---
title: 'X-MACE: Transferable Excited-State Modeling'
url: https://www.emergentmind.com/topics/transferable-excited-state-modeling-x-mace
type: topic
---

# X-MACE: Transferable Excited-State Modeling

Transferable Excited-State Modeling (X-MACE) encompasses a set of computational frameworks that enable high-fidelity, data-efficient, and transferable simulations of excited-state potential energy surfaces (PESs) and their intersections in molecular systems. X-MACE refers to both a family of neural network architectures—rooted in message-passing atomic cluster expansion (MACE) and its variants—as well as transferable excited-state quantum Monte Carlo (QMC) methods that collectively address the challenges of excited-state modeling: smooth interpolation across conical intersections, chemical transferability, scalability to large environments, and efficient reuse of ground-state knowledge [2502.12870, 2503.19847, 2502.21045, 2108.04879].

## 1. Theoretical Foundations and Core Models

Transferable excited-state modeling with X-MACE is grounded in the Born–Oppenheimer approximation, where the molecular Hamiltonian at a fixed nuclear geometry $R$ admits stationary electronic eigenstates:
\[
\hat{H}(R)\,\Psi^i(r|R) = E^i(R)\,\Psi^i(r|R)
\]
with
\[
\hat{H}(R) = -\frac{1}{2}\sum_{i}\nabla_i^2 - \sum_{i,I}\frac{Z_I}{|r_i-R_I|} + \sum_{i<j}\frac{1}{|r_i-r_j|}
\]
The archetypal quantum Monte Carlo instantiation employs a parameterized Slater–Jastrow form:
\[
\Psi_\theta(r|R) = e^{J_\theta(r,R)}\sum_{k=1}^K \det[A_\theta^{(k)}(r,R)]
\]
The loss functional at a given geometry incorporates the Rayleigh quotient, spin penalties, and orthogonality constraints to hierarchically optimize $N_s$ target states:
\[
\mathcal{L}_R(\Psi_{\theta,1}, \ldots, \Psi_{\theta,N_s}) = \sum_{i=1}^{N_s}\Big[ E_\theta^i(R) + \beta\,\langle S^2 \rangle_\theta^i(R) + \sum_{j < i} \alpha_{ij} |\langle \Psi_{\theta,i}(R)|\Psi_{\theta,j}(R)\rangle|^2 \Big]
\]
Geometric and electronic state transferability is realized by sharing parameters $\theta$ across all sampled $R\in\mathcal{R}$, while dynamic state ordering ensures adiabatic surface continuity through conical intersections [2503.19847].

X-MACE as machine-learned PES architecture extends the MACE formalism. Each molecule is represented as a graph $G=(V,E)$ of atom positions and atomic numbers. Local atomic environments are encoded by equivariant message passing, culminating in atom-wise features $h_i^{(T)}$, which were classically mapped to adiabatic state energies. X-MACE, however, deploys DeepSets to obtain permutation-invariant latent descriptors $I$, which are decoded back to energies through a learned Hermitian "companion" matrix. This ensures smoothness even through conical intersections [2502.12870].

## 2. Treatment of Conical Intersections and Nonadiabatic Couplings

Conical intersections, regions where two or more adiabatic PESs become degenerate and exhibit cusp-like non-smoothness, are crucial for simulating nonradiative photochemical transitions. Standard electronic structure and ML PES models struggle near these topological features. In X-MACE, the DeepSets latent invariant ensures the final decoded eigenvalues ($\hat{E}_i$) are smooth functions of geometry—even as adiabatic state labels permute. The reconstruction loss enforces
\[
L_{\mathrm{recon}} = \frac{1}{n}\sum_{i=1}^n (\hat{E}_i - E_i)^2
\]
By fitting not to adiabatic states but to permutation-invariant summarizations, cusps are replaced by smooth manifolds in the latent space [2502.12870].

For quantum Monte Carlo approaches, state continuity through intersections is maintained by dynamic ordering and overlap terms. The interstate overlap,
\[
O^i_{mn} = \langle \Psi_{\theta,i}(R_m) | \Psi_{\theta,i}(R_n) \rangle
\]
remains near unity on an adiabatic sheet, allowing for precise detection of true crossings.

Nonadiabatic couplings, essential for surface hopping and other nonadiabatic molecular dynamics, are treated in X-MACE using a smoothed observable:
\[
S_{mn} = C_{mn}\cdot (E_n - E_m), \qquad C_{mn} = \langle \psi_m | \nabla_R \psi_n \rangle \approx \frac{\langle \psi_m | \nabla_R H | \psi_n \rangle}{E_n - E_m}
\]
This enables robust training and evaluation around degenerate points without singularities [2502.12870].

