For which orbital angular probability distribution is maximum at an angle of `45^@` to the axial direction ?

Options

A. `d_(x^(2))-y^(2)`
B. `d_(Z^(2))`
C. `d_(xy)`
D. `P_(x)`

Explanation

To determine the orbital angular probability distribution that is maximum at an angle of \(45^\circ\) to the axial direction, we need to analyze the angular probability distribution of different orbitals.

### Step-by-Step Solution:

1. **Understanding Angular Probability Distribution**:
The angular probability distribution for an orbital is described by the spherical harmonics \(Y_{l}^{m}(\theta, \phi)\), where \(l\) is the azimuthal quantum number and \(m\) is the magnetic quantum number. The angle \(\theta\) is the polar angle (angle from the z-axis), and \(\phi\) is the azimuthal angle (angle in the xy-plane).

2. **Identifying the Angle**:
We are interested in the case where the maximum probability occurs at an angle of \(45^\circ\) to the axial direction (z-axis). At \(45^\circ\), \(\theta = 45^\circ\) or \(\theta = \frac{\pi}{4}\) radians.

3. **Evaluating Different Orbitals**:
- For \(s\) orbitals (\(l=0\)), the probability distribution is spherically symmetric and does not depend on the angle.
- For \(p\) orbitals (\(l=1\)), the distribution has lobes along the axes (x, y, z) and does not have a maximum at \(45^\circ\).
- For \(d\) orbitals (\(l=2\)), particularly the \(d_{xy}\) orbital, the lobes are oriented in the xy-plane and can show significant probability at \(45^\circ\) to the z-axis.

4. **Conclusion**:
The orbital angular probability distribution that has a maximum at an angle of \(45^\circ\) to the axial direction is the \(d_{xy}\) orbital. This orbital has lobes that extend in the xy-plane, and at \(45^\circ\), the probability density is maximized.

### Final Answer:
The orbital angular probability distribution that is maximum at an angle of \(45^\circ\) to the axial direction is the \(d_{xy}\) orbital.

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