Chapter 5, Problem 50AP

(a) The nitrogen atom has one electron in each of the $2p_x$ , $2p_y$ , and $2p_z$ orbitals. By using the form of the angular wave functions, show that the total electron density, ${\psi }^2\left(2p_x\right)+{\psi }^2\left(2p_y\right)+{\psi }^2\left(2p_z\right)$ , is spherically symmetric (that is, it is independent of the angles $\theta$ and $\varphi$ ). The neon atom, which has two electrons in each 2p orbital, is also spherically symmetric.

(b) The same result as in part (a) applies to d orbitals, thus a filled or half-filled subshell of d orbitals is spherically symmetric. Identify the spherically symmetric atoms or ions among the following: ${\text{F}}^{-}$ , Na, Si, ${\text{S}}^{\text{2}-},{\text{Ar}}^{+}$ , Ni, Cu, Mo, Rh, Sb, W, Au.