9.2 Required symmetries¶
Before we choose any regression model we must decide what function we are trying to fit. The potential energy of a configuration of atoms is not an arbitrary function of \(3N\) Cartesian coordinates: it inherits a collection of symmetries from the physics of the problem. Every successful MLIP architecture enforces these symmetries exactly, in the structure of its representation, rather than hoping the regressor will learn them from data. This section catalogues the five symmetries that matter — translation, rotation, permutation, smoothness, compactness — and introduces the distinction between invariant and equivariant features that organises the modern literature.
9.2.0 Invariance and equivariance — the central distinction¶
Before walking through the individual symmetries, we make sharp the single most important distinction in the modern MLIP literature: that between invariance and equivariance. Many beginners conflate the two and then find Section 9.5 mysterious. The conceptual investment here pays back many times over.
Definitions
Let \(G\) be a group (e.g. \(\mathrm{O}(3)\), rotations and reflections in 3D) acting on the input space of a function \(f\). Let \(\rho_\mathrm{in}\) denote that action on the input and \(\rho_\mathrm{out}\) its action on the output. The function \(f\) is
- invariant under \(G\) if \(f(\rho_\mathrm{in}(g) \cdot \mathbf{x}) = f(\mathbf{x})\) for every group element \(g \in G\). The output never changes.
- equivariant under \(G\) if \(f(\rho_\mathrm{in}(g) \cdot \mathbf{x}) = \rho_\mathrm{out}(g) \cdot f(\mathbf{x})\) for every \(g \in G\). The output transforms in a predictable, group-consistent way.
Invariance is the special case of equivariance in which \(\rho_\mathrm{out}\) is the trivial representation, i.e. the output is a scalar.
The energy \(U(\{\mathbf{r}_i\})\) is invariant under translations, rotations, reflections, and same-species permutations. Forces and stresses are not invariant: they transform predictably under those operations. Specifically, if we apply a rotation \(R\) to every position in the system, then
and the stress tensor transforms as \(\sigma'_{\alpha\beta} = R_{\alpha\gamma} R_{\beta\delta} \sigma_{\gamma\delta}\). Energy is scalar, forces are vectors (\(\ell = 1\)), stress is a symmetric rank-2 tensor (\(\ell = 0\) plus \(\ell = 2\)).
A concrete check
Take three atoms at \(\mathbf{r}_1 = (0, 0, 0)\), \(\mathbf{r}_2 = (1, 0, 0)\), \(\mathbf{r}_3 = (0, 1, 0)\) and any rotation, say a \(90^\circ\) rotation \(R\) about the \(z\)-axis, \(R: (x, y, z) \mapsto (-y, x, z)\). The rotated configuration is \(R\mathbf{r}_1 = (0, 0, 0)\), \(R\mathbf{r}_2 = (0, 1, 0)\), \(R\mathbf{r}_3 = (-1, 0, 0)\).
The pairwise distances are unchanged: \(|r_{12}| = |R r_{12}| = 1\), \(|r_{13}| = |R r_{13}| = 1\), \(|r_{23}| = |R r_{23}| = \sqrt{2}\). The energy, being a function of these distances, is unchanged (invariance). If atom \(1\) originally felt force \(\mathbf{F}_1 = (F_x, F_y, F_z)\), in the rotated frame it feels \(R\mathbf{F}_1 = (-F_y, F_x, F_z)\) — the force has rotated alongside the configuration (equivariance).
The whole point of this chapter is that one can choose to build an MLIP whose internal features are either invariant or equivariant. Both yield invariant energies (just take a final inner product to collapse equivariant features to scalars). But equivariant features preserve more information per dimension and therefore yield more data-efficient potentials — the empirical lesson of §9.5.
The slogan to memorise: invariance throws away information, equivariance keeps it. Both are correct; one is wasteful.
Pause and recall
Before reading on, try to answer these from memory:
- State the definitions of invariance and equivariance, and explain why invariance is a special case of equivariance.
- Which physical quantities of an atomic system are invariant under rotation, and which are equivariant — give an example of each.
