What do we get by integrating over the charge density data of a material system calculated with DFT?

Question

I am confused as to whether integrating over the charge density gets us the number of valence electrons in the material system or the total number of valence electrons in the material system.

Let's take an example: say I have the charge density of Al2O3, which contains exactly 2 atoms of Aluminim and 3 atoms of Oxygen. Now we all know that Al has 3 valence electrons and O has 2 valence electrons. So this charge density has a total of 2*3 + 3*2 = 12 valence electrons right? So if I integrate over the charge density should I get 12 as the result?

It would help all of us understand this better, if one would write out the formula to obtain the valence electrons from the charge density array (rho).

Is no. of valence electrons = round(vol*np.sum(rho)/rho.size) in python?

Please provide suitable references to your answer.

Answer 1

The integration of the charge density over the entire system gives you the total charge of the system. This value should be equal to the total number of electrons in the system, accounting for both the occupied and unoccupied states. Integrating over the charge density of Al2O3 will not directly give you the number of valence electrons in the system. The charge density represents the distribution of electron density in space, but it does not provide a direct count of the number of valence electrons. The formula you provided in Python calculates an estimate of the number of valence electrons. Here's a breakdown of each component of the formula: np.sum(rho),this calculates the sum of all the elements in the charge density array. It sums up the electron density values across all grid points in the system. vol, this represents the volume of the system. rho.size, this returns the total number of elements in the charge density array. It represents the total number of grid points or voxels used to discretize the system. np.sum(rho)/rho.size,this calculates the average electron density by dividing the sum of the electron density values by the total number of grid points. It gives you an estimate of the average electron density in the system. vol * np.sum(rho)/rho.size, this multiplies the average electron density by the volume of the system. The result is an estimate of the total number of electrons in the system. round(vol * np.sum(rho)/rho.size), is the round function applied to round the calculated estimate of the total number of electrons to the nearest whole number. This is done because the number of valence electrons must be an integer value. By multiplying the average charge density by the volume of the system and rounding the result to the nearest whole number, you obtain an estimate of the total number of electrons in the system. In the context of valence electrons, this estimate assumes that the charge density represents the electron density associated with the valence electrons. However, it's important to note that this formula provides an estimate and assumes a uniform charge density distribution throughout the system. It may not accurately capture the precise number of valence electrons, especially in more complex or non-uniform systems. The actual determination of valence electrons typically involves considering the electronic configuration of the individual atoms in the material or performing population analysis methods.