Products Of Spherical Harmonics Research Articles

Computer modeling of hydrogen-like atoms in quantum mechanics with nonnegative distribution function

An algorithm for describing operator structures and calculating energy levels for hydrogen-like atoms in the quantum mechanics with nonnegative distribution function (Kuryshkin's quantum mechanics) is developed. The algorithm is implemented in Maple. An original library of necessary functions and quantities (Coulomb wave functions, Sturm functions, their Fourier images and kernels of integral transformations, Clebsch-Gordan coefficients, products of spherical harmonics, etc.) has been created. In accordance with the quantization rules in Kuryshkin's quantum mechanics, operators of observable quantities are computed. Based on the Hamiltonian obtained, energy levels of a hydrogen-like atom are calculated by the Ritz method (in the basis of Sturm functions). Numerical values of model parameters are determined by comparing results obtained with experimental data for hydrogen and alkali metals taken from the NIST Atomic Spectra Database Levels Data library.

New vibration–rotation code for tetraatomic molecules exhibiting wide-amplitude motion: WAVR4

A general computational method for the accurate calculation of rotationally and vibrationally excited states of tetraatomic molecules is developed. The resulting program is particularly appropriate for molecules executing wide-amplitude motions and isomerizations. The program offers a choice of coordinate systems based on Radau, Jacobi, diatom–diatom and orthogonal satellite vectors. The method includes all six vibrational dimensions plus three rotational dimensions. Vibration–rotation calculations with reduced dimensionality in the radial degrees of freedom are easily tackled via constraints imposed on the radial coordinates via the input file. Program summary Title of program: WAVR4 Catalogue number: ADUN Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUN Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: Persons requesting the program must sign the standard CPC nonprofit use license Computer: Developed under Tru64 UNIX, ported to Microsoft Windows and Sun Unix Operating systems under which the program has been tested: Tru64 Unix, Microsoft Windows, Sun Unix Programming language used: Fortran 90 Memory required to execute with typical data: case dependent No. of lines in distributed program, including test data, etc.: 11 937 No. of bytes in distributed program, including test data, etc.: 84 770 Distribution format: tar.gz Nature of physical problem: WAVR4 calculates the bound ro-vibrational levels and wavefunctions of a tetraatomic system using body-fixed coordinates based on generalised orthogonal vectors. Method of solution: The angular coordinates are treated using a finite basis representation (FBR) based on products of spherical harmonics. A discrete variable representation (DVR) [1] based on either Morse-oscillator-like or spherical-oscillator functions [2] is used for the radial coordinates. Matrix elements are computed using an efficient Gaussian quadrature in the angular coordinates and the DVR approximation in the radial coordinates. The solution of the secular problem is carried through a series of intermediate diagonalisations and truncations. Restrictions on the complexity of the problem: (1) The size of the final Hamiltonian matrix that can be practically diagonalised; (2) The DVR approximation for a radial coordinate fails for values of the coordinate near zero—this is remedied only for one radial coordinate by using analytical integration. Typical running time: problem-dependent Unusual features of the program: A user-supplied subroutine to evaluate the potential energy is a program requirement. External routines: BLAS and LAPACK are required.

Overlap integrals with respect to quantum numbers over Slater‐type orbitals via the Fourier‐transform method

AbstractA unified analytical treatment of overlap integrals over Slater‐type orbitals, using the Fourier transform method, is presented. Our approach leads to considerable simplification of the derivation of analytical representations for the overlap integrals over Slater type orbitals. In this approach, the representations of overlap integrals with both the same and different screening parameters have been obtained with the help of the Rayleigh expansion of a plane wave in terms of spherical Bessel functions and spherical harmonics. Analytical expressions obtained by this method have been expressed in terms of Gaunt coefficients, which are the product of spherical harmonics, Gegenbauer, and binomial coefficients. These overlap integrals have also been calculated with respect to the differences in n1 − l1 and n2 − l2 numbers that are even or odd. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Plane waves and radial polynomials: a new mixed basis

A new mixed basis for quantum-chemical calculations is presented. It consists of plane waves and functions strictly localized within disjoint spheres around the nuclei. The additional functions are products of spherical harmonics with radial polynomials defined using a radial grid. Nearly linear scaling is achieved by projecting the plane-wave components onto the localized basis in mixed products. The required one- and two-electron integrals are calculated, and an energy optimization scheme using preconditioned conjugate-gradient search is derived. Results of Hartree-Fock calculations on small and medium-sized molecules are presented. The total energy is shown to converge exponentially with respect to the cut-off parameters of the basis. A corresponding extrapolation yields total energies with micro-Hartree accuracy. The Hartree-Fock geometry of the water molecule is calculated to femtometre accuracy.