## 3. Transferability Strategies

A core innovation of X-MACE is transferability across (i) geometry, (ii) state, and (iii) chemical space. This is accomplished through:

- **Geometry:** Neural network architectures receive nuclear coordinates $R$ and encode them via nuclear encoders and self-attention modules. Thus, a single set of parameters $\theta$ models the entire relevant domain of $R$ [2503.19847].
- **State:** Extensive parameter sharing across $N_s$ electronic state nets, followed by synchronized averaging after each training step, facilitates efficient joint learning of correlated PESs.
- **Chemical space:** Transfer learning leverages foundational ground-state potentials ("MACE-OFF") as the initialization for excited-state PES training. Only the final DeepSets readout is reinitialized, and all earlier parameters (message-passing kernels, radial basis functions, and equivariant features) are preserved, yielding strong performance even with minimal excited-state data [2502.12870, 2502.21045].

In the context of long-range interactions and QM/MM environments, FieldMACE incorporates the multipole expansion (up to $\ell_{\max}=3$) via an equivariant, attention-weighted message-passing scheme, enabling linear scaling with system size and explicit solvent effects [2502.21045].

## 4. Computational Efficiency and Benchmark Performance

X-MACE methods are characterized by dramatic reductions in wall-clock and sample complexity relative to traditional single-point approaches. Empirical benchmarks include:

| System                      | Metric/Reference    | Cost Reduction         | Energy MAE                |
|-----------------------------|---------------------|-----------------------|---------------------------|
| Ethylene S₀/S₁ torsion      | MAE_rel (meV)       | ~5× fewer iterations  | 29.4(1) (torsion), 4.7(7) (pyramidal) [2503.19847] |
| Carbon dimer (C₂), 8 states | MAE_rel (meV)       | ~3× faster            | 69(1) vs. SHCI [2503.19847]   |
| CH₂NH₂⁺ (3-state PES grid)  | MAE_rel (meV)       | ~100×                 | 96(2) vs. MR-CISD [2503.19847]|
| Butene (ML)                 | MAE (eV)            | —                     | 0.0151 (X-MACE+AE) [2502.12870]|
| Furan in water (FieldMACE)  | Population curves   | Transfer: ~30 points  | Reproduces QM, versus failure from scratch [2502.21045] |

Fine-tuning from ground-state MACE-OFF models with only 1–10% excited-state data yields lower MAEs (e.g., energies < 0.01 eV) than training de novo; when >30% data is available, converged accuracy is equal [2502.12870]. Linear scaling ($O(N)$ in atom number for NN PES evaluation) enables large-scale explicit-environment excited-state dynamics.

## 5. Case Applications: Photochemistry and Large-System Dynamics

X-MACE has demonstrated robust performance across a diverse suite of photochemical benchmarks:

- **Ethylene S₀/S₁ torsion:** Correct conical intersection at $\phi \approx 98.5^\circ$ resolved, with $\Delta E_{ij}(R)$ matching high-level references.
- **Carbon dimer dissociation:** Simultaneous optimization of eight lowest states over bond lengths, all crossings and adiabatic gaps are captured.
- **Methylenimmonium cation ($CH_2NH_2^+$):** 2D PES with conical intersections at $\tau=0^\circ$; surface features and relative gaps resolved over the grid [2503.19847].
- **Butene/ethene/propene/fulvene:** ML X-MACE with DeepSets improves energy-gap accuracy near conical intersections by 45% relative to conventional MACE.
- **Chromophore dataset:** Transferable ML potentials exhibiting 30% lower energy MAE and 15% lower force MAE on held-out molecules [2502.12870].
- **Furan in water:** FieldMACE transfer models reproduce excited-state surface-hopping population curves using only 30 QM points [2502.21045].

Diabatic neural network approaches ("DANN") extended X-MACE principles to the explicit modeling and screening of 3,100 photoswitchable azobenzene derivatives, achieving $\sim 10^5$–$10^6\times$ acceleration over quantum chemistry [2108.04879].

## 6. Implications, Generalization, and Outlook

Transferable excited-state modeling with X-MACE enables routine *ab-initio*-level simulation of photochemical phenomena, including on-the-fly nonadiabatic dynamics without external fitting. Critical implications include:

- Generation of training data for excited-state force fields with high-fidelity reference labeling.
- Direct computation of nonadiabatic coupling elements via overlaps or smooth NAC predictors, facilitating surface hopping and Ehrenfest dynamics.
- Scalability to large systems (peptides, protein chromophores, nanomaterials) due to approximately linear cost in system size and number of geometries.
- Chemical-space extrapolation and transfer: fine-tuning from ground-state foundational models results in improved data efficiency and generalization to unseen chemical motifs [2502.12870, 2502.21045].
- Explicit environmental and solvent modeling via long-range multipole expansions (FieldMACE) [2502.21045].

A plausible implication is the use of X-MACE-derived PESs and couplings as drop-in modules for ML-enhanced surface-hopping packages, enabling controlled studies of photodynamics in previously inaccessible chemical regimes. These advances position X-MACE as a principal tool for high-accuracy, data-efficient, and transferable excited-state simulations in modern theoretical chemistry.

Source: https://www.emergentmind.com/topics/transferable-excited-state-modeling-x-mace