**Figure 1.** Geometry parameters for the XCN (X=Cl, Br) molecules as used in equations (25) and (27).

**Abstract**

In this work, we present the four-component quadratic vibronic coupling model for the description of the Renner–Teller effect (RTE) in the presence of the spin–orbit coupling. The interaction of the two potential energy surfaces emerging from the cationic ^{2}Π states of singly ionized linear triatomic molecules is described by the quadratic coupling constant *c* for the genuine RT repulsion and the second parameter, *d*, for a nonconstant spin–orbit coupling varying with the bond angle of the triatomic. The emergence of a linear RT constant in the presence of the spin–orbit operator was originally shown by Poluyanov and Domcke (2004 *Chem. Phys.* **301** 111–27) and is based on the application of the Breit–Pauli Hamiltonian in combination with nonrelativistic wavefunctions. In contrast to this methodology, we generate the diabatic RT Hamiltonian in a 4-spinor basis where the symmetry transformation properties of the electronic and vibrational wavefunctions completely determine the RT matrix structure. Explicit access to highly correlated wavefunctions is not required in our approach. In addition, the four-component vibronic coupling model takes into account the full spatial orbital relaxation upon the inclusion of the spin–orbit coupling and is therefore well suited for heavy systems. The third parameter, *p*, accounting for a possible pseudo-Jahn–Teller interaction is not considered here, but it does not introduce a principal difficulty. As the initial systems for this study, we considered the BrCN^{+} and ClCN^{+} cations and determined the *c* and *d* parameters by a numerical fit to accurate adiabatic potential energy surfaces obtained by the relativistic Fock-space coupled-cluster method. New values for the computed linear RT parameter *d* amount to 14.7 ± 0.5 cm^{−1} for ClCN^{+} and 73.2 ± 0.7 cm^{−1} for BrCN^{+}.