Optimized basis of covariant density functional theory: point coupling functionals and excited states

This paper demonstrates that optimizing the harmonic oscillator basis frequency and size within covariant density functional theory using point coupling functionals significantly enhances the accuracy of calculated binding energies, fission barriers, and single-particle states for moderately sized fermionic systems, including the successful reproduction of neutron halo densities.

The Gist

Imagine you are trying to paint a perfect portrait of a complex, three-dimensional object (like a nuclear atom) using a limited set of building blocks. In the world of nuclear physics, scientists use a mathematical tool called the Harmonic Oscillator (HO) basis to build these "portraits" of atomic nuclei. Think of this basis as a set of Lego bricks of a specific size.

For decades, scientists have been using a standard, "one-size-fits-all" size for these bricks. However, just like trying to build a detailed model of a giant castle with tiny, standard bricks, you either need a massive number of them (which takes forever to build and breaks your computer) or the final picture looks a bit blurry and inaccurate.

This paper is about finding the perfect brick size for different types of nuclear "castles" so scientists can build accurate models much faster and with fewer bricks.

Here is a breakdown of what the researchers discovered, using simple analogies:

1. The Problem: The "Standard Brick" is Too Small

In the past, scientists used a fixed rule to determine the size of their mathematical "bricks" (the oscillator frequency). This rule was based on looking at just two or three specific atoms (like Oxygen and Lead) over 25 years ago.

• The Analogy: Imagine you are baking a cake. You've been using a measuring cup that was calibrated only for a tiny cupcake. Now you are trying to bake a giant wedding cake. If you keep using that tiny cup, you have to measure out thousands of scoops, and even then, the cake might not rise correctly.
• The Result: When scientists tried to use this old rule for larger or more complex atoms, they had to use huge numbers of "bricks" to get an accurate answer, and even then, the results weren't perfect.

2. The Solution: Custom-Sized Bricks (Optimization)

The authors developed a new method to "tune" the size of these bricks for every single type of atom they study. They call this the optimal scaling factor.

• The Analogy: Instead of using the same tiny cup for everything, they now have a smart measuring tool that automatically adjusts the cup size based on whether you are baking a cupcake, a loaf of bread, or a wedding cake.
• The Discovery: By adjusting this "cup size" (specifically, making it slightly larger than the old standard), they found that they could get the same high-quality results using far fewer bricks. For some heavy atoms, they reduced the number of bricks needed by nearly 20 layers, saving a massive amount of computer time.

3. The "Odd-Even" Wobble

The researchers noticed something strange: when they added bricks one by one, the accuracy of the model didn't go up smoothly. It wobbled up and down.

• The Analogy: Imagine walking up a staircase where every other step is slightly higher than the one before it. If you stop on an "odd" step, you feel a bit off-balance. If you stop on an "even" step, you feel different. This is called odd-even staggering.
• The Cause: This happens because of how the particles inside the atom interact with each other. The researchers found that by adjusting the "brick size" (the scaling factor), they could smooth out these wobbles, making the staircase flat and easy to climb. This makes it much easier to predict what the "perfect" infinite model would look like without actually building the infinite model.

4. The "Halo" Nuclei (The Fuzzy Edges)

Some atoms have a "halo"—a fuzzy cloud of particles (neutrons) that drifts far away from the center, like a fuzzy halo around a saint's head.

• The Challenge: Standard models with small "bricks" act like a cage with hard walls. They can't capture particles that drift too far out because the cage is too small.
• The Breakthrough: The researchers showed that if they use a very large number of bricks (a huge cage) and tune the size correctly, they can perfectly reproduce these fuzzy halos.
• The Limit: They found that for spherical (round) atoms, they can model these halos up to a certain size (about 80 particles). For oddly shaped (deformed) atoms, the limit is smaller (about 40 particles), but it is still a huge improvement over previous methods which couldn't do this at all.

