# 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 ML and MS for each microstate, with where mi is either ml or ms for the i-th electron, and M represents the resulting ML or MS respectively:
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• Third, count the number of microstates for each MLMS possible combination
ml +1 0 −1 ML 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
MS +1 1 1 2 1 1 3 1 1 2 1 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 ML ranging from −2 to +2 (so L = 2), with a single value for MS (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 ML and MS 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 1 1 1 1 1

S=1, L=1, J=2,1,0

3P2, 3P1, 3P0

Ms +1 1 1 1 1 1 1 1 1 1

S=0, L=0, J=0

1S0

Ms 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.)