Spherical harmonics table

In spite of the apparent complexity, the functional form of the spherical harmonics (Table 3.2) is quite simple. The (0,0) harmonic is a constant. The (1,0), (1,1), and (1,-1) harmonics have a nodal plane each, the z = 0, x = 0, and y = 0 plane (the xy, yz, and xz plane) respectively (Fig. 3.2). A study of the harmonics for Z = 2 reveals that they have two nodal planes each. The higher harmonics for Z = 3 are a bit more complicated, but they have three nodal planes each. 

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

Cauchy function 276 Cauchy s ratio test 35-36 central forces 107,132-135 spherical harmonics 134-135 spherical polar coordinates 132-133 chain rule 37, 57, 160 character 153,195,197 orthogonality 197, 204 tables 198-200... 

Taking into account that Bq parameters represent the coefficient of an operator related to the spherical harmonic ykq then the ranges of k and q are limited to a maximum of 27 parameters (26 independent) Bq with k = 2,4,6 and q = 0,1,. .., k. The B°k values are real and the rest are complex. Due to the invariance of the CF Hamiltonian under the operations of the symmetry groups, the number of parameters is also limited by the point symmetry of the lanthanide site. Notice that for some groups, the number of parameters will depend on the choice of axes. In Table 2.1, the effect of site symmetry is illustrated for some common ion site symmetries. 

Figure 5.9 Least-square fit of the 206Pb/204Pb ratios listed in Table 5.9 on basalts from different oceanic islands with spherical harmonics to the degree 4. The results are reported as lines of constant values. Results in continental areas are not shown. 

Table 5.10. Spherical harmonic expansion of the data listed in Table 5.9.

Table I also contains an analysis of the orbital character of these five energy levels. These were determined from the four-component spinors by neglecting the two lower, "small," components, and by assuming that the radial functions depend only upon , i.e. that the radial functions for pi/2 and p3/2> or for da/2 and ds/2> are the same. The orbitals may then be written in "Pauli" form as products of (complex) spherical harmonics and spin functions. Populations are equal to the squares of the absolute magnitudes of the coefficients listed in Table I. [For all but 17e3g, an additional orbital (not shown) is occupied which has the same energy but the opposite spin pattern (i.e. a and 3 are interchanged).]...

Here, the permutations of j, k,l,... include all combinations which produce different terms. The multivariate Hermite polynomials are listed in Table 2.1 for orders < 6. Like the spherical harmonics, the Hermite polynomials form an orthogonal set of functions (Kendal and Stuart 1958, p. 156). 

The first terms of PJm cos (8) are listed in Table 3.3. The real spherical harmonics are given by... 

For a given value of n, the functions httk are identical to a sum of spherical harmonics with l = n, n — 2, n — 4,..., (0,1) for n > 1. The relationships are summarized in Table 3.8. For n = 0,1, the Hirshfeld functions are identical to the spherical harmonics with / = 0, 1, but, starting with the n = 2 functions, lower-order spherical harmonics are included for each n value. Unlike the spherical harmonics, the hnl functions are therefore not mutually orthogonal. As the radial functions in Eq. (3.48) contain the factor r", quite diffuse s, p, and d functions are included in the n = 2, 3, and 4 sets. For n <4 there are 35 deformation functions on each atom, compared with 25 valence-shell density functions with / < 4 in the multipole expansion of Eq. (3.35). 

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... 

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 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 angular functions 7<(0, ), called spherical harmonics, are common to all atoms. They are listed in Table 11-2.1 (in this table the normalizing constants have been omitted) together with the well-known symbols for the orbitals to which they correspond, i.e. s, pr, pr, p0, etc. The subscripts in these symbols are directly related to the angular functions if, for example, the angular function is ain 0 sin 2, then changing to the Cartesian coordinates x, y, and z where x r sin Q cos y = r sin 6 sin and z — r cos 0 gives us ... 

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... 

In either case it is helpful to have a table of transformation properties of spherical harmonics like that given in the Supplementary Notes. We shall illustrate the procedure by finding symmetry-adapted basis functions arising from an s function on each F atom in BF3, using full matrix projection operators. We already know that the three atomic basis functions transform as a 0 s, and since the behaviour with respect to the horizontal plane is already known, we can, without loss of generality, work with the subgroup C3u only. We denote the s functions on Fi, F2, and F3 as si, S2, and s3, respectively, and apply our C3u projection operators... 

A nondegenerate irrep that is symmetric with respect to the principl axis is denoted A, while B indicates antisymmetry with respect to this axis. In point groups with a horizontal plane of reflection, primes and " respectively indicate symmetry and antisymmetry with respect to the plane, while g and u indicate symmetry and antisymmetry with respect to inversion. For doubly degenerate irreps a subscript m indicates which spherical harmonics VJ, m form basis functions for that irrep. Numerical subscripts are used on nondegenerate irreps to distinguish them where necessary the numbers indicate the first of the vertical planes or perpendicular twofold axes (in the order specified in the character table) with respect to which the irrep is antisymmetric. 

We list here full matrix representations for several groups. Abelian groups are omitted, as their irreps are one-dimensional and hence all the necessary information is contained in the character table. We give C3v (isomorphic with D3) and C4u (isomorphic with D4 and D2d). By employing higher 1 value spherical harmonics as basis functions it is straightforward to extend these to Cnv for any n, even or odd. We note that the even n Cnv case has four nondegenerate irreps while the odd n Cnv case has only two. 

In Table 2.2 we have listed the first few spherical harmonics, for the s, p, and d states. It is worth noting that some authors introduce a factor of [—l]m in defining the associated Legendre polynomials, producing a corresponding difference in the spherical harmonics.2 There are i-m nodes in the 6 coordinate, and none in the <(> coordinate. 

An equivalent form is given by Englefield.11 It is possible to find quite a variety of phases for the transformation coefficients of Eq. (6.18).10-13 The phase depends on the phase conventions established for the spherical and parabolic states. The choice of phase in Eq. (6.18) is for spherical functions with an /, as opposed to (-r)e, dependence at the origin and the spherical harmonic functions of Bethe and Salpeter. A few examples of the spherical harmonics are given in Table 2.2. The parabolic functions are assumed to have an ( n) ml/2 behavior at the origin and an e m angular dependence. This convention means, for example, that for all Stark states with the quantum number m, the transformation coefficient (nni>i2m nmm) is positive. To the extent that the Stark effect is linear, i.e. to the extent that the wavefunctions are the zero field parabolic wavefunctions, the transformation of Eqs. (6.17) and (6.18) allows us to decompose a parabolic Stark state in a field into its zero field components, or vice versa. 

Fig. 3.14 Representation of the parfde-on-a-sphere wavefunctions (spherical harmonics) with 1=0,1, and 2, given in Table 3.1.

Table 2.1.1 lists the first few, and the most often encountered, spherical harmonics. 

Table 2.1.1. Explicit expressions for the spherical harmonics with i = 0,1,2,3...

Table 5.1. Explicit forms of the first few spherical harmonics Tfa (0,(p)...

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