Auxiliary Basis Research Articles

Fast periodic Gaussian density fitting by range separation.

We present an efficient implementation of periodic Gaussian density fitting (GDF) using the Coulomb metric. The three-center integrals are divided into two parts by range-separating the Coulomb kernel, with the short-range part evaluated in real space and the long-range part in reciprocal space. With a few algorithmic optimizations, we show that this new method-which we call range-separated GDF (RSGDF)-scales sublinearly to linearly with the number of k-points for small to medium-sized k-point meshes that are commonly used in periodic calculations with electron correlation. Numerical results on a few three-dimensional solids show about ten-fold speedups over the previously developed GDF with little precision loss. The error introduced by RSGDF is about 10-5Eh in the converged Hartree-Fock energy with default auxiliary basis sets and can be systematically reduced by increasing the size of the auxiliary basis with little extra work.

Resolution of the identity approximation applied to PNOF correlation calculations.

In this work, the required algebra to employ the resolution of the identity approximation within the Piris Natural Orbital Functional (PNOF) is developed, leading to an implementation named DoNOF-RI. The arithmetic scaling is reduced from fifth-order to fourth-order, and the memory scaling is reduced from fourth-order to third-order, allowing significant computational time savings. After the DoNOF-RI calculation has fully converged, a restart with four-center electron repulsion integrals can be performed to remove the effect of the auxiliary basis set incompleteness, quickly converging to the exact result. The proposed approach has been tested on cycloalkanes and other molecules of general interest to study the numerical results, as well as the speed-ups achieved by PNOF7-RI when compared with PNOF7.

Calculation of environmental capacity and pollutant discharge in Changxing county based on sustainable development

Due to the improvement of the understanding of the natural environment and social service function of rivers, many countries in the world have started the research and practice of water environment protection. Lots of Chinese cities also paid most of their attentions to the water environment especially in the river network area. Systematic calculation and analysis of the water environment status becomes famous in most of the Chinese cities, which not only shows the current status of water environment but also indicates the methods to reduce the domestic pollution. In this study, Changxing county is selected as a typical Chinese city, COD (Chemical Oxygen Demand) and TN (Total Nitrogen) are considered as two typical pollutants, both water environmental capacity and current pollution are calculated. From the comparison between the present pollution load and the discharge limit into the river, it can be seen that TN is the key influencing factor of the water environment carrying capacity in Changxing County, and its overload ratio is up to 5.0. A series of measures to reduce the pollution load into the river are put forward in order to provide an auxiliary decision-making basis for further protecting “clear water and green mountains”.

All-electron periodic G0W0 implementation with numerical atomic orbital basis functions: Algorithm and benchmarks

We present an all-electron, periodic ${G}_{\text{0}}{W}_{\text{0}}$ implementation within the numerical atomic orbital (NAO) basis framework. A localized variant of the resolution-of-the-identity (RI) approximation is employed to significantly reduce the computational cost of evaluating and storing the two-electron Coulomb repulsion integrals. We demonstrate that the error arising from localized RI approximation can be reduced to an insignificant level by enhancing the set of auxiliary basis functions, used to expand the products of two single-particle NAOs. An efficient algorithm is introduced to deal with the Coulomb singularity in the Brillouin zone sampling that is suitable for the NAO framework. We perform systematic convergence tests and identify a set of computational parameters, which can serve as the default choice for most practical purposes. Benchmark calculations are carried out for a set of prototypical semiconductors and insulators, and compared to independent reference values obtained from an independent ${G}_{\text{0}}{W}_{\text{0}}$ implementation based on linearized augmented plane waves (LAPWs) plus high-energy localized orbitals (HLOs) basis set, as well as experimental results. With a moderate (FHI-aims tier 2) NAO basis set, our ${G}_{\text{0}}{W}_{\text{0}}$ calculations produce band gaps that typically lie in between the standard LAPW and the $\mathrm{LAPW}+\mathrm{HLO}$ results. Complementing tier 2 with highly localized Slater-type orbitals (STOs), we find that the obtained band gaps show an overall convergence towards the $\mathrm{LAPW}+\mathrm{HLO}$ results. The algorithms and techniques developed in this work pave the way for efficient implementations of correlated methods within the NAO framework.

