Gaussian-type Functions Research Articles

Reconstruction algorithm for impenetrable rough surface profile under Neumann boundary condition

In this paper, an algorithm to reconstruct one-dimensional impenetrable rough surface from the knowledge of scattering field is presented. The rough surface is considered as locally perturbed and the scattering field data are collected above the roughness in a simple non-magnetic medium considering Neumann boundary condition. First, the surface integral equation constituted via the Neumann boundary condition is solved and scattering field data are observed synthetically. Then, the same surface integral equation together with the data equation are solved in an iterative fashion to reconstruct the surface variation. In the numerical implementation, the so-called ill-posed inverse problem is regularized with Tikhonov method and a least-squares solution is obtained by using Gaussian-type basis function. Finally, numerical examples are carried out to illustrate effectiveness of the method.

Theory of impact sensitivity revisited: mechanical-to-vibrational energy transfer phenomenon

A revision of the mechanism of mechanical-to-vibrational energy transfer in crystals of energetic materials undergone impact loading is proposed. The new approach takes into account previous inaccuracies of normalization of the number of couplings between phonon overtones and conformational vibrational fundamentals (ζ), which is critical for comparison of molecules that differ greatly in the number of atoms. Moreover, it introduces two very important damping factors, namely a and b. The a factor allows differentiation of the phonon overtones by their coupling strength; the lower the overtone interacts, the stronger the coupling is. Meanwhile, the b factor is aimed to distinguish the coupling strength itself. This factor is the denominator in the exponent of a Gaussian-type function and determines the rate at which the coupling strength decays with the rise of difference between the interacting phonon overtones and conformational vibrational fundamentals. After a careful regression analysis of ζ against impact sensitivities (h50) of 30 common nitroexplosives, we have determined the numerical values of these damping factors as a = 2.5 and b = 40 cm–1. The proposed approach is extrapolated on the vibrational spectra of 21 crystalline energetic materials with Z' ≥ 1 and a general applicability along with limitations of the method are discussed for the family of nitroexplosives and nitrogen-rich (cyclo-pentazolate and 5,5′-bitetrazole) energetic salts as an example.

Gaussian-type activation function for complex-valued CNN and its application in polar-SAR image classification

Activation functions play increasingly important roles in deep learning. The activation functions give the neural network the ability to learn complex patterns by introducing non-linearity. Complex-valued activation functions for complex-valued neural networks are extended from real-valued activation functions like sigmoid, tanh, rectified linear unit, and exponential linear unit. Recently, the most widely used complex-valued activation function is the real-imaginary-type activation function (RIAF), which applies separate real-valued activation functions on both the real and the imaginary parts of a neuron. This separate activation method ignores the internal relationship between the real and imaginary parts of the complex input and loses the integrity of activation. In this work, a complex-valued activation function called Gaussian-type activation function (GTAF) is proposed to activate the real and imaginary parts of a neuron jointly and enhance the learning ability and learning speed. GTAF consists of an ordinary real-valued activation function and a Gaussian function and is suitable for any given complex-valued neural network. Two polar-SAR image classification experiments on complex-valued convolutional neural networks show that GTAF converges faster than RIAF and has higher accuracy.

Cough Classification and Recognition with Mel-Frequency Cepstral Coefficients and Gaussian-type Evaluation Function

Cough Classification and Recognition with Mel-Frequency Cepstral Coefficients and Gaussian-type Evaluation Function

Cough Classification Method with Mel-Frequency Cepstral Coefficients and Gaussian-type Evaluation Function

Cough Classification Method with Mel-Frequency Cepstral Coefficients and Gaussian-type Evaluation Function

Projector Augmented Wave Method with Gauss-Type Atomic Orbital Basis: Implementation of the Generalized Gradient Approximation and Mesh Grid Quadrature.

