What Is? 

Electron density refers to the probability of finding an electron at a certain point in space. In quantum mechanics, the uncertainty principle describes a constraint on an exact simultaneous measurement of the electron's position and momentum. This means that the position of the electron can only be represented statistically through a probability. In general, to calculate the electron density, the wave function Ψ, a mathematical represention that describes the behavior of the electron, is multiplied by its complex conjugate Ψ* and then integrated over the spatial positions of all other electrons. The product of this multiplication is represented as which gives us the probability of finding an electron in a given space.  

History 

There were many discovers in quantum mechanics before we were able to calculate electron density. In 1900, Max Planck argued that radiation must consist of discrete packets of energy called 'quanta.' In 1905, Albert Einstein proposed that the energy of these quantized packets of light is equivalent to the frequency of light multiplied by a constant. By the 1920s, Louis de Broglie and Erwin Schrodinger established the mathematical backing of particle-wave duality. Around the same time, Werner Heisenberg developed his theory on indeterminateness, also known as the uncertainty principle, which describes how any observation of the position of an electron would alter the momentum by an indeterminable amount. Through this picture of electrons, one can no longer view atoms as electrons on orbits with a defined radii but rather a picture of electron probability around the nuclei. In 1928, Max Born established his ideas of statistical quantum mechanics, especially how electron density can be calculated by squaring the wave function. With the ideas of statistical quantum mechanics, we can now predict a probability distrubuition of electrons in a system and start to learn more about the quantum world around us. For example, molecular orbital shapes are the result of electron density calculations. From calculating the electron density of electrons in certain regions of space, we can visualize a 3D spatial distribition of an orbital.  

Detail 

In general, a probability density distribution can be found by 

where  

In a multi-electron system, the particle density distribution of an N-electron system is found by 

where {} and can be defined as 

where  

The Born interpretation of finding electron density requires the wave function for a single particle to be normalized to ensure that the probability of finding that particle is 1 over all space.  

This can be applied to a system of N  particles with positions {} where the normalization would be the following  

Example 

Let's graph the electron density of a molecule! We will do so using the Quantum chemistry toolbox  in Maple. 

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First, let's pick a molecule. I picked Vanillin because it has 'Van' in it. We can find the atomic coordinate of the molecule by the following command. 

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(4.1)

 

And, Vanillin looks like the following below in ball and stick form.  

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The following operation lets you find a variety of information about your molecule including molecular orbital energy, populations, and symmetry. This sequence can be used with a wide range of electron structure methods including HarteeFock, DensityFunctional, and FullCI.  

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(4.2)

 

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(4.3)

 

The electron density plot can be done using the following sequence where we can vary the densitycutoff to gain more insight on electron density of the molecule.  

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When we used a higher densitycutoff parameter, we can see that a greater amount of electron density comes from the oxygen molecules (red spheres). This makes sense due to the higher electronegativity that oxygen demonstrates compared to carbon atoms.  

Selected References 

1. W.Heisenberg, The Physical Principles of the Quantum Theory (University of Chicago Press, 1930)