Which of the following cannot exist on the basic of `MO` theory ? .

Options

A. `H_(2)^(o+)`
B. `He_(2)^(o+)`
C. `He_(2)`
D. `O_(2)`

Explanation

To determine which of the given molecules cannot exist based on Molecular Orbital (MO) theory, we need to analyze the bond order of each molecule. The bond order is calculated using the formula:

\[ \text{Bond Order} = \frac{1}{2} (N_b - N_a) \]

where \(N_b\) is the number of bonding electrons and \(N_a\) is the number of antibonding electrons. A bond order of zero indicates that the molecule cannot exist.

### Step-by-Step Solution:

1. **Analyze H2+**:
- **Electronic Configuration**: \(\sigma 1s^1\)
- **Bond Order Calculation**:
\[ N_b = 1, \quad N_a = 0 \quad \Rightarrow \quad \text{Bond Order} = \frac{1}{2} (1 - 0) = \frac{1}{2} \]
- **Conclusion**: H2+ can exist.

2. **Analyze Hg2+**:
- **Electronic Configuration**: \(\sigma 1s^2 \sigma^* 1s^1\)
- **Bond Order Calculation**:
\[ N_b = 2, \quad N_a = 1 \quad \Rightarrow \quad \text{Bond Order} = \frac{1}{2} (2 - 1) = \frac{1}{2} \]
- **Conclusion**: Hg2+ can exist.

3. **Analyze He2**:
- **Electronic Configuration**: \(\sigma 1s^2 \sigma^* 1s^2\)
- **Bond Order Calculation**:
\[ N_b = 2, \quad N_a = 2 \quad \Rightarrow \quad \text{Bond Order} = \frac{1}{2} (2 - 2) = 0 \]
- **Conclusion**: He2 cannot exist.

4. **Analyze O2**:
- **Electronic Configuration**: \(\sigma 1s^2 \sigma^* 1s^2 \sigma 2s^2 \sigma^* 2s^2 \sigma 2p^2 \pi 2p^2 \pi^* 2p^1\)
- **Bond Order Calculation**:
\[ N_b = 10, \quad N_a = 6 \quad \Rightarrow \quad \text{Bond Order} = \frac{1}{2} (10 - 6) = 2 \]
- **Conclusion**: O2 can exist.

### Final Answer:
The molecule that cannot exist based on Molecular Orbital theory is **He2**.

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