POLYATOMIC COMPLEXES: A TOPOLOGICALLY INFORMED LEARNING REPRESENTATION FOR ATOMISTIC SYSTEMS

## Abstract

Abstract Developing robust representations of chemical structures that enable models to learn topological inductive biases is challenging. In this manuscript, we present a representation of atomistic systems. We begin by proving that our representation satisfies all structural, geometric, efficiency, and generalizability constraints. Afterward, we provide a general algorithm to encode any atomistic system. Finally, we report performance comparable to state-of-the-art methods on numerous tasks. We open-source all code and datasets. The code and data are available at https://github.com/rahulkhorana/PolyatomicComplexes .

## 1 Introduction

Recent advances in machine learning have enabled us to leverage representations of chemical state at different levels of chemical scale to learn meaningful patterns in chemical data. This is a particular kind of representation learning.
In cheminformatics and bioinformatics numerous representations of chemical structures exist. The most popular representations are SMILES, ECFP fingerprints, SELFIES, and Graphs. The following section briefly presents existing molecular representations and their construction.

##### SMILES

Simplified molecular-input line entry systems are obtained by printing symbol nodes encountered in a depth first traversal of a chemical graph []. SMILES are suitable for small molecules, but not large molecules. Additionally, they generate a substantial number of invalid molecules and fail to efficiently handle rings, branches, and bonds between atoms [].

##### DeepSMILES

DeepSMILES was created to resolve some of the issues with SMILES. In particular, DeepSmiles addresses concerns with unmatched parentheses and ring closure symbols []. However, DeepSMILES still allows for semantically incorrect strings [].

##### ECFP Fingerprints

ECFP fingerprints are derived using a variant of the Morgan algorithm []. Specifically, in the iterative updating stage, ECFP fingerprints encode the numbering invariant atom information to an atom identifier. During the duplicate removal stage, ECFPs reduce occurrences of the same feature []. It is important to note that ECFPs are not ideal for substructure searching, as they are slow when applied to large databases. In addition, ECFP fingerprints are non-invertible because the hash function maps randomly to integers. Further, ECFPs do not contain the chemical properties of each atom [, , ].

Figure: Figure 1: We graphically describe, with the example of $\mathrm{H_{2}O}$, how to construct a Polyatomic complex. The first step is to encode each individual atom in detail (protons, neutrons, electrons). We view protons, neutrons and electrons as $n$-spheres. Afterward we combine our representations of each individual atom together to form molecules/atomistic systems. One can choose to further augment the representation at this step. Finally, one can feed the representation into a machine learning model.
Refer to caption: extracted/5879907/graphical_abs_1.png

##### SELFIES

SELFIES follow a specific set of derivation rules, which enable the representation to satisfy semantic and syntactic constraints, while also avoiding syntactic mistakes []. SELFIES cannot fully represent macromolecules, crystals, and molecules with complicated bonds []. Similarly, SELFIES suffer from issues with substructure control, as described in [].

##### GroupSELFIES

GroupSELFIES are a more recent modification of SELFIES []. They remedy some of the issues with SELFIES by cleverly adding group tokens. Such an approach ensures that structures, like benzene rings, are more often preserved if shuffling occurs []. The primary limitation with GroupSELFIES is that groups cannot overlap. Consequently, one cannot represent polycyclic compounds [].

##### Graphs

A graph $\mathcal{G}$ is an ordered pair $(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ comprises a set of vertices and $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ is composed of a set of edges []. For molecules, the vertices $\mathcal{V}=\{v_{1},\ldots,v_{n}\}$ represent the atoms of a molecule and the edges $\mathcal{E}$ represent covalent bonds between the atoms []. Additional vertex and edge labels may be incorporated. The simplicity of graph representations poses some weaknesses, however. For example, there is no natural way to represent 3-dimensional structures of molecules and many other chemical properties that are essential to the molecule’s functionality [].

##### Atomic Cluster Expansion (ACE)

Atomic Cluster Expansion is a technique which enables one to efficiently parameterize many atom interactions []. The basis functions of ACE are
complete and can represent other local descriptors such as SOAP. In fields such as high energy physics, ACE is utilized to construct many-body interaction models which respect physical symmetries []. However, standard ACE is known to be inefficient, thereby leading to numerous proposed variants with different trade-offs [, ].

##### Behler-Parrinello Descriptor

Behler-Parrinello is an ANN architecture that describes atomic environments with symmetry functions and relies on an element-specific neural network for atomic energies []. Behler-Parrinello belongs to a class of machine learning potentials wherein the potential-energy surface (PES) is learned by adjusting parameters. The goal of fitting the PES is usually to accurately reproduce reference electronic structure data [].

##### Bartók/SOAP Descriptor

Smooth Overlap of Atomic Positions (SOAP), sometimes called the Bartók descriptor, is a technique to encode regions of atomic geometries []. SOAP relies on locally expanding a gaussian smeared atomic density with orthonormal functions []. The descriptor has been proven to be invariant to the basic symmetries of physics namely rotation, reflection, translation, and permutation of atoms of the same species [].

##### e3nn/Euclidean Neural Networks

e3nn is a generalized framework for creating euclidean neural networks []. e3nn naturally operates on geometric tensors that describe systems in three dimensions []. Equivariant operations such as spherical harmonics functions can be composed to create more complex modules. The neural network has proven to be E(3) equivariant and well suited to objects in euclidean space [].

### 1.1 Motivation and Representation Criteria

All representations listed above violate a non-empty subset of these criteria: invariances, uniqueness, continuity, differentiability, generality/generalizability, computational efficiency, topological accuracy, ability to consider long-range interactions, and chemical informedness [, , , ]. In this work, we propose a new representation, polyatomic complexes, that mathematically satisfies the above constraints, consequently addressing the limitations of all discussed representations.

##### Invariances

We consider invariance under changes in atom indexing and those fundamental to physics. These invariances are rotation, reflection, and translations. We prove that polyatomic complexes satisfy all these invariances ( 2.14).

##### Uniqueness

As described by Langer et al. (2022), uniqueness refers to the idea that two systems differing in properties should be mapped to different representations. Systems with equal representations that differ in property induce errors. Uniqueness is necessary and sufficient for reconstruction, up to invariant transformations, of an atomistic system from its representation []. We prove that polyatomic complexes satisfy uniqueness( 2.14).

##### Continuity and Differentiability

Representations of atomistic systems should be continuous and differentiable with respect to atomic coordinates []. Moreover, discontinuities work against regularity assumptions of many machine learning models. We prove that polyatomic complexes are continuous and differentiable with respect to atomic coordinates ( 2.14).

##### Generalizability/Generality

We use the words generalizability and generality interchangeably in this paper. We say a representation of atomistic systems or molecules is generalizable only if it can encode any atomistic system. SMILES, SELFIES, and GroupSELFIES are not generalizable, as they cannot represent certain kinds of molecule (crystals, polycyclic compounds, etc.) or they generate invalid molecules. However, we prove that polyatomic complexes are generalizable ( 2.14).

##### Efficiency

Traditionally, computational efficiency is a measure of how well an algorithm utilizes memory or time when completing a task. In the context of molecular representations, efficient representations run in polynomial time and with polynomial space. Ideally, representations are linear in the number of elements in a molecule, $O(S)$, as is the case with molecular graphs.
In practice, polyatomic complexes are linear in the number of elements in a molecule, essentially $O(S)$. We provide a complete proof of time complexity in Theorem 2.14. However, in the case of Atomic Cluster Expansions (ACE), the algorithm described is not computationally efficient []. More specifically, the basis constructed in the original paper leads to $N!$ many terms which arise in the summation over all permutations. Additionally, evaluating the sum over all the $N$-neighbor clusters and $J$ particle positions brings an additional $\binom{J}{N}$ cost [].
In the case of the Bartók/SOAP descriptor, we observe a rapid increase in descriptor size for environments composed of multiple elements []. The SOAP power spectrum scales quadratically with the number of elements $S$, while the length of the bispectrum scales as $S^{3}$. As a result, SOAP descriptors are significantly less efficient than graph based methods which run in $O(S)$.
As a result of the design of Behler-Parrinello, they are slower than $O(S)$ [].
It should be noted that ACE, SOAP, and Behler-Parrinello are traditionally used for quantum-chemistry applications and are not designed for more typical computational chemistry tasks.

##### Topological Accuracy

A representation is deemed topologically accurate if it can correctly represent the geometry of any molecule or atomistic system. Correctness requires representing the shape, bond-angles, dihedrals/torsion, and electronic structure aspects accurately. Only polyatomic complexes, SOAP, and ACE are “reasonably” topologically accurate. Polyatomic complexes are not necessarily topologically accurate when it comes to electronic structure. Our representation, in practice, uses s-orbitals to represent electronic wave functions, which is an oversimplification. However, this can be altered at the cost of additional time complexity. Inclusion of a complete set of commuting observables (CSCO) would potentially enable perfect topological accuracy.
SOAP enforces differentiability with respect to the atoms and invariance with respect to the basic symmetries
of physics. Additionally, SOAP considers the potential energy surfaces (PESs) and electrostatic multipole moment surfaces.
Similarly, polyatomic complexes enforce differentiability, are invariant to the basic symmetries of physics, and can be augmented with electronic structure aspects and forces.
ACEs are invariant under the basic symmetries of physics and systematically describe the local environments of particles at any body order.

##### Long-range interactions

The term long-range interactions refers to electrostatic potential energies between atoms and molecules, with mutual distances ranging from a few tens to a few hundreds Bohr radii []. Interactions can only be evaluated up to a certain distance. The maximum distance applied in a simulation is usually referred to as the cut-off radius, $r_{c}$, because the Lennard-Jones potential is radially symmetric.
Behler-Parinello neglects long-range interactions, which are electrostatics beyond the cutoff radius []. Contrastingly, polyatomic complexes can adjust for varying
definitions of cutoff radius. Similarly all string based representations and graph representations neglect long-range interactions. It should be noted that the precise definition of cutoff radius is dependent on the force field. The value of $r_{c}$ is generally obtained empirically.

##### Chemical and Physical Informedness

We say a representation is well-informed by chemistry or physics if it contains information about the chemical properties of each individual atom. ECFP fingerprints are not well-informed under this definition. If one compares ECFP fingerprints, Graphs, SMILES, or SELFIES to representations like ACE, SOAP, or polyatomic complexes, it is apparent that the former group does not contain the same level of chemical information as the latter. In essence, the representations should encode electronic structure, radial functions, spherical harmonics, wave-functions, and long-range interactions effectively. ACE utilizes basis functions to efficiently parameterize many-atom interactions []. SOAP relies on the local expansion of a Gaussian smeared atomic density with orthonormal functions []. Polyatomic complexes describe the geometry of individual atoms and encode radial functions in an efficient manner. They are flexible enough to potentially encode information about spherical harmonics as well. Additionally, they are compatible with force fields and can calculate the RDF as outlined in Section 2.3. We discuss in detail, how one can integrate our model with the AMBER Force Field in the Appendix A.18. It is theoretically shown that such an integration can be done. In practice, encoding spherical harmonics is left for future work.

In order to address the limitations of other representations, we develop Polyatomic Complexes. We propose a computationally efficient sampling-based approach for representing atomistic systems as CW-complexes. Our proposed sampling-based approach is computationally efficient and compatible with traditional physics-based methods. We delve into the mathematical construction in subsections 2.1 and 2.2. In appendix A.18, we discuss how our representation can integrate with techniques in chemistry and physics. The experiments can be found in section 3. Our conclusion is in section 4. All proofs of theorems, lemmas, and additional definitions are found in the appendix A. Our contributions are as follows.

### 1.2 Contributions

Our theoretical contributions are as follows. We develop a representation that:

- •
satisfies all constraints outlined in Section 1.1.
- •
is generalizable (unlike SELFIES, SMILES, etc.).
- •
is simultaneously computationally efficient and can consider long-range interactions (unlike traditional representations in quantum chemistry).
- •
is geometrically and topologically accurate, regardless of the molecule/atomistic system one tries to model.
- •
is well-informed by physics and chemistry.

Contributions from our experiments are as follows. We provide/demonstrate:

- •
strong statistical evidence that our representation is competitive (in terms of accuracy) with the most commonly used representations in computational chemistry.
- •
our representation works well on crystal and materials datasets (Matbench, Materials Project) and smaller molecule datasets (Photoswitches etc.).
- •
a Gaussian process in tandem with our representation is competitive with GNN’s (deep learning) on Matbench.

In summary, we develop a representation that provides theoretical guarantees (invariance with respect to the fundamental symmetries of physics, etc.) while being computationally efficient, competitive in terms of accuracy, and agnostic concerning the atomistic system one chooses to model.

