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More specifically, why does s-p energy separation increase:

  1. Across the period
  2. Down a group

I do know that penetration of electrons increase with lower angular momentum and lower principal quantum number. An increase in penetration means that it has a higher shielding ability.

If we have two elements in period 2 and period 3 respectively, than the $\ce{3s}$ orbital would be less effective at shielding than the $\ce{2s}$ orbital, and thus the $\ce{3p}$ orbital would be higher in energy than the $\ce{2p }$ orbital would be relative to their s-orbitals? Is this the right justification?

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