Vibrational motion in isotopomers of the HeH⁺ molecular ion: An application of the electron nuclear dynamics method

The importance of isotopic substitution as a tool for elucidation of chemical reaction events originates in the fact mat the Coulombic Hamiltonian is isotopically invariant except for the nuclear kinetic energy term. Thus, in theories of isotope effects based on the Born - Oppenheimer scheme, the basic presumption is the invariance of the potential energy surface (PES). We use, however, a fully dynamic approach, called Electron Nuclear Dynamics (END), which does net require a preconstructed PES. Since the END formalism is rather different from commonly used procedures, we study the anharmonic nuclear vibration in isotopic species of the HeH⁺ molecular ion as a model problem. A single time-dependent complex parametrized determinantal wave function is used for the electrons and the nuclei are treated classically. The time evolution of the nuclear and electronic dynamical variables obtained by integration of equations of motion are reported as bond length, nuclear kinetic energy, and Mulliken populations. The molecule vibrates as a classical object. The product of the reduced mass and the square of the vibrational frequency is isotopomer invariant for any common total energy. The difference between the total energy and the nuclear kinetic energy as a function of the intemudear distance is interpreted as the average dynamic potential.