• Journal of the Korean Chemical Society
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• v.23 no.3
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• pp.125-131
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• 1979

A method for calculation of two center overlap integrals for a pair of Slater type orbitals was developed by Mulliken et al. In this method the spherical polar coordinates for a pair of Slater type orbitals located at two different points are required to be transformed into a spheroidal coordinate set for calculation of two center overlap integrals. A new method, the expansion method for spherical harmonics, in which Slater type orbitals, located at two different points, are expressed in a common coordinate system has been applied for computation of two center overlap integrals. The new method for computation of two center overlap integrals is required to translate Slater type orbitals centered at two different points into the reference point for computation of two center overlap integrals. This work has been expanded the expansion method for spherical harmonics for computation of two center overlap integrals to $|3s{\g}$, $|5s{\g}$ and $|5s{\g}$. Master formulas for two center overlap integrals are derived for these orbitals, using the general expansion formulas. The numerical values of the two center overlap integrals evaluated for a hypothetical NO molecule are in agreement with those of the previous works.

• Nuclear Engineering and Technology
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• v.53 no.12
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• pp.3861-3878
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• 2021

The steady-state simplified spherical harmonics equations (SP_N equations) are a higher order approximation to the neutron transport equations than the neutron diffusion equation that also have reasonable computational demands. This work extends these results for the analysis of transients by comparing of two formulations of time-dependent SP_N equations considering different treatments for the time derivatives of the field moments. The first is the full system of equations and the second is a diffusive approximation of these equations that neglects the time derivatives of the odd moments. The spatial discretization of these methodologies is made by using a high order finite element method. For the time discretization, a semi-implicit Euler method is used. Numerical results show that the diffusive formulation for the time-dependent simplified spherical harmonics equations does not present a relevant loss of accuracy while being more computationally efficient than the full system.

• Journal of the Korean Chemical Society
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• v.22 no.3
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• pp.117-127
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• 1978

Slater type orbitals, located at two different points A and B, are expressed in a common coordinate system by expanding the spherical harmonics and the radial part of these orbitals in terms of the reference point A. Master formulas for two center overlap integrals are derived, using the general expansion formulas of slater type atomic orbitals. Two center overlap integrals for $CH_4,\;H_2O,\;NH_3,\;C_2H_6\;and\;PH_3$molecules are evaluated, using master formulas for two center overlap integrals. The results are in agreement with those of two center overlap integrals of Mulliken.

• Nuclear Engineering and Technology
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• v.54 no.9
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• pp.3478-3487
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• 2022

In this study, we incorporate an anisotropic scattering scheme involving spherical harmonics into the method of characteristics (MOC). The neutron transport solution in a light water reactor can be significantly improved because of the impact of an anisotropic scattering source with the MOC flat source approximation. Several problems are selected to verify the proposed scheme and investigate its effects and accuracy. The MOC anisotropic scattering source is based on the expansion of spherical harmonics with Legendre polynomial functions. The angular flux, scattering source, and cross section are expanded in terms of the surface spherical harmonics. Later, the polynomial is expanded to achieve the odd and even parity of the source components. Ultimately, the MOC angular and scalar fluxes are calculated from a combination of two sources. This paper presents various numerical examples that represent the hot and cold conditions of a reactor core with boron concentration, burnable absorbers, and control rod materials, with and without a reflector or baffle. Moreover, a small critical core problem is considered which involves significant neutron leakage at room temperature. We demonstrate that an anisotropic scattering source significantly improves solution accuracy for the small core high-leakage problem, as well as for practical large core analyses.

• Bulletin of the Korean Chemical Society
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• v.7 no.4
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• pp.299-304
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• 1986

Master formulas for overlap integrals and the dipole moments involving |4p > atomic orbitals have been derived by the expansion method for spherical harmonics. The radial integrals for the hyperfine interaction have also been derived for |4p > atomic orbitals. The calculated values of the overlap integrals and dipole moment matrix elements by the expansion method for spherical harmonics for a hypothetical NO molecule are exactly in agreement with those of Mulliken's method. The radial integrals for the hyperfine interaction may be used to calculate the chemical shift for |4p > atomic orbitals.

