Spherical harmonics regular

Solid-harmonic solutions J1/ and N j of the homogeneous Helmholtz equation in cell pi are products of spherical Bessel functions and spherical harmonics. Specific functional forms for the regular and irregular solid harmonics, respectively, are [188]... 

As usual, there are the spherical harmonics or the complex conjugates (identified by the ). This expression can be inserted into the intermediate scattering function where one gets the following expression containing the regular Bessel functions ji ... 

The general spherical harmonics are familiar, in low order, as the mutually orthonormal angular components of valence atomic orbitals. Now, the sufficient number of these functions to provide basis functions for the regular representations of the molecular point groups, in... 

The expansion of the electrostatic potential into spherical harmonics is at the basis of the first quantum-continuum solvation methods (Rinaldi and Rivail, 1973 Tapia and Goschinski, 1975 Hylton McCreery et al., 1976). The starting points are the seminal Kirkwood s and Onsager s papers (Kirkwood 1934 Onsager 1936) the first one introducing the concept of cavity in the dielectric, and of the multipole expansion of the electrostatic potential in that spherical cavity, the second one the definition of the solvent reaction field and of its effect on a point dipole in a spherical cavity. The choice of this specific geometrical shape is not accidental, since multipole expansions work at their best for spherical cavities (and, with a little additional effort, for other regular shapes, such as ellipsoids or cylinders). 

The least-squares procedure used to evaluate the coefficients involves sampling r (r ) on a grid of points, typically a radial distribution based on that of Herman and Skillman (every tenth point) coupled with an angular mesh consisting of the 12 vertices of a regular icosahedron or 14 points defined by the corners and face centers of a cube. Both of these grids are accurate through fifth-order spherical harmonics. 

E. O. Steinbom, K. Ruedenberg, Rotation and Translation of Regular and Irregular Solid Spherical Harmonics, Adv. Quantum Chem. 7 (1973) 1-81. 

For reasons of notation we have included a phase factor i, and the spherical harmonics Y (r) have the phase defined by Condon and Shortley [5.3]. Inside the muffin-tin well the radial part p (E,r) must be regular at the... 

Sometimes it is useful to represent the multipole electrical moments in a spherical form. The spherical form of these moments allows us to apply effectively the theory of irreducible spherical tensor formalism. For this aim these 2 -pole moments may be written in terms of the regular spherical harmonics using their both complex Rlm r) and real (Rimc r) and Rims r)) forms defined, for m > 0, as... 

External electric fields transform as the regular solid spherical harmonics, (p), for which the Laplacian vanishes, V r Yemi, v) = 0. There is therefore no spin-free correction of 0(c ) to the property operator, and the operator can be written as... 

One can go further by using the following expansion in terms of regular / /, and irregular Ii m normalized spherical harmonics [74—76] ... 

EMTO s onto the spherical harmonics are substituted by the partial waves, mUj>rR) (thick black line in Fig. 1), defined as the regular solutions of the radial Schrodinger equation for potential vn rR) and energy tj... 

The angular dependence of the coefficients C R, < a. < b) can be expressed in a closed form. The relevant formulae are obtained by asymptotic expansion of the polarization series truncated at some finite order. In practice such an asymptotic expansion is best performed by evaluating the polarization energies (as given by equations 9, 18, and 21) using the multipole expansion of the electrostatic potential l/ ri — r2. The latter expansion can be written in terms of either the Cartesian or the spherical tensors. The spherical formulation appears to be more popular because it leads much more easily to closed formulae and only this formulation will be considered in this article. Denoting by (r) the regular solid harmonic r Cim(0,), where Cim 0,) is the spherical harmonic in the Racah normalization and with the Condon and Shortley phase, one can write ... 

In the DIRAC program, molecular spinors are expressed as a sum of regular spherical harmonics (Rif). [33] As an example the p functions are expressed as follows ... 

A binomial Taylor expansion of IR + r2 — and subsequent application of an addition theorem for regular spherical harmonics [82] factorise the electronic (rj, T2) and geometric (R) coordinates as follows ... 

The regular and radiating spherical vector wave functions can be expressed as integrals over vector spherical harmonics [26]... 

In this section we shall state some exact results for the spherically confined isotropic harmonic oscillator inside impenetrable walls. The eigenspectral regularities and the characterization of energy states in terms of the electron density and its derivatives at the equilibrium position will be considered. 

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