Quantum theory and quantum numbers

Orbital shapes

The S-orbital

The s-orbitals are spherical in shape. They all have a value of 0 for the azimuthal or angular momentum quantum number(l). The main difference between the s-orbitals found in different shells is the size, as shown in the diagram below. shapes of the s-orbitals

It is easy to imagine the nucleus at the centre of the s-orbital with the electron density spread throughout the orbital.

Differences between the s-orbitals

description of the differences between s-orbitals and introduction of the term nodes


There are 3 p-orbitals and they all have a dumb-bell like shape as shown below, but they are arranged differently in 3d space:

the shapes of the p-orbitals

The electron density in the p-orbitals is concentrated in two lobes on either side of the nucleus. There are 3 p-orbitals; they are identical in size and shape but are simply arranged differently along each of the x, y and z-axes in 3d-space. The three p-orbitals are called Px, Py and Px, the labels simply indicate the axis along which the lobes of the p-orbital are located. The first p-orbitals are found in the second shell; the p-orbitals in the next shells are similar in shape and orientation but are simply larger. Each p-orbital can hold 2 electrons, meaning that the 3 p-orbitals hold 6 electrons in total if they are full.

The d-orbitals

There are 5 d-orbitals. Three of them have lobes orientated between the axes, for example the dxz orbitals has its lobes orientated between the x and z axes. Two of the orbitals are arranged differently, the dx2-y2 orbital has its lobes orientated along the x and y axes while the dz2 orbital resembles a p-orbital with its 2 lobes but it has a doughnut like ring of electron density around its centre. The shapes of the d-orbitals is shown in the image below:

shapes of the d-orbitals

Effective nuclear charge

energy level diagram for single lectron and multielectron atoms The orbitals in each principal energy level in hydrogen and other one electron atoms such as He+ as we have seen are degenerate (same energy) but in multi-electron atoms the sub- shells or sub-levels and hence the orbitals in each principal energy level are no longer degenerate, for example the 3s, 3p and 3d sub-shells in hydrogen are all degenerate but in multi-electron atoms the repulsion between the electrons leads to a splitting in the energy levels, for example the 2s sub-shell is lower in energy than the 2p sub-shell. The reason for the splitting in the energy levels is due to the repulsion between the electrons and also the effective nuclear charge that each electron feels.

In atoms with many electrons we need to consider how each of the electrons interact with each other in terms of repulsions and also the attractive forces that each electron will experience from the positively charged nucleus. This is no easy task and it is much simpler to just consider a model where we consider each electron individually and try to estimate the forces it is experiencing. Any electrons between the nucleus and the electron we are considering will effective screen or reduce the size of the attractive forces it experiences from the nucleus. This net charge that an electron experiences is called the effective nuclear charge, it is simply the number of protons in the nucleus minus the average number of electrons between it and the nucleus. This means that electrons in the outer shells will feel a much smaller force of attraction from the nucleus because the inner shell electrons will screen or shield them from the positive charge in the nucleus. The effective force that each electron feels will depend also on the shapes orbitals as this will clearly have an effect on how well the nucleus is shielded or screened, for example an electron in an s-orbital is more likely to closer to the nucleus than an electron in a p-orbital, this means it will experience a large attraction from the nucleus and will be pulled in closer to the nucleus, so it will be lower in energy than the p-orbital electrons. This leads to a diagram similar to the one below which shows the relatives energies of the sub-shells in a hydrogen atom compared to those in multi-electron atoms: