ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals
Authors: Jonas Elsborg
Luca Thiede
Alan Aspuru-Guzik
Tejs Vegge
Arghya Bhowmik
Venue: NeurIPS 2025 spotlight ()
Keywords: charge density
quantum chemistry
equivariance
graph neural networks
molecular representation
Primary area: machine_learning_for_sciences
TL;DR: Efficient charge density prediction using floating orbitals

Abstract: We present the Electronic Tensor Reconstruction Algorithm (ELECTRA) - an equivariant model for predicting electronic charge densities using floating orbitals. Floating orbitals are a long-standing concept in the quantum chemistry community that promises more compact and accurate representations by placing orbitals freely in space, as opposed to centering all orbitals at the position of atoms. Finding the ideal placement of these orbitals requires extensive domain knowledge, though, which thus far has prevented widespread adoption. We solve this in a data-driven manner by training a Cartesian tensor network to predict the orbital positions along with orbital coefficients. This is made possible through a symmetry-breaking mechanism that is used to learn position displacements with lower symmetry than the input molecule while preserving the rotation equivariance of the charge density itself. Inspired by recent successes of Gaussian Splatting in representing densities in space, we are using Gaussian orbitals and predicting their weights and covariance matrices. Our method achieves a state-of-the-art balance between computational efficiency and predictive accuracy on established benchmarks. Furthermore, ELECTRA is able to lower the compute time required to arrive at converged DFT solutions - initializing calculations using our predicted densities yields an average 50.72 % reduction in self-consistent field (SCF) iterations on unseen molecules.

ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals
Jonas Elsborg!.211 Luca Thiede
Alán Aspuru-Guzik" 3,4.5 Tejs Vegge Arghya Bhowmik 1.2
Techhical University of Denmark
CAPPe Pioneer Center for Accelerating P2X Materials Discovery University of Toronto
Vector Institute for Artificial Intelligence
CCnadiaa Institute for Advanced Research (CIFAR)
Abstract
We present the Electronic Tensor Reconstruction Algorithm (ELECTRA) - an equivariant model for predicting electronic charge densities using floating orbitals. Floating orbitals are a long-standing concept the quantum chemistry community that promises more compact and accurate representations by placing orbitals freely in space, as opposed centering all orbitals at the position of atoms. Finding the ideal placement of these orbitals requires extensive domain knowledge, though, which thus far has prevented widespread adoption. We solve this in a driven manner by training Cartesian tensor network predict the orbital positions along with orbital coefficients. This is made possible through a symmetry- -bbeecty--brakiig mechanism that is used to learn position displacements with lower symmetry than the input molecule while preserving the rotation equivariance of the charge density itself. Inspired by recent successes of Gaussian Splatting in representing densities in space, we are using Gaussian orbitals and predicting their weights and covari- ance matrices. Our method achieves state-of-the-arr balance between computa- tional efficiency and predictive accuracy on established benchmarks. Furthermore, ELECTRA is able lower the compute time required arrive at converged DFT solutions initializing calculations using our predicted densities yields an average 50.72 % reduction in self-consistent field (SCF) iterations on unseen molecules.
Introduction
High-accuracy simulations for the design of materials and molecules at the atomic scale most often done with density functional theory (DFT) based simulations (Kohn & Sham 1965), as DFT provides good balance between cost and accuracy for quantum mechanical simulations of matter Maazzrii et al. 2021). However, the O(z²)) scaling of DFT still limits the system sizes and time scales that can be simulated. Linear scaling ML surrogates, such as neural network potentials trained with large number of DFT simulations, can alleviate this problem by learning a direct mapping between atomic structure and corresponding energy, forces, and other properties with accuracy similar those from DFT simulations (artoonsFFFFeeee (Friederich et al. 2021). This approach, although first envisioned three decades ago (Blank et al. 1995). has become successful and popular in recent years based on multiple seminal developments (Behler 2021 Deringer et al. 2021 Unke et al. 2021c).
'These authors contributed equally.
Corresponding authors: jelsëdtu.dk, dk. arbh@dtu.dt
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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An alternative underlying fundamental variable of the DFT simulations, the electron density, is predicted directly from atomic structures, without self-consistent field (SCF) iterations. Following the Hohenberg Kohn theorem (Hohenberg & Kohn| 1964), all ground-state properties can be calculated once this ground- state electron density is known (Grisafi et al. [022] Bogojeski et al., 2020). In recent years, researchers have addressed this task in multiple ways, differentiated by the representation of density data, the molecular representation, and the ML architecture itself (Grisafi et al. 2018 Chandrasekaran et al. 2019 Jørgensen & Bhowmik] 2022] Rackers et al. (023). The target electron density is commonly predicted on real space grids (Chandrasekaran et 22119 Jørgensen & Bhowmik 2020 Li et al.
ML-accelerated physics simulation approach can be taken where the
2024) or as an expansion of atom ceom-centered basis functions (Grisafi et 2011 Unke et al. 2021a
Cuevas-Zuvirf & Pacios 2021: Rackers et al. 2023) which usually take the form
Pt.m (r) (r (0, 1= 0,.,..,L
(1)
(r) represents the radial dependence relative to center, while (f,(8, Φ) captures the angular dependencies. Larger quantum numbers correspond to higher-frequency components. The accuracy of the represented density is dependent on the "quality" of the basis set the grid density. Two key attributes define the quality of a basis set: The number of basis functions per angular quantum number and the maximum angular momentum quantum number L included in the expansion.
Different systems and properties necessitate varying levels of basis set complexity. There is no universal basis set that provides both, high accuracy and optimal computational efficiency for all types of systems. Instead, the selection of an appropriate basis set depends on the specific requirements of the system under investigation and requires deep domain expertise.
For example, accurate descriptions of systems involving highly polarizable molecules or those with diffuse electron distributions far away from atom centers may require augmented basis sets like aug-cc-pVTTZ (Kendall et al., 1992). which include functions with high angular momentum and diffuse components that have long-tailed radial functions designed for modeling long-range dependencies. For smaller systems or those dominated by core-clectron interactions, these basis sets lead to unnecessarily large compute costs. In particular, basis functions with higher angular quantum numbers incur significant costs.
A more compact representation of densities can be achieved by putting extra basis functions at locations of presumed interest, particularly in areas far away from atoms, with rapidly varying densities. These basis functions are called "floating" orbitals, and their utility is well-established in electronic structure theory (Tao & Pan 1992: Tao 1993 Tasi & Császár 2007). They date back to the floating spherical Gaussian orbital (FSGO) model (Frost 1968). When chosen wisely, floating orbitals can lead to significant improvements calculation speed and accuracy (Lorinez & Nagy 2024 by reducing the need for diffuse and high angular momentum basis functions.
Well-placed floating orbitals can represent densities more efficiently, using lower maximal angular quantum numbers L. ELECTRA is the first model to predict floating orbital positions without human input.
However, the optimal locations of floating orbitals are often hard to determine Zheng et al. (202) and picking good locations therefore requires deep electronic structure domain expertise (Lorinez & Nagy 2024). Our core contribution is data-driven solution to this problem. We are training a model that, given a molecular graph, accurately reconstructs ground truth charge densities by predicting the 3D position of floating orbitals as well as the coefficients and parameters that define them.
Since charge densities are rotation invariant, we use rotation equivariant neural network as the backbone of our model. However, naive implementation of equivariant neural networks is destined to fail, since good placements of floating orbitals can have lower symmetry than the input molecular graph, as we will discuss in later sections. We address this problem by developing a symmetry- breaking mechanism that retains rotational equivariance. We call the resulting model the Electronic Tensor Reconstruction Algorithm (EL ECCCARA). We test ELECTRA on the widely used QM9 charge density dataset (Jorgensen & Bhowmik 2022) and achieve results that are competitive with of the-art while being consistently faster.

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2 Related work
2.1 Charge density prediction
Prior work on machine learning prediction of charge density (CD) generally falls into two main approaches, inspired by earlier non-ML methods. Orbital-based methods are rooted in linear combi- nations of atom-centers orbitals (LCAO), which take the form
N
4,
p(r) = Σ
Σ
Ci.j. Q Banddaas (r);
m
where the first sum runs over all atoms, and the other two sum index into all basis functions per atom. & usually takes a form as in [ In the ML community, methods based on this construction typically predict coefficients C..im extracted from ground truth DFT calculations (Fabrizio et al.
[ 2019 Qiao et al. 2022, Rackers et al., 2023 Cheng & Pengl 2024 del Rio et al. [023 [eebrer et al. 2024 as well as refined radial functions (r (Fu al. (024). This is computationally efficient at interence, and orbital- -ecitaasde density representations can offer enhanced accuracy in describing both total and orbital energies by utilizing flexible, orbital-specifie potentials that align closely with many-body spectral properties (Ferretti et al., 2014). However, the fixed choice of basis set often limits representation power unless a large, expensive basis set is used, particularly for complex inter-atomic electronic features. By placing additional orbitals on bond midpoints, (Fu et al.) (024) achieved higher expressivity, albeit higher computational costs, and the additional requirement of determining bonds. The latter point sounds trivial, but bonds are not always well defined making this difficult.
The second method is inspired by viewing the charge density as a numerical grid (Cerjan 2013). which must be probed at each point construct the density. By inserting a graph node that receive messages from the atomic graph representation (Gong et al.] 2019 Jørgensen & Bhowmik 2022
Koker al., 2024 Pope & Jacobs 2024 Li al.]| 20024) in each grid point, these models directly predict scalar charge values at grid points, offering high expressiveness and accuracy. Even for small molecules, charge density data contains hundreds of thousands of points, and thus, nobe-based
models are generally more computationally intensive than orbital-basee models.
Once we have the density, two broad strategies emerge:
Use p(r) as an initial guess: Plane-wave based KS-DFT can take p(r) given on a grid and cheaply map it to plane-wave coefficients using fast Fourier transform (FFT), which allows to resume SCF, This approach was demonstrated by Pørgensen Bhowmik (2020) 2022). The same idea applies to orbital-free DFT (OF-DFT) Weizsäcker (1935) y Mi et al. (2023) which natively operates grids and allows for much faster, albeit less accurate, SCF and energy predictions.
Use p(r) directly: A potential alternative route is to train KS-DFI densities and only evaluate the energy with OF-DFT without extra SCF, an approach that standard for benchmarking OF-DFT functionals and that was shown give better predictions than self-consistent OF-DFT densities Constantin & Ruzsinszky (2009); [Iyenaa et al. (2001) Perdew et al. (1988). Other properties that be evaluated directly without extra SCF Bader's Theory of Atoms in Molecules, which shows how a topological analysis of p(r) can be used to define bonds rigorously. This allows us to identify covalent, ionic, and noncovalent interactions Bader & Nguyen-Dang' (1981); Bader & Essén (1944); Booto et al. (017)) and construct charge partitioning schemes with widespread use in quantum chemistry. Densities also provide qualitative insights into electrophilic and nucleophilic regions Bader et al. [1984). as well as weaker interactions such as hydrogen bonding Koch & Popelier (1995). and allow us calculate multipole moments and field tensors. Recently developed usage of p(r) includes density-based generative 3D drug design models Ragoza et al. (2022); Wang et al.
(2022) and density-based descriptors to study redox processes in electrochemical systems (Laubach
et al. 2009 de Blasio et al. 2023 [Shang et al. (2022) and electrocatalysss Zheng et al. (2014); Koch al.
2021 Densities themselves are also often used build cheap surrogates for observables like
ionic diffusion [Kalee et al. 2018).
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2.2 Equivariance and Cartesian tensors
Many objects in physics transform predictably under symmetry transformations. This property is called equivariance. Formally, a function f X + Y is equivariant with respect to group & whose elements g act on X and Y, if
fgxx) - gr
For example, we are often interested in the case where the group of translations, rotations, and reflections, see (Thomas et al., 2018 Geiger & Smidt] [022] [Simeon & De Fabritiss (2024) for more details. Constructing a machine learning model with the equivariance property (3) provides a strong
inductive bias that usually leads increased data efficiency (Brehmer et al. 2024).
If Y is function space, P Y, gY acts on P via the left-regular representation, defined as (Brandstetter et al. 2021)
gy = p(g
(4)
In this work, are learning model that maps from molecular geometry mol {rx) ) the electron density
[f(mol]] p(r(
(5)
This mapping is equivariant under rotation R
f(R mol) = p(r) R mol) = p(R (r(ol)
(6)
Equivalently, the electron density is rotation invariant under joint rotation of both electron and molecular geometry:
p Rr R mol) = p(
= p(r[mol)
For notational convenience, we will drop the conditioning on mol our notation from now on and only write p(r).
Cartesian tensors provide a systematic way to handle rotation equivariance. An Ith-rank Cartesian tensor T is an /th-rank tensor that transforms under rotation
T Tiig
R
(Risj) (Ruzjz) (R (Royji T Tnie
where R an orthogonal matrix in Cartesian coordinates. Equivariant graph networks can be built to leverage operations Cartesian tensors (Simeon & De Fabritiis 2024 Wang et al. 2024 such as linear combinations, tensor contractions, and partial derivatives (Simeon & De Fabritiis 2024; Wang et al., 2024) that ensure that the network's 's outputs equivariant networks that operates on Cartesian tensors is the High-order Tensor Passing Potential (HotPP) <Wang al. 2024). HotPP's node features and messages arbitrary order Cartesian tensors, and the operations are constrained such that the outputs remain Cartesian tensors. This allows predictions higher order physical quantities like dipole moments (rank 1 tensors, i.e., vectors) and polarizability tensors (rank 2 tensors, i.c., 3x3 matrices), and similarly allows for more complex atomic environments
equivariant. One example of
be distinguished. In HotPP, an atomistic system is represented as a graph G = (V, E),, where V is the set of atoms (nodes) and É is the set of edges (defined up to cutoff radius) in molecule. Each atom is characterized by feature vector UA, and each edge (CA,,A2 E) between atoms A₁ and A, is associated with an edge vector UA, and scalar distance dA,,A2
Methods
3.1 The ELECTRA model
For ELECTRA we make simplified ansatz compared to the LCAO ansatz. Inspired by the idea in Gaussian Splatting (errbl al. 2023) of approximating complex 3D field with many inexpensive Gaussians, we here use Gaussians an efficient density basis, because they are closed-form

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separable, and cheap to evaluate on grids (Kerbl et al. 2023 Charlier et al. 2021 Feydy et al. 2020).
Concretely, we represent the charge density as 3D Gausstan mixture model:
NA (ArN(rl Ens)
(r) = ReLU
ΣΣ AEM
(9)
where ReLU is the standard rectified linear unit function used to prevent negative densitie A E M represents the atoms in the molecule, and NA is the number of Gaussians for each atom, which can depend on the atom type. The weights w.j. are signed, which improves expressivity. This can, e.g., be used construct shell-like structures by inserting a negative density at the center of larger sphere. In the following sections, we elaborate on how all Gaussians way, using the model output specific to each atom. Gaussians are equivalent to traditional Cartesian orbital functions with angular quantum number / = 2, an extra nonlinearity. and simplified radial dependency; see appendix B for why. This is the reason we are saying that ELECTRA uses / orbitals. We will see that, if made "floating", these simplified orbitals are enough to achieve strong performance. in contrast to atom-centered basis functions, which require a high maximum angular quantum number L for good performance. In principle, we could also use more conventional basis functions and make them "floating".
constructed in a per-atom
To enforce rotational invariance 7 of the predicted electron density 9 we need the weights WAJ to rotation invariant, while the means A, and covariance matrices ΣA,j need to be rotation equivariant. In particular. the Gaussian means and covariance matrices need to transform like Cartesian tensor, see appendix Cfoo details.
Equivariant backbone neural network. To enforce the constraints W... Ma.. and ΣA.- use a modified version of the HotPP (Wang et al., (024) equivariant message-passe passing network to represent atomistic systems in ELECTRA. We initialize the scalar features as well as the first three rank-1 1 features in each atom using a tailored embedding function. The initialization is important to the final model and is detailed in the paragraphs below. The graph is updated through series of HotPP's update layers. We then use the resulting features to predict the parameterization WA, A, and A, for the Gaussians in (9) using readout head layer. Other important changes to the default HotPP implementation are detailed in the paragraphs below, and ELECTRA otherwise follows the reference implementation.
Atomic embeddings and variable basis set size. In quantum chemistry, different atoms require differently sized basis sets, since the complexity of the electronic structure generally depends on the atomic number (Weigend & Ahlrichs, 2005) and the number of valence electrons. Inspired by this, ELECTRA predicts a variable number of Gaussians depending on the number of valence electrons. This is achieved by assigning each output channel of each atom in HotPP to one Gaussian. Denoting Me as the number of Gaussians per valence electron, we can use the first n. M, channels of each atom to represent the Gaussians, where n. is the number of valence electrons for that atom. For this to work, a channel width of N, = 8 M, is sufficient in HotPP. Each atom A then uses only its first M, A channels. For example, oxygen (n,, from the 2s and 2p shells) utilizes 6M, output channels, while hydrogen (n,, = 1 from the 1s shell) uses only M,, output channels. We use an atomic embedding function inspired by SpookyNet (Unke et al. 2021b) to represent the atom in terms of its nuclear charge and electron configuration (details are in Appendix G
semmetry-breakinn Since ELECTRA's orbitals are not as expressive as standard spherical harmonics-bas bbssed orbitals, the model needs to have maximum freedom in placing the orbitals in space. However, equivariant networks prohibit their outputs from having a lower symmetry than their inputs (Smidt et al. 2021: Xie & Smidi 2024). In particular, local reflection symmetries lead to strong restrictions on the outputs of equivariant models. For example, in Figure Ia we plot all vector-valued outputs of randomly initialized equivariant model with planar molecule as input. Clearly, the outputs are constrained to the reflection plane, which would severely limit a density constructed based on a Gaussian centered the predicted locations. See appendix D for detailed
'Sicee the ReLU clamp enforces non-negative non-negativity, it could in principle create zero- cero-density plateaus with weak local gradients. In practice. this did not impede training: we trained with and without the ReLU clamp and observed no measurable differences, and thus include it here solely out of principle.
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(a)
(b)
(c)
(a) Model output without symmetry-breaking Equivariant neural networks constrain their output have the same symmetry the input. If the input molecule is highly symmetric, this leads to highly constrained Gaussian positions, (b) To solve this issue, ELECTRA initializes each atom's 's I = vector features with the eigenvectors of the moment of inertia tensor as calculated in that specific atom, which breaks the input symmetry but retains rotational equivariance. after first linear layer with commetry-breakinne The model can learn its own set of -bbeakkin vectors, allowing output to not be constrained by the symmetry of the input molecule.
Figure 1:
(c) Model output
mathematical explanation. Since many ground-state geometries are highly symmetric, this poses big issue for equivariant networks. Previous work has investigated both indirect and direct ways of breaking symmetries. Indirect methods typically relax the equivariance constraints (van der Ouderaa al., 2022 Kaba Ravanbakhsh [2023] Huang et al., 2024), which undesirable for electron
densities since these are exactly equivariant (Rackers et al. [0233 Other methods break symmetries
by constructing mammetry-breakkeee inputs (Liu et al. 2019 Locatello al.) 2020 Xie & Smidt
2024) or by learning order-breaking parameters during training (Smidt et al. 2021).
To construct expressive method for placing floating orbitals, the model must allow for output vectors that belong to lower symmetry group than the input structures. Thus, symmetry-breaking mechanism is needed.
Our approach to symmetry- breakkng with ELECTRA broadly falls into the category of symmetry- breaking inputs. For our construction, we are first calculating local moment of inertia (MOI) tensor for each atom:
flatom) ig
Σ m. (Irxll
2 8ig 2000 (K)
(10)
Where A: = 1,.... N' runs over all atoms inside local atomic neighborhood defined up to a cutoff radius from the current atom, and the vectors Tk (2111,, (k) xz are calculated relative to the
current atom. The three eigenvectors of (0) are also rotation equivariant. Thus, we use them to initialize the first three 1 vector features of ELECTRA'S GNN, from largest to smallest eigenvalue (Figure (b). while maintaining rotational equivariance. They define local coordinate system on which the model can learn its own set of brreakiny objects using a linear embedding layer, the output of which is depicted in Figurq Similar ideas of using the moment of inertia vectors have been explored in slightly different contexts in previous work, for example Puny et al. (202T); Duval et al. (003)): ; Gao & Günnemann (021); ) Taniai al. (2024). The eigenvectors of matrix only defined up to sign flip. One can resolve this issue for example by averaging all possible sign combinations Duval et al. (2023). However, in our case, this averages out all meaningful anisotropies. Instead, we opt for canonicalizing the eigenvectors. Canonicalization of eigenvectors is a research topic that is studied independently (Ma al., (024).. In this work, we use a sign convention that maximizes the dot product of each eigenvector with the position vector of the center of mass (COM) of the molecule. Mathematically for eigenvector v, we switch the sign according to:
v.
U TCOM 2 0,
C Ccanon
(11)
if TCOM < 0.
There are transformations where this canonicalization will lead to sign flip. and the model has to learn to compensate for them. However, we find, empirically, that this does not hinder strong performance. There are also some systems in which the moments of inertia degenerate. However, except for some special cases, for example, completely linear molecules, this rarely happens in practice. This is why many successful models rely on the moments of inertia.

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(a)
(b)
(c)
Figure 2: (a) The initial symmetry- -ymmetry-breaking objects of the NH₃ molecule (Ammonia). (b) The output of HotPP model with symmetry-breaking but without debiasing layers: The message passing induces a directional bias that concentrates vectors along certain directions, (c) The output of our model with debiasing layers: The output vectors don't 't show any visible bias.
Debiasing layers. Even though our symmetry-breaking mechanism allows ELECTRA theoretically break any symmetry (see appendix D). we observe that the pesssag--psssing mechanism of HotPP induces a directional bias of the I = features, particularly in highly symmetric molecules. In appendix we discuss the origin of this bias mathematically.
For example, in Figure 26) we plot the output vectors from a randomly initialized HotPP model with the NH₃ molecule (Ammonia) and symmetry-breakinn tend to be parallel to the bond axis of the molecule, which is problematic if we were to use these vectors as Gaussian positions because the ground truth density has a lot of density around the bond axis. Empirically, the model was not able to overcome this bias and place the Gaussians efficiently in space, which led to low performance. To address this issue, we designing layer that learns to dynamically remove directional biases in the vector features. Our debiasing layer, which we place after every message passing layer, first calculates the covariance matrix of all the node features associated with each atom:
objects input. We see, that the vectors
1 1 C D _ VAJVAA
(12)
where VAi are the / = features for atom A, and D is the channel dimension. We denote A1.A as the eigenvector of C with the largest eigenvalue Å₁. If there is a directional bias in the features of an atom, u₁ points in the direction of the largest variation. The stronger the directional bias, the larger the magnitude of U1. We calculate the projection of each = 1 feature onto this principal axis uj
(13) Note that the sign ambiguity of 41,4 is not important in this case, as U1 appears twice in the projection. By subtracting vh, from VA, can reduce the directional bias. To let the model decide how much to subtract, we predict weight w A.j using small neural network, conditioned on / = 0 features. The output W E [0, 1] a number that determines how much of the principal direction to remove in each vector, such that the debiased vector is updated as:
VA (VAg U1.A) U1.A
VAj < VAJ
W A.j
A.
(14)
|V A.j
A.
Where : is the dot product. We normalize the vectors in [4] such that the = 1 features only handle directionality, while scales are handled in readout layer by | predictions. This mechanism, therefore, provides way to determine and remove bias in the 1 = features.
We note that the symmetry and bias limitations were not unique to the HotPP architecture, and we did initially try e.g. TensorNet (Simeon & De Fabrittis 2024 and Equiformer v2 (Liao et al., (023). However, these backbones faced the same issues as HotPP in terms of ultimately, we opted for HotPP initial experiments suggested that worked best.
and
3.2 Density construction
After several message-passing layers, we have a set of features for each atom that we feed into three readout heads. Each readout head produces set of / = 0 (s), / = 1 (v) and [ = 2 (M) features,
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(s1, V1, V1,Mi)..i (s2. V2. M2) A.5 and (s3, V3. Ms). where A indexes the atoms in the molecule, and the channel. Depending the atom type, the channel index is j E [0. [O., (A) x Me]. where Ne is the number of valence electrons of that atom, and MM. is the number of Gaussians per valence electrons. Intuitively, each of the three heads is specialized on different distance scale away from the atoms. We use these predictions for the parameterization of the Gaussians in our ansatz 9 Each mean position is calculated each covariance matrix is a weighted sum of three symmetrized matrices based on the | features. The full details available in Appendix F
weighted sum of three 1 = features (one from each head). Similarly,
Normalization. Prior work has shown that density prediction models whose output does not integrate to the number of electrons can lead to errors in downstream property predictions (Briling et al. 202T). Thus, as a final step, the densities predicted by ELECTRA are normalized to the number of valence electrons in the system:
Ppeed(t) p(r)
Teeler
(15)
p(r) dv
where dV represents the differential volume element on the grid. This ensures that
JeL PPrred [pprea (r)ldV
Telees
(16)
Since Relee is simply the number of valence electrons, this number is already provided an input to standard DFT codes or can easily be obtained via summation over the valence electrons of each atom.
Objective function absolute error:
We train ELECTRA on loss function L based on the normalized mean
Sgs |erer (r) (roea(())dd)
C C= NMAE (ppred. Pret)
(17)
SRa |Pee (r)
is not necessary to compute the denominator in (17 during training since the reference grid must integrate the number of valence electrons, i.e., Jgs Preet (r))dV Relec, and thus the denominator integral can be replaced with Nelee during training.
Experiments
Dataset and implementation. We train ELECTRA using the method outlined in Section 3 reference densities from the QM9 density files. This charge density dataset is the most commonly used benchmark for charge density prediction models. The dataset was generated in VASP (Kresse &
Hafner 1993) using the PBE (Perdew et al.) 1996) functional and the Aoreettt-- Wave
(PAW) Blöchl 1994) method (Jørgensen & Bhowmik 2022). We the full split consisting of
123,835 training molecules, 50 validation molecules, and 10,000 test molecules, following the same
scheme in previous related work (Jørgensen & Bhowmik 2022 Koker al. 2024 Kim & Ahn
2024 Cheng & Peng 2024 Fu et al. 2024). On the QM9 dataset, we train ELECTRA for 864 GPU hours on a single NVIDIA RTX 3090-24GB GPU. We train using batch size of 1, and do inference on the full grid for both training, validation and test. In comparison to our training protocol, the similar current of-th----he-ar method from Fu et al. (024)) was trained for around 1152 GPU hours on NVIDIA A 1100-80GB GPUs. To further validate ELECTRA, we also perform experiments on the MD dataset consisting of different conformations of six different molecules Bogojeski et al.) (2020); Brockherde et al. (007)). 7). A full list of our hyperparameters is given in Table in the Appendix. As an example of how ELECTRA distributes Gaussians around a molecule, we provide Figure 3 which
shows the predicted and ground truth densities for C,HyNO from the QM9 test set, along with the individual placement of all Gaussians and an error isosurface that visualizes the density errors.
Benchmark results. In Table 1 we report the mean accuracy in NMAE [%] (Equation 17 and average inference time per molecule (tanf) on the QM9 test dataset of the main ELECTRA model and concurrent charge density prediction models, using results from the original papers as well as results from the model testing carried out in Fu et al. (2024). The SCDP models are the current state-of-thh-ant and thus, to ensure a fair comparison, we tested them on our own hardware, i.e., on RTX 3090-24GB GPUs. In Appendix ! we report ablated versions of EI ELECTRA, which demonstrate the influence of the floating orbitals themselves, well as symmetry-breakkin and debiasing layers.
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Table 1: QM9 test set performance various models. Top: Results reproduced from Fu et al. 2024
(A100-80GB GPU). t: This version of InfGCN only used Gaussian Type Orbitals (GTOs). Bottom: ELECTRA and SCDP (RTX 3090-24GB GPU, this work)
Metric
i-DeepDFT
e-DeepDFT
GPWNO
InfGCNt
InfGCN
ChargE3Net
NMAR [
0.357
0.284
0.730
3.720
0.869
0.196
linf [s]
0.833
15.18
Metric
ELECTRA (ours)
SCDP (L 3)
SCDP+BO (L 6)
NMAE [%]]
0.176 0.089
0.432
0.178
linf [s] +
0.395
1.022
(a)
(b)
(c)
(d)
Figure (a) Predicted density for CyH,NO using ELECTRA (NMAE 0.19 %, red high density,
blue lower density). (b) Ground-truth density. (c) Gaussian placements (red: wA. >> 0, blue:
WAj 0). (d) 0.001 e/bohr" error isosurfaces (blue = over-prediction, red = under-prediction)
Table shows that ELECTRA is significantly more accurate than the DeepDFT, IncGCN and GPWNO models. Additionally, compared to all timed models in prior publications well this work, ELECTRA is faster by at least an order of magnitude. Compared to ChargE3Net, ELECTRA is 0.020 percentage points more accurate (0.176 O. 0.196), and roughly 170 times faster inference, even when evaluated on inferior hardware (3090-24GB vs A current SOTA, ELECTRA is about 2.4 times more accurate while being 4.4 times faster compared to the fastest model (L = 3), and slightly more accurate than the best model (L = + bond-centered orbitals (BO)) while being over 11 times faster. On the MD dataset (Table 2p. ELECTRA sets new state-of-the-art on all six molecules by halving the NMAE % error compared to SCDP. In Figure 4 we also show how ELECTRA distributes the floating orbitals of Benzene molecule, and how ELECTRA recreates the central density hole of the molecule. These fine details far away from any atom center are hard model using only atom-centered orbitals. To test how close we are the optimal solution allowed by our basis, we conduct overfitting experiments (Table 2 for details see H
Compared to SCDP, the
DFT initialization experiment. Because the iterations of self-consistent field calculations (SCF) dominate wall-clock time in routine electronic-structue DFT workflows, we assessed whether ELECTRA'S predicted densities accelerate SCF convergence unseen molecules. Using VASP with the original dataset settings (Appendix K), we ran paired calculations initialized from (i) atomic superposition and (ii) ELECTRA-predicte densities, recording SCF iterations required to reach convergence at the same tolerance. The results showed that ELECTRA initialization reduces SCF steps by 50.72% on average. Furthermore, note that savings track density accuracy: the worst- error molecule N,, NMAF = 1.43%), which is a significant outlier, shows only 17.4% reduction, whereas the hest (C,H,O, NMAF 0.153%) achieves reduction in SCF steps of 60.87%
ELECTRA uses floating orbitals and lower-order representations to achieve state-of-the-art density prediction accuracies while being an order of magnitude faster on inference than competing methods. This results in significant reductions in the number of SCF steps required to reach convergence when initializing DFT calculations with predicted densities.

---

Table 2: [M on MD test sets for ELECTRA, SCDP (Fu et al. 2024 GPWNO Kim &
Ahn 2024) and InfGCN (Cheng & Peng 2024). as well as the N Gaussian Overfit experiment. ELECTRA EERR models were trained for 10 GPU hours.
Molecule
ELECTRA
SCDP
GPWNO
InfGCN
(N
Gaussian Overfit)
MD-ethanol
1.02
2.34
4.00
8.43
(0.18)
DD-benzene
0.45
1.13
2.45
5.11
(0.08)
MD-phenol
0.56
1.29
2.68
5,51
(0.08)
MD-resorcinol
0.62
1.35
2.73
5,95
(0.24)
MD-ethane
0.91
2.05
3.67
7.01
(0.28)
mDooaloonadeeyyde
0.80
2.71
5.32
10.34
(0.16)
(a) Gaussian centers predicted by ELECTRA for Benzene molecule in the MD dataset.
(b) Closeup (a): central density hole Benzene.
(c) Ground truth density (red high, blue low).
Figure 4: Gaussian placements vs. ground-trute truth electron density for Benzene.
5 Discussion
Since no existing basis sets or handcrafted orbital placements are used, ELECTRA fully data-driven. It represents shift from human-designed to machine-learmed density representation, unlike the traditional DFT paradigm, where users must make heuristic element- based choices regarding the basis function type, size, and placement. However, there are still promising opportunities explore.
Orbital placement. Fu et al. (2024) show that adding bond-centered orbitals increases expressivity, For this method work, bonds must be identified in real-time during training and inference, and may fail for complex systems with non-classical bonding and delocalized interactions, such as partially formed or broken bonds, variable bonding radii, weak interactions like π T-backbonding, and coordination variability. For crystal lattices, issues would arise for, e.g., color centers, where vacancies occupied by unpaired atomic electrons (Seitz 1946). Since the vacancy itself does not contain bond or atom, centered orbitals would likely fail. Similarly, in electrides, the electrons effectively function anions, requiring non-centered positions (Dye 2003). In all the above cases, using freely placable orbitals originating from atoms is still viable in theory. ELECTRA would theoretically be able to learn the non-local behavior of the density through (39) or variations thereof. However, this would require specialized methods to replicate floating orbitals periodically. Additionally, floating orbitals scale with the number of atoms rather than the number of bonds, making them computationally more efficient.
Hybrid Atom-Centered and Ploating-Orbital Models. Naturally, may be possible construct floating orbitals using spherical coordinates, which was also suggested in Fu et al. (2024). We thus believe that ELECTRA complementary to their work and that the benefits of floating orbitals could be combined with the flexibility of the spherical harmonics-basse SCDP models. A model with both atom-centered and floating orbitals likely represents the most efficient use of computational resources, since even in ELECTRA, many orbitals are placed around atomic centers (Figures la and 3a
10

---

Acknowledgements
The authors acknowledge financial support from the Independent Research Foundation Denmark, grant number 0217-00326B (DELIGHT), the Novo Nordisk Foundation grant number 87929 (Ni- troScale), and the Pioneer Center for Accelerating P2X Materials Discovery (CAPeX), DNRF grant number P3.
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15

