🔬 physics

Radial models of helium and atomic hydrogen in two and three dimensions

This paper presents a unified radial Laguerre-basis method that solves atomic hydrogen and helium in both two and three dimensions by mapping dimensional changes to a single parameter, revealing distinct electron–electron interaction kernels for intrinsic 2D versus planar 2D models and reporting the first radial energies for intrinsic Gauss-law 2D helium with high accuracy.

Original authors: Dmitry Konovalov

Published 2026-07-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dmitry Konovalov

Original paper licensed under CC BY 4.0 (https://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 you are trying to understand how tiny particles, like electrons, behave around a nucleus (the center of an atom). Usually, we think of these particles moving in our normal, three-dimensional world (up/down, left/right, forward/backward). But sometimes, in special materials or extreme environments, these particles are forced to move as if they are stuck on a flat sheet of paper, existing in only two dimensions.

This paper is like a master key that unlocks the secrets of how atoms behave in both 3D and this flat 2D world, using a single, unified method.

Here is the breakdown of the paper's story, using simple analogies:

1. The "Universal Remote Control" for Atoms

The author, Dmitry Konovalov, built a computer program that acts like a universal remote control for atoms.

The Problem: Usually, to study an atom in 3D, you need one set of math rules. To study it in 2D, you need a completely different set of rules. It's like needing a different remote for your TV and a different one for your stereo.
The Solution: The author found a way to write the math so that the "remote" only needs one button press to switch modes. By changing just one number (called $L$ $L$ ), the same code can solve the problem for a 3D atom or a 2D atom.
- Set the number to 0, and it solves for our normal 3D world.
- Set the number to -0.5, and it instantly switches to the flat 2D world.
- This means the author didn't have to write two different programs; he wrote one that does both perfectly.

2. The "Flat World" Mystery: Two Different Rules

The paper highlights a confusing point about "2D atoms." Imagine you have a flat sheet of paper. There are two ways to imagine electrons living there:

Scenario A (The "Mixed" World): The electrons are stuck on the paper, but the invisible force field (electricity) still comes from the 3D world above and below. This is like having a 2D character in a 3D video game. The math here is messy and complicated.
Scenario B (The "Pure" World): Both the electrons and the force field are strictly 2D. The force field follows the rules of a flat universe. In this world, the force doesn't get weaker like $1/r$ $1/ r$ (as it does in 3D); instead, it behaves like a logarithm (a specific mathematical curve).
- The Analogy: In 3D, if you shout, the sound gets quieter quickly as you move away. In this "Pure 2D" world, the sound gets quieter much more slowly, following a specific logarithmic pattern.

The author's code can handle both scenarios. It can simulate the messy "Mixed" world and the "Pure" world, allowing scientists to compare them directly.

3. The "Magic Kernel" (The Secret Shortcut)

One of the most interesting discoveries in the paper is about how electrons talk to each other.

In 3D: When two electrons interact, the math simplifies beautifully. The complex interaction depends only on the distance of the electron that is farther away from the center. It's as if the closer electron is invisible to the math, and only the "outer" one matters.
In Pure 2D: The author found that the "Pure 2D" world has a magic shortcut too! Just like in 3D, the interaction simplifies to depend only on the outer electron's distance.
The Contrast: However, in the "Mixed" world (Scenario A), this shortcut doesn't work. The math becomes a complicated, wiggly curve (an elliptic integral) that is much harder to calculate.
The Takeaway: Nature is elegant when the rules match the dimension. When you mix 3D rules with 2D space, you get messy math. When the rules match the space (Pure 2D), the math becomes simple and clean again.

4. Testing the Engine

To prove his "Universal Remote" works, the author tested it on two types of atoms:

Hydrogen (1 electron): He compared his results against known exact answers and found they matched perfectly.
Helium (2 electrons): This is harder because the two electrons push against each other.
- He found that by only looking at the "radial" part (how far the electrons are from the center) and ignoring their side-to-side spinning, he could get 99% accuracy compared to the full, complex calculation.
- Think of it like estimating the weight of a suitcase. You don't need to know the exact shape of every sock inside; just knowing the total volume gets you 99% of the way there.

5. The New Discovery: The "Pure" 2D Helium

The biggest news in the paper is that the author calculated the energy levels for Helium in a "Pure" 2D world (where the force field is also 2D) for the very first time.

Before this paper, no one had published these specific numbers.
Because the force field in this world is different (logarithmic), the electrons bind together differently. The author's code provided these new, never-before-seen numbers, giving scientists a new benchmark to test their theories against.

Summary

The paper is a tour de force of efficiency. It shows that by changing just one parameter, you can switch between 3D and 2D physics. It clarifies that "2D atoms" can mean two very different things (mixed vs. pure), and it proves that when the physics matches the dimension (Pure 2D), the math simplifies into a beautiful, predictable pattern. Finally, it provides the first-ever map of energy levels for a helium atom living in a strictly 2D universe.

Technical Summary: Radial Models of Helium and Atomic Hydrogen in Two and Three Dimensions

Problem Statement
The paper addresses the modeling of few-electron atomic systems (hydrogen and helium) where the effective spatial dimensionality is reduced from three dimensions (3D) to two dimensions (2D). A central ambiguity in "two-dimensional Coulomb problems" is whether the reduction applies only to the electron motion (planar model) or also to the mediating electrostatic field (intrinsic Gauss-law model).

