Normalized spherical harmonics

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

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... 

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. 

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

Thus, we obtain the following normalized spherical harmonics functions ... 

The following list provides some explicit formulae of the (normalized) spherical harmonics for the small angular momentum quantum numbers. 

Using Eqs. (4-51) and (4-43), write the normalized spherical harmonic function 73, 2(6>, 4>)- For which type of hydrogenlike AO does this function give the angular dependence ... 

The Wigner rotation matrix elements are related to the modified (or normalized) spherical harmonics by... 

One can go further by using the following expansion in terms of regular / /, and irregular Ii m normalized spherical harmonics [74—76] ... 

Table 17.1 gives the normalized spherical harmonic functions for / = 0, / = 1, and I = 2. Additional functions can be derived from formulas in Appendix F. 

Table 17.1 Normalized Spherical Harmonic Functions V/m(0, functions, eigenfunctions of L. ...

Oa is the product wavefunction for the combined (generally nu-rovibronic) internal states of the collision partners (i.e., a) = (vA. 7a. wa ) vb. ye. wibI). using various collective quantum numbers such as the vibrational quantum number va in a formal notation), Yt,mt is the normalized spherical harmonic for angular momentum quantum numbers i and me of relative rotation of the collision partners... 

The explicit expressions for the normalized associated Legendre polynomials are given in Table 41 for Z = 0, 1, 2, 3. The normalized spherical harmonics [Pg.58]

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