A simple approximation for inverse distance with application to the one‐center and two‐center Coulomb integrals

1985 ◽  
Vol 83 (5) ◽  
pp. 2616-2618 ◽  
Author(s):  
S. Noor Mohammad
Keyword(s):  
Simple Approximation ◽  
The One ◽  
Coulomb Integrals ◽  
Inverse Distance

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 ESPACIALIZAÇÃO DA TEMPERATURA DO AR PARA A REGIÃO DO SUBMÉDIO SÃO FRANCISCO

Irriga ◽  
2018 ◽  
Vol 22 (1) ◽  
pp. 177-193 ◽  
Author(s):  
Iug Lopes ◽  
Juliana Maria M De Melo ◽  
Brauliro Gonçalves Leal
Keyword(s):  
Spatial Interpolation ◽  
Daily Variation ◽  
Maximum Temperature ◽  
Country Club ◽  
The Mean ◽  
Maximum Temperatures ◽  
Continuous Fields ◽  
One Year ◽  
The One ◽  
Inverse Distance

ESPACIALIZAÇÃO DA TEMPERATURA DO AR PARA A REGIÃO DO SUBMÉDIO SÃO FRANCISCO  IUG LOPES¹; JULIANA MARIA MEDRADO DE MELO² E BRAULIRO GONÇALVES LEAL³ ¹ Departamento de Engenharia Agrícola, Universidade Federal Rural de Pernambuco, Rua Dom Manoel de Medeiros, Dois Irmão, CEP: 52171-900 – Recife, PE. [email protected]² Departamento de Agronomia, Universidade do Estado da Bahia, Rua Edgar Chastinet, s/n - São Geraldo, BA, 48905-680 – Juazeiro, BA. [email protected]³ Colegiado de Engenharia da Computação, Universidade Federal do Vale do São Francisco – Campus Juazeiro, Av. Antonio Carlos Magalhães, 510 Country Club, CEP: 48.902-300 – Juazeiro, BA. [email protected]  1 RESUMO  Dentre as variáveis meteorológicas requeridas para o cálculo do balanço hídrico destacam-se as temperaturas mínimas, médias e máximas do ar, que apresentam uma continuidade no quantitativo de distância e assim permitem de uma maneira mais simples a criação de campos contínuos utilizando métodos de interpolação espacial. O objetivo deste trabalho foi avaliar potências para o método de interpolação do Inverso da Potência da Distância (IPD) na espacialização de valores diários da temperatura no Submédio São Francisco, para os períodos de um ano, das estações do ano (inverno, primavera, verão e outono). Foram obtidos os parâmetros de potência do interpolador Inverso da Potência da Distância das temperaturas mínimas, médias e máximas a partir dos dados medidos em 14 estações meteorológicas automáticas do INMET em operação no Pólo de Desenvolvimento Petrolina-Juazeiro. Foram realizadas interpolações para as épocas: anual, inverno, primavera, verão e outono. A variação diária do erro relativo médio obtida, para a época ano, calculado utilizando os dados de temperatura mínima, média e máxima utilizando o valor da potência do interpolador foram iguais a 3,3; 3,4; e 3,4, respectivamente. Os valores de erro médio foram pequenos quando comparados com o erro instrumental. Palavras-chave: interpolação, validação cruzada, estação meteorológica  LOPES, I; MELO, J. M. M.; LEAL, B. G. SPATIALIZATION OF AIR TEMPERATURE TO THE REGION OF SUBMEDIO SÃO FRANCISCO  2 ABSTRACT Among the meteorological variables required for the calculation of the water balance are the temperatures, which present a continuity in the quantitative distance and thus allow in a simpler way the creation of continuous fields using spatial interpolation methods. The objective of this work was to evaluate the power of the Inverse Distance Power (IPD) in the spatialization of daily values of temperature in the Submedia of São Francisco, for the one-year periods of the seasons (winter, spring, summer it's fall). The power parameters of the Inverse Distance Power Interpolator were obtained from the minimum, average and maximum temperatures from the data measured in 14 INMET automatic meteorological stations operating at the Petrolina-Juazeiro Development Pole. Interpolations were performed for annual, winter, spring, summer and fall seasons. The daily variation of the average relative error obtained for the year time, calculated using the data of minimum, average and maximum temperature using the value of the power of the interpolator were equal to 3.3; 3.4; and 3.4, respectively. The mean error values were small when compared to instrumental error. Keywords: interpolation, validation cross, meteorological station

RELIABILITY OF 3D MODELLING OF THE PB AND MAIN METALS CONTENT IN THE SIEROSZOWICE CU-AG ORE DEPOSIT (LUBIN-GŁOGÓW COPPER DISTRICT)

