Showing papers on "Basis (linear algebra) published in 2022"

Best‐Practice DFT Protocols for Basic Molecular Computational Chemistry

Abstract: Abstract Nowadays, many chemical investigations are supported by routine calculations of molecular structures, reaction energies, barrier heights, and spectroscopic properties. The lion's share of these quantum‐chemical calculations applies density functional theory (DFT) evaluated in atomic‐orbital basis sets. This work provides best‐practice guidance on the numerous methodological and technical aspects of DFT calculations in three parts: Firstly, we set the stage and introduce a step‐by‐step decision tree to choose a computational protocol that models the experiment as closely as possible. Secondly, we present a recommendation matrix to guide the choice of functional and basis set depending on the task at hand. A particular focus is on achieving an optimal balance between accuracy, robustness, and efficiency through multi‐level approaches. Finally, we discuss selected representative examples to illustrate the recommended protocols and the effect of methodological choices.

One decade of quantum optimal control in the chopped random basis

Abstract: The chopped random basis (CRAB) ansatz for quantum optimal control has been proven to be a versatile tool to enable quantum technology applications such as quantum computing, quantum simulation, quantum sensing, and quantum communication. Its capability to encompass experimental constraints-while maintaining an access to the usually trap-free control landscape-and to switch from open-loop to closed-loop optimization (including with remote access-or RedCRAB) is contributing to the development of quantum technology on many different physical platforms. In this review article we present the development, the theoretical basis and the toolbox for this optimization algorithm, as well as an overview of the broad range of different theoretical and experimental applications that exploit this powerful technique.

Systematic Evaluation of Counterpoise Correction in Density Functional Theory

Abstract: A widespread belief persists that the Boys-Bernardi function counterpoise (CP) procedure "overcorrects" supramolecular interaction energies for the effects of basis-set superposition error. To the extent that this is true for correlated wave function methods, it is usually an artifact of low-quality basis sets. The question has not been considered systematically in the context of density functional theory, however, where basis-set convergence is generally less problematic. We present a systematic assessment of the CP procedure for a representative set of functionals and basis sets, considering both benchmark data sets of small dimers and larger supramolecular complexes. The latter include layered composite polymers with ∼150 atoms and ligand-protein models with ∼300 atoms. Provided that CP correction is used, we find that intermolecular interaction energies of nearly complete-basis quality can be obtained using only double-ζ basis sets. This is less expensive as compared to triple-ζ basis sets without CP correction. CP-corrected interaction energies are less sensitive to the presence of diffuse basis functions as compared to uncorrected energies, which is important because diffuse functions are expensive and often numerically problematic for large systems. Our results upend the conventional wisdom that CP "overcorrects" for basis-set incompleteness. In small basis sets, CP correction is mandatory in order to demonstrate that the results do not rest on error cancellation.

Computation of numerical solutions to variable order fractional differential equations by using non-orthogonal basis

Abstract: In this work, we present some numerical results about variable order fractional differential equations (VOFDEs). For the said numerical analysis, we use Bernstein polynomials (BPs) with non-orthogonal basis. The method we use does not need discretization and neither collocation. Hence omitting the said two operations sufficient memory and time can be saved. We establish operational matrices for variable order integration and differentiation which convert the consider problem to some algebraic type matrix equations. The obtained matrix equations are then solved by Matlab 13 to get the required numerical solution for the considered problem. Pertinent examples are provided along with graphical illustration and error analysis to validate the results. Further some theoretical results for time complexity are also discussed.

Quantum algorithms for electronic structures: basis sets and boundary conditions.

Abstract: The advantages of quantum computers are believed to significantly change the research paradigm of chemical and materials sciences, where computational characterization and theoretical design play an increasingly important role. It is especially desirable to solve the electronic structure problem, a central problem in chemistry and materials science, efficiently and accurately with well-designed quantum algorithms. Various quantum electronic-structure algorithms have been proposed in the literature. In this article, we briefly review recent progress in this direction with a special emphasis on the basis sets and boundary conditions. Compared to classical electronic structure calculations, there are new considerations in choosing a basis set in quantum algorithms. For example, the effect of the basis set on the circuit complexity is very important in quantum algorithm design. Electronic structure calculations should be performed with an appropriate boundary condition. Simply using a wave function ansatz designed for molecular systems in a material system with a periodic boundary condition may lead to significant errors. Artificial boundary conditions can be used to partition a large system into smaller fragments to save quantum resources. The basis sets and boundary conditions are expected to play a crucial role in electronic structure calculations on future quantum computers, especially for realistic systems.

