Realistic nuclear charge distribution model function for analytic nuclear attraction integrals in Gaussian basis functions.

A procedure is presented which allows a more economical representation of the potential produced by orbital charge distributions in which the orbitals are expanded in terms of a finite set of polynomial Gaussian functions. The basic idea is that the products of pairs of Gaussian basis functions, on which the charge distributions are expanded, are expressed in terms of a new basis set of optimally chosen single Gaussian functions. Such a procedure has been tested in a particular case and a few possible applications have been suggested.

A new model potential for one-valence-electron atoms is introduced. It consists [Eq. (6)] of a core Coulomb potential modified by the addition of a Gaussian screened Coulomb potential. This potential has the desired features outlined for a model potential, and it has particularly convenient and simple mathematical properties when used with Gaussian basis functions. Since the smooth valence orbitals are sought, Gaussian functions are a good basis set because their deficiencies in nuclear regions do not enter the problem. The potential is calibrated to experimental energies for the 2s and 2p states of Li and 3s and 3p states of Na, using extended basis sets. The model Hamiltonians so defined are used with a variety of more modest basis sets to determine pseudowavefunctions. Based on comparisons with ab initio orbitals, energy analyses, radial density calculations, and overlap and expectation value calculations, the conclusion is that good valence pseudowavefunctions can be obtained by this approach with relatively small basis sets. The approach looks promising for valence-only molecular studies.

Using self-consistent field theory (SCFT), we study the conformational properties of polyelectrolyte dendrimers. We compare results for three different models of charge distributions on the polyelectrolytes: (1) a smeared, quenched charge distribution characteristic of strong polyelectrolytes; (2) a smeared, annealed charge distribution characteristic of weak polyelectrolytes; and (3) an implicit counterion model with Debye-Huckel interactions between the charged groups. Our results indicate that an explicit treatment of counterions is crucial for the accurate characterization of the conformations of polyelectrolyte dendrimers. In comparing the quenched and annealed models of charge distributions, annealed dendrimers were observed to modulate their charges in response to the density of polymer monomers, counterions, and salt ions. Such phenomena is not accommodated within the quenched model of dendrimers and is shown to lead to significant differences between the predictions of quenched and annealed model of dendrimers. In this regard, our results indicate that the average dissociated charge α inside the dendrimer serves as a useful parameter to map the effects of different parametric conditions and models onto each other. We also present comparisons to the scaling results proposed to explain the behavior of polyelectrolyte dendrimers. Inspired by the trends indicated by our results, we develop a strong segregation theory model whose predictions are shown to be in very good agreement with the numerical SCFT calculations.

The study of the charge distribution in deformed nuclei by muonic hyperfine structure

To develop, train and test neural networks for predicting heart surface potentials (HSPs) from body surface potentials (BSPs). The method re-frames traditional inverse problems of electrocardiography into regression problems, constraining the solution space by decomposing signals with multidimensional Gaussian impulse basis functions. Impulse HSPs were generated with single Gaussian basis functions at discrete heart surface locations and projected to corresponding BSPs using a volume conductor torso model. Both BSP (inputs) and HSP (outputs) were mapped to regular 2D surface meshes and used to train a neural network. Predictive capabilities of the network were tested with unseen synthetic and experimental data. A dense full connected single hidden layer neural network was trained to map body surface impulses to heart surface Gaussian basis functions for reconstructing HSP. Synthetic pulses moving across the heart surface were predicted from the neural network with root mean squared error of 9.1±1.4%. Predicted signals were robust to noise up to 20 dB and errors due to displacement and rotation of the heart within the torso were bounded and predictable. A shift of the heart 40 mm toward the spine resulted in a 4% increase in signal feature localization error. The set of training impulse function data could be reduced, and prediction error remained bounded. Recorded HSPs from in-vitro pig hearts were reliably decomposed using space-time Gaussian basis functions. Activation times calculated from predicted HSPs for left-ventricular pacing had a mean absolute error of 10.4±11.4 ms. Other pacing scenarios were analyzed with similar success. Impulses from Gaussian basis functions are potentially an effective and robust way to train simple neural network data models for reconstructing HSPs from decomposed BSPs. The HSPs predicted by the neural network can be used to generate activation maps that non-invasively identify features of cardiac electrical dysfunction and can guide subsequent treatment options.

Self-consistent electron–nucleus cusp correction for molecular orbitals

Nanoparticles in natural systems II: The natural oxide fraction at interaction with natural organic matter and phosphate

An embedded cluster model with either a large number of point charges (PCs) or periodic electrostatic (PE) potential was proposed to incorporate the electrostatic effects by the bulk surface and applied to non-transition-metal oxide supports such as Al2O3 and AlPO4. A large number of PCs are placed on several layers of surface. The PC values were taken to be the same as the Bader charges obtained from periodic DFT calculation of slab model. The PE potential was derived so as to consider the infinite three-dimensional PC distribution obtained by the calculation of slab model. One electron integral of the PE potential in the Gaussian basis function was evaluated using Poisson’s equation, Fourier transformation within a supercell approach, and the Ewald summation method. These embedded cluster models were applied to Rh2-adsorbed Al2O3 and AlPO4. A bare cluster model with neither PC nor the PE potential presented very poor computational results for the interaction energies of Rh2 with Al2O3 and AlPO4 surfaces...