Compaction Quality Inspection Method of Soil-Rock Filled Embankment Based on Continuous Compaction Control Technology

Affected by the site construction conditions, the measurement passes of the Taihang Expressway K8 + 105 ∼ K8 + 341 (K8 worksite) in the Taihang Expressway did not meet the requirements of data analysis, and the quantity of the control points was insufficient so that the linear correlation between the dynamic deformation modulus () and the vibratory compaction value (VCV) was not strong. Therefore, the target value of VCV cannot be used to diagnose the E v d compaction quality of soil-rock filler. This paper analyzes the roller measurement VCV value and in situ measurements E v d value separately. Results reveal the difference between the VCV mean measured in the last two passes and the standard deviation of the measured VCV mean in the last pass are used as the main basis for the actual compaction quality. In addition, the E v d mean in the last rolling can be used as an auxiliary judgment basis for the quality control of the compaction.

Quantum chemical calculations for elucidating chiral resolution mechanism of chiral covalent organic frameworks 6 chromatographic stationary phase

With the aim of exploring the chiral resolution mechanism of the chiral covalent organic frameworks 6 (CCOF6) chromatographic stationary phase, the ORCA program was used to optimize the structures of CCOF6 and four pairs of enantiomers. Then the molecular docking of CCOF6 with each enantiomer was performed by AutoDock program to obtain the initial interaction configurations of CCOF6 with four pairs of enantiomers. Energy calculations of the initial configurations were performed by ORCA program (B3LYP functional with DFT-D3 correction, def2-TZVP orbital basis set, def2/J auxiliary basis set, and RIJCOSX used to accelerate the calculation) to determine the interaction configurations of CCOF6 with four pairs of enantiomers and to obtain the corresponding binding free energy and binding free energy difference. Wave function analyses of ORCA calculation results were performed by Multiwfn program, and the weak interactions between CCOF6 and the four pairs of enantiomers were visualized by Visual Molecular Dynamics program. The results showed the following:① for calculating the binding free energies of CCOF6 and the four pairs of enantiomers, the ORCA method with the solvation effect was more accurate than the AutoDock method as well as the ORCA method without the solvation effect; ② the greater the absolute value of the binding free energy difference between the CCOF6 stationary phase and enantiomers, the greater was the selectivity factor of the enantiomers but not the resolution of the enantiomers; ③ the hydroxyl group of S-1-phenyl-1-propanol interacted with the ether bond of CCOF6, but the hydroxyl groups of the other enantiomers all interacted with the carbonyl groups of CCOF6, and the binding force between S-1-phenyl-1-propanol and CCOF6 was the weakest; ④ from the peak time of the enantiomer and its binding free energy with CCOF6, it was confirmed that the elution ability of n-hexane/isopropanol for 1-phenyl-1-propanol was the weakest, followed by the elution ability for 1-phenyl-2-propanol; ⑤ the peak time of S-1-phenyl-1-propanol was longer than that of R-1-phenyl-1-propanol, while for the other enantiomers, the peak time of the R-enantiomers was longer than that of the S-enantiomers.

An improved seminumerical Coulomb and exchange algorithm for properties and excited states in modern density functional theory.

A seminumerical algorithm capable of performing large-scale (time-dependent) density functional theory (TD-DFT) calculations to extract excitation energies and other ground-state and excited-state properties is outlined. The algorithm uses seminumerical integral techniques for evaluating Coulomb and exchange parts for a set of density matrices as occurring in standard TD-DFT or similar methods for the evaluation of vibrational frequencies. A suitable optimized de-aliasing procedure is introduced. The latter does not depend on further auxiliary quantities and retains the symmetry of a given density matrix. The algorithm is self-contained and applicable to any orbital basis set available without the need for further auxiliary basis sets or optimized de-aliasing grids. Relativistic two-component excited-state TD-DFT calculations are reported for the first time using the developed seminumerical algorithm for standard and local hybrid density functional approximations. Errors are compared with the widely used "resolution of the identity" (RI) approximations for Coulomb (RI-J) and exchange integrals (RI-K). The fully seminumerical algorithm does not exhibit an enlarged error for standard DFT functionals compared to the RI approximation. For the more involved local hybrid functionals and within strong external fields, accuracy is even considerably improved.

Electrostatic Potential Fitting Method Using Constrained Spatial Electron Density Expanded with Preorthogonal Natural Atomic Orbitals.