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A
Group theory
We provide a short introduction to the relevant group theoretical background, mainly adapted from
(Liao al. 2023 Liao & Smidi 2022).
Definition of Groups. A group is an algebraic structure consisting of a set G together with a binary
operation G G G. A group, satisfies the following four axioms:
1. Closure: For all g, h E G, we have g h G.
2. Identity: There exists an element E G such that g e for all E G.
3. Inverse: For every 9 G, there exists element g-1 G such that g g*g****** 9 ==-l==
4. Associativity: For all g, h G. have g (h * = (g +
Symmetry Groups in Three Dimensions. In this work, focus ftrre-dimensional Euclidean symmetry, where the relevant groups are:
The Euclidean group in three dimensions E(3): rotations, translations, and inversion in R
2. The special Euclidean group in three dimensions SE(3): rotations and translations R$.
3. The orthogonal group in three dimensions 0(3): rotations and inversion in IR
4. The special orthogonal group in three dimensions S0(3): rotations in R-3
Group Representations.
Let X be vector space. A representation of a group G on X is map D G > End(X),
where for each g G, (x X X is linear transformation. In practice, Dx (g) is represented as a matrix acting on X, so that group actions are matrix multiplications. These representations respect the group structure: for all g, E G,
D(g) D(g * h))
Two representations (g) and (g) said be equivalent if there exists an invertible N N change-of-basis matrix P such that
P D(g)P = (')g) for all g E G.
representation D(g) called reducible if can be transformed into block-diagonal form, i.e.,
Di(g)
P-ID(g)P=
D2(g)
so that independently on multiple subspaces of Otherwise, the representation is irreducible. Such irreducible representations (irreps) are especially useful for building and classifying group representations.
In the case of SO(3), the Wigner D-matrices provide a family of irreducible representations. Any representation of (3(3) be expressed direct sum (concatenation) of these irreps:
D(lo(g))
D (1o)
D(g) = p-1
+
P p-1
D (
P.
(18)
where D())g) denotes the Wigner D-matrix of degree 1.
The degree can be thought of frequencies in in Fourier decomposition; the higher the 7 the higher frequency.
16

---

B Gaussians as Cartesian basis functions
The most often found form of basis functions in quantum chemistry is
La..m( (r) (lr((r))) (0,
(19)
where is the angular quantum number. To build our densities, we contract the basis functions with set of coefficients Cr.m such that the contribution of all basis functions centered at the same spot can be written in Einstein notation C. 2 a,,
However, a popular alternative in quantum chemistry is Cartesian basis functions. With r (Ir y, z)
Cartesian basis functions can be written as:
L Po.ne:fyy"h (x, V, 2) = N(a, 1z, Ro (1))a"""""""
(20)
where Na, fa, Bly, Cartesian basis functions is defined as = T2 + My 72.. Note that there is a strong relation between the angular momentum quantum number for spherical basis functions and Cartesian basis functions. For each I, there is an invertible transform between the spherical [-shell and the harmonic part the cartesian I-shell (Ribaldone & Desmarais 2025).
R2)) R22) is a normalization factor. The angular quantum number in the case of
We can collect all the terms above belonging to the same in one tensor:
(,, v, = (a, 1) (o(l) r 8 ... 69
(21)
I times
In particular, for 1 we get
N(,, 2) (ar))rr
(22)
When we contract this with a coefficient matrix Cty. calculate the contributions of the basis functions to our density, we get
Ci Ra((r))r (N(a,1)-C)
(23)
22
If we set the radial term (,(()) 1, wrap the remaining term in exponential function and choose C' such that E is positive definite, we get, up to a normalization constant, a Gaussian:
exp -
1 - x Nr(00, E)
(24)
C Equivariance of Gaussians leads to invariant density
The electron density rotation invariant. the main text, we claimed that our ansatz 9
NA
(r) = Σ WA.Warl EA.))
(25)
AEM
rotation invariant, if the weights WAj rotation invariant, and the position AAj and covariance matrices Ea.j transform as / = 1 and / = 2 Cartesian tensors. It is clear, that the entire ansatz is invariant if each Gaussian is individually invariant. So we need to show, that
N(rlp, μ. Σ) = ((R Ru, RER
(26)
for a rotation matrix R. For simplicity, we omit the normalization constant of the Gaussian, since is rotation invariant. Then write
(Rr) Ru, RER = exp
(R(r - p)) (RER)) RRr 11)
(27)
- (r 11T TR R RE R R(r - m1)
exp
(28)
- (r- - 1)) T - m1)
= exp
(29)
NN(ru, E)
(30)
where we have used that R R
This shows claim.
17

