QuMorpheus: Topological Quantum Chemistry Tool

• QuMorpheus is an open-source computational package that employs algebraic topology and symbolic methods to map non-adiabatic conical intersections.
• It replaces unstable numerical root-tracking with exact topological invariants to resolve coupled-cluster bifurcations and correctly classify seam topology.
• The package integrates symbolic algebra, monodromy phase tracking, and visualization modules to analyze models such as Ethylene and Previtamin D photoisomerization.

Searching arXiv for QuMorpheus and directly related papers. QuMorpheus is an open-source computational package for non-adiabatic quantum chemistry that addresses conical intersections by replacing fragile numerical root-tracking with exact topological and algebraic invariants. In the formulation presented in “Topological resolution of conical intersection seams and the coupled cluster bifurcation via mixed Hodge modules” (Saurabh, 23 Dec 2025), conical-intersection seams are treated not merely as numerically troublesome degeneracies but as singular algebraic objects whose topology can be resolved systematically. The package is designed to resolve coupled-cluster root bifurcations near ground-state conical intersections, compute monodromy-related invariants, and automate global seam mapping in systems including the Köhn–Tajti model, Ethylene, the chloronium ion $\mathrm{H_2Cl^+}$, and a reduced model of Previtamin D photoisomerization (Saurabh, 23 Dec 2025).

1. Problem setting and conceptual basis

Conical intersections are regions of exact electronic-state degeneracy on a molecular potential-energy surface where the Born–Oppenheimer approximation breaks down. Near such intersections, nuclear and electronic motion are strongly coupled, and the adiabatic description becomes singular. The package is motivated by a specific failure mode of standard coupled-cluster theory near ground-state conical intersections: root bifurcation. In this regime, the nonlinear coupled-cluster amplitude equations can split the physically relevant root, exchange it with spurious branches, produce complex-valued solutions, or trigger solver divergence.

The underlying issue is framed as a mismatch between a single-reference ansatz and a multi-sheeted energy landscape. The coupled-cluster state is written as

$|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$

but around a conical intersection the exact state acquires nontrivial geometric-phase structure, whereas standard coupled-cluster algorithms attempt to force a single-valued branch where the physical solution resides on a branched covering. The paper describes this as a “coordinate crisis.”

A central geometric object is the Yarkony seam, the generic degeneracy manifold associated with a conical intersection. For a molecule with $N$ nuclei, its dimension is

$3N - 8.$

Accordingly, the degeneracy set is generically a codimension-2 manifold in nuclear configuration space rather than an isolated point. QuMorpheus is built around the claim that this seam must be resolved globally, not merely located locally, and that the “Yarkony Problem” is better posed as a problem in algebraic geometry and sheaf-theoretic topology than as a purely numerical optimization task.

2. Algebraic and topological framework

QuMorpheus is based on the Dissipative Mixed Hodge Module framework. Its operational philosophy is to view the Hamiltonian or electronic-structure equations as defining an algebraic variety, encode the singularity structure in a sheaf-theoretic object, and compute exact invariants of that object instead of relying on path-dependent numerical continuation.

The coupled-cluster equations are treated as a polynomial system

$F(\mathbf{t}) = 0,$

with cluster amplitudes $\mathbf{t} = (t_1,\dots,t_m)$. Bifurcation is signaled algebraically when the Jacobian becomes singular:

$$\det\!\left(\frac{\partial F}{\partial \mathbf{t}}\right) = 0.$$

The workflow constructs a Jacobian ideal,

$$J = \langle \nabla E \rangle \quad \text{or more generally} \quad J_H = \langle \partial_1 H,\dots,\partial_n H \rangle,$$

and then applies Gröbner-basis reduction to obtain a finite-dimensional quotient algebra.

The resulting polynomial data are “mapped” to a spectral sheaf $\mathcal{F}_H$ over the nuclear configuration manifold

$M = \mathbb{R}^{3N-6},$

equipped with a differential connection

$|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$0

The construction also introduces a Liouvillian sheaf $|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$1 and a Brieskorn lattice $|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$2, whose local cohomology captures the singularity class.

Two invariants organize the analysis. The first is the Milnor number $|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$3, defined locally by

$|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$4

or, in the polynomial setting,

$|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$5

In Gröbner-basis form, this becomes the count of standard monomials,

$|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$6

The second invariant is the geometric or monodromy phase,

$|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$7

equivalently encoded by the monodromy representation

$|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$8

which, for the conical-intersection examples discussed, is effectively reduced to a $|\Psi_{\mathrm{CC}}\rangle = e^T|\Phi_0\rangle,$9-valued action,

$N$0

A notable feature of this framework is that $N$1 and $N$2 are treated as exact and discrete rather than as numerically inferred surrogates. This is the technical basis for the package’s claim to robust seam classification near singularities.