• Journal of the Korean Chemical Society
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• v.22 no.6
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• pp.357-364
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• 1978

The dipole moments for $NH_3$, HF, CO, HCHO, HCN, PO, $PO^-\;and\;H_2O$ molecules are calculated, using the method for evaluation of the dipole moment matrix elements by the expansion method for spherical harmonics. The calculated dipole moments in this work are closer to the experimental values than those of the other work.

• Journal of the Korean earth science society
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• v.34 no.5
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• pp.393-401
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• 2013

Spherical harmonics power spectrum of the geopotential field of Gaussian-bell type on the sphere was investigated using integral formula that is associated with Legendre polynomials. The geopotential field of Gaussian-bell type is defined as a function of sine of angular distance from the bell's center in order to guarantee the continuity on the global domain. Since the integral-formula associated with the Legendre polynomials was represented with infinite series of polynomial, an estimation method was developed to make the procedure computationally efficient while preserving the accuracy. The spherical harmonics power spectrum was shown to vary significantly depending on the scale parameter of the Gaussian bell. Due to the accurate procedure of the new method, the power (degree variance) spanning over orders that were far higher than machine roundoff was well explored. When the scale parameter (or width) of the Gaussian bell is large, the spectrum drops sharply with the total wavenumber. On the other hand, in case of small scale parameter the spectrum tends to be flat, showing very slow decaying with the total wavenumber. The accuracy of the new method was compared with theoretical values for various scale parameters. The new method was found advantageous over discrete numerical methods, such as Gaussian quadrature and Fourier method, in that it can produce the power spectrum with accuracy and computational efficiency for all range of total wavenumber. The results of present study help to determine the allowable maximum scale parameter of the geopotential field when a Gaussian-bell type is adopted as a localized function.

• The Journal of Korean Institute of Electromagnetic Engineering and Science
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• v.26 no.1
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• pp.88-93
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• 2015

To estimate RCS(Radar Cross Section) at arbitrary incident angles for large objects, an interpolation method is required based on the pre-calculated RCS database at finite discrete sampling points. It is numerically difficult to compute the RCS by a large object at all required sampling points, since the computation time may be very long for one sampling point and many sampling points are required to satisfy the exact sampling condition. Therefore, it may be required to accurately estimate the RCS at any incident angles based on a database whose size is as small as possible. In this paper, the accuracy of two interpolation methods base on the sinc-and VSH(Vector Spherical Harmonics) functions are numerically compared.

• Journal of the Korean Chemical Society
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• v.22 no.4
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• pp.229-238
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• 1978

Two methods for evaluation of the dipole moment matrix elements are developed, one using the expansion method for spherical harmonics and the other the transformation method of the dipole moment matrix elements into overlap integrals for Mulliken. The numerical values of the dipole moment matrix elements evaluated by two methods are in agreement with each other.

• Investigative Magnetic Resonance Imaging
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• v.11 no.2
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• pp.95-102
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• 2007

Purpose: To acquire high-resolution spiral-scan images at higher magnetic field, high homogeneous magnetic field is needed. Field inhomogeneity mapping and in-vivo shimming are important for rapid imaging such as spiral-scan imaging. The rapid scanning sequences are very susceptible to inhomogeneity. In this paper, we proposed a higher-order shimming method to obtain homogeneous magnetic field. Materials and Methods: To reduce measurement time for field inhomogeneity mapping, simultaneous axial/ sagittal, and coronal acquisitions are done using multi-slice based Fast Spin echo sequence. Acquired field inhomogeneity map is analyzed using the spherical harmonic functions, and shim currents are obtained by the multiplication of the pseudo-inverse of the field pattern with the inhomogeneity map. Results: Since the field inhomogeneity is increasing in proportion to the magnetic field, higher order shimming to reduce the inhomogeneity becomes more important in high field imaging. The shimming technique in which axial, sagittal, and coronal section inhomogeneity maps are obtained in one scan is developed, and the shimming method based on the analysis of spherical harmonics of the imhomogenity map is applied. The proposed technique is applicable to a localized shimming as well. High resolution spiral-scan imaging was successfully obtained with the proposed higher order shimming. Conclusion: Proposed pulse sequence for rapid measurement of inhomogeneity map and higher order shimming based on the inhomogeneity map work very well at 3 Tesla MRI system. With the proposed higher order shimming and localized higher order shimming techniques, high resolution spiral-scan images are successfully obtained at 3 T MRI system.