Relativistic multimode pseudo-Jahn-Teller effect in planar molecules Y X 3 with D 3 h ′ symmetry

Highlights: • Relativistic multimode pseudo-Jahn-Teller effect $\left({}^2{E}^{\prime }+{}^2{A}_2^{″}\right)\times \left(e^{\prime }+a_2^{″}\right)$. • Planar tetratomic molecules $Y{X}_3$ with the odd number of electrons. • Double symmetry group $D_{3h}^{\prime }$. • Symmetry properties of electronic Hamiltonian. • Spin-orbit coupling terms. • Second order terms for electrostatic and spin-orbit contributions. • $6\times 6$ vibronic matrix. • $e^{\prime }$ and $a_2^{″}$ vibration normal modes. • 9 electrostatic and 10 spin-orbital real constant parameters. -- Abstract: Relativistic multimode pseudo-Jahn-Teller effect $\left({}^2{E}^{\prime }+{}^2{A}_2^{″}\right)\times \left(e^{\prime }+a_2^{″}\right)$ is considered for planar tetratomic molecules $Y{X}_3$, which have the odd number of electrons and are characterized by the double symmetry group $D_{3h}^{\prime }$. Our analysis is based on the symmetry properties of electronic Hamiltonian, which includes spin-orbit coupling terms. Taylor series expansion of electronic Hamiltonian in terms of vibration normal modes $e^{\prime }$ and $a_2^{″}$ is restricted by second order terms for electrostatic and spin-orbit contributions. The result of this research is $6\times 6$ vibronic matrix, which depends on $e^{\prime }$ and $a_2^{″}$ vibration normal modes and includes nine electrostatic and ten spin-orbital real constant parameters.