The Single Surface Beyond Born-Oppenheimer Equation for the Excited States of Sodium Trimer

Summary

We consider a three state diabatic potential matrix to represent the excited states of Na₃ cluster [J. Chem. Phys, 88, 6068 (1988)] at the pseudo - Jahn - Teller model situation, and analytically calculate the non - adiabatic coupling (NAC) elements and their Curls to explore the nature of the beyond Born - Oppenheimer effect on the adiabatic single surface equation. When the energy gap between the third state (1² A ₁ ^′/2² A ₁ ^′) and the doubly degenerate states (2² E ^′/3² E ^′) of the model Hamiltonian for Na₃ cluster is considered to be identically zero, i.e., accidental degeneracy, the Curl for each NAC element clearly vanishes leading to a theoretically valid Extended Born - Oppenheimer (EBO) equation. We calculate the non - adiabatic effects on the photoabsorption spectrum for the ground electronic state by using EBO equation and demonstrate it’s validity by comparing with the corresponding diabatic spectrum (locating the initial wavefunction on the same state), when the three states are degenerate at a point. Finally, we calculate the vibrational eigenspectrum of the ground adiabatic state by employing theoretically and numerically valid EBO equation and demonstrate substantial difference from those eigenspectrum obtained within BO approximation.