$l=2$ orbitals

The $l=2$ orbitals are known as $d$ (``$d$'' for ``diffuse'') orbitals. Again, we seek combinations of the spherical harmonics that give us real orbitals. The combinations we arrive at are known as $Y_{xy}$, $Y_{xz}$, $Y_{yz}$, $Y_{z^2}$, $Y_{x^2-y^2}$, which gives us the required 5 orbitals we need for $m=-2,-1,0,1,2$. The notation again reflects the angular dependence we would have if we took products $xy$, $xz$, $yz$, $z^2$, and $x^2-y^2$ using the spherical coordinate transformation equations.

Fig. 8) shows the 5 $d$ orbitals. Note the presence of two nodal planes in most of the orbitals. The exception is the $3d_{z^2}$ orbital which has a nodal cone, as shown in Fig. 8). The radial probabilities are shown in the last row of Fig. 5. Here, the number of radial nodes is still $n-l-1$, and, as noted earlier, the overall number of nodes remains the same, so as $l$ increaes, radial nodes are exchanged for angular nodes.

For any of the wave functions $\psi_{nlm}(r,\theta,\phi)$, the result of measuring the distance of the electron from the nucleus many times yields the average value or expectation value of $r$, which can be shown to be

$$\langle r \rangle = \int_0^{2\pi}\int_0^{\pi}\int_0^{\infty} \vert\psi_{nlm}(r,\theta,\phi)\vert^2 r^3 sin\theta dr d\theta d\phi$$

$$= {n^2 a_0 \over Z} \left[1 + {1 \over 2}\left(1-{l(l+1) \over n^2}\right)\right]$$