atomview is an application used to visualize atomic orbitals.
atomview supports visualization of a large number of complex and
real atomic orbitals and includes probability contour, multi-probability
contour, and volume density visualizations of the 3D atomic orbitals.
atomview visualizations of the (n, l, m) = (3, 2, 0)
orbital.
Left: solid contour visualization enclosing 50% probability.
Center: Transparent multi-contour visualization enclosing 20%, 50%,
and 80% probability.
Right: Transparent cloud volume visualization.
Currently atomview only runs on Windows.
The executable can be
:download:`downloaded here <../../dist/AtomView.exe>`.
What are atomic orbitals? Consider a single Hydrogen atom. This atom consist of a small positively charged nucleus consisting of a single proton along with a small negatively charged electron which "orbits" the proton. Because these particles are so small their dynamics are described by quantum mechanics, and in particular, by the Schrödinger equation. For a Hydrogen atom, the (time-independent) Schrödinger equation is given by
.. TODO:: Citation
E \psi = -\frac{\hbar^2}{2m} \nabla^2 \psi - \frac{e^2}{4\pi \epsilon_0 r} \psi.
Solutions \psi to the Schrödinger differential equation are called wavefunctions. In the same way the state of a classical billiards ball is specified by its position and velocity, the state of a quantum electron is described by the wavefunction. In the expression above E is the energy of the system when the electron's state is specified by the corresponding \psi. \hbar is Planck's reduced constant, m is the mass of the electron, e is the charge of the electron, and \epsilon_0 is the permittivity of free space.
In the case of the Hydrogen atom, the wavefunctions are called atomic orbitals. In general, these wavefunctions are complex-valued functions of 3D space \psi(x, y, z) = \psi(r, \theta, \phi). Mathematically, the wavefunctions are just specific solutions to the Schrödinger differential equation. Physically, the squared amplitude of the wavefunction for a particle at some point in space, |\psi(x, y, z)|^2, is interpreted as the probability of finding the particle at that location in space.
There are an infinite number of solutions to Schrödinger's equation for the Hydrogen atom. These solutions are enumerated by three integer indices, n, l, and m. n, l, and m are called the Hydrogen quantum numbers. n is any (non-zero) positive integer, l is any non-negative integer satisfying 0 \le l < n and m is any integer satisfying -l \le m \le l. The corresponding solutions are given by
E_{n, l, m} =& - \frac{me^4}{2(4\pi \epsilon_0)^2 \hbar^2} \frac{1}{n^2}\\
\psi_{n, l, m}(r, \theta, \phi) =& \sqrt{\left(\frac{2}{2 a_0}\right)^3 \frac{(n - l - 1)!}{2n(n+l)!}}
e^{-\rho/2} L_{n-l-1}^{2l+1}(\rho) Y_l^m(\theta, \phi)
- a_0 is the Bohr radius, equal to a_0 = \frac{4\pi \epsilon_0 \hbar^2}{me^2}.
- \rho is a rescaled radial coordinate given by \frac{2}{na_0} r.
- L_{n-l-1}^{2l+1}(\rho) is a generalized Laguerre polynomial of degree n-l-1.
- Y_l^m(\theta, \phi) is the spherical harmonic of degree l and order m. \theta is the polar coordinate angle and \phi is the azimuthal coordinate angle.
Note that the energy E_{n, l, m} only depend on the n quantum number. The square root term in the expression for \psi_{n, l, m} is a pre-factor to ensure the squared wavefunction integrates to unity when integarting over all space (this is critical for the probability interpretation of the wavefunction to be sensible). The radial behavior is given by e^{-\rho/2}L_{n-l-1}^{l+1}(\rho) and the angular behavior is given by Y_l^m(\theta, \phi).
atomview is dedicated to visualizing the Hydrogen atomic orbitals described above. It can be challenging to visualize a complex-valued scalar function on 3D space but there are a few strategies we can use. In this section we will build up our understanding of different visualization techniques to build up to an understanding of the visualizations provided by atomview.
The simplest Hydrogen atomic orbital is the \psi_{1, 0, 0} orbital. This wavefunction is given by
\psi_{1, 0, 0}(r, \theta, \phi) = \frac{1}{\sqrt{\pi} a_0^{3/2}} e^{-r/a_0}
We see that this orbital is purely a function of r with no angular \theta or \phi dependence. This means we can simply visualize it's behavior on a regular 1D plot:
We see that the wavefunction is maximal at the origin and then the amplitude decreases exponentially as the radius increases.
