Set Convergence Research Articles

Explicitly Correlated Double-Hybrid DFT: A Comprehensive Analysis of the Basis Set Convergence on the GMTKN55 Database.

Double-hybrid density functional theory (DHDFT) offers a pathway to accuracy approaching composite wavefunction approaches such as G4 theory. However, the Görling–Levy second-order perturbation theory (GLPT2) term causes them to partially inherit the slow ∝L–3 (with L the maximum angular momentum) basis set convergence of correlated wavefunction methods. This could potentially be remedied by introducing F12 explicit correlation: we investigate the basis set convergence of both DHDFT and DHDFT-F12 (where GLPT2 is replaced by GLPT2-F12) for the large and chemically diverse general main-group thermochemistry, kinetics, and noncovalent interactions (GMTKN55) benchmark suite. The B2GP-PLYP-D3(BJ) and revDSD-PBEP86-D4 DHDFs are investigated as test cases, together with orbital basis sets as large as aug-cc-pV5Z and F12 basis sets as large as cc-pVQZ-F12. We show that F12 greatly accelerates basis set convergence of DHDFs, to the point that even the modest cc-pVDZ-F12 basis set is closer to the basis set limit than cc-pV(Q+d)Z or def2-QZVPPD in orbital-based approaches, and in fact comparable in quality to cc-pV(5+d)Z. Somewhat surprisingly, aug-cc-pVDZ-F12 is not required even for the anionic subsets. In conclusion, DHDF-F12/VDZ-F12 eliminates concerns about basis set convergence in both the development and applications of double-hybrid functionals. Mass storage and I/O bottlenecks for larger systems can be circumvented by localized pair natural orbital approximations, which also exhibit much gentler system size scaling.

Basis Set Selection for Molecular Core-Level GW Calculations.

The GW approximation has been recently gaining popularity among the methods for simulating molecular core-level X-ray photoemission spectra. Traditionally, Gaussian-type orbital GW core-level binding energies have been computed using either the cc-pVnZ or def2-nZVP basis set families, extrapolating the obtained results to the complete basis set limit, followed by an element-specific relativistic correction. Despite achieving rather good accuracy, it has been previously stated that these binding energies are chronically underestimated. In the present work, we show that those previous studies obtained results that were not well-converged with respect to the basis set size and that, once basis set convergence is achieved, there seems to be no such underestimation. Standard techniques known to offer a good cost-accuracy ratio in other theories demonstrate that the cc-pVnZ and def2-nZVP families exhibit contraction errors and might lead to unreliable complete basis set extrapolations for absolute binding energies, often deviating about 200-500 meV from the putative basis set limit found in this work. On the other hand, uncontracted versions of these basis sets offer vastly improved convergence. Even faster convergence can be obtained using core-rich property-optimized basis set families like pcSseg-n, pcJ-n, and ccX-nZ. Finally, we also show that the improvement observed for core properties using these specialized basis sets does not degrade their description of valence excitations: vertical ionization potentials and electron affinities computed with these basis sets converge as fast as the ones obtained with the aug-cc-pVnZ family, thus offering a balanced description of both core and valence regions.

MP2-F12 Basis Set Convergence near the Complete Basis Set Limit: Are h Functions Sufficient?

We have investigated the title question for the W4-08 thermochemical benchmark using l-saturated truncations of a large reference (REF) basis set, as well as for standard F12-optimized basis sets. With the REF basis set, the root-mean-square (RMS) contribution of i functions to the MP2-F12 total atomization energies (TAEs) is about 0.01 kcal/mol, the largest individual contributions being 0.04 kcal/mol for P2 and P4. However, even for these cases, basis set extrapolation from {g,h} basis sets adequately addresses the problem. Using basis sets insufficiently saturated in the spdfgh angular momenta may lead to exaggerated i function contributions. For extrapolation from spdfg and spdfgh basis sets, basis set convergence appears to be quite close to the theoretical asymptotic ∝ L–7 behavior. We hence conclude that h functions are sufficient even for highly demanding F12 applications. With one-parameter extrapolation, spdf and spdfg basis sets are adequate, aug-cc-pV{T,Q}Z-F12 yielding a RMSD = 0.03 kcal/mol. A limited exploration of CCSD(F12*) and CCSD-F12b suggests our conclusions are applicable to higher-level F12 methods as well.

