Let's consider the example of the carbon atom, which has an atomic number of 6.
First, we need to determine the electronic configuration of the ground state of the carbon atom. The electron configuration of carbon is 1s2 2s2 2p2. This means that there are two electrons in the 1s orbital, two in the 2s orbital, and two in the 2p orbital.
Next, we can determine the total orbital angular momentum, L, and the total spin angular momentum, S, of the electrons in the ground state.
For the 1s and 2s orbitals, each has a single subshell, so the total angular momentum is simply the angular momentum of the subshell. The 1s subshell has l=0, so L1s=0, while the 2s subshell has l=1, so L2s=1.
For the 2p orbitals, there are three subshells (px, py, and pz) with l=1. The total angular momentum is the vector sum of the three subshell angular momenta, which can be represented using the vector addition diagram:
```
pz
|
|
py -----+------ px
|
|
```
From this diagram, we can see that the three p orbitals have the same magnitude of angular momentum, so the total angular momentum can be represented as L2p=√2.
Next, we need to determine the total spin angular momentum, S. Since there are two electrons in each orbital (except for the 1s orbital, which has only one), the total spin angular momentum for the 1s and 2s orbitals is S=1, since there are two ways to combine the electron spins (↑↓ and ↓↑) to give a total spin of 1. For the 2p orbitals, there are four electrons, so there are six possible combinations of electron spins, giving a total spin of S=1 or S=0.
Finally, we can determine the possible term symbols for the ground state of carbon. The term symbol is represented as 2S+1L_J, where S is the total spin angular momentum, L is the total orbital angular momentum, and J is the total angular momentum (which is the vector sum of S and L).
For carbon, we have S=1 and L=0 (for the 1s orbital), S=1 and L=1 (for the 2s orbital), and S=0 or 1 and L=1 (for the 2p orbitals). Using these values, we can construct the possible term symbols as follows:
- For the 1s orbital: 2S+1L_J = 3P0, 3P1, 3P2
- For the 2s orbital: 2S+1L_J = 1S0, 3S1, 3D1, 3D2, 3D3
- For the 2p orbitals: 2S+1L_J = 1P1, 3P0, 3P1, 3P2, 1D2, 3D1, 3D2, 3D3, 1S0
Therefore, the possible term symbols for the ground state of carbon are: 3P0, 3P1, 3P2, 1S0, 3S1, 1P1, 1D2, 3D1, 3D2, 3D3.
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