- Why is the slogan "invariance throws away information, equivariance keeps it" correct, even though both yield invariant energies?
If any of these is shaky, re-read the preceding section before continuing.
9.2.0a Why differentiability matters¶
The forces that drive molecular dynamics are not measured; they are computed as gradients of the energy with respect to positions:
This identity, banal in classical mechanics, has sharp consequences for MLIP design.
Consequence 1: continuity. The energy must be a continuous function of every atomic coordinate. A discontinuity at any configuration produces a delta-function in the force, which integrates to a finite-momentum kick in MD. The kick injects energy on every neighbour-list update, and energy conservation in NVE fails catastrophically.
Consequence 2: differentiability. The energy must be differentiable. A continuous but kinked function (think \(|r - r_0|\) as a stand-in bond term) has finite forces on either side but no well-defined force at the kink, and the integrator oscillates pathologically near it.
Consequence 3: smooth derivatives for thermodynamic integration and free-energy methods. Many post-MD analyses — free-energy perturbation, thermodynamic integration, response-function evaluation — require second derivatives of the energy (the Hessian). Models with \(C^1\) but not \(C^2\) smoothness will produce noisy phonon spectra and unreliable enhanced-sampling free-energy estimates.
The implication for design: we will require not just invariance and equivariance of the energy, but smooth invariance and equivariance. This rules out hard-cutoff descriptors, requires smooth cutoff envelopes (§9.2.4), and forbids \(\mathrm{ReLU}\)-like non-smooth activations in the network (we use \(\tanh\), SiLU, softplus instead).
A useful heuristic: the energy should be at least \(C^2\) for honest phonon work and \(C^1\) at the very least for stable MD. Best-of-breed MACE potentials are \(C^\infty\) inside the cutoff sphere and \(C^p\) at the cutoff with \(p \approx 5\).
9.2.0b On the body-order hierarchy¶
The atomic energy \(E_i\) depends on the geometry of all neighbours of \(i\) inside the cutoff. With \(n\) neighbours one can in principle construct features of body order \(1, 2, 3, \dots, n + 1\) (the \(+1\) counts atom \(i\) itself). A two-body feature depends on \(i\) and one neighbour; a three-body feature depends on \(i\) and two neighbours (and thus involves an angle); a four-body feature involves three neighbours; and so on.
Why truncate? Two reasons:
- Combinatorial explosion. With \(n = 12\) neighbours (a typical first coordination shell), there are \(n\) two-body terms, \(\binom{n}{2} = 66\) three-body terms, \(\binom{n}{3} = 220\) four-body terms, \(\binom{n}{4} = 495\) five-body terms. The number of distinct feature parameters grows correspondingly.
- Diminishing physical relevance. Most chemistries are well described by terms up to four-body order: covalent bonds (two-body), bond angles (three-body), dihedrals (four-body). The five-body and higher contributions to the energy are small corrections.
The practical consequence is the body-order truncation: Behler descriptors capture two- and three-body, ACE and MACE go to four- or five-body, and that is essentially sufficient. Where it is not — in some metallic systems with strong electronic-structure non-locality, in conjugated \(\pi\)-systems with delocalised electrons — one needs deeper message-passing networks whose effective receptive field grows beyond a single cutoff sphere.
We will return to body order constantly in §9.3 (it is the organising principle of the ACE expansion) and §9.5 (it sets MACE apart from NequIP).
9.2.1 Translation invariance¶
The first symmetry is the simplest. Translate every atom by the same vector \(\mathbf{t}\) and nothing physical changes:
This rules out functional forms that depend on absolute positions. Concretely, a network that takes \(\mathbf{r}_i\) as input and learns biases on the position coordinates will memorise the simulation box and fail the moment you translate. The fix is also simple: depend only on relative coordinates,
or on functions of relative coordinates such as scalar distances \(r_{ij} = \|\mathbf{r}_{ij}\|\) and angles \(\cos\theta_{ijk} = \hat{\mathbf{r}}_{ij}\cdot\hat{\mathbf{r}}_{ik}\).
