Sum rules for variational wavefunctions.
Sum rule properties of finite set of wave functions obtained from application of variational principles to basis functions linear combinations
Dalgarno, A.↗
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Extending GPU-accelerated Gaussian integrals in the TeraChem software package to f type orbitals: Implementation and applications
Here, the increasing availability of graphics processing units (GPUs) for scientific computing has prompted interest in accelerating quantum chemical calculations through their use. However, the complexity of integral kernels for high angular momentum basis functions often limits the utility of GPU implementations with large basis sets or for metal containing systems. In this work, we report the implementation of f function support in the GPU-accelerated TeraChem software package through the development of efficient kernels for the evaluation of Hamiltonian integrals. The high efficiency of the resulting code is demonstrated through density functional theory (DFT) calculations on increasingly large organic molecules and transition metal complexes, as well as coupled cluster singles and doubles calculations on water clusters. Preliminary investigations into Ni(I) catalysis with DFT and the photochemistry of MnH(CH 3 ) with complete active space self-consistent field are also carried out. Overall, our GPU-accelerated software appears to be well-suited for fast simulation of large transition metal containing systems, as well as organic molecules.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Non-oscillatory spectral Fourier methods for shock wave calculations
A non-oscillatory spectral Fourier method is presented for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order.
Cai, Wei↗
Essentially nonoscillatory spectral Fourier methods for shock wave calculations
An essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations is presented. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution.
Cai, Wei↗
Time-dependent treatment of scattering. II - Novel integral equation approach to quantum wave packets
The novel wave-packet propagation scheme presented is based on the time-dependent form of the Lippman-Schwinger integral equation and does not require extensive matrix inversions, thereby facilitating application to systems in which some degrees of freedom express the potential in a basis expansion. The matrix to be inverted is a function of the kinetic energy operator, and is accordingly diagonal in a Bessel function basis set. Transition amplitudes for various orbital angular momentum quantum numbers are obtainable via either Fourier transform of the amplitude density from the time to the energy domain, or the direct analysis of the scattered wave packet.
Sharafeddin, Omar A.↗
A Meshless Local Petrov-Galerkin Method for Euler-Bernoulli Beam Problems
An accurate and yet simple Meshless Local Petrov-Galerkin (MLPG) formulation for analyzing beam problems is presented. In the formulation, simple weight functions are chosen as test functions. The use of these functions shows that the weak form can be integrated with conventional Gaussian integration. The MLPG method was evaluated by applying the formulation to a variety of patch test and thin beam problems. The formulation successfully reproduced exact solutions to machine accuracy when test functions with C2 continuity and an appropriate order of basis functions are used.
Raju, I. S.↗
Spatial Statistical Data Fusion (SSDF)
As remote sensing for scientific purposes has transitioned from an experimental technology to an operational one, the selection of instruments has become more coordinated, so that the scientific community can exploit complementary measurements. However, tech nological and scientific heterogeneity across devices means that the statistical characteristics of the data they collect are different. The challenge addressed here is how to combine heterogeneous remote sensing data sets in a way that yields optimal statistical estimates of the underlying geophysical field, and provides rigorous uncertainty measures for those estimates. Different remote sensing data sets may have different spatial resolutions, different measurement error biases and variances, and other disparate characteristics. A state-of-the-art spatial statistical model was used to relate the true, but not directly observed, geophysical field to noisy, spatial aggregates observed by remote sensing instruments. The spatial covariances of the true field and the covariances of the true field with the observations were modeled. The observations are spatial averages of the true field values, over pixels, with different measurement noise superimposed. A kriging framework is used to infer optimal (minimum mean squared error and unbiased) estimates of the true field at point locations from pixel-level, noisy observations. A key feature of the spatial statistical model is the spatial mixed effects model that underlies it. The approach models the spatial covariance function of the underlying field using linear combinations of basis functions of fixed size. Approaches based on kriging require the inversion of very large spatial covariance matrices, and this is usually done by making simplifying assumptions about spatial covariance structure that simply do not hold for geophysical variables. In contrast, this method does not require these assumptions, and is also computationally much faster. This method is fundamentally different than other approaches to data fusion for remote sensing data because it is inferential rather than merely descriptive. All approaches combine data in a way that minimizes some specified loss function. Most of these are more or less ad hoc criteria based on what looks good to the eye, or some criteria that relate only to the data at hand.