## 2 Core methods and representation

In this section, we outline how we model protons, neutrons, and electrons. These assumptions are, to an extent, simplifications. We assume that an atom consists of a positively-charged nucleus, containing protons and neutrons, and is surrounded by negatively-charged electrons. Additionally, a proton $p$ is isomorphic to a sphere with radius $r\neq 1$. Similarly, a neutron $n$ is isomorphic to a sphere with radius $r\neq 1$. Further, an electron $e$ is isomorphic to a sphere and is paired up with a collapsed wave-function $w_{e}$, wherein the position is not definite; instead, the probability of finding an electron in a certain region can be calculated (Section 2.1). An atom, then, is a particular CW-complex formed by gluing together protons, neutrons, and electrons. We view protons, neutrons and electrons as objects in the category of topological spaces, and the gluing maps between spaces as an implicit force model.

### 2.1 Mathematical representation of an atom

We now develop formal mathematical definitions that can be used to reason about the objects we construct in our computational representation.

###### Definition 2.1 .

We define the set $ee^{i}:=\{x\in\mathbb{R}^{i+1}~{}|~{}\|x\|<2.8fm\}$
where $1fm=1\times 10^{-15}~{}m$.

###### Definition 2.2 .

We define the set $PD^{i}:=\{x\in\mathbb{R}^{i}~{}|~{}\|x\|\leq 1fm\}$
where $1fm=1\times 10^{-15}~{}m$.

###### Definition 2.3 .

We define the set $ND^{i}:=\{x\in\mathbb{R}^{i}~{}|~{}\|x\|\leq 0.8fm\}$
where $1fm=1\times 10^{-15}~{}m$.

An atom is composed of protons, neutrons, and electrons.
We construct protons, neutrons, and electrons using the sets $ee^{i},PD^{i}$ and $ND^{i}$.

- (1)
Electrons: We know from above that an electron is isomorphic to a sphere and is paired up with a collapsed wave-function $w_{e}$. We know that $ee^{i}$ is a set and $w_{e}$ is a function. Functions can be represented as sets of ordered pairs $(x,y)$ such that $x\in X$ and $y\in Y$ when $w_{e}:X\to Y$. We can then let $Z=\{ee^{i},\{(x,y)~{}|~{}x\in X,y\in Y\}\}$. Give $Z$ the indiscrete topology $\tau$. A single electron $e$ is then $(Z,\tau)$. Therefore electron $e\in\operatorname{Top}$. Consequently, we can view $e$ as an object in the category of topological spaces.
- (2)
Protons: A proton is isomorphic to a sphere therefore we represent a single proton $p$ as a set $PD^{i}$ for $i\in\mathbb{N}$. Geometrically this set is a filled $i-1$-sphere with a radius of $1fm$ for instance realized up to homeomorphism as a closed ball in Euclidean space with the induced subspace topology. Traditionally, we choose $i=3$; however, one can consider higher dimensional spheres.
- (3)
Neutrons: A neutron is isomorphic to a sphere therefore we represent a single neutron $n$ as a set $ND^{i}$ for $i\in\mathbb{N}$. Geometrically this set is a filled $i-1$-sphere with a radius of $0.8fm$ for instance realized up to homeomorphism as a closed ball in Euclidean space with the induced subspace topology.

It should be noted that one can change the values we have set for all the radii at will. We mention this because the exact radius of a proton is contested []. We now proceed to develop our representation in a manner following the usual conventions in Algebraic Topology [].

###### Definition 2.4 .

We define the Atomic Complex with A.17 in mind.
Suppose that atom $A$ has $\mathcal{N}\in\mathbb{N}$ many neutrons $\mathcal{P}\in\mathbb{N}$ many protons and $\mathcal{E}\in\mathbb{N}$ many electrons.
Let $I_{n}=\{1,\ldots,\mathcal{N}\}$, $I_{p}=\{1,\ldots,\mathcal{P}\}$, $I_{e}=\{1,\ldots,\mathcal{E}\}$ be index sets enumerating protons, neutrons and electrons respectively. Additionally we assert that $K=\mathcal{P}+\mathcal{N}+\mathcal{E}$. (^1^11We explicitly make a design choice to let $K=\mathcal{P}+\mathcal{N}+\mathcal{E}$ primarily for dimensionality reasons.)
Let $\mathcal{T}=\{0,1,\ldots,K\}$ such that $\mathcal{T}$ is ordered and $\tau_{i}\in\mathcal{T}$.
Then for any $\tau_{i}$ we can generalize A.17 by attaching many protons, neutrons and electrons.
We let $P$ be our complex of protons, $N$ be our complex of neutrons, $E$ our complex of electrons.
We then construct sets of attaching maps for each complex
$\chi_{p}=\{\phi_{p,i}:\partial PD^{\tau_{i}}\to P_{i}~{}|~{}\forall i\in I_{p}\}$,
$\chi_{n}=\{\phi_{n,i}:\partial ND^{\tau_{i}}\to N_{i}~{}|~{}\forall i\in I_{n}\}$, and
$\chi_{e}=\{\phi_{e,i}:\partial ee^{\tau_{i}}\to E_{i}~{}|~{}\forall i\in I_{e}\}$.
We can think of these sets as continuous functions between the disjoint union spaces.
More formally we can write
$(\phi_{p,i})_{i\in I_{p}}:\bigsqcup_{i\in I_{p}}\partial PD^{\tau_{i}}\to P_{i}$,
$(\phi_{n,i})_{i\in I_{n}}:\bigsqcup_{i\in I_{n}}\partial ND^{\tau_{i}}\to N_{i}$, and
$(\phi_{e,i})_{i\in I_{e}}:\bigsqcup_{i\in I_{e}}\partial ee^{\tau_{i}}\to E_{i}$.
Recalling from above we know that $PD^{\tau_{i}},ND^{\tau_{i}},ee^{\tau_{i}}\in\operatorname{Top}$. Therefore we have three sequences of topological spaces $P=P_{0}\hookrightarrow P_{1}\hookrightarrow\cdots\hookrightarrow P_{\mathcal{P}}$,
$N=N_{0}\hookrightarrow N_{1}\hookrightarrow\cdots\hookrightarrow N_{\mathcal{N}}$, and
$E=E_{0}\hookrightarrow E_{1}\hookrightarrow\cdots\hookrightarrow E_{\mathcal{E}}$.
(^2^22The reasoning for why we need three separate complexes that we combine, is that $\mathcal{P}$ may not equal $\mathcal{N}$ which may not equal $\mathcal{E}$. Note that each of the inclusions $P_{i}\hookrightarrow P_{i+1}$, $N_{i}\hookrightarrow N_{i+1}$, and $E_{i}\hookrightarrow E_{i+1}$ induces an isomorphism on $\pi_{i}$ for all $i\leq n$. Additionally note that $\pi_{i}$ denotes the $i$-th fundamental group.)
For each $P_{i}\hookrightarrow P_{i+1}$, $N_{i}\hookrightarrow N_{i+1}$, and $E_{i}\hookrightarrow E_{i+1}$ we have that the following commute:
${\bigsqcup_{i\in I_{p}}\partial PD^{\tau_{i}}}$${P_{i}}$${\bigsqcup_{i\in I_{p}}PD^{\tau_{i}}}$${P_{i+1}}$$\scriptstyle{(\phi_{p,i})_{i\in I_{p}}}$
${\bigsqcup_{i\in I_{n}}\partial ND^{\tau_{i}}}$${N_{i}}$${\bigsqcup_{i\in I_{n}}ND^{\tau_{i}}}$${N_{i+1}}$$\scriptstyle{(\phi_{n,i})_{i\in I_{n}}}$
${\bigsqcup_{i\in I_{e}}\partial ee^{\tau_{i}}}$${E_{i}}$${\bigsqcup_{i\in I_{e}}ee^{\tau_{i}}}$${E_{i+1}}$$\scriptstyle{(\phi_{e,i})_{i\in I_{e}}}$
We combine the spaces $P$,$N$,$E$ together to get our atomic complex $A$.
We let $\phi_{p}:\partial P\to\partial N$, $\phi_{n}:\partial N\to\partial E$, and $\phi_{e}:\partial E\to A$.
Thus $A=A\cup_{\phi_{p}}P\cup_{\phi_{n}}N\cup_{\phi_{e}}E\in\operatorname{Top}$. (^3^33$A$ is initially the empty space $A=\varnothing$)

We now introduce the idea of a monad of atomic complexes. Fundamentally, in probability theory, one can consider the category of topological spaces $\operatorname{Top}$ and equip it with a monad whose functor assigns a space $PA$ of probability measures on $X$ to each element $A$. In the case of $\operatorname{Ato}$, a monad, we define a functor that assigns atomic complexes, elements of $\operatorname{Top}$, to a space of probability measures.

###### Definition 2.5 .

$\operatorname{Ato}$ is a monad whose objects are atomic complexes. Atoms are contained within $ob(\operatorname{Top})$. The morphisms are measurable functions encoding the probability that an atom forms a chemical bond. More explicitly:
Let $A_{1}$ and $A_{2}$ be atomic complexes and $f:A_{1}\to A_{2}$ be a measurable function. Let $f_{*}$ denote the pushforward, $\eta$ be a natural transformation, $T$ be a functor, and $\mu:A\to TA$.
Then the diagrams commute:

###### Proof.

The full proof is found in the Appendix A.6. At a high level, since atomic complexes are topological spaces, they can be equipped with a $\sigma$-algebra, and the morphisms are measurable; the result follows mutandis mutatis from the Giry monad on $\operatorname{Meas}$ [].
∎

###### Remark 2.6 .

This property allows a user to encode and make statements about the probability of two atomic complexes forming a chemical bond in a compact way. In essence, the $\frac{1}{r}$ Coulomb potential that gives the attraction between charges.

###### Definition 2.1 .

###### Definition 2.2 .

###### Definition 2.3 .

###### Definition 2.4 .

###### Definition 2.5 .

###### Proof.

###### Remark 2.6 .

### 2.2 Mathematical representation of polyatomic systems

In this section, we provide a generalized mathematical algorithm for modeling any polyatomic system. An atom, from this point onward, is represented by an atomic complex $A\in\operatorname{Top}$ with the monad structure from $\operatorname{Ato}$.

###### Definition 2.7 .

We define the Polyatomic Complex with A.5 and 2.4 in mind.
Suppose that atomistic system $M$ has $\mathcal{K}$ many atoms.
Let $I_{a}=\{1,\ldots,\mathcal{K}-1\}$ be an index set enumerating atoms.
We can generalize A.5 by attaching $\mathcal{K}$ many atoms, the corresponding force field, and electronic structure spaces.
We let $\phi_{A_{\mathcal{K}}}:\partial A_{\mathcal{K}}\to M$ and construct our set of attaching maps:
$\chi_{a}=\{\phi_{a,i}:\partial A_{i}\to A_{i+1}~{}|~{}\forall i\in I_{a}\}$.
We can think of these sets as continuous functions between the disjoint union spaces.
In essence,
$(\phi_{a,i})_{i\in I_{a}}:\bigsqcup_{i\in I_{a}}\partial A_{i}\to A_{i}$.
Recalling from above, we know that $\forall i\in I_{a}$ that $A_{i}\in\operatorname{Top}$.
Therefore, we have a sequence of topological spaces:
$M=A_{0}\hookrightarrow A_{1}\hookrightarrow\cdots\hookrightarrow A_{\mathcal{K}}$.(^4^44Note that each inclusion $A_{i}\hookrightarrow A_{i+1}$ induces an isomorphism on $\pi_{i}$ for $i\leq n$.)

Thus, $M=M\cup_{(\phi_{a,i})_{i\in I_{a}}}\left(\bigsqcup_{i\in I_{a}}A_{i}\right)
\cup_{\phi_{A_{\mathcal{K}}}}A_{\mathcal{K}}$. (^5^55$M$ is initially the empty space $M=\varnothing$)

###### Remark 2.8 .

It is apparent, at this point, that Polyatomic Complexes and Atomic Complexes are, by construction, finite $n$-connected CW-complexes. In any case, we prove each statement.

###### Lemma 2.9 .

Atomic complexes are finite $n$-connected CW-complexes.
The full proof can be found in the Appendix A.7.

###### Lemma 2.10 .

Polyatomic complexes are finite $n$-connected CW-complexes.
The full proof can be found in the Appendix A.8.

###### Theorem 2.11 .

Every Polyatomic complex has a smooth approximation.
The full proof can be found in the Appendix A.9.

###### Remark 2.12 .

Note that the atomic coordinates are contained in a linear subspace of $\mathbb{R}^{3}$. Therefore, the atomic coordinates are a smooth embedded submanifold of $\mathbb{R}^{3}$ by A.15.

###### Theorem 2.13 .

Atomic complexes are unique, continuous and differentiable with respect to atomic coordinates.
The proof for uniqueness can be found in the Appendix A.10. The proof of continuity and differentiability is found in the Appendix A.11.