Why distances are automatically translation- and rotation-invariant
The pairwise distance \(r_{ij} = \|\mathbf{r}_j - \mathbf{r}_i\|\) is invariant under any global translation \(\mathbf{r}_i \mapsto \mathbf{r}_i + \mathbf{t}\), because the translation cancels in the difference. It is also invariant under any rotation \(R \in \mathrm{O}(3)\), because
using the fact that rotations preserve the Euclidean norm. So any function \(f\) built from pairwise distances alone satisfies \(f(\{R\mathbf{r}_i + \mathbf{t}\}) = f(\{\mathbf{r}_i\})\) for every \((R, \mathbf{t})\) — translation and rotation invariance come for free.
The corresponding statement for angles, which are inner products of unit vectors \(\hat{\mathbf{r}}_{ij} \cdot \hat{\mathbf{r}}_{ik}\), is the same: inner products are rotation-invariant scalars. Two- and three-body invariants built from distances and angles are automatically \(\mathrm{O}(3) \times \mathbb{R}^3\)-invariant.
This is why §9.3.1's Behler symmetry functions, built only from \(r_{ij}\) and \(\cos\theta_{ijk}\), are invariant by inspection — no extra machinery is needed.
In a periodic cell with lattice vectors \(\mathbf{L}_a\) the relative coordinate must be taken modulo the lattice: we use the minimum-image convention or, more generally, list all neighbours \(j\) such that \(\|\mathbf{r}_j + \mathbf{L} - \mathbf{r}_i\| < r_\mathrm{c}\) for any lattice translation \(\mathbf{L}\). Every MLIP architecture in this chapter operates on a neighbour list of relative vectors, never on absolute coordinates.
Common mistake
Beginners sometimes encode the periodic cell into the model by concatenating the lattice matrix into the input. This is unnecessary and counter-productive: it breaks the local decomposition of energy and gives the network a route to overfit on box size. The correct treatment is to build the neighbour list using the periodic boundary conditions and feed only relative neighbour vectors to the model.
9.2.2 Rotation invariance — and rotation equivariance¶
Rotate the entire configuration about a point: again nothing physical changes. Letting \(R \in \mathrm{SO}(3)\) act on each position,
The same statement extends to \(\mathrm{O}(3)\) if the energy is parity invariant, as it is for non-chiral systems. Energy is a scalar; it transforms trivially under rotation.
Forces, however, are vectors. Under a rotation they pick up the rotation:
A function that transforms in this predictable way is called equivariant: rotate the input, the output rotates correspondingly. Equivariance generalises invariance — an invariant is an equivariant that happens to transform as the trivial (scalar) representation.
Internally, an MLIP can carry either invariant or equivariant features. Both choices yield an invariant energy (you can always project an equivariant feature down to a scalar at the end), but they differ in how much information they preserve about the local geometry.
Consider two carbon atoms in a tetrahedral environment, one at the centre and another at the apex of a methylene bridge. The scalar description of the central atom — interatomic distances and bond angles to its four neighbours — does not by itself reveal where any one of those neighbours sits in space; that information is collapsed the moment you take inner products. An equivariant feature, by contrast, keeps the direction information explicitly, as a vector indexed by \(\mathrm{O}(3)\) irreducible-representation labels. Later layers can combine those vectors, taking dot products to form invariants when they are needed and keeping the directionality when they are not.
The mathematics of equivariance lives in representation theory. The irreducible representations of \(\mathrm{O}(3)\) are labelled by an integer \(\ell = 0, 1, 2, \dots\) and a parity. The \(\ell = 0\) irrep is the scalar, the \(\ell = 1\) irrep is the vector (three components transforming as \(\mathbf{v} \mapsto R\mathbf{v}\)), the \(\ell = 2\) irrep is the symmetric traceless rank-2 tensor (five components transforming under the rotation matrices \(D^{(2)}(R)\)), and so on. A general equivariant feature is a list of vectors \(\mathbf{x}^{(\ell)}\) each of which transforms as \(\mathbf{x}^{(\ell)} \mapsto D^{(\ell)}(R)\,\mathbf{x}^{(\ell)}\). Sections 9.3 and 9.5 will make this concrete via the spherical harmonics \(Y_\ell^m\), which are the canonical basis for the \(\ell\)-th irrep on the unit sphere.
