Spherical harmonic normalization

Y"(0,4>) = spherical harmonic normalized by integration over sphere of unit radius... 

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. 

Since the spherical harmonics are normalized, the value of the double integral is unity. 

In Equation 1.3, the radial function Rnl (r) is defined by the quantum numbers n and l and the spherical harmonics YJ" depend on the quantum numbers l and W . When the spin of the electron is taken into account, the normalized antisymmetric function is written as a Slater determinant. The corresponding eigenvalues depend only on n and l of each single electron, which determine the electronic configuration of the system. 

Spherical harmonics closely resemble normal Fourier harmonics except that they are functions of both the latitude and the longitude instead of the linear abscissa on a standard axis. Bi-dimensional Fourier analysis on a plane exists but is inadequate since the most desirable property of the requested expansion is the orthogonality of its components upon integration over the surface of the Earth, assumed to be spherical for most practical purposes. 

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. 

Coefficients multiply a normalized radial functions (not shown), complex spherical harmonics Yj jjj, and spin functions as indicated. Values for the ligand are for a single atom. Coefficients smaller than 0.01 are not shown. 

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. 

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

For our purpose, it is preferable to express the right-hand side of this equation in terms of the spherical harmonic density functions. Use of the ratio of orbital- and density-function normalization factors gives the result... 

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 ... 

Equations (4.30) and (4.31) have been developed and dehned within a time-dependent framework. These equations are identical to Eqs. (35) and (32), respectively, of Ref. 80. They differ only in that a different, more appropriate, normalization has been used here for the continuum wavefunction and that the transition dipole moment function has not been expanded in terms of a spherical harmonic basis of angular functions. All the analysis given in Ref. 80 continues to be valid. In particular, the details of the angular distributions of the various differential cross sections and the relationships between the various possible integral and differential cross sections have been described in that paper. 

We start our derivation by writing down the explicit form of the vacuum asymptote of a tip wavefunction (as well as its vacuum continuation in the tip body). As we have explained in Section 5.3, for the simplicity of relevant mathematics, the rather complicated normalization constants of the spherical harmonics are absorbed in the expression of the sample wavefunction. Up to 1=2, we define the coefficients of the expansion by the following expression ... 

The general expression for the tunneling current can be obtained using the explicit forms of tunneling matrix elements listed in Table 3.2. To put the five d states on an equal footing, normalized spherical harmonics, as listed in Appendix A, are used. The wavefunctions and the tunneling matrix elements are listed in Table 5.1. 

Table 5.1. Wavefunctions and tunneling matrix elements for different d-type tip states. The tip is assumed to have an axial symmetry. For brevity, the common factor in the normalization constant of the spherical harmonics and a common factor (2TT VKm,) in the expressions for the tunneling matrices is omitted.

The spherical harmonics in real form have explicit nodal lines on the unit sphere. Morse and Feshbach (1953) have given a detailed description of those real spherical harmonics, and gave them special names. Here we list those real spherical harmonics in normalized form. In other words, we require... 

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. 

The eigenfunctions belonging to these energy levels are the spherical harmonics Yl5m(9,4>) which are normalized according to... 

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