###### Theorem 2.14 .

Polyatomic complexes satisfy all requirements for representations of atomistic systems, as outlined by Langer et al. in [].
In essence, we want to prove invariances, uniqueness, continuity, differentiability, generalizability/generality, and efficiency are all satisfied. All the aforementioned conditions are defined in Section 1.1. The full proofs for every condition are found in the Appendix.

Invariances: The full proof of satisfying all invariances can be found in the Appendix A.12.

Uniqueness: Uniqueness is necessary and sufficient for reconstruction, up to invariant transformations, of an atomistic system from its representation. The full proof of uniqueness can be found in the Appendix A.13.

Continuity and Differentiability: The full proof of continuity and differentiability with respect to atomic coordinates can be found in the Appendix A.14.

Generality/Generalizability: The full proof of generality can be found in the Appendix A.15.

Regressions: An important condition to demonstrate is that the structure of of our representation is suitable for regression. We provide statistical evidence of this in our experiments.

Computational efficiency: A proof of computational efficiency relative to the reference simulations is found in the Appendix A.16.

###### Definition 2.7 .

###### Remark 2.8 .

###### Lemma 2.9 .

###### Lemma 2.10 .

###### Theorem 2.11 .

###### Remark 2.12 .

###### Theorem 2.13 .

###### Theorem 2.14 .

### 2.3 Algorithms

In this section, we provide the pseudo-code for Atomic and Polyatomic complexes. Additionally we elaborate on each algorithmic method in subsections 2.3.1 and 2.3.2. We refer to these algorithms throughout the paper as Algorithm 1 and 2 respectively.

Figure: Algorithm 1 Atomic Complexes

#### 2.3.1 Algorithmic method for atomic complexes

In Algorithm 1, we provide our approach. In practice, we sample from the surface of each $n$-sphere uniformly when $i>0$ for $PD^{i},ND^{i},ee^{i}$. In Algorithm 1, we let $D_{F}$ denote a random matrix encoding pairwise forces and energetics. $D_{F}$ may be derived from a known force field model (QMDFF, CHARMM, ReaxFF) or be randomly initialized (Gaussian Unitary Ensemble) [, , ]. The method $update\text{\textunderscore}distances$ uses information from the electron wave function to update distance information (radial contribution). The matrix $D_{E}$ keeps track of this distance information. The function $glue(T,o,\phi)$ glues object $o$ to space $T$ by using gluing map $\phi$. Gluing can be as simple as an append or as complex as one likes. Using a dictionary allows one to specify an entire gluing scheme.

Figure: Algorithm 2 Polyatomic Complexes

#### 2.3.2 Algorithmic method for polyatomic systems

In Algorithm 2, we assume that all atoms are connected. This makes intuitive sense, as there are electromagnetic forces, strong nuclear forces, and weak nuclear forces that hold the atom together. Additionally, there are strong attractive forces, or intermolecular forces acting on all constituents.

In Algorithm 2, the functions $update\text{\textunderscore}radial$ and $update\text{\textunderscore}forces$ can optionally be provided by the user or default to standard behavior. If utilized, $update\text{\textunderscore}radial$ calculates the $RDF$ by picking a particular atom and calculating the density within the sphere. This can be implemented via Monte-Carlo methods as described by Lyubartsev [, ]. Alternatively, users can provide x-ray diffraction data. The $update\text{\textunderscore}forces$ method by default assumes Coulomb’s Law.

## 3 Benchmarks and experimentation

### 3.1 Experimental setup

In each experiment, we do the following. First, we fix a number of trials $n_{trials}$, number of epochs per trial $n_{epochs}$, and choose a training/test split ratio. Then, we process our dataset. We choose a kernel and fit an exact Gaussian process to our data. At the end of each trial, we compute the test set MAE, RMSE, and CRPS and store these values. Upon completion, we report the average MAE, RMSE, and CRPS across all trials. We calculate $1$-sigma error bars using bootstrapping, which represent the standard error across trials. For reference, choosing SELFIES as a representation with the Tanimoto Kernel is state-of-the-art for molecular learning tasks []. See Appendix A.19 for compute cost.

##### Fast Complex

In this approximation algorithm, we disconnect all cells in complex $C$ and project the entire representation onto the real plane to get a corresponding matrix $M\in\text{GL}_{n}(\mathbb{R})$. We zero-pad to ensure all matrices in our dataset have the same shape. We do not use any force field model, $update\text{\textunderscore}forces$ or $update\text{\textunderscore}distances$.

##### Deep Complex

In this approximation algorithm, we project all cells in $C$ onto the real plane to get a corresponding matrix $M\in\text{GL}_{n}(\mathbb{R})$. We do, however, preserve connectedness information. We zero-pad to ensure all matrices in our dataset have the same shape. We do not use any force field model, $update\text{\textunderscore}forces$, or $update\text{\textunderscore}distances$.

### 3.2 Photoswitches dataset

The photoswitches dataset consists of photoswitchable molecules reported as SMILES, experimental transition wavelengths reported in nanometers, and DFT-computed transition wavelengths reported in nanometers among others []. It should be noted that, for particular columns of the Photoswitches dataset, there are some missing values. Since we believe the values to be missing at random, we utilize mean imputation, instead of discarding experimental data. We convert all SMILES strings to all other representations prior to fitting, as part of the data processing step. For all representations, $n_{trials}=20$, $n_{epochs}=5$, and the train/test split is 67/33. Note that WL refers to the Weisfeiler Lehman Kernel for graphs. We report all values rounded up to one decimal place. Numerous additional tables are found in the Appendix A.17.

**Table 1: Photoswitches benchmark**
|  |  | Z isomer $\pi$-$\pi$* wavelength (nm) | DFT Z isomer $\pi$-$\pi$* wavelength (nm) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Kernel | Representation | RMSE | MAE | CRPS | RMSE | MAE | CRPS |
| Tanimoto | SELFIES | $7.7\pm 0.3$ | $3.9\pm 0.1$ | $2.9$ | $4.3\pm 0.2$ | $1.1\pm 0.1$ | $0.8$ |
| Tanimoto | SMILES | $6.9\pm 0.3$ | $3.4\pm 0.1$ | $2.8$ | $4.2\pm 0.2$ | $1.0\pm 0.1$ | $0.8$ |
| Tanimoto | ECFP | $7.0\pm 0.2$ | $3.3\pm 0.1$ | $2.8$ | $4.3\pm 0.3$ | $1.1\pm 0.1$ | $0.8$ |
| Tanimoto | Fast Complex | $8.0\pm 0.3$ | $2.9\pm 0.1$ | $2.9$ | $4.3\pm 0.3$ | $0.9\pm 0.1$ | $0.9$ |
| Tanimoto | Deep Complex | $8.0\pm 0.3$ | $2.9\pm 0.1$ | $2.9$ | $4.3\pm 0.3$ | $0.9\pm 0.1$ | $0.9$ |
| WL | Graphs | $7.9\pm 0.3$ | $3.6\pm 0.1$ | $2.7$ | $4.7\pm 0.2$ | $1.3\pm 0.1$ | $0.8$ |

### 3.3 ESOL

ESOL describes a simple method for estimating the solubility of a compound directly from its structure []. We convert all SMILES strings to all
other representations prior to fitting as part of the data processing step. For all representations, $n_{trials}=20$, $n_{epochs}=5$, and the train/test split is 67/33. The predicted log solubility column contains theoretical values. The measured log solubility column contains values obtained through experimentation. We report all values rounded up to one decimal place and the confidence limits to two places.

**Table 2: ESOL benchmark**
|  |  | Predicted log solubility (mol/L) | Measured log solubility (mol/L) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Kernel | Representation | RMSE | MAE | CRPS | RMSE | MAE | CRPS |
| Tanimoto | SELFIES | $0.5\pm 0.01$ | $0.4\pm 0.01$ | $0.9$ | $0.8\pm 0.01$ | $0.6\pm 0.01$ | $1.2$ |
| Tanimoto | SMILES | $0.6\pm 0.01$ | $0.4\pm 0.00$ | $1.0$ | $0.8\pm 0.01$ | $0.6\pm 0.00$ | $1.2$ |
| Tanimoto | ECFP | $0.7\pm 0.01$ | $0.5\pm 0.01$ | $1.0$ | $1.0\pm 0.01$ | $0.8\pm 0.01$ | $1.2$ |
| Tanimoto | Fast Complex | $1.7\pm 0.01$ | $1.3\pm 0.01$ | $1.3$ | $2.1\pm 0.01$ | $1.7\pm 0.01$ | $1.7$ |
| Tanimoto | Deep Complex | $1.7\pm 0.01$ | $1.3\pm 0.01$ | $1.3$ | $2.1\pm 0.01$ | $1.7\pm 0.01$ | $1.7$ |
| WL | Graphs | $0.4\pm 0.00$ | $0.3\pm 0.02$ | $0.9$ | $0.8\pm 0.01$ | $0.6\pm 0.01$ | $1.2$ |

### 3.4 FreeSolv

The Free Solvation database (FreeSolv) consists of experimental and calculated hydration free energies for small neutral molecules in water, along with molecular structures, input files, references, and annotations []. We convert all SMILES strings to all other representations prior to fitting as part of the data processing step. For all representations, $n_{trials}=20$, $n_{epochs}=5$, and the train/test split is 67/33. The experimental and calculated values correspond to hydration free energies reported in kcal/mol. The calculated column contains theoretical values. The experimental column contains values obtained through experimentation.

**Table 3: FreeSolv benchmark**
|  |  | experimental (kcal/mol) | calculated (kcal/mol) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Kernel | Representation | RMSE | MAE | CRPS | RMSE | MAE | CRPS |
| Tanimoto | SELFIES | $1.7\pm 0.10$ | $1.2\pm 0.03$ | $2.0$ | $1.6\pm 0.06$ | $1.2\pm 0.05$ | $2.3$ |
| Tanimoto | SMILES | $1.9\pm 0.05$ | $1.4\pm 0.02$ | $2.1$ | $1.8\pm 0.02$ | $1.3\pm 0.02$ | $2.3$ |
| Tanimoto | ECFP | $2.0\pm 0.06$ | $1.4\pm 0.02$ | $2.1$ | $2.2\pm 0.04$ | $1.5\pm 0.02$ | $2.3$ |
| Tanimoto | Fast Complex | $3.8\pm 0.05$ | $2.8\pm 0.03$ | $2.8$ | $4.2\pm 0.05$ | $3.1\pm 0.04$ | $3.1$ |
| Tanimoto | Deep Complex | $3.9\pm 0.06$ | $2.8\pm 0.04$ | $2.8$ | $4.2\pm 0.05$ | $3.1\pm 0.04$ | $3.1$ |
| WL | Graphs | $1.4\pm 0.02$ | $1.0\pm 0.01$ | $2.0$ | $1.2\pm 0.02$ | $0.8\pm 0.01$ | $2.2$ |

### 3.5 ChEMBL/lipophilicity

ChEMBL is an Open Data database containing binding, functional, and ADMET information for a large number of drug-like bioactive compounds []. We convert all SMILES strings to all other representations prior to fitting as part of the data processing step. For all representations, $n_{trials}=20$, $n_{epochs}=5$, and the train/test split is 67/33. The experimental values contain log octanol–water partition coefficients $\log(K_{ow})$ for bioactive compounds.

**Table 4: ChEMBL benchmark**
|  |  | experimental |  |  |
| --- | --- | --- | --- | --- |
| Kernel | Representation | RMSE | MAE | CRPS |
| Tanimoto | SELFIES | $0.9\pm 0.01$ | $0.7\pm 0.01$ | $0.7$ |
| Tanimoto | SMILES | $0.7\pm 0.01$ | $0.6\pm 0.01$ | $0.7$ |
| Tanimoto | ECFP | $0.7\pm 0.01$ | $0.6\pm 0.00$ | $0.7$ |
| Tanimoto | Fast Complex | $1.2\pm 0.01$ | $1.0\pm 0.00$ | $1.0$ |
| Tanimoto | Deep Complex | $1.2\pm 0.01$ | $1.0\pm 0.00$ | $1.0$ |
| WL | Graphs | $0.7\pm 0.01$ | $0.5\pm 0.00$ | $0.7$ |

### 3.6 Materials Project

Materials Project is a large database containing $\approx 173,000$ complex materials []. We run our GP using a fixed batch size. For our representation, $n_{trials}=20$, $n_{epochs}=5$, and the train/test split is 10/90. The experimental values are equilibrium reaction energy reported in eV/atom and magnetization normalized vol. reported in $\mu$B. It should be noted that we ran our algorithm on the entire Materials Project database, not a distinct subset. Some example materials are $\mathrm{U_{3}(HO_{5})_{2}}$, $\mathrm{LuTaO_{4}}$, and $\mathrm{Dy(SiPd)_{2}}$ referenced by their materials project codes mp-626062, mp-5489, and mp-3301 respectively. It should be noted that e3nn is currently state of the art on Materials Project []. A visualization of the materials are provided in Figure 2.

