Gaussian‐Transform Method for Molecular Integrals. I. Formulation for Energy Integrals

1965 ◽  
Vol 43 (2) ◽  
pp. 398-414 ◽  
Author(s):  
I. Shavitt ◽  
M. Karplus
Keyword(s):  
Transform Method ◽  
Energy Integrals ◽  
Molecular Integrals ◽  
Gaussian Transform

scholarly journals Computing Four-Center Two-Electron Coulomb Integrals Using Exponential Transformations and Trapezoidal Rule

2020 ◽  
Vol 226 ◽  
pp. 01009
Author(s):  
Jordan Lovrod ◽  
Hassan Safouhi
Keyword(s):  
Convergence Properties ◽  
Molecular Electronic ◽  
Molecular Physics ◽  
Transform Method ◽  
New Approach ◽  
Double Exponential ◽  
Molecular Integrals ◽  
Molecular Electronic Structure ◽  
Coulomb Integrals ◽  
Structure Calculations

The numerical evaluations of the four-center two-electron Coulomb integrals are among the most time-consuming computations involved in molecular electronic structure calculations. In the present paper we extend the double exponential (DE) transform method, previously developed for the numerical evaluation of threecenter one-electron molecular integrals [J. Lovrod, H. Safouhi, Molecular Physics (2019) DOI:10.1030/0026867.2019.1619854], to four-center two-electron integrals. The fast convergence properties analyzed in the numerical section illustrate the advantages of the new approach.

ANALYTICAL TREATMENT OF NUCLEAR MAGNETIC SHIELDING TENSOR INTEGRALS OVER EXPONENTIAL-TYPE FUNCTIONS

2008 ◽  
Vol 07 (06) ◽  
pp. 1215-1225 ◽  
Author(s):  
LILIAN BERLU ◽  
HASSAN SAFOUHI
Keyword(s):  
Second Order ◽  
Basis Set ◽  
Magnetic Shielding ◽  
Analytical Treatment ◽  
Transform Method ◽  
Analytic Expressions ◽  
Molecular Integrals ◽  
Shielding Tensor ◽  
The Fourier Transform ◽  
Nuclear Magnetic

The present work concerns the analytical and numerical development of three-center molecular integrals over Slater-type functions (STFs) and B functions of the second order involving [Formula: see text] in the operator. These integrals appear in the analytic expression of the nuclear magnetic shielding tensor. The basis set of STFs is used to represent atomic orbitals. These STFs are expressed in terms of B functions, which are better suited to apply the Fourier transform method thoroughly developed by Steinborn group. Analytic expressions are obtained for the integrals of the second order involved in nuclear magnetic resonance shielding tensor over B functions. These expressions turned out to be similar to those obtained for the usual molecular multi-center integrals. Consequently, the numerical evaluation of the integrals under consideration will benefit from the work previously done on the molecular multi-center integrals.

scholarly journals Molecular integrals over Laguerre Gaussian-type functions of real spherical harmonics

1992 ◽  
Vol 70 (2) ◽  
pp. 388-392 ◽  
Author(s):  
Osamu Matsuoka
Keyword(s):  
Spherical Harmonics ◽  
Distribution Models ◽  
Molecular Integral ◽  
Gaussian Type ◽  
Quantum Numbers ◽  
Integral Formulas ◽  
Energy Integrals ◽  
Molecular Integrals ◽  
Gaussian Type Function ◽  
Nuclear Attraction

Molecular integrals are formulated over the Laguerre Gaussian-type functions (LGTF) of real spherical harmonics. They include the overlap integrals and the energy integrals of kinetic, nuclear attraction, and electron repulsion. For the nuclear-attraction integrals the formulations based on the point as well as the Gaussian nuclear charge distribution models are presented. Integral formulas over the LGTFs of real spherical harmonics are found a little more complicated than those of the LGTFs of complex spherical harmonics due to the summations over magnetic quantum numbers. Keywords: molecular integral, Gaussian-type function, spherical harmonic, solid harmonic, Sonine polynomial.

Application of electron holography to material science studies on magnetic domain states of small particles

1993 ◽  
Vol 51 ◽  
pp. 1028-1029
Author(s):  
T. Hirayama ◽  
Q. Ru ◽  
T. Tanji ◽  
A. Tonomura
Keyword(s):  
Magnetic Field ◽  
Material Science ◽  
Science Studies ◽  
Phase Distribution ◽  
Carbon Film ◽  
Objective Lens ◽  
Electron Holography ◽  
Transform Method ◽  
Transmission Electron ◽  
Thin Carbon

The observation of small magnetic materials is one of the most important applications of electron holography to material science, because interferometry by means of electron holography can directly visualize magnetic flux lines in a very small area. To observe magnetic structures by transmission electron microscopy it is important to control the magnetic field applied to the specimen in order to prevent it from changing its magnetic state. The easiest method is tuming off the objective lens current and focusing with the first intermediate lens. The other method is using a low magnetic-field lens, where the specimen is set above the lens gap.Figure 1 shows an interference micrograph of an isolated particle of barium ferrite on a thin carbon film observed from approximately [111]. A hologram of this particle was recorded by the transmission electron microscope, Hitachi HF-2000, equipped with an electron biprism. The phase distribution of the object electron wave was reconstructed digitally by the Fourier transform method and converted to the interference micrograph Fig 1.

NUMERICAL AND STABILITY ANALYSIS OF THE TRANSMISSION DYNAMICS OF SVIR EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE

2019 ◽  
Vol 4 (2) ◽  
pp. 349 ◽  
Author(s):  
Oluwatayo Michael Ogunmiloro ◽  
Fatima Ohunene Abedo ◽  
Hammed Kareem
Keyword(s):  
Reproduction Number ◽  
Disease Spread ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Transform Method ◽  
Globally Asymptotically Stable ◽  
Differential Transform ◽  
Mathematical Techniques ◽  
Fourth Order Method ◽  
Theoretical Results

In this article, a Susceptible – Vaccinated – Infected – Recovered (SVIR) model is formulated and analysed using comprehensive mathematical techniques. The vaccination class is primarily considered as means of controlling the disease spread. The basic reproduction number (Ro) of the model is obtained, where it was shown that if Ro<1, at the model equilibrium solutions when infection is present and absent, the infection- free equilibrium is both locally and globally asymptotically stable. Also, if Ro>1, the endemic equilibrium solution is locally asymptotically stable. Furthermore, the analytical solution of the model was carried out using the Differential Transform Method (DTM) and Runge - Kutta fourth-order method. Numerical simulations were carried out to validate the theoretical results. 

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