Atomistic Graph

This document fixes the graph language used to describe atomic structures and their associated features. We first define an atomistic graph for a finite set of atoms. A periodic atomistic graph then admits two equivalent pictures:

an infinite periodic graph in which every atomic image is an ordinary node;
a finite unit-cell graph in which an edge records which periodic image of its endpoint is involved.

The infinite picture makes local environments and geometric operations easy to interpret. The unit-cell picture is the finite graph used for storage and computation.

This point of view leads to a useful separation:

graph construction selects local atomic relations, including relations to periodic images;
the unit-cell graph records those image-resolved relations using relative cell shifts;
message passing applies the same local update in either picture;
the Physical Interface interprets node, edge, and graph outputs as task-specific physical quantities.

Consequently, most message-passing formulas apply to finite and periodic systems without modification. Periodicity changes how local neighbors are identified and stored, but not the basic form of a local graph update.

The lattice and Fourier conventions follow Periodic Systems.

1. A Finite Atomistic Graph

Consider a finite structure containing \(N\) atoms at positions \(\boldsymbol r_i\). Its atomistic graph is

\[ \mathcal G=(V,E), \qquad V=\{1,\ldots,N\}. \]

Each node \(i\in V\) represents one atom. Each directed edge \((i,j)\in E\) runs from source atom \(i\) to target atom \(j\). It represents a selected atomic relation, not necessarily a chemical bond.

The reverse directed edge \((j,i)\) is distinct and must be included or reconstructed explicitly when it is needed. The edge direction fixes the source and target; the architecture still determines how information is read from or written to them.

1.1 Edge displacements and neighborhoods

For each directed edge \((i,j)\), the displacement from its source to its target and the corresponding distance are

\[ \boldsymbol d_{ij} = \boldsymbol r_j-\boldsymbol r_i, \qquad r_{ij} = \left|\boldsymbol d_{ij}\right|. \]

A model may use the displacement, distance, direction, radial embedding, or other fixed spatial data derived from the relative positions of the endpoints. We denote this information collectively by \(g_{ij}\).

The global edge set \(E\) is chosen for the task rather than implied by the atomic positions alone. Given \(E\), the neighborhood (or neighbor set) of node \(i\) is

\[ \mathcal N(i) = \left\{ j: (i,j)\in E \right\}. \]

For the simplest cutoff graph, this becomes

\[ \mathcal N(i) = \left\{ j: r_{ij}<r_{\mathrm c} \right\}. \]

The precise rule may also exclude self-edges, cap the number of neighbors, or encode additional chemical or physical constraints. Different roles may use different edge sets. For example, \(E^{\mathrm{int}}\) may define the local interactions used by a Backbone, while \(E^{\mathrm{out}}\) contains the pairs on which a two-center quantity must be predicted.

2. Two Pictures of a Periodic Atomistic Graph

Let a unit cell contain atoms \(i=1,\ldots,N\) at positions \(\boldsymbol\tau_i\). The crystal contains a translated image of each atom in every lattice cell.

2.1 The infinite periodic picture

In the infinite picture, a node is written \((i,\boldsymbol T)\), where \(\boldsymbol T\) is the cell-shift vector of its cell. Its position is

\[ \boldsymbol r_{i,\boldsymbol T} = \boldsymbol\tau_i+\boldsymbol T. \]

A periodic edge may then be written

\[ (i,\boldsymbol T) \longrightarrow (j,\boldsymbol T'). \]

Its displacement depends only on the relative translation between the two cells:

\[ \boldsymbol d_{(i,\boldsymbol T)(j,\boldsymbol T')} = \left(\boldsymbol\tau_j+\boldsymbol T'\right) - \left(\boldsymbol\tau_i+\boldsymbol T\right) = \boldsymbol\tau_j +\left(\boldsymbol T'-\boldsymbol T\right) -\boldsymbol\tau_i. \]

This is an ordinary geometric graph with infinitely many nodes and edges. Its usefulness is conceptual: every periodic image is an ordinary neighboring atom, so local geometry and local graph operations have the same meaning as they do in a finite structure.

2.2 The unit-cell picture

Translation-equivalent images of atom \(i\) have the same role, so they can be represented by one stored node \(i\). The finite unit-cell graph obtained by identifying translation-equivalent nodes and edges is often called a quotient graph in the periodic-graph literature. Here we use the more concrete term unit-cell graph.

