A close look at the intricacies of the simultaneous triple valued analytical function having the potential to account for the presence of higher order seams, namely three usually seen in any intersecting multiple PESs of polyatomic molecular systems

Abstract

In the previous work, a novel and yet complicated mathematical model to predict the presence of triply degenerate seam was presented and was illustrated its efficiency using some physical models. Here in this work, a discussion on the ability of the model to account for the cusping features (needle like sharp spike) found in the multiple intersecting potential energy surfaces (PESs) is presented in addition to a discussion on their derivatives with discontinuities and infiniteness at a specific point. Analytical evaluation of derivatives are possible in principle but in practice, numerical approximation to the derivatives is resorted to overcome this bottleneck. Two examples (one being physical and the other being the chemically realistic) are considered to the probe the above mentioned properties of the simultaneous analytical model. Integration of this model function gives finite values everywhere disregarding the presence or the absence of degeneracies. This is just opposite to derivative evaluation where at the degenerate point evaluation runs into numerical instability.