A Quantum Chemistry Approach to Linear Vibro-Polaritonic Infrared Spectra with Perturbative Electron–Photon Correlation

Abstract

In the vibrational strong coupling (VSC) regime, molecular vibrations and resonant low-frequency cavity modes form light–matter hybrid states, vibrational polaritons, with characteristic infrared (IR) spectroscopic signatures. Here, we introduce a molecular quantum chemistry-based computational scheme for linear IR spectra of vibrational polaritons in polyatomic molecules, which perturbatively accounts for nonresonant electron–photon interactions under VSC. Specifically, we formulate a cavity Born–Oppenheimer perturbation theory (CBO-PT) linear response approach, which provides an approximate but systematic description of such electron–photon correlation effects in VSC scenarios while relying on molecular ab initio quantum chemistry methods. We identify relevant electron–photon correlation effects at the second order of CBO-PT, which manifest as static polarizability-dependent Hessian corrections and an emerging polarizability-dependent cavity intensity component providing access to transmission spectra commonly measured in vibro-polaritonic chemistry. Illustratively, we address electron–photon correlation effects perturbatively in IR spectra of CO₂ and Fe(CO)₅ vibro-polaritonic models in sound agreement with nonperturbative CBO linear response theory.

Vibrational polaritons are light–matter hybrid states formed when molecular vibrational modes strongly interact with quantized modes of optical low-frequency cavities¹ underlying the emerging field of vibro-polaritonic chemistry.^2,3 Experimentally, vibrational polaritons exhibit characteristic spectroscopic signatures, specifically transitions to upper and lower vibro-polaritonic states, which have been probed by both linear⁴⁻⁹ and nonlinear¹⁰ infrared (IR) spectroscopic techniques.

Computationally, linear vibro-polaritonic IR spectra are commonly obtained from linear response theory, where two distinct routes exist: (1) linear response approaches based on effective ground state Pauli–Fierz Hamiltonians, which are fully characterized by purely molecular properties, e.g., the molecular ground state potential energy surface (PES) and dipole moment, that can be easily accessed by means of standard quantum chemistry methods,¹¹⁻¹⁴ and (2) a linear response approach formulated in the cavity Born–Oppenheimer (CBO) approximation,¹⁵⁻¹⁸ which relies on an extended electronic structure problem accounting for electron–photon correlation due to nonresonant interactions between electrons and low-frequency cavity modes.¹⁸ The CBO formulation is more complete but also more involved because generalized cavity PES and dipole moments depend on molecular and cavity coordinates. Both approaches rely on a linear response formulation in double-harmonic approximation, i.e., harmonic molecular and cavity modes in combination with a linearized ground state dipole moment. This is particularly beneficial for treating the vibrations and spectra of large molecules or molecular ensembles quantum mechanically.

Recently, we showed that effective ground state and CBO formulations differ in their description of electron–photon correlation.¹⁹ Specifically, effective ground state models can be understood from the perspective of a crude CBO approximation relying on the adiabatic electronic ground state, which completely neglects electron–photon correlation in the VSC regime.¹⁹ We proposed a perturbative connection between crude CBO and correlated CBO formulations, denoted CBO perturbation theory (CBO-PT), motivated by distinct excitation energy scales of electronic and low-frequency cavity mode subsystems.¹⁹ CBO-PT solves the extended CBO electronic structure problem perturbatively and allows for systematic electron–photon correlation corrections of the crude CBO approach.¹⁹ We note, CBO-PT is conceptually similar to other perturbative approaches,^20,21 which however constitute a further approximation^19,20 or do not directly connect to the CBO formulation.²¹

In this work, we introduce an approximation to the nonperturbative CBO linear response framework of Bonini and Flick¹⁵ by combining CBO-PT with linear response theory. This approach allows us to systematically correct vibro-polaritonic IR spectra for electron–photon correlation effects while fully relying on molecular properties that can be accessed by ab initio quantum chemistry methods. We derive explicit expressions for electron–photon correlation-corrected vibro-polaritonic Hessian matrix elements and IR intensities, which are both shown to be determined by the molecule’s static polarizability, whose possible relevance has been noted only recently.²¹⁻²³ We illustratively apply the CBO-PT linear response approach to vibro-polaritonic model systems of CO₂ and Fe(CO)₅ (cf. Figure 1) in line with ref (15) and discuss electron–photon correlation effects in connection to experimentally relevant transmission spectra.^4,7

Figure 1.

