In this chapter, we will learn about shapes of orbitals. We know that an atomic orbital is the three-dimensional region or area in space around the nucleus in which the tendency of finding an electron is maximum.
The square of the wave function (ψ2) represents the probability density, which describes the probability of finding an electron in a particular region around the nucleus of an atom. It explains the spatial distribution of electron cloud in the space around the nucleus. The electron cloud represents the shapes of orbitals.
The density of electron cloud at any point is proportional to the probability of finding the electron at that point. However, the probability of finding an electron is not uniform everywhere in an atom. Therefore, the density of electron cloud is also not identical everywhere in an atom.
The electron cloud is more dense in some regions (where the probability is higher) and less dense in some other (where the probability is lower). This variation in electron density defines the shape and characteristics of atomic orbitals.
Ways to Draw Shapes of Atomic Orbitals
In chemistry, there are two ways to draw the plot of total probability density, which gives the shapes of orbitals. They are as follows:
- Charge cloud or electron cloud diagrams
- Boundary surface diagrams
Charge Cloud Diagrams
In the charge cloud diagrams, the probability density (ψ2) is shown as the collection of dots. The density of dots in any region indicates the probability of electron density in that region, as shown in the below figure.
In the electron cloud diagrams, the probability of finding an electron is a particular region of space is directly proportional to the density of such dots in that region. The higher the density of dots, the greater the probability of finding an electron in that space. In regions where the dots are sparse, the probability of finding an electron is low.
Boundary Surface Diagrams
The boundary surface diagrams provide an accurate representation of the shapes of atomic orbitals. In this representation, a boundary surface, also known as contour surface, is drawn in space for an orbital on which the value of probability density (ψ2) is constant. This boundary surface encloses the regions where the probability of finding an electron is maximum around 90 – 95%.
The shape of the boundary surface best describes the shape of the atomic orbital because it outlines the region where an electron is most likely to be found. The boundary surface diagrams for s-orbitals are spherical in shape. However, the size of s-orbital increases with the increase in the value of principal quantum number (n). For example, 1s < 2s < 3s . . . . .
The boundary surface diagrams for three 2p-orbitals are not spherical. Each p-orbital contains two lobes of electron density, which are separated by a nodal plane. Nodal plane is a region where the probability of finding an electron is zero. The boundary surface diagrams for s-orbitals, p-orbitals, and d-orbitals are discussed below.
Shapes of s-Orbitals
For s-orbitals, l = 0, m = 0, which indicates that s-orbitals have only one orientation. This means that the probability of finding an electron is uniform in all directions. In other words, the probability of finding an electron for a given distance is identical at all angles. Therefore, s-orbitals are spherically symmetrical about the nucleus. Each s-orbital is symmetrical along all three axes, x, y, and z.
As the value of principal quantum number (n) increases, the size of s-orbital also increases, while retaining the spherical symmetry. For example, 2s-orbital (n = 2) is bigger in size than 1s-orbital (n = 1). The total number of the concentric spheres at any main energy level is equal to the principal quantum number (n). For example, 1s orbital has only one sphere, 2s-orbital has two concentric spheres, 3s-orbital consists of three concentric spheres, and so on.
A spherical shell within an orbital where the probability of finding an electron is zero is called a radial node or simply node. These nodes occur in s-orbitals and higher energy orbitals. Nodes represent regions where the electron density is exactly zero.
The number of radial nodes in an orbital is given by the formula:
Radial nodes = n – ℓ – 1, where n is the principal quantum number and ℓ is the azimuthal quantum number.
For example:
- 1s orbital has 0 radial nodes (no empty spherical shell).
- 2s orbital has 1 radial node (i.e. one spherical shell with zero probability of finding electrons).
- 3s orbital has 2 radial nodes, and so on.
Hence, the value of n increases, the number of radial nodes also increases, leading to more complex electron distributions within the s-orbitals.
The energy of the s-orbital increases as the value of principal quantum number (n) also increases. Thus, the energies of the various s-orbitals follow the below order:
1s < 2s < 3s < 4s . . . . .
Note that s-orbital does not consist of any directional property.
Shapes of p-Orbitals
For p-orbitals, the azimuthal quantum number (ℓ) is 1, and the magnetic quantum number (m) can have three possible values: -1, 0, and +1. These values indicate that p-orbitals have three different orientations in space. Therefore, there are three p-orbitals named px, py, and pz, which are oriented along x-axis, y-axis, and z-axis, respectively.
The shape of each p-orbital has a dumbbell shaped and are mutually perpendicular to the other two. Each p-orbital has two lobes which are separated by a plane of zero probability known as nodal plane. The spatial distributions of 2p orbitals are shown in the below figure.
Node and Nodal Plane
For 2p-orbital, the electron density is maximum along the orbital axis, meaning that the probability of finding an electron is highest in this direction. At the nucleus and in the plane perpendicular to the orbital axis passing through nucleus, the electron density is zero. The nucleus at which the electron density is zero is called node.
The plane passing through the node and perpendicular to the orbital axis is called nodal plane. A 2p-orbital has one nodal plane because it has one angular node. Each p-orbital (px, py, pz) has a different nodal plane:
- px has a nodal plane in the yz-plane.
- py has a nodal plane in the xz-plane.
- pz has a nodal plane in the xy-plane.
The lobes of 3p orbitals are bigger than those of the 2p orbitals because the size of an atomic orbital goes on increasing with the increase of the principal quantum number (n).
Note that p-orbital has directional properties.
Degenerate Orbitals
In the absence of an external electric or magnetic field, all the three p-orbitals (px, py, and pz) have the same energy. The orbitals having the same energy are called degenerate orbitals. Thus, px, py, and pz orbitals are equivalent in energy under normal conditions.
However, in the presence of external electric or magnetic field, this degeneracy gets lost because the three p-orbitals are oriented differently with respect to applied field. This effect is known as Zeeman splitting (in a magnetic field) or Stark effect (in an electric field).
Shapes of d-Orbitals
For d-orbitals, the azimuthal quantum number (ℓ) is 2, and the magnetic quantum number (m) can have five possible values: -2, -1, 0, +1, and +2. These values indicate that d-orbitals have five different orientations in space. These five orientations or orbitals are designated as dxy, dyz, dzx, dx2 – dy2, and dz2. Three of these orbitals, dxy, dyz, and dzx are identical in shape but orientations are different. Each orbital has four lobes of electron density which bisects the angles between principal axes. The dx2 – dy2 orbital also has four lobes of electron density which lie along x and y axes. The dz2 orbital has two lobes lying along z-axis and a ring of high electron density in the xy-plane as shown in the below figure.
All five d-orbitals have equivalent energy in the absence of an external field, meaning that they are degenerate orbitals. However, they differ in their spatial orientation and shape. The d-orbital has directional properties. The shape of d-orbital has a double dumbbell shape.
The shapes of 4d and 5d orbitals are identical to 3d-orbital but differ in size and energy. This is because principal quantum number (n) increases.