The projector augmented wave (PAW) method is a powerful numerical algorithm that serves as a backend, enabling efficient density functional theory (DFT) calculations through the smoothing of valence electronic descriptions. Although it is mainly used in conjunction with plane-wave basis for solid-state systems, its generality permits the combination with other types of basis functions. In the previous study, we proposed a scheme to incorporate the PAW method into the conventional quantum chemical DFT implementation based on Gauss-type function (GTF) basis (Xiong et al., J. Chem. Theory Comput. 2017, 13, 3236-3249). The potentially high usability of the GTF-based PAW method, referred to as GTF-PAW, was previously shown, while its implementation was limited to the local density approximation (LDA). Here, we present a development of two technical extensions in this method toward practical DFT calculations. The GTF-PAW-based formulation and implementation to raise the level of the functional treatment to the generalized gradient approximation (GGA) is presented for improving reliability. In addition, we attempt to use the uniform mesh grid for DFT's quadrature in place of the conventional Becke grid, which was previously used. With the test calculations performed on illustrative molecules, it is confirmed that the conventional approach to implement GGA within GTF basis code can be straightforwardly integrated into the GTF-PAW method, allowing for the numerically stable treatment of the gradients of density. It is demonstrated that the uniform mesh grid can be used as an efficient numerical quadrature approach, which may be advantageous for handling larger systems.

Distribution of oscillator strengths and correlated electron dynamics in artificial atoms

The electronic structure of the quasi-two-dimensional disk-like artificial atom involving holes inside has been studied by the full configuration interaction (FCI) wave function employing mixed basis sets consisting of standard atomic orbitals and anisotropic Gaussian-type functions. The one-particle confinement potential for electrons has been modeled by a sum of an anisotropic harmonic-oscillator potential and attractive Coulomb potentials, the latter being responsible for the hole positive charges. The oscillator strengths from the ground state to a number of low-lying excited states have been calculated by the FCI energies and wave functions for different strength of the harmonic confinement ω. The results for two electrons with a hole at the center, namely, the He-like system, have shown accumulation of the oscillator strengths distributed among different dipole-allowed transitions in the excitation into the center-of-mass modes or the plasmon modes for increasing confinement strength ω. The system of two electrons with two impurity holes, namely, the H2-like system, has been also studied, which has shown a far more complicated trend of the oscillator strengths with respect to ω owing to the additional degree of freedom of the internuclear distance r. The observed trend has been rationalized on the basis of the nodal patterns in the molecular orbitals and in the electron density distributions.

Use of Gaussian-type functions for flux-based dose calculations in carbon ion therapy.

In radiation therapy, it is very important to ensure that the radiation dose is correctly delivered to the patient. This is achieved by obtaining quantitative dose measurements for beam calibration in the treatment planning system. Dose calculations should be performed with the required accuracy to a degree of uncertainty of less than 1%. The measurement of the absorbed dose in and around body tissues irradiated with carbon ions requires careful use of materials selected from established phantom and radiation detectors. The main advantage of such materials is that when information on the energy and nature of charged particles at the desired point is incomplete or inaccurate, they can allow determination of the absorbed dose. In general, radiation interactions in a tissue representation caused by carbon ions can be characterized by calculating the linear stopping power. Carbon ions have a limited penetration depth within human tissues that depends on the energy and stopping power of these ions as they penetrate into the body. The purpose of the present study was to calculate the stopping power, range and dose to intestinal and prostate tissues of carbon ions. The stopping power values of these tissues were specified by the effective charge approach method. The 5ZaPa-NR-CV, pcemd-4 and pcSseg-4 sets of Gaussian-type functions were employed for the calculation of electronic charge density. Range calculations were made by means of the Gaussian quadrature method, making use of the continuous slowing down approximation. Flux-based dose calculations were also carried out in accordance with the Bragg-Gray theorem using the Geant4 and FLUKA simulation toolkits. The results were compared with each other and with the SRIM and CasP datasets. Then, depth-dose distributions and range values were verified by positron emission activity using the GATE toolkit. Among the different types of Gaussian functions used here, the best semi-analytical result was found for the 5ZaPa-NR-CV set. The results obtained in the present study can be used for dose verification and dose reconstruction in charged particle radiotherapy and for radiation research on the interaction of radiation with matter. The results calculated here will be useful for quantifying uncertainties associated with stopping power, range, and reconstruction of dose in charged particle therapy.