5. Fission Barriers (The Mountain Pass)

To understand how atoms split (fission), scientists need to map the "energy landscape" of the atom. This is like mapping a mountain range to find the lowest pass to cross.

• The Risk: If your map is slightly off (even by a tiny amount), you might think a mountain pass is safe to cross when it's actually a cliff. In nuclear physics, a small error in calculating this "pass" (the fission barrier) can change the predicted lifespan of an atom by millions of years.
• The Fix: The researchers found that to get a map accurate enough to see these passes clearly, you need at least 20 layers of bricks and the correct "brick size" tuning. With this setup, they can predict the energy of these "passes" with extreme precision (within 100 keV), which is accurate enough to trust the predictions for heavy elements like those used in nuclear energy or weapons.

6. Single Particles (The Solo Dancers)

The paper also looked at the energy of individual particles dancing inside the nucleus.

• The Result: Using the optimized "brick size," the accuracy of predicting these individual energies doubled compared to the old method.
• The Exception: There is one group of dancers that is hard to catch: the very weakly bound neutrons (the ones on the very edge of the halo) with low momentum. For these specific particles, the "standard" brick size actually works better than the optimized one. It's like a specific type of shoe that fits a specific foot better than a custom-made pair.

Summary

In short, this paper is a "user manual update" for nuclear physicists. It tells them:

1. Don't use the old, fixed size for your mathematical building blocks.
2. Tune the size based on the specific atom you are studying.
3. Do this, and you can get super-accurate results (for binding energy, fission, and halo structures) using much less computer power.
4. Be careful with the "fuzzy edge" particles, as they sometimes need a different approach.

This allows scientists to study the heaviest and most complex atoms with a level of detail that was previously too expensive or impossible to calculate.

Technical Summary

Technical Summary: Optimized Basis of Covariant Density Functional Theory: Point Coupling Functionals and Excited States

Problem Statement
The basis set expansion method, particularly using the harmonic oscillator (HO) basis, is a standard tool in low-energy nuclear physics for solving many-body problems within Covariant Density Functional Theory (CDFT). However, practical applications require truncating the infinite basis to a finite size ($N_F$), introducing numerical errors that are difficult to quantify without exact solutions. Historically, CDFT calculations have relied on an oscillator frequency $\hbar\omega_0$ defined by a fixed scaling factor $f=1.0$ (based on $\hbar\omega_0 = 41A^{-1/3}$ MeV), a convention established over 25 years ago based on limited analysis of spherical nuclei. While a recent study [1] optimized this basis for meson exchange (ME) functionals, point coupling (PC) covariant energy density functionals (CEDFs) and the description of excited states remained unoptimized. Furthermore, the convergence behavior of PC functionals differs from ME functionals, converging from above rather than exhibiting the specific convergence patterns of ME functionals, necessitating a systematic re-evaluation.

Methodology
The authors employ the Relativistic Hartree-Bogoliubov (RHB) framework using both spherical and axially deformed computer codes. The study utilizes a large set of spherical and deformed even-even nuclei distributed across the nuclear chart (Fig. 1) to benchmark solutions.

• Benchmarking: Exact solutions corresponding to an infinite HO basis (or extrapolated to such a basis) serve as the reference. For spherical nuclei, exact infinite basis solutions are obtained directly. For deformed nuclei, where direct calculation to infinite $N_F$ is computationally prohibitive (limited to $N_F \approx 40$), extrapolation techniques (Shanks transformations) are applied to the monotonic parts of convergence curves.
• Optimization: The study systematically varies the scaling factor $f$ in the oscillator frequency definition $\hbar\omega_0 = f \times 41A^{-1/3}$ MeV. The goal is to identify an optimal scaling factor $f_{opt}(A)$ and the minimum basis size $N_F^\epsilon$ required to reproduce infinite basis results within a specified error tolerance $\epsilon$ (specifically 30 keV and 100 keV for binding energies).
• Scope: The investigation covers ground state properties (binding energies, charge radii), excited states (single-particle energies, fission barriers, fission isomers), and the description of halo nuclei. Both PC-Z and DD-MEZ functionals are utilized.

