The Angular Wave Functions

Shapes of the three p orbitals. [Modified from http //en. wikipedia.org/wiki/Atomic orbital (accessed November 30, 2013).] 

Illustration of how a linear combination of the 2p, and 2p, orbitals can be used to construct the more familiar 2py orbital. [Images by Lisa M. Goss. Used by permission.] 

Normalization means that the integral of Y Y over all space must equal unity. Because each of the original wave functions Y( I, I) and Y( I,— I) are normalized and have integrals of one, the integral of the positive linear combination must equal two. Therefore, N = 2 in the normalization equation and the normalizing coefficient is c= l/2 l 

For normalization of the negative linear combination, the i factors out in the determination of the normalizing coefficient because of the complex conjugate in Equation (3.54), such that  

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The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3,...,oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is... 

Figure 1.8 Polar diagrams for Is, 2p and 3d atomic orbitals showing the distributions of the angular wave functions...

Because no symmetry operation can alter the value of R(n, r), we need not consider the radial wave functions any further. Symmetry operations do alter the angular wave functions, however, and so we shall now examine them in more detail. It should be noted that, since A(0, 0) does not depend on n, the angular wave functions for all s, all / , all d, and so on, orbitals of a given type are the same regardless of the principal quantum number of the shell to which they belong. Table 8.1 lists the angular wave functions for sy p, d, and / orbitals. 

The angular part of the wave function determines the shape of the electron cloud and varies depending upon the type of orbital involved (s. p, tl, or /) and its orientation in space. However, for a given type of orbital, such as s or />., the angular wave function is independent of the principal quantum number Or energy level. Some... 

The first Tour letters originate in spectroscopic notation (see page 26) and the remainder follow alphabetically. In the previous section we have seen the various angular wave functions and the resulting distribution or electrons. The nature or the angular wave Function is determined by the value or the quantum number i... 

Surfaces may be drawn to enclose the amplitude of the angular wave function. These boundary surfaces are the atomic orbitals, and lobes of each orbital have either positive or negative signs resulting as mathematical solutions to the Schrodinger wave equation. 

Theoretically, the radius of an ion extends from the nucleus to the outermost orbital occupied by electrons. The very nature of the angular wave function of an electron, which approaches zero asymptotically with increasing distance from the nucleus, indicates that an atom or ion has no definite size. Electron density maps compiled in X-ray determinations of crystal structures rarely show zero contours along a metal-anion bond. 

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... 

Figure 6-3. Representations of the hydrogen l v and 2pz orbitals (a) Plot of the angular wave function,. 4(0,

As mentioned before, the symmetry properties of the one-electron wave function are shown by the simple plot of the angular wave function. But, what are the symmetry properties of an orbital and how can they be described We can examine the behavior of an orbital under the different symmetry operations of a point group. This will be illustrated below via the inversion operation. 

Figure 6-4. Shapes of one-electron orbitals. They are representations of the angular wave function, A((-). ) (a) s,p, and d orbitals (b)/orbitals.

At this point we can, again, appreciate the possibility of separating the total wave function into a radial and an angular wave function. The angular wave function does not depend on n and r, so it will be the same for every atom. This is why the shapes of atomic orbitals are always the same. Hence, symmetry operations can be applied to the orbitals of all atoms in the same way. The differences occur in the radial part of the wave function the radial contribution depends on both n and r and it determines the energy of the orbital, which is, of course, different for different atoms. 

FIGURE 5.5 Two representations of hydrogen p orbitals, (a) The angular wave function for the orbital. The Px and Py orbitals are the same, but are oriented along the x- and y-axis, respectively, (b) The square of the angular wave function for the p orbital. Results for the p and Py orbitals are the same, but are oriented along the X- and y-axis, respectively. 

The angular wave functions Yn(0, (f>) for = 1, m = 1 and Yi i(0, (f>) for = 1, wr = -1 do not have a simple geometrical interpretation. However, their sum and their difference, which are also allowed solutions of the Schrodinger equation for the hydrogen atom, do have simple interpretations. Therefore, we form two new angular wave functions ... 

However, since the angular wave functions, the Y m(0,4>), are normalized, as defined in Equation 1.8, with... 

Note, the surface area of a spherical shell at radius r is 4nr-, but the normalization constant for the angular wave function reduces this to the factor in equation 1.14. 

The Is atomic orbital for the hydrogen atom results as an exact solution, for the choice of the first Laguerre polynomial (n = 1) for the radial wave function and the lowest spherical harmonic (/ = 0) Yqo, for the angular wave function. Thus, from Table 1.1, the normalized Is atomic orbital for the hydrogen atom is. 

Now allow for the integration over the angular wave function, for example, COS0, for the 2p. orbital, for which the normalization constant is (3/4 r) /, which requires changes in the master formula for the TABLE macros, with... 

The first step in the symmetry determination of the dynamic properties is the selection of the appropriate basis. Appropriate here means the correct representation of the changes in the properties examined. In the investigation of molecular vibrations (Chapter 5), either Cartesian displacement vectors or internal coordinate vectors are used. In the description of the molecular electronic structure (Chapter 6), the angular components of the atomic orbitals are frequently used bases. Since the angular wave function changes its sign under certain symmetry operations, its behavior will be characteristic of the spatial symmetry of a particular orbital. Molecular orbitals can also be used as basis of representation. The simple scheme below shows some important areas in chemistry where group theory is indispensable, and the most convenient basis functions are also indicated ... 

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