Comparative analysis of the Coulomb barrier in heavy-ion collisions by the double-folding method

The double-folding formalism has been applied to calculate the nucleus-nucleus potential by the use of the effective nucleon-nucleon (Migdal) potential and the nuclear densities of the interacting nuclei presented as the Gaussian-type functions and polynomials. The Coulomb barrier heights obtained by this way and by different types of the proximity potentials have been compared. The deviations of the theoretical values of the Coulomb barrier from the values extracted from the experimental data are discussed. The capture cross section calculated for the reactions with the light projectiles is in good agreement with experimental values. The theoretical results obtained for the 16O+144Sm and 17O+144Sm reactions could reproduce a significant difference in the experimental results for these reaction, which is related with the extra neutron in 17O. The overestimation of the experimental data obtained from different sources by the curve of the theoretical results of the 40Ca+96Zr reactions is explained by the appearance of the slight hindrance to complete fusion, i.e the capture and fusion cross sections are not equal in this reaction.

Polarization consistent basis sets using the projector augmented wave method: a renovation brought by PAW into Gaussian basis sets.

A recently introduced framework incorporating the Projector Augmented Wave method and Gauss-type function (GTF-PAW) [X.-G. Xiong and T. Yanai, J. Chem. Theory Comput., 2017, 13, 3236-3249] opens alternative possibilities for performing low-cost molecular computational chemistry calculations. In this work, we present our first attempt to expand the applicability of this method by developing a family of compact general contracted polarization consistent basis sets (PAW-Ln) as an optimized GTF basis in combination with PAW. The results show that PAW-Ln, despite having small numbers of primitives, can provide not only better performance than effective core potential (ECP) but also good accuracy and desirable systematic convergence compared to larger all-electron basis sets. This demonstrates that GTF-PAW using the PAW-Ln basis sets could be a better alternative to both conventional all-electron- and ECP-based approaches for routine DFT calculations.

NONPARAMETRIC CORRELOGRAM TO IDENTIFY THE GEOGRAPHIC DISTANCE OF SPATIAL DEPENDENCE ON LAND PRICES

The spatial autocorrelation measurement of land prices uses a covariance function to describe the spatial dependence and it can be identified as a geographic distance on the correlogram. The geographic distance of spatial dependence can state that land prices are interdependent to each other and scattered in the research area. Therefore, the purpose of this research is to define the geographic distance of spatial dependence on land prices using a nonparametric correlogram. A nonparametric approach to covariance functions using the composition of Bessel and Gaussian-type functions are adopted because they correspond to the positive definite characteristics. The cubic spline interpolation is used to refine the curve fitting, while the intersection between the nonparametric correlogram value C(h) against the horizontal axis is determined using the Jenkins Traub algorithm. The results showed that the nonparametric correlogram identified a geographic distance of land prices smaller than the correlogram used so far. A small distance means that the land price in a location is greatly affected by the neighbors compared to a larger distance.

Effect of Bispectrum on Radar Backscattering From Non-Gaussian Sea Surface

The upwind-downwind asymmetry in normalized radar backscattering cross section (NRBCS) from ocean surface is well known; one acceptable and convincing reasoning is explained by the fact that the surface height distribution deviates from Gaussian one, which causes a nonzero skewness function, and consequently affects the radar cross section in up and down wind directions. Specific forms of skewness function, in between Gaussian and exponential, have been proposed in previous studies to account for the upwind/downwind asymmetry in radar backscatter. Attempt is made, through numerical simulation, in this article to examine the impact of these two types of skewness functions on NRBCS. The simulated NBRCSs, with and without skewness contributions, are compared with measured data in upwind and downwind directions at L-, C-, and Ku-bands for different wind speeds. The results indicate that the Gaussian-type skewness function works better to account for the upwind/downwind asymmetry of NBRCS by choosing an appropriate root mean square (RMS) height appearing in skewness function and as the wind speed or radar frequency increases, the RMS height increases but regardless of wind direction and radar incidence angle.