MRSCloud: A cloud‐based MRS tool for basis set simulation

Abstract: The purpose of this study is to present a cloud‐based spectral simulation tool “MRSCloud,” which allows MRS users to simulate a vendor‐specific and sequence‐specific basis set online in a convenient and time‐efficient manner. This tool can simulate basis sets for GE, Philips, and Siemens MR scanners, including conventional acquisitions and spectral editing schemes with PRESS and semi‐LASER localization at 3 T.

Constraint Preserving Mixers for the Quantum Approximate Optimization Algorithm

Abstract: The quantum approximate optimization algorithm/quantum alternating operator ansatz (QAOA) is a heuristic to find approximate solutions of combinatorial optimization problems. Most of the literature is limited to quadratic problems without constraints. However, many practically relevant optimization problems do have (hard) constraints that need to be fulfilled. In this article, we present a framework for constructing mixing operators that restrict the evolution to a subspace of the full Hilbert space given by these constraints. We generalize the “XY”-mixer designed to preserve the subspace of “one-hot” states to the general case of subspaces given by a number of computational basis states. We expose the underlying mathematical structure which reveals more of how mixers work and how one can minimize their cost in terms of the number of CX gates, particularly when Trotterization is taken into account. Our analysis also leads to valid Trotterizations for an “XY”-mixer with fewer CX gates than is known to date. In view of practical implementations, we also describe algorithms for efficient decomposition into basis gates. Several examples of more general cases are presented and analyzed.

Correlation-Consistent Gaussian Basis Sets for Solids Made Simple

Abstract: The rapidly growing interest in simulating condensed-phase materials using quantum chemistry methods calls for a library of high-quality Gaussian basis sets suitable for periodic calculations. Unfortunately, most standard Gaussian basis sets commonly used in molecular simulation show significant linear dependencies when used in close-packed solids, leading to severe numerical issues that hamper the convergence to the complete basis set (CBS) limit, especially in correlated calculations. In this work, we revisit Dunning's strategy for construction of correlation-consistent basis sets and examine the relationship between accuracy and numerical stability in periodic settings. We find that limiting the number of primitive functions avoids the appearance of problematic small exponents while still providing smooth convergence to the CBS limit. As an example, we generate double-, triple-, and quadruple-ζ correlation-consistent Gaussian basis sets for periodic calculations with Goedecker-Teter-Hutter (GTH) pseudopotentials. Our basis sets cover the main-group elements from the first three rows of the periodic table. Especially for atoms on the left side of the periodic table, our basis sets are less diffuse than those used in molecular calculations. We verify the fast and reliable convergence to the CBS limit in both Hartree-Fock and post-Hartree-Fock (MP2) calculations, using a diverse test set of 19 semiconductors and insulators.

Nonlinearity Characteristic of High Impedance Fault at Resonant Distribution Networks: Theoretical Basis to Identify the Faulty Feeder

Abstract: Feeder identification is indispensable for distribution networks to locate faults at a specific feeder, especially when measuring devices are insufficient to locate faults more precisely. For the high impedance fault (HIF), the feeder identification is much more challengeable and related approaches are still in the early stage. This paper thoroughly reveals the nonlinearity characteristics of different feeders when a HIF happens at a three-wire system that is with the resonant grounded neutral (RGN). Firstly, the diversity of nonlinearity existing in HIFs is explained from the perspective of energy. Then, the nonlinearity of zero-sequence current that differs between healthy and faulty feeders are deduced theoretically. Effects of the detuning index of Petersen coil, the damping ratio of system, and the length of feeder are all considered. Afterward, these theoretical conclusions are verified by the HIF cases experimented in a real 10kV system. Finally, after indicating the problems of a classic phase-relationship-based algorithm, we suggest an improved method based on the phase differences between the harmonic currents of different feeders. The effectiveness of the method has been verified.