The next more sophisticated way we can visualize this wavefunction is by plotting the amplitude of the wavefunction on a 2D slice of space using a density plot where the brightness of the plot corresponds to the amplitude.
In both the z and y slices the wavefunction appears as a circle that is bright at the middle and whose brightness decreases as the radius increases. This begins to show the spherical symmetry of this wavefunction. In fact, the wavefunction look the same no matter which 2D slice plane passing through the origin was chosen.
Let us now consider 3D visualization techniques. First, we can visualize the wavefunction as a 3D cloud where each voxel of space is transparent, with an opacity proportional to the probability of finding a particle there. This is similar to viewing a regular cloud where the opacity of each voxel of space is proportional to the density of cloud-stuff in that region.
We see that this looks like a spherical cloud that is most dense in the center.
We now turn to 3D iso-probability contour surface visualizations. If we have a wavefunction \psi then the squared magnitude |\psi|^2 is related the probability of finding a particle at a given location. Suppose we pick a value p < \text{max}\left(|\psi|^2\right). There will be a closed and bounded 2D surface of points in 3D space which satisfy |\psi|^2 = p. If we integrate up the probability contained inside this surface then can determine the probability P that an electron is found inside the surface.
P = \int_{|\psi|^2 < p} |\psi|^2 dV
For any chosen probability P we can numerically determine the required value for p such that the corresponding iso-probability contour |\psi|^2 = p contains P probability. Below we plot two types of iso-probability contour plots for the \psi_{1,0,0} waveform.
Left: solid iso-probability contour plot for \psi_{1, 0, 0} corresponding to P=0.5. Right: Multiple transparent iso-probability contours for \psi_{1, 0, 0} corresponding to P=[0.2, 0.4, 0.6]. Each iso-probability has an opacity equal to the relative squared magnitude of the wavefunction on that surface. This plot gives a similar effect to the volume desnity plot.
We will find that iso-probability contour surface visualizations can give us good intuitions for the general shape of an orbital even though they don't technically give us information about the value of the function at all points in 3D space.
The next most complicated orbital is the (2, 1, 0) orbital. This wavefunction is given by
\psi_{2, 1, 0} = \frac{\sqrt{2}}{8\sqrt{\pi}a_0^{3/2}}
\left(r/a_0\right) e^{-\frac{1}{2}(r/a_0)}
\cos(\theta)
This orbital has a few important features beyond those of the \psi_{1, 0, 0} orbital. The first is that it now has dependence on the polar angle \theta given by \cos(\theta) (though there is no dependence on the azimuthal angle \phi). The second is that the wavefunction is positive in some regions of space (0 \le \theta < \pi/2) and negative in others (\pi/2 < \theta <= \pi). Below, we will introduce strategies to incorporate this information into our visualizations.
First, as before, even though this wavefunction has angular dependence, we can still visualize the radial dependence on a 1D plot.
We see that wavefunction is now zero at the origin, then has a finite lobe of amplitude before decaying exponentially at large radii. Note also that this wavefunction has a larger radial extent than the \psi_{1, 0, 0} wavefunction. Indeed, the wavefunction radial extent scales as n^2.
We can again plot 2D density plots of z and x slices:
We see there is no density along the z=0 plane because this plane corresponds to \theta=\pi/2 and \cos(\pi/2)=0. However, in the x=0 plane we now see two colors. We see red for \theta<\pi/2 where the wavefunction is positive, but we see that blue has been used for \theta>\pi/2 where the wavefunction is negative. We see there is a positive red lobe above the z=0 plane and a negative blue lobe below the z=0 plane.
Finally, we can utilize the same three 3D visualization techniques from above, simply adopting the red/blue convention for positive/negative parts of the wavefunction.
Left: volume density visualization for \psi_{2,1,0}. Center: Solid iso-probability contour visualization for \psi_{2, 1, 0} corresponding to P=0.5. Right: Transparent iso-probability contour visualization for \psi_{2, 1, 0} corresponding to P=[0.2, 0.4, 0.6].