  Path integral Monte Carlo approach is used to study the coupled quantum
dynamics of the electron and nuclei in hydrogen molecule ion. The coupling
effects are demonstrated by comparing differences in adiabatic
Born--Oppenheimer and non-adiabatic simulations, and inspecting projections of
the full three-body dynamics onto adiabatic Born--Oppenheimer approximation.
  Coupling of electron and nuclear quantum dynamics is clearly seen. Nuclear
pair correlation function is found to broaden by 0.040 a_0 and average bond
length is larger by 0.056 a_0. Also, non-adiabatic correction to the binding
energy is found. Electronic distribution is affected less, and therefore, we
could say that the adiabatic approximation is better for the electron than for
the nuclei.