2017 ◽  
pp. 0-0
Author(s):  
Monika WASILEWSKA-BŁASZCZYK ◽  
Mateusz KRUK ◽  
Jacek Mucha
Keyword(s):  
3D Model ◽  
Probability Distributions ◽  
3D Modelling ◽  
Training Set ◽  
Estimation Errors ◽  
Sampling Network ◽  
Metal Contents ◽  
Distance Weighting ◽  
The One ◽  
Inverse Distance

The paper attempts to evaluate the possibility of reliable 3D modelling of lead content against the reliability of 3D modelling of Cu and Ag (main metals in the Sieroszowice Cu-Ag deposit) contents. The 3D modelling of the content of these elements was performed using interpolation (ordinary kriging in two variants, and squared inverse distance weighting) and simulation methods (Turning Bands Simulation). The assessment of the accuracy of estimates of metal contents determined from the 3D model, based on both spot samples and a training set, has shown that the estimated contents of Cu and Ag are acceptable for geological and mining applications (medians of estimation errors in the order of 40%), while the reliability of the Pb model is low (medians of estimation errors in the order of 70%). This is due to the unfavourable statistical features of the majority of the accompanying elements and, in particular, the highly asymmetric probability distributions, their high variability, the presence of numerous anomalous values, and the much lower density of the sampling network in the mining excavations when determining accompanying elements other than Cu. In general, the calculated medians of interpolation errors for the four methods do not allow determining the one providing (in each case) higher accuracy when assessing the content of the elements.

scholarly journals Averaging of Earth-Crossing Orbits

1999 ◽  
Vol 172 ◽  
pp. 433-434
Author(s):  
G.F. Gronchi ◽  
A. Milani
Keyword(s):  
Equations Of Motion ◽  
Generalized Solution ◽  
Elementary Functions ◽  
Averaged Equations ◽  
Major Planet ◽  
The One ◽  
Polar Singularity ◽  
Straight Lines ◽  
Method Of Extraction ◽  
Inverse Distance

The orbits of planet-crossing asteroids (and comets) can undergo close approaches and collisions with some major planet. This introduces a singularity in the N-body Hamiltonian, and the averaging of the equations of motion, traditionally used to compute secular perturbations, is undefined. We have shown (Gronchi and Milani, 1998) that it is possible to define in a rigorous way some generalised averaged equations of motion, in such a way that the generalised solutions are unique and piecewise smooth, with corners on the node crossing lines.The model is the averaged equations of motion first introduced by Kozai (1962): the perturbing planets are assumed to move in circular, coplanar orbits, and the equations of motion are averaged over the anomalies of the asteroid and of the planets. In the non-crossing case the averaging is integrable; in the planet-crossing case there is a polar singularity of order two in the equations of motion, and averaging is not possible. To define a generalized solution, we decrease the order of the polar singularity by the method of extraction of the singularities by Kantorovich. The singularity of the perturbing function is approximated by a modified inverse distance, the one between the straight lines tangent to the two orbits at the nodal points. In this approximation the averaged perturbing function has an analytical expression, allowing explicit computation with elliptic integrals and elementary functions.

scholarly journals Dendrochronology as a source of data for landslide activity maps – an example from Beskid Żywiecki Mountains (Western Carpathians, Poland)

2017 ◽  
Vol 5 (3) ◽  
pp. 40-46 ◽  
Author(s):  
Katarzyna Łuszczyńska ◽  
Małgorzata Wistuba ◽  
Ireneusz Malik
Keyword(s):  
Sampling Site ◽  
Abies Alba ◽  
Western Carpathians ◽  
Landslide Activity ◽  
Level Of Activity ◽  
Distance Weighting ◽  
Hazard And Risk ◽  
The One ◽  
Landslide Slope ◽  
Inverse Distance

AbstractWe applied dendrochronological methods for dating landslide activity in the study area (3.75 km2), on the slopes of Sucha Mountain (1040 m a.s.l.), in the Beskid Żywiecki Mountains, in the Western Carpathians. 46 sampling sites were distributed throughout the study area. At each site we sampled 1-3 coniferous trees: Norway spruces (Picea abies Karst.) and/or silver firs (Abies alba Mill.). From each tree 2 cores were sampled: one from the upslope and the other from the downslope side of the stem. Based on tree-ring widths measured for opposite sides of stems we have calculated eccentricity index values and dated past landslide events. Mean frequency of landslides was obtained for each sampling site. Finally, the data was interpolated into a map of landslide activity. Inverse Distance Weighting (IDW) interpolation has been applied. For most of the study area we found medium (19 sites) and low (23 sites) levels of landslide activity. The highest level of activity was recorded for the largest landslide slope and for the one small landslide. The study conducted on Sucha Mountain has shown that dendrochronology can be an effective method for analysing landslide activity and may be useful in further studies, including those for landslide hazard and risk assessments.

