Products of Spherical Harmonic Functions

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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In the Arthurs-Dalgarno (31) space-fixed representation, a= j,i and (R,r) is the total angular momentum function, defined as an a pTop riate linear combination of products of spherical harmonic functions for the rotation of R and r (10). Alter-... 

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). 

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... 

The antisymmetrized state function for N electrons is the sum of products such as j/ i(1) 02(2). .. i7n N) and similar terms with the variables permuted between the same set of one-electron eigenfunctions i/ i Vb /VI- Thus each term contains the same product of spherical harmonics and the state therefore has parity... 

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]... 

In this paper we have taken the Green function for the Fues model potential from [8,9]. The angular part of G (ri, V2) is simply the product of spherical harmonics, while for the radial part we have taken an expansion in Sturm functions, which have only a discrete spectrum ... 

In the case of the spherical approximation applied to atoms, the three-dimensional overlap and energy integrals which must be evaluated separate very neatly into an integral involving a product of spherical harmonics and a radial integral involving the exponential function. Both types of integral are very well researched and the results are well known. 

It is possible to expand the total wave function as in Equation 1.36 using simple products of spherical harmonics. 

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. 

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]

Table E.3 lists the products of the real spherical harmonic functions in terms of the density-normalized spherical harmonic functions dlmp.

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

The 4>) function turns out to be an exponential and the ( ) function consists of Legendre polynomials. Their product () ( ) gives the spherical harmonic functions which Arfken writes as Y 6, ). Then, from Eq. 20.56,... 

While we have chosen to proceed here by reducing representations for the full group D3h, it would have been simpler to take advantage of the fact that D3h is the direct product of C3u and C where the plane in the latter is perpendicular to the principal axis of the former. The behaviour of any atomic basis functions with respect to the C3 subgroup is trivial to determine, and there are only two classes of non-trivial operations in C3v. In more general cases, it is often worthwhile to look for such simplifications. It is seldom useful, for instance, to employ the full character table for a group that contains the inversion, or a unique horizontal plane, since the symmetry with respect to these operations can be determined by inspection. With these observations and the transformation properties of spherical harmonics given in the Supplementary Notes, it should be possible to determine the symmetries spanned by sets of atomic basis functions for any molecular system. Finally, with access to the appropriate literature the labour can be eliminated entirely for some cases, since... 

The product of the ground-state wavefunction and the transition dipole function is also expanded in terms of spherical harmonics,... 

All of the information that was used in the argument to derive the >2/1 arrangement of nuclei in ethylene is contained in the molecular wave function and could have been identified directly had it been possible to solve the molecular wave equation. It may therefore be correct to argue [161, 163] that the ab initio methods of quantum chemistry can never produce molecular conformation, but not that the concept of molecular shape lies outside the realm of quantum theory. The crucial structure-generating information carried by orbital angular momentum must however, be taken into account. Any quantitative scheme that incorporates, not only the molecular Hamiltonian, but also the complex phase of the wave function, must produce a framework for the definition of three-dimensional molecular shape. The basis sets of ab initio theory, invariably constructed as products of radial wave functions and real spherical harmonics [194], take account of orbital shape, but not of angular momentum. 

For covalently bonded atoms the overlap density is effectively projected into the terms of the one-center expansion. Any attempt to refine on an overlap population leads to large correlations between p>arameters, except when the overlap population is related to the one-center terms through an LCAO expansion as discussed in the last section of this article. When the overlap population is very small, the atomic multipole description reduces to the d-orbital product formalism. The relation becomes evident when the products of the spherical harmonic d-orbital functions are written as linear combinations of spherical harmonics ( ). 

There are now two independent angular coordinates, 6 and cp (besides the fixed distance R to the centre of the sphere) and wavefunctions that can be expressed as products of 0(0) and (p). The wave equation can be split into two, one for each variable 6 and cp. Each one of the functions 0(0) and (0) is subject to boundary conditions, in a similar way to the motion on a circular ring. Accordingly, two quantum numbers arise. The complete solutions Q 6) (p) are known as spherical harmonic functions and the allowed energies are given by an expression that resembles Eq. (2.72) ... 

The analysis is for concentric confinement of the atom in a sphere of radius r = ro, at which the wave function must vanish. The complete wave functions are products of the radial function of Equation (38) and spherical harmonics of Equations (36) and (34) or spheroconal harmonics of Equation (57). The boundary condition depends only on the radial function, becoming... 

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