**Table 5: Materials Project Benchmarks**
| Quantity | Fast Complexes |  |  |
| --- | --- | --- | --- |
|  | RMSE | MAE | CRPS |
| Equilibrium Reaction Energy (eV/atom) | $0.3296\pm 0.0005$ | $0.1017\pm 0.0005$ | $0.1017\pm 0.0005$ |
| Total Magnetization Normalized Vol ($\mu$B) | $0.0196\pm 0.0000$ | $0.0101\pm 0.0000$ | $0.0101\pm 0.0000$ |

Figure: Figure 2: Materials project: materials mp-626062, mp-5489, and mp-3301 []
Refer to caption: extracted/5879907/Styles/mp-626062.png

### 3.7 Matbench: JDFT2D

Matbench is a collection of benchmarks for materials science []. Matbench: JDFT2D is a task in which one must predict exfoliation energies from crystal structure. The exfoliation energies are computed with the OptB88vdW and TBmBJ functionals []. For our representation, $n_{trials}=20$, $n_{epochs}=5$, and the train/test split is 67/33. The experimental values are reported in meV/atom.

**Table 6: Matbench: JDFT2D Benchmarks**
|  |  | Exfoliation Energy (meV/atom) |  |
| --- | --- | --- | --- |
| ML Model | Representation | RMSE | MAE |
| MODNet (v0.1.12) [] | Graph | 96.7332 | 33.1918 |
| coNGN | Graph | 95.4766 | 36.1698 |
| SchNet | Graph | 111.0187 | 42.6637 |
| DeeperGATGNN (v0.1.12) | Graph | 113.4192 | 45.4611 |
| ALIGNN | Graph | 117.4213 | 43.4244 |
| GP + Tanimoto | Fast Complexes | 117.5536 | 60.8629 |
| GP + Tanimoto | Deep Complexes | 117.7818 | 60.8381 |
| Finder_v1.2 (structure-based) | Graph | 120.0917 | 46.1339 |
| CrabNet | Graph | 120.0088 | 45.6104 |
| MegNet | Graph | 129.3267 | 54.1719 |

## 4 Discussion

Polyatomic complexes are a generalizable, topologically-informed method for encoding atomistic systems. We have introduced the mathematical underpinnings of our representation, proven we satisfy numerous important criteria, provided an algorithm to construct them, and benchmarked our representation on numerous tasks. Polyatomic complexes satisfy all discussed criteria while being computationally efficient. In addition to providing invariance under coordinate transformations, our approach is suitable for regression. The scope of any claims with regard to experimental performance made are limited by our experiments.

##### Limitations and Future Work

Our results raise several important questions for future work. How can we develop better learning methods and kernels that operate on CW-complexes? Our results suggest that the Tanimoto kernel is not well-suited to our representation. We justified using the Tanimoto kernel as it enabled us to make an apples-to-apples comparison. We wanted a standard kernel that was well-defined for most representations, making comparisons more fair. The Tanimoto kernel is not well suited to our representation as it does not consider geometric information. The primary limitation of our method is that the observed accuracy is only comparable with existing methods and not significantly better. Future work may involve investigating how to improve the accuracy we observe. Improving accuracy involves developing better learning methods and introducing wave functions beyond the s-radial function. We plan to encode diffusion functions (chemistry) and s,p,d, and f orbitals. We leave this as a future direction. Discovering the ideal balance of computational cost and accuracy when comparing a precise wave function to an approximated representation is an exciting direction.

## Acknowledgements

Rahul Khorana acknowledges support from the University of California, Berkeley and Imperial College London. Dr. Marcus Noack acknowledges support by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under U.S. Department of Energy Contract No. DE-AC02-05CH11231. Dr. Jin Qian was supported by the DOE Office of Science Energy Earthshot Initiatives as part of the Center for Ionomer-based Water Electrolysis at Lawrence Berkeley National Laboratory under contract number DE-AC02-05CH11231.

## Appendix A Appendix

### A.1 Molecular kernels for Gaussian processes

In this section we provide some common kernels used with Gaussian processes for molecular machine learning tasks.

###### Definition A.1 .

Tanimoto: Let $d\in\mathbb{N}_{\geq 1}$ and $x,x^{\prime}\in\{0,1\}^{d}$ be binary vectors where $\|\cdot\|$ is the Euclidean norm and $\sigma^{2}_{f}$ is the scalar signal variance hyperparameter. Then define

$$ $k_{tanimoto}=\sigma^{2}_{f}\cdot\frac{\langle x,x^{\prime}\rangle}{\|x\|^{2}+ \|x^{\prime}\|^{2}-\langle x,x^{\prime}\rangle}$ $$

###### Definition A.2 .

String Kernel: Let $S$ be a SMILES or SELFIES string, $d\in\mathbb{N}_{\geq 1}$ and $\phi:S\to\mathbb{R}^{d}$ be a bag-of-characters representation then define for any strings $S,S^{\prime}$

$$ $k_{string}(S,S^{\prime})=\sigma^{2}\langle\phi(S),\phi(S^{\prime})\rangle$ $$

###### Definition A.3 .

Let $\mathcal{G}$ be a graph domain and $\mathcal{H}$ be a reproducing kernel Hilbert space (RKHS) in which an inner product between $g,g^{\prime}\in\mathcal{G}$ is well defined. Let $\phi_{\lambda}:\mathcal{G}\to\mathcal{H}$ where $\lambda$ is a kernel specific hyperpameter and $\sigma^{2}$ is a scale factor. then define

$$ $k_{graph}(g,g^{\prime})=\sigma^{2}\cdot\langle\phi_{\lambda(g)},\phi_{\lambda} (g^{\prime})\rangle_{\mathcal{H}}$ $$

###### Definition A.1 .

###### Definition A.2 .

###### Definition A.3 .

### A.2 Definitions: CW-Complexes

We utilize the standard definitions of CW-complexes as defined by Whitehead [].

###### Definition A.4 .

A cell complex $K$, or alternatively a complex, is a Hausdorff space which is the union of disjoint open cells $e,e^{n},e_{i}^{n}$ subject to the condition that the closure $\bar{e}^{n}$ of each $n$-cell, $e^{n}\in K$ is the image of a fixed $n$-simplex in a map $f:\sigma^{n}\to\bar{e}^{n}$ such that

- (1)
$f|\sigma^{n}-\partial\sigma^{n}$ is a homeomorphism onto $e^{n}$
- (2)
$\partial e^{n}\subset K^{n-1}$, where $\partial e^{n}=f\partial\sigma^{n}=\bar{e}^{n}-e^{n}$ and $K^{n-1}$ is the $(n-1)$-section of $K$ consisting of all the cells whose dimension do not exceed $n-1$.

###### Definition A.5 .

A complex $K$, can be described as closure finite $\iff$ $K(e)$ is a finite subcomplex, for every cell $e\in K$. Moreover since $K(p)=K(e)$ if $p\in e$ this is equivalent to the condition that $K(p)$ is finite for each point $p\in K$ [].

###### Lemma A.6 .

If $L\subset K$ is a subcomplex and $e\in L$ then $L(e)=K(e)$. As a result any subcomplex of a closure finite complex is closure finite [].

###### Definition A.7 .

A complex $K$ has the weak topology $\iff$ a subset $X\subset K$ is closed provided $X\cap\bar{e}$ is closed for each cell $e\in K$ [].

###### Definition A.8 .

A CW-Complex is a complex which is closure finite and has the weak topology [].

Figure: Algorithm 3 Construction of a Cell Complex or CW-Complex $|$ Hatcher (2002) []

In essence one is inductively forming the $n$-skeleton $K^{n}$ from $K^{n-1}$ by attaching $n$-cells via attaching maps $\varphi_{\alpha}:S^{j-1}\to K^{j-1}$. Therefore $K^{n}$ is the quotient space of the disjoint union $K^{n-1}\coprod_{\alpha}D_{\alpha}^{n}$ of $K^{n-1}$ with a collection of $n$-disks $D_{\alpha}^{n}$ under identifications $k\sim\varphi_{\alpha}(k)$ for $k\in\partial D_{\alpha}^{n}$. Thus $K^{n}=K^{n-1}\coprod_{\alpha}e_{\alpha}^{n}$. One can stop the induction at a finite state setting $K=K^{n}$ for $n<\infty$ or one can continue indefinitely setting $K=\bigcup_{n}K^{n}$. In the second case $K$ has the weak topology [].

###### Definition A.9 .

Given a CW-Complex $X$, one denotes the $j$-th cell of dimension $k$ as $e_{j}^{k}$. Traditionally, one lets the relation $\prec$ denote incidence []. If two cells $e_{j}^{k-1}$ and $e_{i}^{k}$ are incident, we write $e_{j}^{k-1}\prec e_{i}^{k}$. For not incident, we write $e_{j}^{k-1}\nprec e_{i}^{k}$. Let the relation $\sim$ denote orientation. We write $e_{j}^{k-1}\sim e_{i}^{k}$ if the cells have the same orientation. For the opposite orientation we write $e_{j}^{k-1}\nsim e_{i}^{k}$.

###### Definition A.10 .

The Hodge Laplacian $\Delta_{k}:C^{k}(X)\to C^{k}(X)$ on the space of $k$-cochains is then $\Delta_{k}:=d_{k-1}\circ d^{*}_{k-1}+d^{*}_{k}\circ d_{k}$. The matrix representation is then $\Delta_{k}:=B_{k}^{\top}W_{k-1}^{-1}B_{k}W_{k}+W_{k}^{-1}B_{k+1}W_{k+1}B_{k+1}
^{\top}$. Here, $W_{k}=\text{diag}(w_{1}^{k},\ldots,w_{N_{k}}^{k})$ is the diagonal matrix of cell weights and $B_{k}$ is the order $k$ incidence matrix, whose $j$-th column corresponds to a vector representation of the cell boundary $\partial e_{j}^{k}$ viewed as a $k-1$ chain [].

###### Definition A.4 .

###### Definition A.5 .

###### Lemma A.6 .

###### Definition A.7 .

###### Definition A.8 .

###### Definition A.9 .

###### Definition A.10 .

### A.3 Theorems: Manifolds and Embeddings

###### Theorem A.11 .

(Whitney) The strong Whitney Embedding Theorem states that any smooth real $m$-dimensional manifold (Hausdorff and second-countable) can be smoothly embedded in $\mathbb{R}^{2m}$, if $m>0$ [].

###### Theorem A.12 .

(Whitney) The weak Whitney Embedding Theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m>2n$. Moreover, such a map can be approximated by an immersion provided $m>2n-1$ [].

###### Theorem A.13 .

(Lazarus) Any finite simplical complex of dimension $n$ embeds linearly into $\mathbb{R}^{2n+1}$.
This follows from the Whitney embedding theorem extended to simplical complexes. We reproduce the full proof given by Lazarus [].

###### Proof.

Define a linear mapping $f$ of the $n$-dimensional complex $K$ into $\mathbb{R}^{2n+1}$ by mapping the vertices of $K$ to points in general position in $\mathbb{R}^{2n+1}$, such that no hyperplane contains more than $2n+1$ points. Then $f$ is an embedding. This is apparent when restricted to any simplex of $K$: the simplex has at most $n+1$ vertices which are sent to affinely independent points by the general position assumption. To see that $f$ is injective we need to prove that distinct simplicies have their interior sent to disjoint sets. So let $\sigma=(v_{1},\ldots,v_{k})$ and $\tau=(w_{1},\ldots,w_{\ell})$ be two distinct simplicies of $K$. Since $k+\ell\leq 2n+2$, the general position assumption implies that the image points $f(v_{1}),\ldots,f(v_{k}),f(w_{1}),\ldots,f(w_{\ell})$ span a simplex of dimension $k+\ell-1$ and that $f(\sigma)$ and $f(\tau)$ are two distinct faces of this simplex. It follows that $f(\tau)$ and $f(\sigma)$ are disjoint, Then by injectivity we have embedding for finite simplical complexes.
∎

###### Definition A.14 .

(Fushida) If $f:\mathbb{R}^{n}\to\mathbb{R}$ is a smooth function then its gradient is the vector field grad $f$ defined by $\text{grad}_{x}f=\left(\frac{\partial f}{\partial x_{1}}(x),\cdots,\frac{
\partial f}{\partial x_{n}}(x)\right)$. Equivalently the vector field is defined by $g(\text{grad}f,Y)=df(Y)$ for all vector fields $Y$ [].

###### Definition A.15 .

A subset $M$ of linear space $E$ of dimension $d$ is a smooth embedded submanifold of dimension $n$ if $\forall x\in M$ there exists neighborhood $U$ of $x$ in $E$ and open set $V\subseteq\mathbb{R}^{d}$ and diffeomorphism $\varphi:U\to V$ such that $\varphi(U\cap M)=V\cap E$ where $E$ is a linear subspace of dimension $n$ [].

