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

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 nonadiabatic coupling elements arising from both “slow” nuclei and cavity modes compared to “fast” electrons via the generalized Hellmann–Feynman theorem, discuss their properties, and reevaluate 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. Those 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 correlation relative to the CBO ground state due to neglected cavity-induced nonadiabatic transition dipole couplings to excited states. A perturbative connection between both ground state approximations is proposed, which identifies the crude CBO ground state as a first-order approximation to its CBO counterpart. We provide an illustrative numerical analysis of the cavity Shin–Metiu model with a focus on nonadiabatic coupling under VSC and electron–photon correlation effects on classical activation barriers. We finally discuss the potential shortcomings of the electron-polariton Hamiltonian when employed in the VSC regime.

Ab Initio Vibro-Polaritonic Spectra in Strongly Coupled Cavity-Molecule Systems

“…In all cases studied, we neglect the spatial structure of the cavity field, assuming that all molecules experience the same field. If the quantized cavity modes are coupled via their characteristic frequencies to vibrational degrees of freedom of molecules, the situation is described as VSC, for which CBOA ,− is a well-suited theoretical approach. Within CBOA, the cavity modes are grouped with the nuclei in a generalized Born–Huang expansion, , and then one can subsequently solve the quantum problem of the electrons and then of the nuclei and photons.…”

“…In our recent work, we have introduced a Hartree–Fock ansatz in the cavity Born–Oppenheimer approximation (CBOA), ,− capable of describing the electronic ground state of single molecules as well as of an ensemble of molecules coupled to an optical cavity. Within the framework of this cavity Born–Oppenheimer Hartree–Fock (CBO-HF) ansatz, we now derive analytic expressions for the first derivatives of the energy with respect to the nuclear and photonic degrees of freedom.…”

Cavity-Mediated Molecular Entanglement and Generation of Non-classical States of Light

“…The dipole-self-energy has been demonstrated to be important in eq 1 for it to be bounded from below (which is a necessary prerequisite for a groundstate to exist), 31 and it is important to obtain physically meaningful results 35 and accounts for the explicit transverse interactions between the vibrational and electronic degrees of molecules in the strong-coupling regime. 36,37 Next, we define the electron−photon coupling from the bilinear interaction term of eq 1 as…”

“…In our simulations, we include the dipole-self-energy term, which is 1 2 ∑ α = 1 M ( bold-italicλ α · boldR̂ ) 2 of eq . The dipole-self-energy has been demonstrated to be important in eq for it to be bounded from below (which is a necessary prerequisite for a ground-state to exist), and it is important to obtain physically meaningful results and accounts for the explicit transverse interactions between the vibrational and electronic degrees of molecules in the strong-coupling regime. , Next, we define the electron–photon coupling from the bilinear interaction term of eq as g α false( i j false) = λ ℏ ω α 2 false⟨ φ i | e α · R̂ | φ j false⟩ where |φ j ⟩ are the vibrational eigenstates and false⟨ ϕ i | e α · R̂ | ϕ j false⟩ is the transition dipole moment. We define the dimensionless ratio η = g α /ℏω α which characterizes the strength of the light-matter interaction in relation to the matter excitations.…”

“…For the single photon mode considered in this work, we use the Fock state representation for and given in eq . We note that, as discussed in ref , proper treatment for vibrational strong coupling must be taken into account due to nonvanishing nonadiabatic couplings. We sampled 30 photon Fock states in order to correctly represent the coherent state with an amplitude of β = 2 to obtain converged results.…”