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.

1 Answer
Nov 24, 2016

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


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")