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D Local reflection symmetry constraints
We claim that "local", vide infra, reflection symmetries lead to restrictions on the expressivity of equivariant neural networks. Equivariant neural networks are constrained such that the output transforms with the input:
g()))
(31)
where G describes our molecules and g is the action of a group. In particular, we are interested in g E 03)), so equivariance to rotations and reflections, in which case we have:
ft (RG)- det(R))DD(R)))) (t.
(32)
where (R) the Wigner-D matrix of rotation, R' det(R.) = R the rotation part of an (im)proper rotation and p is the parity of the output, with p = 1 called odd and p 0 called even. Next, we define local invariance:
Definition D.I. Let G
{(r, - ry), Zi, Zj J..= be the description of molecule using pairwise displacement vectors and associated atom types. Let G; = ((re-Ty), Zi, Z;hje C G be the subset of the graph with nodes within some neighborhood Nii,r, L) of node i, defined by L- hop message passing with cutoff distance r. Let Rp be the representation of some reflection symmetry through some arbitrary plane. We call G; locally symmetric/invariant under Rp if RpG; = G.
Importantly, part of molecule can be locally symmetric, even if the molecule as a whole does not have the same symmetry globally. Take, for example, water in its ground state geometry with O being located at (0. 0, 0) and H1 and H2 lying in the xz-plane. Then GH, {(TH, ro. 0), (rH, TH H2)} is locally symmetric to reflections along the xz- xz-plane, represented by the matrix
1
0
0 0
0
-1
(33)
0
0
1
and we have GHGHH = GHH so GH. is locally symmetric under Te. Local symmetry leads to constraints on the outputs of f(G,). In particular, the output of an equivariant function needs to be an eigenfunction of the symmetry operator Rp with eigenvalue 1:
(RpG,) (C,)
(34)
For example, in our water molecule example, we have Rp = a.. For I we have DR'R') = R'' and therefore for our p 1,7 = 1 get
f. (RRG. = (G; =
= RPf1,
(35)
Reflection operators three dimensions always have exactly one eigenvalue of -1 and two eigenvalues of 1. Therefore, the requirement above restricts the output of the equivariant network to live in the degenerate subspace with eigenvalue 1. For example, the eigenspace of a, for the eigenvalue is spanned by (1,0,0) and (0,0.1). This means any vector in the xz-plane is an eigenvector of o.. This is exactly what we are seeing in Figure Tal where all output vectors from the hydrogen atom lie in the xz-plane. It becomes even more restrictive for neighborhoods with more than one reflection symmetry. For example, Go. the descriptor around the Oxygen atom in our water example, has two symmetries. In general, the highest number of symmetries we can have in molecule is 9 (the full octahedral group 0x). However, for most organic molecules it will be at most 6, for example, CHa with the tetrahedral group Ta- If we have more than one reflection symmetry, we have more eigenvalue equations of the form 34 one for each symmetry. Our output has to fulfill all of them simultancously. This lowers the dimension of the allowed subspace for the output of our equivariant functions by one, since the output needs to live in the union of the allowed subspaces from each reflection individually. For example, if we have both a, (x2) and (,.2) our = 1,p 1 outputs need to live in the intersection span((1 (1, O, 0), (0. O, 1)) O span((1, O. 0), (0, 1, 0)) = span((1 (1, 0, 0)). which again limits the expressivity of our function. An example of this can be seen on the oxygen atom in Figure Ia
The discussion shows that local reflection symmetries lead to restrictions on the outputs of equivariant networks. This is a problem since we also need to place Gaussians outside the xz-plane to model the density faithfully.
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By augmenting GH, with a set of vectors fsk I GH. GH, U not cigenvectors to de. we get o, H, # HHH, and therefore break the symmetry and increase never threefold degenerate. such vectors always exist. Given an
that are
expressivity. Since reflections orthonormal coordinate system in Figure ((b), we can learn any vector with simple linear layer, including (sk Sko.......K and therefore break any local reflection symmetry.
E
Directional bias
As described in the main text 3.1 we observe that message passing in equivariant networks induces directional bias, see figure 2b While this is mainly an empirical observation, by examining the message construction in equivariant networks, we get an idea of the origin of this bias. Consider the vector (1 1) valued messages mij.e from node node i with channel dimension c, displacement vector between nodes Tıy. neural network fi, (-) and node features fu.jj.i
1=1 mij.c
fi,=o((ri)))
(u, Fij + f1,=1 DM, + Higher body terms
(36)
Scalars=18e
randomly initialized
= Sc
Tt
+random directions,
(37)
observed hias direction
So in each layer, we add the displacement vectors while everything else is randomly initialized, leading to the observed bias.
F Density construction details
Scalar factors. As a first step, we use the scalar features together with the node embedding fş to construct an input for three different MLPs: Simp A.j [f,, S1 A.j. S: Ss.. The MLPs then
predict the Gaussian mixture weights WA.j together with two other sets of scalars to use in the mean and covariance predictions:
SPAJ E R MLP P (Simp A.3):
SmAj E Rd = MLP (Simp A.J
(38)
WAJ R MLP w (Sim
Gaussian positions. ELECTRA places Gaussians (i.e., predicts the mean positions (A,)) equiv- ariantly by using the 1 outputs V1.A,J: V2 V3,.5 and the position scaling factors 8p of the framework displacement vectors to the atomic positions:
AAj FA + exp (Spa V1A.j
(39)
A2 V24.j + Sps V3AJ
Therefore, each Gaussian associated with position equal to the position r. of the atom originates from, plus three displacement vectors multiplied by scaling factors. We transform the scaling factors in different ways (exponential, square, and identity) to provide different seales of position displacement, thereby aiming capture different levels of detail of the output density with each readout head.
Density prediction using Gaussian mixture models. To construct the Gaussian's 's covariance matrices Σn., we calculate a weighted sum of the = 2 outputs M,, M2. M3. To ensure symmetry and positive semi-definiteness of the covariance matrix, we symmetrize the matrices by constructing the Gram matrices of M, i.e. using the transformation M MM For notational simplicity, we omit the A, subscript for all matrices and scalars in the equations below:
SG, S.g: SG, = softmax ([sm) Smy, Sma)
M,
M2M2
MaM3
(40)
Σ SG1
M, SGa
SGs
M
M Ir
The Gram matrices are normalized by the Frobenius norm the original output matrices to preserve the scale for the symmetrized matrices.
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G
Embedding function
To encode information about the electronic and nuclear properties of each atom into the scalar features, ELECTRA uses an embedding function similar to SpookyNer(Unke et al. 2021b), which uses nuclear and electronic embeddings encode information about the ground state electron configuration of each element. In ELECTRA, each atom's 's electronic and nuclear properties are encoded as [P, N, V, Ei,..... En, Fi,.... FrI, where P, N and are the numbers of protons, neutrons and valence electrons while E; and P; denote orbital occupancies and free-electron counts, respectively, for each orbital. multi-layer perceptron (MLP) maps the embedding space, Sinit MLP Emb (f), which is used as the initial scalar features.
H
Basis limitation experiment.
Errors are markedly higher on the MD dataset than on QM9 for ELECTRA and baselines. Specifically for ELECTRA, tested whether this reflects basis limit of our simplified Gaussian ansatz (Eq. ( Thus, for each molecule, we overfit an unconstrained Gaussian mixture with the same number of components ELECTRA, initializing means atom-center displacements. Using identical loss and optimizer, we trained for 100,000 steps on single snapshot per molecule. As Table 2 shows, this overfit GMM achieves much lower errors, indicating that the gap is predominantly model error rather than limitation of the Gaussian basis. Given GMM non-convexity and local minima, these fits are conservative lower bound on basis expressivity, and we expect that longer or stabilized schedules could do better, and the results are consistent with the theoretical universality of Gaussian mixtures in L (RR CCalcaterra Boldt 2008
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I
Hyperparameter table
Summarized in Table beloo are standard hyperparameters used for all experiments.
Table 3: Hyperparameters for the main ELECTRA model.
Hyperparameter
Value
Graph radius cutoff
8.0 A
HotPP body layers
Gaussians per electron (M.)
90
Graph network channel width
750
GNN Lmaz Body layers
GNN Imaz Head layers and inference
2
Body order maaa))
Precision
Float32
Optimizer
Adam (Default parameters) (Kingma
2014
Weight decay
Initial learning rate LRiiitiall
3.5 x 10-5
Learning rate scheduler
Linear (LR LRanitial X Epoch
Learning rate gamma (t)
0.85
Ablation studies
ablate the main ELECTRA model in five different ways. Not allowing for negative Gaussians ("No negative Gaussians" in the Table) means that the weights WA,j in Equation are restricted to being positive. Removal of scaling ("No scaling") means removing the predicted scaling factors from the displacement vectors in Equation 39 and from the weighted sum of symmetrized covariance matrices in Equation 40 Removal of the debiasing layers is seff-explanatory. while removal the symmetry-breaking mechanism ("No conmetry--reakingg means that we do not initialize the S Finally, removal of floating orbitals ("No floating orbitals") means that the displacement vectors of Equation 33 are set to zero such that for the the mean position of the j'th Gaussian of the A'th atom, MA..j r.. Le., all Gaussians become atom-centered.
In Table 4
features of the GNN with the moment of inertia eigenvectors
described in
Table Ablation studies of ELECTRA.
Model
NMAE [%] 1
ELECTRA
0.176
ELECTRA No negative Gaussians
2.027
ELECTRA No scaling
0.356
ELECTRA No debiasing layers
0,584
ELECTRA No debiasing layers and no symmetry-breakien
0.705
ELECTRA No floating orbitals
6.069
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K QM9 DFT initialization experiment parameters
The parameters used initialize fresh DFT runs with without initial density follow the original parameters used by Jørgensen & Bhhomik (022 to create the QM9 dataset. They are reported in [
Table 5: VASP parameters used to reproduce the DFT initialization experiment on QM9. Baseline uses ACHARG=2 (atomic superposition). ELECTRA-initiaaizee runs use ICHARG=I to read predicted CHGCAR the starting density; all other tags are identical.
Setting (shorthand)
Value
Exchange-corrllation functional (XC)
PBE
Job start mode (ISTART)
Electronic minimization algorithm (ALGO)
Normal
Charge density initialization (ICHARG)
Spin treatment (ISPIN)
Max electronic steps (NELM)
180
Initial non-SCF steps (NELMDL)
Symmetry usage (IS YM) Correction output (LCORR)
0
TRUE
Ionic time step (POTIM)
0.1
Min electronic steps (NELMIN)
k-point mesh (KPOINTS) Smearing method (ISMEAR)
1x1x1
0
Electronic conv, criterion (EDIFF)
1.0 X 10-6
Smearing width (SIGMA)
0.1
lonic steps (NSW)
Subspace diagonalization (LDIAG)
TRUE
Real-space projectors (LREAL)
Auto
Write wavefunctions (LWAVE)
FALSE
Write charge density (LCHARG)
TRUE
Plane-wave cutoff (ENCUT)
400
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L
Broader impact
This paper and the ELECTRA model presented in it can dramatically speed up the prediction of charge densities and can potentially be applied to model other 3D densities. Charge density prediction foundational problem computational chemistry, and by accelerating it our approach can advance scientific discovery, though its impact positive or negative depends on how it's applied. The greatest benefit in materials design and drug-discovery workflows that address real-world challenges in e.g. green transition or medicine.
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Reviews:
[{"review_id": "jOcExjkeDZ", "reviewer": "Reviewer_qQZb", "summary": "The paper introduces ELECTRA, a model for predicting electronic charge density using floating orbitals, a concept rooted in quantum chemistry but adapted in a data-driven manner via an Equivariant backbone neural network. Incorporates a symmetry-breaking mechanism using eigenvectors of the moment of inertia tensor to enhance expressivity while preserving rotational equivariance, addressing limitations of highly symmetric inputs. Compared to prior work (e.g., Fu et al., 2024; Rackers et al., 2023), ELECTRA’s use of floating orbitals and symmetry-breaking is distinct, as it avoids reliance on fixed basis sets or bond-centered orbitals, offering a fully data-driven solution (Page 9). This is a significant departure from traditional DFT methods and other ML-based approaches like SCDP or Charge3Net, which often use atom-centered or spherical harmonic representations.", "questions": "Can you also address some examples of molecules where this approach fails or has less advantages?", "limitations": "N/A", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths - Incorporating floating orbitals with an Equivariant backbone neural network is novel. \nWeaknesses - The author should also address when this would fail in certain cases.", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for recognizing the novelty and strong performance of ELECTRA. Below, we address the question of whether our model struggles with some molecules.\n\n>**_\"Can you also address some examples of molecules where this approach fails or has less advantages?\"_**\n\nThank you for the question!\nWe were looking through the samples of the QM9 test dataset with the largest errors. The highest NMAE error is 1.43 %. For comparison, the conventional SAD (superposition of atomic densities) guess, which DFT is usually initialized with, is around 13% on average. The molecule (C3H5N3O2) is a significant outlier containing an Oxadiazole-type structure and an NH3 side group. These two motifs have heavily distorted symmetry in the charge density and are rare and underrepresented in the data. The error then drops quickly for the next worst molecules. We did not find any particular pattern between these bad performing molecules apart from having some motives that are rare in the train set. \nGenerally, the error is higher around heavier atoms, which is also expected, as there are fewer heavy atoms in the train set, and the electron density is more complex around them. Overall, the model seems robust to most molecules in the QM9 dataset, which tests for molecule generalization, and the MD dataset, which tests generalization to conformal changes. \n\nSome current limitations of our model from a practical point are that it is only trained on neutral, organic molecules. Expanding this to more complicated settings, such as charged systems or highly correlated and diffuse systems, will require extra work. We see this as orthogonal to the subject of our paper, which tries to fundamentally explore charge density representation using floating orbitals. There is nothing in principle preventing us from training on different kinds of systems (and the same limitations hold for competing published methods as well).