Question

How would I determine all the possible term symbols for an `s^1p^2` electron configuration (such as first-excited-state boron)?

They are `""^4 P_"1/2"`, `""^4 P_"3/2"`, `""^4 P_"5/2"`, `""^2 D_"3/2"`, `""^2 D_"5/2"`, `""^2 P_"1/2"`, `""^2 P_"3/2"`, and `""^2 S_"1/2"`.

I'll answer this question, for people who eventually may run into this topic.

Answer 1

DISCLAIMER: This is a long process! If you want to try this, set aside about 1-2 hours.

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Let's say you wanted to find each possible term symbol for an `s^1p^2` configuration. The general notation is:

`bb(""^(2S + 1) L_J)`

where

• `S` is the total spin.
• `L` is the total orbital angular momentum.
• `J` is the total angular momentum, taking on the range `{|L - S|, |L - S + 1|, . . . , |L + S - 1|, |L + S|}`.
• `2S + 1` is the spin multiplicity.

For this, I would first identify all the possible values of `m_l` and `m_s` for the `s` and `p` electrons:

• `s^1: m_l = 0`, `m_s = pm1/2`
• `p^2: m_l = {-1,0,+1}`, `m_s = pm1/2`

ELECTRON CONFIGURATION "OUTLINE"

To outline the possible electron configurations, let us list each possible electron configuration out. We call them microstates.

The way I think makes sense to organize them is doing all the spins for some lefthand `m_l`, and then restricting the lowest lefthand `m_l`.

• Without electron pairing, and with a spin-up `s` electron (`L_max = sum_i l_i = 0 + 1 = 1`):

• Without electron pairing, and with a spin-down `s` electron (`L_max = sum_i l_i = 0 + 1 = 1`):

• With electron pairing, with a spin-up or spin-down `s` electron (`L_max = sum_i l_i = 0 + 1 + 1 = 2`):

That gives us a total of `30` electron configuration "microstates".

CONSTRUCTING A MICROSTATE TABLE

Each microstate has its corresponding total spin angular momentum `S` and total orbital angular momentum `L` in the `z` direction, which are called `M_S` and `M_L`, respectively. These are defined as:

`M_L = sum_i m_(l)(i)`
`M_S = sum_i m_(s)(i)`

meaning the sum of the `m_l` or `m_s` values for electron `i`.

Earlier, we said that we had an `L_max` of `1` or `2`. Well, that gives the allowed range of `M_L` to be `color(green)({-2,-1,0,+1,+2})`, just like how `m_l = {-l,-l+1,...,l-1,l}`.

That will be the number of rows of our table.

Also, with `3` electrons, the total spin could be `S = 1/2,3/2`. Therefore, the range of `M_S` is `color(green)({-3/2,-1/2,+1/2,+3/2})`.

That will be the number of columns of our table.

From this, the blank microstate table that organizes our electron configurations is:

`M_Luarr" "" "larr M_S rarr`
`ul(" "" "" "" "" "" "" "" "" "" "" "" "" "" ")`
`color(white)([(color(black)(""),color(black)(-3/2),color(black)(-1/2),color(black)(+1/2),color(black)(+3/2)),(color(black)(+2),color(black)(""),color(black)(""),color(black)(""),color(black)("")),(color(black)(+1),color(black)(""),color(black)(""),color(black)(""),color(black)("")),(color(black)(0),color(black)(""),color(black)(""),color(black)(""),color(black)("")),(color(black)(-1),color(black)(""),color(black)(""),color(black)(""),color(black)("")),(color(black)(-2),color(black)(""),color(black)(""),color(black)(""),color(black)(""))])`

The outline we did above is how we can keep track of which ones we've accounted for already.

As an example of the notation we'll put into the table,

`ul(color(white)(uarr darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))`
`ul(darr color(white)(uarr))`

would be written as `0^(-) 0^(+) 1^(+)`, to indicate that:

• the `s` electron went into an orbital of `m_l = 0` as spin-down `(-)`
• a `p` electron went into an orbital of `m_l = 0` as spin-up `(+)`,
• a `p` electron went into an orbital of `m_l = 1` as spin-up `(+)`.

So,

• `bb(M_S) = sum_i m_(s)(i) = -1/2 + 1/2 + 1/2 = bb(+1/2)`
• `bb(M_L) = sum_i m_(l)(i) = 0 + 0 + 1 = bb(1)`

Therefore, it goes into the cell that is indicated by `M_S = +1/2` and `M_L = +1`.

Give yourself maybe half an hour to an hour, and you should get:

SEPARATING INTO INDIVIDUAL MICROSTATE TABLES FOR EACH FREE-ION TERM

Now, to find each term symbol, we first make the table easier to manage by setting each microstate as `x`. That gives:

Above, I've highlighted the microstates as follows:

1. Starting at the maximum number of `M_L` rows, and then the maximum number of those `M_S` columns, and choose the first term in each cell.
2. Then, decrease the range of `S` symmetrically (thus going from 4 columns to 2 columns) and find the new maximum number of `M_L` rows out of the available microstates.
3. Then, decrease the range of `L` once you've reached the minimum number of `M_S` columns.

Each color of `x` is placed into a separate microstate table.

• The first table would be the `color(blue)("blue")` `x`'s.
• The second would be the `color(red)("red")` `x`'s.
• The third would be the `color(orange)("orange")` `x`'s.
• The fourth would be the `color(green)("green")` `x`'s.

Here's a GIF illustrating how to do it:

FINDING EACH FREE-ION TERM SYMBOL (NO J)

This is how I knew which free-ion term symbols to write for the above microstate tables:

• The number of `M_L` rows is the range of `L` in the `+z` and `-z` directions, so `|M_(L,max)| = L_max`, which tells you what letter the term symbol is (`0,1,2,3,4,... harr S,P,D,F,G,...`).
• The number of `M_S` columns is the range of `S` in the `+z` and `-z` directions, so `|M_(S,max)| = S_max`, which tells you what the total spin for the term symbol is.

Once you work it out, you should confirm that your initial term symbols are:

• `""^(2(3/2) + 1) (L = 1) = ""^4 P` (blue `x`'s)
• `""^(2(1/2) + 1) (L = 2) = ""^2 D` (red `x`'s)
• `""^(2(1/2) + 1) (L = 1) = ""^2 P` (orange `x`'s)
• `""^(2(1/2) + 1) (L = 0) = ""^2 S` (green `x`'s)

FINDING EACH "MULTIPLET" TERM SYMBOL (INCLUDING J)

Finally, find `J` by using the `L` and `S` values you have available. For each `L` and `S`, take the largest `|M_L|` and use each `|M_S|`, respectively:

`""^4 P: L = 0,bb(1); S = 1/2,3/2`
`=> color(green)(J) = (1-1/2),(1+1/2),(1+3/2) = color(green)(1/2,3/2,5/2)`

`""^2 D: L = 0,1,bb(2); S = 1/2`
`=> color(green)(J) = (2-1/2),(2+1/2) = color(green)(3/2,5/2)`

`""^2 P: L = 0,bb(1); S = 1/2`
`=> color(green)(J) = (1-1/2),(1+1/2) = color(green)(1/2,3/2)`

`""^2 S: L = bb(0); S = 1/2`
`=> color(green)(J = 1/2)`

So, we finally have:

`color(blue)(""^4 P_"1/2", ""^4 P_"3/2", ""^4 P_"5/2", ""^2 D_"3/2", ""^2 D_"5/2", ""^2 P_"1/2", ""^2 P_"3/2", ""^2 S_"1/2")`