Calculation of Quantum Chemical Two-Electron Integrals by Applying Compiler Technology on GPU.

In this article, we present an effective approach to calculate quantum chemical two-electron integrals over basis sets consisting of Gaussian-type basis functions on graphical processing unit (GPU). Our framework generates several different variants called routes to the same integral problem with different integral algorithms (McMurchie-Davidson, Head-Gordon-Pople, and Rys) and precision. Each route is benchmarked on more GPU architectures, and with this data, a model is fitted to select the best available route for an integral task given a GPU architecture. Moreover, this approach supports the computation of high angular momentum orbitals up to g effectively on GPU, tested up to cc-pVQZ-sized basis sets. Rigorous analysis is shown regarding the effectiveness of our method. Molecule simulations with several basis sets are measured using NVIDIA GTX 1080 Ti, NVIDIA P100, and NVIDIA V100 cards.

All-electron fully relativistic Kohn-Sham theory for solids based on the Dirac-Coulomb Hamiltonian and Gaussian-type functions

We present the first full-potential method that solves the fully relativistic 4-component Dirac-Kohn-Sham equation for materials in the solid state within the framework of atom-centered Gaussian-type orbitals (GTOs). Our GTO-based method treats one-, two-, and three-dimensional periodic systems on an equal footing, and allows for a seamless transition to the methodology commonly used in studies of molecules with heavy elements. The scalar relativistic effects and spin-orbit interaction are handled variationally. The full description of the electron-nuclear potential in the core region of heavy nuclei is straightforward due to the local nature of the GTOs and does not pose any computational difficulties. We show how the time-reversal symmetry and a quaternion algebra-based formalism can be exploited to significantly reduce the increased methodological complexity and computational cost associated with multiple wave-function components coupled by the spin-orbit interaction. We provide a description of how to employ the matrix form of the multipole expansion and an iterative renormalization procedure to evaluate the conditionally convergent lattice sums arising for periodic systems. We investigate the problem of inverse variational collapse that arises if the Dirac operator containing a repulsive periodic potential is expressed in a basis containing diffuse functions, and suggest a possible solution. We demonstrate the validity of the method on silver halide crystals with large relativistic effects, and two-dimensional honeycomb structures (silicene and germanene) exhibiting the spin-orbit-driven quantum spin Hall effect. Our results are well-converged with respect to the basis limit using standard bases developed for molecular calculations, and indicate that the rule of removing basis functions with small exponents should not be applied when transferring the molecular basis to solids.

Ab Initio Calculation of Total X-ray Scattering from Molecules.

We present a method to calculate total X-ray scattering cross sections directly from ab initio electronic wave functions in atoms and molecules. The approach can be used in conjunction with multiconfigurational wave functions and exploits analytical integrals of Gaussian-type functions over the scattering operator, which leads to accurate and efficient calculations. The results are validated by comparison to experimental results and previous theory for the molecules H2 and CO2. Importantly, we find that the inelastic component of the total scattering varies strongly with molecular geometry. The method is appropriate for use in conjunction with quantum molecular dynamics simulations for the analysis of new ultrafast X-ray scattering experiments and to interpret accurate gas-phase scattering experiments.

Sharp Results on Sampling with Derivatives in Shift-Invariant Spaces and Multi-Window Gabor Frames

We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions.

