Solid Spherical Harmonic Function

Higher order oscillator orbitals, not used in the present work, can be generated in an analogous manner, using higher order solid spherical harmonic functions. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

The distribution of orientation of the structural units can be described by a function N(0, solid angle sin 0 d0 dtp d Jt. It is most appropriate to expand this distribution function in a series of generalised spherical harmonic functions. 

Evidently, correlation functions for different spherical harmonic functions of two different vectors in the same molecule are also orthogonal under equilibrium averaging for an isotropic fluid. Thus, if the excitation process photoselects particular Im components of the (solid) angular distribution of absorption dipoles, then only those same Im components of the (solid) angular distribution of emission dipoles will contribute to observed signal, regardless of the other Im components that may in principle be detected, and vice versa. The result in this case is likewise independent of the index n = N. Equation (4.7) is just the special case of Eq. (4.9) when the two dipoles coincide. 

Cockroft, J. K., Fitch, A. N., and Simon, A. Powder neutron diffraction studies of orientational order-disorder transitions in molecular and molecular-ionic solids use of symmetry-adapted spherical harmonic functions in the analysis of scattering density distributions arising from orientational disorder. In Collected Papers. Summerschool on Crystallography and its Teaching. Tianjin, China. Sept 15-24, 1988. (Ed., Miao, F.-M.) p. 427. Tianjin Tianjin Normal University (1988). 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

Even-tempered basis sets. - Large basis sets can be efficiently generated by utilizing the concept of an even-tempered basis set. Such a basis set consists of pure exponential or pure Gaussian functions multiplied by a solid spherical harmonic, that is a spherical harmonic multiplied by rl. Thus an even-tempered basis set consists of Is, 2p, 3d, 4/,. .. functions. A set of even-tempered basis functions is thus defined by... 

Point nuclear electric fields transform as the irregular solid spherical harmonics for which the Laplacian is a generalized delta function. For finite... 

The potential within the volume of the solute molecule is a smooth function and can be expanded in an appropriate set of expansion functions gifr). This is a generalization of the usual multipole expansion. The latter uses Cartesian monomials 1, x, y, z, xy, y, xz, yz, x, ..., or the corresponding solid spherical harmonics to expand the potential. However, for reasons explained in [17], the origin-centered multipole expansion is unsuitable for most systems. We experimented with several expansion sets. One important requirement is that the expansion functions should not diverge to infinity like the solid harmonics do. We finally settled on a sine function expansion of the potential. It is conveniently defined in an outer box that is larger than the extent of the molecular electronic density, to avoid problems with the periodic nature of the sine expansion. Other... 

The quantities r Yij 9,) and Yi,m(9,)/r + appearing in equations (67) and (68) are known as solid spherical harmonics. Because of their central role in multipole approximations, it is important to have optimized procedures for their generation and manipulation. An equivalent and efficient reformulation of the solid spherical harmonics has been published recently, with the interesting property of having very simple derivatives with respect to Cartesian coordinates, which are required for the computation of forces, or for obtaining useful recurrence relations for the integrals of solid spherical harmonic with Gaussian basis functions. 

Equations (4.329) for a solid assembly and (4.332) for a magnetic suspension are solved by expanding W with respect to the appropriate sets of functions. Convenient as such are the spherical harmonics defined by Eq. (4.318). In this context, the internal spherical harmonics used for solving Eq. (4.329) are written Xf (e, n). In the case of a magnetic fluid on this basis, a set of external harmonics is added, which are built on the angles of e with h as the polar axis. Application of a field couples [see the kinetic equation (4.332)] the internal and external degrees of freedom of the particle so that the dynamic variables become inseparable. With regard to this fact, the solution of equation (4.332) is constructed in the functional space that is a direct product of the internal and external harmonics ... 

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]... 