A HANDY CALCULATING METHOD OF HIGGS POTENTIAL IN SUSY MODEL

1993 ◽  
Vol 08 (29) ◽  
pp. 2729-2736 ◽  
Author(s):  
MASAKO BANDO ◽  
NOBUHIRO MAEKAWA ◽  
HIROAKI NAKANO ◽  
JOE SATO
Keyword(s):  
Standard Model ◽  
Supersymmetric Standard Model ◽  
Simple Approximation ◽  
Higgs Potential ◽  
Calculating Method ◽  
The One ◽  
Susy Model

We examine a simple approximation to the one-loop Higgs potential in the supersymmetric standard model. We study the conditions under which our approximation can well simulate the full one-loop results. The results by our method are compared with those by the full one-loop potential as well as the tree one in several examples. It turns out that our method works quite well in many cases and thus provides a useful calculating method, instead of a complicated evaluation of one-loop potential.

scholarly journals A simple approximation of the one-loop corrected cross section for e+e− → W+W− at LEP 2

1997 ◽  
Vol 409 (1-4) ◽  
pp. 405-411 ◽  
Author(s):  
M. Kuroda ◽  
I. Kuss ◽  
D. Schildknecht
Keyword(s):  
Cross Section ◽  
Simple Approximation ◽  
The One

scholarly journals Distribusi Spasial Total Padatan Tersuspensi Puncak Musim Hujan Di Permukaan Perairan Teluk Benoa, Bali

2017 ◽  
Vol 3 (2) ◽  
pp. 223 ◽  
Author(s):  
I Gede Surya Risuana ◽  
I Gede Hendrawan ◽  
Yulianto Suteja
Keyword(s):  
Spatial Distribution ◽  
Interpolation Method ◽  
Total Suspended Solid ◽  
Suspended Solid ◽  
Mouth Area ◽  
High Concentration ◽  
Distance Weighted ◽  
Enclosed Bay ◽  
The One ◽  
Inverse Distance

Benoa Bay is semi- enclosed bay, which has embouchures. The existence of embouchures are able to contribute the Total Suspended Solid (TSS) to the Bay. As the one of pollution parameter, TSS could be influencing the ecosystem of coral reefs, seagrass, and the existence of dissolved oxygen. This research was aimed to know the spatial distribution of TSS on the surface layer in Benoa Bay. Sample was taken on January 2016, in flood and ebb. TSS was analyzed using gavimetry method based on SNI No. 06-6989.3-2004. Spatial distribution was analyzed by interpolation method- Inverse Distance Weighted (IDW). The concentration range of TSS on January in ebb to tide and tide to ebb was 22.44 mg/L to 261.33 mg/L. The highest distribution in flood was dominated in around of embouchures. It was suspected by inputs of suspended matter from land to the bay. In tide to ebb condition, TSS concentration was high in some areas, such as around the embouchures, midlle and mouth area of bay. The high concentration was suspected by pattern of tidal current in Benoa Bay. The pattern of current in ebb was leaving to the mouth of Benoa Bay, therefore the TSS concentration dispersed to the mouth of the bay.

Molecular Orbitals Calculation on LiH with Algebraic Treatment of the Integrals

1997 ◽  
Vol 08 (05) ◽  
pp. 1159-1168 ◽  
Author(s):  
A. F. M. Righi ◽  
C. A. Kuhnen
Keyword(s):  
Ground State ◽  
Variational Calculation ◽  
Molecular Orbitals ◽  
Computational Method ◽  
Slater Type Orbitals ◽  
Atomic Orbitals ◽  
Algebraic Treatment ◽  
Nuclear Separation ◽  
The One ◽  
Coulomb Integrals

In this work we employ an algebraic computational method to solve the integrals which arise in the study of diatomic molecules. Using the Slater-type orbitals (STO), we obtain analytical solutions for the one-center and two-center hybrid and Coulomb integrals. The exchange integrals are considered, as much as possible, in a similar manner. These results are used to calculate the electronic properties of the ground state of the LiH molecule, by means of a variational calculation using a basis of molecular orbitals. The behavior of the Slater exponents of the atomic orbitals are studied as functions of internuclear distance. We show that the STO undergo a lengthening of their shape when the nuclear separation increases.

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