Braverman, Amy J.↗
Practical method for highly accurate large-scale surface calculations
An accurate and efficient film linearized muffin-tin orbital (FLMTO) technique for surface electronic-structure calculations is presented which uses only 60-70 basis functions, as opposed to the 300 functions used in the linear augmented plane-wave method. Calculations for three different (3d and 4d) transition-metal films resulted in high quality results for five-layer slabs of Cu(001), Fe(001), and Ru(001), in addition to good results for the work functions and projected density of states. By retaining the LMTO small basis size, computer time and memory are reduced, making practical the study of systems with a larger number of atoms in the two-dimensional unit cell.
Fernando, G. W.↗
Parameter dimension of turbulence-induced phase errors and its effects on estimation in phase diversity
The method of phase diversity has been used in the context of incoherent imaging to estimate jointly an object that is being imaged and phase aberrations induced by atmospheric turbulence. The method requires a parametric model for the phase-aberration function. Typically, the parameters are coefficients to a finite set of basis functions. Care must be taken in selecting a parameterization that properly balances accuracy in the representation of the phase-aberration function with stability in the estimates. It is well known that over parameterization can result in unstable estimates. Thus a certain amount of model mismatch is often desirable. We derive expressions that quantify the bias and variance in object and aberration estimates as a function of parameter dimension.
Thelen, Brian J.↗
A quasi-meshfree method for constructing boundary-aware reproducing bases on geometrically complex domains using manifold geodesics
In many applications, physical domains are geometrically complex making it challenging to perform coarse-scale approximation. A defeaturing process is often used to simplify the domain in preparation for approximation and analysis at the coarse scale. In this paper, a methodology is presented for constructing a coarse-scale reproducing basis on geometrically complex domains given an initial fine-scale mesh of the fully featured domain. The initial fine-scale mesh can be of poor quality and extremely refined. The construction of the basis functions begins with a coarse-scale covering of the domain and generation of weighting functions with local support. Manifold geodesics are used to define distances within the local support for general applicability to non-convex domains. Conventional moving least squares is used to construct the coarse-scale reproducing basis. Applications in quasi-interpolation and linear elasticity are presented.
97 MATHEMATICS AND COMPUTING↗
Toward machine-learning-assisted PW-class high-repetition-rate experiments with solid targets
We present progress in utilizing a machine learning (ML) assisted optimization framework to study the trends in a parameter space defined by spectrally shaped, high-intensity, petawatt-class (8 J, 45 fs) laser pulses interacting with solid targets and give the first simulation-based overview of predicted trends. A neural network (NN) incorporating uncertainty quantification is trained to predict the number of hot electrons generated by the laser–target interaction as a function of pulse shaping parameters. The predictions of this NN serve as the basis function for a Bayesian optimization framework to navigate this space. For post-experimental evaluation, we compare two separate neural network (NN) models. One is based solely on data from experiments, and the other is trained only on ensemble particle-in-cell simulations. Reviewing the predicted and observed trends across the experiment-capable laser parameter search space, we find that both ML models predict a maximal increase in hot electron generation at a level of approximately 12%–18%; however, no statistically significant enhancement was observed in experiments. On direct comparison of the NN models, the average discrepancy is 8.5%, with a maximum of 30%. Since shot-to-shot fluctuations in experiments affect the observations, we evaluate the behavior of our optimization framework by performing virtual experiments that vary the number of repeated observations and the noise levels. Here, we discuss the implications of such a framework for future autonomous exploration platforms in high-repetition-rate experiments.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Theoretical study of the nitrogen atom hyperfine coupling constant
The nitrogen-atom isotropic hyperfine coupling constant A(iso) is studied as a function of improvements in both the one-particle and n-particle basis sets. The study underscores the importance of diffuse basis functions. For example, the (9s 5p) primitive set of Huzinaga (1965) augmented with an even-tempered diffuse s function yields values for A(iso) that are virtually identical to an energy-optimized (23s 12p) even-tempered set. The A(iso) constant is found to converge relatively quickly with increasing l quantum numbers: d, f, and g functions are estimated to contribute 2.5 + or - 0.2, 0.4 + or - 0.1, and 0.05 + or - 0.05 MHz, respectively. Full CI calibration calculations indicate that very high levels of correlation treatment are required for quantitative results. In addition, a strong coupling is observed between the one-particle and n-particle requirements. The best result, 10.4 MHz, is in excellent agreement with the accurate experimental value of 10.4509 MHz.