The extension of the ROZ formalism to confined molecular fluids has recently been carried out for adsorbed diatomic molecules [6] and dipolar fluids confined in hard sphere matrices [18, 19], In the case of ionic matrix, new features of the system have to be taken into account. On one hand, we have now a two component matrix (with positive and negative ions). This case was already considered in [14, 15] for the primitive model electrolyte adsorbed in an electroneutral charged matrix. On the other hand, we have to deal with two different temperatures the matrix temperature, (h (at which the ionic fluid is equilibrated before quenched) and the fluid temperature fi, at which the fluid is adsorbed in the solid matrix. As usual when dealing with molecular fluids one starts with an expansion of the correlation functions in terms of spherical harmonics as follows,... 

In the present context, their importance attaches to their role in describing the angular dependence of the hydrogenic wave functions. When we take up bonding in solids, we will see that the angular character implied by the spherical harmonics makes itself known in macroscopic observables as fundamental as those of the elastic moduli. We now take up the specific nature of the radial wave function. 

To relate Dff to the anisotropic motion of a molecule in a liquid crystalline solvent, we employ the function P(d, < ), defined as the probability per unit solid angle of a molecular orientation specified by the angles 6 and <3>, the polar coordinates of the applied magnetic field direction relative to a molecule-fixed Cartesian coordinate system. We expand P(0, ) in real spherical harmonics ... 

Sokolowski and Steele have adapted the spherical harmonic expansion technique described in Section III.A to the calculation of the density profiles of nonspherical molecules in contact with solid surfaces. They have used the results to investigate molecular orientation effects in high temperature adsorption from the gas phase. The RAM theory described in Section III.E has been extended to the adsorption of fluids on solid surfaces by Smith et al. ° for hard-sphere interactions and by Sokolowski and Steele to the case of more realistic fluid-solid interactions. The principal deficiency of the approach is the accuracy of the predicted correlation functions for the bulk fluid which are required as input to the theory. 

Here we use as basis functions the spherical harmonics on a real solid harmonic form. They are eigenfunctions of the orbital angular momentum operator,... 

The real standard basis functions are chosen as the usual orthonormal spherical harmonics 21). They will be denoted and in solid harmonic form ) they may be given by the analytical expressions... 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

The ab initio potentials used in solid nitrogen are from Refs. [31] and [32]. They have been respresented by a spherical expansion, Eq. (3), with coefficients up to = 6 and Lg = 6 inclusive, which describe the anisotropic short-range repulsion, the multipole-multipole interactions and the anisotropic dispersion interactions. They have also been fitted by a site-site model potential, Eq. (5), with force centers shifted away from the atoms, optimized for each interaction contribution. In the most advanced lattice dynamics model used, the TDH or RPA model, the libra-tions are expanded in spherical harmonics up to / = 12 and the translational vibrations in harmonic oscillator functions up to = 4, inclusive. 

In nmr spectroscopy the solute orientation is often described by an orientational probability function P(0,0) [45]. TO,0) is a measure for the probability (per unit solid angle) that the applied magnetic field (or the optic axis of the liquid crystal) assumes the spherical coordinates (dy 0) in the molecule-fixed coordinate system (x, y, z). P(0, 0) can be expressed in terms of spherical harmonics of second order as follows [45] ... 

A suitable order parameter r) should distinguish locally between the ordered (solid-like) and disordered (liquid-like) environment of an atom. Usually it is taken to be a function adapted to the symmetry of the solid phase, so that it takes nonzero values in the crystal-like configurations and zero in the liquid-like ones (ideal liquids are isotropic). One possible choice is a linear combination of spherical harmonics in a similar fashion as the popular Steinhardt order parameter Qi [26] defined in the Sect. 3.2.1 Angioletti-Uberti et al. instead chose polynomials adapted to the face-centered cubic (fee) symmetry of Ar, since they are cheaper to compute than spherical harmonics. Their polynomials have the form (using the same notation as in [27]) ... 

Nonsimple molecular fluids and solids can also be analyzed with density functional theory. Here, the trick is to. regard molecules as objects composed of atoms. (An alternative view in which molecules are made up of spherical harmonics is possible, but then the density fields involve orientational variables as well as points in space.) With the atomistic view of molecules, as we have illustrated in the first part of this lecture, one can use the site or atomic density fields, p (r), to describe configurational states. Furthermore, the following generalization of atomic density functional theory has been derived (Chandler et al., 1986a) ... 

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