Ab initio Description of Nuclear Motion in Nonrigid M k XY n ( k ≥1) Molecules with Quasirigid XY n Fragments

For nonrigid MkXYn (k≥1) molecules, a model is constructed that describes the motion of k M nuclei relative to the quasirigid XYn fragment taking into account 3*k degrees of freedom. The parameters of the potential and kinetic terms of the model Hamiltonian are determined from results of ab initio calculations of the properties of a molecule and its fragments. Solutions of the corresponding Schrodinger equations are obtained by a variational method using bases constructed from products of spherical harmonics and harmonic–oscillator eigenfunctions. The form of model Hamiltonians for nonrigid MXY4 and M2XY4 molecules with quasitetrahedral XY4 fragments are discussed in detail. Group–theoretical analysis of the symmetry of the Hamiltonians is performed. It is shown that the molecular symmetry groups of nonrigid MXY4 and M2XY4 molecules are the G24 and G48 groups, which are isomorphic to the Td and Oh point groups, respectively.

Implementation of exact non-reflecting boundary conditions in the finite element method for the time-dependent wave equation

When solving the wave equation in infinite regions using finite element methods, the domain must be truncated. We investigate the accuracy of time-dependent non-reflecting boundary conditions (NRBC) derived in Grote, Keller (1995), when implemented in the finite element method. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary. High-order temporal derivatives are formulated as a system of first-order ordinary differential equations. Several versions of implicit and explicit multi-step, time-integration schemes are presented for solution of the finite element equations concurrently with the first-order system appearing in the NRBC. An alternative scaling of the boundary variables is introduced which leads to a well-conditioned coefficient matrix. Although the boundary conditions are global over the boundary, when implemented in the finite element method, they only require inner products of spherical harmonics within the force vector, and as a result, they are easy to implement and do not disturb the banded/sparse structure of the matrix equations. Several numerical examples are presented which demonstrate the improvement in accuracy over standard finite element methods.

Calculation of two-center two-electron integrals in the slaters,p,d-basis set for the Ono electron-electron potential

This paper deals with the problems of extending semiempirical MNDO methods to compounds with d-elements. The problem is solved by estimating two-center two-electron integrals (TTI) with the Ono potential modeling interaction between two electrons in a molecule. A scheme for calculating TTI is suggested which uses the expansion of the Ono potential in a series of pairwise products of spherical harmonics centered on two atoms in the molecule. The scheme is stable and efficient for calculations in arbitrary Slater basis sets (including the s,p,d-basis set) and seems to be useful for development of NDDO methods.

Spectral analysis using orthonormal functions with a case study on the sea surface topography

b Spherical harmonics are orthonormalized using the Gram-Schmidt process in a function space. The problem of linear dependence of spherical harmonics over the oceans is studied using the Gram matrices and consequently three sets of orthonormal (ON) functions have been constructed. For the process an efficient formula for computing inner products of spherical harmonics has been developed. Important spectral properties of the ON functions are addressed. The ON functions may be used for representing the sea surface topography (SST) in the analysis of satellite altimeter data. The geoid error can be transformed to a representation by the ON functions and hence the comparison of powers of the geoid error and the SST signal only over the oceans is possible, leading to a better way of determining the cut-off frequency of the SST in the simultaneous solution using satellite altimeter data. As a case study, the modified Levitus SST is expanded into the ON functions. The results show that 99.90 per cent of that signal's energy is contained within degree 24 of the orthonormal functions. Such expansions also render better spectral behaviour of oceanic signals as compared to that from spherical harmonic expansions. The study shows that these generalized Fourier functions are suitable for spectral analyses of oceanic signals and they can be applied to future altimetric mission where the geoid and the SST are to be recovered.

The torsional dependence of an interaction potential: the CH 3OHHe system

An interaction potential for the methanol-helium system was calculated, consisting of an SCF part plus a damped dispersion contribution. A double-zeta basis set was used for the methanol, augmented with one set of polarization functions on each atom. The He basis was a ( 6s 3s ) set augmented with two p-functions. The counterpoise method was used to help correct the basis set superposition error. Interaction energies were computed as a function of the relative position of the HE ( R, θ, φ) as well as of the methanol conformation, described by the single angle γ. The SCF potential was found to be relatively more repulsive at the methyl end of the molecule compared to the Billing site-site potential. A φ-shift approximation was found to predict fairly accurately the γ-dependence of the potential from the staggered conformation interaction alone. The anisotropy of the potential was analyzed using an expansion in products of spherical harmonics in θ, φ and exp(inγ) functions, in which it was shown that only n values which are multiples of three appear. The resulting expansion coefficients are those needed for the matrix elements of the potential between two rotational-torsional states in the close-coupling formation of the scattering problem. It was found that the torsional anisotropy is much weaker than the overall anisotropy and that the φ and γ anisotropies are strongly coupled.

