Complex spherical harmonics

Here are some normalized spherical harmonics (complex) ... 

The orbitals d. and dy can be expressed in terms of the complex forms di and whose angular parts are given by the spherical harmonics and respectively. The matrix of orbital angular momentum about the z axis in the complex basis is... 

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. 

This can be seen by expressing Ull2l>l3 in terms of complex spherical harmonics and effective Slater integrals Fk as in Eq. [63]70... 

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).]...

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. 

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

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. 

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]

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

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

For magnet configurations in which coils are coaxial and symmetric about the illustrated xy-plane, such as the magnet configurations in Figure 2A and C, the spherical harmonic expansion results in the elimination of all even order terms within the expansion. To further reduce computational complexity, the strategy employed here considers only one quarter of the magnet domain, and thus, the constraints in Equation (5) simplify to ... 

Since spherical harmonics are functions from the sphere to the complex numbers, it is not immediately obvious how to visualize them. One method is to draw the domain, marking the sphere with information about the value of the function at various points. See Figure 1.8. Another way to visualize spherical harmonics is to draw polar graphs of the Legendre functions. See Figure 1.9. Note that for any , m we have F ,m = , the Legendre function carries all the information about the magnitude of the spherical harmonic. 

For each nonnegative integer f, the space of spherical harmonics of degree f (see Dehnition 2.6) is the vector space for a representation of 50(3). These representations appear explicitly in our analysis of the hydrogen atom in Chapter 7. Recall the complex scalar product space L (S ) from Definition 3.3. 

In general, the Q m are complex quantities.1- The Y(m are spherical harmonics an asterisk designates the conjugate complex. Equation 2.41... 

Let us return then to the problem of low symmetry in transition metal complexes. The most direct and unassuming approach would be to write a symmetry-based expansion of the ligand field potential in terms of spherical harmonics. For a completely unsymmetrical molecule (C,) this would be written as... 

Note that the quantum number mi appears in the exponential function c"" in the spherical harmonics. The Yim functions, being complex, cannot be conveniently drawn in real space. However, we can linearly combine them to make... 

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