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# Visualization of Atomic Orbitals

## d Orbitals

As the value of the angular momentum quantum number increases, the number of values of *m* increases (there are 2 *l* + 1 values of *m*) and the complexity of the orbital geometry increases. The d orbitals all possess *l* = 2. For this value of *l*, the magnetic quantum number may have values of *m* = -2, -1, 0, +1, and +2. Only orbitals with *m* = 0 are real; all other values of *m* give rise to complex wave functions. As is the case with p orbitals, chemists combine the *m* = -1 and +1 wave functions (which are complex) to obtain two new functions that are both real. Similarly, the wave functions with *m* = -2 and +2 are also combined to yield two new real wave functions.

The d orbital with *m* = 0 is designated z^{2}. The two orbitals created from the *m* = -1 and +1 orbitals are designated xz and yz. The two orbitals created from the *m* = -2 and +2 orbitals are designated xy and x^{2}-y^{2}. These designations arise from the mathematical formulas for the wave functions and indicate the orientation of the orbital.

### Exercise

Carefully examine the d orbitals for various values of *n* and the various orientations (d_{z2}, d_{xz}, d_{yz}, d_{xy}, d_{x2-y2} ) and answer the following questions.

- What are the shapes of a d orbitals?
- For a given value of
*n*, how many nodal surfaces are present?
- What is(are) the shape(s) of the nodal surface(s) for the 3d
_{xz}, 3d_{yz}, 3d_{xy}, and 3d_{x2-y2} orbitals?
- What is(are) the shape(s) of the nodal surface(s) for the 4d
_{xz}, 4d_{yz}, 4d_{xy}, and 4d_{x2-y2} orbitals?
- What is(are) the shape(s) of the nodal surface(s) for the 3d
_{z2} orbital?
- What is(are) the shape(s) of the nodal surface(s) for the 4d
_{z2} orbital?
- Why does the program prevent you from using
*n* = 1 and *n* = 2 in this exercise?

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