Sketch of CO₂ and Fe(CO)₅ molecules strongly coupled (yellow wavy lines) to the low-frequency cavity mode (yellow) with a cavity IR response (right red pulse) and a molecular IR response (bottom red pulse).

We first recapitulate the main aspects of CBO linear response theory,¹⁵ which fully accounts for electron–photon correlation in vibro-polaritonic IR spectra. We consider a molecular system with N_n nuclei under a VSC with 2N_c quantized transverse field modes of an IR cavity. We furthermore assume the CBO approximation to be valid; i.e., non-adiabatic coupling to the excited state manifold is negligible.^18,19 The CBO electronic ground state problem is described by an electron–photon time-independent Schrödinger equation (TISE)^17,18

1

with the adiabatic electron–photon ground state, |Ψ^(ec)₀⟩, and ground state cavity potential energy surface (cPES), E^(ec)₀, which parametrically depend on both nuclear, , and cavity displacement coordinates, . The electron–photon Hamiltonian reads¹⁸

2

with electronic Hamiltonian composed of the electronic kinetic energy, , and the molecular Coulomb potential, V_coul. In addition, is the harmonic cavity potential characterized by harmonic frequencies, ω_k, and displacement coordinates, x_λk, with polarization λ and mode index k. The third term on the right-hand side of eq 2 constitutes the light–matter interaction potential

3

with polarization-projected molecular dipole operator , cavity polarization vector , and light–matter interaction constant , which is determined by the cavity volume V_cav and the permittivity ε₀, and relates to a mode specific equivalent, (cf. section S1 of the Supporting Information). The first term in eq 3 is the bare light–matter interaction (g₀), and the second term (g₀²) resembles the dipole self-energy (DSE).²⁴ couples the electronic subsystem (in addition to the nuclear one) to low-frequency cavity modes and contains a DSE-induced electron–electron interaction, which is independent of interparticle distance.^15,18,19

In nonperturbative CBO linear response theory, the ground state cPES in eq 1 is harmonically approximated¹⁵

4

around a minimum configuration, , with coordinate vector , which collects N_vib mass-weighted Cartesian displacement coordinates, , with atomic mass M_i and 2N_c cavity displacement coordinates, Δx_λk = x_λk – x⁰_λk. Vectors and collect all reference values, R_i,0 and x⁰_λk, respectively. The vibro-polaritonic Hessian, , contains second-order derivatives of the ground state cPES with respect to both mass-weighted molecular Cartesian and cavity displacement coordinates, and specifies a matrix eigenvalue problem¹⁵

5

where is written as a 2 × 2-block matrix with molecular (QQ), cavity (CC), and light-matter interaction blocks (QC). Eigenvectors resemble vibro-polaritonic normal modes with molecular, , and cavity, , components characterized by corresponding harmonic vibro-polaritonic frequencies, Ω_m. The eigensystem of eq 5 provides access to linear vibro-polaritonic IR spectra

6

for N_p = N_vib + 2N_c vibro-polaritonic normal modes. IR intensities, I_m = ∑_κ|Z_mκ|², are determined by Cartesian components of mode effective charges

7

with molecular and cavity mode contributions¹⁵

8

9

where and with q_mi and c_mλk being elements of and , respectively, in eq 5. On the basis of eq 7, each intensity component decomposes into

10

such that σ_IR exhibits a molecular, σ^(Q)_IR, a cavity, σ^(C)_IR, and a mixed contribution, σ^(X)_IR. Finally, Z_mκ is determined by Cartesian components, D^(ec)_00,κ, of a generalized ground state dipole moment

11

which is evaluated with respect to the electron–photon ground state, |Ψ^(ec)₀⟩, where integration over electronic coordinates is indicated by .

