Charge asymmetry in HD⁺ ion and HD molecule

Within the framework of the variational method without assuming the Born-Oppenheimer (BO) approximation we consider vibrational states of the HD⁺ and HD molecular species in the ground rotational state. The three- and four-particle non-BO wave functions of these systems are expanded in terms of two types of all-particle explicitly correlated Gaussian (ECG) basis functions – the complex ones and real ones with prefactors in the form of powers of the internuclear distance. The non-BO variational wave functions are used to calculate the proton, deuteron, and electron probability densities and pair correlation functions, which are then compared with the corresponding probability densities obtained from pure BO calculations and from BO calculations that include the diagonal BO corrections. The charge asymmetry in the two systems that originates from the isotopic H/D mass difference are analyzed and it is found that, for the HD molecule, the asymmetry is the most significant for the v = 7-9 states, where v is the vibrational quantum number. Note that in the non-BO calculation, v is not, strictly speaking, a good quantum number, as in such a calculation the vibrational motion couples with the electronic motion. In the case of HD⁺ molecular ion, the particle distributions obtained from the BO calculations agree well with their non-BO counterparts up to the v = 20 state where only a tiny amount of charge asymmetry can be observed. However, for the v = 21 and v = 22 states, a complete breakdown of the Born–Oppenheimer approximation is observed. In these two states, the electron almost entirely localizes at the deuteron.