Lagrangian Discretization of Variational Mean Field Games

In this article, we introduce a method to approximate solutions of some variational mean field game problems with congestion, by finite sets of player trajectories. These trajectories are obtained by solving a minimization problem, similar to the initial variational problem. In this discretized problem, congestion is penalized by a Moreau envelop with the 2-Wasserstein distance. Study of this envelop as well as efficient computation of its values and variations is done using semi-discrete optimal transport. We show convergence of the discrete sets of trajectories toward a solution of the mean field game, as well as conditions on the discretization in order to get this convergence. We also show strong convergence (in particular almost everywhere) in some cases for the underlying densities resulting from the Moreau projection of solution to the discrete problem.

Multi objective ant Colony Optimisation to obtain efficient metro speed profiles

Obtaining efficient speed profiles for metro trains is a multi-objective optimisation problem where energy consumption and travel time must be balanced. Automatic Train Operation (ATO) systems may handle a great number of possible speed profiles; hence optimisation algorithms are required to find efficient ones in a timely manner. This paper aims to assess the performance of a particular meta-heuristic optimisation algorithm, a variation of the traditional Ant Colony (ACO) modified to deal with multi-objective problems with continuous variables: MOACOr. This algorithm is used to obtain efficient speed profiles in up to 32 interstation sections in the metro network of Valencia (Spain), and the convergence and diversity of these solution sets is evaluated through metrics such as Inverse Generational Distance (GD) and Normalised Hypervolume (NH). The results are then compared to those obtained with a conventional genetic algorithm (NSGA-II), including a statistical analysis to identify significant differences. It has been found that MOACOr shows a better performance than NSGA-II in terms of convergence, regularity and diversity of the solution. These results indicate that MOACOr is a good alternative to the widely used genetic algorithm and could be a better tool for rail operation managers trying to improve energy efficiency.

Stability Analysis for Semi-Infinite Vector Optimization Problems under Functional Perturbations

This paper aims to study the stability of a class of semi-infinite vector optimization problems (SVOP) under functional perturbations. By using an important hypothesis a necessary and sufficient condition of Hausdorff continuity for weak efficient solution mappings and certain sufficient conditions for Painlevé-Kuratowski convergence of weak efficient solution sets for SVOP are established under the perturbations of both constraint sets and objective functions.

ANALYSIS OF INFLATION CONVERGENCE IN THE ECONOMIC COMMUNITY OF WEST AFRICAN STATES (ECOWAS) ZONE (1987-2021)

This study examines the convergence of inflation (country annual average inflation rate) through it determinants vis-?-vis the set convergence criteria in the Economic Community of West African States during the period 1987-2020. The study adopted ex-post factor design in order to answer the research questions. The study undertook unit root test employing Levin, Lin, and Chu (LLC) panel unit root method to check the stationarity of the variables, Auto Regressive Distributed Lag (ARDL) model and Error Correction Model (ECM) to determine whether or not they are co-integrated and a dynamic fixed effect panel data model covering 15 countries over the period under review. The study found that while West African Economic Monetary Union (WAEMU) countries inflation rates fall below the threshold, West African Monetary Zone (WAMZ) inflation rates are beyond the ECOWAS inflation convergence rate of less than or equal to 5 percent, that cross?country differences in inflation rates are attributed to the variation of the output gap, the increase in liquidity, the variation of exchange and interest rates, and the persistent increase in the price level. In the view of targeting the five percent (5 %) percent convergence criterion, the study recommended that Changes of traditional country monetary approaches to a supra-national monetary authority that compels compliance to a regional monetary and inflation targets, stimulation of economic growth, lowering of interest rates as well as adoption of a strong and efficient exchange rate mechanism becomes imperative for the ECOWAS member states

On Computing the Nonlinearity Interval in Parametric Semidefinite Optimization

This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the set of transition points is finite and the continuity of the optimal set mapping, on the basis of Painlevé–Kuratowski set convergence, might fail on a nonlinearity interval. Under a local nonsingularity condition, we then develop a methodology, stemming from numerical algebraic geometry, to efficiently compute nonlinearity intervals and transition points of the optimal partition. Finally, we support the theoretical results by applying our procedure to some numerical examples.

