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Radial Part (radial + part)
Selected AbstractsHalf-numerical evaluation of pseudopotential integralsJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 9 2006Roberto Flores-Moreno Abstract A half-numeric algorithm for the evaluation of effective core potential integrals over Cartesian Gaussian functions is described. Local and semilocal integrals are separated into two-dimensional angular and one-dimensional radial integrals. The angular integrals are evaluated analytically using a general approach that has no limitation for the l -quantum number. The radial integrals are calculated by an adaptive one-dimensional numerical quadrature. For the semilocal radial part a pretabulation scheme is used. This pretabulation simplifies the handling of radial integrals, makes their calculation much faster, and allows their easy reuse for different integrals within a given shell combination. The implementation of this new algorithm is described and its performance is analyzed. © 2006 Wiley Periodicals, Inc. J Comput Chem 27: 1009,1019, 2006 [source] An evaluation of the radial part of numerical integration commonly used in DFTJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 11 2004Aisha El-Sherbiny Abstract Recently, Gill and Chien introduced a new radial quadrature for multiexponential integrands (MultiExp grid) to deal with the radial part of the numerical integration. In this article, the MultiExp grid is studied and used to integrate the charge density. The MultiExp grid, along with an optimal pruning scheme, performed very well both in terms of accuracy and efficiency compared to other radial mappings commonly used in Density Functional Theory. © 2004 Wiley Periodicals, Inc. J Comput Chem 11: 1378,1384, 2004 [source] Molecular shapes from small-angle X-ray scattering: extension of the theory to higher scattering anglesACTA CRYSTALLOGRAPHICA SECTION A, Issue 2 2009V. L. Shneerson A low-resolution shape of a molecule in solution may be deduced from measured small-angle X-ray scattering I(q) data by exploiting a Hankel transform relation between the coefficients of a multipole expansion of the scattered amplitude and corresponding coefficients of the electron density. In the past, the radial part of the Hankel transform has been evaluated with the aid of a truncated series expansion of a spherical Bessel function. It is shown that series truncation may be avoided by analytically performing the radial integral over an entire Bessel function. The angular part of the integral involving a spherical harmonic kernel is performed by quadrature. Such a calculation also allows a convenient incorporation of a molecular hydration shell of constant density intermediate between that of the protein and the solvent. Within this framework, we determine the multipole coefficients of the shape function by optimization of the agreement with experimental data by simulated annealing. [source] Dependence of s -waves on continuous dimension: The quantum oscillator and free systemsFORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 12 2006K.B. Wolf Abstract Wavefunctions with rotational symmetry (i.e., zero angular momentum) in D dimensions, are called s -waves. In quantum quadratic systems (free particle, harmonic and repulsive oscillators), their radial parts obey Schrödinger equations with a fictitious centrifugal (for integer D , 4) or centripetal (for D = 2) potential. These Hamiltonians close into the three-dimensional Lorentz algebra so(2,1), whose exceptional interval corresponds to the critical range of continuous dimensions 0 < D < 4, where they exhibit a one-parameter family of self-adjoint extensions in ,2(,+). We study the characterization of these extensions in the harmonic oscillator through their spectra which , except for the Friedrichs extension , are not equally spaced, and we build their time evolution Green function. The oscillator is then contracted to the free particle in continuous- D dimensions, where the extension structure is mantained in the limit of continuous spectra. Finally, we compute the free time evolution of the expectation values of the Hamiltonian, dilatation generator, and square radius between three distinct sets of ,heat'-diffused localized eigenstates. This provides a simple group-theoretic description of the purported contraction/expansion of Gaussian-ring s -waves in D > 0 dimensions. [source] | ||||||||||