scholarly journals Semi-Analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian Expansion of Distance Operators W= RC1-nRD1-M, RC1-Nr12-M, R12-Nr13-M

2019 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Gaussian Function ◽  
Approximate Solutions ◽  
Equation System ◽  
Least Square ◽  
Position Vector ◽  
Least Square Fit ◽  
Analytic Expressions ◽  
The One ◽  
Analytical Expressions ◽  
Coulomb Integrals

The equations derived help to evaluate semi-analytically (mostly for k=1,2 or 3) the important Coulomb integrals Int rho(r1)…rho(rk) W(r1,…,rk) dr1…drk, where the one-electron density, rho(r1), is a linear combination (LC) of Gaussian functions of position vector variable r1. It is capable to describe the electron clouds in molecules, solids or any media/ensemble of materials, weight W is the distance operator indicated in the title. R stands for nucleus-electron and r for electron-electron distances. The n=m=0 case is trivial, the (n,m)=(1,0) and (0,1) cases, for which analytical expressions are well known, are widely used in the practice of computation chemistry (CC) or physics, and analytical expressions are also known for the cases n,m=0,1,2. The rest of the cases – mainly with any real (integer, non-integer, positive or negative) n and m - needs evaluation. We base this on the Gaussian expansion of |r|^-u, of which only the u=1 is the physical Coulomb potential, but the u≠1 cases are useful for (certain series based) correction for (the different) approximate solutions of Schrödinger equation, for example, in its wave-function corrections or correlation calculations. Solving the related linear equation system (LES), the expansion |r|^-u about equal SUM(k=0toL)SUM(i=1toM) Cik r^2k exp(-Aik r^2) is analyzed for |r| = r12 or RC1 with least square fit (LSF) and modified Taylor expansion. These evaluated analytic expressions for Coulomb integrals (up to Gaussian function integrand and the Gaussian expansion of |r|^-u) are useful for the manipulation with higher moments of inter-electronic distances via W, even for approximating Hamiltonian.

scholarly journals Semi-Analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian Expansion of Distance Operators W= RC1-nRD1-M, RC1-Nr12-M, R12-Nr13-M

2019 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Gaussian Function ◽  
Approximate Solutions ◽  
Equation System ◽  
Least Square ◽  
Position Vector ◽  
Least Square Fit ◽  
Analytic Expressions ◽  
The One ◽  
Analytical Expressions ◽  
Coulomb Integrals

The equations derived help to evaluate semi-analytically (mostly for k=1,2 or 3) the important Coulomb integrals Int rho(r1)…rho(rk) W(r1,…,rk) dr1…drk, where the one-electron density, rho(r1), is a linear combination (LC) of Gaussian functions of position vector variable r1. It is capable to describe the electron clouds in molecules, solids or any media/ensemble of materials, weight W is the distance operator indicated in the title. R stands for nucleus-electron and r for electron-electron distances. The n=m=0 case is trivial, the (n,m)=(1,0) and (0,1) cases, for which analytical expressions are well known, are widely used in the practice of computation chemistry (CC) or physics, and analytical expressions are also known for the cases n,m=0,1,2. The rest of the cases – mainly with any real (integer, non-integer, positive or negative) n and m - needs evaluation. We base this on the Gaussian expansion of |r|^-u, of which only the u=1 is the physical Coulomb potential, but the u≠1 cases are useful for (certain series based) correction for (the different) approximate solutions of Schrödinger equation, for example, in its wave-function corrections or correlation calculations. Solving the related linear equation system (LES), the expansion |r|^-u about equal SUM(k=0toL)SUM(i=1toM) Cik r^2k exp(-Aik r^2) is analyzed for |r| = r12 or RC1 with least square fit (LSF) and modified Taylor expansion. These evaluated analytic expressions for Coulomb integrals (up to Gaussian function integrand and the Gaussian expansion of |r|^-u) are useful for the manipulation with higher moments of inter-electronic distances via W, even for approximating Hamiltonian.

scholarly journals Shell model calculations in the A=80-100 mass region and study of double β transitions

2020 ◽  
Vol 1 ◽  
pp. 156
Author(s):  
J. Sinatkas ◽  
L. D. Skouras ◽  
D. Strottman ◽  
J. D. Vergados
Keyword(s):  
Shell Model ◽  
Satisfactory Agreement ◽  
Effective Interaction ◽  
Model Space ◽  
Mass Region ◽  
Least Square ◽  
Matrix Elements ◽  
Model Calculations ◽  
Least Square Fit ◽  
The One

The structure of the Ζ,Ν < 50 nuclei is examined in a model space consisting of the 0g9/2, 1p1/2, 1p3/2 and the 0f5/2 hole orbitals outside the doubly closed 100Sn core. The effective interaction for this model space is derived by introducing second order corrections to the Sussex matrix elements, while the one-hole energies are deduced by a least square fit to the observed levels. The results of the calculation are found to be in very satisfactory agreement with experiment for all nuclei with 38<Ζ<46 but for Ζ<38 this agreement begins to deteriorate. Such a feature possibly indicates the appearance of deformation and the breaking of the Ν=50 core. The wavefunctions of the calculation are used to determine double β matrix elements in the Ge, Se, Sr and Kr isotopes.

