampled concurrently using standard dipper (350 ml) or 10 L bucket in previously identified aquatic habitats in selected villages in southern Tanzania. Predators and mosquito larvae were identified using standard identification keys. All positive habitats were geo-located and their physical features characterized. Water physicochemical parameters such as dissolved oxygen (DO), pH, electrical conductivity (EC), total dissolved solids (TDS) and temperature were also recorded. A total of 85 previously identified An. funestus aquatic habitats in nine villages were sampled for larvae and potential predators. A total of 8,295 predators were sampled. Of these Coenagrionidae 57.7% (n = 4785), Corixidae 12.8% (n = 1,060), Notonectidae 9.9% (n = 822), Aeshnidae 4.9% (n = 405), Amphibian 4.5% (n = 370), Dytiscidae 3.8% (n = 313) were common. A total of 5,260 mosquito larvae were sampled, whereby Anopheles funestus group were 60.3% (n = 3,170), Culex spp. 24.3% (n = 1,279), An. gambie s.l. 8.3% (n = 438) and other anophelines 7.1% (n = 373). Permanent and aquatic habitats larger than 100m2 were positively associated with An. funestus group larvae (P<0.05) and predator abundance (P<0.05). Habitats with submerged vegetation were negatively associated with An. funestus group larvae (P<0.05). Only dissolved oxygen (DO) was positively and significantly affect the abundance of An. funestus group larvae (P<0.05). While predators' abundance was not impacted by all physicochemical parameters. Six potential predator families were common in aquatic habitats of An. funestus group larvae. Additional studies are needed to demonstrate the efficacy of different predators on larval density and adult fitness traits. Interventions leveraging the interaction between mosquitoes and predators can be established to disrupt the transmission potential and survival of the An. funestus mosquitoes.

Vertical Ionization Potentials and Electron Affinities at the Double-Hybrid Density Functional Level.

The double-hybrid (DH) time-dependent density functional theory is extended to vertical ionization potentials (VIPs) and electron affinities (VEAs). Utilizing the density fitting approximation, efficient implementations are presented for the genuine DH ansatz relying on the perturbative second-order correction, while an iterative analogue is also elaborated using our second-order algebraic-diagrammatic construction [ADC(2)]-based DH approach. The favorable computational requirements of the present schemes are discussed in detail. The performance of the recently proposed spin-component-scaled and spin-opposite-scaled (SOS) range-separated (RS) and long-range corrected (LC) DH functionals is comprehensively assessed, while popular hybrid and global DH approaches are also discussed. For the benchmark calculations, up-to-date test sets are selected with high-level coupled-cluster references. Our results show that the ADC(2)-based SOS-RS-PBE-P86 approach is the most accurate and robust functional. This method consistently outperforms the excellent SOS-ADC(2) approach for VIPs, although the results are somewhat less satisfactory for VEAs. Among the genuine DH functionals, the SOS-ωPBEPP86 approach is also recommended for describing ionization processes, but its performance is even less reliable for electron-attached states. In addition, surprisingly good results are attained by the LC hybrid ωB97X-D functional, where the corresponding occupied (unoccupied) orbital energies are retrieved as VIPs (VEAs) within the present formalism.

Impacts of Max-Stable Process Areal Exceedance Calculations to Study Area Sampling Density, Surface Network Precipitation Gage Extent and Density, and Model Fitting Method

Max-stable process (MSP) models can be fit to data collected over a spatial domain to estimate areal-based exceedances while accounting for spatial dependence in extremes. They have theoretical grounding within the framework of extreme value theory (EVT). In this work, we fit MSP models to three-day duration cool season precipitation maxima in the Willamette River Basin (WRB) of Oregon and to 48 h mid-latitude cyclone precipitation annual maxima in the Upper Trinity River Basin (TRB) of Texas. In total, 14 MSP models were fit (seven based on the WRB data and seven based on the TRB data). These MSP model fits were developed and applied to explore how user choices of study area sampling density, gage extent, and model fitting method impact areal precipitation-frequency calculations. The impacts of gage density were also evaluated. The development of each MSP involved the application of a recently introduced trend surface modeling methodology. Significant reductions in computing times were achieved, with little loss in accuracy, applying random sample subsets rather than the entire grid when calculating areal exceedances for the Cougar dam study area in the WRB. Explorations of gage extent revealed poor consistency among the TRB MSPs with modeling the generalized extreme value (GEV) marginal distribution scale parameter. The gauge density study revealed the robustness of the trend surface modeling methodology. Regardless of the fitting method, the final GEV shape parameter estimates for all fourteen MSPs were greater than their prescribed initial values which were obtained from spatial GEV fits that assumed independence among the extremes. When two MSP models only differed by their selected fitting method, notable differences were observed with their dependence and trend surface parameter estimates and resulting areal exceedances calculations.

