Function spherical harmonic

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

1 Symbol v Angular function, Ctop Normalization for Wave Functions, Mlmc Normalization for Density Functions, Llmd  

Wave Functions, Mlmc Density Functions, Llmy 

Symmetry Choice of Coordinate Axes Indices of Allowed dlmp 

4 Transformation of Real Spherical Harmonic Density Functions on Rotation of the Coordinate System 

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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. 

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. 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

The solutions of the inner and outer fields can now be written as expansions in these spherical harmonic functions or vector eigenfunctions, once the incident irradiation and the boundary conditions are specified. 

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. 

For sites of cubic symmetry the point-group symmetry elements mix the spherical harmonic basis functions. As a result, linear combinations of spherical harmonic functions, referred to as Kubic harmonics (Von der Lage and Bethe 1947), must be used. 

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. 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

Fourier transformation of the spherical harmonic functions is accomplished by expanding the plane wave exp(27r/ST) in terms of products of the spherical harmonic functions. In terms of the complex spherical harmonics Ylm 6, [Pg.68]

Combining terms with m — —l and m = l, gives the expansion in terms of the real spherical harmonic functions, which we will use to evaluate the Fourier transform of the real density functions ... 

Expressions (3.42) and (3.43) show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same /, in. This is summarized in the statement that the spherical harmonic functions are Fourier-transform invariant. It means, for example, that a dipolar density described by the function dl0, oriented along the c axis of a unit cell, will not contribute to the scattering of the (hkO) reflections, for which H is in the a b plane, which is a nodal plane of the function dU)((l, y). 

The Hirshfeld functions give an excellent fit to the density, as illustrated for tetrafluoroterephthalonitrile in chapter 5 (see Fig. 5.12). But, because they are less localized than the spherical harmonic functions, net atomic charges are less well defined. A comparison of the two formalisms has been made in the refinement of pyridinium dicyanomethylide (Baert et al. 1982). While both models fit the data equally well, the Hirshfeld model leads to a much larger value of the molecular dipole moment obtained by summation over the atomic functions using the equations described in chapter 7. The multipole results appear in better agreement with other experimental and theoretical values, which suggests that the latter are preferable when electrostatic properties are to be evaluated directly from the least-squares results. When the evaluation is based on the density predicted by the model, both formalisms should perform well. 

Not surprisingly, formalisms with very diffuse density functions tend to yield large electrostatic moments. This appears, in particular, to be true for the Hirshfeld formalism, in which each cos 1 term in the expansion (3.48) includes diffuse spherical harmonic functions with / = n, n — 2, n — 4,... (0, 1) with the radial factor rn. For instance when the refinement includes cos4 terms, monopoles and quadrupoles with radial functions containing a factor r4 are present. For pyridin-ium dicyanomethylide (Fig. 7.3), the dipole moment obtained with the coefficients from the Hirshfeld-type refinement is 62.7-10" 30 Cm (18.8 D), whereas the dipole moments from the spherical harmonic refinement, from integration in direct space, and the solution value (in dioxane), all cluster around 31 10 30 Cm (9.4 D) (Baert et al. 1982). 

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

The spherical harmonic functions constitute a complete set of functions in the spherical point group. A product of two spherical harmonics such as ytyj must therefore be a linear combination of spherical harmonic functions. An example of such an expression is... 

Symmetry restrictions for spherical harmonic functions are given in appendix D. 

The complex spherical harmonic functions, defined by Eq. (3.22), transform under rotation according to (Rose 1957, Arfken 1970)... 

E.1 Expressions for the Integrals over Products of Three Real Spherical Harmonic Functions... 

The integral over the product of three real spherical harmonic functions (Su 1993)... 

The integrals C can be expressed in terms of the integrals of the product of three complex spherical harmonic functions ... 

Expressions for the products of two spherical harmonic functions are given in Tables E.l and E.2. Multiplication of both sides of the expressions by a spherical harmonic function appearing on the right-hand side, and subsequent integration, leads to equations of the type of Eq. (E.l). Thus, coefficients in Tables E.l and... 

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