Shapes of Atomic Orbitals

Atomic orbitals are mathematical functions that represent the wave-like behavior of electrons within an atom. Each type of orbital has a distinct shape, highlighting regions where the probability of finding an electron is high or zero in certain directions around the nucleus.

1.0Introduction

The orbital wave function, denoted as ψ, for an electron in an atom, does not have a direct physical meaning; it is purely a mathematical function based on the electron's coordinates. However, the shape of the wave function varies for different orbitals, and the plots of these wave functions as a function of r (the distance from the nucleus) differ for each orbital. 

Electrons in orbitals are arranged in specific ways, and the orbitals are classified into four main groups: s, p, d, and f. Each group contains distinct subsets, each with a unique shape. These atomic orbitals vary in shape, meaning the electrons they describe have different probability distributions around the nucleus. This difference in electron distribution affects the energy levels of the orbitals because electrons occupying various orbitals are likely to be found in different regions relative to the nucleus. As a result, they experience the nucleus's attraction with varying strengths, influencing the orbital's energy.

2.0What are Atomic Orbitals?

Atomic orbitals are three-dimensional regions around an atom's nucleus where the likelihood of finding an electron is greatest.

According to the quantum atomic model, an atom can have infinite orbitals. These orbitals are classified based on their size, shape, and orientation. A smaller orbital indicates a higher probability of finding an electron close to the nucleus. The orbital wave function is a mathematical expression that represents the coordinates of an electron, and the square of this wave function gives the probability of locating an electron. This wave function is also used to create boundary surface diagrams. 

The shapes of orbitals vary: an s-orbital has a spherical shape with the nucleus at the centre, a p-orbital is dumbbell-shaped, and four of the five d orbitals resemble a cloverleaf. The fifth d orbital is shaped like an elongated dumbbell with a ring around its center. In an atom, orbitals are arranged in layers or electron shells.

3.0Shape of s-Orbital

The boundary surface diagram of an s orbital resembles a sphere with the nucleus at its centre, appearing as a circle in a two-dimensional view. s orbitals are spherically symmetric, meaning the probability of finding an electron is uniform in all directions at a given distance from the nucleus.

As the principal quantum number (n) increases, the size of the s orbital also increases, following the order: 4s > 3s > 2s > 1s.

Nodes are regions with zero probability of finding an electron. They are classified as radial nodes, which depend on distance from the nucleus, and angular nodes, which relate to the orbital's orientation.

4.0Shape of p-Orbital

p-orbitals are atomic orbitals with two lobes aligned along the x, y, or z axes. Each 2p orbital has a unique orientation, with one along the x-axis (σ-orbital), one along the y-axis (π-orbital), and one along the z-axis (π*-orbital). These orbitals have a nodal plane through the nucleus, where the probability of finding an electron is zero. According to the Pauli exclusion principle, each p orbital can hold a maximum of two electrons with opposite spins.

5.0Shape of d-Orbital

The shape of d orbitals is associated with the magnetic quantum number, which can take values from -2 to +2, leading to five distinct d orbitals. These are labelled dxy, dyz, dxz, dx²–y², and dz². The first four orbitals (dxy, dyz, dxz, dx²–y²) have similar shapes, typically consisting of four lobes, while the dz² orbital has a distinct shape, resembling a doughnut around the nucleus with a lobe along the z-axis. Despite their different shapes, all five d orbitals have the same energy in a given atom.

6.0Shape of f-Orbital

The shape of f orbitals is complex and dispersed. With l=3l = 3l=3, the minimum value of the principal quantum number (n) is 4. The corresponding values of the magnetic quantum number (mlm_lml​) for f orbitals are -3, -2, -1, 0, +1, +2, and +3, meaning there are seven distinct f orbitals. These orbitals have intricate shapes, often described as containing multiple lobes, and are more complex than the s, p, and d orbitals.