In this study, an electrostatic potential (ESP) fitting method using constrained spatial electron density (cSED) expanded with preorthogonal natural atomic orbitals (pNAOs) was proposed. In this method, the electron density of a molecule is divided into spherical atom-centered electron densities and the expansion coefficient is determined to reproduce the ESP around the molecule. Our method was then applied to two systems: (i) a hydration reaction of cis-platin and (ii) a variety of organic/inorganic molecules. By evaluating the atomic charges along the hydration reaction, our method showed good conformational transferability, which cannot be obtained using conventional ESP fitting methods. Moreover, we successfully obtained the hydration structure along the reaction by coupling our method with a reference interaction site model (RISM). Reasonable data were obtained not only for organic molecules but also for inorganic molecules. This success came from the introduction of pNAOs as auxiliary basis sets in the charge fitting.

Optimization of atomic density-fitting basis functions for molecular two-electron integral approximations.

A general procedure for the optimization of atomic density-fitting basis functions is designed with the balance between accuracy and numerical stability in mind. Given one-electron wavefunctions and energies, weights are assigned to the product densities, modeling their contribution to the exchange and second-order correlation energy, and a simple weighted error measure is minimized. Generally contracted Gaussian auxiliary basis sets are optimized to match the wavefunction basis sets [D. N. Laikov, Theor. Chem. Acc. 138, 40 (2019)] for all 102 elements in a scalar-relativistic approximation [D. N. Laikov, J. Chem. Phys. 150, 061103 (2019)].

Minimal auxiliary basis set for time-dependent density functional theory and comparison with tight-binding approximations: Application to silver nanoparticles.

The modeling of optical spectra of plasmonic nanoparticles via first-principles approaches is computationally expensive; thus, methods with high accuracy/computational cost ratio are required. Here, we show that the Time-Dependent Density Functional Theory (TDDFT) approach can be strongly simplified if only one s-type function per atom is employed in the auxiliary basis set, with a properly optimized exponent. This approach (named TDDFT-as, for auxiliary s-type) predicts excitation energies for silver nanoparticles with different sizes and shapes with an average error of only 12 meV compared to reference TDDFT calculations. The TDDFT-as approach resembles tight-binding approximation schemes for the linear-response treatment, but for the atomic transition charges, which are here computed exactly (i.e., without approximation from population analysis). We found that the exact computation of the atomic transition charges strongly improves the absorption spectra in a wide energy range.

Constraint energy minimizing generalized multiscale finite element method for nonlinear poroelasticity and elasticity

In this paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to first solving a nonlinear poroelasticity problem. The arising system consists of a nonlinear pressure equation and a nonlinear stress equation in strain-limiting setting, where strains keep bounded while stresses can grow arbitrarily large. After time-discretization of the system, to tackle the nonlinearity, we linearize the resulting equations by Picard iteration. To handle the linearized equations, we employ the CEM-GMsFEM and obtain appropriate offline multiscale basis functions for the pressure and the displacement. More specifically, first, auxiliary multiscale basis functions are generated by solving local spectral problems, via the GMsFEM. Then, multiscale spaces are constructed in oversampled regions, by solving a constraint energy minimizing (CEM) problem. After that, this strategy (with the CEM-GMsFEM) is also applied to a static case of the above nonlinear poroelasticity problem, that is, elasticity problem, where the residual based online multiscale basis functions are generated by an adaptive enrichment procedure, to further reduce the error. Convergence of the two cases is demonstrated by several numerical simulations, which give accurate solutions, with converging coarse-mesh sizes as well as few basis functions (degrees of freedom) and oversampling layers.

A self-consistent coulomb bath model using density fitting.

A self-consistent Coulomb bath model is presented to provide an accurate and efficient way of performing calculations for interfragment electrostatic and polarization interactions. In this method, a condensed-phase system is partitioned into molecular fragment blocks. Each fragment is embedded in the Coulomb bath due to other fragments. Importantly, the present Coulomb bath is represented using a density fitting method in which the electron densities of molecular fragments are fitted using an atom-centered auxiliary basis set of Gaussian type. The Coulomb bath is incorporated into an effective Hamiltonian for each fragment, with which the electron density is optimized through an iterative double self-consistent field (DSCF) procedure to realize the mutual many-body polarization effects. In this work, the accuracy of interfragment interaction energies enumerated using the Coulomb bath is tested, showing a good agreement with the exact results from an energy decomposition analysis. The qualitative features of many-body polarization effects are visualized by electron density difference plots. It is also shown that the present DSCF method can yield fast and robust convergence with near-linear scaling in performance with increase in system size.

Resolution-of-the-identity second-order Møller-Plesset perturbation theory with complex basis functions: Benchmark calculations and applications to strong-field ionization of polyacenes.