An edge must still record which periodic image of its target is involved. Its relative cell shift is

\[ \boldsymbol R = \boldsymbol T'-\boldsymbol T, \]

and \(\boldsymbol R\) forms part of the edge label. The complete edge identity is therefore

\[ (i,j;\boldsymbol R), \]

and different shifts may label multiple distinct edges between the same unit-cell nodes \(i\) and \(j\). A graph that allows multiple edges between the same pair of nodes is called a multigraph. Under our directed-edge convention, the unit-cell graph is therefore generally a directed multigraph.

The edge represents the infinite family of relations

\[ (i,\boldsymbol T) \longrightarrow (j,\boldsymbol T'), \qquad \boldsymbol T'-\boldsymbol T=\boldsymbol R, \]

for every pair of cells with the same relative shift \(\boldsymbol R\). The corresponding displacement and distance are

\[ \boxed{ \boldsymbol d_{ij}^{\boldsymbol R} = \boldsymbol\tau_j+\boldsymbol R-\boldsymbol\tau_i }, \qquad r_{ij}^{\boldsymbol R} = \left| \boldsymbol d_{ij}^{\boldsymbol R} \right|. \]

Implementations commonly store \(\boldsymbol R\) as three integer shifts along the lattice vectors. The fixed spatial data associated with the edge are written \(g_{ij}^{\boldsymbol R}\).

The reverse of \((i,j;\boldsymbol R)\) is \((j,i;-\boldsymbol R)\). When \(i=j\), a nonzero shift still reaches a distinct periodic copy of the same unit-cell atom. Only \((i,i;\boldsymbol 0)\) is a zero-displacement self-edge.

2.3 Periodic cutoff neighborhoods and edge sets

As in the finite case, the atomic positions do not by themselves determine the global edge set. A cutoff is one common edge-selection rule. In the infinite picture, the neighborhood of unit-cell node \(i\) is the set of image nodes

\[ \mathcal N(i) = \left\{ (j,\boldsymbol R): r_{ij}^{\boldsymbol R}<r_{\mathrm c} \right\}, \]

subject to the same task-specific self-edge and neighbor-selection rules as in the finite case. The same unit-cell atom \(j\) may occur several times with different shifts because \((j,\boldsymbol R)\) denotes a different node for each periodic image.

In the unit-cell picture, these image nodes are represented by the same stored node \(j\), while the shift becomes part of the edge label. The corresponding object is therefore the set of outgoing multigraph edges

\[ \mathcal E(i) = \left\{ (i,j;\boldsymbol R): r_{ij}^{\boldsymbol R}<r_{\mathrm c} \right\}. \]

Thus \(\mathcal N(i)\) is a set of image nodes, whereas \(\mathcal E(i)\) is the equivalent set of unit-cell edges.

The cutoff makes the infinite graph locally finite: although the crystal has infinitely many nodes, each node has only finitely many selected neighbors. It also makes every \(\mathcal E(i)\) finite and hence the stored unit-cell edge set finite. Implementations construct these finite image-resolved edge lists directly; they do not need to instantiate the infinite graph.

The lattice, unit-cell positions, relative shifts, and pair displacements remain explicit geometry. The unit-cell graph omits the redundant absolute cell label without discarding relations between different cells.

The two pictures can be summarized as follows:

	Infinite periodic picture	Unit-cell picture
Node	One node \((i,\boldsymbol T)\) for each atomic image	One stored node \(i\) for each unit-cell atom
Edge	\((i,\boldsymbol T)\to(j,\boldsymbol T')\)	\((i,j;\boldsymbol R)\) with \(\boldsymbol R=\boldsymbol T'-\boldsymbol T\)
Graph type	Ordinary directed graph: different images are different nodes	Directed multigraph: different shifts may connect the same stored nodes
Local environment of \(i\)	Image-node neighborhood \(\mathcal N(i)\)	Outgoing edge set \(\mathcal E(i)\)
Size with a cutoff	Infinite graph with finite neighborhoods	Finite node and edge sets
Main use	Interpreting local environments and spatial relations	Storage and computation

3. Features Attached to an Atomistic Graph

3.1 Graph features and fixed spatial data

Once the graph and its fixed spatial data have been defined, features may be attached to its nodes, edges, or the whole graph:

Location	Finite graph	Periodic unit-cell graph
Node	\(v_i\)	\(v_i\), stored once for each unit-cell atom
Edge	\(e_{ij}\)	\(e_{ij}^{\boldsymbol R}\), stored for each image-resolved edge
Graph	\(u\)	\(u\), describing the complete structure or unit cell

These symbols may denote multi-component, matrix-valued, or irrep-typed features; plain typeface does not imply that they are scalars. Graph features should be distinguished from positions, cell shifts, displacements, and distances, which specify the spatial structure of the graph.