###### Remark A.16 .

Rotations, reflections, and translations are coordinate transformations [].

###### Theorem A.11 .

###### Theorem A.12 .

###### Theorem A.13 .

###### Proof.

###### Definition A.14 .

###### Definition A.15 .

###### Remark A.16 .

### A.4 Example: Atomic Complex construction for Deuterium

###### Lemma A.17 .

For the sake of illustration we describe a simple case, namely how we encode deuterium which contains $1$ proton, $1$ neutron, and $1$ electron. Let $\operatorname{Top}$ be the category of topological spaces.
Let $p\cong PD^{3}$ be a proton, $n\cong ND^{3}$ be a neutron and $e=(Z,\tau)$ be an electron. Then we know that $p,n,e,\partial p,\partial n,\partial e\in\operatorname{Top}$.
Let $\varphi_{p}:\partial p\hookrightarrow p$, $\varphi_{n}:\partial n\hookrightarrow n$, $\varphi_{e}:\partial e\hookrightarrow e$ be the generating cofibrations for protons, neutrons and electrons.
Let the $n$-cell attachment to space $X$ be the result of gluing either $p,n,$ or $e$ along a prescribed image of it’s boundary.
In essence $\phi_{p}:\partial p\to\partial n$, $\phi_{n}:\partial n\to\partial e$, $\phi_{e}:\partial e\to X$ and all are continuous functions.
The attaching space is then $X\cup_{\phi_{p}}p\cup_{\phi_{n}}n\cup_{\phi_{e}}e\in\operatorname{Top}$ which makes the following diagram commute:

###### Lemma A.17 .

### A.5 Example: Polyatomic Complexes (two-atoms)

###### Lemma A.18 .

For the sake of illustration we describe the simplest possible case of connecting $2$ atoms.
Let $A_{1}$ and $A_{2}$ be atoms.
We know that $A_{1},A_{2}\in\operatorname{Top}$.
Let $\varphi_{A_{1}}:\partial A_{1}\hookrightarrow A_{1}$, $\varphi_{A_{2}}:\partial A_{2}\hookrightarrow A_{2}$ be the generating cofibrations for each complex.
Let the $n$-cell attachment to space $X$ be the result of gluing either $A_{1}$, or $A_{2}$, along a prescribed image of its boundary.
In essence $\phi_{A_{1}}:\partial A_{1}\to\partial A_{2}$, $\phi_{A_{2}}:\partial A_{2}\to\partial X$ such that all are continuous functions.
The attaching space is then $X\cup_{\phi_{A_{1}}}A_{1}\cup_{\phi_{A_{2}}}A_{2}$
which makes the following diagram commute:

###### Lemma A.18 .

### A.6 Proof: Definition 3.6

###### Proof.

We want to show that $\operatorname{Ato}$ defines a monad structure.
Since atomic complexes are topological spaces that you can equip with a $\sigma$-algebra and the morphisms are measurable the result follows mutandis mutatis from the Giry monad on
$\operatorname{Meas}$ []. You can generate the $\sigma$-algebra by the integration functions $\varepsilon_{f}:PA\to\mathbb{R}$ by

$$ $p\mapsto\int fdp,$ $$

for all $f:A\to[0,1]$ measurable.
You can define multiplication in the usual way:

$$ $E_{\pi}(X):=\int_{PA}p(X)d\pi(p)$ $$

when given $A$ and measure $\pi\in PPA$. By composition of morphisms you can reproduce the Chapman-Kolmogorov equation for general measures given $f:A_{1}\to PA_{2}$ and $g:A_{2}\to PA_{3}$ we see:

$$ $(f\circ g)(a_{1})(Z)=\int_{A_{2}}g_{a_{2}}(Z)df_{a_{1}}(a_{2})$ $$

for each $a_{1}\in A_{1}$ and $Z\subseteq A_{3}$. This composition is associative and unital. Thus we have a monad structure.
∎

###### Proof.

### A.7 Proof: Lemma 2.9

###### Proof.

We want to show atomic complex $A$ is finite.
By 2.4 an atomic complex is a union of $P_{i},N_{j}$, and $E_{k}$. The frequency of each is governed by $\mathcal{N},\mathcal{P},\mathcal{E}$ which correspond to the number of protons, neutrons and electrons found in an atom. Since individual atoms have a finite number of protons neutrons and electrons, we know each cell is finite and there are finitely many cells in each dimension. Therefore $A$ must be finite.
We want to show that an atomic complex $A$ is $n$-connected.
For every $i\in I_{a}$, let $g_{i}$ be a map from a one point space to $P_{i}$ or $N_{j}$ or $E_{k}$. Since both spaces have all homotopy groups trivial, $\forall i$ $g_{i}$ induces an isomorphism on all the homotopy groups. Therefore by Whitehead’s Theorem for every $i$ we know $g_{i}$ is a homotopy equivalence and so $P$ is contractible $\implies P$ is $n$-connected.
Thus atomic complex $A$ is finite and $n$-connected.
∎

###### Proof.

### A.8 Proof: Lemma 2.10

###### Proof.

We want to show polyatomic complex $P$ is finite.
By 2.7 an polyatomic complex is a union of atomic complexes $A_{i}$. The frequency of atomic complexes is governed by $\mathcal{K}$ which corresponds to the number of atoms in the system. Since the atomistic systems we model are finite, we know each cell is finite and there are finitely many cells in each dimension. Therefore $P$ must be finite.
We want to show that an polyatomic complex $P$ is $n$-connected.
For every $i\in I_{a}$, let $g_{i}$ be a map from a one point space to $A_{i}$. Since both spaces have all homotopy groups trivial, $\forall i$ $g_{i}$ induces an isomorphism on all the homotopy groups. Therefore by Whitehead’s Theorem for every $i$ we know $g_{i}$ is a homotopy equivalence and so $P$ is contractible $\implies P$ is $n$-connected.
Thus polyatomic complex $P$ is finite and $n$-connected.
∎

###### Proof.

### A.9 Proof: Theorem 2.11

###### Proof.

Let $P$ be a polyatomic complex. By 2.10 $P$ is a finite CW complex.
It is then implied that each $P$ is homotopy equivalent to a finite dimensional locally finite simplical complex $S$ (Theorem 2C.5 Hatcher []). By A.13, we know $S$ has linear embedding $X\in\mathbb{R}^{k}$. Pick regular neighborhood $M_{x}$ of $X\in\mathbb{R}^{k}$. It will be homotopy equivalent to $X$ and a complex PL submanifold with boundary. Take the interior $\text{int}(M_{x})=\mathcal{M}$. Then, $\mathcal{M}$ is a smooth manifold.
∎

###### Proof.

### A.10 Proof: Theorem 2.13 Uniqueness

###### Proof.

We want to show that an atomic complexes are unique. In essence two systems differing in property should be mapped to different representations.
This immediately follows from Hatcher Corollary 4.19 [], an $n$-connected CW model is unique up to homotopy equivalence $\text{rel}A$. By lemma 2.10 a polyatomic complex $P$ is $n$-connected.(^6^66All chemical elements are distinguished by their number of protons. In the case of isotopes the number of neutrons would differ. You would have a different number of cells.)
∎

###### Proof.

### A.11 Proof: Theorem 2.13 Continuity and Differentiability

###### Proof.

We want to show that atomic complexes are continuous and differentiable with respect to atomic coordinates.
We want to show that atomic complexes are continuous with respect to atomic coordinates.
Let $\mathcal{M}_{A}$ be the embedded smooth submanifold of atomic coordinates as in 2.12. Similarly let the smooth manifold $\mathcal{M}_{C}$ correspond to an atomic complex $C$ as in 2.11.
Claim: The map $F:\mathcal{M}_{C}\to\mathcal{M}_{A}$ is smooth and therefore continuous. It is sufficient to show that $F$ is locally continuous i.e. $\forall x\in\mathcal{M}_{C}$ $\exists$ neighborhood $U_{x}$ such that $F|_{U_{x}}$ is continuous. Then since both $\mathcal{M}_{A}$ and $\mathcal{M}_{C}$ are smooth manifolds there exist local coordinate systems $(U,\xi)$ at $x$ and $(V,\chi)$ at $F(x)$ such that the coordinate expression $\Phi=\chi\circ F\circ\xi^{-1}:\xi(U\cap F^{-1}(V))\to\mathbb{R}^{n}$ is smooth. Pick $U_{x}=U\cap F^{-1}(V)$. Then it follows that $U_{x}\subset F^{-1}(V)\implies F(U_{x})\subset F(F^{-1}(V))\subset V$. Thus the maps $\xi^{-1}$ and $\chi$ are homeomorphisms and $\Phi$ is a smooth map between euclidean spaces and $F|_{U_{x}}$ is a composition of continuous maps $\implies$ $F$ is continuous. Then let $g:C\to\mathcal{M}_{C}$ be continuous and $h:M_{A}\to A$ be continuous. Then composition of continuous functions are continuous $g\circ F\circ h$. Therefore we have continuity.
We want to show that atomic complexes are differentiable with respect to atomic coordinates.
Let $\mathcal{M}_{A}$ be the embedded smooth submanifold of atomic coordinates as in 2.12. Similarly let the smooth manifold $\mathcal{M}_{C}$ correspond to an atomic complex $C$ as in 2.11. Then the differential of the map $F:\mathcal{M}_{C}\to\mathcal{M}_{A}$ is well defined. The differential of smooth map $F$ at $x$ $dF(x):T_{x}\mathcal{M}_{C}\to T_{F(x)}\mathcal{M}_{A}$ is defined by: $dF(x)[v]=(F\circ c)^{\prime}(0)=\lim_{t\to 0}\frac{F(c(t))-F(x)}{t}$ where $c:\mathbb{R}\to\mathcal{M}_{C}$ satisfies $c(0)=x$ and $c^{\prime}(0)=v$. This map is linear and we have a chain rule. If $\bar{F}$ is the smooth extension of $F$, then $dF(x)=d\bar{F}(x)|_{T_{x}\mathcal{M}_{C}}$. In general by A.14 we just write $g(\text{grad}F,X)=dF(X)$ for vector field $X$. We can let $dF(x)$ be our (approximate) derivative for $C$ with respect to atomic coordinates. We can alternatively let $f:C\to\mathcal{M}_{C}$ be a continuous differentiable map, let $g:\mathcal{M}_{A}\to A$ be continuous and differentiable, and rely on chain rule. Therefore we can define the derivative with respect to atomic coordinates.
∎

###### Proof.

### A.12 Proof: Theorem 2.14 Invariances

###### Proof.

We want to show polyatomic complexes are invariant with respect to rotations, changes in atom indexing, translations and reflections.
It suffices to choose an atlas such that all the coordinate change maps are smooth. Let the smooth manifold $\mathcal{M}_{C}$ correspond to an polyatomic complex $C$ as in 2.11. Additionally, let $C^{\infty}([a,b])$ denote the space of smooth functions $f:[a,b]\to\mathcal{M}_{C}$. Then, for functionals $S:C^{\infty}([a,b])\to\mathbb{R}$ of the form $S[f]=\int_{a}^{b}(L\circ\dot{f})(x)dx$ where $L:T\mathcal{M}_{C}\to\mathbb{R}$ is the Lagrangian, we know $dS_{f}=0\iff\forall t\in[a,b]$ each coordinate frame trivialization $(x^{i},X^{i})$ of a neighborhood of $\dot{f}(t)$ yields the following $\dim\mathcal{M}_{C}$ equations: $\forall i:\frac{d}{dx}\frac{\partial L}{\partial X^{i}}|_{\dot{f(x)}}=\frac{
\partial L}{\partial x^{i}}|_{\dot{f(x)}}$. This recovers the Euler-Lagrange equations (^7^77We can use the lie derivative $\mathcal{L}$ and local charts $(q^{\alpha},\dot{q^{\alpha}})$ such that $\theta_{L}=\frac{\partial L}{\partial\dot{q}^{\alpha}}dq^{\alpha}$ and $\Delta=\frac{d}{dx}$ [].) $\mathcal{L}\theta_{L}=dL$ which are invariant under coordinate transformations and therefore rotationally, reflectionally, and translationally invariant. (^8^88A change of coordinates is a diffeomorphism between manifolds.)
We want to show polyatomic complexes are invariant w.r.t changes in atom indexing.
Let $I=\{1,\ldots,i,i+1,\ldots,n\}$ and $n\in\mathbb{N}$ s.t. $1\leq i<i+1\leq n$. Suppose that polyatomic complex $P$ is composed of atomic complexes $[A_{1},\ldots,A_{i},A_{i+1},\ldots,A_{n}]$. Suppose that we swap $A_{i}$ and $A_{i+1}$ and construct polyatomic complex $O=[A_{1},\ldots,A_{i+1},A_{i},\ldots,A_{n}]$. Then, it suffices to show that $O=P$.
We know that $O$ and $P$ are homeomorphic with isomorphic fundamental groups. In essence $\forall~{}i$ we have $\pi_{i}(P)\ \raisebox{2.15277pt}{$\stackrel{{\scriptstyle\sim}}{{\rightarrow}}
$\,}\pi_{i}(O)$. Since polyatomic complexes are contractible $\implies$ all homotopy groups are trivial $\implies\forall~{}i~{}\pi_{i}(P)=\pi_{i}(O)=0$.
Thus $O=P$. In essence, they encode the same information regardless of whether one re-indexes.
∎