Target Plane Confidence Boundaries: Mathematics of The 1997 XF11 Scare

1999 ◽  
Vol 172 ◽  
pp. 369-370
Author(s):  
Andrea Milani ◽  
Giovanni B. Valsecchi
Keyword(s):  
Linear Approximation ◽  
Close Approach ◽  
Confidence Regions ◽  
Least Square ◽  
Computational Load ◽  
Approach Distance ◽  
The Monte Carlo Method ◽  
Hazardous Object ◽  
Least Square Fit ◽  
The One

The uncertainty of the close approach distance of a Potentially Hazardous Object (PHO), either an asteroid or a comet, can be represented on the Modified Target Plane (MTP), a modification of the one used by Öpik. The MTP is orthogonal to the geocentric velocity at the closest approach along the nominal orbit, solution of the least square fit to the observations. The confidence regions of this solution in the 6-D space of orbital elements (for an epoch close to the observations) are well approximated by a family of concentric ellipsoids, if the observed arc is not too short. In the linear approximation these ellipsoids are mapped on the MTP into concentric ellipses, which can be computed by solving for the state transition matrix.For a PHO observed at only one opposition, with a close approach expected after many revolutions, the ellipses on the MTP become extremely elongated and the linear approximation may fail. In this case the confidence boundaries on the MTP, i.e. the nonlinear images of the confidence ellipsoids, may not be well approximated by the ellipses. The Monte Carlo method (Muinonen and Bowell, 1993) can be used to find nonlinear confidence regions, but the computational load is very heavy: to estimate a low probability event the number of test cases must be larger than the inverse of the probability. We propose a new method to compute semilinear confidence boundaries on the MTP (Milani and Valsecchi, 1998), based on the theory developed to compute confidence boundaries for predicted observations (Milani, 1999). This method is a good compromise between reliability and computational load, and can be used for real time risk assessment.

Method for CCD Edge Signal Processing

2013 ◽  
Vol 441 ◽  
pp. 695-698
Author(s):  
Guo Sheng Xu
Keyword(s):  
Least Square ◽  
Digital Data ◽  
Edge Point ◽  
Precise Position ◽  
Order Curve ◽  
Least Square Fit ◽  
Edge Location ◽  
Edge Detection Method ◽  
The One ◽  
Fitting Part

In order to improve the measuring precision of the one-dimension line-matrix CCD, a new sub-pixel edge detection method is proposed. Firstly the date collected by the line-matrix CCD is converted to digital data by virtual oscillograph. Base on threshold comparison, the fitting part of the edge signal is picked up for further processing; secondly, the pixels are expanded following the edge direction of the edge point, and the edge signal is fitted through the two order multinomial; finally, comparing the two order curve with the threshold voltage through the least square fit method, the precise position of the edge point can achieve the sub-pixel edge location precision.

Analytic evaluation of Coulomb integrals for one, two and three-electron distance operators

2017 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Laplace Transformation ◽  
Numerical Solutions ◽  
State Of The Art ◽  
Gaussian Function ◽  
Higher Moments ◽  
Integral Evaluation ◽  
Analytic Expressions ◽  
Dimensional Version ◽  
Coulomb Integrals ◽  
Boys Function

<p> The state of the art for integral evaluation is that analytical solutions to integrals are far more useful than numerical solutions. We evaluate certain integrals analytically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the (n,m)=(0,0) case is trivial, the (n,m)=(1,0) and (0,1) cases are well known, fundamental milestone in integration and widely used in computation chemistry, as well as based on Laplace transformation with integrand exp(-a<sup>2</sup>t<sup>2</sup>). The rest of the cases are new and need the other Laplace transformation with integrand exp(-a<sup>2</sup>t) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations. </p>

Analytic evaluation of Coulomb integrals for one, two and three-electron distance operators

2017 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Laplace Transformation ◽  
Numerical Solutions ◽  
State Of The Art ◽  
Gaussian Function ◽  
Higher Moments ◽  
Integral Evaluation ◽  
Analytic Expressions ◽  
Dimensional Version ◽  
Coulomb Integrals ◽  
Boys Function

