Atomic Orbitals Density

Density plots of atomic orbitals as electron positions are sampled.

Open the Atomic Orbitals Density simulation

This simulation builds up density plots for the wavefunctions of different atomic orbitals to show their spatial distribution around an atom. The phase of the wavefunctions are indicated by the colour of the points: blue is positive, pink is negative. You can show the Cartesian axes using the checkbox, as well as the location of any radial or angular nodes.

What is an orbital?

An orbital is a mathematical function (wavefunction) that describes the behaviour of electrons, including their location around an atom. They also encode other properties of the electrons, including their momentum and energy.

The most intuitive understanding of an orbital is where the electron is likely to be found around the atom. It is possible to think of the orbital (or more strictly the square of the orbital's wavefunction) as a probability density function; the higher the value of the orbital's squared wavefunction, the more likely the electron will be found in that region. This means we can ignore the sign of the wavefunction for now. In terms of the simulation plots, the higher the density of points, the more likely the electron will be found there. This behaviour stems from the quantum mechanical nature of electrons: they exhibit wave-particle duality and will behave more like waves in some contexts and more like particles in others. Trying to 'find' the electron around the atom means it will behave more like a particle.

Shapes of atomic orbitals

You will notice that the orbitals have different shapes. How these shapes vary can be broken down into two parts: a radial component (based on the distance from the nucleus) and the angular component (based on the angles around the nucleus).

Angluar component

The angular component depends entirely on what type of orbital it is: $s$, $p$, $d$ or $f$.

$s$ orbitals are spherically symmetric and look the same in every direction. Their wavefunctions do not depend on any angles, only the distance from the nucleus $r$.

$p$ orbitals have two 'lobes' on opposite sides. Between the two lobes is a plane where electrons are not found; this is known as a node (analagous to a node on a vibrating string where there is no movement). These are called angular nodes. You can see from the plane slices that the three p orbitals have nodes in one of the three planes perpendicular to one of the Cartesian axes. This is where their names come from; $p_x$ has a node in the axis perpendicular to the $x$-axis, for example.

$d$ orbitals have more 'lobes'. The $xy$, $yz$, $xz$ and $x^2-y^2$ orbitals have four lobes arising from two angular nodes in perpendicular planes. The $z^2$ orbital is different; it has a torus (doughnut) shaped ring with two lobes above and below. This is because its angular nodes are cones rather than planes.

Radial component

The radial component depends on both the type of orbital ($s$, $p$, $d$ or $f$) and the shell ($1s$, $2s$ etc). Let's consider the $s$-orbitals.

All orbital wavefunctions decay exponentially from the nucleus. The $1s$ does so simply, decreasing from a maximum value at the nucleus. The $2s$ is more involved; it has a maximum at the nucleus and decays exponentially at large distances, but has a 'gap' at an intermediate distance. This is called a radial node and is a set distance from the nucleus where the electron is not found.

Mathematical Basis

Wavefunctions are fundamentally mathematical expressions. Looking at these expressions can help us see why they have their different shapes. You may have encountered expressions in three dimensions that use Cartesian coordinates ($x$, $y$ and $z$). These are less helpful for orbital wavefunctions so we use spherical polar coordinates ($r$, $\phi$, $\theta$) instead. $r$ is the distance from the nucleus (centred at the origin), $\phi$ is the angle in the $xy$-plane measured from the $x$ axis (similar to longitude) and $\theta$ is the angle measured from the $z$ axis (similar to latitude). $\phi$ takes values from $0\degree$ to $360\degree$ and $\theta$ takes values from $0\degree$ to $180\degree$.

Orbitals are defined by three numbers:

the principal quantum number, $n$, which corresponds to the shell,
the azimuthal quantum number, $\ell$, which corresponds to the type of orbital ($s$ is $\ell=0$, $p$ is $\ell=1$ etc),
the magnetic quantum number, $m_{\ell}$, which corresponds to the orientation of the orbital ($p_x$, $p_y$ etc).

$n$ can take integer values starting from 1. $\ell$ can take integer values from 0 up to the value of $n-1$, hence why the first shell ($n=1$) only has an $s$ orbital ($\ell$=0) and the second shell ($s=2$) only has $s$ and $p$ orbitals ($\ell=0$ or $1$). $m_\ell$ can take integer values from $-\ell$ to $+\ell$, so there is only one $s$ orbital, three $p$ orbitals ($m_\ell = -1,0,+1$), five $d$ orbitals ($m_\ell = -2,-1,0,1,2$), and so on.

Let's take the $2p$ orbitals ($n=2$, $\ell=1$) as an example.

$$\psi_{2p_x} = \frac{1}{4\sqrt{2\pi}\,a_0^{3/2}}\;\frac{r}{a_0}\,e^{-r/(2a_0)}\,\sin\theta\cos\phi$$ $$\psi_{2p_y} = \frac{1}{4\sqrt{2\pi}\,a_0^{3/2}}\;\frac{r}{a_0}\,e^{-r/(2a_0)}\,\sin\theta\sin\phi$$ $$\psi_{2p_z} = \frac{1}{4\sqrt{2\pi}\,a_0^{3/2}}\;\frac{r}{a_0}\,e^{-r/(2a_0)}\,\cos\theta$$

$a_0$ is a constant (the Bohr radius, about 53 pm), so the fist fraction in each expression is simply a constant that we can ignore for now. The second part, $\frac{r}{a_o}\,e^{-r/(2a_0)}$ is the radial component. The final part with trigonometric functions are the angular component.

All three orbitals have the same radial component as they are all $2p$ orbitals. The $2p$ radial component is plotted below.

The radial component increasing from zero to a maximum at $r=2a_0$ before decaying exponentially. As the radial component never crosses the $r$ axis, it has no radial nodes.

Now consider the angular components. The $2p_z$ orbital has a $\cos\theta$ term, meaning that the wavefunction is zero when $\theta=90\degree$, regardless of the values of $r$ and $\phi$. From the above diagram of spherical polar coordinates, every point with $\theta=90\degree$ lies on the $xy$ plane, meaning that the $xy$ plane is an angular node as the wavefunction is zero at any point on the plane. The $p_x$ and $p_y$ orbitals both have a $\sin\theta$ term, which is zero when $\theta=0\degree$ and $180\degree$, which corresponds to the $z$-axis and is not too insightful. However, they both have a $\phi$ term. For $p_x$, $\cos\phi=0$ when $\phi=90\degree$ and $270\degree$ which corresponds to the $yz$ plane. For $p_y$, $\sin\phi=0$ when $\phi=0\degree$ and $180\degree$, which corresponds to the $xz$ plane.