Sketch the shape and orientation of the following types of orbitals: \((\mathbf{a}) p_{x},(\mathbf{b}) d_{z^{2}},(\mathbf{c}) d_{x^{2}-\gamma^{2}}\)

Understanding the shape of atomic orbitals is fundamental in grasping the basics of quantum chemistry and how electrons behave in atoms. Orbitals are regions in space where there is a high probability of finding an electron. Each type of orbital has a distinctive shape that indicates the spatial distribution of electron density.

The simplest type is the s orbital, which is spherical. As we move to more complex orbitals, such as p and d orbitals, the shapes become more elaborate. The shapes of these orbitals are not just for theoretical interest; they have direct consequences on the chemical behavior of atoms, including the angles at which atoms bond and the types of bonds that can form.

P orbitals are one of the first steps beyond the simplest s orbitals in understanding electron configuration. They have a peculiar dumbbell shape, with each lobe representing the area where an electron is likely to be found. There are three p orbitals in every energy level above the first, denoted as p_x, p_y, and p_z, oriented along the three axes of 3D space.

The p_x orbital, for instance, extends along the x-axis on both sides of the nucleus, having two lobes with a node at the nucleus where the probability of finding an electron is zero. These p orbitals can hold a maximum of six electrons in total, and their orientation in space influences the shape and physical properties of molecules.

D orbitals, present from the third energy level onwards, introduce even more complexity to the understanding of electron distribution within an atom. There are five d orbitals, labeled as d_z², d_x2-y2, d_xy, d_xz, and d_yz. These labels give us an indication of the orbital shapes and the axes along which they are aligned.

The d_z² orbital is characterized by a clover shape along the z-axis with a doughnut-shaped region in the xy plane. In contrast, the d_x2-y2 orbital has four lobes that lie in the xy plane and extend along the x and y axes, forming a shape similar to a four-leaf clover. These complex shapes allow for a variety of bonding patterns, which are crucial in fields such as coordination chemistry.