Engineering and Analytical Method for Estimating the Parametric Reliability of Products by a Low Number of Tests

Abstract: The paper provides an overview of methods for determining reliability indicators and, on the basis of the analysis, proposes a new method for assessing the parametric reliability of products based on a small number of tests. The determination of the parameters and double logistic distribution based on the test results is considered, a statistical experiment was carried out, which was based on the method of statistical modeling of Monte Carlo. An example of evaluating parametric reliability by a new method is also given, on the basis of which an engineering technique is proposed. In the conclusion, remarks are made regarding the advantages of the novel method.

Discrete Lehmann representation of imaginary time Green's functions

Abstract: We present an efficient basis for imaginary time Green's functions based on a low-rank decomposition of the spectral Lehmann representation. The basis functions are simply a set of well-chosen exponentials, so the corresponding expansion may be thought of as a discrete form of the Lehmann representation using an effective spectral density which is a sum of $\ensuremath{\delta}$ functions. The basis is determined only by an upper bound on the product $\ensuremath{\beta}{\ensuremath{\omega}}_{max}$, with $\ensuremath{\beta}$ the inverse temperature and ${\ensuremath{\omega}}_{max}$ an energy cutoff, and a user-defined error tolerance $\ensuremath{\epsilon}$. The number $r$ of basis functions scales as $O(log(\ensuremath{\beta}{\ensuremath{\omega}}_{max})log(1/\ensuremath{\epsilon}))$. The discrete Lehmann representation of a particular imaginary time Green's function can be recovered by interpolation at a set of $r$ imaginary time nodes. Both the basis functions and the interpolation nodes can be obtained rapidly using standard numerical linear algebra routines. Due to the simple form of the basis, the discrete Lehmann representation of a Green's function can be explicitly transformed to the Matsubara frequency domain, or obtained directly by interpolation on a Matsubara frequency grid. We benchmark the efficiency of the representation on simple cases, and with a high-precision solution of the Sachdev-Ye-Kitaev equation at low temperature. We compare our approach with the related intermediate representation method, and introduce an improved algorithm to build the intermediate representation basis and a corresponding sampling grid.

Efficient ordering of the Hadamard basis for single pixel imaging.

Abstract: Single-pixel imaging is a technique that can reconstruct an image of a scene by projecting a series of spatial patterns on an object and capturing the reflected light by a single photodetector. Since the introduction of the compressed sensing method, it has been possible to use random spatial patterns and reduce its number below the Nyquist-Shannon limit to form a good quality image but with lower spatial resolution. On the other hand, Hadamard pattern based methods can reconstruct large images by increasing the acquisition measurement time. Here, we propose an efficient strategy to order the Hadamard basis patterns from higher to lower relevance, and then to reconstruct an image at very low sampling rates of at least 8%. Our proposal is based on the construction of generalized basis vectors in two dimensions and then ordering in zigzag fashion. Simulation and experimental results show that the sampling rate, image quality and computational complexity of our method are competitive to the state of the art methods.

A Mountaineering Strategy to Excited States: Revising Reference Values with EOM-CC4

Abstract: In the framework of the computational determination of highly accurate vertical excitation energies in small organic compounds, we explore the possibilities offered by the equation-of-motion formalism relying on the approximate fourth-order coupled-cluster (CC) method, CC4. We demonstrate, using an extended set of more than 200 reference values based on CC including up to quadruples excitations (CCSDTQ), that CC4 is an excellent approximation to CCSDTQ for excited states with a dominant contribution from single excitations with an average deviation as small as 0.003 eV. We next assess the accuracy of several additive basis set correction schemes, in which vertical excitation energies obtained with a compact basis set and a high-order CC method are corrected with lower-order CC calculations performed in a larger basis set. Such strategies are found to be overall very beneficial, though their accuracy depends significantly on the actual scheme. Finally, CC4 is employed to improve several theoretical best estimates of the QUEST database for molecules containing between four and six (nonhydrogen) atoms, for which previous estimates were computed at the CCSDT level.

