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

TL;DR

This paper introduces QuMorpheus, a topological computational framework based on Dissipative Mixed Hodge Modules, to accurately characterize conical intersections in quantum chemistry, overcoming numerical instabilities of traditional methods.

Contribution

It presents a novel algebraic geometry approach that resolves singularities in coupled cluster theory and automates the mapping of intersection seams in complex molecules.

Findings

Successfully identifies physical ground state topology in models and real molecules

Resolves intersection seams in Ethylene and Chloronium ion

Links topological invariants to experimental photoisomerization rules

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 ($\mu$) invariants of the intersection. We demonstrate that this automated algebraic geometry approach correctly identifies the physical ground state topology in the K\"{o}hn-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'' ($\mu=1, \gamma=\pi$) 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)].