We study the performance of the resolution-of-the-identity (RI) approximation for complex basis functions that we recently introduced [M. Hernández Vera and T.-C. Jagau, J. Chem. Phys. 151, 111101 (2019)] for second-order Møller-Plesset (MP2) perturbation theory as well as for the Coulomb and exchange contributions in Hartree-Fock theory. The sensitivity of this new RI-MP2 method toward the basis set and the auxiliary basis set is investigated, and computation times are analyzed. We show that the auxiliary basis set can be chosen purely real, that is, no complex-scaled functions need to be included. This approximation enables a further speedup of the method without compromising accuracy. We illustrate the application range of our implementation by computing static-field ionization rates of several polyacenes up to pentacene (C22H18) at the RI-MP2 level of theory. Pronounced anisotropies are observed for the ionization rates of these molecules.

Comparison of China-ASHRAE HVAC Design Criteria in Coal-fired Power Plants

This paper investigates the requirements of Chinese and ASHRAE HVAC design manuals and codes for various types of buildings within the power plants. By analysing the characteristics of HVAC basic design criteria for different building areas, the summary of similarities and differences between these two design systems are found. This paper studies the applicability and adaptability of Chinese HVAC design system as an auxiliary design basis in foreign power plant project, aiming to provide the basis for the selection of HVACdesignfor foreign design of power plant projects.

Accuracy of Localized Resolution of the Identity in Periodic Hybrid Functional Calculations with Numerical Atomic Orbitals.

We present an implementation of hybrid density functional approximations for periodic systems within a pseudopotential-based, numerical atomic orbital (NAO) framework. The two-electron Coulomb repulsion integrals (ERIs) are evaluated using the localized resolution-of-the-identity (LRI) approximation. The accuracy of the LRI approximation is benchmarked unambiguously against independent reference results obtained via a computational scheme whereby the ERIs are accurately evaluated by expanding the products of NAOs in terms of plane waves. An alternative strategy for constructing auxiliary basis sets is proposed, and its accuracy is assessed and compared to the previously used procedure. Finally, the reliability of our algorithm and implementation is benchmarked against other established implementations within different numerical frameworks in terms of the calculated band gap values of a set of semiconductors and insulators.

Efficient evaluation of three-centre two-electron integrals over London orbitals

The nonperturbative calculation of molecular properties in magnetic fields requires the evaluation of integrals over complex-valued Gaussian-type London atomic orbitals (LAOs). With these orbitals, the calculation of four-centre electron-repulsion integrals (ERIs) is particularly demanding, because their permutational symmetry is lowered, and because complex algebra is required. We have implemented the resolution-of-the-identity (RI) approximation for LAOs in the TURBOMOLE program package. With respect to LAOs, employing the RI approximation is particularly beneficial, because the auxiliary basis set may always be chosen to be real-valued. As a consequence, the two-centre integrals in the RI approximation remain real-valued, and the three-centre integrals possess the same permutational symmetry as their real-valued counterparts. Compared to a direct calculation of four-centre ERIs over LAOs, using the RI approximation thus not only reduces the scaling of the integral evaluation, but also increases the efficiency by an additional factor of at least two. By using other well-established methods such as Cauchy–Schwarz screening, the difference-density approach, and Pulay's direct inversion in the iterative subspace (DIIS), the efficiency of nonperturbative calculations in magnetic fields can be increased even further.

Accelerating Time-Dependent Density Functional Theory and GW Calculations for Molecules and Nanoclusters with Symmetry Adapted Interpolative Separable Density Fitting

Computing integrals over orbital pairs is one of the most costly steps in many popular first-principles methods used by the quantum chemistry and condensed matter physics community. Here, we employ a recently proposed interpolative separable density fitting method (ISDF) to significantly reduce the cost of steps involving orbital pairs in linear response time-dependent density functional theory and GW calculations. In our implementation, we exploit the symmetry property of a system to effectively reduce the number of interpolation points and thus the computational cost. The performance of ISDF is illustrated with calculations on the GW100 set and silicon nanoclusters. We demonstrate the cost for constructing auxiliary basis and interpolation coefficients are negligible compared to the total cost. Compared to the conventional brute-force approach, the computation cost for evaluating all kernel matrix elements is reduced by nearly 3 orders of magnitude. The total cost for GW calculations can be reduced by four to eight times, depending on the system size.

