d orbitals
For the l = 2 case, ml =0, ±1, ±2 and so there are five d-orbitals. The simplest is the 3dz2 wavefunction, whose angular dependence is given by
| Y2, 0 (θ φ) = | √ | 5 | (3 cos2θ - 1) |
| 16π |
This angular part is presented as a polar plot in the figure on the right.
For other values of ml we can take linear combinations like we did for the 2p functions. The customary linear combinations are:
| 3dxz = | 1 | (Y2, +1 + Y2, -1) = | √ | 15 | sinθ cosθ cosφ |
| √2 | 4π |
| 3dyz = | 1 | (Y2, +1 + Y2, -1) = | √ | 15 | sinθ cosθ sinφ |
| √2i | 4π |
| 3dx2 - y2 = | 1 | (Y2, +2 + Y2, -2) = | √ | 15 | sin2θ cos2φ |
| √2 | 16π |
| 3dxy2 - y2 = | 1 | (Y2, +2 + Y2, -2) = | √ | 15 | sin2θ sin2φ |
| √2i | 16π |
The angular parts of all five real d orbitals are also given on p.441 of Engel and Reid. Note that the last four orbitals differ only in their orientations, although they look quite distinct mathematically.
Other representations of the various real s, p, and d orbitals include probability densities proportional to the density of dots:
