Question

In a hydrogenic atom, s- and p-type orbitals of the same shell are degenerate. Explain why the 2s and 2p
orbitals would not be degenerate in a carbon atom, and justify their relative energy ordering. (Hint: consider
the radial distribution functions for these orbitals).

Answer 1

When we consider Hydrogenic atom we exclusively consider atoms as unielectron system which is not the real cas4e for most of the atoms as well as ions.

In case of Carbon a higher ℓ value implies a stronger centrifugal force, that pushes electrons away from the nucleus. This concept

of centrifugal force can be seen in the radial Schroedinger equation for the radial part R(r) of the wave function Ψ(r,θ,φ)=R(r)Y_ℓ,m(θ,φ).

The radial part experiences an additional ℓ dependent potential that pushes the electron away. The core electrons are contracted compared to the outer electrons because there are less inner electron-shells screening the nuclear attraction from the core shells than from the valence electrons. Since the inner electron shells are more contracted than the outer ones, the core repulsion is weakened whereas the effects due to the centrifugal force remain unchanged. The reduced core repulsion in turn stabilizes the states with lower angular momenta, i.e. lower ℓ values. So, 2s states are lower in energy than 2p states.