Bauschlicher, Charles W., Jr.↗
Linear Scaling Density Functional Calculations with Gaussian Orbitals
Recent advances in linear scaling algorithms that circumvent the computational bottlenecks of large-scale electronic structure simulations make it possible to carry out density functional calculations with Gaussian orbitals on molecules containing more than 1000 atoms and 15000 basis functions using current workstations and personal computers. This paper discusses the recent theoretical developments that have led to these advances and demonstrates in a series of benchmark calculations the present capabilities of state-of-the-art computational quantum chemistry programs for the prediction of molecular structure and properties.
Scuseria, Gustavo E.↗
An Algorithm for Atom-Centered Lossy Compression of the Atomic Orbital Basis in Density Functional Theory Calculations
Large atomic-orbital (AO) basis sets of at least triple and preferably quadruple-ζ (QZ) size are required to adequately converge Kohn–Sham density functional theory (DFT) calculations toward the complete basis set limit. However, incrementing the cardinal number by one nearly doubles the AO basis dimension, and the computational cost scales as the cube of the AO dimension, so this is very computationally demanding. Here, in this work, we develop and test a threshold-based natural atomic orbital (NAO) scheme in which ϵ-NAOs are obtained as eigenfunctions of atomic blocks of the density matrix in a one-center orthogonalized representation. This enables compression of the AO basis that is optimal for a given threshold, 10 –ϵ , by discarding NAOs with occupation numbers below that threshold. Extensive pilot test calculations using the Hartree–Fock functional and taking the converged density matrix as input suggest that a threshold of 10 –5 can yield a compression factor (ratio of AO to compressed ϵ-NAO dimension) between 2.5 and 4.5 for the QZ pc-3 basis. The errors in relative energies are typically less than 0.1 kcal/mol when the compressed basis is used instead of the uncompressed basis. Between 10 and 100 times smaller errors (i.e., usually less than 0.01 kcal/mol) can be obtained with a threshold 10 –7 , while the compression factor is typically between 2 and 2.5.
basis sets↗
Utility of a finite element solution algorithm for initial-value problems
The Galerkin criterion within a finite element Weighted Residuals formulation is employed to establish an implicit solution algorithm for an initial-value partial differential equation. Numerical solutions of a transient parabolic and a hyperbolic equation, obtained using linear, quadratic and two cubic finite element basis functions, are employed to quantize accuracy and confirm and refine theoretical convergence rate estimates. The linear basis algorithm for the hyperbolic equation displays excellent accuracy on a coarse computational grid and a high-order convergence rate with discretization refinement. Good accuracy and a strong convergence rate in surface flux are determined for a nonhomogeneous Neumann boundary constraint applied to a parabolic equation. The results amply demonstrate the impact of the nondiagonal finite element initial-value matrix structure on solution accuracy and/or convergence rate.