Revisiting the Performance of Time-Dependent Density Functional Theory for Electronic Excitations: Assessment of 43 Popular and Recently Developed Functionals from Rungs One to Four.

In this paper, the performance of more than 40 popular or recently developed density functionals is assessed for the calculation of 463 vertical excitation energies against the large and accurate QuestDB benchmark set. For this purpose, the Tamm-Dancoff approximation offers a good balance between computational efficiency and accuracy. The functionals ωB97X-D and BMK are found to offer the best performance overall with a root-mean square error (RMSE) of around 0.27 eV, better than the computationally more demanding CIS(D) wave function method with a RMSE of 0.36 eV. The results also suggest that Jacob's ladder still holds for time-dependent density functional theory excitation energies, though hybrid meta generalized-gradient approximations (meta-GGAs) are not generally better than hybrid GGAs. Effects of basis set convergence, gauge invariance correction to meta-GGAs, and nonlocal correlation (VV10) are also studied, and practical basis set recommendations are provided.

Efficient maximal biclique enumeration for large sparse bipartite graphs

Maximal bicliques are effective to reveal meaningful information hidden in bipartite graphs. Maximal biclique enumeration (MBE) is challenging since the number of the maximal bicliques grows exponentially w.r.t. the number of vertices in a bipartite graph in the worst case. However, a large bipartite graph is usually very sparse, which is against the worst case and may lead to fast MBE algorithms. The uncharted opportunity is taking advantage of the sparsity to substantially improve the MBE efficiency for large sparse bipartite graphs. We observe that for a large sparse bipartite graph, a vertex u may converge to a few vertices in the same vertex set as u via its neighbours, which reveals that the enumeration scope for a vertex could be very small. Based on this observation, we propose novel concepts: unilateral coreness for individual vertices, unilateral order for each vertex set and unilateral convergence (ζ) for a large sparse bipartite graph, ζ could be a few thousand for a large sparse bipartite graph with hundreds of million edges. Using the unilateral order, every vertex with τ unilateral coreness only needs to check at most 2 τ combinations so that all maximal bicliques can be enumerated and τ is bounded by ζ, which leads to a novel MBE algorithm running in O * (2 ζ ). We then propose a batch-pivots technique to eliminate all enumerations resulting in non-maximal bicliques, which guarantees that every maximal biclique is reported in O (ζ e )-delay, where e is the number of edges. We devise novel data structures that allow storing subgraphs at omissible space for further speeding up MBE. Extensive experiments are conducted on synthetic and real large datasets to justify that our proposed algorithm is faster and more scalable than the existing algorithms.

Basis Set Dependence of Optical Rotation Calculations with Different Choices of Gauge.

In this work, the basis set dependence of optical rotation (OR) calculations is examined for various choices of gauge/level of theory. The OR is calculated for a set of 50 molecules using B3LYP and CAM-B3LYP and 17 molecules using coupled cluster with single and double excitations (CCSD). The calculations employ the correlation-consistent basis sets, aug-cc-pVζZ with ζ = D, T, Q. An inverse-power extrapolation formula is then utilized to obtain OR values at the complete basis set (CBS) limit. We investigate the basis set convergence for these methods and three choices of gauge: length gauge (with gauge-including atomic orbitals, LG(GIAOs), for DFT), the origin-invariant length gauge [LG(OI)], and the modified velocity gauge (MVG). The results show that all methods converge smoothly to the CBS limit and that the LG(OI) approach has a slightly faster convergence rate than the other choices of gauge. While the DFT methods reach gauge invariance at the CBS limit, CCSD does not. The significant difference between the MVG and LG(OI) results at the CBS limit, 26%, indicates that CCSD is not quite at convergence in the description of electron correlation for this property. On the other hand, gauge invariance at the CBS limit for DFT does not lead to the same OR values for the two density functionals, which is also due to electron correlation incompleteness. A limited comparison to gas-phase experimental OR values for the DFT methods shows that CAM-B3LYP seems more accurate than B3LYP. Overall, this study shows that the LG(OI) approach with the aug-cc-pVTZ basis set for DFT, and with the CBS(DT) extrapolation for CCSD, provides a good cost/accuracy balance.