The empirical lesson, which we will revisit, is that throwing away direction information at every layer is wasteful. Equivariant networks need far less data to reach a given accuracy because their inductive bias matches the symmetry of the underlying physics. We will see in §9.5 that MACE on rMD17 achieves SchNet-level accuracy with roughly \(1/20\) the training data, a difference attributable almost entirely to the choice of equivariant features.
Tip
The slogan: Energy is invariant. Forces are equivariant. Internal features can be either, but equivariant features waste less information.
9.2.3 Permutation invariance¶
If atoms \(i\) and \(j\) are of the same chemical species, swapping their labels cannot change the energy. Letting \(\pi\) be any permutation of atom indices that respects element identity,
This rules out architectures that index atoms by their position in a list. It is the reason MLIPs are built as atom-centred sums: the energy is
where each atomic contribution \(E_i\) is a function of the unordered set of neighbours of atom \(i\), partitioned by element. A function of an unordered set can be implemented in two ways. The first is to sum a function of each element of the set:
This deep set construction is permutation-invariant by inspection. Most descriptors in §9.3 — Behler symmetry functions, SOAP power spectra, ACE basis functions — are sums over neighbours and inherit permutation invariance for free.
The second route is message passing. A graph neural network defines features on each atom and updates them by aggregating messages from neighbours,
where the aggregation is sum, mean, or max — all permutation-invariant. NequIP and MACE are message-passing networks of this kind, with equivariant features playing the role of \(h\).
Permutation invariance — three atoms by hand
Let us verify permutation invariance for three identical atoms explicitly. Take three hydrogens at \(\mathbf{r}_1 = (0, 0, 0)\), \(\mathbf{r}_2 = (1, 0, 0)\), \(\mathbf{r}_3 = (0, 1, 0)\), and a Behler radial descriptor
With the original labelling, \(G_1 = g(1) + g(1) = 2e^{-1}\); \(G_2 = g(1) + g(\sqrt{2}) = e^{-1} + e^{-2}\); \(G_3 = g(1) + g(\sqrt{2}) = e^{-1} + e^{-2}\).
Now swap labels \(1 \leftrightarrow 2\), so the same three atoms are relabelled as \(\mathbf{r}'_1 = (1, 0, 0)\), \(\mathbf{r}'_2 = (0, 0, 0)\), \(\mathbf{r}'_3 = (0, 1, 0)\). \(G'_1 = g(1) + g(\sqrt{2}) = e^{-1} + e^{-2}\); \(G'_2 = g(1) + g(1) = 2e^{-1}\); \(G'_3 = g(1) + g(\sqrt{2}) = e^{-1} + e^{-2}\).
The descriptor values are permuted along with the labels, but the multiset \(\{G_1, G_2, G_3\}\) is identical. The sum \(E = E_1 + E_2 + E_3 = f(G_1) + f(G_2) + f(G_3)\) — for any choice of regression function \(f\) — is therefore invariant under the relabelling. The atom-centred sum structure is what makes permutation invariance automatic.
Now consider what would go wrong with a non-atom-centred model. Suppose we tried \(E = \mathrm{NN}(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3)\) as a single function of nine ordered coordinates. The neural network would treat \((0, 0, 0; 1, 0, 0; 0, 1, 0)\) and \((1, 0, 0; 0, 0, 0; 0, 1, 0)\) as different inputs, and would return different energies unless trained to symmetrise — which it will only learn approximately, never exactly. Atom-centred sums are the architectural commitment that makes permutation invariance exact rather than approximate.
9.2.4 Smoothness and the cutoff function¶
Molecular dynamics integrators rely on forces that are continuous and differentiable functions of position. A potential whose forces jump or diverge will break energy conservation in the NVE ensemble and produce artefacts in any thermostatted ensemble. Smoothness is therefore not a luxury but a hard requirement.