3.2 Translation and physical interpretation

Storing features only on the unit-cell graph normally means that translation-equivalent nodes and edges share the same feature:

\[ v_{i,\boldsymbol T}=v_i, \qquad e_{(i,\boldsymbol T)(j,\boldsymbol T')} = e_{ij}^{\boldsymbol R}, \qquad \boldsymbol R=\boldsymbol T'-\boldsymbol T. \]

For node features, this is strict lattice periodicity. For edge features, it means invariance under a simultaneous translation of both endpoints: the feature may depend on their relative cell shift \(\boldsymbol R\), but not on their absolute cells. This convention is used for the latent features of the Backbone and is appropriate for many real-space physical quantities.

A periodic local field, such as the electron density, can be represented through coefficients attached to the unit-cell atoms. These coefficients are periodic node features. Likewise, for a lattice-translation-invariant one-electron operator \(\hat X\), the real-space block

\[ X_{\mu s,\nu s'}(\boldsymbol R) \equiv \left\langle \phi_{\mu\boldsymbol 0};s \middle| \hat X \middle| \phi_{\nu\boldsymbol R};s' \right\rangle \]

depends on the relative cell shift and can be attached to the corresponding edge \((i,j;\boldsymbol R)\). Hamiltonian and overlap blocks both fit this cell-shift-indexed edge data. A total energy is a graph feature, possibly formed from periodic node contributions.

Not every physical quantity follows the strictly periodic node convention. A Bloch state at \(\boldsymbol k\) obeys \(\psi_{a\boldsymbol k}(\boldsymbol r+\boldsymbol T)=\mathrm e^{+\mathrm i\boldsymbol k\cdot\boldsymbol T}\psi_{a\boldsymbol k}(\boldsymbol r)\). It can still be represented by a unit-cell node feature \(v_i(\boldsymbol k)\), provided that its value on each image node is reconstructed with the same Bloch phase:

\[ v_{i,\boldsymbol T}(\boldsymbol k) = \mathrm e^{+\mathrm i\boldsymbol k\cdot\boldsymbol T} v_i(\boldsymbol k). \]

The stored feature is therefore a unit-cell representative rather than a value copied unchanged to every cell. The graph states where a feature is stored; its physical meaning and translation law are supplied by the Physical Interface.

The physical definitions of the node and edge examples are developed in Auxiliary-Basis Expansion and Projection of Density and Localized Atomic-Orbital Basis. Their transformation laws are developed in Real-Space Fields and LCAO Operators.

4. One Message-Passing Rule for Finite and Periodic Graphs

A message-passing layer only requires a center node \(i\) and its neighborhood \(\mathcal N(i)\). If \(a\) denotes a neighboring node, the local update has the generic form

\[ m_{ia} = M\left(v_i,v_a,e_{ia},g_{ia}\right), \]

\[ M_i = \sum_{a\in\mathcal N(i)} m_{ia}, \qquad v_i' = U\left(v_i,M_i\right). \]

For a finite graph, \(a=j\) is an ordinary neighboring atom. For a periodic graph in the infinite picture, \(a=(j,\boldsymbol R)\) is an ordinary neighboring image. The same formula therefore applies in both cases. The cutoff makes \(\mathcal N(i)\) finite even though the entire periodic graph is infinite.

The unit-cell picture provides an equivalent edge indexing:

\[ (j,\boldsymbol R)\in\mathcal N(i) \quad\longleftrightarrow\quad (i,j;\boldsymbol R)\in\mathcal E(i). \]

Because the image node \((j,\boldsymbol R)\) carries the periodic feature \(v_j\), its message is

\[ m_{ij}^{\boldsymbol R} = M\left( v_i, v_j, e_{ij}^{\boldsymbol R}, g_{ij}^{\boldsymbol R} \right). \]

The periodic aggregation can then be written equivalently in the two pictures:

\[ M_i = \sum_{(j,\boldsymbol R)\in\mathcal N(i)} m_{ij}^{\boldsymbol R} = \sum_{(i,j;\boldsymbol R)\in\mathcal E(i)} m_{ij}^{\boldsymbol R}. \]

The left-hand side uses the node neighborhood of the infinite picture; the right-hand side uses the outgoing edges of the unit-cell multigraph. They sum the same messages. The unit-cell graph therefore changes how the periodic graph is stored and indexed, not the local message-passing architecture.

This is the convention used in Message Passing, where a generic edge is written as \((i,j)\). For a periodic graph, that notation represents the complete image-resolved edge \((i,j;\boldsymbol R)\).