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

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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.

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

##### 1 Answer

**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

#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

#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

**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

**CONSTRUCTING A MICROSTATE TABLE**

Each microstate has its corresponding total spin angular momentum

#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

That will be the number of

rowsof our table.

Also, with

That will be the number of

columnsof 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

- 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

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

Above, I've highlighted the microstates as follows:

- 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. - 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. - 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

#""^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")#