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Structure and information measures of few-electron systems under a spherically symmetric Gaussian potential within a density functional approach

This study employs a density functional approach with the generalized pseudospectral method to investigate the energies and information-theoretic measures of few-electron systems under a spherically symmetric Gaussian potential, revealing how variations in potential width and depth influence electron correlation, density localization, and the behavior of specific exchange-correlation functionals.

Original authors: Raveena Arya, Santanu Mondal, Amlan K. Roy

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Raveena Arya, Santanu Mondal, Amlan K. Roy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tiny, invisible ballroom where electrons dance. Usually, these electrons are free to roam the entire universe, but in this study, the scientists put them inside a special, soft-walled cage called a Quantum Dot. Think of this cage not as a hard steel box, but as a soft, invisible Gaussian cloud that gently pushes the electrons back if they try to wander too far.

The researchers wanted to see how changing the size and "tightness" of this soft cage affects the energy and behavior of the electrons inside. They looked at systems with just one electron (Hydrogen), two electrons (Helium), and even a few more (Lithium and Beryllium), including some excited states where the electrons are jumping around more energetically.

Here is a breakdown of their findings using simple analogies:

1. The Soft Cage (The Gaussian Potential)

The scientists used a mathematical shape called a "Gaussian potential" to represent the cage.

The Analogy: Imagine the cage is like a soft, rubbery bowl.
- The Width ( $R$ ): This is how wide the bowl is. If the bowl is very narrow, the electrons are squished tight against the center. If it's wide, they have room to spread out.
- The Depth ( $V_0$ ): This is how deep the bowl is. A deeper bowl holds the electrons more tightly.

2. The Energy Dance

The researchers calculated the energy of these electrons under different bowl conditions.

Squishing the Ball: When they made the bowl very narrow (small width), the electrons were forced into a tiny space. Just like trying to squeeze a large beach ball into a shoebox, the "pressure" (kinetic energy) skyrocketed, making the total energy very high.
Letting it Loose: When the bowl was very wide, the electrons acted like they were free in open space. Their energy settled down to what we see in normal atoms.
The Sweet Spot: The most interesting changes happened in the middle. As they adjusted the bowl's width, the energy didn't just go up or down in a straight line; it shifted dramatically in a specific range, showing that the "cage" has a powerful influence only when it's neither too tight nor too loose.

3. The "Fuzziness" of the Dancers (Information Theory)

To understand the electrons better, the scientists didn't just look at energy; they looked at "information measures." Think of this as measuring how "spread out" or "fuzzy" the electron's position is.

Shannon Entropy (The "Spread" Meter):
- Position ( $S_r$ ): This measures how spread out the electron is in space. The study found that as the cage got tighter, the electrons got more "localized" (less spread out), causing this number to hit a minimum.
- Momentum ( $S_p$ ): This measures how spread out the electron's speed is. Because of a rule in quantum mechanics (Heisenberg's Uncertainty Principle), if you squeeze the electron's position (make it less fuzzy), its speed becomes more fuzzy. So, as the cage got tighter, the position spread went down, but the speed spread went up.
- The Result: The study showed a seesaw effect: as the cage tightened, the electrons became very precise about where they were, but very uncertain about how fast they were moving.
Fisher Information (The "Sharpness" Meter):
- This measures how "sharp" or "jagged" the electron's density is. The study found that as the cage tightened, the electron cloud became sharper and more concentrated near the center, creating a peak in this "sharpness" measurement.

4. The "Loop" of Correlation

One of the most fascinating discoveries was about how electrons interact with each other (correlation).

The Analogy: Imagine two dancers trying to avoid bumping into each other.
The Loop: When the researchers plotted the energy of this interaction against the "spread" of the electrons while changing the cage size, the data didn't just go in a straight line. It formed a loop.
- As they tightened the cage, the system went through a cycle: the electrons got squeezed, their behavior changed, and then as the cage got even tighter, the system started to look like it was going back to the beginning, creating a circular path on their graph. This "loop" suggests that the electrons' behavior is complex and non-linear, reacting differently depending on exactly how tight the squeeze is.

5. Heavy vs. Light Atoms

The study also looked at heavier atoms (like Oxygen ions) compared to lighter ones (like Helium).

The Finding: For heavy atoms, the electrons are already pulled very tightly to the center by the nucleus (the core of the atom). It's like the dancers are already hugging the center of the room. Adding the "cage" didn't change much because the electrons were already squished tight by the nucleus itself. The cage only had a big effect on the lighter atoms where the electrons were more relaxed to begin with.

Summary

In short, this paper is a detailed map of what happens to tiny electrons when you put them in a soft, adjustable cage. The main takeaways are:

Tightening the cage makes electrons more precise in position but wilder in speed.
There is a "Goldilocks" zone (intermediate cage sizes) where the effects are most dramatic.
Heavy atoms are less affected by the cage because they are already tightly bound by their own nuclei.
The relationship between energy and uncertainty creates a unique "loop" pattern, showing that these quantum systems behave in complex, non-linear ways.

The researchers used advanced computer math (Density Functional Theory) to solve these puzzles, providing a new way to understand how we might control the properties of tiny materials by simply adjusting the "shape" of the space they live in.