A comparison between molecular-dynamics and theoretical results for the structure of fluids of hard ellipsoids

For fluids of hard ellipsoids of revolution the pair distribution function can be expressed as an expansion in products of spherical harmonics. In this paper molecular-dynamics results are given for eight coefficients in this expansion and detailed comparisons are made with the hypernetted-chain (HNC) and Percus-Yevick (PY) approximations. Ellipsoids with length-to-breadth ratios of ⅓, 2, 3 and 5 are considered, and in some cases results are reported for different densities. It is shown that both the HNC and PY theories are in reasonable agreement with molecular-dynamics results for prolate particles. The accuracy of the theories improves with decreasing density as expected, but does not depend strongly upon the length-to-breadth ratio for the prolate systems considered. For oblate ellipsoids the HNC theory remains in very good agreement with the molecular-dynamics results but the PY approximation is less accurate, tending to be rather poor for some coefficients.

Continuous-wave laser oscillation in a Nd:YAG sphere

I report the observation of continuous-wave laser oscillation in a Nd:YAG sphere. The spherical laser is excited by a dye laser that is focused to pump selectively certain regions of the sphere. With careful mode matching, single-frequency oscillation is obtained. The observed emission patterns agree well with the modes predicted by a classical treatment of a spherically symmetric resonator. These modes are described by products of spherical harmonics and Bessel functions, in contrast to the standard Gaussian-beam modes of a linear cavity.

On series expansions of orientation-dependent properties of pairs of non-spherical molecules

The accuracies of expansions of pair-wise molecular properties in angle-dependent functions are re-examined. The usual expansions in orthogonal polynomials such as products of spherical harmonics are those which, for a given truncation of the infinite series, minimize the mean square error. It is argued therefore, that the magnitude of the mean square deviations is a good criterion for determining the accuracy of this kind of expansion. Numerical illustrations are given for expansions of a site-site intermolecular interaction and for two functions of it: the Boltzmann factor and the product of the Boltzmann factor and the potential. As an alternative to the spherical harmonics, expansions based on products of Chebyshev polynomials are also considered. It is shown that coefficients in these expansions can be calculated either by minimizing the mean square error (with a different weight factor than for the spherical harmonics) or by utilizing them as interpolating polynomials. An alternative criterion for accuracy is also explored, which is to determine the absolute magnitude of the maximum difference between a given approximation and the exact function. The various Chebyshev expansions and the spherical harmonic expansion are examined in the light of both of the criteria suggested. It is concluded that the Chebyshev interpolating polynomials possess some attractive features which make them useful in particular applications.

Evaluation of a class of integrals by summing Legendre series. II. Integrands containing spherical harmonics

For pt. I See Abstr. A930 of 1971. It is shown how the integral integral f(r)YLM( theta , phi ) mod r-R mod -1 mod r-R' mod -1dr can be reduced to a single integral on the interval (0, 1) which is amenable to numerical computation. The analysis does not depend on the functional form of f(r). Integrals involving products of spherical harmonics are also discussed, in particular the case where the integrand contains a product of two spherical harmonics.

Volume integrals of the products of spherical harmonics and their application to viscous dissipation phenomena in fluids

The equations governing the conversion of kinetic energy into heat in moving viscous media are formulated as volume integrals of products of spherical harmonics. Although the formulation of the fundamental equations is classical, difficulties in the integration of certain products of generalized spherical harmonics over a sphere have permitted heretofore the treatment of only two cases. The closed, form evaluation of eight fundamental types of definite integrals of the product of spherical harmonics, some of them new, or at least missing in the literature, makes possible for the first time the evaluation of these volume integrals in closed form for arbitrary order and index. Explicit details are given for the rates of energy dissipation produced by viscous motions characterized by spheroidal as well as toroidal symmetry.

A Multi-dimensional Linear Growth Birth and Death Process

In [5] Karlin and McGregor analyzed a class of linear growth birth and death processes admitting a representation formula of the transition probability function in terms of classical orthogonal polynomials. Birth and death processes belong to the category of reversible stochastic processes (see [12] and [13]). In this case by invoking the spectral resolution of the identity for Hermitian operators one achieves the representation formula of Karlin and McGregor described in [3] and [4]. This theory is mostly restricted to the case of one-dimensional birth and death processes and diffusion processes. Some special higher dimensional birth and death processes motivated by certain applications in studies of population growth are reversible. For these processes there are representation formulae for the transition probability function that can be explicitly determined. In [9] and [10] Karlin and McGregor develop a discretized version of the classical technique of solving Laplace's equation in terms of spherical harmonics. This paper applies their method to determine the representation formula for the transition probability function of two two-dimensional (Sections 2 and 3) and the corresponding higher dimensional (Sections 4 and 5) linear growth birth and death processes. In each case the representation formula is expressed explicitly in terms of semi-direct products of classical orthogonal polynomials somewhat reminiscent of such products of spherical harmonics. In the Appendix the properties of the orthogonal polynomials are summarized and the method of derivation of the explicit form of the representation formula is outlined.

A Table for the Evaluation of Slater Coefficients and Integrals of Triple Products of Spherical Harmonics

A Table for the Evaluation of Slater Coefficients and Integrals of Triple Products of Spherical Harmonics