Knowledge Bank

The shifts of the molecular vibrational frequencies when going from the ground electronic state to the lowest excited electronic states pose some problems for the mutual co-assignment of the calculated vibrational frequencies in the different excited states. The \textit{trans}-\chem{C_2 O_2 F_2} shift of the frequency of the symmetrical $\nu$(C=O) stretching vibration between the S$0$ and T$1$ is 373~\wn. The feasibility of mutual co-assignments of the vibrational frequencies in these electronic states has been demonstrated for \textit{trans}-\chem{C_2 O_2 F_2}. Matrices analogous to the Duschinsky matrix, \textbf{7},(4), 551--566 (1937).} were used to juxtapose the $ag$ vibrational frequencies of this molecule calculated at the CASPT2/cc-pVTZ level in the ground S$0$ and excited triplet T$1$ and singlet S$1$ electronic states. The analog of the Duschinsky matrix $D$ was obtained for this molecule using the equation $D=(LI)-1LII$ where $LI$ and $LII$ are the matrices of the vibrational modes (normalized atomic displacements) obtained by solving the vibrational problems for the S$0$ and T$1$ electronic states, respectively. Choosing the dominant elements in columns of the $D$ matrix and permuting these columns to arrange these elements along the diagonal of the transformed matrix $D\ast$ makes it possible to establish the correct mutual co-assignments of the calculated $ag$ vibrational frequencies of the \textit{trans}-\chem{C_2 O_2 F_2} molecule in the S$0$ and T$1$ electronic states. The analogous procedure was performed for the \textit{trans}-\chem{C_2 O_2 F_2} molecule in the T$1$ and S$1$ excited electronic states. The recent reassignments of the \nub{2} and \nub{3} calculated vibrational frequencies in the \textit{trans}-\chem{C_2 O_2 F_2} molecule in the ground state were also obtained for the triplet T$1$ and singlet S$1$ excited electronic states. The approach set forth in this text makes it possible to juxtapose the calculated vibrational frequencies of the same molecule in the different electronic states and to refine the assignments of these frequencies. This is essential in correctly analyzing the vibronic spectra of a molecule under investigation.