We now introduce a systematic approximation to the CBO linear response approach¹⁵ based on cavity Born–Oppenheimer perturbation theory (CBO-PT).¹⁹ In CBO-PT, the light–matter interaction potential, , in eq 3 is treated as perturbation of the electronic subsystem¹⁹

12

with zeroth-order Hamiltonian and formal perturbation parameter λ. In , the cavity potential, V_c, is a constant with respect to the bare electronic problem, such that zeroth-order states are given by bare adiabatic electronic states, |Ψ^(e)_μ⟩. CBO-PT approximates nonresonant interactions between electrons and low-frequency cavity modes, i.e., electron–photon correlation, by perturbatively solving the electron–photon TISE (1).¹⁹ At the nth order of CBO-PT, i.e., CBO-PT(n), the approximate cPES, E^(ec)₀ ≈ ∑ⁿ_k=0λ^kE^(k)₀ (cf. section S1), provides access to the corresponding Hessian

13

with eigenvalues (Ω⁽ⁿ⁾_m)² and eigenvectors . In contrast to nonperturbative CBO linear response theory,¹⁵ we work directly with mass-weighted molecular normal modes with coordinates , which constitute the CBO-PT(0) reference. CBO-PT(n) linear vibro-polaritonic IR spectra are given by

14

with approximate vibro-polaritonic normal mode frequencies Ω⁽ⁿ⁾_m and IR intensities . Here, Z⁽ⁿ⁾_mκ = Z^(Q,n)_mκ + Z^(C,n)_mκ are approximate mode effective charges with (cf. section S1A)

15

16

which result from the linearized CBO-PT(n) ground-state dipole moment

17

where is determined by |Φ^(n–1)₀⟩ = ∑^n–1_k=0λ^k|Ψ^(k)₀⟩, which is the perturbatively corrected adiabatic ground state with |Ψ^(k)₀⟩ being defined in section S1. Note, in contrast to eq 7, both Z^(Q,n)_mκ and Z^(C,n)_mκ depend on prefactors with molecular normal mode, ω_i, and cavity, ω_k, frequencies that follow from the normal mode representation (cf. section S1A). Notably, CBO-PT relies exclusively on quantities that can be obtained from established ab initio quantum chemistry methods.

In the following, we provide explicit expressions up to perturbation order n ≤ 2, i.e., up to CBO-PT(2), which is sufficient to capture leading-order electron–photon correlation corrections.¹⁹ In CBO-PT(1), which does not account for electron–photon correlation, Hessian matrix elements read (cf. sections S1B and S1C)

18

19

20

The DSE term in eq 18 induces a normal mode frequency shift on the diagonal (i = j) and inter-normal mode (i ≠ j) couplings. The light–matter interaction matrix element in eq 19 contains a polarization-projected ground state dipole moment derivative, . The CBO-PT(1) IR spectrum, σ⁽¹⁾_IR, is fully determined by the molecular ground state dipole moment , leading to a first-order molecular IR intensity

21

22

since the cavity mode effective charge vanishes, Z^(C,1)_mκ = 0, such that vibro-polaritonic contributions enter only via cavity-induced linear combinations of molecular normal modes. Thus, σ⁽¹⁾_IR, exclusively probes the effective matter response of the light–matter hybrid system under VSC.

Next, CBO-PT(2) Hessian matrix element corrections are given by (cf. sections S1D and S2)

23

24

25

which are all determined by polarization-projected static polarizability tensor elements . The leading-order correction (g²₀) enters the cavity block in eq 25, which leads to a redshift of cavity mode frequencies and introduces a matter-mediated coupling of distinct cavity modes¹⁹ in agreement with nonperturbative results.¹⁵ The CBO-PT(2) IR spectrum, σ⁽²⁾_IR(ℏω), is determined by second-order vibro-polaritonic frequencies, Ω⁽²⁾_m, and intensities derived from second-order mode effective charges as

26

27

28

which contains both molecular and cavity contributions with dipole derivatives explicitly given by (cf. section S3)

29

30

In both cases, we find static polarizability-dependent corrections with D^(C,2)_00,κ being linear in g₀ and x_λk (cf. section S3) in agreement with numerical results reported in the Supporting Information of ref (15). In contrast to CBO-PT(1), the nonvanishing CBO-PT(2) cavity intensity component allows us to address transmission spectra commonly measured in vibro-polaritonic chemistry,^4,7,8 which exclusively probe the cavity response, σ^(C)_IR, of the light–matter hybrid system.