Predicting Drug Risk Level from Adverse Drug Reactions Using SMOTE and Machine Learning Approaches

Adverse drug reactions (ADRs) are the major source of morbidity and mortality. The prediction of drug risk level based on ADRs is few. Our study aims at predicting the drug risk level from ADRs using machine learning approaches. A total of 985,960 ADR reports from 2011 to 2018 were attained from the Chinese spontaneous reporting database (CSRD) in Jiangsu Province. Among them, there were 887 Prescription (Rx) Drugs (84.72%), 113 Over-the-Counter-A (OTC-A) Drugs (10.79%) and 47 OTC-B Drugs (4.49%). An over-sampling method, Synthetic Minority Oversampling Technique (SMOTE), was applied to the imbalanced classification. Firstly, we proposed a multi-classification framework based on SMOTE and classifiers. Secondly, drugs in CSRD were taken as the samples, ADR signal values calculated by proportional reporting ratio (PRR) or information component (IC) were taken as the features. Then, we applied four classifiers: Random Forest (RF), Gradient Boost (GB), Logistic Regression (LR), AdaBoost (ADA) to the tagged data. After evaluating the classification results by specific metrics, we finally obtained the optimal combination of our framework, PRR-SMOTE-RF with an accuracy rate of 0.95. We anticipate that this study can be a strong auxiliary judgment basis for experts on the status change of Rx Drugs to OTC Drugs.

SOME ISSUES GRAPHIC MOVEMENT. USE MIRRORING IN RESEARCH M. PYSANKA

The matter of the graphic image of the motion, taking into account the laws of the picture plane, as well as the study of patterns of motion as one of the leading composition factor, especially in terms of creating and perceiving the artistic image, is extremely important for the instructional program of the academic drawing. The formula of the work involves the presence of students with a very specific knowledge and skills related to the properties of the picture plane, the nature and character of the motion within the composition, as well as the patterns of its visual perception.The purpose of the article is an attempt to determine the main issues of graphic image of motion and the peculiarities of its visual perception in the structure of the picture plane, as well as to demonstrate the expediency of use a mirroring in work on the composition, mainly on the basis of research of the famous native artist-educator Mykola Pysanko. The author collected his searches in the book “Motion, Space, and Time in the Figurative arts”. We can make a conclusion that the study of the influence of the basic laws of the picture plane, in particular the areas of its right and left sides, on the mode and contextual aspect of the motion in the middle of the picture field is a fundamental issue for the instruction of academic drawing. This predetermines the need for a broader coverage of theoretical issues of the graphic image of motion and its research using the mirroring.This topic is relevant in terms of using the article's material as an auxiliary methodological basis for the instruction of vocational subjects to students of creative specialties. The action of the laws of the right and left sides of the picture plane reveals the internal pattern of the artist's concept.Knowledge and understanding of the nature of the sides of the picture field, as well as its connection with the mode and content aspect of the motion within the composition is important for students of any creative specialty.

Systematically Improvable Tensor Hypercontraction: Interpolative Separable Density-Fitting for Molecules Applied to Exact Exchange, Second- and Third-Order Møller-Plesset Perturbation Theory.

We present a systematically improvable tensor hypercontraction (THC) factorization based on interpolative separable density fitting (ISDF). We illustrate algorithmic details to achieve this within the framework of Becke's atom-centered quadrature grid. A single ISDF parameter cISDF controls the trade-off between accuracy and cost. In particular, cISDF sets the number of interpolation points used in THC, NIP = cISDF × NX with NX being the number of auxiliary basis functions. In conjunction with the resolution-of-the-identity (RI) technique, we develop and investigate the THC-RI algorithms for cubic-scaling exact exchange for Hartree-Fock and range-separated hybrids (e.g., ωB97X-V) and quartic-scaling second- and third-order Møller-Plesset theory (MP2 and MP3). These algorithms were evaluated over the W4-11 thermochemistry (atomization energy) set and A24 noncovalent interaction benchmark set with standard Dunning basis sets (cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ). We demonstrate the convergence of THC-RI algorithms to numerically exact RI results using ISDF points. Based on these, we make recommendations on cISDF for each basis set and method. We also demonstrate the utility of THC-RI exact exchange and MP2 for larger systems such as water clusters and C20. We stress that more challenges await in obtaining accurate and numerically stable THC factorization for wave function amplitudes as well as for the space spanned by virtual orbitals in large basis sets and implementing sparsity-aware THC-RI algorithms.