Complex spherical harmonic functions

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

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 ... 

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. 

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]

For the simplest case of a one-electron configuration dl the term functions are identical with the d-orbitals, and thus the formulae for the pertinent matrix elements listed in Table 8.10 are directly applicable. Then the 5 x 5 secular determinant is solved. For the case of an octahedral complex the matrix elements of the crystal field potential in the basis set of spherical harmonic functions Yi m form the matrix... 

If we choose the complex <5 functions, the Y functions are called spherical harmonic functions. 

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

Before we begin our discussion of Gaussian basis sets, let us briefly review the one-electron basis functions studied in Chapter 6. The complex spherical-harmonic GTOs are given by... 

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. 

Proposition 7.2 is crucial to our proof in Section 7.2 that the spherical harmonics span the complex scalar product space L (S ) of square-integrable functions on the two-sphere. 

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... 

The wave functions (6.8) are known as atomic orbitals, for / = 0, 1,2, 3, etc., they are referred to as s, p, d, f, respectively, with the value of n as a prefix, i.e. Is, 2s, 2p, 3s, 3p, 3d, etc., From the explicit forms ofthe wave functions we can calculate both the sizes and shapes of the atomic orbitals, important properties when we come to consider molecule formation and structure. It is instructive to examine the angular parts of the hydrogen atom functions (the spherical harmonics) in a polar plot but noting from (6.9) that these are complex functions, we prefer to describe the angular wave functions by real linear combinations of the complex functions, which are also acceptable solutions of the Schrodinger equation. This procedure may be illustrated by considering the 2p orbitals. From equations (6.8) and (6.9) the complex wave functions are... 

An alternative solution to the gauge-independence problem in molecular calculations is to attach the complex phase factors directly to the atomic basis functions or atomic orbitals (AOs) rather than to the MOs. Thus, each basis function—which in modern calculations usually corresponds to a Gaussian-type orbital (GTO)—is equipped with a complex phase factor according to Eq. 87. A spherical-harmonic GTO may then be written in the from... 

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. 

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