<p> The state of the art for integral evaluation is that analytical solutions to integrals are far more useful than numerical solutions. We evaluate certain integrals analytically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the (n,m)=(0,0) case is trivial, the (n,m)=(1,0) and (0,1) cases are well known, fundamental milestone in integration and widely used in computation chemistry, as well as based on Laplace transformation with integrand exp(-a<sup>2</sup>t<sup>2</sup>). The rest of the cases are new and need the other Laplace transformation with integrand exp(-a<sup>2</sup>t) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations. </p>

scholarly journals Numerical solutions of one-dimensional parabolic convection-diffusion problems arising in biology by the Laguerre collocation method

BIOMATH ◽  
2017 ◽  
Vol 6 (1) ◽  
pp. 1706047 ◽  
Author(s):  
Burcu Gürbüz ◽  
Mehmet Sezer
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Equation System ◽  
Diffusion Problem ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Laguerre Series ◽  
The Matrix ◽  
The One

In this work, we present a numerical scheme for the approximate solutions of the one-dimensional parabolic convection-diffusion model problems. Diffusion models form a reasonable basis for studying insect and animal dispersal and invasion, which arise from the question of persistence of endangered species, biodiversity, disease dynamics, multi-species competition so on. Convection diffusion problem is also a form of heat and mass transfer in biological models. The presented method is based on the Laguerre collocation method used for these problems of differential equations.In fact, the approximate solution of the problem in the truncated Laguerre series form is obtained by this method. By substituting truncated Laguerre series solution into the problem and by using the matrix operations and the collocation points, the suggested scheme reduces the problem to a linear algebraic equation system. By solving this equation system, the unknown Laguerre coecients can be computed. The accuracy and the efficiency of the method is showed by numerical examples and the comparisons by the other methods.

Least-square refinement of structure parameters from CBED integrated intensities: application to determination of temperature factors in Y BA2CU3O7

1992 ◽  
Vol 50 (2) ◽  
pp. 1456-1457
Author(s):  
Kjersti Gjønnes ◽  
Jon Gjønnes
Keyword(s):  
Least Square ◽  
Measured Intensity ◽  
Structure Information ◽  
Accurate Temperature ◽  
Least Square Fit ◽  
Case Application ◽  
Integrated Intensities ◽  
Temperature Factors ◽  
Narrow Bands

Electron diffraction intensities can be obtained at large scattering angles (sinθ/λ ≥ 2.0), and thus structure information can be collected in regions of reciprocal space that are not accessable with other diffraction methods. LACBED intensities in this range can be utilized for determination of accurate temperature factors or for refinement of coordinates. Such high index reflections can usually be treated kinematically or as a pertubed two-beam case. Application to Y Ba2Cu3O7 shows that a least square refinememt based on integrated intensities can determine temperature factors or coordinates.LACBED patterns taken in the (00l) systematic row show an easily recognisable pattern of narrow bands from reflections in the range 15 < l < 40 (figure 1). Integrated intensities obtained from measured intensity profiles after subtraction of inelastic background (figure 2) were used in the least square fit for determination of temperature factors and refinement of z-coordinates for the Ba- and Cu-atoms.

scholarly journals APPLICATIONS OF DENSITY MATRICES IN A TRAPPED BOSE GAS

2006 ◽  
Vol 20 (15) ◽  
pp. 2189-2221 ◽  
Author(s):  
K. CH. CHATZISAVVAS ◽  
S. E. MASSEN ◽  
CH. C. MOUSTAKIDIS ◽  
C. P. PANOS
Keyword(s):  
Quantum Information ◽  
Bose Gas ◽  
Einstein Condensation ◽  
Density Distributions ◽  
Occupation Numbers ◽  
The One ◽  
Bose Einstein ◽  
Jensen Shannon Divergence ◽  
Analytical Expressions ◽  
Short Range Correlations

An overview of the Bose–Einstein condensation of correlated atoms in a trap is presented by examining the effect of interparticle correlations to one- and two-body properties of the above systems at zero temperature in the framework of the lowest order cluster expansion. Analytical expressions for the one- and two-body properties of the Bose gas are derived using Jastrow-type correlation function. In addition numerical calculations of the natural orbitals and natural occupation numbers are also carried out. Special effort is devoted for the calculation of various quantum information properties including Shannon entropy, Onicescu informational energy, Kullback–Leibler relative entropy and the recently proposed Jensen–Shannon divergence entropy. The above quantities are calculated for the trapped Bose gases by comparing the correlated and uncorrelated cases as a function of the strength of the short-range correlations. The Gross–Piatevskii equation is solved, giving the density distributions in position and momentum space, which are employed to calculate quantum information properties of the Bose gas.

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