Moment Tensor Potential

• Moment Tensor Potentials (MTP) are systematically improvable machine-learning models providing quantum-level accuracy for atomistic simulations.
• Unlike computationally expensive DFT, MTPs scale linearly with system size, making large-scale atomistic simulations with quantum-level accuracy feasible.
• The potential uses a linear expansion of basis functions derived from atomic environment descriptors, ensuring symmetry compliance and systematic improvability.

Moment Tensor Potentials (MTP) are a class of systematically improvable, symmetry-adapted machine-learning interatomic potentials developed to provide quantum-level accuracy for large-scale atomistic simulations with computational efficiency rivaling traditional empirical models. By employing a linearly parametrized functional form based on tensorial descriptors of atomic neighborhoods, MTPs bridge the gap between first-principles accuracy and the scalability required for practical materials modeling.

1. Definition and Core Properties

Moment Tensor Potentials (MTP) are interatomic potentials constructed to approximate quantum-mechanical interaction models—such as those emerging from density functional theory (DFT)—using a basis of scalar polynomials built from moment tensors of atomic environments. MTPs are characterized by several key features:

• Systematic Improvability: The potential can be made arbitrarily accurate by systematically extending the functional basis.
• Symmetry Compliance: The formulation ensures invariance to permutation, rotation, and reflection of atoms.
• Computational Efficiency: Linear regression in parameter fitting and localized evaluation allow fast computation and linear scaling with system size.
• Generality: The approach is extensible to multi-component systems by using radial functions dependent on chemical species.

These properties set MTPs apart from both traditional empirical potentials—which lack systematic improvability—and more general nonparametric machine-learning potentials, which may suffer from computational inefficiency or incomplete symmetry encoding (1512.06054).

2. Mathematical Framework

The MTP formalism expresses the total energy as a sum of local (atomic-neighborhood) energies:

$E(\{x_i\}) = \sum_{k=1}^N V(Dx_k)$

where each neighborhood $$Dx_k = \{x_i - x_k: \; 0 < |x_i - x_k| \leq R_{\text{cut}}\}$$.

Basis Construction via Moment Tensors

For an atomic environment $u = (u_1, ..., u_n)$, the moment tensors are defined for integers $\mu, \nu \ge 0$:

$$M_{\mu,\nu}(u) = \sum_{i=1}^n |u_i|^{2\mu} u_i^{\otimes \nu}$$

where $u_i^{\otimes \nu}$ denotes the $\nu$-fold tensor product (Kronecker power).

These moment tensors are then combined into scalar basis polynomials via contractions determined by a symmetric integer matrix $\alpha$:

$$B_\alpha(u) = \bigotimes_{i=1}^k M_{\alpha_{ii}, \alpha'_i}(u)$$

with $\alpha'_i = \sum_{j \neq i} \alpha_{ij}$.

The local potential is a linear expansion in these basis functions: $V(u) = \sum_{\alpha \in A} c_\alpha B_\alpha(u)$ with the coefficients $c_\alpha$ optimized by a regularized least-squares fit to quantum-mechanical data.

For practical efficiency and the inclusion of physically meaningful cutoff behavior, the original polynomial moment tensors are replaced by radial-function-weighted variants:

$$\tilde{M}_{\mu, \nu}(u) = \sum_{i=1}^n f_{\mu, \nu}(|u_i|) u_i^{\otimes \nu}$$

with suitably chosen radial functions $f_{\mu, \nu}$ (1512.06054).

3. Systematic Improvability and Convergence

A central theoretical property of MTP is systematic improvability: by increasing the set and degree of basis polynomials, MTP can approximate any regular, symmetric function of atomic environments to arbitrary precision (Theorem 1). For analytic quantum-mechanical site energies, the approximation error decays exponentially with the basis polynomial degree (Theorem 2):

$$\sup_{u} |V^\text{QM}(u) - V^{(N)}(u)| < \epsilon(N)$$

with $\epsilon(N) \to 0$ as basis size $N$ increases.

Regularization ($\ell_0$, $\ell_1$, or $\ell_2$ penalties) can be applied during linear regression to prevent overfitting and promote compact, effective representations (1512.06054).

4. Practical Application and Testing

MTP was validated by fitting to a large set of DFT data for tungsten comprising 9,693 configurations (≈150,000 atomic environments). Key practical outcomes include:

• Error Convergence: Root mean square (RMS) force fitting error decreased algebraically with the number of basis functions, in line with theoretical predictions for systematic improvability.
• Efficiency: MTP provided orders-of-magnitude faster potential evaluation than Gaussian Approximation Potentials (GAP), while delivering comparable or better accuracy with the same or fewer fitting parameters.
• Generalization: Rigorous 16-fold cross-validation showed that, with appropriate regularization, MTPs generalized well to unseen atomic environments.

	GAP	MTP$_1$	MTP$_2$
CPU ms/atom	134.2	2.9	0.8
Basis functions	10,000	11,133	760
Force RMS error (eV/Å)	0.0633	0.0427	0.0633

This highlights MTP's capability for efficient, DFT-quality large-scale molecular dynamics and property calculations in metals and other materials (1512.06054).

5. Comparison to Density Functional Theory (DFT) and Other Potentials

• Accuracy: For sufficient basis complexity, MTPs achieve errors that approach the DFT "noise floor", e.g., RMS force error <0.05 eV/Å.
• Scaling: DFT scales as $O(N^3)$ with system size, limiting its applicability to small samples; MTP scales as $O(N)$, enabling million-atom simulations.
• Versus Empirical Potentials: MTP outperforms empirical models, which use fixed forms and cannot be systematically improved.
• Versus GAP: MTP matches GAP's accuracy but is significantly more computationally efficient.

MTP, therefore, enables previously intractable atomistic simulations with DFT-level fidelity.

6. Extensions and Future Directions

• Multicomponent Systems: The MTP framework can be generalised to multi-species materials by introducing chemical species dependence in the radial basis:

$$M_{\mu, \nu} = \sum_j f_{\mu,\nu}(|x_j - x_i|, t_i, t_j) (x_j - x_i)^{\otimes \nu}$$

• Sparse Optimization: Ongoing work focuses on sparse basis selection using advanced optimization to further enhance efficiency.
• Practical Code Integration: Integration into packages such as QUIP targets the broader community.
• Transferability and Validation: Further investigation is ongoing regarding the predictive power of MTP when extrapolating outside the scope of the training database.

This suggests the suitability of MTP as a general-purpose machine-learned potential for complex, chemically diverse materials, with further advancements likely tied to optimization and validation in heterogeneous systems.

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Significance

The Moment Tensor Potential framework enables the construction of physical, symmetry-respecting, and systematically controllable machine-learned potentials. These potentials achieve quantum-mechanical accuracy for materials property prediction and molecular dynamics—with a clear strategy for balancing speed, accuracy, and transferability. MTP thus represents a foundational advance in computational materials science, unifying efficiency and precision.