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 z². 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 x²-y². 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.

1. What are the shapes of a d orbitals?
2. For a given value of n, how many nodal surfaces are present?
3. What is(are) the shape(s) of the nodal surface(s) for the 3d_xz, 3d_yz, 3d_xy, and 3d_x²-y² orbitals?
4. What is(are) the shape(s) of the nodal surface(s) for the 4d_xz, 4d_yz, 4d_xy, and 4d_x²-y² orbitals?
5. What is(are) the shape(s) of the nodal surface(s) for the 3d_z² orbital?
6. What is(are) the shape(s) of the nodal surface(s) for the 4d_z² orbital?
7. Why does the program prevent you from using n = 1 and n = 2 in this exercise?

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