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 (dz2, dxz, dyz, dxy, dx2-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 3dxz, 3dyz, 3dxy, and 3dx2-y2 orbitals?
  4. What is(are) the shape(s) of the nodal surface(s) for the 4dxz, 4dyz, 4dxy, and 4dx2-y2 orbitals?
  5. What is(are) the shape(s) of the nodal surface(s) for the 3dz2 orbital?
  6. What is(are) the shape(s) of the nodal surface(s) for the 4dz2 orbital?
  7. Why does the program prevent you from using n = 1 and n = 2 in this exercise?

n:  
dz2
dxz   dyz
dxy   dx2-y2



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