The challenge arises at the cutoff. To keep the neighbour list finite we discard atoms beyond a distance \(r_\mathrm{c}\). If we simply truncate the sum at \(r_\mathrm{c}\), every time an atom drifts across the cutoff boundary the energy jumps by a finite amount and the force acquires a delta-function spike. The remedy is a smooth cutoff function \(f_\mathrm{c}(r)\) that decays smoothly to zero at \(r_\mathrm{c}\):
This is the standard Behler cutoff. It is \(C^1\): continuous with continuous first derivative at \(r = r_\mathrm{c}\), which is what molecular dynamics requires. Higher-order alternatives — polynomial \((1 - r/r_\mathrm{c})^p\) envelopes with \(p \ge 4\), or the \(1/r^p\)-style envelopes used in MACE — buy additional smoothness at the cost of slightly more arithmetic.
Why this step? Derivative of the cosine cutoff
Let us check that the cosine cutoff vanishes smoothly. Set \(u = \pi r / r_\mathrm{c}\) for brevity. Then \(f_\mathrm{c}(r) = \tfrac{1}{2}(\cos u + 1)\) and
At \(r = r_\mathrm{c}\): \(\cos(\pi) = -1\), so \(f_\mathrm{c}(r_\mathrm{c}) = \tfrac{1}{2}(-1 + 1) = 0\). Good — the value goes to zero.
Also at \(r = r_\mathrm{c}\): \(\sin(\pi) = 0\), so \(f_\mathrm{c}'(r_\mathrm{c}) = 0\). The first derivative also vanishes at the cutoff. This is the \(C^1\) property: both \(f_\mathrm{c}\) and its first derivative are continuous at \(r = r_\mathrm{c}\) (with the convention \(f_\mathrm{c}(r) = 0\) for \(r > r_\mathrm{c}\)). Forces, which involve \(\partial r_{ij}/\partial \mathbf{r}_i\) multiplied by \(f_\mathrm{c}'(r_{ij})\) (chain rule through a descriptor), therefore go smoothly to zero at the cutoff and there is no jump.
Sanity check: \(f_\mathrm{c}(0) = \tfrac{1}{2}(\cos 0 + 1) = 1\). The cutoff is unity at the origin and decays smoothly to zero at \(r_\mathrm{c}\) — exactly the envelope we want.
The second derivative does not vanish at \(r_\mathrm{c}\): \(f_\mathrm{c}''(r_\mathrm{c}) = -\pi^2/(2 r_\mathrm{c}^2) \cos(\pi) = \pi^2 / (2 r_\mathrm{c}^2) \neq 0\). So the cosine cutoff is \(C^1\) but not \(C^2\) at the boundary. For phonon work and for sensitive free-energy estimates one prefers a polynomial cutoff \((1 - r/r_\mathrm{c})^p\) with \(p \ge 4\), which is \(C^{p-1}\) at the boundary.
Every descriptor in this chapter applies \(f_\mathrm{c}\) at every place where a neighbour enters a sum:
so that adding or removing a neighbour at the cutoff has vanishing effect.
Common mistake
A symmetric variant — multiplying by \(f_\mathrm{c}(r_{ij})\) inside the radial part and inside the angular part of a three-body descriptor — is required, not optional. If you forget to apply \(f_\mathrm{c}\) to the angular triple \(\{ij, ik\}\) you will see spurious oscillations in the radial distribution function and the NVE drift will become visible after a few picoseconds.
9.2.5 Compactness¶
The descriptor of atom \(i\) should be a fixed-length vector, independent of how many neighbours \(i\) has. A copper atom in the bulk has twelve neighbours within a typical cutoff; the same atom on a surface might have nine; an interstitial copper near a dislocation might have fifteen. The regression model downstream — a feed-forward neural network, a Gaussian process — expects inputs of a fixed dimension and cannot accommodate a variable-length list directly.