A new method for the evaluation of one- and two-centre magnetic and electric multipole integrals for Slater-type functions is presented. The method is strictly analytical in that no approximations of any kind are involved. Two simple functions, ℐ1aug and ℐ2aug, are introduced, which employ only functions that are well known in electronic structure theory. With the use of augmentation exponents these functions apply to multipole integrals as well as other one-electron integrals, such as nuclear attraction integrals. The proposed method includes the analytic determination of derivatives of the integrals with respect to atomic displacements. Some illustrative test calculations are presented and compared to results from the literature.

The advantages of a Gaussian amplitude distribution at the surface of a transducer are similar to the advantages of a Gaussian amplitude distribution in a light beam: the diffraction pattern is both simpler to measure and simpler to calculate. Further, the simplification remains when one considers nonlinear terms in the wave equation. If one can achieve a truly Gaussian amplitude distribution, then comparison of experimental data with theoretical calculation also is facilitated. Previously we used electrical fringing to produce a one‐dimensional Gaussian amplitude distribution for schlieren photography. Now, we have used an electrical fringing to produce a two‐dimensional Gaussian distribution in an ultrasonic beam. The sidelobes are undetectable, and the maxima and minima in the Fresnel region have disappeared. Design and testing of the transducer are discussed. Both advantages and limitations are delineated. The mathematical description of the sound field is discussed. Analytical expressions are given for the amplitude in the nearfield (Fresnel?) as well as in the farfield (Fraunhofer?). In both cases the distribution is described by a Gaussian function. [Research supported in part by ONR.]

In this paper, an adaptive controller for a class of uncertain nonlinear systems is presented using a combination of Legendre polynomials and brain emotional learning-based intelligent controller (BELBIC). Recently, some versions of BELBIC have been presented with the aim of satisfying the universal approximation property using Gaussian basis function. However, the size of regressor vector is too large that imposes a heavy computational load to the processor. The novelty of this paper is presenting a new version of BELBIC with less computational burden using Legendre polynomials. Moreover, there are very few tuning parameters in Legendre polynomials. Another contribution of this paper is editing the stability analysis presented in recent related works. Due to the intrinsic non-differentiability of the adaptation rules of BELBIC, the second time derivative of Lyapunov function is undefined and thus, the Barbalat’s lemma cannot be applied to verify the asymptotic convergence of the error function. Therefore, bounded-input-bounded-output (BIBO) stability can only be claimed for this controller. Simulation results on different case studies show that Legendre polynomials can improve the universal approximation property of BELBIC with less tuning parameters. Moreover, in the absence of the robust control term in the control law, the performance Legendre polynomials will not deteriorate, while the performance degrade in Gaussian basis function is quite considerable.

In this study, a new prediction method is suggested for sound transmission loss (STL) of multilayered panels of infinite extent. Conventional methods such as random or field incidence approach often given significant discrepancies in predicting STL of multilayered panels when compared with the experiments. In this paper, appropriate directional distributions of incident energy to predict the STL of multilayered panels are proposed. In order to find a weighting function to represent the directional distribution of incident energy on the wall in a reverberation chamber, numerical simulations by using a ray-tracing technique are carried out. Simulation results reveal that the directional distribution can be approximately expressed by the Gaussian distribution function in terms of the angle of incidence. The Gaussian function is applied to predict the STL of various multilayered panel configurations as well as single panels. The compared results between the measurement and the prediction show good agreements, which validate the proposed Gaussian function approach.

SummaryThe reaction yields of 63 radionuclides with the mass numberA= 56 − 135 produced in the photofission of209Bi by bremsstrahlung of end-point energies (E0) from 450 to 1100 MeV have been measured using a catcher foil technique with the aid of intensive chemical separation. The charge distribution was well reproduced by a Gaussian function with the most probable charge (Zp) expressed by a linear function ofA,i.e.,Zp=RA+S, and with theA-independent full width at half maximum (FWHMCD). The charge distribution parametersR,Sand FWHMCDwere independent ofE0above 600 MeV, reflecting the resonance nature in photonuclear reactions at intermediate energies. The weighted mean values atE0≥ 600 MeV wereR= 0.421 ± 0.001,S= 0.6 ± 0.1 and FWHMCD= 2.1 ± 0.1 charge unit (c.u.). The numbers of pre- and post-fission neutrons were deduced to be νpre= 12 ± 1 and νpost= 1.4 ± 0.3, respectively, assuming the Unchanged Charge Distribution (UCD). Based on the charge distribution parameters, the symmetric mass yield distributions with the most probable massApof 96 ± 1 mass unit (m.u.) and the width FWHMMDof 33 ± 1 m.u. were also obtained. The characteristics of the charge and mass yield distributions are discussed by referring to those for197Au based on the results of calculations using the Photon-induced Intranuclear Cascade Analysis code combined with the Generalized Evaporation Model (PICA3/GEM).

This paper proposed clustering algorithms applied Gaussian basis function neural network compensator with fuzzy control for magnetic bearing system (MBS). The nonlinear MBS improved traditional bearing friction losses, and nonlinear system with fuzzy controller and neural network does not require precise MBS mathematical model. We used clustering algorithms which are fuzzy c-means and k-means adjusted Gaussian basis function in neural network. Finally, we used the Lyapunov stability to guarantee MBS convergence, and the experimental results shows proposed algorithm has satisfactory performance in MBS.

Determination of the isotopic shift of the first resonance line in Cs(I) spectrum for the isotopes 131Cs, 132Cs, and 136Cs