Technical Summary: Structure and Information Measures of Few-Electron Systems under a Spherically Symmetric Gaussian Potential

Problem Statement
Quantum confinement in nano-structures, such as quantum dots (QDs), significantly alters the electronic and optical properties of atoms compared to their free counterparts. While various model potentials (harmonic, Woods-Saxon, etc.) have been proposed to describe these systems, the Gaussian potential (GP) offers a soft, continuous boundary useful for modeling effective interactions in nuclear reactions, molecular processes, and atoms confined within fullerene cages. Despite existing literature on the energetics of two-electron systems in GPs, there is a notable lack of comprehensive studies involving systems with more than two electrons, excited states, and information-theoretic measures. This work addresses these gaps by investigating the structural and information properties of H, He-like ions ( $Z=2-18$ ), Li, and Be confined within a spherically symmetric Gaussian potential.

Methodology
The study employs a time-independent Density Functional Theory (DFT) framework applicable to arbitrary atomic states. The core of the approach involves solving the radial Kohn-Sham (KS) equation:
$\hat{H}(\vec{r})\psi_i(\vec{r}) = \epsilon_i(\vec{r})\psi_i(\vec{r})$
where the effective potential includes electron-nuclear attraction, classical Coulomb repulsion, exchange-correlation (XC) potentials, and the confining Gaussian potential $V_{conf}(r) = V_0 [1 - \exp(-r^2/R^2)]$ .

Key methodological components include:

Exchange Potential: A physically motivated, work-function-based exchange potential is utilized, treating exchange energy as an interaction between an electron and its associated Fermi-Coulomb hole charge density.
Correlation Functionals: Electron correlation is analyzed using two distinct functionals: a local, parameterized Wigner functional and a non-linear, gradient- and Laplacian-dependent Lee-Yang-Parr (LYP) functional.
Numerical Solution: The Generalized Pseudospectral (GPS) method is employed to solve the KS equations. This basis-set-free approach utilizes a non-uniform, optimally distributed spatial grid to satisfy Dirichlet boundary conditions ( $\psi(0) = \psi(r_c) = 0$ ), ensuring high accuracy in eigenfunctions and eigenvalues.
Information Measures: The study calculates Shannon entropy in position ( $S_r$ ) and momentum ( $S_p$ ) spaces, and Fisher information in position space ( $I_r$ ). These are derived from the $r$ -space density and its Fourier transform. Correlated contributions are estimated by comparing results from XC functionals against exchange-only (XX) results.

Key Results
The investigation covers ground states of H, He, Li, and Be, as well as low-lying excited states ( $1s2s$ $^1S$ and $^3S$ ) of He. The primary findings regarding the variation of confinement parameters (depth $V_0$ and width $R$ ) and atomic number ( $Z$ ) are:

Energetics:
- Energy decreases monotonically as the confinement width $R$ increases, approaching the free-atom limit as $R \to \infty$ .
- As $R \to 0$ , energy becomes positive due to increased kinetic energy.
- For a fixed $V_0$ , the transition from negative to positive energy occurs at $R \approx 5$ for He, but at much smaller $R$ ( $\approx 1$ ) for Li and Be.
- As $Z$ increases, the nuclear attraction ( $-Z/r$ ) dominates the Gaussian confinement. Consequently, the influence of the GP diminishes for high- $Z$ ions ( $Z > 8$ ), and the system behaves more like a free atom even at moderate confinement.
- Correlation energies ( $|E_{corr}|$ ) exhibit a maximum in the intermediate $R$ region (around $R \sim 0.5-0.6$ for $V_0=5$ ), where the competition between the GP and nuclear potential is most significant.
Information-Theoretic Measures:
- Shannon Entropy ( $S_r, S_p$ ): $S_r$ exhibits a minimum in the intermediate $R$ region, indicating maximum spatial localization, while $S_p$ shows a corresponding maximum (momentum delocalization). As $Z$ increases, $S_r$ decreases (stronger localization) and $S_p$ increases.
- Fisher Information ( $I_r$ ): $I_r$ attains a maximum in the intermediate $R$ region, reflecting the sharpness of the electron density gradient. The peak shifts to lower $R$ values as $Z$ increases.
- Correlation Effects: The magnitude of correlated contributions to energy and information measures ( $E_{corr}, S_{corr}, I_{corr}$ ) is generally larger for lower $Z$ systems and diminishes as $Z$ increases. The Wigner functional generally yields larger correlation magnitudes than the LYP functional.
Fisher-Shannon Plane and Collin's Conjecture:
- The Fisher-Shannon plane ( $J_r$ vs. $I_r$ ) reveals a progressive localization and compression of electronic density as $Z$ increases, indicating a weakening of relative electron-correlation effects.
- Analysis of the relationship between correlated Shannon entropy ( $S_{corr}$ ) and correlated energy ( $E_{corr}$ ) as functions of $R$ and $V_0$ reveals a non-linear, loop-like feature (Collin's conjecture). This loop structure is more pronounced for higher $V_0$ , reflecting the system's transition between strong confinement and free-atom behavior.

Significance and Claims
The authors claim that this work provides the first comprehensive evaluation of ground- and excited-state energies, along with information entropies ( $S_r, S_p, I_r$ ), for atoms and ions confined within a Gaussian potential, specifically extending beyond two-electron systems to include Li, Be, and excited He states.

The study demonstrates that the Gaussian potential allows for the manipulation of energy and information measures by tuning dot parameters ( $V_0$ and $R$ ). The results highlight that while confinement significantly alters properties for low- $Z$ systems, the nuclear attraction dominates for high- $Z$ ions, rendering the confinement effects minimal. The identification of the loop-like feature in the $S_{corr}$ vs. $E_{corr}$ plane offers a multidimensional characterization of electron correlation that extends beyond traditional energy-based descriptions. The authors suggest these findings may motivate future investigations using other model confining potentials and the extension of such studies to complexity and entanglement measures.