For a molecule strongly coupled to a cavity mode, the Born-Oppenheimer approximation (BOA) neglects the nonadiabatic coupling between the lower and upper polaritonic potential surfaces. Fábri, Halász, Cederbaum and Vibók [Chem. Sci. 12, 1251 (2021)] showed for conical-intersection-free formaldehyde that this BOA breaks down even in a one-dimensional treatment, for every coupling strength, because at symmetric (planar C2vC_{2v}) geometries the transition dipole moment between the two electronic states vanishes and the cavity-induced nonadiabatic coupling grows linearly with the coupling strength; in two dimensions the failure follows from the light-induced conical intersection (LICI) itself. Their study deliberately excluded molecules with inherent (field-free) nonadiabaticity and used coupling strengths far beyond experimental reach. A companion paper by the same group [J. Chem. Phys. 153, 234302 (2020)] already computed the exact full-dimensional (six-vibrational-coordinate) field-dressed spectra with all nonadiabatic couplings at representative realistic coupling, and rotational, collective and lossy-cavity extensions have since been substantially developed. What remains open is the regime the showcase could not address: how the BOA and reduced-dimensional strategies perform when a molecule's natural conical intersections and cavity-induced LICIs coexist and interact in the same configuration space. The problem is to delineate this validity domain for cavity-coupled molecules with inherent nonadiabaticity.

Background

The Born-Oppenheimer approximation (BOA) underlies most of molecular physics and quantum chemistry: because electrons are much lighter than nuclei, the electronic problem is solved at fixed nuclear geometry and the nuclei then move on a single potential energy surface (PES). The approximation fails in nonadiabatic processes, where the nuclear kinetic energy couples different electronic surfaces. The strongest such coupling occurs at conical intersections (CIs) - degeneracy points of two PESs at which the nonadiabatic coupling becomes singular - which provide ultrafast (femtosecond) radiationless-decay channels that dominate photochemistry. Diatomic molecules, having only one vibrational coordinate, cannot host natural CIs (only avoided crossings).

A molecule strongly and resonantly coupled to a quantized mode of an optical or plasmonic cavity is described by cavity quantum electrodynamics (cQED). In the single-mode, two-electronic-state model the Hamiltonian is

H^=H^0+ℏωca^†a^−g μ⃗^⋅e⃗ (a^†+a^),\hat{H}=\hat{H}_0+\hbar\omega_c\hat{a}^{\dagger}\hat{a}-g\,\hat{\vec{\mu}}\cdot\vec{e}\,(\hat{a}^{\dagger}+\hat{a}),

where H^0\hat{H}_0 is the field-free molecular Hamiltonian, a^†\hat{a}^{\dagger} and a^\hat{a} create and annihilate a photon of the cavity mode of frequency ωc\omega_c, μ⃗^\hat{\vec{\mu}} is the molecular dipole operator, e⃗\vec{e} the field polarization and gg the coupling strength. Diagonalizing the potential part of this Hamiltonian at each nuclear geometry yields hybrid light-matter (polaritonic) adiabatic surfaces, lower V−V_- and upper V+V_+, which can cross conically even when the bare molecule has no CI: a light-induced conical intersection (LICI), the cavity analogue of the LICIs known from intense classical laser fields. At least two nuclear coordinates are needed to form a LICI; in diatomics the second coordinate is a rotational angle, while polyatomics supply several vibrational modes. The BOA in this context groups the nearly resonant cavity mode with the electrons and neglects the nonadiabatic coupling between V−V_- and V+V_+.

