Abstract
Linear molecules in degenerate electronic states are influenced by the Renner-Teller (RT) effect. Currently the formula to quantify this vibronic interaction only considers the harmonic term. However, such a harmonic formula cannot well describe molecules that have double-well potential along the bending coordinate. In this work, we propose a new formula to quantify the RT effect by explicitly including the anharmonic term. A sign function and a delta function are additionally included to distinguish different scenarios. It is mathematically proved that the new formula is capable of differentiating different degrees of RT splitting. Representative molecules of different types are calculated. According to the new formula, molecules experiencing a weak RT effect have a positive parameter smaller than 1. For those experiencing a medium RT effect, the parameter will be negative. For others experiencing a large RT effect, the parameter will be larger than 1.