Beyond Cavity Born-Oppenheimer: On Non-Adiabatic Coupling and Effective Ground State Hamiltonians in Vibro-Polaritonic Chemistry

TL;DR

The paper addresses how vibro-polaritonic chemistry transcends the cavity Born-Oppenheimer approximation by explicitly deriving non-adiabatic couplings arising from nuclei and cavity modes, and by contrasting fully correlated VSC theory with crude, adiabatic-based approaches. It introduces a crude VSC framework and a perturbative scheme (cVSC-PT) that connects crude CBO to the fully correlated CBO, while highlighting how electron-photon entanglement is altered in crude models. Through analytic NAC expressions and a numerical study of the cavity Shin-Metiu model, the work shows cavity-induced non-adiabatic effects can modify activation barriers and reaction paths, and demonstrates the limitations and potential of crude models for capturing essential physics in VSC. Overall, the results provide guidance on when effective ground-state models suffice and how perturbative corrections can systematically incorporate electron-photon correlation in vibro-polaritonic chemistry, with broader implications for modeling light-mmatter interactions in chemical reactivity.

Abstract

The emerging field of vibro-polaritonic chemistry studies the impact of light-matter hybrid states known as vibrational polaritons on chemical reactivity and molecular properties. Here, we discuss vibro-polaritonic chemistry from a quantum chemical perspective beyond the cavity Born-Oppenheimer (CBO) approximation and examine the role of electron-photon correlation in effective ground state Hamiltonians. We first quantitatively review ab initio vibro-polaritonic chemistry based on the molecular Pauli-Fierz Hamiltonian in dipole approximation and a vibrational strong coupling (VSC) Born-Huang expansion. We then derive non-adiabatic coupling elements arising from both ``slow'' nuclei and cavity modes compared to ``fast'' electrons via the generalized Hellmann-Feynman theorem, discuss their properties and re-evaluate the CBO approximation. In the second part, we introduce a crude VSC Born-Huang expansion based on adiabatic electronic states, which provides a foundation for widely employed effective Pauli-Fierz Hamiltonians in ground state vibro-polaritonic chemistry. The latter do not strictly respect the CBO approximation but an alternative scheme, which we name crude CBO approximation. We argue that the crude CBO ground state misses electron-photon entanglement relative to the CBO ground state due to neglected cavity-induced non-adiabatic transition dipole couplings to excited states. A perturbative connection between both ground state approximations is proposed, which identifies the crude CBO ground state as first-order approximation to its CBO counterpart. We provide an illustrative numerical analysis of the cavity Shin-Metiu model with a focus on non-adiabatic coupling under VSC and electron-photon correlation effects on classical activation barriers. We finally discuss potential shortcomings of the electron-polariton Hamiltonian when employed in the VSC regime.

Figures (6)

• Figure 1: Schematic representation of the cavity Shin Metiu model with a moving positively charged nucleus (blue) with coordinate, $R$, and a moving negatively charged electron (yellow) with coordinate, $r$, both interacting with a single cavity mode. Fixed positively charged nuclei are colored in red and located at a distance $L$ from each other, while mobile particles move along the related molecular axis connecting fixed nuclei.
• Figure 2: Contour plots of reduced adiabatic electron-photon densities, $\rho^{(ec)}_\nu(r,R)$, (a$_1$-d$_1$), reduced adiabatic electron densities, $\rho^{(e)}_\nu(r,R)$, (a$_2$-d$_2$) and adiabatic difference densities, $\Delta \rho^{(ec)}_\nu(r,R)$, (a$_3$-d$_3$) in electron-nuclear coordinate space for adiabatic ground and first excited states ($\nu=0,1$) in weak and strong non-adiabatic coupling (NAC) regime of the CSM model under vibrational strong coupling with $\eta=0.04$.
• Figure 3: Weakly non-adiabatic CSM model under VSC with $\eta=0.04$: a) Ground, $V_0(R,x_c)$, and d) first excited state cPES, $V_1(R,x_c)$, as function of nuclear, $R$, and cavity displacement, $x_c$, coordinates with energies in atomic units (a.u.). b) Mass-weighted nuclear derivative NAC element, $-\frac{1}{M_a}\mathcal{F}^{(n)}_{10}(R,x_c)$, and e) cavity derivative NAC element, $-\,\mathcal{F}^{(c)}_{10}(R,x_c)$, in a.u. as function of coordinates under VSC. c) Molecular, $-\frac{1}{M_a}\mathcal{F}^{(n)}_{en}(R,x_c)$, and f) DSE-induced nuclear NAC contributions, $-\frac{1}{M_a}\mathcal{F}^{(n)}_{dse}(R,x_c)$, to mass-weighted nuclear NAC element in b) with same parameters.
• Figure 4: Strongly non-adiabatic CSM model under VSC with $\eta=0.04$: a) Ground, $V_0(R,x_c)$, and c) first excited state cPES, $V_1(R,x_c)$, as function of nuclear, $R$, and cavity displacement, $x_c$, coordinates with energies in atomic units (a.u.). b) Mass-weighted nuclear derivative NAC element, $-\frac{1}{M_a}\mathcal{F}^{(n)}_{10}(R,x_c)$, and d) cavity derivative NAC element, $-\,\mathcal{F}^{(c)}_{10}(R,x_c)$, in a.u. as function of coordinates under VSC as a) and d).
• Figure 5: Differences between ground state cPES and ground state crude cPES for the weakly non-adiabatic CSM model under VSC with $\eta=0.04$. (a) Energy difference, $\Delta E_0(R,x_c)$, in wave numbers $(\mathrm{cm}^{-1})$ and (b) numerical approximations to cavity minimum energy paths for $E^{(ec)}_0(R,x_c)$ in red and for $V_0(R,x_c)$ in blue (contour plot shown in $\mathrm{cm}^{-1}$).