\n\nOne potential downside of our approach specifically, is that chemists have built up a lot of intuition to reason about spherical harmonics-based orbitals. Our gaussians are less interpretable, which can be a downside in some situations.\n\nWe hope this answers your question. If not, can you tell us more about the kind of disadvantages you were thinking about?\n\n\n---\n\nMore generally, could you please let us know if there are parts of the paper that are still unclear or would benefit from more polishing to warrant an increased score? Considering that our model beats the current state-of-the-art, which was published at last year's Neurips by wide margins (10x faster with better accuracy and trained with less memory), we do not understand where we can improve more without more detailed feedback."}, {"review_id": "08Pc8d913K", "reviewer": "Reviewer_9wy9", "summary": "This work proposes predicting electronic charge density using floating orbitals inspired by Gaussian splatting. By leveraging an equivariant neural network architecture, symmetry-breaking initialization, and debiasing layers, the model aims to predict charge densities directly from molecular structures.", "questions": "1.\tInterpretation of Figures 1 and 2: What is the frame or vector space represented in these figures? Specifically, are the vectors shown in Figures 1a and 1c the predicted mean positions $\\boldsymbol{\\mu}_{A,j}$ in Equation (6)? A detailed explanation of what each vector represents in these figures would greatly improve clarity.\n2.\tGPU Memory Requirements: How much GPU memory was used during training and inference? Are there specific trade-offs in inference performance compared to prior work?\n3.\tChannel Dimension Limitation: Line 184 specifies $N_c = 8 \\cdot M_e$. Does this imply that the model can only handle atoms with fewer than 8 valence electrons? How generalizable is the model to elements beyond the first two rows of the periodic table?\n4.\tDensity Grid Sampling: How did you handle sampling of the density during training? Did you employ specific sampling algorithms over the reference density grids in the dataset?\n5.\tPractical Use Cases: Could you elaborate on whether ELECTRA’s predicted electron densities can improve computational physics simulations, such as providing initial guesses for SCF iterations or being directly used in orbital-free DFT calculations?\n\n---\nupdate Quality 1 -> 2, Clarity 1 -> 2", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- Utilizing a Gaussian splatting-inspired idea for electron density prediction is clever and leads to superior performance and significantly reduced inference time.\n\nWeaknesses:\n- The manuscript makes a major assumption regarding the rotational invariance of the electron density, which is conceptually flawed.\n  - If the charge density were strictly rotationally invariant, then $\\rho(\\mathbf{r}_1) = \\rho(\\mathbf{r}_2)$ would hold for all vectors with the same magnitude, i.e., $\\|\\mathbf{r}_1\\|=\\|\\mathbf{r}_2\\|$. This is clearly not the case for most molecular charge densities.\n  - In my view, the authors appear to confuse invariance of the physical quantity under rigid-body rotation of the molecule with the concept of group action on function spaces (regular representations). For example, as in [1] (see Equation 2), a rotated density transforms as $\\rho(\\mathbf{R} \\mathbf{r}) = (T_{\\mathbf{R}} \\cdot \\rho)(\\mathbf{r})$, which is not invariance.\n- A more detailed introduction to Cartesian tensors (e.g., covariances) is necessary, especially given that the topic lies at the intersection of quantum chemistry and machine learning. Many ML researchers may not be familiar with the geometric tensor concepts; the background and appendices do not sufficiently support understanding.\n- While using Cartesian tensor-based basis functions is an interesting idea, the manuscript does not provide adequate justification for why decomposing functions in this way is natural. Additional explanations comparing conventional basis choices—such as Gaussian-type orbitals, Slater-type orbitals, or plane-wave basis functions—to Cartesian basis functions would greatly aid comprehension.\n- The statement that “Gaussians are equivalent to traditional Cartesian orbital functions with angular quantum number $l=2$, an extra nonlinearity, and simplified radial dependency” is unconvincing. The relationship between spherical harmonics-based basis functions and Cartesian functions is not clearly described. Moreover, while the manuscript refers to higher-order tensors (e.g., vectors for dipoles or matrices for polarizabilities), the connection between geometric tensor rank and quantum angular momentum is fundamentally different and requires clarification.\n  - Specifically, the relationship between Equation (15) and Equation (16) is ambiguous. There is no explanation of why the angular quantum number $l$ in both equations should be interpreted identically.\n\n\nTypos and Minor Issues:\n- Equation (2): $\\mathbf{r}_i$ is not defined.\n- Inconsistent notation between \\mathbf{} and \\boldsymbol{} for vectors, e.g., $\\mathbf{f}$ vs. $\\boldsymbol{f}$, and $\\mathbf{v}$ vs. $\\boldsymbol{v}$.\n- Duplicated results for InfGCN in Table 1.\n- Line 192: Inconsistent use of “figure” and “fig”; consistent referencing with \\ref or \\Cref in LaTeX is recommended.\n- Equation (11): The definition of $\\hat{\\cdot}$ is missing.\n- Line 208: The notation k=1..N is informal; it should be $k=1,\\ldots,N$.\n- Line 538: Inconsistent use of $\\top$ and $T$ for transposition.\n- Line 555: Potential typo, where $g$ should be $G$.\n- Equation (18): $\\Phi_{l=2}$ should be $\\Phi_{\\alpha, l=2}$ for clarity.", "was_revised": true, "final_justification": "First, I would like to commend the results on SCF cycle reduction—this is a remarkable contribution that significantly enhances the practical value of the work. I also appreciate the overall improvement in density estimation achieved by the proposed framework.\n\nHowever, I would like to clarify that some concepts are misused in the manuscript. Specifically, I find the use of the term invariant in the context of electron density somewhat awkward—even when conditioned on the molecular configuration. In group theory, invariance under a group action is typically defined as $f(Rx) = f(x)$. Given that the motivation and architecture of your work are grounded in handling symmetries, I recommend using group-theoretic terminology more deliberately and precisely. A more rigorous explanation—supported by clear definitions and formal justifications—of how symmetry is handled in your framework would substantially strengthen the paper.\n\nRegarding the use of Cartesian tensors, I appreciate the references you provided, which helped me better understand your intentions. However, I still feel that further explanation of how Cartesian tensors are used for representing electron density is needed in the manuscript. Additionally, there appears to be some confusion between geometrical tensor rank (ex. l=1 for force, l=2 for stress 3x3 matrice) and angular momentum quantum numbers, particularly in Equation (5) and lines 140–141. While I agree that using CGTOs (Cartesian Gaussian Type Orbital) makes sense, the proposed model “HotPP” does not appear to be a Cartesian tensor in the conventional sense, but rather a tensor used in mathematical physics—often referred to as a geometrical tensor. I strongly encourage the authors to improve the consistency and rigor of their mathematical formulations. Despite the strong empirical performance, such conceptual imprecisions may mislead readers and weaken the scientific impact of the work.\n\nFinally, regarding the reported reduction in SCF steps, to the best of my knowledge, this metric is sensitive to various hyperparameter choices—such as the functional, pseudopotentials, and the integration grid level. I hope that these choices were adequately and fairly reported in future manuscripts.\n\nOther aspects of the rebuttal addressed my concerns well. While I am hesitant to recommend outright acceptance at this stage, I believe this work is both timely and promising. With improved **mathematical rigor around equivariance**, **clearer distinction between geometric tensors and Cartesian basis functions**, **consistent notation**, and **careful correction of minor issues and typos**, this paper has the potential to make a significant contribution to the ML community. Accordingly, I have raised my score.", "rebuttal": "We thank the reviewer for the thorough review. We believe a notational oversight may have given the wrong impression about symmetry properties, which led to the primary concern raised. We address this and all other concerns below.\n\n>**_\"The manuscript makes a major assumption regarding the rotational invariance of the electron density, which is conceptually flawed.\n    - If the charge density were strictly rotationally invariant...\"_**\n\nWe apologize for the confusion. You are right that there was an error in our notation in equation 4. It should have been \n$$\\rho(Rr | \\{R r_\\text{nuclei,1},..., R r_\\text{nuclei,N}\\}) = \\rho(r | \\{r_\\text{nuclei,1},...,r_\\text{nuclei,N}\\})\n$$\nIn other words, the electron density is invariant under simultaneous rotation of the molecule and electron/point of evaluation.\n\nThus, we want to stress that this is not a conceptual flaw, but simply a **notation mistake**.\n\nThe construction of our model encodes the correct symmetries both in theory and in practice. This is also evidenced by the state-of-the-art accuracy of our model, which would not have been possible if we had assumed the wrong transformation properties and thereby erroneously overconstrained the model.\n\n>**_A more detailed introduction to Cartesian tensors (e.g., covariances) is necessary, ..._**\n\nThank you for the feedback. We will expand on the background knowledge in the appendix. Due to the character limit, we unfortunately can't post the section here.\n\n\n>**_While using Cartesian tensor-based basis functions is an interesting idea, the manuscript does not provide adequate justification for why decomposing functions in this way is natural. Additional explanations comparing conventional basis choices—such as Gaussian-type orbitals, Slater-type orbitals, or plane-wave basis functions—to Cartesian basis functions would greatly aid comprehension._**\n\nYou are right that more background will benefit the understanding, as the different basis choices will be less known in the ML community. Please let us know if the following helps.\n\nCartesian GTOs, as written in equation (16), are a standard choice of basis functions in quantum chemistry [1], implemented in most major quantum chemistry codes, such as PySCF (see the pyscf.gto package) and QChem (See the PURECART function of QCHEM). They are tightly related to spherical GTOs (we will elaborate on this in the answer to the next question). Both Cartesian and spherical GTOs mostly share the same benefits and challenges, with spherical GTOs being preferred recently due to higher numerical stability and compactness. The plane wave basis is preferred for solids but also used for molecules due to their simplicity, numerical stability, and absence of basis-set superposition errors.\n\nIn the context of ML models, Cartesian basis functions are natural for the same reason spherical harmonics-based basis functions are natural: The known transformation of the angular part allows for the construction of rotation equivariant architectures. This is also the reason that many equivariant graph neural networks, e.g. TensorNet[2] and indeed HotPP[3] (the backbone for ELECTRA), use Cartesian tensors as the basis for the equivariant operations.\n\nThe justification for our choice of using Gaussians is that the reduced complexity makes the model faster, while the connection to Cartesian basis functions allows us to enforce the symmetry properties. \n \nWe will add a paragraph based on the discussion above to the paper to better motivate our choices.\n\n>**_... the connection between geometric tensor rank and quantum angular momentum is fundamentally different and requires clarification.\nSpecifically, the relationship between Equation (15) and Equation (16) is ambiguous. There is no explanation of why the angular quantum number in both equations should be interpreted identically._**\n\nThank you for the feedback. We agree that the link between tensor rank and angular momentum is less known in the ML community and will provide more background:\nWe can write any solid spherical harmonic as a linear combination of monomials in the following way (see for example, [4], equation 2):\n$$\nr^lY_{l,m}(\\theta, \\phi) = \\sum_{n_x + n_y + n_z = l} C_{n_x,n_y,n_z}^{l,m} x^{n_x} y^{n_y} z^{n_z}\n$$\nSimilarly, the traceless symmetric component of the Cartesian tensor of rank $l$ can be spanned by solid harmonics of degree $l$.\n\nTherefore, the angular momentum quantum number $l$ and $n_x + n_y + n_z = l$ are indeed very related and also both commonly called \"angular momentum\" in the quantum chemistry literature.\n\nThe equation above is unfortunately somewhat involved to show, so in our paper, we will refer the interested reader to [5].\n\n>**_Typos and Minor Issues:..._**\n\nThank you for the in-depth feedback. We will address all notation issues and typos and make sure to re-check for any potential typos we missed!\n\n>**_Interpretation of Figures 1 and 2: What is the frame or vector space represented in these figures? Specifically, are the vectors shown in Figures 1a and 1c the predicted mean positions in Equation (6)? A detailed explanation of what each vector represents in these figures would greatly improve clarity._**\n \nYes, these are the vector-valued ($l=1$) outputs which we use, after multiplication with predicted scalar factors, as our mean predictions in equation (6). We will add this to the caption.\n\n>**_GPU Memory Requirements: How much GPU memory was used during training and inference? Are there specific trade-offs in inference performance compared to prior work?_**\n\nEverything (both training and inference) runs on a single NVIDIA RTX 3090 with 24GB, with batch size 1, see line 285.  For comparison, SCDP was trained on 4xNVIDIA A100 80 GB with a batch size of 4. There are no specific trade-offs we are aware of; on our tests, ELECTRA is consistently both much faster and more accurate compared to competing methods.\n\n\n>**_Channel Dimension Limitation: Line 184 specifies $N_c = 8 \\cdot M_e$. Does this imply that the model can only handle atoms with fewer than 8 valence electrons? How generalizable is the model to elements beyond the first two rows of the periodic table?_**\n\nThis is not a fundamental limitation. $N_c$ is a hyperparameter, and $N_c = 8 \\cdot M_e$ was just a heuristic choice to allocate more expressivity to atoms with more valence electrons. This is analogous to traditional basis sets allocating more basis functions to heavier atoms. However, nothing prevents us from breaking this rule of thumb. If we were to include new atom types, we would of course have to retrain the network using different parameters, but this is the same for any atomistic model, not just density prediction models.