Galego, Garcia-Vidal and Feist [Phys. Rev. X 5, 041022 (2015)] found that this BOA is valid for one-dimensional models of larger organic molecules at sufficiently strong coupling, which encouraged BOA-based reduced-dimensional simulation of cavity chemistry. Fábri, Halász, Cederbaum and Vibók [Chem. Sci. 12, 1251 (2021)] tested this for formaldehyde (H2CO), a four-atomic molecule chosen precisely because it has no natural CI in the studied region, comparing exact and BOA spectra of 1D, 2D and 6D quantum-dynamical models. Their findings: (i) the BOA fails even in the 1D(ν4\nu_4 out-of-plane-bend) model for every coupling strength gg, because the transition dipole moment between the two electronic states vanishes at planar C2vC_{2v} geometries, so the nonadiabatic coupling (NAC) at the symmetric point grows linearly with gg (a two-harmonic analytical model gives NAC(0)=gα/(Δ−ℏωc)(0)=g\alpha/(\Delta-\hbar\omega_c) for a linear transition moment αx\alpha x and detuning Δ−ℏωc\Delta-\hbar\omega_c); (ii) in the 2D(ν2\nu_2 C=O stretch, ν4\nu_4) model the BOA fails because of the LICI itself (located at Q2=8.84Q_2=8.84, Q4=0Q_4=0 for ωc=29957.23\omega_c=29957.23 cm−1^{-1}, at energy 30776 cm−1^{-1}) whenever the spectrum extends above the LICI energy, and the BOA survives only for the lowest group of polaritonic states whose wavefunctions have negligible amplitude near the LICI; (iii) they argued the BOA likewise fails in the full 6D model, but computed the exact field-dressed spectra only in 1D and 2D. A companion study by the same authors [J. Chem. Phys. 153, 234302 (2020)] then computed exact full-dimensional (all six vibrations) field-dressed spectra of H2CO - diagonalizing the full molecule-cavity Hamiltonian in a vibronic-by-Fock basis, hence including all nonadiabatic couplings - at the realistic coupling g=8.44×10−4g=8.44\times10^{-4} a.u. (classical-field equivalent 101110^{11} W cm−2^{-2}), observing LICI-induced splittings and their polarization dependence and concluding that at least two vibrational coordinates are needed, while exact 6D calculations at strong coupling remain difficult.

Both works fix the molecular orientation (rotations omitted), treat a single molecule (no collective coupling), and use for illustration couplings up to g=2.67×10−1g=2.67\times10^{-1} a.u. (equivalent to 1000010000 TW cm−2^{-2}), far beyond experimental reach; real single-molecule strong coupling is achieved only in plasmonic nanocavities whose modes decay within a few tens of femtoseconds, so cavity loss must be included (e.g. via Lindblad or non-Hermitian stochastic dynamics). Reduced-dimensional model studies of CI-bearing molecules in cavities do exist - cavity manipulation of pyrazine conical-intersection dynamics, cooperative CI dynamics of two pyrazine molecules, collective CI-mediated decay, and the butatriene cation, where a natural CI between cationic states and a nearby LICI jointly reshape the ionization spectrum - but they are all reduced-dimensional, and none establishes the validity of the BOA or of reduced-dimensional strategies when inherent and cavity-induced nonadiabaticity coexist.

Problem Statement

For a polyatomic molecule that possesses inherent nonadiabaticity - natural conical intersections or strong vibronic coupling between close-lying electronic states - placed in strong resonance with a cavity mode, determine whether and under what quantitative conditions the Born-Oppenheimer approximation and reduced-dimensional quantum-dynamical models yield correct spectra and dynamics when natural conical intersections and light-induced conical intersections coexist and interact within the same nuclear configuration space. Because "coexist and interact within the same nuclear configuration space" has no unique formal meaning, the solver must fix the regime explicitly: the interaction is given by a quantitative or topological statement - adopted and justified by the solver - of how the light-induced conical intersection is situated with respect to the natural one (for example, lying inside the natural intersection's branching space or within a stated energy-geometry window of it, as against the two merely sharing the same coordinate set), the criterion reported for the validity domain must be stated relative to that declared regime, and any such quantitatively defined and justified notion of interaction is acceptable. Resolving the task requires, at minimum: (i) testing the Born-Oppenheimer approximation and the dimensional truncation separately and as their combination - the BOA-based reduced model - so that each of the three is independently confirmed or refuted; (ii) spanning the light-matter coupling strength, the cavity detuning and the number of retained vibrational coordinates, and treating "yield correct spectra and dynamics" quantitatively, namely that the approximate field-dressed spectra and dynamics reproduce the exact full-nonadiabatic reference within a tolerance stated explicitly by the solver; and (iii) supporting the resulting criterion with a demonstration for at least one identified polyatomic molecule that possesses a documented natural conical intersection, a general criterion for the class being the target level and an enumeration of all photochemically relevant molecules not being required. The model class is that of the formaldehyde showcase - a single molecule at fixed orientation coupled to the single-mode cavity of the Hamiltonian above - so that the rotational, collective-coupling and lossy-cavity extensions developed since then are separate questions rather than part of this validity domain. The overarching objective is to delineate the validity domain of BOA-based reduced models for real cavity-coupled molecules, extending the conical-intersection-free formaldehyde showcase to the generality of photochemically relevant molecules.

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