Reduced-Cost Second-Order Algebraic-Diagrammatic Construction Method for Core Excitations

Our reduced-cost scheme [J. Chem. Phys. 2018, 148, 094111] based on the frozen virtual natural orbital and natural auxiliary function approaches is extended to core excitations. The efficiency of the approximation is presented for the second-order algebraic-diagrammatic construction [ADC(2)] method invoking the core-valence separation (CVS) and density fitting approaches. The errors introduced by the present scheme are comprehensively analyzed for more than 200 excitation energies and 80 oscillator strengths, including C, N, and O K-edge excitations, as well as 1s → π* and Rydberg transitions. Our results show that significant savings can be gained in computational requirements at the expense of a moderate error. That is, the mean absolute error for the excitation energies, being lower than 0.20 eV, is an order of magnitude smaller than the intrinsic error of CVS-ADC(2), while the mean relative error for the oscillator strengths is between 0.06 and 0.08, which is still acceptable. As significant differences for different types of excitations cannot be observed, the robustness of the approximation is also demonstrated. The improvements in the computational requirements are measured for extended molecules. In this case, an overall 7-fold speedup is obtained in the wall-clock times, while dramatic reductions in the memory requirements are also achieved. In addition, it is also proved that the new approach enables us to perform CVS-ADC(2) calculations within reasonable runtime for systems of 100 atoms using reliable basis sets.

Low-rank approximations for accelerating plane-wave hybrid functional calculations in unrestricted and noncollinear spin density functional theory.

The adaptively compressed exchange (ACE) operator combined with interpolative separable density fitting (ISDF) decomposition has been utilized to accelerate plane-wave hybrid functional calculations for restricted Kohn-Sham density functional theory (DFT), but the neglect of spin degree of freedom has limited its application in the exploration of systems where the spin property of the electron is critical. Herein, we derive the ACE-ISDF formulation for hybrid functional calculations in both unrestricted and noncollinear spin DFT with plane waves and periodic boundary conditions. We proposed an improved ISDF algorithm for the sum of Kohn-Sham orbital pairs to further reduce the computational cost for the spin-noncollinear case. Numerical results demonstrate that these improved ACE-ISDF low-rank approximations can not only significantly reduce the computational time by two orders of magnitude compared with conventional plane-wave hybrid functional calculations but also lead to a good convergence behavior when a moderate rank parameter is set, even for complex periodic magnetic systems. By using these ACE-ISDF approximations, we investigate the electronic and magnetic properties of two-dimensional periodic ferromagnetic semiconductors consisting of triangular zigzag graphene quantum dots and transition metal atoms. Our computational results showcase that hybrid functional calculations in spin DFT can provide not only accurate electronic structures but also accurate magnetic order temperature of ferromagnetic semiconductors compared to local or semilocal functional calculations.

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.

Low-rank approximations to accelerate hybrid functional enabled real-time time-dependent density functional theory within plane waves

Real-time time-dependent density functional theory (RT-TDDFT) is a powerful tool for predicting excited-state dynamics. Herein, we combine the adaptively compressed exchange (ACE) operator with interpolative separable density fitting (ISDF) algorithm to accelerate the hybrid functional calculations in RT-TDDFT (hybrid RT-TDDFT) dynamics simulations for molecular and periodic systems within plane waves. Under this low-rank representation, we demonstrate that the ACE-ISDF enabled hybrid RT-TDDFT can yield accurate excited-state dynamics, but much faster than conventional calculations. Furthermore, we describe a massively parallel implementation of ACE-ISDF enabled hybrid RT-TDDFT dynamics simulations containing thousands of atoms (1728 atoms), which can scale up to 3456 central processing unit cores on modern supercomputers.