Control of Jump Markov Uncertain Linear Systems With General Probability Distributions

This paper introduces a method to control a class of jump Markov linear systems with uncertain initialization of the continuous state and affected by disturbances. Both types of uncertainties are modeled as stochastic processes with arbitrarily chosen probability distributions, for which however, the expected values and (co-)variances are known. The paper elaborates on the control task of steering the uncertain system into a target set by use of continuous controls, while chance constraints have to be satisfied for all possible state sequences of the Markov chain. The proposed approach uses a stochastic model predictive control approach on moving finite-time horizons with tailored constraints to achieve the control goal with prescribed confidence. Key steps of the procedure are (i) to over-approximate probabilistic reachable sets by use of the Chebyshev inequality, and (ii) to embed a tightened version of the original constraints into the optimization problem, in order to obtain a control strategy satisfying the specifications. Convergence of the probabilistic reachable sets is attained by suitable bounding of the state covariance matrices for arbitrary Markov chain sequences. The paper presents the main steps of the solution approach, discusses its properties, and illustrates the principle for a numeric example.

All-Electron BSE@GW Method for K-Edge Core Electron Excitation Energies.

We present an accurate computational approach to calculate absolute K-edge core electron excitation energies as measured by X-ray absorption spectroscopy. Our approach employs an all-electron Bethe-Salpeter equation (BSE) formalism based on GW quasiparticle energies (BSE@GW) using numeric atom-centered orbitals (NAOs). The BSE@GW method has become an increasingly popular method for the computation of neutral valence excitation energies of molecules. However, it was so far not applied to molecular K-edge excitation energies. We discuss the influence of different numerical approximations on the BSE@GW calculation and employ in our final setup (i) exact numeric algorithms for the frequency integration of the GW self-energy, (ii) G0W0 and BSE starting points with ∼50% of exact exchange, (iii) the Tamm-Dancoff approximation and (iv) relativistic corrections. We study the basis set dependence and convergence with common Gaussian-type orbital and NAO basis sets. We identify the importance of additional spatially confined basis functions as well as of diffuse augmenting basis functions. The accuracy of our BSE@GW method is assessed for a benchmark set of small organic molecules, previously used for benchmarking the equation-of-motion coupled cluster method [Peng et al., J. Chem. Theory Comput., 2015, 11, 4146], as well as the medium-sized dibenzothiophene (DBT) molecule. Our BSE@GW results for absolute excitation energies are in excellent agreement with the experiment, with a mean average error of only 0.63 eV for the benchmark set and with errors <1 eV for the DBT molecule.

Δ‐SCF calculations of core electron binding energies in first‐row transition metal atoms

AbstractCore electron binding energies (CEBE) of nickel and copper atoms have been calculated using single‐configuration energy differences, the so‐called Δ‐self‐consistent field (Δ‐SCF) method. Basis set convergence has been examined for calculated L‐shell and M‐shell core electron binding energies, and a wide array of density functionals have been evaluated. Scalar relativistic corrections have been estimated using the popular Douglas‐Kroll‐Hess (DKH) approximation. While basis set convergence and functional dependence mirror the behavior reported in the literature for main group elements, the simplest Δ‐SCF calculations with pure Hartree‐Fock (HF) exchange and no correlation surprisingly outperform all density functionals in reproducing free‐atom CEBE values for Ni and Cu.

Convergence of the solution sets for set optimization problems

Convergence of the solution sets for set optimization problems

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

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

Novel PIO Algorithm with Multiple Selection Strategies for Many-Objective Optimization Problems

With the increase of problem dimensions, most solutions of existing many-objective optimization algorithms are non-dominant. Therefore, the selection of individuals and the retention of elite individuals are important. Existing algorithms cannot provide sufficient solution precision and guarantee the diversity and convergence of solution sets when solving practical many-objective industrial problems. Thus, this work proposes an improved many-objective pigeon-inspired optimization (ImMAPIO) algorithm with multiple selection strategies to solve many-objective optimization problems. Multiple selection strategies integrating hypervolume, knee point, and vector angles are utilized to increase selection pressure to the true Pareto Front. Thus, the accuracy, convergence, and diversity of solutions are improved. ImMAPIO is applied to the DTLZ and WFG test functions with four to fifteen objectives and compared against NSGA-III, GrEA, MOEA/D, RVEA, and many-objective Pigeon-inspired optimization algorithm. Experimental results indicate the superiority of ImMAPIO on these test functions.