In the remainder of this work, we discuss electron–photon correlation effects in IR spectra of CO₂ and Fe(CO)₅ vibro-polaritonic models already studied in ref (15) as obtained from CBO-PT(2) linear response theory. We consider polarization-averaged IR spectra

31

to mimic the random molecular orientation, where the sum runs over all unique polarization components, α = x, y, or z, with K = 1, 2, or 3. Molecular structures, harmonic frequencies, dipole derivatives, and static polarizability tensor elements are obtained from density functional theory (DFT) using the TPSSh^25,26 and B3LYP²⁷ functionals and a Def2-TZVP basis set.^28,29 Computational details are provided in section S4A. Spontaneous emission-induced Lorentzian peak broadening is assumed (we neglect other dissipative channels) with a full width at half-maximum κ = 41 cm^–1 for a cavity quality factor with experimentally motivated Q values of 50 for CO₂ and 59 for Fe(CO)₅^8,30,31 (cf. section S4B).

We first discuss the antisymmetric stretching mode of CO₂ under VSC with a single resonant cavity mode, ℏω_c = ℏω_as = 2400 cm^–1 (TPSSh/Def2-TZVP), at coupling strength . Here, σ⁽ⁿ⁾_IR(ℏω) = σ⁽ⁿ⁾_z(ℏω) because only the dipole component along the molecular z-axis contributes. Figure 2a shows the lower and upper vibro-polaritonic transitions in CBO-PT(1) and CBO-PT(2) IR spectra with Rabi splittings, Ω⁽¹⁾_R = 121 cm^–1 and Ω⁽²⁾_R = 124 cm^–1, which indicate a slightly stronger effective light–matter interaction in the presence of electron–photon correlation. In the CBO-PT(1) spectrum both vibro-polaritonic states exhibit nearly identical molecular and photonic contributions (cf. section S4B), in contrast to the CBO-PT(2) scenario, which is characterized by a dominantly photonic lower and dominantly molecular upper polariton state in agreement with nonperturbative CBO results.¹⁵ This asymmetry in state composition translates into a strongly asymmetric CBO-PT(2) spectrum in favor of the lower polariton transition, which is also found for larger values of the light–matter coupling parameter in ref (15), e.g., in Figure 2 there, equivalent to g₀ employed here. (Note that a more quantitative comparison cannot be made because of differences in computational protocols and methods.) We will discuss peak asymmetry in more detail below. Turning to the details of σ⁽²⁾_IR(ℏω) (Figure 2b), we compare molecular, σ^(Q,2)_IR, cavity, σ^(C,2)_IR, and mixed contributions, σ^(X,2)_IR (cf. eq 10). Although σ^(Q,2)_IR is dominant relative to σ^(C,2)_IR, σ^(X,2)_IR determines the peak asymmetry of σ⁽²⁾_IR by “shifting” intensity from the upper to the lower vibro-polaritonic transition. This effect directly translates to the sign of the photonic contribution in linear combinations forming upper (+) and lower (−) polariton states (cf. section S4B). Mixed intensity contributions satisfy , because Z^(C,2)_∓κ is negative due to the dipole derivative in eq 30. In addition, we recall that vibro-polaritonic IR transmission spectra probe the cavity response of the light–matter hybrid system,^4,7,8 which translates here into the CBO-PT(2) cavity component, σ^(C,2)_IR, not accounted for in uncorrelated CBO-PT(1) models. In Figure 2c, we compare σ⁽²⁾_IR and σ^(C,2)_IR normalized to the lower polariton intensity and find different relative intensities for upper polariton peaks. This observation can be traced back to the mixed component σ^(X,2)_IR in σ⁽²⁾_IR, which reduces the intensity of the upper polariton transition. Note that a larger frequency range (cf. section S4B) reveals in addition a purely molecular contribution of the CO₂ bending modes at low frequencies of σ⁽²⁾_IR, which is absent in σ^(C,2)_IR displaying only vibro-polaritonic spectral features.

