\[\Large \hat{\mathcal{L}} \xi_n = -\omega_n^2 \rho_0 \xi_n\]
\[\Large \hat{\mathcal{L}} = {\color{red}\hat{\mathcal{L}}_\gamma} + \hat{\mathcal{R}}_\gamma\]
\[\Large {\color{red}\hat{\mathcal{L}}_\gamma \xi_{\gamma,n} = -\omega^2_{\gamma,n}\xi_{\gamma,n}}\]
\[\Large \hat{\mathcal{L}} = \hat{\mathcal{R}}_\pi + {\color{grey}\hat{\mathcal{L}}_\pi}\]
\[\Large {\color{grey}\hat{\mathcal{L}}_\pi \xi_{\pi,n} = -\omega^2_{\pi,n}\xi_{\pi,n}}\]
\[\Large \hat{\mathcal{L}} = {\color{grey}\hat{\mathcal{L}}_\pi} + \hat{\mathcal{R}}_\pi = \hat{\mathcal{R}}_\gamma + {\color{red}\hat{\mathcal{L}}_\gamma}\]
\[\Large \xi_\text{mixed} \sim {\color{grey}c_\pi \xi_\pi} + {\color{red}c_\gamma \xi_\gamma}\]
Mixed Modes as
Acoustic “Molecular Orbitals”
\[\xi_\text{mixed} \sim {\color{grey} \sum_i c_{\pi, i} \xi_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j} \xi_{\gamma,j}}\]
Mixed mode
frequencies and mixing coefficients can likewise be found by
solving a Generalised Hermitian Eigenvalue Problem: \[
\scriptsize
\begin{bmatrix}
{\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\
\mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}}
\end{bmatrix}
\begin{bmatrix}
{\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma}
\end{bmatrix}
= -\omega^2 \begin{bmatrix}
\mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1}
\end{bmatrix}
\begin{bmatrix}
{\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma}
\end{bmatrix}.
\] \[
\small
L_{{\color{grey}i}{\color{red}j}} = \langle {\color{grey}\xi_i},
\hat{\mathcal{L}}{\color{red}\xi_j}\rangle;~~~~D_{{\color{grey}i}{\color{red}j}}
= \langle {\color{grey}\xi_i}, {\color{red}\xi_j}\rangle
\]
(Ong & Basu 2020)