The deep-set construction discussed above solves this. By writing the descriptor as a sum of one-neighbour or two-neighbour contributions, \(\sum_j f(\mathbf{r}_{ij})\) or \(\sum_{j,k} g(\mathbf{r}_{ij}, \mathbf{r}_{ik})\), the dimension of the result is determined by the parameterisation of \(f\) or \(g\) and not by the number of neighbours. Bartók's SOAP power spectrum (§9.3.2), for example, is a vector of dimension \(N_\mathrm{rad}^2 (\ell_\mathrm{max} + 1)/2\), independent of neighbour count.
Compactness sits alongside locality: the radial cutoff \(r_\mathrm{c}\) bounds the number of neighbours physically (by the average atomic density times \(\tfrac43 \pi r_\mathrm{c}^3\)), and the descriptor parameterisation bounds the dimension mathematically. Together they make \(E_i\) a function whose computational cost per atom is constant in \(N\), which is what gives MLIPs their favourable scaling.
9.2.6 Invariant versus equivariant features: a worked example¶
To make the invariant/equivariant distinction concrete, consider a single atom with two neighbours at relative positions \(\mathbf{r}_1, \mathbf{r}_2\). We will build two two-body features and ask what each tells us about the geometry.
The invariant feature is the pair of distances and the cosine of the included angle:
Under any rotation \(R\), these three numbers are unchanged. Good: that is what invariance means. But the information lost is also clear: the absolute orientation of the pair in space is gone.
The equivariant feature, in the spirit of NequIP, keeps the vectors themselves but groups them by irreducible representation:
The \(\ell = 0\) part has two components and is scalar; the \(\ell = 1\) part has six components (two vectors of three components each) and transforms as \(\mathbf{x} \mapsto R\mathbf{x}\) under rotation. The total dimensionality is larger, eight numbers rather than three, but no geometric information has been thrown away. From the equivariant feature we can reconstruct the original \(\mathbf{r}_1, \mathbf{r}_2\) up to a single global rotation; from the invariant feature we cannot.
This loss matters. Consider three neighbours at the vertices of an equilateral triangle versus three neighbours at the vertices of an isoceles triangle of the same edge length sum. Their distance multisets are different — three equal versus two equal and one different — and an invariant descriptor distinguishes them. But there are pairs of configurations that have identical distance and angle multisets while being not related by rotation. These are the so-called degenerate environments of Pozdnyakov et al. (2020). Strictly two-body and three-body invariant descriptors cannot distinguish them; one needs either higher body-order or equivariant features that propagate direction information through the network.
This is the deep reason equivariant networks outperform invariant ones at small training-set sizes. They have a representation rich enough to distinguish geometries that invariant descriptors collapse, so they need fewer examples to learn the right mapping.
9.2.7 Putting the symmetries together¶
A correct MLIP architecture, then, must:
- Operate on a neighbour list of relative vectors \(\mathbf{r}_{ij}\) (translation invariance).
- Construct features that either are invariant under \(\mathrm{O}(3)\) or transform as irreps of \(\mathrm{O}(3)\) (rotation invariance/equivariance), with the final energy a scalar.
- Build atomic contributions \(E_i\) as functions of an unordered set of neighbour features (permutation invariance), and sum atomic contributions to obtain the total energy.
- Multiply every neighbour contribution by a smooth cutoff \(f_\mathrm{c}(r_{ij})\) (smoothness).
- Produce descriptors of fixed dimension independent of the number of neighbours (compactness).
The remainder of the chapter is a catalogue of architectures that satisfy these constraints in different ways, and an exploration of the trade-offs between them. We begin with the oldest and simplest of the modern descriptors — Behler–Parrinello symmetry functions — which makes every constraint visible in a few lines of code.
9.2.8 Locality and the cutoff radius¶
A symmetry not yet stated but implicit throughout is locality: the atomic energy \(E_i\) depends only on neighbours within a finite cutoff \(r_\mathrm{c}\). This is not a symmetry of nature — Coulomb interactions are long-ranged — but a modelling decision. Three considerations justify it.