Figure 2.

Linear vibro-polaritonic IR spectra for selected single-molecule models under VSC with a single cavity mode at coupling strength . The top row shows z-polarized IR spectra of the antisymmetric CO₂ stretching mode under VSC with a single cavity mode, ℏω_c = ℏω_as = 2400 cm^–1, with (a) CBO-PT(1) and CBO-PT(2) IR spectra, σ⁽¹⁾_IR(ℏω) and σ⁽²⁾_IR(ℏω), in addition to the bare molecular spectrum, σ⁽⁰⁾_IR(ℏω), (b) molecular, cavity, and mixed CBO-PT(2) contributions, σ^(Q,2)_IR(ℏω), σ^(C,2)_IR(ℏω), and σ^(X,2)_IR(ℏω), respectively, and (c) a comparison of normalized σ⁽²⁾_IR(ℏω) and σ^(C,2)_IR(ℏω) (cf. the text for details). The bottom row shows polarization-averaged linear vibro-polaritonic IR spectra, , for the CO stretching band of a single Fe(CO)₅ molecule under VSC with a single cavity mode, ℏω_c = ℏω_e′ = 2052 cm^–1, with panels d–f providing information analogous to that in panels a–c, respectively. Stick spectra, σ⁽²⁾_α(ℏω), in panel d resemble individual polarization-dependent contributions to in eq 31 with α = x, y, or z (cf. the text for details). The respective cavity mode frequency is indicated by a gray vertical line in all panels.

As a second example, we consider the CO stretching band of Fe(CO)₅ under VSC, which is a both experimentally and theoretically relevant molecular multimode system.^8,15,33 The CO stretching band contains a doubly degenerate equatorial normal mode (symmetry e′ in molecular point group D_3h) and a single axial normal mode (a^″₂ in D_3h) with harmonic frequencies ℏω_e′ = 2052 cm^–1 and (TPSSh/Def2-TZVP, cf. section S4A)). We consider a single cavity mode tuned resonant to the equatorial e′ modes, ℏω_c = ℏω_e′, at light–matter interaction . Here, e′ modes are (x, z)-polarized, while the a^″₂ mode is y-polarized (cf. Figures 1 and 2d). Thus, depending on the cavity mode polarization, either all molecular modes couple strongly to the cavity or some remain decoupled. In Figure 2d, we show polarization-averaged CBO-PT(1) and CBO-PT(2) IR spectra and the bare molecular CBO-PT(0) spectrum, where the average in eq 31 was performed with respect to polarization directions equivalent to the Cartesian components (α = x, y, or z). The latter are resolved by stick spectra, σ⁽²⁾_α, for individual α values to highlight the polarization dependence of the different spectral contributions. Each σ⁽²⁾_α contains four peaks in the spectral region of interest due to the presence of e′ and a^″₂ modes in addition to the cavity mode. Note that the a^″₂ peak in σ⁽⁰⁾_IR is only visible as a blue-shifted shoulder relative to the dominant e′ transition for the herein chosen peak broadening. Under VSC, the effective e′ peak splits and we find in contrast to CO₂ a pronounced molecular contribution related to uncoupled molecular modes (cf. σ⁽⁰⁾_IR). The correlated CBO-PT(2) IR spectrum exhibits a strong redshift and significantly more intense vibro-polaritonic transitions relative to the uncorrelated CBO-PT(1) result, whereas both share a slight peak asymmetry in favor of the upper polariton peak. An analysis of in terms of , and reveals a trend similar to that of the CO₂ example (cf. Figure 2e). is the dominant contribution relative to , and the cross term again “shifts” the intensity from the upper to the lower vibro-polaritonic transition. Importantly, in contrast to the CO₂ model, contains here in addition to the matter response of the light–matter hybrid system bare molecular contributions, which relate to uncoupled CO stretching modes. A comparison of Rabi splittings, Ω⁽¹⁾_R = 169 cm^–1 and Ω⁽²⁾_R = 195 cm^–1, where we extract the latter from to avoid dominant molecular signatures, indicates also here a slightly enhanced effective light–matter interaction in the presence of electron–photon correlation. In Figure 2f, we compare and normalized relative to the lower polariton intensity. In contrast to the CO₂ model, we find here a dominant “upper” polariton contribution and a frequency mismatch of both peaks. Both observations relate to the fact that the “upper” polariton peak in actually contains a dominant molecular contribution, which hides the vibro-polaritonic transition clearly visible in . Thus, correlation-corrected CBO-PT(2) IR spectra allow us to avoid artificial molecular intensity contributions in vibrational multimode systems under VSC, which are not probed in vibro-polaritonic transmission spectroscopy.^4,7,8

