Term Symbol of Carbon & Configuration Explanation

[insanul]

Homework Statement

further physical chemistry questions

Relevant Equations

none

A ground state configuration table for carbon is shown below. I am wondering why although configuration 4 and 5 share the same ML and MS ,they belong to different energy levels 1D2 and 3P2. It is just randomly chosen or they do have some rules? What cause the difference of energy levels of No.3 and NO.4 when they have similar configuration?(just 2 electrons with opposite spins exchange their position)

 

[DrClaude]

Homework Statement: further physical chemistry questions
Homework Equations: none

A ground state configuration table for carbon is shown below. I am wondering why although configuration 4 and 5 share the same ML and MS ,they belong to different energy levels 1D2 and 3P2. It is just randomly chosen or they do have some rules? What cause the difference of energy levels of No.3 and NO.4 when they have similar configuration?(just 2 electrons with opposite spins exchange their position)

It is "random" in the sense that the process used here is one of accounting only. There is one $M_L = 1$ and $M_S = 0$ component in $^1$D and there is one in $^3$P, so that accounts for the two microstates that lead to $M_L = 1$, $M_S = 0$.

This doesn't mean that microstate 3 is the one leading to $^1$D and not $^3$P. Actually, this can't be the case because valid multi-electron states must have a definite parity with respect to the exchange of two electrons. So in reality, it is a linear combination of the microstates 3 and 4 that lead to $^1$D, and another to $^3$P. These linear combinations can have different energies (look up Fermi hole and Fermi heap) and therefore the terms have different energies.