Which one of the following can not exist on the basis of molecular orbital theory?

Options

A. `H_2^(+)`
B. `He_2^(+)`
C. `C_2`
D. `He_2`

Answer

To determine which molecule cannot exist based on molecular orbital theory (MOT), we need to calculate the bond order of the given molecules. The bond order is calculated using the formula:

\[ \text{Bond Order} = \frac{1}{2} (\text{Number of electrons in bonding MOs} - \text{Number of electrons in anti-bonding MOs}) \]

A bond order of zero indicates that the molecule cannot exist.

### Step-by-Step Solution:

1. **Identify the Molecules**: The question asks us to evaluate specific molecules. Let's consider common diatomic molecules such as H2, He2, H2+, and He2+.

2. **Calculate Bond Order for H2**:
- H2 has 2 electrons (1 from each hydrogen atom).
- Electronic configuration: σ1s (2 electrons in bonding).
- Bonding electrons = 2, Anti-bonding electrons = 0.
- Bond Order = \( \frac{1}{2} (2 - 0) = 1 \).
- H2 exists.

3. **Calculate Bond Order for H2+**:
- H2+ has 1 electron (1 from one hydrogen, 1 removed due to positive charge).
- Electronic configuration: σ1s (1 electron in bonding).
- Bonding electrons = 1, Anti-bonding electrons = 0.
- Bond Order = \( \frac{1}{2} (1 - 0) = 0.5 \).
- H2+ exists.

4. **Calculate Bond Order for He2**:
- He2 has 4 electrons (2 from each helium atom).
- Electronic configuration: σ1s (2 electrons) and σ*1s (2 electrons).
- Bonding electrons = 2, Anti-bonding electrons = 2.
- Bond Order = \( \frac{1}{2} (2 - 2) = 0 \).
- He2 does not exist.

5. **Calculate Bond Order for He2+**:
- He2+ has 3 electrons (4 from He2, 1 removed due to positive charge).
- Electronic configuration: σ1s (2 electrons) and σ*1s (1 electron).
- Bonding electrons = 2, Anti-bonding electrons = 1.
- Bond Order = \( \frac{1}{2} (2 - 1) = 0.5 \).
- He2+ exists.

### Conclusion:
The molecule that cannot exist based on molecular orbital theory is **He2** because its bond order is zero.