Spherical harmonic functions normalized

We recall that in the multipolar expansion, the 3d density is expressed in terms of the density-normalized spherical harmonic functions dlmp as... 

Thus, we obtain the following normalized spherical harmonics functions ... 

Using Eqs. (4-51) and (4-43), write the normalized spherical harmonic function 73, 2(6>, 4>)- For which type of hydrogenlike AO does this function give the angular dependence ... 

Table 17.1 gives the normalized spherical harmonic functions for / = 0, / = 1, and I = 2. Additional functions can be derived from formulas in Appendix F. 

Table 17.1 Normalized Spherical Harmonic Functions V/m(0, functions, eigenfunctions of L. ...

A is a normalization constant and T/.m are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. 

When the size parameter x is sufficiently small, that is, when the particle is small compared with the wavelength of light, only the leading term in the normal mode expansion for the spherical harmonic functions is needed. In this case Eq. (76) reduces to Rayleigh s result, Eq. (47), for the ratio of the scattered irradiance to the incident irradiance. 

Atomic density functions are expressed in terms of the three polar coordinates r, 6, and multipole formalism, the density functions are products of r-dependent radial functions and 8- and -dependent angular functions. The angular functions are the real spherical harmonic functions ytm (8, ), but with a normalization suitable for density functions, further discussed below. The functions are well known as they describe the angular dependence of the hydrogenic s, p, d,f... orbitals. 

The functions ylmp are linear combinations of the complex spherical harmonic functions Ylm. Including normalization, the latter are defined as... 

FIG. 3.5 Definition of the normalization coefficients for the spherical harmonic functions. Relations such as yimp — Am d,mp are implied by the direction of the arrows. 

A more detailed discussion of the complex and real spherical harmonic functions, with explicit expressions and numerical values for the normalization factors, can be found in appendix D. 

D.1 Real Spherical Harmonic Functions and Associated Normalization Constants (x, y, and z are Direction Cosines)... 

Let f, P and f, P be (2/ + 1) x 1 matrices representing the density-function normalized spherical harmonics and their population parameters, before and after rotation, respectively. Then, by using Eq. (D.10), we construct a (21 + 1) x (21 + 1) matrix M such that... 

TABLE E.3 Products of Two Real Spherical Harmonic Functions ylmp, with Normalization Defined in Appendix D ... 

The parameters Pim , Pcore, and k can be refined within a least square procedure, together with positional and thermal parameters of a normal refinement to obtain a crystal structure. In the Hansen and Coppens model, the valence shell is allowed to contract or expand and to assume an aspherical form [last term in (11)], as it is conceivable when the atomic density is deformed by the chemical bonding. This is possible by refining the k and k radial scaling parameters and population coefficients Pim of the multipolar expansion. Spherical harmonics functions yim are used to describe the deformation part. Several software packages [68-71] are available for multipolar refinement of the electron density and some of them [68, 70, 72] also compute properties from the refined multipolar coefficients. 

Orbitals (GTO). Slater type orbitals have the functional form e, if) = NYi, d, e- -- (5.1) is a normalization constant and T are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. centre of a bond. 5.2 Classification of Basis Sets Having decided on the type of function (STO/GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smallest number of functions... 

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). 

A is a normalization constant and are the spherical harmonic functions. STOs lack radial nodes, which are introduced by making linear combinations of STOs. They are primarily used for high-accuracy atomic and diatomic calculations and with semiempirical methods, which neglect all three- and four-center integrals (which cannot... 

The transformations of the standard vector form the fundamental irrep of spherical symmetry. All other irreps can be constructed by taking direct products of this vector. In particular, the spherical harmonic functions can be constructed by taking fully symmetrized powers of the vector. The symmetrized direct square of the / -functions yields a six-dimensional function space with components z, x, yz, xz, xy. This space is not orthonormal the components are not normalized, and the first... 

The traditional labels for the spatial d orbitals are dp, d y, dxx, dy,, and dp-p.. As in the case of the labels p, py, and pj, the subscripts given to these d orbitals are such that the orbital has the same nodes and phases (signs) as the function in the subscript. Find a normalized expression for each of these in terms of the spherical harmonic functions YT B,4>) in Table 3.1. 

In order to further describe the molecular wavefunctions or the molecular orbitals. Linear Combinations of Atomic Orbitals (LCAO) are normally used (LCAO method). Such a method of solution is possible since the directional dependence of the spherical-harmonic functions for the atomic orbitals can be used. The Pauli principle can be applied to the single-electron molecular orbitals and by filling the states with the available electrons the molecular electron configurations are attained. Coupling of the angular momenta of the open shell then gives rise to molecular terms. 

The imaginary functions were normalized so that the integration over all space of the corresponding probability gives 1, but this will not be the case for the linear combinations in Equation (A9.78). So we have included normalization constants Nc and Ns which must be determined. This can be done quite neatly by making use of the orthogonality of the spherical harmonic functions. We require... 

These functions are called the normalized spherical harmonic and, for m > 0, are given by ... 

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