Qx–Qy Vibronic Mixing

• Qx–Qy vibronic mixing is the coupling of two orthogonal components—vibrational coordinates and electronic states—through nuclear motion, observed in systems like Jahn–Teller solids and porphyrin aggregates.
• The phenomenon is modeled via specialized Hamiltonians that incorporate both linear and quadratic coupling terms, reflecting symmetry-selected vibrational modes and electronic doublets.
• Applications range from order–disorder phase transitions in layered solids to modulated energy transfer in molecular aggregates, highlighting the role of adiabatic potential topology and Berry phase effects.

Searching arXiv for the specified papers and closely related work on Qx–Qy vibronic mixing. Qx–Qy vibronic mixing denotes the entanglement of two orthogonal degrees of freedom through nuclear motion, but the meaning of “Qx” and “Qy” depends on the literature. In Jahn–Teller and pseudo–Jahn–Teller treatments of solids, $Q_x$ and $Q_y$ are usually the two components of an $E_g$ vibrational doublet, and their direction in the distortion plane fixes the admixture of an electronic doublet. In porphyrins, chlorins, and related aggregates, by contrast, Qx and Qy usually denote two orthogonally polarized electronic transitions within the Q band, and vibronic mixing refers to nuclear-coordinate-dependent coupling between those electronic manifolds. Across both usages, the common structure is a two-component electronic sector, a symmetry-selected vibrational sector, and an adiabatic potential-energy surface whose topology controls intensity borrowing, pseudorotation, order–disorder behavior, or energy-flow pathways (Bercha et al., 2015, Iwahara et al., 2024, Arsenault et al., 2021, Thomas et al., 15 Jul 2025).

1. Terminological scope and definitions

The term is not tied to a single physical system. It designates a pattern of coupling in which two orthogonal components become mixed by vibronic interaction, but those components may be vibrational coordinates, electronic basis states, or both.

Context	Meaning of $Q_x,Q_y$	Character of mixing
CuInP$_2$S$_6$	$Q_x \equiv Q_1$, $Q_y \equiv Q_2$ of a $\Gamma_5(E_g)$ mode	An $E_g$ distortion mixes a $Q_y$0 electronic doublet
Ba$Q_y$1CaReO$Q_y$2	$Q_y$3, $Q_y$4	An $Q_y$5 Jahn–Teller mode mixes pseudoorbital states in the SOC-adapted manifold
LHCII and porphyrin aggregates	Qx and Qy are orthogonally polarized electronic transitions	Herzberg–Teller activity and vibronic resonance mix Qx and Qy manifolds
SrOPh / SrOPh-d$Q_y$6	No true Qx–Qy degeneracy	Two distinct non-degenerate bends generate a second-order analogue rather than degenerate-mode Qx–Qy mixing

In CuInP$Q_y$7S$Q_y$8, the near-gap copper $Q_y$9-derived states form an elementary energy band whose $E_g$0-point doublets transform as $E_g$1 and $E_g$2 of $E_g$3, and the active vibrational mode is likewise $E_g$4. In Ba$E_g$5CaReO$E_g$6, the Jahn–Teller-active $E_g$7 coordinates are written as $E_g$8 and $E_g$9, which can be mapped directly to $Q_x,Q_y$0 and $Q_x,Q_y$1. In LHCII and porphyrin nanotubes, Qx and Qy are the two orthogonally polarized transitions of the Q band, and the mixing is assigned to Herzberg–Teller activity rather than to a purely Franck–Condon mechanism. In SrOPh and SrOPh-d$Q_x,Q_y$2, the analogy is explicitly limited: the in-plane and out-of-plane bends are distinct $Q_x,Q_y$3 and $Q_x,Q_y$4 modes, so the observed state mixing is not a true degenerate Qx–Qy problem (Bercha et al., 2015, Iwahara et al., 2024, Arsenault et al., 2021, Wojcik et al., 25 Oct 2025).

2. Symmetry structure and model Hamiltonians

In the trigonal protostructure of CuInP$Q_x,Q_y$5S$Q_x,Q_y$6, the $Q_x,Q_y$7 electronic doublet is represented in an orthonormal basis $Q_x,Q_y$8, with Pauli matrices $Q_x,Q_y$9 and $_2$0. The symmetry content of the symmetric square is

$_2$1

so the scalar $_2$2 contributes the harmonic term and the $_2$3 component contributes the linear and quadratic anisotropy terms. The resulting vibronic matrix is

$_2$4

This is a crystal realization of the linear $_2$5 problem supplemented by symmetry-allowed warping terms, and it is explicitly tied to the elementary energy band induced from the copper Wyckoff position $_2$6 (Bercha et al., 2015).

In Ba$_2$7CaReO$_2$8, the onsite model is written as

$_2$9

with

$_6$0

and a linear $_6$1 Jahn–Teller interaction. In the standard $_6$2 form, the coupling is written as $_6$3. The authors use $_6$4 eV and $_6$5 meV, and emphasize that SOC quenches the Jahn–Teller coupling in the $_6$6 states by half (Iwahara et al., 2024).

