# The true shape of p orbitals

I'm trying to model $$\mathrm{p}$$ orbitals using a 3D program. What confuses me is that there are 2 shapes of $$\mathrm{p}$$ orbitals. I'm not sure which one is the accurate model. And what is the reason to have 2 types of shape?

Shape 1

Shape 2

• The second one is more accurate. The first one is easier to comprehend. May 28, 2020 at 9:12
• Out of curiosity, what 3D program are you using for modelling orbitals? The renders you've posted look pretty good. May 28, 2020 at 9:44
• @andselisk those are not modelled by me they are 2d images i found on internet.not sure what program they have used. but i guess any popular 3d program can do that.i'm using blender by the way May 28, 2020 at 9:58
• The Wikipedia Page on atomic orbitals (en.wikipedia.org/wiki/Atomic_orbital) is actually quite good. One should keep in mind that the electron density of an atom is spherical, so the orbitals also need to reflect that. May 28, 2020 at 11:04
• These are not the shape of p orbital. These shapes are the place around the subshell which tells you the possibility of electron around that subshell.( Probability of finding electron in some area is greater than 60% is mostly considered as a shape of that subshell. P orbital whose l=1 has dumbbell shape which is represented by both the figures May 25, 2021 at 8:46

There is a misconception here. A p orbital is a 3D-function, and these functions don't have shapes, they have values at any point in space. If you describe an electron distribution with one of these functions, you can plot contours at a chosen value, and these contours have shapes. For example, you could choose a contour level such that an electron has a 90% chance of being inside the shape defined by the contour level.

[OP...] What confuses me is that there are 2 shapes of p orbitals.

There are actually four common graphs depicting p orbitals, and a lot of conceptual drawings that don't have an exact mathematical shape. The graphs differ in whether they show the wave function or the square of it (the latter is proportional to the probability of finding the electron in that location). They also differ in whether they use a cartesian coordinate system, or plot the angular dependence in a polar plot.

The one you probably want is a contour plot of the probability density (you could also say the electron density). Here is an example from an article in the Journal of Chemical Education (center panel):

The shape of the contour lines of the wavefunction are the same, but you can have positive and negative values (and because of the squaring involved, the contour lines also have different values). Your "shape 2" is a contour plot of the wavefunction, with two contours of equal magnitude and opposite sign.

## Polar plots

The other type of depiction is a polar plot, where the distance from the origin is the value of a function. This is different from a Cartesian coordinate system, where the nucleus is at the origin, and the distance from the origin corresponds to the distance from the nucleus.

If you plot the angular component of the wavefunction (you can think of this as keeping the distance from the nucleus constant, and sampling different angles), you get what looks like two spheres. A polar plot has no good way of dealing with negative values, so often one part of the curve is shown in a different color.

If you plot the square of the angular component, you get a shape closer to your "shape 1". For technical reasons, the plots are rotated by 90 degrees (the angle zero is defined along the x-axis in Wolfram Alpha, which I used to plot these).

So what does your "shape 1" represent? The left-most panel looks like it wants to show an electron density, but it is very inaccurate. There are other artists' interpretations floating around, some that look like they are actual graphs of functions, such as this one:

It is not clear which function, if any, this shows a graph of.

[OP...] The true shape of p orbitals

There is none. You have a choice of representations, and it depends on the context which one, if any, makes sense.

Which representation you would use depends on what kind of information you would want to represent. The p-orbitals (and all orbitals with a finite angular momentum quantum number $$l$$, for that matter) have a spatially dependent phase: The nodal planes in your bottom representation separate the parts of the p-orbitals that have opposite phases. In other words, there is a phase difference of $$\pi$$.

Whether you would want to display the phase difference depends on the property of the p-orbitals that you would like to represent. If you just want to display the part of the space that captures e.g. 90% of the probability density to find an electron, then the top representation will suffice. However, once you start to consider interactions with other orbitals, the phase will be important, as it will determine whether the interaction is bonding or antibonding. In that case, the colour of the p-orbital and other involved orbitals can be included to emphasise the type of interaction.