Confusion about electron energy levels

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The discussion centers on the confusion regarding the filling order of electron orbitals, specifically why the 4s orbital is filled before the 3d orbital despite the principal quantum number suggesting otherwise. It emphasizes that an atom's energy levels are influenced by multiple factors, including electron spin, screening effects, and special relativity, which complicate the straightforward application of the Aufbau principle. The n+l rule is introduced as a method to determine the relative energy of orbitals, where a higher n+l value indicates higher energy, and if n+l values are equal, the orbital with the higher principal quantum number is favored. The conversation also touches on the contradiction between the expectation that electrons farther from the nucleus should have higher potential energy and the observed filling order. Overall, the complexities of electron configurations in multi-electron atoms challenge simplistic interpretations of energy levels.
Yashbhatt
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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?
 
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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 ...
 
What are the arguments?
 
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.
 
Murtuza Tipu said:
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.
 
Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?

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