The asymmetry in intensities and splittings can be understood from the CBO-PT(2) molecular normal and cavity mode frequencies in eqs 18 and 25. From a series expansion up to leading order in g₀, one obtains dressed frequencies

32

33

where the dressed molecular frequency, , is slightly blue-shifted and the dressed cavity mode frequency, , is subject to a polarization-induced redshift. Thus, for the bare frequency resonance condition, ω_c = ω_i, the CBO-PT(2) Hessian is in general not degenerate in the respective subspace, i.e., , which manifests in asymmetric intensities and/or splittings. A recent work³² proposed to replace ω_c = ω_i by a dressed resonance condition, , an idea already formulated in ref (15), which takes the matter feedback onto the vacuum cavity mode frequency, ω_c, into account. Following this argument, we compare in Figure 3 CBO-PT IR spectra obtained from ω_c = ω_i with the cavity response, , calculated for a blue-detuned cavity frequency, ℏω_c + δ_c (dashed vertical line), such that is satisfied. For a single CO₂ molecule under VSC with g₀ as before (Figure 3a), we observe a shift of δ_c = 30 cm^–1 leading to nearly identical intensities for lower and upper polariton transitions in in line with ref (32). The cavity response of Fe(CO)₅ under VSC for the dressed resonance condition is obtained for , where we averaged over different polarization-dependent δ_c values in analogy to IR spectra. For this multimode model, the peak asymmetry is also significantly reduced in , which in combination with the avoided molecular intensity contributions leads to a spectrum qualitatively closer to experimental results^8,33 and highlights the potential relevance of feedback mechanisms between electrons and low-frequency cavity modes.

Figure 3.

Effect of matter-induced cavity frequency renormalization on vibro-polaritonic transmission spectra, , with ℏω_c + δ_c (dashed gray vertical line in all panels) blue-detuned by δ_c from ω_c = ω_i (bold gray vertical line in all panels) such that . Shown is a comparison of with σ⁽¹⁾_IR(ℏω) and σ⁽²⁾_IR(ℏω) obtained from the bare resonance condition, ω_c = ω_i, for (a) the single CO₂ molecule under VSC with ℏω_c + δ_c = 2430 cm^–1 and (b) the single Fe(CO)₅ molecule under VSC with polarization-averaged both at coupling strength and (c) the explicit CO₂ ensemble composed of M = 20 molecules with ℏω_c + δ_c = 2462 cm^–1 at coupling strength .

