Variational free complement method with Gaussian-expanded complement functions: convergence with fixed Gaussian expansion length

This paper investigates the energy convergence of the Variational Free Complement method using Gaussian-expanded complement functions in the limit where the number of basis functions approaches infinity while the Gaussian expansion length ($n_\mathrm{G}$) remains fixed.

The Gist

Imagine you are trying to paint a perfect portrait of a hydrogen atom (the simplest atom in the universe). To do this, you are using a special digital brush called the Variational Free Complement Method. This brush is designed to get closer and closer to the "true" picture (the exact energy of the atom) by adding more and more layers of detail.

In this paper, the author, Cong Wang, is testing a specific version of this brush that uses Gaussian functions. Think of Gaussian functions as "soft, fuzzy clouds" of paint. They are very easy to work with mathematically, but they have a specific shape: they are smooth and fade away quickly.

Here is the core experiment the author ran, explained simply:

The Two Experiments

The author wanted to see if this "fuzzy cloud" brush could eventually paint a perfect picture, even if he was forced to use a fixed, limited number of cloud shapes (let's call this number $n_G$). He asked: If I keep adding more layers of these specific clouds forever, will I eventually get the perfect energy value?

He ran two different scenarios:

Scenario 1: The "One-Cloud" Limit (Fixed $n_G = 1$)

• The Setup: The author started with a basic "Slater-type" wave (a specific mathematical shape for the atom) and tried to improve it using only one single Gaussian cloud to represent the corrections. He kept adding more layers of this same single cloud shape over and over again.
• The Problem: Gaussian clouds are "stubborn." They fade away too fast compared to the actual atom. If you only have one type of cloud, you can never paint the very "fuzzy" edges (diffuse parts) of the atom correctly.
• The Result: The author ran the math up to 1,200 layers. The picture got better and better, but it stopped short. It got very close to the perfect energy (-0.5), but it got stuck at about -0.4998. It was like trying to fill a bucket with a cup that has a tiny hole in the bottom; no matter how many times you pour, you never quite reach the top.
• The Conclusion: With a fixed, small number of cloud shapes, the method does not converge to the perfect answer. It hits a "ceiling" it cannot break through.

Scenario 2: The "Infinite-Cloud" Limit (Increasing $n_G$)

• The Setup: In the second experiment, the author started with a "Gaussian-type" initial wave (a cloud to begin with) and allowed the number of cloud shapes ($n_G$) to grow infinitely large.
• The Result: This time, the picture did get perfect. As he added more and more different cloud shapes, the energy value converged exactly to the true answer (-0.5).
• The Conclusion: If you allow the variety of your "clouds" to grow, the method works perfectly.

The Big Takeaway

The paper answers a specific question: "If I am stuck with a fixed, small number of Gaussian shapes, will the method eventually work if I just keep going forever?"

The answer is No.

The author uses a mathematical concept called the Müntz–Szász theorem (which is like a rulebook for whether a collection of shapes can build any possible curve) to explain why. He shows that when you are stuck with a fixed number of Gaussian shapes, you are missing the "diffuse" parts of the atom (the parts that stretch far out). No matter how many times you stack those specific shapes, you can't create the missing pieces.

What This Means (and Doesn't Mean)

• What it means: If you are using this specific method with a fixed, small set of Gaussian functions, you will never get the exact mathematically perfect energy, no matter how much computing power you throw at it. You will always be slightly off.
• What it doesn't mean: The author is not saying the method is useless. In real-world chemistry, scientists usually use many different Gaussian shapes (a large $n_G$) and a reasonable number of layers. In those practical cases, the method works very well and is fast. This paper only warns that if you try to be too stingy with your "clouds" (keeping $n_G$ fixed and small), the method has a hard limit it cannot cross.

In a nutshell: You can't build a perfect house using only one type of brick, no matter how many times you stack them. You need a variety of brick sizes (diffuse functions) to fill in all the gaps. This paper proves that if you refuse to use more brick sizes, your house will always have a tiny, unfixable gap.

Technical Summary

Technical Summary: Variational Free Complement Method with Gaussian-Expanded Complement Functions

Problem Statement
This work investigates the convergence properties of the Variational Free Complement (FC) method when employing Gaussian-expanded complement functions. Specifically, it addresses the question of whether the method converges to the exact eigenenergy of the Hamiltonian under the constraint of a fixed and finite Gaussian expansion length ($n_G$), even as the order of the free complement method ($n$) and the number of complement functions ($M_n$) approach infinity.

The study is motivated by the necessity of obtaining numerically exact solutions to the time-independent Schrödinger equation to serve as reference values for other computational approaches and machine learning datasets. While the Free Complement theory (recently renamed "free complete-element") offers a framework for such solutions, previous implementations using Gaussian expansions faced challenges with integral difficulties and exponential complexity. Although these were mitigated by hierarchical decontraction, a theoretical question remains: does a fixed $n_G$ (the number of Gaussian functions in the STO-$n$G expansion) suffice for convergence to the exact ground state energy as $n \to \infty$?

