Question

Not considering the electronic spin, the degeneracy of the second excited state (n = 3) of H atom is 9, while the degeneracy of the second excited state of $H^{-}$ is:___

Answer 1

Degeneracy: Degeneracy is defined as the number of orbitals having the same energy.

For the mono electronic species: The penetration effect and shielding effect do not play any significant role in calculating the degeneracy.

Example: Hydrogen

Ground state: $1s$

First excited state: $2s,2p$

Second excited state: $3s,3p,3d$

Hence, the degeneracy in the second excited state $=9$

For the poly electronic species: The penetration effect and shielding effect play a significant role in calculating the degeneracy.

Example: $H^{-}$ is a 2 electron system.

Ground state: $1s^2$
First excited state: $1s^1,2s^1$
Second excited state: $1s^1,2p^1$
In the second excited state, the electron is in the $p$ orbital, it has 3 degenerate orbitals $3p_x,3p_y,3p_z$.
Therefore, the degeneracy of the second excited state is 3.