Machine learning Hamiltonian enables scalable and accurate defect calculations: The case of oxygen vacancies in amorphous SiO$_2$

This paper introduces a machine learning Hamiltonian (MLH) method that achieves linear-scaling computational cost and high accuracy for defect simulations in large supercells, overcoming the transferability limitations of traditional machine learning potentials by successfully predicting oxygen vacancy formation energies in amorphous SiO$_2$ with deviations below 50 meV from density functional theory.

The Gist

Imagine you are trying to fix a tiny, invisible crack in a massive, complex glass sculpture (like a smartphone screen made of amorphous silica). To understand how this crack affects the whole screen, you need to know exactly how the atoms around the crack are moving and how much energy it takes to make that crack.

In the world of science, this is called studying "point defects."

The Problem: The "Super-Computer" Bottleneck

Traditionally, scientists use a method called Density Functional Theory (DFT) to simulate these atoms. Think of DFT as a hyper-accurate, high-definition camera that captures every single atom's movement perfectly.

• The Catch: It's incredibly slow and expensive. If you want to study a tiny crack in a small piece of glass, it takes a supercomputer a few hours. But if you want to study that same crack in a larger piece of glass (to make sure the results are real and not just a fluke of the small size), the time required explodes. It's like trying to paint a masterpiece by hand; it's accurate, but you can't paint a whole city in a day.

The Old Shortcut: The "Cheap Camera"

To speed things up, scientists started using Machine Learning Interatomic Potentials (MLIPs). Think of this as a "smart filter" or a cheap camera that guesses what the atoms are doing based on patterns it learned from a few photos.

• The Catch: These "smart filters" are great at guessing what happens in small, simple rooms. But if you try to use them in a giant cathedral (a large supercell), they get confused. They start making systematic mistakes, like thinking the walls are made of jelly instead of glass. They might say the crack is stable when it's actually collapsing, or vice versa. They lack transferability—they can't handle new, bigger situations well.

The New Solution: The "Universal Blueprint" (MLH)

This paper introduces a new method called the Machine Learning Hamiltonian (MLH).

Here is the best way to understand it:

• DFT is like calculating the physics of a building from scratch every time you want to know if a wall will hold.
• MLIPs are like hiring a contractor who memorized the blueprints for one specific type of house. If you ask them to build a skyscraper, they get it wrong because they only know houses.
• The MLH (Machine Learning Hamiltonian) is like giving the contractor the fundamental laws of physics (the "Hamiltonian") and a small set of examples. Instead of just memorizing the result (the energy), the MLH learns the rules of how atoms talk to each other.

Because it learns the underlying "rules of the game" rather than just memorizing specific outcomes, it can apply those rules to a tiny room or a massive skyscraper with equal accuracy.

How They Tested It: The Oxygen Vacancy

The researchers tested this on Oxygen Vacancies in Amorphous SiO2 (basically, missing oxygen atoms in glass).

1. Training: They taught the MLH model using data from a relatively small 95-atom "room." They only showed it 120 examples of missing oxygen atoms.
2. The Test: They then asked the model to predict what happens in much larger "rooms" (up to 576 atoms) that it had never seen before.
3. The Result:
   • The old "cheap camera" (MLIP) failed miserably in the big rooms, getting the energy wrong by huge amounts.
   • The new "Universal Blueprint" (MLH) was spot on. It predicted the energy and forces with almost the same accuracy as the slow, expensive DFT method, but much faster.

The Magic Trick: Error Cancellation

Here is the clever part. Even though the MLH model isn't perfectly 100% identical to the super-accurate DFT (it has a tiny error), it makes the same tiny error for both the "perfect glass" and the "glass with a crack."

When you calculate the Formation Energy (how much energy it costs to make the crack), you subtract the energy of the perfect glass from the energy of the cracked glass. Because the errors are the same in both, they cancel each other out.

• Analogy: Imagine you are weighing two bags of apples. Your scale is slightly off by 1 pound. If you weigh a bag of 10 apples and a bag of 11 apples, both weigh 1 pound too much. But when you calculate the difference (the weight of the one extra apple), the 1-pound error cancels out, and you get the exact weight of that single apple.

Why This Matters

This method is a game-changer because:

1. Speed: It scales linearly. If you double the size of the material, the calculation time only doubles, not explodes.
2. Accuracy: It gives results as good as the slow, expensive methods.
3. Versatility: It works for complex, messy materials (like amorphous glass) where other methods fail.

In summary: The researchers built a "smart physics engine" that learns the fundamental rules of atomic interactions from a small dataset. This engine can now simulate massive, complex materials with high speed and high accuracy, allowing scientists to design better electronics and more reliable devices without waiting years for a computer to finish its calculations.

Technical Summary

1. Problem Statement

Point defects, such as oxygen vacancies ($V_O$), critically influence the performance and reliability of amorphous materials like silicon dioxide (a-SiO$_2$) used in microelectronics. Accurately modeling these defects requires Density Functional Theory (DFT) calculations on large supercells to capture statistical behavior and minimize finite-size effects. However, DFT is computationally expensive (scaling poorly with system size), making large-scale sampling and structural relaxation prohibitive.