Baker, A. J.↗
Development of deterministic transport methods for low energy neutrons for shielding in space
Transport of low energy neutrons associated with the galactic cosmic ray cascade is analyzed in this dissertation. A benchmark quality analytical algorithm is demonstrated for use with BRYNTRN, a computer program written by the High Energy Physics Division of NASA Langley Research Center, which is used to design and analyze shielding against the radiation created by the cascade. BRYNTRN uses numerical methods to solve the integral transport equations for baryons with the straight-ahead approximation, and numerical and empirical methods to generate the interaction probabilities. The straight-ahead approximation is adequate for charged particles, but not for neutrons. As NASA Langley improves BRYNTRN to include low energy neutrons, a benchmark quality solution is needed for comparison. The neutron transport algorithm demonstrated in this dissertation uses the closed-form Green's function solution to the galactic cosmic ray cascade transport equations to generate a source of neutrons. A basis function expansion for finite heterogeneous and semi-infinite homogeneous slabs with multiple energy groups and isotropic scattering is used to generate neutron fluxes resulting from the cascade. This method, called the FN method, is used to solve the neutral particle linear Boltzmann transport equation. As a demonstration of the algorithm coded in the programs MGSLAB and MGSEMI, neutron and ion fluxes are shown for a beam of fluorine ions at 1000 MeV per nucleon incident on semi-infinite and finite aluminum slabs. Also, to demonstrate that the shielding effectiveness against the radiation from the galactic cosmic ray cascade is not directly proportional to shield thickness, a graph of transmitted total neutron scalar flux versus slab thickness is shown. A simple model based on the nuclear liquid drop assumption is used to generate cross sections for the galactic cosmic ray cascade. The ENDF/B V database is used to generate the total and scattering cross sections for neutrons in aluminum. As an external verification, the results from MGSLAB and MGSEMI were compared to ANISN/PC, a routinely used neutron transport code, showing excellent agreement. In an application to an aluminum shield, the FN method seems to generate reasonable results.
Ganapol, Barry↗
Accurate parameterization of the kinetic energy functional
The absence of a reliable formulation of kinetic energy density functional has hindered the development of orbital free density functional theory. Using the data-aided learning paradigm, we propose a simple prescription to accurately model the kinetic energy density of any system. Our method relies on a dictionary of functional forms for local and nonlocal contributions which have been proposed in the literature and the appropriate coefficients are calculated via a linear regression framework. To model the nonlocal contributions, we explore two new nonlocal functionals - a functional that captures fluctuations in electronic density and a functional that incorporates gradient information. Since, the analytical functional forms of the kernels present in these nonlocal terms are not known from theory, we propose a basis function expansion to model these seemingly difficult nonlocal quantities. This allows us to easily reconstruct kernels for any system using only a few structures. The proposed method is able to learn kinetic energy densities and total kinetic energies of molecular and periodic systems, such as H 2 , LiH, LiF and a one-dimensional chain of 8 hydrogens using data from Kohn-Sham density functional theory calculations for only a few structures. For the ease of reproduction, codes used to generate the models are provided in the supporting materials.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Analysis of Stub Loaded Microstrip Patch Antennas
A microstrip patch antenna fed by a coaxial probe and reactively loaded by a open circuited microstrip line has been used previously to produce circular polarization and also as a building block for a series fed microstrip patch array. Rectangular and circular patch antennas loaded with a microstrip stub were previously analyzed using the generalized Thevenin theorem. In the Thevenin theorem approach, the mutual coupling between the patch current and the surface current on the stub was not taken into account. Also, the Thevenin theorem approach neglects continuity of current at the patch-stub junction. The approach in this present paper includes the coupling between the patch and stub currents as well as continuity at the patch-stub junction. The input impedance for a stub loaded microstrip patch is calculated by the general planar dielectric dyadic Green's function approach in the spectral domain, as was initiated much earlier and has been extensively expanded upon and utilized successfully throughout the literature for microstrip antenna configurations. Using the spectral domain dyadic Green s function derived earlier with the electric field integral equation (EFIE), the problem is formulated by using entire domain basis functions to represent the surface current densities on the patch, the loading stub and the attachment mode at the junction. Galerkin's procedure is used to reduce the EFIE to a matrix equation, which is then solved to obtain the amplitudes of the surface currents. These surface currents are then used for calculating the input impedance of stub loaded rectangular and circular microstrip patches. Numerical results are compared with measured results and with previous results calculated by the Thevenin's theorem approach.
Deshpande, M. D.↗