DFT Study of Structure, IR and RAMAN Spectra for Betulinic Acid Solvatomorphs

Combined experimental and theoretical studies on molecular structure of the betulinic acid solvatomorph of ethanol, 2-butanol and isopropyl alcohol solvents, grown by recrystallization method, are reported. Theoretical investigations, in the frame of DFT, are done using the new method of the betulinic acid solvatomorph molecule extraction from the corresponding molecular crystal, based on typical fragmentation scheme and chemical connectivity. For all electrons, a localized basis set of Gaussian type functions TZVP and local gradient corrected functional BP86 were used. Good correlation was found between the calculated and experimental data. Slight differences in the lattice modes spectral range (0-200 cm-1) from experimental Raman spectra suggest slight differences in the crystalline network and highlight the powerful Raman approach in evidencing BA pseudo-polymorphs or solvatomorphs. Data obtained suggest that incorporated solvents to the solvates generaly induced changes in crystal symmetry, intermolecular arrangements, stoichiometry and hydrogen bonding interactions of the analysed solvatomorphs.

Three-Dimensional Simulation of Aggregate and Asphalt Mixture Using Parameterized Shape and Size Gradation

Aggregate occupies at least three-quarters of the volume of asphalt mixture and can significantly affect the performance of pavement. The geometrical morphology influences the slippage and interlock among aggregates for resisting and distributing applied loads. In recent years, the discrete-element method (DEM) has been employed for simulation of asphalt mixture structure. This paper introduces an approach for simulation of aggregate and asphalt mixtures using parameterized shape and size gradation. Both the plane geometry factor (PGF) and the section aspect ratio (SAR) were employed to describe the three-dimensional (3D) geometric characteristics of aggregates. A numerical technique of aggregate models was implemented with probabilistic parameters depending on statistical results of PGFs and SARs. The 3D numerical model of asphalt mixtures was assembled with three different components, and was validated by uniaxial compression tests via comparison with the laboratory result. It was found that the PGF and SAR are appropriate to describe the three-dimensional features of aggregate shapes, because a simplified space object can be described by a two-dimensional (2D) graphical projection and a vector scalar corresponding to the space vector. Probability distribution curves of PGFs and SARs between coarse aggregates were in concordance with the Gauss-type function, because their correlation coefficients were all greater than 95%. It was verified that the developed clumping algorithm of aggregates was reasonable in terms of the shape and size gradation. Based on the parallel-bond model and Burger's model, the results of virtual tests were in good agreement with those of laboratory uniaxial tests. The angularity (PGF) of aggregates has a beneficial effect on the strength and stability of asphalt mixtures, whereas the flat-elongated feature (SAR) has a negative effect on the strength and stability of asphalt mixtures.

Numerical calculation of the Spherical Bessel Transform from Gaussian quadrature in the complex-plane

A numerical method for the calculation of spherical Bessel transforms via Gaussian quadrature in the complex plane is presented. The method is evaluated by transforming Slater and Gaussian-type spherical coordinate basis functions, used in electronic structure calculations, from position space to momentum space. The feasibility and efficiency of the method is explored for different regions of momenta and different orders of the spherical Bessel transform. The results illustrate that in general the method performs very well in the large p regions when applied to the Slater-type functions. On the other hand, a parameter has to be introduced for the application to Gaussian-type functions in order to avoid cancellation. In this case, the method performs well but accuracy is lost at larger p. Use of the parameter in application to Slater functions yields accurate results for all regions of p.

Solid harmonic wavelet scattering for predictions of molecule properties.

We present a machine learning algorithm for the prediction of molecule properties inspired by ideas from density functional theory (DFT). Using Gaussian-type orbital functions, we create surrogate electronic densities of the molecule from which we compute invariant "solid harmonic scattering coefficients" that account for different types of interactions at different scales. Multilinear regressions of various physical properties of molecules are computed from these invariant coefficients. Numerical experiments show that these regressions have near state-of-the-art performance, even with relatively few training examples. Predictions over small sets of scattering coefficients can reach a DFT precision while being interpretable.