Electronic Structure Analysis:How to Analyze Charge Density Difference and Charge Population?

Electronic Charge Density

Whether we are talking about charge density difference or charge population analysis, they both originate from the electronic charge density of a system.

But how is electronic charge density obtained?

We know that in first-principles calculations, it is necessary to solve the system’s electronic Schrödinger equation. The solution yields the ground-state energy $E$ and the wavefunction $\psi(\mathbf{r})$ of the electrons in the system.

The electron probability density of the system is given by the square of the wavefunction:

|\psi(\mathbf{r})|^2

The electronic charge density in a region is proportional to the probability density of electrons being present there. Therefore, the system’s electronic charge density is:

\rho(\mathbf{r}) = -e|\psi(\mathbf{r})|^2

where $e$ is the absolute value of the electron charge. When we define the unit charge of the system as $e = 1$ , the electronic charge density becomes:

\rho(\mathbf{r}) = -|\psi(\mathbf{r})|^2

What is the relationship between the electronic charge density obtained by solving the Schrödinger equation and the actual charge density of a system?

First, it is evident that the positively charged nuclei are not considered when calculating the electronic charge density. Second, when solving the Schrödinger equation, we usually solve it only for the outer valence electrons, treating the inner electrons as core electrons, which are not included in the calculation. Therefore, the calculated charge density differs from the system’s actual charge density. The good news is that the atomic core, composed of the nucleus and core electrons, changes very little during chemical reactions. As a result, their contribution to analyzing electron transfer is negligible. Hence, calculations of valence electrons can often represent the electron transfer behavior during chemical reactions well.

Charge Density Difference

With the electronic charge density in hand, let’s first discuss the charge density difference.

The so-called charge density difference refers to the difference between the charge density of a system after an operation such as adsorption or substitution and the charge density before the operation.

Why adsorption and substitution? Because in these operations, the positions of the atomic cores can remain unchanged.

For example, adsorption involves subtracting the charge density of the surface and the molecule before adsorption (in their post-adsorption geometries) from the charge density of the adsorbed system. Even though structural changes would occur during optimization, for charge density difference calculations, single-point calculations are performed using the post-adsorption geometry to ensure that the geometries remain the same.

The same principle applies to substitution, where one atom is replaced by another while keeping all atomic positions unchanged. In this case, the positions of the nuclei remain fixed, and the distribution of core electrons also does not change. Therefore, the variation in charge distribution equals the change in valence electron distribution.

Charge Density Difference 2D Heatmaps and 3D Isosurfaces

Figure 1:Charge Density Difference 2D Heatmaps and 3D Isosurfaces.

Charge density difference can be visualized in different ways:

As a colored 2D heatmap on a particular plane
As a 3D isosurface

As shown in Figure 1, these are the charge density difference maps (2D and 3D) for the same two systems.

In the left heatmaps, red indicates regions of charge accumulation and blue indicates depletion.
In the 3D isosurface plots on the right, yellow represents charge increase and light blue represents charge decrease.

These maps are often used to discuss bonding in systems. For example, Figure 1 shows the charge density difference maps for COOH adsorption on FeN₄/C and N/C. From the left heatmaps, both systems exhibit charge accumulation between the adsorbate and the surface. However, the color scales differ: in the left image, the highest charge density is 0.02, while in the right, it is 0.005. This indicates that in the FeN₄/C system, more electrons accumulate in the bonding region, leading to stronger adsorption.

Bader Charge Population

While charge density difference maps allow for intuitive visualization of electron transfer during chemical processes, they are qualitative. When regions of charge increase and decrease have complex geometries and are intertwined, meaningful conclusions are difficult to extract from the images. Therefore, quantitative methods are sometimes used to count the number of electrons associated with each atom, which helps analyze atomic-level charge transfer during a chemical process. These methods fall under charge population analysis.

There are many types of charge population analyses—Bader, Mulliken, Löwdin, Hirshfeld, etc. They may count not only the charge on each atom but also the electron numbers in atomic orbitals. Here we focus only on Bader charge analysis.

Bader Charge Analysis Principle

Bader charge analysis is based on the concept of zero-flux surfaces in the charge density.

How are boundaries between atoms defined?

Since the electron density is highest near the nucleus, it decreases with increasing distance from the nucleus. The direction in which the density decreases the fastest is the gradient direction. Moving along this direction causes continuous electron density decrease—until one reaches a region influenced more by another atom. Across this boundary, although moving further from atom A, one moves closer to atom B, and the density increases. On this boundary, the gradient of charge density is zero—it’s called the zero-flux surface.

The region around each nucleus enclosed by zero-flux surfaces is defined as that atom’s "domain" in Bader analysis. Integrating the electron density within this domain gives the number of electrons assigned to that atom—this is the basis of Bader charge analysis.

A Typical Charge Population Table

Figure 2:A Typical Charge Population Table.

Charge population results are often tabulated.

Sometimes they report the number of valence electrons per atom (first row in Figure 2).
Other times they add the nuclear charge to report the total atomic charge (second row in Figure 2).

Bader charge analysis is commonly used to determine charge transfer during adsorption and explain the strength of the adsorption interaction.

If significant charge transfer occurs, it suggests strong interaction between the surface and adsorbate.
If little charge transfer occurs, interaction may be weak.

However, there are exceptions to the latter. For example, O–O or N–N bonds can be strong due to identical electronegativity, leading to little net charge transfer.

Thus, while substantial electron transfer usually indicates strong interaction, lack of charge transfer does not necessarily mean weak bonding.

Conclusion

This article introduced the concepts of charge density difference and Bader charge analysis.

Both are based on the spatial distribution of valence electrons:

Charge Density Difference: Defined as the difference in charge density before and after a process involving fixed nuclei.
Bader Charge Analysis: Defines the charge associated with each atom using the zero-flux surfaces of the charge density.

Both methods can be used to analyze electron transfer during a process and thus infer the presence and strength of chemical interactions.