Confusion about electron energy levels

[Yashbhatt]

If 4s orbital is in the 4th shell and 3d is in the 3rd shell and if the Principal Quantum Number is the direct indication of the amount of energy in of the electrons in that shell. Then, why according to Aufbau principle is 4s filled before 3d?

 

[dextercioby]

Because an atom's energy doesn't depend on one quantum number only. In other words, if you model the electronic configuration of the multi-electron atom/ion as to resemble the hydrogen's, then there are other effects to take into account (spin, special relativity, screening of the nuclear charge by the inner electrons). There's no 100% accurate theoretical description of the Aufbau Principle, but there are rigorous arguments in favor of it.
If all was simple, then 1s < 2s < 2p < 3s < 3p < 3d < 4s < 4p < 4d < 4f < 5s < 5p < 5d < 5f < 5g < 6s ... would be the normal energy diagram (as for the plain simple H-atom/hydrogenic ions in Schrödinger's description), but due to a myriad of effects (symmetry of the spin-spin couplings between electrons, special relativity), we have these inversions: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s ...

 

[Yashbhatt]

What are the arguments?

 

[Murtuza Tipu]

For seeing whether that which orbital has higher energy there a formula in which we have to calculate n+l where n is Principal quantum number and l is Azimuthal quantum number.
Orbital having more value of n+l has high energy.
If value of n+l is equal for two orbitals than orbital having higher principal quantum number has higher energy.

 

[Yashbhatt]

For seeing whether that which orbital has higher energy there a formula in which we have to calculate n+l where n is Principal quantum number and l is Azimuthal quantum number.
Orbital having more value of n+l has high energy.
If value of n+l is equal for two orbitals than orbital having higher principal quantum number has higher energy.

Okay. But that seems to be against the notion: The farther you pull an electron from the nucleus, the more potential energy it has.