Transcorrelated coupled cluster methods.

Transcorrelated coupled cluster and distinguishable cluster methods are presented. The Hamiltonian is similarity transformed with a Jastrow factor in the first quantization, which results in up to three-body integrals. The coupled cluster with singles and doubles equations on this transformed Hamiltonian are formulated and implemented. It is demonstrated that the resulting methods have a superior basis set convergence and accuracy to the corresponding conventional and explicitly correlated methods. Additionally, approximations for three-body integrals are suggested and tested.

Approaching the basis set limit in Gaussian-orbital-based periodic calculations with transferability: Performance of pure density functionals for simple semiconductors.

Simulating solids with quantum chemistry methods and Gaussian-type orbitals (GTOs) has been gaining popularity. Nonetheless, there are few systematic studies that assess the basis set incompleteness error (BSIE) in these GTO-based simulations over a variety of solids. In this work, we report a GTO-based implementation for solids and apply it to address the basis set convergence issue. We employ a simple strategy to generate large uncontracted (unc) GTO basis sets that we call the unc-def2-GTH sets. These basis sets exhibit systematic improvement toward the basis set limit as well as good transferability based on application to a total of 43 simple semiconductors. Most notably, we found the BSIE of unc-def2-QZVP-GTH to be smaller than 0.7 mEh per atom in total energies and 20 meV in bandgaps for all systems considered here. Using unc-def2-QZVP-GTH, we report bandgap benchmarks of a combinatorially designed meta-generalized gradient approximation (mGGA) functional, B97M-rV, and show that B97M-rV performs similarly (a root-mean-square-deviation of 1.18eV) to other modern mGGA functionals, M06-L (1.26eV), MN15-L (1.29eV), and Strongly Constrained and Appropriately Normed (SCAN) (1.20eV). This represents a clear improvement over older pure functionals such as local density approximation (1.71eV) and Perdew-Burke-Ernzerhof (PBE) (1.49eV), although all these mGGAs are still far from being quantitatively accurate. We also provide several cautionary notes on the use of our uncontracted bases and on future research on GTO basis set development for solids.

Property-optimized Gaussian basis sets for lanthanides

Property-optimized Gaussian basis sets of split-valence, triple-zeta valence, and quadruple-zeta valence quality are developed for the lanthanides Ce-Lu for use with small-core relativistic effective core potentials. They are constructed in a systematic fashion by augmenting def2 orbital basis sets with diffuse basis functions and minimizing negative static isotropic polarizabilities of lanthanide atoms with respect to basis set exponents within the unrestricted Hartree-Fock method. The basis set quality is assessed using a test set of 70 molecules containing the lanthanides in their common oxidation states and f electron occupations. 5d orbital occupation turns out to be the determining factor for the basis set convergence of polarizabilities in lanthanide atoms and the molecular test set. Therefore, two series of property-optimized basis sets are defined. The augmented def2-SVPD, def2-TZVPPD, and def2-QZVPPD basis sets balance the accuracy of polarizabilities across lanthanide oxidation states. The relative errors in atomic and molecular polarizability calculations are ≤8% for augmented split-valence basis sets, ≤ 2.5% for augmented triple-zeta valence basis sets, and ≤1% for augmented quadruple-zeta valence basis sets. In addition, extended def2-TZVPPDD and def2-QZVPPDD are provided for accurate calculations of lanthanide atoms and neutral clusters. The property-optimized basis sets developed in this work are shown to accurately reproduce electronic absorption spectra of a series of LnCp3 '- complexes (Cp' = C5H4SiMe3, Ln = Ce-Nd, Sm) with time-dependent density functional theory.