Key Contributions and Results

1. Odd-Even Staggering in Convergence:
   The paper identifies a previously unreported odd-even staggering in the convergence curves of binding energies when calculations are performed in steps of $\Delta N_F = 1$. This effect, most pronounced at low $N_F$ ($10-20$), arises from non-regular changes in neutron densities caused by the sequential addition of shells, which feed back self-consistently into binding energies. The magnitude of this staggering is suppressed by increasing the scaling factor $f$ (e.g., from 1.0 to 1.5) and is reduced by nuclear deformation. This effect complicates extrapolation procedures at low $N_F$ but diminishes at very large basis sizes.

2. Optimization of Point Coupling Functionals:
   For PC functionals, the study demonstrates that optimizing the scaling factor $f$ drastically improves convergence compared to the standard $f=1.0$.

   • Scaling Factor: An approximate global expression for the optimal scaling factor is proposed: $f_{opt}(A) \approx 1.394 + 0.00161A$.
   • Basis Size: To achieve a global accuracy of 30 keV in binding energies, PC functionals require significantly larger basis sizes than ME functionals. While ME functionals can achieve this with $N_F \approx 20$, PC functionals require $N_F$ up to 34 for superheavy nuclei.
   • Deformed Nuclei: For deformed actinides and superheavy nuclei, the standard $f=1.0$ basis fails to allow reliable extrapolation to infinite basis solutions due to non-monotonic convergence. The optimized $f_{opt}$ enables the definition of extrapolated infinite basis solutions, resolving previous issues in defining fission barriers for these systems.

3. Halo Nuclei in the HO Basis:
   The study demonstrates that very large fermionic HO bases can reproduce neutron halo densities obtained in coordinate space calculations. By increasing the effective cavity radius $L_0$ (via large $N_F$ or reduced $f$), the HO basis can describe halo structures. The authors estimate that spherical halo nuclei up to $A \approx 80$ and axially deformed halo nuclei up to $A \approx 40$ (depending on pairing treatment) can be studied using large HO bases, offering a computationally cheaper alternative to coordinate space methods for these systems.

4. Excited States and Fission Barriers:

   • Single-Particle Energies: Optimization improves the accuracy of bound single-particle state energies by a factor of approximately two, achieving accuracies better than 10 keV for doubly magic nuclei. However, weakly bound neutron states with low orbital angular momentum ($l=0, 1, 2$) remain an exception; they are better described with $f < f_{opt}(A)$ due to their extended spatial distribution.
   • Fission Barriers: The accuracy of potential energy curves (PECs), fission barriers, and fission isomers is highly sensitive to basis size and $f$. A combination of $N_F = 20$ and $f = 1.5$ is identified as necessary to reproduce infinite basis PECs with an accuracy better than 100 keV for actinides and superheavy nuclei. Smaller bases ($N_F < 20$) yield errors exceeding 0.5–1.0 MeV, which is significant given the sensitivity of spontaneous fission half-lives to barrier heights.

Significance
The paper establishes that the global optimization of the HO basis is essential for achieving high-precision results in CDFT with point coupling functionals. It provides specific, globally applicable scaling factors and basis size recommendations ($N_F^\epsilon$) that allow for controllable numerical errors in binding energies, charge radii, and fission properties. The work extends the applicability of the HO basis expansion method to halo nuclei and excited states, demonstrating that with appropriate optimization and sufficient basis size, the HO basis can match the accuracy of coordinate space calculations and infinite basis solutions at a fraction of the computational cost. This resolves long-standing issues regarding the definition of fission barriers in superheavy nuclei within the PC framework and clarifies the convergence behavior of CEDFs, including the previously unaddressed odd-even staggering in convergence curves.