Methodology
The author employs the non-relativistic electronic hydrogen ground state as a benchmark system to test convergence under two distinct scenarios:

1. Fixed $n_G = 1$ with $n \to \infty$:

   • Initial Wavefunction: Slater-type orbital $\psi_0 = e^{-\zeta r}$ (with $\zeta = 0.5$).
   • Complement Functions: Generated using a $g$-function of the form $g = 1 - e^{-\gamma r}$ (where $\gamma = 1 - \zeta$).
   • Expansion: The complement functions are expanded into a single Gaussian function ($n_G=1$). The exponents of these Gaussians follow a scaling relation derived from STO-$n$G basis sets, resulting in a sequence $\alpha(k) \propto (1 + k\gamma/\zeta)^2$.
   • Theoretical Analysis: The convergence is analyzed using the Müntz–Szász theorem and its generalizations regarding basis completeness in $L^2(\mathbb{R}^3)$ and Sobolev spaces $W^{(1)}_2(\mathbb{R}^3)$. The summation condition for completeness is evaluated against the specific sequence of exponents generated by the $n_G=1$ constraint.
   • Numerical Implementation: Rayleigh-Ritz variational calculations are performed using Python (for integral evaluation and fitting) and Julia (for generalized eigenvalue problems). High-precision arithmetic (1200 to 1500 decimal places) is utilized to handle the extreme ill-conditioning of the overlap matrices as $n$ increases.

2. Limit $n_G \to \infty$ with $n \to \infty$:

   • Initial Wavefunction: Gaussian-type orbital $\psi_0 = e^{-\alpha r^2}$.
   • Complement Functions: Generated using the same $g$-function form but without Gaussian expansion (effectively $n_G \to \infty$), leading to functions of the form $e^{-k\gamma r}\psi_0$.
   • Comparison: This scenario serves as a control to compare against the fixed $n_G$ case and to verify convergence behavior when the lower bound of Gaussian exponents is not restricted by a single initial Gaussian.

Key Results

• Case 1 ($n_G = 1$):

  • Theoretical Finding: The sequence of Gaussian exponents generated by the fixed $n_G=1$ expansion fails to satisfy the sufficient conditions of the Müntz–Szász theorem for completeness in $L^2(\mathbb{R}^3)$ and $W^{(1)}_2(\mathbb{R}^3)$. The summation term related to the completeness condition converges to a finite value rather than diverging to infinity, suggesting the basis is incomplete.
  • Numerical Finding: Variational calculations show that the energy converges slowly to a value of approximately $-0.4997995$ a.u., which is strictly above the exact ground state energy of $-0.5$ a.u. Extrapolation fitting ($E = A + B/M_n^C$) confirms a limit that does not reach the exact eigenvalue.
  • Conclusion: For a fixed $n_G=1$, the method does not converge to the exact eigenenergy as $n \to \infty$.

• Case 2 ($n_G \to \infty$):

  • Numerical Finding: When the initial wavefunction is Gaussian and the expansion allows for an arbitrary number of exponents (simulating $n_G \to \infty$), the energy converges to the exact value of $-0.5$ a.u. as $n$ increases.
  • Conclusion: The lack of convergence in the first case is attributed to the restriction of the Gaussian exponents (specifically the lack of diffuse functions and the specific accumulation point behavior) imposed by a fixed, small $n_G$.

Significance and Claims
The paper provides a definitive negative answer to the central question: for a fixed and finite $n_G$, the Gaussian-expanded free complement method does not converge to the exact eigenenergy as the order $n$ approaches infinity.

The author makes several modest but significant claims regarding the implications of these findings:

1. Limitations of Fixed $n_G$: The Gaussian-expanded FC method with a fixed $n_G$ (specifically $n_G=1$ in this study) lacks the computational capacity to capture the ground state energy of the hydrogen atom to arbitrary precision compared to a basis set that satisfies the Müntz–Szász completeness conditions.
2. Role of Diffuse Functions: The failure to converge is linked to the absence of diffuse functions and the specific behavior of the exponent accumulation point. However, the author notes that the lack of diffuse functions does not necessarily imply non-convergence in all contexts; it can be compensated by an appropriate accumulation point of exponents (as seen in the $n_G \to \infty$ case).
3. Practicality: The non-convergence observed in the theoretical limit ($n \to \infty$ with fixed $n_G$) does not imply a disadvantage for practical calculations where finite $n > 0$ and $n_G > 1$ are chosen.
4. Potential Remedies: The paper suggests that convergence issues in the fixed $n_G$ scenario could potentially be cured by optimizing exponents beyond the zeroth order (similar to explicitly correlated Gaussian methods), though this is computationally expensive. Alternatively, using Gaussian-type $g$-functions or introducing kinetic operators (the $p+k$ approach) might generate basis sets that satisfy completeness conditions more effectively.

In summary, the work clarifies the theoretical boundaries of the Gaussian-expanded free complement method, demonstrating that while it can provide high-accuracy results for finite orders, a fixed Gaussian expansion length imposes a fundamental limit on asymptotic convergence to the exact eigenenergy.