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## Main Question or Discussion Point

Let's say I'm considering the [tex]3p^2[/tex] electrons. From the Pauli Exclusion Principle, we know that two electrons cannot have the same state, which in this case means m

What this means is that the following 6 terms must not be allowed:

[tex]m_{l1} \hspace{0.1 in} m_{l2} \hspace{0.1 in} m_{s1} \hspace{0.1 in} m_{s2}[/tex]

[tex]-1 \hspace{0.1 in} -1 \hspace{0.1 in} \downarrow \hspace{0.3 in} \downarrow[/tex]

[tex]-1 \hspace{0.1 in} -1 \hspace{0.1 in} \uparrow \hspace{0.3 in} \uparrow[/tex]

[tex]0 \hspace{0.4 in} 0 \hspace{0.3 in} \downarrow \hspace{0.3 in} \downarrow[/tex]

[tex]0 \hspace{0.4 in} 0 \hspace{0.3 in} \uparrow \hspace{0.3 in} \uparrow[/tex]

[tex]+1 \hspace{0.1 in} +1 \hspace{0.1 in} \downarrow \hspace{0.3 in} \downarrow[/tex]

[tex]+1 \hspace{0.1 in} +1 \hspace{0.1 in} \uparrow \hspace{0.3 in} \uparrow[/tex]

These correspond to [tex]M_L=\sum m_{li}

= -2, -2, 0, 0, 2, 2[/tex] and [tex]M_S = \sum m_{si} = -1, 1, -1, 1, -1, 1[/tex] respectively.

My question is this - how does this lead to the conclusion that the allowed terms are

Also, what leads us to disallow

_{l}and m_{s}cannot both be the same for each electron.What this means is that the following 6 terms must not be allowed:

[tex]m_{l1} \hspace{0.1 in} m_{l2} \hspace{0.1 in} m_{s1} \hspace{0.1 in} m_{s2}[/tex]

[tex]-1 \hspace{0.1 in} -1 \hspace{0.1 in} \downarrow \hspace{0.3 in} \downarrow[/tex]

[tex]-1 \hspace{0.1 in} -1 \hspace{0.1 in} \uparrow \hspace{0.3 in} \uparrow[/tex]

[tex]0 \hspace{0.4 in} 0 \hspace{0.3 in} \downarrow \hspace{0.3 in} \downarrow[/tex]

[tex]0 \hspace{0.4 in} 0 \hspace{0.3 in} \uparrow \hspace{0.3 in} \uparrow[/tex]

[tex]+1 \hspace{0.1 in} +1 \hspace{0.1 in} \downarrow \hspace{0.3 in} \downarrow[/tex]

[tex]+1 \hspace{0.1 in} +1 \hspace{0.1 in} \uparrow \hspace{0.3 in} \uparrow[/tex]

These correspond to [tex]M_L=\sum m_{li}

= -2, -2, 0, 0, 2, 2[/tex] and [tex]M_S = \sum m_{si} = -1, 1, -1, 1, -1, 1[/tex] respectively.

My question is this - how does this lead to the conclusion that the allowed terms are

^{1}S,^{1}D and^{3}P ? For example, there's a M_{L}= 0, M_{S}= -1 term in both^{3}S and^{3}P - why do we disallow one and not the other?Also, what leads us to disallow

^{1}P (for which M_{L}=-1, 0, 1 and M_{S}=0)? Surely the only way to have M_{S}= 0 is to have [tex]\downarrow_1 \hspace{0.1 in} \uparrow_2[/tex] or vice versa, and hence [tex]m_{s1} \neq m_{s2}[/tex] and we have no violation of the Pauli Exclusion Principle?