\n\n>**_Density Grid Sampling: How did you handle sampling of the density during training? Did you employ specific sampling algorithms over the reference density grids in the dataset?_**\n\nThank you for the question! We did not sample. We will clarify this by adding the following to our paper:\n_\"ELECTRA was trained on the standard QM9 benchmarking dataset using the full grid for both training and inference.\"_\n\n\n>**_Practical Use Cases: Could you elaborate on whether ELECTRA’s predicted electron densities can improve computational physics simulations, such as providing initial guesses for SCF iterations or being directly used in orbital-free DFT calculations?_**\n\nYes, they can. A density given on a regular grid can be efficiently transformed into plane wave basis using fast Fourier transformation to initialize SCF cycles. This follows previous work, for example, ChargE3Net [4], and is also how plane wave codes conventionally initialize their calculations from SAD guesses. This is straightforward in practice; in the VASP code base, you can provide custom initializations provided on the grid using the ICHARG=1 setting. (we are not allowed to post links to the documentation, but it can easily be found)\n\nTo demonstrate this point, we used the density predictions of ELECTRA to initialize DFT calculations for our QM9 test molecules and achieved an **average reduction of 50.72 %SCF cycles**  compared to conventional initialization. We will add these numbers and a better description of how this is done to our paper to demonstrate practicality! \n\nOrbital-free DFT provides additional motivation as it should work in principle, but we did not try it out in practice.\n\n\n---\n\nWe hope we were able to address all concerns. Since the primary concern arose from a misunderstanding due to notational oversight, we hope that the reviewer agrees that this and other issues have been resolved.\nIn that regard, we would also like to add that we sincerely appreciate that the reviewer acknowledges the novelty and superior performance and efficiency of our approach. Thus, we hope that our responses have further reinforced this assessment and convinced the reviewer to revise their final recommendation accordingly.\nPlease let us know if there are remaining problems or improvements that could be addressed before the paper should be accepted. \n\n---\n\n[1] (Helgaker, Trygve, Poul Jørgensen, and Jeppe Olsen. 2000. Molecular Electronic-Structure Theory. John Wiley & Sons, Ltd. - Chichester. Section 9.2.1)\n\n[2] Simeon, G., & De Fabritiis, G. (2023). Tensornet: Cartesian tensor representations for efficient learning of molecular potentials. Advances in Neural Information Processing Systems, 36, 37334-37353.\n\n[3] Wang, J., Wang, Y., Zhang, H., Yang, Z., Liang, Z., Shi, J., ... & Sun, J. (2024). E (n)-Equivariant cartesian tensor message passing interatomic potential. Nature Communications, 15(1), 7607.\n\n[4] Koker, Teddy, et al. \"Higher-order equivariant neural networks for charge density prediction in materials.\" npj Computational Materials 10.1 (2024): 161.\n\n[5] Ribaldone, Chiara, and Jacques Kontak Desmarais. \"Spherical to Cartesian Coordinates Transformation for Solid Harmonics Revisited.\" arXiv preprint arXiv:2412.16733 (2024)."}, {"review_id": "DEAjhVsuoZ", "reviewer": "Reviewer_amF4", "summary": "This paper introduces the Electronic Tensor Reconstruction Algorithm (ELECTRA), which uses floating orbitals to predict electronic charge densities. Inspired by the Gaussian Splatting representation, ELECTRA replaces costly traditional linear combinations of fixed basis functions with floating orbitals. Since designing such orbitals usually requires expert intuition, ELECTRA learns them in a purely data-driven manner, enabling high-resolution charge density prediction. Experiments on well-known benchmarks demonstrate its high accuracy.", "questions": "- Why does the method compute directions via eigenvectors? In highly symmetric cases, these eigenvectors are unstable, and noise can easily switch their order or reverse their orientation. How is this handled?\n- Debiasing is an important contribution, yet the origin of the bias itself is unclear. Which HotPP component introduces this bias? Understanding the root cause may be more valuable than an ad hoc debiasing fix.\n- This approach seems heavily tied to HotPP. While choosing an appropriate backbone is reasonable, what about other equivariant GNNs—or even backbones that ignore symmetry? Since the core idea is to break symmetry within an otherwise symmetric network, broader validation would strengthen the claim.", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The narrative is clearly explained. The paper introduces floating orbitals and justifies their data-driven prediction via machine learning, which makes the paper easy to follow.\n- The authors pinpoint the directional bias that arises in HotPP message passing and propose a dedicated Debiasing Layer to address it.\n- Experimental results show that ELECTRA achieves fast and accurate charge-density prediction.\n\nWeaknesses:\n- The model largely inherits purely the equivariance property from the existing HotPP framework.\n- L.224: \"However, we find, empirically, that this does not hinder strong performance.\" Since equivariance is essential to the method, substantial empirical or theoretical justification is necessary.\n- A key contribution of Gaussian Splatting is its tight link to efficient, differentiable rendering and optimization. However, ELECTRA does not exploit this property; it merely outputs a set of Gaussians. Alternative representations, such as point clouds or sparse voxels, should be considered. Alternatively, a loss function tailored to the Gaussian representation must be provided to justify this design choice.", "was_revised": true, "final_justification": "The authors addressed my concerns with clarity and rigor. I support the acceptance of the paper.", "rebuttal": "We thank the reviewer for thoughtful critique and clear summary of our contributions. While ELECTRA builds on the HotPP framework, its floating-orbital parameterization, symmetry-breaking initialization, and debiasing layers are what sets it apart from standard applications of equivariant models. Below, we address the concerns regarding these design choices and clarify any other issues.\n\n>**_The model largely inherits purely the equivariance property from the existing HotPP framework._**\n\nIt is very common for authors to repurpose backbone architectures for different tasks. Most previously published competing methods did so as well, most recently SCDP which was published in last year's NeurIPS proceedings[1]. For our purpose, the HotPP model is well-suited since it outputs Cartesian tensors of varying ranks, which have exactly the properties needed to construct our densities. Could you expand on why this is interpreted as a weakness of our method?\n\n>**_L.224: \"However, we find, empirically, that this does not hinder strong performance.\" Since equivariance is essential to the method, substantial empirical or theoretical justification is necessary._**\n\nThe empirical justification is provided in the form of the results themselves, which are state-of-the-art on both the QM9 dataset and the MD dataset. The MD dataset is specifically designed for testing model performance on molecular dynamics (MD) snapshots of the same molecules. These snapshots span changes in bond stretches, angle bends and torsional motions[2], and thus presents a benchmark where any weakness due to the above-mentioned assumption would be exposed. However, the results show the contrary; we see the most significant accuracy gap to the prior state-of-the-art on this dataset, and we achieve more than 2x better accuracy on all MD molecules. \n\nIf you have a specific experiment in mind that we should run we can try to do so.\n\n>**_Alternative representations, such as point clouds or sparse voxels, should be considered._**\n\nThank you for the suggestion, however we are not aware of any work that represent charge densities as point clouds or sparse voxels. If you have papers in mind that we should be comparing to please let us know the references and we will do so. \n\nThere are some theoretical reason to believe that these approaches would not work well though: Point clouds can not naturally represent a function $\\rho(\\vec{r}): ℝ^3 \\rightarrow ℝ$. If you mean to encode the atoms with point cloud methods we are unsure how this will be different from the equivariant network. Voxel based methods cannot be built $SO(3)$ equivariant as far as we know, which was shown by many works to be highly beneficial for molecular property prediction. Achieving the sub angstrom resolution neccessary would also be very expensive.\n\n>**_A key contribution of Gaussian Splatting is its tight link to efficient, differentiable rendering and optimization. However, ELECTRA does not exploit this property; it merely outputs a set of Gaussians. Alternatively, a loss function tailored to the Gaussian representation must be provided to justify this design choice._**\n\nThe justification for the choice of Gaussians is that their simplicity allows for very efficient evaluation, as shown by the order of magnitude speedup compared to previous approaches, analogously to the reasoning for Gaussians in Gaussian splatting.\nWhile we informally mention in the paper that our method is inspired by Gaussian Splatting, this is mainly in the sense of placing a large number of cheap-to-evaluate Gaussians in space to approximate an intricate 3D field. This is also why we did not explicitly call our method Gaussian splatting. \nHowever, regardless of the chosen nomenclature for our approach, the method is completely novel in the field of ML-driven density prediction, while achieving state-of-the-art accuracy and speed.\nHowever, if you have ideas for how the gaussian structure could be further exploited, we would be happy and interested to pursue such directions!\n\n\n>**_Why does the method compute directions via eigenvectors? In highly symmetric cases, these eigenvectors are unstable, and noise can easily switch their order or reverse their orientation. How is this handled?_**\n\nWe are using eigenvectors in two parts of the model; for the symmetry breaking and the debiasing module. We will therefore answer this question in two parts:\n\nFor symmetry breaking:\nOur goal is to learn symmetry-breaking, equivariant inputs. For this we need an equivariant local coordinate system. The moment of inertia provide exactly this equivariant orthogonal coordinate system and are for that reason widely used in molecular modeling, for example in stochastic frame averaging [3] and for equivariant wave functions [4]. These methods either learn to become robust to sign flips [3] or use extra canonicalizaton [4]. We choose to canonicalize, see equation 8. While this might not be mathematically elegant, as our results and also the cited methods show, it is a pragmatic solution that does work well in practice.\n\nFor debiasing:\nThe eigenvectors of the covariance matrix are the direction with the largest variances. By subtracting the largest eigenvector we can therefore learn to remove the direction with the most variance, which corresponds to the bias direction. \n\nThe covariance matrix is built from the learned vectors. \nBoth experimentally (see figure 2b) and due to the mathematical argument we provide for the next question below, we know that there is always a biased direction. The largest eigenvector will therefore never be degenerate and therefore not unstable to noise.\nWe also do not see any instabilities either during training nor validation on either of the datasets.\n\nOrientation flips are no problem since the eigenvector appears twice, see line 245.\n\n\n>**_Debiasing is an important contribution, yet the origin of the bias itself is unclear. Which HotPP component introduces this bias? Understanding the root cause may be more valuable than an ad hoc debiasing fix._**\n\nSince submission we have identified the mathematical origin of the bias:\nConsider the vector ($l=1$) valued messages $m_{ij,c}^{l=1}$ from node $j$ to $i$ with channel dimension $c$, displacement vector between nodes $\\vec{r}_{ij}$, neural network $f\\_{l\\_r}()$ and node features $h\\_{l\\_r,j,c}$:\n\n$$\n\\begin{aligned}\nm\\_{ij,c}^{l=1} &= \\underbrace{f\\_{l\\_r=0}(|r\\_{ij}|) h\\_{l\\_r=0,j,c}}\\_{\\text{Scalars} =: s_c} \\vec{r}\\_{ij} + \\underbrace{f\\_{l\\_r=1}(|r\\_{ij}|) h\\_{l\\_r=1,j,c} + \\text{Higher body terms}}\\_{\\text{randomly initialized}} \\\\\\\\\n&= s\\_c \\underbrace{\\vec{r}\\_{ij}}\\_{\\text{Observed bias direction}} + \\text{random directions}\\_c \n\\end{aligned}\n$$\nSo in each layer we add the displacement vectors while everything else is randomly initialized, leading to the observed bias. We will add this explanation in the appendix.\nWe cannot just set $s=0$ and ignore the displacement vector since this vector is what carries the geometric information of the molecule. Therefore we still think that our debiasing solution is very sensible.\nNote also that other equivariant networks like TensorNet [2], Equiformer [5], MACE [6] etc. construct their messages in similar ways and would therefore lead to the same bias.\nPlease let us know if this explanation makes sense to you and we will include it in the paper. \n\n\n>**_This approach seems heavily tied to HotPP. While choosing an appropriate backbone is reasonable, what about other equivariant GNNs—or even backbones that ignore symmetry? Since the core idea is to break symmetry within an otherwise symmetric network, broader validation would strengthen the claim._**\n\nThe approach is not particularly tied to HotPP, as the main differentiating factor for ELECTRA is rather the way it interprets the Cartesian tensor outputs as the building blocks of a Gaussian-based density, as well as the necessary changes that make the task more tractable (symmetry-breaking, debiasing, etc.). This could in principle be done with other models, and we did initially try e.g. TensorNet [2] and Equiformer v2 [5] . However, these backbones faced the same issues as HotPP in terms of symmetry-breaking, and in the end we opted for HotPP as initial experiments suggested that it worked best. We will mention this in the paper.\n\n\n---\n\nWe hope we were able to address all concerns. Please let us know if there are open questions that could be addressed to recommend our paper for acceptance.\n\n---\n \n[1] Fu, Xiang, Andrew Rosen, Kyle Bystrom, Rui Wang, Albert Musaelian, Boris Kozinsky, Tess Smidt, and Tommi Jaakkola. \"A recipe for charge density prediction.\" Advances in Neural Information Processing Systems 37 (2024): 9727-9752.\n\n[2] Simeon, Guillem, and Gianni De Fabritiis. \"Tensornet: Cartesian tensor representations for efficient learning of molecular potentials.\" Advances in Neural Information Processing Systems 36 (2023): 37334-37353.