Planar Model: Electrons are confined to a plane, but the interaction remains the 3D $1/r$ law.
Intrinsic Model: Both matter and the electromagnetic field are 2D, resulting in a logarithmic interaction derived from Gauss's law.
Intermediate Cases: The paper also considers crossover potentials (log-Coulomb) and Rytova–Keldysh potentials, which model embedded 2D materials where screening creates a transition from logarithmic short-range to Coulombic long-range behavior.

Existing literature often treats these models with distinct methods. The goal of this work is to establish a uniform radial formulation that solves the 3D S-wave ( $l=0$ ) and 2D $m=0$ sectors using a single computational framework, allowing for a direct comparison of how dimensionality and interaction laws independently affect binding energies and spectra.

Methodology
The author employs a Laguerre-basis method on the half-line ( $r \geq 0$ ) to solve the radial Schrödinger equation. The core innovation is a "dimensional transcription" that unifies the 3D and 2D problems through a single parameter, $L$ .

Unified Hamiltonian: The effective radial Hamiltonian for both dimensions is expressed as:
$\hat{H} = \hat{K}_{1d} + \frac{L(L+1)}{2r^2} + V(r)$
where $\hat{K}_{1d} = -\frac{1}{2}\frac{d^2}{dr^2}$ .
- For 3D S-wave ( $l=0$ ): $L = 0$ .
- For 2D $m=0$ : $L = -1/2$ .
  This allows the same code to handle both dimensions by simply changing the parameter $L$ .
Interaction Kernels:
- 3D ( $1/r_{12}$ ): The angular projection of the electron-electron interaction for $l_1=l_2=0$ yields the standard outer-radius kernel $1/r_>$ , where $r_> = \max(r_1, r_2)$ .
- Intrinsic 2D ( $-2\ln(r_{12}/r_0)$ ): The angular projection for $m_1=m_2=0$ yields a logarithmic outer-radius kernel $-2\ln(r_>/r_0)$ .
- Planar 2D ( $1/r_{12}$ ): Confining 3D electrons to a plane while keeping the $1/r_{12}$ law results in a mixed-dimensional kernel involving complete elliptic integrals, which lacks the simple outer-radius property.
Radial Projection: For helium, the two electrons are restricted to the matched radial sector ( $l_1=l_2=0$ in 3D; $m_1=m_2=0$ in 2D). This is a deliberate approximation that isolates the effects of radial measure and interaction laws from full angular correlation.
Validation: The solver is validated against:
- Exact planar 2D hydrogen energies.
- Published dimensionless logarithmic spectra for intrinsic 2D hydrogen.
- Full correlated $L=0$ benchmarks for 3D helium.

Key Contributions and Results

Unified Framework: The paper demonstrates that the dimensional difference between 3D and 2D radial problems resides entirely in the parameter $L$ and the specific form of the interaction potential $V(r)$ . A single validated codebase successfully reproduces results for 3D, planar 2D, and intrinsic 2D models.
Outer-Radius Correspondence: A key theoretical observation is that when the interaction law matches the dimension of the field (3D Coulomb in 3D, Logarithmic in 2D), the angularly projected electron-electron kernel collapses to a function of the outer radius $r_>$ alone. This simplifies the 2D logarithmic case to a form structurally identical to the 3D Coulomb case, a property not widely recognized.
Intrinsic 2D Helium Energies: The paper reports, for the first time, radial energies for intrinsic Gauss-law 2D helium ( $m_1=m_2=0$ ). These results are presented in atomic units with a matched reference length $r_0 = e/2$ .
Accuracy of Radial Projections: The restricted radial projection ( $l_1=l_2=0$ $l_{1} = l_{2} = 0$ or $m_1=m_2=0$ $m_{1} = m_{2} = 0$ ) recovers full correlated energies with high accuracy:
- Ground States: Errors are approximately 0.85% in 3D and 1.15% in 2D.
- Excited States: Errors are an order of magnitude smaller ( $\lesssim 0.1\%$ ).
- This confirms that omitted angular correlation is concentrated in the compact ground configuration.
Potential Comparisons:
- Planar vs. Intrinsic: The planar 2D model (using $1/r$ ) yields significantly more compact and strongly bound states than the 3D counterpart, but less bound than the intrinsic logarithmic model at short range.
- Crossover Models: The paper introduces a simple analytic log-Coulomb crossover potential ( $X$ ) and compares it with the Rytova–Keldysh potential ($RK$). The $X$ potential is found to be more attractive at short range due to its matching conditions, leading to deeper binding energies than $RK$ for the states considered.
- Scaling: The dimensionless eigenvalues of the intrinsic logarithmic potential are shown to be independent of the nuclear charge $Z$ , a property verified numerically.

Significance and Claims
The paper claims that its primary value lies not in discovering new quantum mechanics, but in providing a clean separation of generic dimensional reduction effects from dimension-specific interaction laws.

Modularity: By treating the electron-nucleus potential and the projected electron-electron kernel as tabulated inputs, the solver is "interaction-agnostic." It can handle screened, confining, or crossover potentials without code modification.
Benchmarking: The work provides a validated reference set for correlated calculations in 2D systems, particularly for the intrinsic Gauss-law model where no benchmarks previously existed.
Conceptual Clarity: It clarifies that the "two-dimensional Coulomb problem" is ambiguous until the field law is specified. The elliptic-integral kernel of the planar model is identified as an artifact of mixing 3D fields with 2D matter, whereas the logarithmic kernel is the true Green's function of 2D electrostatics.

The author explicitly states that the open-source Python code will be released upon publication to facilitate further study of these radial models and their extensions.