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I know that the exact (OK most of the times not the exact) energy of an electron can be calculated by solving Schrodinger's differential equation, but can we explain an electron's energy of a specific orbital using the Penetration and Shielding effect?

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Can we explain an electron's energy of a specific orbital using the Penetration and Shielding effect?

Yes, to some extent it is possible. Look no further than Slater's rules which specifies the values of the effective nuclear charge $Z^*$, and the effective principal quantum number $n^*$ that can be used to replace the nuclear charge $Z$ and the principal quantum number $n$ in the hydrogen-like atom orbital energy expression. The energy of each and every orbital can be approximated then by $$ E = - \left( \frac{Z^*}{n^{*}} \right)^{2}. $$ But I would be careful calling this an explanation, since it is rather a rationalization. I wrote about the difference between those two here.

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