\n\n[3] Duval, Alexandre Agm, et al. \"Faenet: Frame averaging equivariant gnn for materials modeling.\" International Conference on Machine Learning. PMLR, 2023.\n\n[4] Gao, Nicholas, and Stephan Günnemann. \"Ab-initio potential energy surfaces by pairing GNNs with neural wave functions.\" arXiv preprint arXiv:2110.05064 (2021).\n\n[5] Liao, Yi-Lun, et al. \"Equiformerv2: Improved equivariant transformer for scaling to higher-degree representations.\" arXiv preprint arXiv:2306.12059 (2023).\n\n\n[6] Batatia, Ilyes, et al. \"MACE: Higher order equivariant message passing neural networks for fast and accurate force fields.\" Advances in neural information processing systems 35 (2022): 11423-11436."}, {"review_id": "dKtQIlPpzs", "reviewer": "Reviewer_5buA", "summary": "The paper introduces the novel machine learning model ELECTRA for the prediction of the electron density in molecular systems. Contrary to previous approaches, the density is not represented in an atom-centered basis expressed in spherical harmonics, but through floating orbitals using ordinary Gaussian functions. The positions and covariance matrices of these floating orbitals are learned by a neural network alongside the corresponding coefficients. Since the neural network is equivariant ELECTRA also introduces a symmetry breaking mechanism in order to allow for expressive densities also in molecular systems which exhibit a high degree of symmetry. The model achieves state of the art results, while being an order of magnitude faster than models with similar performance.", "questions": "- On page 5 it is mentioned that ELECTRA uses an atomic embedding function inspired by SpookyNet, but I could not find any details on the exact implementation. Could the authors provide details on this in the Appendix?\n- I would be interested in how well a Gaussian mixture model with the same number of Gaussians as in ELECTRA is able to approximate the target densities in principle in order to know how much of the density error can be attributed to the choice of basis and how much to the limited accuracy of the model.\n- If possible I would like to see a comparison to the SCDP+BO (L = 6) model on the MD dataset.\n\nI am willing to increase my score if the above points are adequately addressed.", "limitations": "Limitations are addressed. Not being able to directly use the density prediction in conventional Kohn-Sham SCF procedures is another point which might still be added.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "- Strengths:\n    - The paper is written in a clear and understandable fashion. I also found the blue boxes highlighting key takeaways helpful.\n    - The model shows very good results in particular with respect to inference speed at comparable accuracy.\n- Weaknesses\n    - Although the need for a symmetry breaking mechanism is convincingly demonstrated by the conducted ablation studies, I am not sure about the claim that this is due to local symmetries within the molecules. The mathematical arguments provided in Appendix C are perfectly fine for global symmetries of molecules, but break down for local symmetries in a message passing framework where messages are propagated beyond the locally symmetric regions.  The problem I see is that in the proof it is assumed that the output features of nodes within a sub graph depend only on input features within the same sub graph. For a general message passing network this is however not the case.\n    - The final densities are constructed using the ReLU function to ensure positivity everywhere. However this can result in regions of unphysical densities where they are constant 0. This could reduce the performance in downstream tasks, but might also be problematic during training, e.g. when the position of a Gaussian is within such a region of vanishing density. In such a case also the gradient signal on the parameters of the respective Gaussian would become very small.\n    - In the comparison of evaluation metrics on the MD dataset, ELECTRA is compared only to the fastest SCDP model, but given the results from the QM9 dataset I would expect the SCDP+BO (L = 6) model to perform much closer (maybe even better) compared to ELECTRA.\n    - Since the densities are not represented in a LCAO basis the resulting densities cannot straightforwardly be used as initial guess in conventional Kohn Sham DFT which is one of the most common use cases for machine learned electron densities for molecular systems.", "was_revised": true, "final_justification": "During the rebuttal the authors addressed all my remaining concerns. I think that exploring alternative density representations besides atom-centered bases in machine learning, is a relevant and promising research topic. This paper makes valuable contributions and presents noteworthy findings in that context.", "rebuttal": "We thank the reviewer for their thoughtful and thorough feedback. Below, we address all raised concerns, and provide new results which we hope further reinforces the reviewer's favorable opinion of our work. This includes showing that ELECTRA predictions can indeed be used as initial guesses, as well as a test of the limitations of the choice of basis, as suggested. \n\n>**_Although the need for a symmetry breaking mechanism is convincingly demonstrated by the conducted ablation studies, I am not sure about the claim that this is due to local symmetries within the molecules. The mathematical arguments provided in Appendix C are perfectly fine for global symmetries of molecules, but break down for local symmetries in a message passing framework where messages are propagated beyond the locally symmetric regions. The problem I see is that in the proof it is assumed that the output features of nodes within a sub graph depend only on input features within the same sub graph. For a general message passing network this is however not the case._**\n\nThank you for taking the time to read our proof in depth! We understand your concern, but the argument still works if we define the symmetric subgraph as the $n$-hop neighborhood of the node, where $n$ is the number of layers. We can upper bound this neighborhood with $n$*cutoff_radius. In that case, no symmetry breaking information from outside the sub-graph would reach the considered node. \nIn practice, we did observe problematic behavior even for not completely symmetric graphs, which would need a refined theory to explain fully rigorously. We think our analysis is a good starting point, though: If we perturb a symmetric input graph to be slightly unsymmetric, we know by the smoothness of the neural network that the output would still be close to the completely restricted case. The network would need to learn a very large Lipschitz constant to \"fight\" these constraints, which is known to be detrimental to model performance.\nPlease let us know if this argument makes sense to you, and we will include it in the appendix.\n\nAs a side note: Many ground state geometries are also globally symmetric, so even if we disregard local symmetries, the argument is still valid.\n\n>**_The final densities are constructed using the ReLU function to ensure positivity everywhere. However this can result in regions of unphysical densities where they are constant 0. This could reduce the performance in downstream tasks, but might also be problematic during training, e.g. when the position of a Gaussian is within such a region of vanishing density. In such a case also the gradient signal on the parameters of the respective Gaussian would become very small._**\n\nWe had the same concern about the training signal and actually tried training both with and without the ReLU, but did not see any difference. We think this is because even if part of the predicted density is zeroed out during training, the Gaussians have support everywhere and therefore still have training signals from all the other points.\nMany other charge density models must contend with the same issue, namely that ML models are not always constrained to predict positive densities, and so choosing whether to use ReLU is essentially choosing whether to risk areas with negative density or areas with zeroed-out densities (as you mention above). \nHowever, this did not matter much in practice, since even without the ReLU the model learns to never predict negative densities, and we only added the ReLU as a safety mechanism.\n\n\n>**_In the comparison of evaluation metrics on the MD dataset, ELECTRA is compared only to the fastest SCDP model, but given the results from the QM9 dataset I would expect the SCDP+BO (L = 6) model to perform much closer (maybe even better) compared to ELECTRA._**\n>**_If possible I would like to see a comparison to the SCDP+BO (L = 6) model on the MD dataset._**\n\nUnfortunately, we could not find the data preparation scripts for running SCDP on the MD dataset, which makes reproduction difficult. However, the choice of model was made by the SCDP authors themselves, and we merely reference their reported results. Note that the SCDP model used for these experiments is **not** the fastest model. While they use a relatively low K=4 (number of graph layers) and L=3 value (tensor order), they also use an intermediate beta value of beta=1.5 (which determines the basis set size), and importantly, they **do** use the extra bond-centered orbitals and a fine-tuning step. Inclusion of the bond-centered orbitals is the model choice that gives SCDP the largest relative performance gain according to their own experiments (see e.g. Figure 3 in the SCDP paper[1]). As the parameters used are not identical to any set used for the SCDP QM9 experiments it appears that the parameters were optimized by the authors.\n\n\n>**_Since the densities are not represented in a LCAO basis the resulting densities cannot straightforwardly be used as initial guess in conventional Kohn Sham DFT which is one of the most common use cases for machine learned electron densities for molecular systems._**\n\nWe actually can use them as an initial guess! A density given on a regular grid can be efficiently transformed into plane wave basis using fast Fourier transformation to initialize SCF cycles. This follows previous work, for example, ChargE3Net [2], and is also how plane wave codes conventionally initialize their calculations from SAD guesses. This is straightforward in practice; in the VASP code base, you can provide custom initializations provided on the grid using the ICHARG=1 setting. (we are not allowed to post links to the documentation, but it can be easily found)\n\nTo demonstrate this point, we used the density predictions of ELECTRA to initialize DFT calculations for our QM9 test molecules and achieved an **average reduction of 50.72 % SCF cycles**  compared to conventional initialization. We will add these numbers and a better description how this is done to our paper to demonstrate practicality! \n\n\n>**_On page 5 it is mentioned that ELECTRA uses an atomic embedding function inspired by SpookyNet, but I could not find any details on the exact implementation. Could the authors provide details on this in the Appendix?_**\n\nWe will add a small paragraph in the appendix on the model details. We reference here a draft of this paragraph:\n\n*“To encode information about the electronic and nuclear properties of each atom into the scalar features, ELECTRA uses an embedding function similar to SpookyNet. Each atom’s electronic and nuclear properties are encoded as $\\mathbf{f}=\\left[P, N, V, E_1, \\ldots, E_n, F_1, \\ldots, F_{\\mathrm{n}}\\right]$, where $P$, $N$ and $V$ are the numbers of protons, neutrons and valence electrons while $E_{i}$ and $F_{i}$ denote orbital occupancies and free-electron counts, respectively, for each orbital. \nA multi-layer perceptron (MLP) maps $\\mathbf{f}$ to the embedding space, $s_{init} = \\mathrm{MLP_{Emb}}(\\mathbf{f})$, which is used as the initial scalar features. “*\n\n>**_I would be interested in how well a Gaussian mixture model with the same number of Gaussians as in ELECTRA is able to approximate the target densities in principle in order to know how much of the density error can be attributed to the choice of basis and how much to the limited accuracy of the model._**\n\nThat is an interesting question! \nWe overfitted an unconstrained Gaussian mixture model with means parameterized as displacements from atoms (for better initialization) on a single snapshot from each of the MD molecules with the same number of Gaussians as ELECTRA used. We directly minimized the error using the same loss function and optimizer as ELECTRA for 100,000 steps. The results are reported in the table below. Note that Gaussian mixture models are known to have many local minima; as such the presented result are only a lower bound on the expressivity of the Gaussian mixture model. Longer and more elaborate training (similar to all the tricks employed by Gaussian splatting) would likely result in better accuracy.\n\n\n| Molecule         | ELECTRA | N=1 Gaussian Overfit |\n| ---------------- | ------: | -------------------: |\n| MD-ethanol       |   1.02% |                0.18% |\n| MD-benzene       |   0.45% |                0.08% |\n| MD-phenol        |   0.56% |                0.08% |\n| MD-resorcinol    |   0.62% |                0.24% |\n| MD-ethane        |   0.91% |                0.28% |\n| MD-malonaldehyde |   0.80% |                0.16% |\n\n\nWe see that the majority of the error comes from the model, not from the basis choice.\nWe will include these experiments in the appendix of our paper\n\nFrom a theoretical point we also know that Gaussian mixture models are universal approximators in L²(ℝ³) (see e.g. [3] for a proof and discussion of this universal approximation result). \n\n---\n\nWe thank the reviewer for generally acknowledging the strength of our method, and we hope that our responses addressed all the concerns. Please let us know if there are any remaining questions.\n\n---\n\n[1] Fu X, Rosen A, Bystrom K, Wang R, Musaelian A, Kozinsky B, Smidt T, Jaakkola T. A recipe for charge density prediction. Advances in Neural Information Processing Systems. 2024 Dec 16;37:9727-52.\n\n[2] Koker, Teddy, et al. \"Higher-order equivariant neural networks for charge density prediction in materials.\" npj Computational Materials 10.1 (2024): 161.\n\n[3] Calcaterra, Craig, and Axel Boldt. \"Approximating with gaussians.\" arXiv preprint arXiv:0805.3795 (2008)."}]

Author rebuttal:

Meta-review: This paper introduces a model for electron charge density prediction that leverages floating basis functions, i.e., Gaussian mixture with non-linearities. The approach demonstrates significant improvements in both predictive accuracy and computational efficiency. In particular, the paper demonstrates practical utility through reductions in SCF iterations . 

The reviewers generally agreed on the novelty and strength of the contributions, especially the data-driven treatment of floating orbitals and the demonstrated efficiency. While some concerns were raised about conceptual precision in symmetry arguments and notational inconsistencies, the authors addressed these points during the rebuttal.

Overall, I believe the paper is timely and impactful, hence recommend acceptance.
Decision: Accept (spotlight)