Statistical Analysis of the Effect of Deuteration on Quantum Delocalization in CH₅

Statistical approaches are applied to an investigation of how deuterated forms of CH₅⁺ sample the potential energy surface for this ion. In its ground state, CH₅⁺ has been shown to have amplitude in all of the 120 equivalent minima in the potential as well as in the region of each of the 180 low-energy saddle points that connect these minima. Although deuteration quenches this delocalization, the multidimensional nature of the isomerization pathways makes it difficult to visualize and quantify the delocalization of the ground state wave function. In this work, the localization of the ground state wave function is explored through an analysis of projections of the ground state probability amplitude onto various atom-atom distances. Specifically, analysis of two-dimensional projections of the ground state probability amplitude on to the distances between pairs of hydrogen atoms reveals that the ground state probability amplitude is localized near structures in which the hydrogen atoms in the ion can be divided into those that make up a CH₃⁺ part and those that make up in a H₂ part. Analysis of fits of projections of the probability amplitude onto distances between pairs of hydrogen atoms, between pairs of deuterium atoms, and between hydrogen and deuterium atoms to sums of Gaussian functions allows us to quantify the increased localization of hydrogen and deuterium atoms in the CH₃⁺ and H₂ parts of the ion. In addition, the amplitude of the wave function in the regions of the potential that correspond to the transition state for exchange of hydrogen or deuterium atoms between these two parts of the ion is found to decrease in the partially deuterated forms of CH₅⁺ and is smallest for CH₂D₃⁺ and CH₃D₂⁺.