Topological resolution of conical intersection seams and the coupled cluster bifurcation via mixed Hodge modules

Abstract

The rigorous description of Conical Intersections (CIs) remains the central challenge of non-adiabatic quantum chemistry. While the ``Yarkony Seam'' -- the $(3N-8)$-dimensional manifold of degeneracy -- is well-understood geometrically, its accurate characterization by high-level electronic structure methods is plagued by numerical instabilities. Specifically, standard Coupled Cluster (CC) theory suffers from root bifurcations near Ground State CIs, rendering the ``Gold Standard'' of chemistry inapplicable where it is needed most. Here, we present \textbf{QuMorpheus}, an open-source computational package that resolves these singularities by implementing a topological framework based on Dissipative Mixed Hodge Modules (DMHM) [P. Saurabh, arXiv:2512.19487 (2025)]. By algorithmically mapping the CC polynomial equations to a spectral sheaf, we compute the exact Monodromy ($μ$) invariants of the intersection. We demonstrate that this automated algebraic geometry approach correctly identifies the physical ground state topology in the Köhn-Tajti model and resolves the intersection seams of realistic chemical systems, including Ethylene and the Chloronium ion ($\mathrm{H_2Cl^+}$). Furthermore, we apply QuMorpheus to the photoisomerization of Previtamin D, proving that the experimentally observed Woodward-Hoffmann selection rules are a direct consequence of a topological ``Monodromy Wall'' ($μ=1, γ=π$) rather than purely energetic barriers. This establishes a general software solution to the ``Yarkony Problem,'' enabling the robust, automated mapping of global intersection seams in complex molecular systems. The topological stability of these intersections allows for the control protocols discussed in Ref.[P. Saurabh, Submitted to Phys. Rev. X (2025)].

Figures (5)

• Figure 1: Topological origin of the Coupled Cluster instability. (a) The Bifurcation Problem: The breakdown of the standard Coupled Cluster (CC) expansion near a conical intersection (Köhn-Tajti model). As the nuclear coordinates encircle the degeneracy ($R \to 0$), the standard iterative solver fails to track the physical root. The energy surface (red) undergoes a square-root bifurcation, resulting in unphysical complex-valued solutions (dashed regions) and a discontinuity in the potential energy surface. (b) The Sheaf-Theoretic Resolution: The global solution manifold constructed by QuMorpheus. By treating the CC polynomial equations as a coherent spectral sheaf, the method reconstructs the full Riemann surface of the problem. This reveals the intersection not as a numerical error, but as a topological branch point. QuMorpheus correctly computes the monodromy ($\mu$) around the singularity, allowing for the smooth, automated analytic continuation of the ground state (cyan \ref{['fig:benchmark']}) across the entire intersection seam.
• Figure 2: Topological Robustness and Algorithmic Stability. (A) Perturbation Stability: Under breaking of spatial symmetry ($C_{2v} \to C_1$) via a cubic term $x^3$, the DMHM invariant $\mu$ (black line) remains robustly integer-quantized at 1, while the gap (blue) opens only away from the seam. (B) Algorithmic Metrology: Comparison of the geometric phase error $\epsilon$ near the singularity. Standard Berry Phase integration (Green) diverges as $1/R$. Exact Factorization (Orange) exhibits spikes due to density nodes ($1/|\chi|^2$). QuMorpheus (Blue) yields machine-precision integer invariants down to $R=10^{-6}$, demonstrating homological protection.
• Figure 3: Universality across Topologies: From Point Seams to Loops. (A) Search on the Ethylene PES (Point Seam). Numerical optimizers (Red) oscillate, while QuMorpheus (Cyan) converges directly. (B) Stability Slice. (C) Search on the H2Cl+ Toroidal Seam. QuMorpheus locks onto the continuous degeneracy loop ($R=2.0$). (D) Inset shows exact tracking of the seam crossing even at numerical singularity.
• Figure 4: Predictive Topochemistry: Topological Selection Rules in Previtamin D. (A) 3D Visualization of the Hula-Twist Bifurcation. Disrotatory (Forbidden) and Conrotatory (Allowed) paths. (B) Monodromy Phase Map ($Q_{rxn}$ vs $Q_{sym}$) visualizing the "Topological Wall" (Branch Cut) where vector alignment flips. (C) Invariant Metrology: Explicit calculation of $\mu=1$ (robustness) and $\gamma=\pi$ (Geometric Phase) along the Disrotatory path, proving destructive interference. (D) Branching Ratio Prediction: QuMorpheus predicts the correct 100:0 selection rule, solving the statistical 80:20 error of standard Energetic Methods.
• Figure 5: The QuMorpheus Computational Pipeline Compared with Standard and Exact Factorization methods. The workflow automates the translation of quantum chemical data into rigorous topological invariants. (a) Input Layer: The system accepts symbolic Hamiltonians $H(\mathbf{R})$ or interfaces with standard Electronic Structure packages (e.g., PSI4, CFOUR) to ingest Coupled Cluster amplitudes via the cc_interop module. (b) Algebraic Engine: The algebraic_analyzer constructs the Jacobian ideal $J = \langle \nabla E \rangle$ of the potential surface. It utilizes the Singular kernel (via a SymPy interface) to compute the Groebner Basis, reducing the continuous energy landscape into discrete algebraic residues. (c) Topological Classification: The engine outputs the precise Milnor ($\mu$) and Tjurina ($\tau$) numbers, classifying the intersection (e.g., as a Conical vs. Glancing intersection) without manual inspection of the wavefunction nodes.