Fast Overlapping Block Processing Algorithm for Feature Extraction

Abstract: In many video and image processing applications, the frames are partitioned into blocks, which are extracted and processed sequentially. In this paper, we propose a fast algorithm for calculation of features of overlapping image blocks. We assume the features are projections of the block on separable 2D basis functions (usually orthogonal polynomials) where we benefit from the symmetry with respect to spatial variables. The main idea is based on a construction of auxiliary matrices that virtually extends the original image and makes it possible to avoid a time-consuming computation in loops. These matrices can be pre-calculated, stored and used repeatedly since they are independent of the image itself. We validated experimentally that the speed up of the proposed method compared with traditional approaches approximately reaches up to 20 times depending on the block parameters.

Tensor-based basis function learning for three-dimensional sound speed fields.

Abstract: Basis function learning is the stepping stone towards effective three-dimensional (3D) sound speed field (SSF) inversion for various acoustic signal processing tasks, including ocean acoustic tomography, underwater target localization/tracking, and underwater communications. Classical basis functions include the empirical orthogonal functions (EOFs), Fourier basis functions, and their combinations. The unsupervised machine learning method, e.g., the K-singular value decomposition (K-SVD) algorithm, has recently tapped into the basis function design, showing better representation performance than the EOFs. However, existing methods do not consider basis function learning approaches that treat 3D SSF data as a third-order tensor, and, thus, cannot fully utilize the 3D interactions/correlations therein. To circumvent such a drawback, basis function learning is linked to tensor decomposition in this paper, which is the primary drive for recent multi-dimensional data mining. In particular, a tensor-based basis function learning framework is proposed, which can include the classical basis functions (using EOFs and/or Fourier basis functions) as its special cases. This provides a unified tensor perspective for understanding and representing 3D SSFs. Numerical results using the South China Sea 3D SSF data have demonstrated the excellent performance of the tensor-based basis functions.

BSSE‐corrected consistent Gaussian basis sets of triple‐zeta valence with polarization quality of the fifth period for solid‐state calculations

Abstract: Revised versions of our published pob‐TZVP basis sets [Laun, J.; Vilela Oliveira, D. and Bredow, T., J. Comput. Chem., 2018, 39 (19), 1285–1290] have been derived for periodic quantum‐chemical solid‐state calculations. They complete our pob‐TZVP‐rev2 series [Vilela Oliveira, D.; Laun, J.; Peintinger, M. F. and Bredow, T., J. Comput. Chem., 2019, 40 (27), 2364–2376 and Laun, J. and Bredow, J. Comput. Chem. 2021; 42 (15), 1064–1072] for the elements of the fifth period and are based on the fully relativistic effective core potentials (ECPs) of the Stuttgart/Cologne group and the def2‐TZVP valence basis of the Ahlrichs group. The pob‐TZVP‐rev2 basis sets are developed to minimize the basis set superposition error (BSSE) in crystalline systems. For the applied PW1PW hybrid functional, the overall performance, transferability, and SCF stability of the resulting pob‐TZVP‐rev2 basis sets are significantly improved compared to the original pob‐TZVP basis sets. After augmentation with single diffuse s‐ and p‐functions, reference plane‐wave band structures of metals can be accurately reproduced.

Compressing Many-Body Fermion Operators under Unitary Constraints

Abstract: The most efficient known quantum circuits for preparing unitary coupled cluster states and applying Trotter steps of the arbitrary basis electronic structure Hamiltonian involve interleaved sequences of Fermionic Gaussian circuits and Ising interaction-type circuits. These circuits arise from factorizing the two-body operators generating those unitaries as a sum of squared one-body operators that are simulated using product formulas. We introduce a numerical algorithm for performing this factorization that has an iteration complexity no worse than single particle basis transformations of the two-body operators and often results in many times fewer squared one-body operators in the sum of squares, compared to the analytical decompositions. As an application of this numerical procedure, we demonstrate that our protocol can be used to approximate generic unitary coupled cluster operators and prepare the necessary high-quality initial states for techniques (like ADAPT-VQE) that iteratively construct approximations to the ground state.

Basis Set Selection for Molecular Core-Level GW Calculations.

Abstract: 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.