###### Proof.

### A.13 Proof: Theorem 2.14 Uniqueness

###### Proof.

We want to show polyatomic complexes are unique.
By Hatcher Corollary 4.19 [], an $n$-connected CW model is unique up to homotopy equivalence $\text{rel}A$. By lemma 2.10, a polyatomic complex $P$ is $n$-connected.
∎

###### Proof.

### A.14 Proof: Theorem 2.14 Continuity and Differentiability

###### Proof.

We want to show that polyatomic complexes are continuous and differentiable with respect to atomic coordinates.
We want to show that polyatomic complexes are continuous with respect to atomic coordinates.
Let $\mathcal{M}_{A}$ be the embedded smooth submanifold of atomic coordinates as in 2.12. Similarly, let the smooth manifold $\mathcal{M}_{C}$ correspond to an polyatomic complex $C$ as in 2.11.
Claim: The map $F:\mathcal{M}_{C}\to\mathcal{M}_{A}$ is smooth and therefore continuous. It is sufficient to show that $F$ is locally continuous i.e. $\forall x\in\mathcal{M}_{C}$ $\exists$ neighborhood $U_{x}$ such that $F|_{U_{x}}$ is continuous. Then since both $\mathcal{M}_{A}$ and $\mathcal{M}_{C}$ are smooth manifolds there exist local coordinate systems $(U,\xi)$ at $x$ and $(V,\chi)$ at $F(x)$ such that the coordinate expression $\Phi=\chi\circ F\circ\xi^{-1}:\xi(U\cap F^{-1}(V))\to\mathbb{R}^{n}$ is smooth. Pick $U_{x}=U\cap F^{-1}(V)$. Then, it follows that $U_{x}\subset F^{-1}(V)\implies F(U_{x})\subset F(F^{-1}(V))\subset V$. Thus, the maps $\xi^{-1}$ and $\chi$ are homeomorphisms and $\Phi$ is a smooth map between euclidean spaces and $F|_{U_{x}}$ is a composition of continuous maps $\implies$ $F$ is continuous. Then we see $g:C\to\mathcal{M}_{C}$ and $h:\mathcal{M}_{A}\to A$ are continuous, since they are smooth embeddings. Essentially, we know $g$ is continuous on the restriction to every space in the diagram $\implies$ by the universal property of a colimit that $g$ is continuous. Then, composition of continuous functions are continuous. Therefore, we have continuity.
We want to show that polyatomic complexes are differentiable with respect to atomic coordinates. Let $\mathcal{M}_{A}$ be the embedded smooth submanifold of atomic coordinates as in 2.12. Similarly, let the smooth manifold $\mathcal{M}_{C}$ correspond to an polyatomic complex $C$ as in 2.11. Then the differential of the map $F:\mathcal{M}_{C}\to\mathcal{M}_{A}$ is well defined. The differential of smooth map $F$ at $x$ $dF(x):T_{x}\mathcal{M}_{C}\to T_{F(x)}\mathcal{M}_{A}$ is defined by: $dF(x)[v]=(F\circ c)^{\prime}(0)=\lim_{t\to 0}\frac{F(c(t))-F(x)}{t}$ where $c:\mathbb{R}\to\mathcal{M}_{C}$ satisfies $c(0)=x$ and $c^{\prime}(0)=v$. This map is linear and we have a chain rule. If $\bar{F}$ is the smooth extension of $F$, then $dF(x)=d\bar{F}(x)|_{T_{x}\mathcal{M}_{C}}$. In general, by A.14, we just write $g(\text{grad}F,X)=dF(X)$ for vector field $X$. We can let $dF(x)$ be our (approximate) derivative for $C$ with respect to atomic coordinates. Alternatively, we know $f:C\to\mathcal{M}_{C}$, and $g:\mathcal{M}_{A}\to A$ are smooth embeddings $\implies$ the derivatives are everywhere injective. Thus, we can compose these functions and apply chain rule. Therefore, we can define the derivative with respect to atomic coordinates.
∎

###### Proof.

### A.15 Proof: Theorem 2.14 Generality/Generalizability

###### Proof.

Suppose for the sake of contradiction that a polyatomic complex cannot encode any atomistic system. This implies, by definition an atomistic system, that there exists a finite collection of atoms $\{A_{1},\ldots,A_{k}\}$ such that $\nexists$ a corresponding polyatomic complex $P$. However, we can construct such a $P$. By 2.4 $\forall A_{i}\in\{A_{1},\ldots,A_{k}\}$ there exists corresponding atomic complex $\mathcal{A}_{i}$. Let our corresponding set of atomic complexes be $\{\mathcal{A}_{1},\ldots,\mathcal{A}_{k}\}$. Then, can define attaching maps $\chi_{a}=\{\phi_{a,i}:\partial\mathcal{A}_{i}\to\mathcal{A}_{i}~{}|~{}a~{}
\text{corresponds to}~{}A_{i}\}$. Then, we have a sequence of topological spaces $\mathcal{A}_{0}\hookrightarrow\ldots\hookrightarrow\mathcal{A}_{K}$ and can form $P$ such that a diagram $D$ isomorphic to the one in 2.7 commutes. Therefore, we have a valid polyatomic complex $P$. As a result, we have a contradiction and the opposite must be true.
∎

###### Proof.

### A.16 Proof: Theorem 2.14 Time complexity

###### Proof.

Suppose the reference simulation is a system with $k=|M_{A}|$ many atoms. By Algorithm 2, we can show that we can construct a polyatomic complex in $O(C)$ which is polynomial time.
For an atomic complex, Algorithm 1 runs in $O(M\cdot(d_{n}\cdot N+d_{e}\cdot E\cdot S+d_{p}\cdot P))=O(A)$ where $P,N,E$ are the number of protons, neutrons, and electrons and $d_{p},d_{n},d_{e}$ correspond to the range of dimensions for protons, neutrons, and electrons; additionally, $S$ is the time complexity of the method in [], and $M$ is the time complexity of constructing an attaching map via helper function. Then, we know that Algorithm 2 runs in $O(k\cdot A\cdot F\cdot M\cdot R)=O(C)$ where $M$ is the time it takes to construct $1$ attaching map, $F$ is the time complexity of $update\text{\textunderscore}forces$, and $R$ is the time complexity of $update\text{\textunderscore}radial$. Thus, Algorithm 2 runs in $O(C)$.
∎

###### Proof.

### A.17 Data: Photoswitches Tables

**Table 7: Photoswitches Benchmark**
|  |  | E isomer $n$-$\pi$* wavelength (nm) | DFT E isomer $n$-$\pi$* wavelength (nm) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Kernel | Representation | RMSE | MAE | CRPS | RMSE | MAE | CRPS |
| Tanimoto | SELFIES | $8.9\pm 0.2$ | $5.7\pm 0.1$ | $6.4$ | $7.3\pm 0.2$ | $3.6\pm 0.1$ | $2.6$ |
| Tanimoto | SMILES | $8.6\pm 0.2$ | $5.1\pm 0.1$ | $6.4$ | $6.5\pm 0.2$ | $3.2\pm 0.04$ | $2.5$ |
| Tanimoto | ECFP | $7.8\pm 0.2$ | $4.4\pm 0.1$ | $6.3$ | $6.2\pm 0.2$ | $3.1\pm 0.1$ | $2.5$ |
| Tanimoto | Fast Complex | $13.8\pm 0.3$ | $7.0\pm 0.1$ | $6.9$ | $8.0\pm 0.2$ | $2.5\pm 0.1$ | $2.5$ |
| Tanimoto | Deep Complex | $13.8\pm 0.3$ | $7.0\pm 0.1$ | $7.0$ | $8.0\pm 0.2$ | $2.5\pm 0.1$ | $2.5$ |
| WL | Graphs | $7.8\pm 0.2$ | $4.4\pm 0.1$ | $6.2$ | $6.2\pm 0.2$ | $3.1\pm 0.1$ | $2.5$ |

**Table 8: Photoswitches Benchmark**
|  |  | E isomer $\pi$-$\pi$* wavelength (nm) | DFT E isomer $\pi$-$\pi$* wavelength (nm) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Kernel | Representation | RMSE | MAE | CRPS | RMSE | MAE | CRPS |
| Tanimoto | SELFIES | $31.8\pm 0.7$ | $22.2\pm 0.5$ | $35.7$ | $26.1\pm 0.5$ | $17.1\pm 0.2$ | $15.1$ |
| Tanimoto | SMILES | $30.8\pm 0.6$ | $22.1\pm 0.3$ | $35.9$ | $22.9\pm 0.5$ | $14.6\pm 0.2$ | $14.6$ |
| Tanimoto | ECFP | $30.5\pm 0.6$ | $21.6\pm 0.3$ | $36.4$ | $22.9\pm 0.5$ | $13.7\pm 0.3$ | $14.6$ |
| Tanimoto | Fast Complex | $64.4\pm 0.7$ | $52.1\pm 0.5$ | $52.1$ | $32.0\pm 0.5$ | $15.7\pm 0.5$ | $15.7$ |
| Tanimoto | Deep Complex | $64.4\pm 0.7$ | $52.1\pm 0.5$ | $52.1$ | $32.0\pm 0.5$ | $15.7\pm 0.5$ | $15.7$ |
| WL | Graphs | $28.1\pm 0.8$ | $17.9\pm 0.4$ | $35.5$ | $23.8\pm 0.6$ | $15.3\pm 0.3$ | $14.9$ |

**Table 9: Photoswitches Benchmark**
|  |  | Z isomer n-$\pi$* wavelength (nm) | DFT Z isomer n-$\pi$* wavelength (nm) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Kernel | Representation | RMSE | MAE | CRPS | RMSE | MAE | CRPS |
| Tanimoto | SELFIES | $7.8\pm 0.2$ | $3.9\pm 0.1$ | $3.3$ | $13.0\pm 0.8$ | $4.1\pm 0.1$ | $2.8$ |
| Tanimoto | SMILES | $7.4\pm 0.2$ | $3.4\pm 0.1$ | $3.3$ | $12.8\pm 1.0$ | $3.2\pm 0.2$ | $2.7$ |
| Tanimoto | ECFP | $6.7\pm 0.2$ | $3.0\pm 0.1$ | $3.2$ | $12.4\pm 0.9$ | $4.0\pm 0.1$ | $2.8$ |
| Tanimoto | Fast Complex | $8.1\pm 0.2$ | $3.6\pm 0.1$ | $3.6$ | $12.8\pm 1.0$ | $2.8\pm 0.3$ | $2.8$ |
| Tanimoto | Deep Complex | $8.1\pm 0.2$ | $3.6\pm 0.1$ | $3.6$ | $12.8\pm 1.0$ | $2.8\pm 0.3$ | $2.8$ |
| WL | Graphs | $7.3\pm 0.2$ | $3.1\pm 0.1$ | $3.2$ | $12.8\pm 1.0$ | $2.8\pm 0.3$ | $2.8$ |

**Table 10: Photoswitches Benchmark**
|  |  | Z Photostationary State ($\%$ isomer) | E Photostationary State ($\%$ isomer) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Kernel | Representation | RMSE | MAE | CRPS | RMSE | MAE | CRPS |
| Tanimoto | SELFIES | $5.0\pm 0.2$ | $2.3\pm 0.1$ | $1.8$ | $6.2\pm 0.3$ | $2.8\pm 0.1$ | $2.0$ |
| Tanimoto | SMILES | $4.8\pm 0.2$ | $2.2\pm 0.1$ | $1.8$ | $6.1\pm 0.3$ | $2.6\pm 0.1$ | $2.0$ |
| Tanimoto | ECFP | $4.6\pm 0.2$ | $2.0\pm 0.1$ | $1.7$ | $5.8\pm 0.3$ | $2.5\pm 0.1$ | $1.9$ |
| Tanimoto | Fast Complex | $5.1\pm 0.2$ | $1.8\pm 0.1$ | $1.8$ | $6.1\pm 0.3$ | $2.1\pm 0.1$ | $2.0$ |
| Tanimoto | Deep Complex | $5.1\pm 0.2$ | $1.8\pm 0.1$ | $1.8$ | $6.1\pm 0.3$ | $2.1\pm 0.1$ | $2.1$ |
| WL | Graphs | $4.9\pm 0.2$ | $2.1\pm 0.1$ | $1.8$ | $6.5\pm 0.2$ | $2.8\pm 0.1$ | $2.0$ |