For molecular excited-state manifolds, a broader linear vibronic coupling representation writes

$_6$7

Here $_6$8 are intrastate couplings and $_6$9 are interstate vibronic couplings. The mode-selection rule is

$Q_x \equiv Q_1$0

which separates gap-tuning totally symmetric modes from non-totally symmetric modes that provide off-diagonal interstate mixing (Fumanal et al., 2018).

3. Adiabatic potentials, mixing angles, and geometric phase

Diagonalization of the CuInP$Q_x \equiv Q_1$1S$Q_x \equiv Q_1$2 vibronic matrix gives

$Q_x \equiv Q_1$3

with electronic mixing angle

$Q_x \equiv Q_1$4

In the linear limit, the lower sheet has the circular trough characteristic of $Q_x \equiv Q_1$5 coupling. The quadratic term $Q_x \equiv Q_1$6 warps that trough and, in the $Q_x \equiv Q_1$7 protostructure, the lower adiabatic surface acquires six equivalent minima with $Q_x \equiv Q_1$8 angular spacing. In the monoclinic paraphase, by contrast, the active vibronic instability is reduced to a one-dimensional double-well,

$Q_x \equiv Q_1$9

and in the ferriphase further symmetry lowering selects one well, which is the basis for the discussion of spontaneous polarization (Bercha et al., 2015).

In Ba$Q_y \equiv Q_2$0CaReO$Q_y \equiv Q_2$1, the adiabatic $Q_y \equiv Q_2$2 form is

$Q_y \equiv Q_2$3

and the local pseudoorbital orientation is fixed by

$Q_y \equiv Q_2$4

The ideal linear problem has a circular trough at $Q_y \equiv Q_2$5, but the work distinguishes three regimes: a strongly warped classical limit without SOC, a shallow almost circular trough with strong SOC, and an anisotropy-lifted trough under static elastic fields. The Berry phase around the conical intersection imposes antiperiodic boundary conditions and gives the vibronic sequence $Q_y \equiv Q_2$6 (Iwahara et al., 2024).

The same geometry reappears in the explicit $Q_y \equiv Q_2$7 treatment of cavity-coupled Jahn–Teller molecules, where

$Q_y \equiv Q_2$8

and the lower adiabatic surface is

$Q_y \equiv Q_2$9

This produces the conical intersection at $\Gamma_5(E_g)$0, the Mexican-hat minimum at $\Gamma_5(E_g)$1, and half-integer vibronic angular momentum associated with pseudorotation. This suggests that Qx–Qy vibronic mixing is governed not only by coupling strength but also by the topology of the adiabatic surface (Pandit et al., 11 Nov 2025).

4. Crystalline and oxide realizations

In CuInP$\Gamma_5(E_g)$2S$\Gamma_5(E_g)$3, Qx–Qy vibronic mixing is embedded in an order–disorder phase-transition scenario. The near-gap elementary energy band is induced from the copper site-symmetry group, and the copper $\Gamma_5(E_g)$4 states at the valence-band top provide the $\Gamma_5(E_g)$5 and $\Gamma_5(E_g)$6 doublets relevant for the Jahn–Teller analysis. The trigonal protostructure supports six equivalent minima on the lower adiabatic surface, the monoclinic paraphase reduces this to a two-minimum double-well associated with order–disorder between Cu1u and Cu1d sites, and the ferriphase selects one minimum and breaks inversion symmetry. The discussion connects this sequence directly to the possibility that spontaneous polarization arises normal to the layers in CuInP$\Gamma_5(E_g)$7S$\Gamma_5(E_g)$8 (Bercha et al., 2015).

In Ba$\Gamma_5(E_g)$9CaReO$E_g$0, Iwahara et al. identify spectroscopic signatures that cannot be reproduced by a purely classical lattice description. At the Re $E_g$1 edge, the main inelastic feature appears at $E_g$2 eV and is accompanied by a high-energy tail and vibronic satellites at $E_g$3, $E_g$4, and $E_g$5 eV. At the O $E_g$6 edge, a distinct low-energy peak occurs at $E_g$7 eV with a broad tail up to $E_g$8 eV. These features persist across the structural transition at $E_g$9 K. The analysis attributes them to orbital–lattice entangled states generated by Qx–Qy vibronic mixing within the $Q_y$00 manifold. The same work reports dynamic vibronic stabilization of approximately $Q_y$01 meV, static Jahn–Teller stabilization of approximately $Q_y$02 meV, and cooperative elastic fields of order several meV to $Q_y$03 meV, concluding that the dynamic effect remains dominant even below the onset of multipolar order (Iwahara et al., 2024).

5. Molecular and aggregate realizations

In photosynthetic light harvesting, Qx–Qy vibronic mixing refers to coupling between orthogonally polarized electronic transitions rather than to a doubly degenerate normal mode. In LHCII, the relevant mechanism is assigned to Herzberg–Teller activity. The note on LHCII identifies a higher-energy side-band in linear absorption and 2DEV spectra that appears only with Herzberg–Teller activity in the model, reproduces higher-lying excitonic states composed mainly of chlorophyll $Q_y$04 in the $Q_y$05–$Q_y$06 cm$Q_y$07 range, and is tracked through a ground-state bleach at $Q_y$08 cm$Q_y$09. Follow-up work cited there attributes green-light absorption across nearly a $Q_y$10 cm$Q_y$11 span of the electronic spectrum to mixed vibronic Qy–Qx states. The note therefore revises earlier interpretations by assigning the dominant role to Herzberg–Teller rather than realistic Franck–Condon-only coupling (Arsenault et al., 2021).

