Draw the shapes of various `p` and `d` orbitals.

Answer

To draw the shapes of various `p` and `d` orbitals, follow these steps:

### Step 1: Draw the `p` Orbitals
1. **Draw the Axes**: Start by drawing the three Cartesian axes: x-axis, y-axis, and z-axis.

2. **Draw the `p_x` Orbital**:
- The `p_x` orbital is oriented along the x-axis.
- Draw a dumbbell shape that extends in the positive and negative x-direction, with the center at the origin.

3. **Draw the `p_y` Orbital**:
- The `p_y` orbital is oriented along the y-axis.
- Draw a dumbbell shape that extends in the positive and negative y-direction, with the center at the origin.

4. **Draw the `p_z` Orbital**:
- The `p_z` orbital is oriented along the z-axis.
- Draw a dumbbell shape that extends in the positive and negative z-direction, with the center at the origin.

### Step 2: Draw the `d` Orbitals
1. **Draw the `d_{xy}` Orbital**:
- The `d_{xy}` orbital has a double dumbbell shape in the xy-plane.
- Draw two lobes in the xy-plane, oriented along the x and y axes.

2. **Draw the `d_{xz}` Orbital**:
- The `d_{xz}` orbital has a double dumbbell shape in the xz-plane.
- Draw two lobes in the xz-plane, oriented along the x and z axes.

3. **Draw the `d_{yz}` Orbital**:
- The `d_{yz}` orbital has a double dumbbell shape in the yz-plane.
- Draw two lobes in the yz-plane, oriented along the y and z axes.

4. **Draw the `d_{x^2-y^2}` Orbital**:
- The `d_{x^2-y^2}` orbital has a double dumbbell shape along the x and y axes.
- Draw two lobes extending along the x and y axes, similar to the `d_{xy}` orbital but positioned differently.

5. **Draw the `d_{z^2}` Orbital**:
- The `d_{z^2}` orbital has a unique shape with a lobe along the z-axis and a torus around the xy-plane.
- Draw a lobe extending along the z-axis and a ring around the xy-plane.

### Summary of Shapes
- The `p` orbitals (`p_x`, `p_y`, `p_z`) are dumbbell-shaped and oriented along the respective axes.
- The `d` orbitals (`d_{xy}`, `d_{xz}`, `d_{yz}`, `d_{x^2-y^2}`, `d_{z^2}`) have more complex shapes, with double dumbbell configurations and unique orientations.