Term Symbol for Carbon

Question

Currently studying how to "compute" term symbols. My book gives the example of carbon. Carbon has the electron configuration $(1s)^2(2s)^2(2p)^2$. We can ignore full orbitals, so we only look at $(2p)^2$.

Now the possible values for $L$ and $S$ are:

$l_1 = 1, \ l_2 = 1 \quad \Rightarrow \quad L = 0, 1, 2$

$s_1 = 1/2, \ s_2 = 1/2 \quad \Rightarrow \quad S= 0, 1$

We get the possible terms: $ \sideset{^3}{}D, \ \sideset{^1}{}D, \ \sideset{^3}{}P, \ \sideset{^1}{}P, \ \sideset{^3}{}S, \ \sideset{^1}{}S $

with $(2l_1 + 1)(2l_2 + 1)(2s_1 + 1)(2s_2 + 1) = 36$ states.

Now they use the following table where they consider the above 36 states as a two electron function with the quantum number $M_L = m_l + m_l^{'}$ and $M_s = m_s + m_s^{'}$

Now they start to check which entries they can cancel and the first argument they bring is:

In the cells on the diagonal, both electrons have identical quantum numbers. That's now allowed according to the Pauli principle. We now have 30 states

They continue to cancel states but that's not important to my questions.

My main problem is: I don't understand the above quote. If I look at the diagonal, I see $(2,1), (2, -1), (0, 1), (0,-1), (-2, 1), (-2, -1)$, how are those quantum numbers the same? I first thought that they meant the diagonal from the left lower corner to the upper right one but that's not the case.

On another note, I'm also not sure I completely understand why we know that $l_1 =1, \ l_2 = 1$ and why the $s_i > 0$.

Edit: Ah I think they actually meant that for the diagonal entries, we have $m_l^{'} = m_l$ and $m_s^{'} = m_s$ and not the values in the cell (the totals).

Answer 1

'In the cells on the diagonal, both electrons have identical quantum numbers. That's not allowed according to the Pauli principle.' [...] I don't understand the above quote.

If two electrons have the same quantum numbers, it means that they are in the same orbital ($m_l$) with the same spin ($m_s$). This is forbidden by the Pauli exclusion principle — you can't have two electrons with the same spin in the same orbital.

how are those quantum numbers the same

The quantum numbers of electron 1 are on the left column, those of electron 2 are on the top column. The diagonal entries correspond to (row, column) pairs where electrons 1 and 2 have the same quantum numbers. So yes, your edit is correct.

I'm also not sure I completely understand why we know that $l_1 = 1$, $l_2 = 1$ and why the $s_i > 0$.

$l$ here refers to the azimuthal quantum number of the orbitals the electrons are in. Since you are considering a $2\mathrm{p}^2$ configuration, both electrons are in p-orbitals, so they both have $l = 1$.

$s$ here is the spin quantum number, it is related to the magnitude of the spin angular momentum. Spin quantum numbers are always non-negative and for electrons is $1/2$. You are probably confusing it with $m_s$ which relates to the projection of the spin angular momentum and takes values between $-s, -s+1, \ldots, s$, i.e. $\pm 1/2$ for electrons.

Why use $l$ and $s$ instead of $m_l$ and $m_s$? It comes down to how angular momentum coupling is worked out. In this case, the objective is to couple two sources of angular momentum: that of electron 1 with that of electron 2.

For any two sources of angular momentum with quantum numbers $J_1$ and $J_2$, the resulting sum can have quantum numbers

$$J = |J_1 - J_2|, |J_1 - J_2| + 1, \ldots, J_1 + J_2$$

$J$ is just a generic symbol for some source of angular momentum. Note that here, $J_1$, $J_2$, and $J$ are the quantum numbers which relate to the magnitude of the angular momentum. It's got nothing to do with the projection quantum numbers $m_J$.

In the case of the two p-orbital electrons, we have that $J_1 = 1/2$ and $J_2 = 1/2$, so the allowed values for $J$ are

$$J = 0, 1, 2$$

which are called $S$, $P$, and $D$ states respectively (because spectroscopists have weird terminology sometimes).

It just so happens that in the case of orbital angular momentum, instead of calling the quantum number a generic $J_1$ and $J_2$, we have a special symbol for it $l_1$ and $l_2$. The resulting summed angular momentum is called $L$ instead of just $J$. And for spin angular momenta, we call it $s_1$, $s_2$, $S$.

Coupling of angular momenta is a rather large topic and can't be covered in detail here, but you can find it in pretty much any good textbook on quantum mechanics.