Term Symbol - Term Symbols For An Electron Configuration

Term Symbols For An Electron Configuration

To calculate all possible term symbols for a given electron configuration the process is a bit longer.

• First, calculate the total number of possible microstates N for a given electron configuration. As before, we discard the filled (sub)shells, and keep only the partially filled ones. For a given orbital quantum number l, t is the maximum allowed number of electrons, t = 2(2l+1). If there are e electrons in a given subshell, the number of possible microstates is

As an example, lets take the carbon electron structure: 1s22s22p2. After removing full subshells, there are 2 electrons in a p-level (l = 1), so we have

different microstates.

• Second, draw all possible microstates. Calculate M_L and M_S for each microstate, with where m_i is either m_l or m_s for the i-th electron, and M represents the resulting M_L or M_S respectively:

	ml
	+1	0	−1	ML	MS
all up	↑	↑		1	1
	↑		↑	0	1
		↑	↑	−1	1
all down	↓	↓		1	−1
	↓		↓	0	−1
		↓	↓	−1	−1
one up one down	↑↓			2	0
	↑	↓		1	0
	↑		↓	0	0
	↓	↑		1	0
		↑↓		0	0
		↑	↓	−1	0
	↓		↑	0	0
		↓	↑	−1	0
			↑↓	−2	0

• Third, count the number of microstates for each M_L—M_S possible combination

		MS
		+1	0	−1
ML	+2		1
	+1	1	2	1
	0	1	3	1
	−1	1	2	1
	−2		1

• Fourth, extract smaller tables representing each possible term. Each table will have the size (2L+1) by (2S+1), and will contain only "1"s as entries. The first table extracted corresponds to M_L ranging from −2 to +2 (so L = 2), with a single value for M_S (implying S = 0). This corresponds to a 1D term. The remaining table is 3×3. Then we extract a second table, removing the entries for M_L and M_S both ranging from −1 to +1 (and so S = L = 1, a 3P term). The remaining table is a 1×1 table, with L = S = 0, i.e., a 1S term.

S=0, L=2, J=2 1D2 Ms 0 Ml +2 1 +1 1 0 1 −1 1 −2 1

S=1, L=1, J=2,1,0 3P2, 3P1, 3P0 Ms +1 0 −1 Ml +1 1 1 1 0 1 1 1 −1 1 1 1

S=0, L=0, J=0 1S0 Ms 0 Ml 0 1

• Fifth, applying Hund's rules, deduce which is the ground state (or the lowest state for the configuration of interest.) Hund's rules should not be used to predict the order of states other than the lowest for a given configuration. (See examples at Hund's rules#Excited states.)