Concept of shapes of s, p & d orbitals

In orbitals, electrons are grouped in different ways. The orbitals can be divided into four main groups: s, p, d and f. Each group is further divided into subsets or orbitals, each characteristic shape.

The s-orbitals have a spherical shape, and their name comes from the letter of the Greek alphabet – sigma. They have a positive charge on their centre and a negative charge on the outside. They can accommodate two electrons by surrounding the nucleus in a spherical manner. The p-orbitals are dumbbell-shaped and have positive charges on both sides of the nucleus. They are capable of accommodating six electrons per orbital. 

The five d-orbitals have a cloverleaf structure with positive charges spread over a large area of space (d1 to d5). Each orbital can accommodate 10 electrons at a time in contrast to the 2 and 6 electrons that occupy s and p orbitals, respectively. However, only three-d-orbitals are used for bonding purposes (d2, d3, and d4). The fourth orbital, d1, is usually empty, or an electron from another atom jumps into it, resulting in an unpaired electron that gives rise to an ionic or covalent bond.

It is a particularly important concept in quantum mechanics. The wave function is a complex-valued function that specifies the overall state of one or more systems. The probability density associated with the wave function is often called the probability current, and this current determines the rate of change of the probability distribution for a system.

The wave function can be visualised as indicating the “shape” of the probability distribution or specifying a viewpoint in space to regard the system. For example, if we look at a quantum system under a microscope and see an electron, this may be because the electron has generated a wave packet; or, if we are very clever at arranging things experimentally, there may be some way to arrange to view the electron from different directions at once.

The wave function does not directly describe any physical properties of the system but only calculable quantities such as probabilities. The probability distribution for one observable may be calculated from the probabilities for other observables and thus corresponds to an eigenvector of those other operators (or their corresponding matrices). Each such calculation provides information about only one observable.

The shape of s orbitals

Orbital is defined as the region in which a given electron can be found at a given time. s orbitals are the spherical region around the nucleus, in which the electron density is uniform like that of the surface of a sphere.

Totally filled subshells have spherical symmetry in two dimensions, hence the name s-orbital (s stands for spherical). The radial distribution of probability density is equal to that of a point charge.

It should be noted that atomic orbitals are mathematical models with no intuitive counterpart. It is impossible to draw an orbital or show it to you on a sheet of paper. We cannot even give you an accurate picture of what an orbital looks like because we can’t see atoms! However, we can describe atomic orbitals very precisely in terms of mathematical equations.

The shape of p orbitals

p-orbitals are a type of molecular orbital used in molecular orbital theory. p orbitals, or pi-orbitals, are the three equivalent orbitals with two lobes pointing along the x axis, y axis and z axis respectively. These orbitals may be thought of as resulting from the overlap of three atomic orbitals when an electron is removed from a neutral atom.

The three p-orbitals have different orientations in space. The 2p-orbital contains two lobes lying on the xy plane (the plane containing both the nucleus and the electron); this is termed as σ-orbital. The 2p-orbital with two lobes lying in the xz plane is called a π-orbital. The 2p-orbital with both lobes lying along the yz plane is a π*-orbital (a “star” orbital), which is similar to an s orbital, but has an overall node at the centre of symmetry.

The Pauli exclusion principle requires that only two electrons can occupy each p-orbital. 

The shape of d orbitals

The quantum number of a d orbital is given as (-2,-1,0, 1,2). Hence we can say that there are five d-orbitals. These orbitals are dxy, dyz, dxz, dx2–y2 and dz2. Out of these five d orbitals, the shapes of the first four d-orbitals are similar, whereas the fifth d-orbital is different from the others. Also, the energy of all five d orbitals is the same.

Understanding the shape of these orbitals will help you to visualise how these orbitals are formed. These five shapes are not identical. The first four orbitals (dxy, dyz, dxz and dx2–y2 ) have a dumbbell shape or two dumbbells connected to each other. But the fifth d-orbital (dz2) is really different from the others. We can say that it looks like a sphere while the other four look like dumbbells.

Conclusion

In this material, we discussed the concept of the shape of s, p and d orbitals. We also discussed how the shapes of these orbitals have been determined due to some formulas. The notes regarding the concept of the shape of s, p and d orbitals have also been discussed.