Physical. In condensed matter at non-trivial density, screening suppresses long-range correlations rapidly. The bonded and short-range non-bonded contributions to the energy decay over a few \(\text{\AA}\); the residual long-range dispersion and Coulomb contributions are small enough to be absorbed into smooth, slowly varying corrections, or, in the case of strongly ionic systems, treated by an explicit long-range term (a learnable charge model or an Ewald sum on top of the local MLIP).
Algorithmic. Locality makes the per-atom evaluation cost constant in system size. Without locality the cost would scale as \(O(N)\) per atom and \(O(N^2)\) for the whole configuration, the same unfavourable scaling that dooms naive DFT. The factor that turns MLIPs into practical tools is exactly the linear total cost in \(N\).
Statistical. Locality is a strong regulariser. By forcing \(E_i\) to be a function of only \(\mathcal{O}(50)\) inputs (the neighbour list within \(r_\mathrm{c}\)), we shrink the hypothesis class enormously and require far less training data. A non-local architecture would have to learn that distant atoms do not influence the energy — wasteful, when we already know this is true to good approximation.
The choice of \(r_\mathrm{c}\) is the main hyperparameter that controls the locality assumption. Typical values are:
- \(r_\mathrm{c} = 3.5\) to \(4.0\,\text{\AA}\) for tightly bound molecular systems (organic chemistry, biomolecules) — captures first-shell covalent bonding and immediate hydrogen bonds.
- \(r_\mathrm{c} = 5\) to \(6\,\text{\AA}\) for general condensed matter — captures the first two coordination shells in most metals and oxides.
- \(r_\mathrm{c} = 6\) to \(8\,\text{\AA}\) for ionic and polar systems where electrostatic screening is partial.
For a message-passing network with \(T\) layers, the effective receptive field is \(T \times r_\mathrm{c}\); a two-layer MACE network with \(r_\mathrm{c} = 5\,\text{\AA}\) sees information up to \(10\,\text{\AA}\) away from each atom, even though no single layer extends beyond \(5\,\text{\AA}\). This is the trick that lets modest cutoffs cover surprisingly long-range correlations.
How to choose \(r_\mathrm{c}\) in practice
A good empirical rule: pick the smallest cutoff such that the radial distribution function \(g(r)\) of your system is essentially unity (\(g(r) \approx 1.0 \pm 0.05\)) at \(r = r_\mathrm{c}\). This ensures the cutoff lies past the structured part of the local environment, in the bulk-like regime where neighbours look like a uniform sea. Pre-computing \(g(r)\) from a short classical-force-field or DFT-MD trajectory costs almost nothing and saves considerable later debugging.
9.2.9 Why the symmetry constraints come first¶
A pedagogical note before we move on. Many introductions to MLIPs present the architecture (Behler network, GAP, MACE) and then add "oh, and it respects rotation/translation/permutation symmetry" as a side remark. The presentation in this chapter inverts that emphasis deliberately. We have stated the symmetry constraints first because:
- The constraints are universal. Any correct MLIP — past, present, future — must satisfy them. Architectures come and go; the symmetries persist.
- The constraints determine the design space. Once you have accepted that you must be translation-, rotation-, and permutation-invariant with smooth cutoffs and bounded body order, the set of possible architectures is greatly narrowed. The architectures we will study are natural answers to the constraints, not arbitrary inventions.
- The constraints are how you debug. If your trained potential fails — energy drifts in NVE, the radial distribution function has spurious sharp features, MD blows up in five picoseconds — the first questions are: does it respect the symmetries exactly? Is the cutoff smooth to the required order? Are the descriptor gradients correct at boundary? Almost every MLIP bug traces back to a violated symmetry constraint.
Reader's compass: as you read §9.3 through §9.5, ask yourself for each architecture which of the five symmetries it satisfies exactly, which it satisfies approximately, and which it relies on the data to teach. The honest architectures (modern equivariant networks) satisfy all five exactly; older architectures cut corners on one or two (e.g. older message-passing nets approximate rotation invariance rather than enforcing it). The trend is clear: enforce more, learn less, generalise better.