In porphyrin nanotube aggregates, polarization-controlled 2DES resolves mixed Qx–Qy pathways at room temperature. The cross-peak-selective polarization sequence $Q_y$12 removes isotropic contributions and suppresses purely Qx–Qx or Qy–Qy pathways. Cross-peaks between the main Q band and vibronic shoulders are already present at $Q_y$13 fs, the lower cross-peak is about $Q_y$14 stronger than the upper one at that time, and the ratio $Q_y$15 changes from $Q_y$16 in all-parallel 2D to $Q_y$17 in the cross-peak-selective experiment. Low-frequency beats at $Q_y$18 and $Q_y$19 cm$Q_y$20 are removed by the polarization filter, whereas anisotropic beats at $Q_y$21 and $Q_y$22 cm$Q_y$23 survive and are assigned to out-of-plane pyrrole-ring deformations that modulate Qx–Qy mixing. Static energetic disorder with $Q_y$24 cm$Q_y$25 spreads Qx character across the Q band and increases simulated $Q_y$26–$Q_y$27 mixing from approximately $Q_y$28 to $Q_y$29 for the $Q_y$30 cm$Q_y$31 mode and from approximately $Q_y$32 to $Q_y$33 for the $Q_y$34 cm$Q_y$35 mode (Thomas et al., 15 Jul 2025).

A more restrictive conclusion emerges from a heterodimer study of vibronic excitons in photosynthetic energy transfer. There, a low-frequency intramolecular vibration with $Q_y$36 cm$Q_y$37 and $Q_y$38 generates robust long-lived beating at cryogenic temperature, but environmentally induced fluctuations eradicate the mixing at physiological temperature. The same work argues that such electronic–vibrational quantum mixtures do not necessarily play a significant role in electronic energy-transfer dynamics even when they enhance long-lived beating in 2D spectra. This suggests that the spectroscopic visibility of Qx–Qy vibronic mixing and its dynamical functionality need not coincide (Fujihashi et al., 2015).

6. Computational parameterization, diagnostics, and limits of the label

One route to quantitative parameterization is the overlap-based extraction of interstate vibronic coupling constants. For a displaced geometry along mode $Q_y$39, the protocol computes

$Q_y$40

constructs a Löwdin-orthogonalized transformation $Q_y$41, forms

$Q_y$42

and estimates the linear coupling through

$Q_y$43

Because it uses overlaps between auxiliary excited-state wavefunctions rather than only excited-state energies, this protocol was proposed as an alternative to Hessian-based extraction in dense excited-state manifolds (Fumanal et al., 2018).

For aggregates, the dimensionality of the vibronic problem can be reduced by exact construction of aggregate normal modes within each equal-frequency set. In the trimer example, the orthonormal coordinates

$Q_y$44

separate a spectator mode from promoter modes that tune energy gaps. The mode $Q_y$45 tunes the $Q_y$46 and $Q_y$47 gaps simultaneously, whereas $Q_y$48 does not tune any pairwise gap. This reduction was used to show how vibronic resonance can convert an electronically uncoupled “trap” into a “conduit” for transport, and it provides a mode-resolved language for identifying which coordinates actually promote Qx–Qy mixing in aggregates (Patra et al., 2020).

The spectroscopic diagnosis of vibronic mixing in 2D coherent spectroscopy is likewise symmetry- and resonance-sensitive. In the Holstein-type dimer analysis, off-resonant cases preserve a clean distinction between electronic beats and vibrational beats, while near-resonant cases with $Q_y$49 generate avoided crossings, multiple oscillation frequencies, and congested Fourier maps. The effective two-level description uses

$Q_y$50

and complete exciton–vibronic mixing occurs when $Q_y$51. The same work formulates vibrational lifetime borrowing as the mechanism by which mixed coherences can outlive purely electronic ones (Butkus et al., 2013).

A recurrent terminological limit is that not every orthogonal two-mode interaction is a genuine Qx–Qy problem. In SrOPh and SrOPh-d$Q_y$52, the observed mixing is between $Q_y$53 and the combination level $Q_y$54, with $Q_y$55 a $Q_y$56 in-plane bend and $Q_y$57 a $Q_y$58 out-of-plane bend. The effective $Q_y$59–$Q_y$60 coupling is deduced to be about $Q_y$61 cm$Q_y$62, the direct linear $Q_y$63 coupling is vanishingly small, and the dominant pathway is second order through the higher $Q_y$64 state. The work explicitly states that this is not Qx–Qy degenerate-mode mixing. This clarification is important because it separates true two-component vibronic angular-momentum physics from combination-mode symmetry pathways that only resemble it at a coarse level (Wojcik et al., 25 Oct 2025).