3. Package architecture and computational workflow

QuMorpheus is described as a Python package with a modular pipeline integrating symbolic algebra, topological classification, and visualization. Its input layer accepts symbolic Hamiltonians $N$3 and interfaces with PSI4 and CFOUR through a cc_interop module to ingest coupled-cluster or EOM-CCSD data. The algebraic-geometry engine, algebraic_analyzer, constructs the Jacobian ideal and uses SymPy together with the Singular kernel to compute Gröbner bases and extract $N$4 and $N$5 from the quotient algebra. A MonodromyIntegrator computes the phase accumulated along specified loops, while qumorpheus.viz produces seam plots, monodromy maps, and Riemann-surface-like branch-structure visualizations.

The basic workflow proceeds in eight stages. A symbolic or discretized Hamiltonian is loaded; the associated sheaf or connection structure is built; the Jacobian ideal

$N$6

is formed; a Gröbner basis $N$7 is computed; the quotient dimension is used to obtain $N$8; the Tjurina number

$N$9

may be computed by adjoining $3N - 8.$0 to the ideal; monodromy is numerically integrated or inferred to obtain $3N - 8.$1; and the seam is classified and plotted.

The implementation is described as open-source and reproducible, with YAML-driven configuration and Lean 4 formalizations of the core theorems. The package’s computational strategy is to avoid unstable iterative root tracking and instead separate expensive symbolic algebra from stable numerical phase tracking.

4. Validation on model systems and molecular seams

The Köhn–Tajti model serves as the benchmark for coupled-cluster bifurcation. In this setting, standard EOM-CCSD is reported to fail because the root structure becomes non-Hermitian and bifurcates near the conical intersection. QuMorpheus reconstructs the correct branch through monodromy-aware continuation and identifies the physical ground-state topology correctly. For the simple CI-like singularity discussed in this model, the computed invariant remains

$3N - 8.$2

which the paper interprets as a basic nodal structure stable under perturbations.

For Ethylene, QuMorpheus finds the canonical point-like conical-intersection seam. The contrast drawn in the source is with numerical optimizers that oscillate near the vanishing gap; QuMorpheus instead converges to the seam by exploiting algebraic structure. For the chloronium ion $3N - 8.$3, the seam is described as a toroidal or circular degeneracy manifold rather than a point, and the reported invariant is again

$3N - 8.$4

The significance assigned to these two examples is methodological rather than merely chemical: the same machinery is used for both point-like and extended seams.

These case studies also clarify a common misconception. QuMorpheus is not presented as a new approximate electronic-structure ansatz. Its stated contribution is a computational language in which singularities are represented as algebraic and sheaf-theoretic objects endowed with exact invariants.

5. Previtamin D, monodromy walls, and selection rules

The most interpretive application in the source concerns a reduced topological model of Previtamin D Hula-Twist photoisomerization (Saurabh, 23 Dec 2025). Along the disrotatory path, the reported invariants are

$3N - 8.$5

The resulting phase factor is

$3N - 8.$6

which is interpreted as destructive interference blocking flux through the disrotatory channel.

On this basis, the paper argues that the experimentally observed Woodward–Hoffmann selection rule is a consequence of a topological “Monodromy Wall” rather than purely energetic barriers. The conrotatory path is described as preserving constructive phase alignment, while the disrotatory path crosses the monodromy wall and acquires a $3N - 8.$7 phase shift. The corresponding prediction is a $3N - 8.$8 branching rule consistent with the selection rule.

This application extends QuMorpheus beyond seam localization into mechanistic interpretation. A plausible implication is that geometric phase, when rendered as an exact computable invariant, can serve as a chemically meaningful diagnostic rather than only a formal property of adiabatic transport.

6. Scope, significance, and nomenclature

QuMorpheus is presented as a general software solution to the Yarkony Problem: the robust, automated mapping of global conical-intersection seams from local electronic-structure data. Its broader implication is that such intersections are topologically protected objects, and the source connects this topological stability to possible control protocols. In practical terms, the package is proposed for robust seam mapping, automated coupled-cluster branch selection, classification of seam type and rigidity, and prediction of photochemical selection rules.

The name should be distinguished from several nearby but different usages in the contemporary arXiv literature. “Morpheus” denotes a benchmark for evaluating the physical reasoning of video generative models with real physical experiments (Zhang et al., 3 Apr 2025). “Quantum Metamorphosis” or “QuMorph” refers to scale-programmable spectral and topological reorganization in hierarchically nested lattices (Mehrabad et al., 17 Nov 2025). “Quantamorphisms” designate a correct-by-construction framework for recursive quantum programming compiled through Quipper, PyZX, Qiskit, and IBM Q hardware (Neri et al., 2020). These projects share lexical proximity, but QuMorpheus in the strict sense is the conical-intersection and coupled-cluster package introduced in (Saurabh, 23 Dec 2025).