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 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.

Interactions of cell volume, membrane potential, and membrane transport parameters

1980 ◽  
Vol 238 (5) ◽  
pp. C196-C206 ◽  
Author(s):  
E. Jakobsson
Keyword(s):  
Membrane Potential ◽  
Cell Volume ◽  
Membrane Transport ◽  
Numerical Solutions ◽  
Transport Equations ◽  
Time Varying ◽  
Transport Parameters ◽  
Animal Cells ◽  
Analytic Expressions ◽  
Solute Species

Equations have been written and solved that describe for animal cells the relationships among membrane transport, cell volume, membrane potential, and distribution of permeant solute. The essential system consists of n + 2 equations, where n is the number of permeant solute species. The n of the equations are the n transport equations for the permeant species, one for each species. The other two equations are statements of 1) the condition for bulk electroneutrality inside the cell and 2) the condition for isotonicity between the interior and exterior of the cell. Numerical solutions have been obtained in both the steady-state and time-varying cases for transport equations that are physically and phenomenologically reasonable. In addition to numerical solutions analytic expressions are presented that show the ranges of membrane parameters essential for volume regulation; for values of membrane parameters beyond explicitly defined bounds, the equations do not have real, positive solutions for cell volume.

Transient Temperature Distributions in a Cylinder Heated by Microwaves

1996 ◽  
Vol 430 ◽  
Author(s):  
H. W. Jackson ◽  
M. Barmatz ◽  
P. Wagner
Keyword(s):  
Numerical Solutions ◽  
Single Mode ◽  
Dielectric Constants ◽  
Microwave Cavity ◽  
Transient Temperature ◽  
Hybrid Heating ◽  
Fine Mesh ◽  
Temperature Distributions ◽  
Analytic Expressions ◽  
Lossy Dielectric

AbstractTransient temperature distributions were calculated for a lossy dielectric cylinder coaxially aligned in a cylindrical microwave cavity excited in a single mode. Results were obtained for sample sizes that range from fibers to large cylinders. Realistic values for temperature dependent complex dielectric constants and thermophysical properties of the samples were used. Losses in cavity walls were taken into account as were realistic thermal emissivities at all surfaces. For a fine mesh of points in time, normal mode properties and microwave power absorption profiles were evaluated using analytic expressions. Those expressions correspond to exact solutions of Maxwell's equations within the framework of a cylindrical shell model. Heating produced by the microwave absorption was included in self-consistent numerical solutions of thermal equations. In this model, both direct microwave heating and radiant heating of the sample (hybrid heating) were studied by including a lossy dielectric tube surrounding the sample. Calculated results are discussed within the context of two parametric studies. One is concerned with relative merits of microwave and hybrid heating of fibers, rods, and larger cylinders. The other is concerned with thermal runaway.

Lubrication Analysis Between Piston and Cylinder in High Pressure Piston Pump Considering Circumferential Grooves and Viscosity Variation With Pressure

2008 ◽  
Author(s):  
Tae-Jo Park
Keyword(s):  
High Pressure ◽  
Pressure Distribution ◽  
Reynolds Equation ◽  
Numerical Solutions ◽  
Lateral Force ◽  
Analytic Expressions ◽  
Large Pressure Gradient ◽  
Viscosity Variation ◽  
Axially Moving ◽  
Tapered Piston

In this study, a theoretical study on the lubrication analysis of an axially moving tapered piston with circumferential grooves subjected to a large pressure gradient within cylindrical bore has been done. Taking into account viscosity variation with pressure, analytic expressions for the pressure distribution leakage are obtained solving modified one-dimensional Reynolds equation for the case that the cylinder and the multigrooved piston axis has uniform eccentricity. It is shown that the viscosity variation with pressure highly affect the pressure distribution, lateral force acting on piston and leakage flow rate. Numerical solutions of two-dimensional Reynolds equation are also presented to confirm the validity of the analytical results. It is recommended that the piston used in various piston pumps should be tapered toward its high pressure end.

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