### A.18 Force Models

As described by Lin et al. (2019), in the context of force fields, the typical potential energy function is as shown in equation (1) [].
{dmath}
V_total = ∑_i=1^N_bond V_bond + ∑_i=1^N_angle V_angle + ∑_i=1^N_dihedral V_dihedral + ∑_i=1^N_nonbonded V_Nb = ∑_bonds k_r(r - r_eq)^2 + ∑_angles k_θ(θ- θ_eq)^2 + ∑_dihedrals k_t[1+cos(nω- γ)] + ∑_i < j [ Aijrij12 - Bijrij6 ] + ∑_i<j qiqj4 πϵrij
For polyatomic complexes one can write the above equations as sums over incident cells. The definitions of the relations $\prec$, $\nprec$, $\sim$ and $\nsim$ are found in the Appendix A.9.
For the $j$-th proton of dimension $\tau_{k}$ we let

$$ $\mathcal{N}_{PD_{j}^{\tau_{k}}}:=\left\{PD_{j^{\prime}}^{\tau_{k-1}}~{}|~{}PD_ {j^{\prime}}^{\tau_{k-1}}\prec PD_{j}^{\tau_{k}}\land(PD_{j^{\prime}}^{\tau_{k -1}}\sim PD_{j}^{\tau_{k}}\lor PD_{j^{\prime}}^{\tau_{k-1}}\nsim PD_{j}^{\tau_ {k}})\right\}$ (1) $$

contain all protons incident to proton $PD_{j}^{\tau_{k}}$.
For the $j$-th neutron of dimension $\tau_{k}$ we similarly let

$$ $\mathcal{N}_{ND_{j}^{\tau_{k}}}:=\left\{ND_{j^{\prime}}^{\tau_{k-1}}~{}|~{}ND_ {j^{\prime}}^{\tau_{k-1}}\prec ND_{j}^{\tau_{k}}\land(ND_{j^{\prime}}^{\tau_{k -1}}\sim ND_{j}^{\tau_{k}}\lor ND_{j^{\prime}}^{\tau_{k-1}}\nsim ND_{j}^{\tau_ {k}})\right\}$ (2) $$

contain all neutrons incident to neutron $ND_{j}^{\tau_{k}}$.
Finally for the $j$-th electron of dimension $\tau_{k}$ we let

$$ $\mathcal{N}_{ee_{j}^{\tau_{k}}}:=\left\{ee_{j^{\prime}}^{\tau_{k-1}}~{}|~{}ee_ {j^{\prime}}^{\tau_{k-1}}\prec ee_{j}^{\tau_{k}}\land(ee_{j^{\prime}}^{\tau_{k -1}}\sim ee_{j}^{\tau_{k}}\lor ee_{j^{\prime}}^{\tau_{k-1}}\nsim ee_{j}^{\tau_ {k}})\right\}$ (3) $$

contain all electrons incident to $ee_{j}^{\tau_{k}}$. For the $i$-th atom of dimension $k$ we similarly let

$$ $\mathcal{N}_{A_{j}^{k}}:=\left\{A_{j^{\prime}}^{k-1}~{}|~{}A_{j^{\prime}}^{k-1 }\prec A_{j}^{k}\land(A_{j^{\prime}}^{k-1}\sim A_{j}^{k}\lor A_{j^{\prime}}^{k -1}\nsim A_{j}^{k})\right\}$ (4) $$

contain all atoms incident to $A_{j}^{k}$. The set $\mathcal{N}_{A_{j}^{k}}$ then describes an atom’s local environment. By fixing a dimension $k$, and taking the intersection over all local environments $\mathcal{N}_{A^{k}}:=\bigcap_{j=1}^{n}\mathcal{N}_{A_{j}^{k}}$ one can consider more complex interactions. In particular, one can choose a particular radius $r>0$, and distance function $d$ such that:

$$ $\mathbf{ENV}_{A_{\ell}^{k}}:=\{A_{i}^{k-1}~{}|~{}\forall(i,j)~{}d(A_{i}^{k-1}, A_{j}^{k-1})\leq r\land A_{i}^{k-1}\prec A_{\ell}^{k}\land A_{j}^{k-1}\prec A_ {\ell}^{k}\}.$ (5) $$

$\mathbf{ENV}_{A_{\ell}^{k}}$ more generally defines local environments in a molecule or polyatomic system.
This setup enables us to write out an equation analogous to (1). In particular, let $X$ be a polyatomic complex. Let $d_{n,k}:A_{i_{1}}^{k}\times A_{i_{2}}^{k}\times\cdots\times A_{i_{n}}^{k}\to
\mathbb{R}$ be a valid metric defined on products of atoms (e.g. $\sup$ metric). Then let $\mathcal{V}_{bond}:=\{(A_{i}^{k},A_{j}^{k})~{}|~{}d_{2,k}(A_{i}^{k},A_{j}^{k})
<r_{bond}~{}\forall k\leq\dim(X)\}$. In essence, $\mathcal{V}_{bond}$ is the set of all pairs of atoms whose separation is only $1$ bond, given by $r_{bond}$. This definition, namely the choice to define bond parameters over pairs of atoms, follows the bond parameterization of the General Amber Force Field []. Let $\mathcal{V}_{angle}:=\{(A_{i}^{k},A_{j}^{k},A_{\ell}^{k})~{}|~{}d_{3,k}(A_{i}^
{k},A_{j}^{k},A_{\ell}^{k})<r_{angle}~{}\forall k\leq\dim(X)\}$. In essence, $\mathcal{V}_{angle}$ is the set of all triples of atoms whose separation is only $2$ bonds, thereby forming an angle, given by $r_{angle}$. This definition, namely the choice to define angle parameters over triples of atoms, follows the angle parameterization of the General Amber Force Field []. Let $\mathcal{V}_{dihedral}:=\{(A_{i}^{k},A_{j}^{k},A_{\ell}^{k},A_{m}^{k})~{}|~{}d
_{4,k}(A_{i}^{k},A_{j}^{k},A_{\ell}^{k},A_{m}^{k})<r_{dih}~{}\forall k\leq\dim
(X)\}$. In essence, $\mathcal{V}_{dihedral}$ is the set of all quadruples of atoms whose separation is only $3$ bonds, thereby forming a dihedral potential or torsion potential, given by $r_{dih}$. This definition, namely the choice to define torsion parameters over quadruples of atoms, follows the torsional angle parameterization of the General Amber Force Field []. Let $\mathcal{V}_{nb}:=\{(A_{i}^{k}\times A_{j}^{k})~{}|~{}d_{2,k}(A_{i}^{k}\times A
_{j}^{k})>r_{nb}~{}\forall k\leq\dim(X)\}$. In essence, $\mathcal{V}_{nb}$ is the set of all tuples of atoms whose separation is greater than $3$ bonds given by $r_{nb}$. This definition, namely the choice to define non-bonded parameters over pairs of atoms, follows the parameterizations given by numerous force fields []. For classification one can apply numerous force fields such as GAFF [, , ]. The function sep describes the separation between the atoms. The function ang determines the angle between atoms. The function dih determines angles formed by the planes defined by atoms.
{dmath}
V_total = ∑_i=1^N_bond V_bond + ∑_i=1^N_angle V_angle + ∑_i=1^N_dihedral V_dihedral + ∑_i=1^N_nonbonded V_Nb
= ∑_v_r ∈V_bond k_r(sep(v_r) - r_eq)^2 + ∑_v_a ∈V_angle k_θ(ang(v_a) - θ_eq)^2
+ ∑_v_d ∈V_dihedral k_t[1+ cos(n ⋅dih(v_d) - γ)] + ∑_v_b ∈V_nb [ Aijsep(vb)12 - Bijsep(vb)6 ] + qiqj4 πϵsep(vb)
More complex interactions can be modeled by considering sets like

$$ $\mathbf{Int}_{n}:=\{(A_{i_{1}}^{k}\times\cdots\times A_{i_{n}}^{k})~{}|~{}d_{n ,k}(A_{i_{1}}^{k}\times\cdots\times A_{i_{n}}^{k})<r~{}\forall k\leq\dim(X)\}$ (6) $$

Note that all parameters can be chosen to reproduce experimental data or quantum mechanical calculations [, ].
If one wishes to consider more complex dynamics, in the context of polyatomic complexes, doing so is possible. Suppose we wish to model the dynamics of evolution of a non-relativistic quantum system determined by the time-dependent Schrödinger equation (TDSE) []. The time-dependent Schrödinger equation is defined as

$$ $i\hbar\frac{\partial}{\partial t}|\Psi(r,R,t)\rangle=\widehat{H}(r,R)|\Psi(r,R ,t)\rangle$ (7) $$

where $R=\{R_{I}\}$ are nuclear positions and $\{r_{i}\}$ are electronic positions. One can write approximations to the TDSE using polyatomic complexes. In physics one usually writes out the Hamiltonian as a sum of kinetic energy, potential energy, nuclear repulsion, electronic repulsion and electron-nuclear interaction [, ].

$$ $\widehat{H}(r,R)=\widehat{T}+\widehat{V}=\widehat{T}_{N}+\widehat{T}_{e}+ \widehat{V}_{NN}+\widehat{V}_{ee}+\widehat{V}_{eN}$ (8) $$

We approximate this quantity in the following way.
Let $\mathbf{NUC}:=\bigcup_{k=1}^{\dim(X)}\bigcup_{j=1}^{n_{k}}\left(\mathcal{N}_{
PD_{j}^{\tau_{k}}}\cup\mathcal{N}_{ND_{j}^{\tau_{k}}}\right)$ and $\mathbf{ELEC}:=\bigcup_{k=1}^{\dim(X)}\bigcup_{j=1}^{n_{k}}\mathcal{N}_{ee_{j}
^{\tau_{k}}}$.
{dmath}
^H(r,R) ≈-∑_I ∈NUC -ℏ22MI Δ_I^2 - ∑_i ∈ELEC -ℏ22me Δ_i^2 + ∑_I ≠J ∈NUCZAZBe24 πε0|pos(I) - pos(J)| + ∑_i ≠j ∈ELEC e24 πε0|pos(i) - pos(j)| - ∑_i ∈ELEC, I ∈NUC ZAe24πε0|pos(I)- pos(i)|
where $\operatorname{pos}$ determines the nuclear or electronic positions and $\Delta_{k}$ is the Hodge-Laplacian as described in Appendix A.10. The Hodge-Laplacian terms approximate the derivatives with respect to the nuclear coordinates and electronic coordinates $\Delta_{I}^{2}\approx\frac{\partial^{2}}{\partial\operatorname{pos}(I)^{2}}$ and $\Delta_{i}^{2}\approx\frac{\partial^{2}}{\partial\operatorname{pos}(i)^{2}}$. The last four terms can be combined into an approximation for the electronic Hamiltonian $\widehat{H}_{e}(r,R)\approx\widetilde{H}_{\mathbf{NUC},\mathbf{ELEC}}$ []. Which enables us to write:

$$ $\widehat{H}(r,R)\approx-\sum_{I\in\mathbf{NUC}}\frac{-\hbar^{2}}{2M_{I}}\Delta _{I}^{2}+\widetilde{H}_{\mathbf{NUC},\mathbf{ELEC}}$ (9) $$

Then by substitution of (12) into (9) we see

$$ $i\hbar\frac{\partial}{\partial t}|\Psi(r,R,t)\rangle\approx\left(-\sum_{I\in \mathbf{NUC}}\frac{-\hbar^{2}}{2M_{I}}\Delta_{I}^{2}+\widetilde{H}_{\mathbf{ NUC},\mathbf{ELEC}}\right)|\Psi(r,R,t)\rangle$ (10) $$

Applying the Born-Huang approximation we can approximate the total wave-function as below [].

$$ $|\Psi(r,R,t)\rangle=\sum_{\ell=0}^{\infty}|\chi_{\ell}(r;R)\Omega_{\ell}(R,t) \rangle\approx\sum_{l=0}^{\infty}|\phi_{l}\psi_{l}\rangle$ (11) $$

Here, the $\phi_{l}$ correspond to the attaching maps $(\phi_{p,i})_{i\in I_{p}}$ and $(\phi_{n,i})_{i\in I_{n}}$ as in Definition 2.4. Similarly, $\psi_{l}$ correspond to the attaching maps $(\phi_{e,i})_{i\in I_{e}}$ as in Definition 2.4. Usually, $\chi_{\ell}(r;R)$ are the time-dependent nuclear wave-functions and $\Omega_{\ell}(R,t)$ are the solutions of the time independent Schrödinger equation.