We finally address collective strong coupling effects in two different molecular ensemble models of CO₂ under VSC (cf. section S5): an explicit ensemble of M molecules in the dilute gas limit³⁴ and an effective “ensemble” model, which contains a single CO₂ molecule under VSC characterized by an effective enhanced interaction constant, .¹⁵ In Figure 4a, we compare ensemble and effective IR spectra, σ⁽ⁿ⁾_ens and σ⁽ⁿ⁾_eff, respectively, for M = 20 parallel aligned CO₂ molecules at with the intensities being scaled by a factor of M^–1 for reasons of comparison. In both cases, we find excellent agreement for σ⁽¹⁾_IR and σ⁽²⁾_IR. In addition, for the ensemble model, we observe qualitatively identical results for the different spectral contributions to σ⁽²⁾_ens in Figure 4b as in the single-molecule model in Figure 2. In addition, a comparison of normalized σ⁽²⁾_ens and σ^(C,2)_ens for the explicit ensemble in Figure 4c reveals a slightly stronger mismatch of intensity ratios that can be observed from the upper polariton peak relative to Figure 2. Eventually, for the dressed resonance condition, we find here a required detuning of δ_c = 62 cm^–1 for M = 20 molecules (cf. section S5), which leads to nearly symmetric intensities of upper and lower vibro-polaritonic transitions (cf. Figure 3c). Thus, approximate transmission spectra of molecular ensembles under VSC obtained from CBO-PT linear response theory can be calculated via effectively scaled single-molecule approaches subject to an effective light–matter interaction constant, , when intermolecular interactions are neglected in agreement with the nonperturbative CBO approach.¹⁵

Figure 4.

Linear vibro-polaritonic IR spectra for CO₂ ensemble models under VSC with a single cavity mode, ℏω_c = ℏω_as = 2400 cm^–1. (a) Ensemble and effective CBO-PT(n) IR spectra, σ⁽ⁿ⁾_ens(ℏω) and σ⁽ⁿ⁾_eff(ℏω) for n = 0 ,1, or 2. (b) Molecular, cavity, and mixed CBO-PT(2) contributions, σ^(Q,2)_ens(ℏω), σ^(C,2)_ens(ℏω), and σ^(X,2)_ens(ℏω), respectively, for the explicit ensemble. (c) Comparison of normalized σ⁽²⁾_ens(ℏω) and σ^(C,2)_ens(ℏω) for the explicit ensemble. The respective cavity mode frequency is indicated by a gray vertical line.

In summary, we introduced a cavity Born–Oppenheimer perturbation theory (CBO-PT) linear response approach for calculating electron–photon correlation-corrected vibro-polaritonic IR spectra of polyatomic molecules under VSC. This approach approximates nonperturbative CBO linear response theory¹⁵ by perturbatively accounting for nonresonant interactions between electrons and low-frequency cavity modes, while fully relying on molecular ab initio quantum chemistry methods. Relative to the bare molecular zeroth-order IR spectrum, electron–photon correlation is accounted for at the second order of CBO-PT linear response theory. It manifests as a static polarizability-dependent cavity intensity component and related corrections of Hessian matrix elements, most notably a matter-induced cavity frequency shift and inter-cavity mode coupling. Those corrections capture the main characteristics of nonperturbative CBO linear response theory¹⁵ and provide additional physical insight due to their explicit relation to molecular and cavity mode properties. A comparison of uncorrelated CBO-PT(1) and correlation-corrected CBO-PT(2) IR spectra for CO₂ and Fe(CO)₅ vibro-polaritonic models reveals the impact of electron–photon correlation on vibro-polaritonic intensity ratios and Rabi splittings. In addition, CBO-PT(2) linear response theory allows to address the cavity response of the light–matter hybrid system related to experimentally relevant transmission spectroscopy. In both model scenarios, CBO-PT(2) linear response theory exhibits significant qualitative agreement with the nonperturbative CBO linear response results of ref (15). The origin of asymmetric intensities and splittings in CBO-PT(2) spectra was analyzed in context of a recently discussed dressed resonance condition,³² which takes into account the matter feedback on the bare cavity mode frequency and significantly reduces peak asymmetries also for molecular multimode and ensemble models. Finally, the CBO-PT linear response approach constitutes a promising since computationally easily accessible path to vibro-polaritonic IR spectra of polyatomic molecules under VSC, which accounts for nontrivial electron–photon correlation effects.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpclett.4c00105.

• Derivation of CBO-PT Hessian matrix elements, mode effective charges, and dipole moments and details of the computational methods for molecular properties and CBO-PT extension to molecular ensemble models in the dilute gas limit (PDF)

The authors declare no competing financial interest.