While Machine Learning Interatomic Potentials (MLIPs) offer a faster alternative, they face two major limitations in defect simulations:

1. Data Scarcity: Training MLIPs requires massive datasets of DFT calculations, which are costly to generate.
2. Poor Transferability: MLIPs trained on small supercells often exhibit systematic energy errors when applied to larger supercells. They struggle to accurately describe the host (defect-free) environment when trained only on defect configurations, leading to inaccurate formation energy predictions due to a lack of error cancellation between host and defect systems.

2. Methodology

The authors propose a Machine Learning Hamiltonian (MLH) approach that directly predicts the electronic Hamiltonian matrix rather than just interatomic potentials. This allows for the derivation of total energies, atomic forces, and electronic structures (band structures, charge densities) with linear scaling computational cost.

Key Methodological Steps:

• Model Architecture: The study utilizes HamGNN, an equivariant graph neural network (E(3)-equivariant). It decomposes the real-space Hamiltonian into on-site and off-site parts using node and edge features derived from atomic structures.
• Dataset Construction:
  • System: Amorphous SiO$_2$ with oxygen vacancies ($V_O$).
  • Training Data: Generated from 120 self-consistent field (SCF) calculations and 12 structural relaxations of $V_O$ defects in 95-atom supercells.
  • Crucial Distinction: The model was trained only on defect configurations in small supercells, without explicit training on large supercells or the defect-free host.
• Workflow:
  1. The MLH model predicts the Hamiltonian ($H$) for a given atomic configuration.
  2. The overlap matrix ($S$) is computed analytically.
  3. The Kohn-Sham equations are solved to obtain wavefunctions and charge densities.
  4. Total energy ($E$) and atomic forces ($F$) are derived from the predicted Hamiltonian and charge density.
  5. Structural relaxation is performed iteratively until forces converge (< 20 meV/Å).
• Formation Energy Calculation: Calculated as $E_f = E_{defect} - E_{host} + \sum n_i \mu_i$. The method relies on the cancellation of systematic errors between the host and defect total energies.

3. Key Contributions

• Direct Energy/Force Derivation from MLH: Unlike previous MLH studies that could only predict band structures, this work demonstrates the first successful derivation of total energies and atomic forces directly from an MLH model, enabling full structural relaxation without DFT.
• Solving the Transferability Issue: The study proves that an MLH trained solely on small-defect supercells can accurately predict properties for larger supercells (up to 383 atoms) and the defect-free host. This overcomes the systematic energy errors typical of MLIPs.
• Error Cancellation Mechanism: The authors demonstrate that while the MLH may have small absolute errors in total energy for both host and defect systems, these errors cancel out during the formation energy calculation, yielding high accuracy.
• Linear Scalability: The computational cost of the MLH method scales linearly with system size, offering a significant efficiency advantage over DFT for large supercells.

4. Results

The method was benchmarked against DFT and MLIP (Allegro) for a-SiO$_2$ systems:

• Accuracy of Predictions:
  • Hamiltonian: Mean Absolute Error (MAE) of 0.72 meV relative to DFT.
  • Charge Density: MAE of $3 \times 10^{-4}$ e/Å$^3$.
  • Total Energy: The MLH-predicted energy is slightly higher than DFT (error ~0.66–1.1 meV/atom), but significantly more accurate than MLIPs, which showed errors of ~1180–1195 meV/atom for hosts due to systematic overestimation.
  • Forces: MLH force MAE is ~25 meV/Å (approaching DFT convergence criteria), whereas MLIP errors were ~115 meV/Å.
• Structural Relaxation:
  • MLH-relaxed structures closely match DFT references, with generalized configuration coordinate deviations ($\Delta Q$) below 2 amu$^{1/2}$Å.
  • MLIP-relaxed structures showed large deviations ($\Delta Q$ up to 18 amu$^{1/2}$Å) and failed to converge to the correct local minima due to potential energy surface softening.
• Formation Energy:
  • For $V_O$ in supercells of 95, 215, and 383 atoms, the MLH predicted formation energies with MAEs of 16, 26, and 45 meV, respectively, relative to DFT.
  • This accuracy was achieved despite the MLH having larger absolute total energy errors, confirming the effectiveness of the error cancellation mechanism.
• Efficiency:
  • DFT calculation time increases sharply with system size (exponential/polynomial scaling).
  • MLH calculation time grows linearly with the number of atoms, making it highly efficient for large-scale simulations.

5. Significance

This work establishes the Machine Learning Hamiltonian as a superior tool for defect simulations in complex, disordered materials compared to traditional MLIPs.

• Practicality: It enables accurate defect property predictions (formation energy, structural relaxation) using modest training datasets (only small supercells), bypassing the need for expensive large-scale DFT training data.
• Electronic Insight: Unlike MLIPs, the MLH provides access to full electronic structure information (band structures, wavefunctions, charge densities), offering deeper physical insights into defect states (e.g., identifying deep trap levels in a-SiO$_2$).
• Scalability: The linear scaling nature of the method opens the door to simulating defect behaviors in large, realistic material systems that were previously computationally inaccessible.

In summary, the paper presents a robust, scalable, and highly accurate framework for defect engineering in amorphous materials, solving the critical transferability and efficiency bottlenecks of current machine learning approaches in computational materials science.