### A.19 Compute Cost

All reported experiments can be reproduced using an AWS m7g.4xlarge instance (16 vCPU, 64 GiB) in under three hours per experiment. The full project required more compute than the experiments reported in the paper. This is because we conducted a wider variety of experiments on different datasets which are not reported. Certain experiments require resources equivalent to that of an AWS p5.48xlarge (192 vCPU, 2TiB, 8 GPU-H100) and approximately ten hours per experiment.

## Peer Reviews

confidence: 2
contribution: 3
initial_rating: 6
presentation: 3
questions: Refer to the questions in the Weaknesses.
soundness: 3
strengths: 1). Theoretical proof and experimental validation are sufficient.
summary: This paper presents a characterization method for polyatomic complexes that meets more properties compared to existing methods, such as Invariances, continuity and differentiability, and generality, generalizability.
It achieves strong results across multiple benchmarks.
weaknesses: 1). The readability of the paper, including its organization and tables, could be improved. Could you provide more explanation for Figure 1?

2). Could you provide more explanations of the datasets and metrics used in the benchmarks section of the experiments? It might be a bit difficult for those without relevant background to understand.

---

confidence: 2
contribution: 2
initial_rating: 3
presentation: 2
questions: - 'Generalizability': I felt that this property could be differently named, in order to avoid confusion with its definition in the ML literature. Perhaps, 'completeness' captures the essence of the property better?
- e3nn is not a chemical representation, it is a framework for training E(3)-equivariant neural networks (which may be used to learn representations of chemical systems).
- Section 3.1: "Since we believe the values to be missing at random, we utilize mean imputation, instead of discarding experimental data". This is a big assumption, do the results change significantly if missing values are removed?
- Please highlight your method in all tables for clarity.
- What exactly is the role of the random matrix denoting proton-electron interactions? Are these forces/energies learned?
- Some of the definitions seem arbitrary. eg. where does the constant 2.8 fm come from in Definition 2.1?
soundness: 3
strengths: - The idea of polyatomic complexes for describing chemical systems is novel as far as I know. I think there are some good ideas in this paper, and it would be nice to improve the writing to flesh them out a bit more.
- The choice of datasets and baselines is quite comprehensive and interesting. I really appreciate the authors' honesty here in showing all results even when their representation is not the best-performing.
summary: This paper proposes a new representation (polyatomic complexes) for chemical systems, aimed at capturing topological properties of molecules. In particular, the representation 'stitches' together representations of different neutrons, protons and electrons. The authors prove that polyatomic complexes satisfy (rotational, translational, permutational) invariances, continuity (with respect to atomic positions), generalizability (in the sense of being well-defined for arbitrary chemical compounds) and topological accuracy (upto electronic structure).
The authors perform experiments on the photoswitches, ESOL, FreeSolv and Matbench JDFT2D dataset with two variants of their method (Fast Complex and Deep Complex), and find comparable performance to state-of-the-art methods.

My main concerns are the readability of the paper together with the lack of convincing empirical results (which the authors also admit to). It would be nice to come up with a task where this representation is provably better than existing methods.
weaknesses: - This paper is quite difficult to read (but this might also be my lack of familiarity with the notation). It is hard to understand what exactly the representation corresponds to. Some more figures describing the construction of polyatomic complexes could help.
- The formatting could be improved significantly. For example, Figure 1 is barely readable due to small text, and there is extra whitespace throughout the paper (eg. "( 2.12)" on page 3 and so forth.)
- The modeling assumption of an electron as a single sphere is quite chemically inaccurate (especially for delocalized systems). It would be good for the authors to discuss this weakness and how to improve this aspect of their representation.
- Experimental results are not super convincing, given that the RMSE of their method is often much higher than state-of-the-art methods. The authors do mention this as a weakness, but I would like to see how improvements can be made.

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 2
questions: - Could the authors discuss the time complexity of the proposed algorithm for constructing the complex, along with an ablation study? This would help clarify the potential trade-offs associated with the method.
- During the construction of the atom complex, is there a way to incorporate neutron and proton arrangements in energy levels (similar to the nuclear shell model)?
- Regarding the uniqueness of the representation, can the method distinguish between chiral molecules or stereoisomers of the same molecule by obtaining distinct representations?
- Given that the method represents an atom complex by decomposing electrons, neutrons, and protons and then reconstructing the molecule, how does it handle bond formation between electron-donating and electron-accepting atoms, where the atoms may carry a charge? or in general, representing a charged molecule?
soundness: 2
strengths: - A novel way of constructing a representation for molecules satisfying key ingredients
- Incorporating structural, topological information to construct representation
summary: The paper proposes a novel way of representing molecules and materials by incorporating topological information through a series of cell complexes that satisfy all structural, geometric, efficiency, and generalizability.
weaknesses: - The paper's writing lacks clarity, making it difficult to follow.
- The latter part of the introduction, which describes and compares various representation approaches, could be more effectively presented in a table. This would also help clarify where the proposed method fits within current methods, with detailed explanations that can be moved to an appendix.
- Including a subsection on notation would improve readability by clearly defining the meaning of each symbol used.
- Although the representation meets vital criteria, it still underperforms in benchmarks (e.g., ESOL and FreeSolv). Could this be due to limitations in the current method's ability to leverage higher-level topological information from this representation, or are there other underlying reasons?

---

confidence: 3
contribution: 2
initial_rating: 3
presentation: 2
questions: It is not very clear to me how the CW complex is implemented. The authors should clarify the form of the final representation (e.g. scalar/vector/tensor/graph, what is the dimension) which is input to the regression model.
soundness: 2
strengths: - The idea of building representation from topological information is novel.
- Numerical experiments have been done and compared comprehensively
summary: This paper proposes a method using CW-complex to build representations for atomistic systems. It consists two stages: in the first stage the CW complex representation is built for single atom based on proton, neutron and electrons; in the second stage, the CW complex representation is built for multiple atoms using the representations from single atoms. Numerical experiments have been performed on multiple benchmarks and compared with other baselines.
weaknesses: - Some statements related to relevant works have factual incorrectness. For example,
    - In line 142-146, in the description of ACE, the computational cost is not N! for permutations, and (J N) for J-neighbors. I suggest the authors to check the original ACE paper (Ralf Drautz, PRB 2019). The permutation invariant within atoms of the same element is achieved by simply summing over basis function of neighbor displacement, and by using clusters it also circumvents the (J N) summation.
    - In line 391, the authors state that the electromagnetic force, strong nuclear force and weak nuclear force hold the atom together, which is not a physically correct statement.
- The construction with proton, neutron, and electron is not very physically meaningful for the tasks of interests. This is just to obtain a representation for a single atom, which basically includes only the element and isotope information. A direct embedding on the element type will be simpler and achieve the same goal and learnable.
- In the comparison with other methods, this method does not show impressive results. E.g. on benchmarks like ESOL, FreeSolv, ChEMBL, the error is much worse than SMILES and SELFIES

---

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: - Time cost of computing representations compared with other methods is not provided to evaluate the **efficiency** constrain.
- The theory behind the model is excellent, but the practical implementation employs many simplifications and approximations to save computational efficiency, raising concerns that it may not achieve the capabilities claimed in theory.

- We notice that the authors have added more stringent constraints of molecular representations based on [1], such as Topological Accuracy, Long-range interactions and Chemical and Physical Informedness. We are skeptical about the necessity of these additional criteria.  Molecular representation is a lossy compression of information, and expecting a perfect representation of all properties may be challenging. Perhaps focusing on certain aspects while neglecting others could be more helpful for specific tasks. For example, electronic structure information and topological accuracy may not be very helpful in predicting macroscopic properties such as solubility, which is also experimentally demonstrated in Table2 and Table 3. Have the authors discussed why these criteria are added and are these additional criteria really useful?



[1] M. F. Langer, A. Goeßmann, and M. Rupp, “Representations of molecules and materials for interpolation of quantum-mechanical simulations via machine learning,” npj Comput Mater, vol. 8, no. 1, p. 41, Mar. 2022, doi: 10.1038/s41524-022-00721-x.
soundness: 3
strengths: This paper demonstrates strong originality at the theoretical level, introducing a method grounded in a rigorous mathematical model that meets stringent constraints.
summary: This paper  developed a new molecular representation encoding atomistic systems called polyatomic complexes, which constructs mathematical models by CW complexes and connects different elementary particles through gluing maps.

Besides, this paper expanded the criteria for good molecular representations and demonstrated that the proposed method satisfies all these constrains.

The authors also demonstrated the performance of this representation on several benchmark datasets in chemistry and materials science.
weaknesses: The method presented in this paper exhibits weak predictive accuracy across multiple benchmark datasets. 
Although the polyatomic complexes method characterizes at the electronic level, its performance is still less than that of the molecular level methods, e.g. SMILES and SELFIES.

---

confidence: 4
contribution: 2
initial_rating: 3
presentation: 1
questions: W2. What new tasks could be uniquely enabled by the proposed representation, and how do the authors plan to validate its performance in these tasks?

For additional concerns, please refer to the "Weaknesses" section above.
soundness: 2
strengths: S1. The author asserts that the proposed representation satisfies several key criteria, including invariance, uniqueness, continuity and differentiability, generalizability, efficiency, topological accuracy, long-range interaction handling, and incorporation of chemical and physical insights.

S2. These claims are supported, at least partially, by mathematical discussions.
summary: This paper introduces a new representation for chemical structures, designed to offer a more robust alternative to existing methods such as SMILES and fingerprints. By incorporating the physical characteristics of the target material, the proposed representation seems to demonstrate several desirable features that are critical for an effective chemical representation.
weaknesses: The paper appears to be in a draft stage, possibly intended to gather feedback before submitting to another conference or journal. This is inferred from several issues:

W1: The overall structure, figures, and explanations appear incomplete. For instance, the contents and sentences in Figure 1 are too small and difficult to understand. Section 2 lacks sufficient explanations connecting the lemmas and theorems, making it challenging to follow the argument. The description of the algorithm in Section 2.3 is brief and does not fully clarify the logic. Additionally, Figure 2 is uninformative and takes up space that could be better used to elaborate on the explanation and discussion.

W2: The experiments do not convincingly demonstrate the effectiveness or utility of the new representation. Across most datasets, the performance of the proposed representation is suboptimal, failing to justify its usefulness. The experiments focus solely on property prediction performance. It would be more informative to include comparisons of computational cost, assessments of generalizability, evaluations of long-range interaction capabilities against machine learning interatomic potential (MLIP) models, and tests on datasets where other representations struggle due to issues such as lack of uniqueness. In such cases, MLIP models could serve as a meaningful baseline due to their ability to represent diverse materials.

W3:  Due to the suboptimal layout, the sections on experiments and discussions (Sections 3 and 4) are very brief and lack insightful analysis that could demonstrate the effectiveness of the new representation.

In conclusion, the current draft requires substantial revisions before it can be considered for acceptance. It may be more appropriate to submit to a journal focused on computational chemistry, where an unlimited number of pages would be allowed for a more thorough explanation of the new representation and its advantages.

Meta-review: **Summary**:
This paper contributes a novel representation for chemical systems. Specifically, the authors introduce 'polyatomic complexes' in order to characterize the structural and topological properties. Polyatomic complexes can account for symmetries (rotation, translation, and permutation invariances), continuity with respect to atomic positions, and topological accuracy upto electronic structure. The method can also accommodate arbitrary chemical compounds, providing well-defined representations. Experimental results are provided for multiple data: photoswitches, ESOL, FreeSolv and Matbench JDFT2D.    

**Strengths**: 
Reviewers acknowledged several strengths of this work including (a) the novelty of polyatomic complexes as an alternative to existing methods such as fingerprints and SMILES, (b) the flexibility to incorporate the physical characteristics of the target material, (c) comprehensive choice of datasets and baselines for empirical validation, and (d) the theoretical analysis.  The authors were also commended for being transparent about the experimental results being not particularly strong.

**Weaknesses**:
Reviewers also raised concerns about the readability of the paper (including lack of clarity in presentation, insufficient coverage of relevant chemistry background, details about baselines, datasets and metrics used for benchmarking), suboptimality of most experimental results compared to even simple baselines such as SMILES, and modeling inaccuracies from a chemical/physics perspective (such as treating electron as a single sphere).  

The reviewers also pointed out some factual inaccuracies and made several suggestions for improvements, including, empirical assessments beyond property prediction such as evaluation of computational costs, generalizability, and ability to model long-range interactions.  

The authors satisfactorily addressed some of these concerns during the response period, but many remained unresolved. 

**Recommendation**:

The reviewers were almost unanimous in appreciating the novelty of the representation, and its theoretical guarantees. However, there is also consensus that experimental performance is not up to the mark and writing could be significantly improved. I agree with these concerns, and believe that this work requires major effort on these aspects. Therefore, I'm unable to recommend the paper positively in its current form.

Decision: Reject